John_Evans,_Peter_Marlow_Quantitative_Methods_in_Maritime_Economics

Text Material Preview

Quantitative Methods
in Maritime
Economics
Second Edition
J J Evans~& P B Marlow
ID
‘II.
\\
.I‘~ 5'
‘I
Ix
It
It
-.-I
.r"'
.-P"
'U".i'
I I
I. IP- I H I
1* j
.I\,,/fix |
i I I,‘ I' I
I ‘I. III‘ I
I E all , Bull
|
I \. I I
I I
D
I a I
I L I
I - I .-_ _ -_._.I
U C11 C12 CI.
I
II \ _//"
I‘ \ -»*
I. xx,
S
I
I
4
KEK: IO;
Ap stony. ...............
S
QUANTITATIVE METHODS
IN
MARITIME ECONOMICS
bv
John Evans and Peter Marlow
i62¥§I
Second Edition
I 5%;-*5 ?'-4:‘lI’fi§E5'
IoryBook Economist E I'l E.
6 CD
' -' -';'-' G I“ i - u1' I e
-é Eiaaymyég-Bpaiogia Bs$1Iimv,Ilsg|.e ixév, I -ROM
I IHITOKPATOYZ 39
I 106 BO ABHNA
I I '1 THl\é3602695-3640735 THl\ 8: FAX: 36453"Mall glorybook@hoI gr
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
l
Published and distributed by
FAIRPLAY PUBLICATIONS LTD.
20 Ullswater Crescent,
Coulsdon Business Park,
Coulsdon, Surrey CR5 2HR,
United Kingdom
Telephone: +44 181 660 2811
Fax: +44 181 660 2824 '
Email: sales@fairplay.co.uk
Web: http: //wwwfairplay-publications.co.uk
ISBN 1870093 31 3
Copyright © 1990 J. J. Evans and P. B. Marlow
First Published 1986
Second Edition April 1990
Reprinted August 1997
All rights reserved. No part of this publication may be reproduced or
transmitted, in any fonn or by any means, electronic, mechanical,
photocopying, recording or any information storage or retrieval system
without permission in writing from the publisher.
Printed by ‘Wednesday Press Ltd.
Unit 3, Baron Court, Chandlers Way,
Temple Farm Industrial Estate, Southend-on-Sea, Essex SS2 SSE.
KONSTANTINOS
Rectangle
About the authors
John Evans
John Evans was born in 1933. He joined the Merchant Navy as an apprentice in I951 and
served as a deck oificer with the New Zealand Shipping Company. After attaining the
qualification of Extra Master in 1963 he began-a career in teaching at what was then the
Welsh College of Advanced Technology. He began to specialise in maritime economics in
1971. He has worked on a consultancy basis for UNCTAD and also for CENSA in
connection with which he was one of the main authors of the 1978 UWIST study, Liner
Shipping in the US Trades. He is currently a lecturer in the Department of Maritime
Studies, University of Wales College of Cardiif. In 1981 he completed his M.Sc. by research
in the field of liner freight rates. He was appointed editor of the Journal of Maritime Policy
and Management in 1984. He plays golf when he has time. He is married with two sons.
Peter Marlow
Peter Marlow is married with three children and was educated at the University College of
Wales, Aberystwyth where he obtained a joint honours degree in Economics and Statistics
prior to working as a Research Assistant in the Department oi‘ Economics. He subsequently
obtained a higher degree by research in the field of quantitative economics. Following a
short period as a school teacher he joined the Department of Maritime Studies of the
University of Wales Institute of Science and Technology (now the University of Wales
College of Cardifl') where he is employed as a lecturer. His main areas of interest include
transport economics and the fiscal treatment of shipping and he has undertaken studies for a
variety of outside bodies including the Commission of the European Communities, the
Organisation for Economic Cooperation and Development, the General Council of British
Shipping, and various government departments. He is a Member of the Chartered Institute of
Transport and of the Institute of Logistics and Distribution Management. In 1989 he was
awarded a doctorate for his work on investment incentives in shipping.
To Ann and Alma
Ill
KONSTANTINOS
Rectangle
Foreword
After spending a number of years teaching university students various aspects of quantitative
methods, especially in relation to maritime economics, it became increasingly evident that the
literature available for students to consult was widely dispersed and, even then, the content
was often not at the level required. Furthermore, many of the text books on quantitative
methods tend to be incomprehensible and raise barriers to those with limited mathematical
ability: seldom can one find applications in the maritime context. '
We decided therefore to attempt to write a book that would, as far as possible, be self-
contained and cater for the general needs of students of maritime economics and possibly
others concemed with shipping andrelated activities. We have not striven to achieve
enormous scholastic heights; rather we have tried to provide a self-supporting volume in
which many cross-references can be made.
We have tried to explain principles, models and methodologies rather than leave the
mathematics to explain itself. Since a knowledge of calculus, for example, is desirable in
order to follow the mathematical developments in certain chapters we decided to include a
chapter containing the rudiments of that subject. Similarly, we have included chapters on
analytical geometry, power series and probability theory which contain fundamental
information suflicient to enable readers with little or no knowledge of those subjects to be
able to follow the main parts of the text; should they wish to acquire a more advanced or
detailed understanding, they can consult a number of texts cited in the bibliography.
We have not attempted to teach economics as such. We assume that readers have a
reasonableqknowledge of this, though one chapter is devoted to a presentation of basic
economic principles. A number of important topics are dealt with, e.g., supply and demand;
optimum speed, size and distance of ships; correlation —- both simple and multiple;
queueing; and decision theory. Examples are given in the maritime context throughout.
In this second edition we have taken the opportunity to include some new topics which with
hindsight might well have been included in the first. These are: the supply of sea transport;
voyage estimates; freight futures; investment appraisal; linear programming; replacement
theory; and the efleet of shipping on the balance of payments. The chapter on optimum
speed has been enlarged; there has been some restructuring; changes made in the order of
chapters; and the opportunity has been taken to make minor changes in the text as necmsary
and/or appropriate.
We acknowledge with gratitude our appreciation for the information so willingly supplied by
the General Council of British Shipping; Worldscale Association (London) Ltd.; and the
Far Eastern Freight Conference without whose help the chapters on freight index numbers
and CAFs, BAFs & EAIDs would not have been possible; we thank them for giving their
permission to reproduce text from some of their oficial publications and other material
provided.
iv
KONSTANTINOS
Rectangle
We also thank HMSO for permission to reproduce data, extracted from the Pink Book 1988,
contained in Table 20.1 (page 230); Taylor 8t Francis Ltd. for permission to reproduce text
(ch. 6 (part); ch. 18 (part); and ch. 19), written by one of the authors, that originally appeared
in the Journal of Maritime Policy and Management.
Our grateful thanks are also due to Mrs. Barbara Fletcher, Director of Education of the
Institute of Chartered Shipbrokers; Mr. R. W. Porter, manager of Worldscale (Association)
Ltd.; Mr. Mark Lewicki of the Far Eastem Freight Conference; and Mr. David Ellis of
Graig Shipping Co., Cardilf for their invaluable assistance in the preparation of certain
sections of the book.
We are indebted to colleagues in the Department of Maritime Studies of the University of
Wales College of Cardiff, especially to Alun Rogers and Andrew Wiltshire of the
Cartography Unit who produced the diagrams and figures; also to Sian Poppleton and Tracy
Shellard for their unenviable task of typing the manuscript.
Any errors or omissions that remain are, of course, the responsibility of the authors.
J. J. Evans
P. B. Marlow
V
KONSTANTINOS
Rectangle
I-._
Contents
Chapter
1
Oifllil-all\-I
7
3"».Elements of calculus
Relevant aspects of analytical geometry
Progressions and series
Probability theory and distributions
Basic economic relationships.
The demand and supply ofsea transport
Optimum speed of ships
Ships’ costs; voyage estimates; Worldscale
E1 Freight futures
10
ll
The optimum size of ships
Liner freight rates
12, Linear programming and transportation
61:3)Regression and correlation
14
15
16
17
18
19
20
21
Annex I
Annex II
Annex III
Annex [V
Appendix A
Appendix B
Appendix C
Bibliography
References
and notes
Index
Decision theory
Queueing theory
The theory and practice of index numbers
Currency, bunker and inflation differential factors
Investment appraisal in shipping
Replacement, obsolescence and modifications of ships
Shipping and the balance of payments
Calculations in shipping economics
Derivation of formulae used in discounting
Tax relief on capital allowances
To find the PV of interest payments on loan outstanding
Analysis of an annuity
Maclaurin’s theorem
Speed and fuel consumption
Conditions for ship total revenue to remain constant
when one cargo is substituted for another
vi
Pass
1
l 3
24
29
43
61
79
92
116
124
ll!
Qi;€i§>
156
167
I 77
191
197
221
229
235
261
26.4
266
268
270
272
275
276
278
281
KONSTANTINOS
Rectangle
Chapter 1
Elements of Calculus
The fundamental purpose of difierential calculus is to determine the slope of any given
function at a specific point on the curve.
2
Consider the parabola y = % + 6 as shown in Figure 1.1.
Figure 1.1
Y
A
.o
The slope at any point A is defined as the t_a_nge_n_t __Qf_ the angle that the tangent to the curve
at A makes with the x axis, i.e. Tan tp in the Figure. Clearly, the task of finding the slope,
by drawing, would be tedious and the result quite inaccurate. The angle ¢ is always
measured anti-clockwise from the direction of the positive abscissa and can have values in
the range 0-180°. A slope of 0° means that the curve is horizontal; from 0-90° the slope is
directed upwards to the right; and from 90°-180° the slope is directed downwards to the
right.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Figure 1.2
Y pi
i5vA ------I Q
bx
0 X
Now consider the curve shown in Figure 1.2 where y is a function of x (f(x)). A is any point
on the curve and B is another point close to A so that AB is a ghord whose slope
approximates to the slope of the curve at A. The incremental values of B relative to A are
expressed by by and bx in the vertical and horizontal directions respectively. Thus the slope
_ _ /\ by
of the chord 15 given by Tan BAC =
If for example, the curve represents a part of the function y = x3 + 6 since point A given by
(x, y) lies on the curve, so then does point B given by (x + bx), (y + by).
Thusy+by=(x+bx)3+6 * "‘
and expanding (x + bx)?‘ binomially (see Chapter 3), E
\ = - 3.2 *-
yj+ by = xi + 3x2bx + Yxbxz + bx3 + 61
or by = 3x2bx + 3xbx2 + bx’ (since y = x3 +
by 2
so that 5 = 3x + 3xbx + bx’.
As bx becomes smaller so will B more closely approach A and the slope of the chord AB
will eventually be equal to the slope of the tangent to the cu1've at A.
Thus, in the limit, as bx —> 0 (Le. as bx closely approximates to 0 without ever reaching that
value):
by 2
ax - 3x .
It will be noticed that even though bx tends towards 0, the ratio %I- remains finite. The
x
I1'
I
'\
_.-1-‘.-"-Qu-£-,-n.h-
— t
|
-II\|$iq.._-
\
.-\._-.-_._
1-:,...
.44:1-—£_2_._i‘i‘»~+A".;'ti
"I
I
-.....__--_-...¢_,_,=-:.__‘.j.:-'-...-____:'_
~I
I.
9
I
I
I
_fiL4
_1aI
I L
44.I.
‘I’
II
I‘
P J
‘*4 -pr"
I
"rt
I
‘I
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
b d
limiting value of F: is called the differential coefficient of x being written ax! and it should
be understood that this is not a fraction in the ordinary sense. ‘The differential coefficient
therefore, gives the precise slope of a curve at any point on it. In this example when x == l
the slope is 3 X 13 = 3 and when
x = 1/\/3 the slope is 3.(1/'\/3): = 1.
In the original equation (y = xi + 6) it may be seen that the effect of the constant is to raise
all the values of y by 6 units without affecting the slope at any point.
The General Case
d
If y = ax“ + b where a and b are constants it may readily be shown that ay = anx"" where
n is any real number either positive or negative.
It follows that in the case of a straight line of the form ax + by + c = 0, after rearranging
ax cy=—?—Eand
d
the slope, FE = -3 which as is already well known is constant. If the straight line is in the
form
y = mx + c
d
then F3 = m which again is known from fundamental analysis of the straight line.
Difierentiation by Substitution
There are some functions that cannot be differentiated by following the general principle
shown above.
For example, if y = \/(x - 2) another method must be found. A useful method is by
substitution:
Letu=x-2.
Theny=\/u
=u1/2
now ix = in"/2du
and it may be accepted (without proof) that
dy d du_=_Y _
dx du'dx
Nov\Isinceu=x-2, %=1
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
and%;- = iu'1/2 .1
= “X ___ 2)-1/2 or _...L...._.
2\/(x —- 2)
Differentiation of e’
It will be shown (Chapter 3) that
xz xi
e"=1+x+-2-T+§+... (whereehasavalue2.7l8...)
so that if y = e"
dy__ 2x 3x2'
1I'lCI1a—_1+-2—!+-5+...
X2
--1+X'I-5+.-.
=ex
This is the only example where the differential coefficient is equal to the function itself.
dIn general: E; (ae‘”‘) = abe""
Differentiation of log,x
If y = Log,x
then e’ = x, and differentiating both sides of the equation with respect to x,
d(e!') Q = 1
dy ' dx
_ dlt . d(e’)..eY dx-1 since dy -eY
H fi-i_1ence dx -ey - X
_ d 1
t.e. dx (logex) - X
This is a most important result because of implications for the reverse process known as
integration (q.v.) -
D;'fi'erentiatz'0n of logax
where a is any positive number.
It can be proved that
log,x = log,x . log,e
<1 1 . .Hence E (logax) = ;. log,e since log,e is a constant.
1
I
i—‘%,-Q-..,
\ .
I
I
I
i
-.'J1--—-I~
-?’-.—-_
-.._.§.__.____...,p_.-.-lg»--—
A‘.-F»‘S,-.---...----»-1-an-pass-r.,=,0-M.-s.--a..-u-u»sun¢-q-hInu-—u-4/Ih-n~n-d¢a?|p¢-iII|#¥1J-nm-Q-an——--v—-'--'~-—
I.
1
i
I
I
I
3"
I
it
‘I
is
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
_-
Differentiation of a‘
Let y --= a"
then x = logay
= 1. log,e = logae . -1-dy y a"
and inverting,
dy a" _
=E or a"log,a (since log,,a . log,e = 1)
G
Differentiation of a Product
Let u and v be functions of x and let
y = uv
Hence y + by = (u + bu)(v + bv)
.'.y + by = uv + vbu + ubv + bubv
.'.by = vbu + ubv + bubv (since y = uv)
.2- 91+ §+@
"bx-vbx ubx bx
Now in the limit as at -> 0
3: * v + u or in words, the difl'erelntial of a product is equal to the first times the
differential of the second plus the second times the differential of the first.
dExample: Let p = (3s + 2)3(s2 — 4)2 and it is required to find the differential coefficient —p
1 . Qt 95‘ dsI $__|/ 1 ?’ S
In this case v = (s2 — 4)2 gs
and u = (3s + 2)3 = §i_">‘*"‘-‘TI ‘I
SdSo that F: = 3(3t +-2)= . 3 = sot + 2)1
d
and —v = 2(s2 - 4) -. 2s = 4s(s2 - 4)ds
d .
Hence FE = (s2 — 4)29(3s + 2)’ + (3s + 2)34s(s2 — 4) which reduces to:
g-E = (s2 - 4)(3s + 2)2{9(s2 — 4) + 4s(3s + 2)} _
= (52 — 4)(3s + 2)2(21s2 + 8s - 36)
5
1
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Differentiation of a Quotient
Let u and v be functions of x so that
IIi'=;
bmeny+by=%-F:
u+bu u
by:-v+bv v
v(u +bu) - u(v + bv)
= v(v + bv)
uv + vbu - uv - ubv
vz + vbv
51- 2!
_b_y_vbx “bx
"bx-' vz+vbv
In the limit as bx-+0
E- 1‘!
__ dl=vdx udx
“dx v2
The use of this relationship is generally unnecessary since the quotient can normally be
transposed and treated as a product.
Maximo, Minima and Points of Inflexion
A quadratic function of x in the form f(x) = ax’ + bx + c has a maximum or minimum value
of x according to the values of a, b and c. A frequently used convention to express the
differential coefficient of f(x) is f'(x)
and so f'(x) = 2ax + b
At the turning point the gradient of the curve is equal to 0 so that f '(x) = 2ax + b = O
_ _—b
..X—-2a
bz b-bandf(x)-=%a-5+(—2a-l+c
b2 bz=———+c4a2a
If°4t
1.8.11 Ctllflllllg POI!) IS at 28 , C 4a~- - (I-i><-1-‘I
In order to determine whether the point is a maximum or minimum it is necessary to find
I
I
I.II
d~
.-._,.---_----{\l—-~=&-—-——1--'-"--
:-.a4-
I
I
‘T
II
I.
It.
‘__:fi._¢.,..;A‘I;-:fli,--1I
IV
..,____-..-
I
I
I
"I
it
I
4
r-!,"'~
H‘.__,.;,..---_-_.__
J:__
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
out whether, at this point, the gradient is increasing or decreasing. If the turning point is a
maximum, increasing values of x cause the gradient to change from positive through zero to
negative i'.e. it is decreasing in magnitude.
z
Hence, if the second difierential % or f"(x) is negative, i'.e. the gradient is decreasing with
increasing values of x at the turning point, then it is a maximum; and vice verse.
It has been seen that
f'(x) = 2ax + b and so, f"(x) == 2a
If 2a is positive, therefore, the turning point is a minimum and, if negative, the turning
point is a maximum value of y.
A point of inflexion is neither a maximum nor a minimum but occurs when the gradient
becomes zero incidentally in a generally upward or downward sloping curve. At a point of
inflexion the second differential is zero.
Example I A parabola is given by the equation y-= 15 + 2x - xi. Find the turning point and
determine whether it is a maximum or minimum.
y=15+2x-xi
dv..dx-2 ZX-0
.'.x=1
andy==15+2-1
=16
d2
% = -2 which is negative. Therefore the tuming point at (1.16) is a maximum.dx I
Example 2. Find the turning points of the equation
y=x3—12x2+44x—48
Differentiating with respect to x and equating to zero
dl=3tt= -24tt+44=0dx
and solving for x,
x _ 24 1 V247 — 4.3.44
6 I
2whence x-= 4 : V5 = 2.85 or 5.15
and the corresponding values ofy are 3.08 and --3.08. It can readily be shown that the first is
a maximum while the second point is a minimum.
Partial Differentiation
If a variable is a function of two or more other variables then the rate at which that variable
changes with one of the other variables while the others remain constant is termed the
partial derivative of that variable.
KONSTANTINOS
Rectangle
14
l
f
I
5
S
--—-.1-...“15..-_..._-
Thus if z = f(x, y) so that 22 = x2 + yz, then the partial derivative of z with respect to x is
5 3given by 22 i = 2x (y being regarded as a constant) 51- = 2
_F_ _ 62 62 ySimilarly 22 ay - 2y and ay - Z
_ ~1-
If x and y are each functions of a third variable (t) then the total differential coefficient of z 7
with respect to t is given by *
0
Pdz dx dy
Z23]:-—ZXEt-+2ya';
l_ dz 62 dx dz dy
“zzdt-2261: dt+2z6y dt
thtdzaz dx+8z dy _
5° aot"att'c|t 6y'dt B
._..c_____-.\t.___..-._,._—-——
This relationship is of general application.
Example 3. The volume of a ship that is box-shaped is given by V = L x B x D where V is
the internal volume, B the breadth, D the depth and L the length.
Thus
._.__---r~ss-fin
av 5V av _’
5-I:-—BD, '(;:E—LD, and '5-[;—LB
_,,_,..__€,__,,_=g~,_;-_‘tn..,__._.-_t4'_,.-..--_.i‘rB-)4-s....-H.--..t...---\¢.._.,u-..._-svqti-|---a--_..-..-elt,t-_---t-.n||u4-no-rs‘__|tp-a‘-an.-¢;..~u»-t.‘fii-HI-I-it-1|---=-nus-P;,o_...--7
These three partial derivatives indicate the rates at which the internal volume of the ship
varies with a change in each dimension respectively. It follows that since L and B are the
two larger dimensions, the volume is changed to the greatest extent by a change in depth
rather than length or breadth.
It does not follow from this however that the volume can be increased at least cost, by
increasing the depth.
Assuming that the cost (C) is proportional to area (A), ._ -- _
. l ‘ -~C = KA where K is a constant of proportionality J: fi“_""““ 5:1 L“
Now A = 2LD + ZLB + 2BD -i.."...-‘H. -l/i '5;-ll 1
whence I r Z5?
.__
1..- Q t-2.V.D 2v.B “L 'A=- B5 +-BD—+2BD
rv _,I. {:3 £3
3&1 o2V 2V /I ,.= -5- + -5 + 2BD “
-' ' is ‘N’
t1ttaA - 2 + 2 we i»~;J€5° a av ‘ B _o (with B and D constant) . . . (I) qffgs
5- -,, 1o..2.,a"“'“ Y’ av'L D (with LandD constant)... (11)
d 8A - 2. + Z. .an av ‘ L B (with L and B constant) . . . (111) 
It should be clear that the option that increases the volume at ‘least cost is the last -- since B
is normally greater than D.
Iq.u-m-l--n-
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Integration: The Indefinite Integral
In economics, marginal revenue (MR) is usually defined as the increase in revenue resulting
from the sale of one more unit. Marginal Revenue can be defined altematively as the ratio
of the change in Total Revenue to the (small) change in output that causes such change in
revenue.
Thus, if Average Revenue (AR) is a function of output Q
AR = f(Q)
and total revenue (TR)
= Qf(Q)
Hence MR =- = Qf'(Q) + f(Q)
_ co_1
SIIICC dQ —
If the relationship between MR and Q is known, Le. MR = F(Q), then to determine TR it is
only necessary to perform a process known as integration which is the inverse of
differentiation. The result of integration, i.e. the integral, is denoted by:
TR = I F(Q)dQ i. e. total revenue is given by the integral of F(Q) with respect to Q.
In general, if y = ax"
axn+1 d(xn)
=i--— —-i: "*1thenJ'ydx n +1 (cf. dx nx )
1Note however that I -IE dx = log,x
Since the process of differentiation eliminates constants, the inverse process of integration
must include an arbitary constant thus:
33 22
((3tt1+2tt-5)t1tt=%+%-sx+c
=.x3+x2 -5x+C
where C is known a_s the constant of integration. Note that this integral is equivalent to
I 3x2dx + I 2xdx I Sdx where each term is integrated separately.
Unlike differentiation, which generally poses few problems, integration can be very
complex. In some cases it is necessary to know the result of differentiation before the
inverse process can be accomplished. Space does not permit a detailed account of the
methods of integration but one example will be given for illustrative purposes.
1Example. Find the indefinite integral of V(—1-l?)
Method: Substitute sin 6 = x
dx . .so that E = cos 6 (given without proof)
KONSTANTINOS
Rectangle
1 1 dx
H¢nwI M=I .a§.dB
1==J’§.cos0.d6
=Iae=e+c
=sin'1x+C
It can readily be shown that the differential coefficient of
Sill“ X = V(T1_T)
The Definite Integral
The definite integral is used generally for finding the area between a curve and the
horizontal axis between two specified points on that axis.
Figure 1.3
Y
r-9'1!
He go
o sn x
Consider an element of area 6A, (PMRS) contained between the curve and the x axis.
P is any point (x, y) on the curve f(x)
Now if SR (bx) is a small element of the abscissa, and PL(6y) a small element of the
ordinate, it is clear that
y6x>6A>(y-6y)6x
andas6x—=-0
ydx=dA
J
1t-
)-t-“*'_i_‘*“"—"'““""r'i‘_
I
q.
v
I
I
I
%
‘,_
qrutuin,-I-Li;o_*,4‘hnqp-an-pram--pan-n-aw—"'"-"-'-‘"-
i
‘ r
l
it
-t.‘--.-5,,-=--4
T
‘V
l
2
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOSRectangle
KONSTANTINOS
Rectangle
_ dA
t.e. -3-; = y = f(x)
Hence A = I f(x)dx + C
This gives the total area between the curve and the x axis to the left of a given value of x.
However, since C is unknown the solution is indeterminate. Where, however, it is required
to find the area between two given values of x it is not necessary to know the value of C
since if the two values of x are a and b, C disappears from the expression
A = f(x)dx + CY“ - f(x)dx + CT“ and a > b
This is normally abbreviated to A_= I l f(x)dx.b .
Example. Find the area under the curve y = —3x2 + 24x — 24.
This is a parabola as shown approximately in Figure 1.4.
Figure 1.4
Y
0
5.-1-.5:
Before this can be solved it is necessary to find the x coordinates where the curve cuts this
axis.
Sincey=0
-3x2+24x — 24=0
i.e.x2-8x+8=0
KONSTANTINOS
Rectangle
8 1-. \/"86-t_lT'§§
so that x =
8 : 4\/'2
='—2.'_
= 4 1 2\/2
= 4 + 2.82 = 6.82
or 4 — 2.82 = 1.18
o.s2
Hence area = I (—3x2 + 24x — 24)dx
us
...3x3 24x2 6.82
= [T * T " 24*]
= [-6.823 + 1Z(6.8Z)2 - Z4-(5.82)] -' ['-1.183 + l2(|.l8): - 21-l>(1.18)]
= ['/7.25] - [-13.25]
= 90.5 units
It should be noted that the area of -13.25 units contained between the second pair of
brackets represents the shaded area in Figure 1.4. i.e. the area under the x axis between
x = 0 and x = 1.18.
____,__‘_,_____,,_,__...-.._-‘,---H.._,
Y
I
:-
.,-s-t.4-.-_s_-s.-
O
..',-ts
_;--._.,‘.-..-.-_-q,
__-_-4-|-.--.q,--.-----
=1
|
‘1.
t
F
t
€=
set
I
-'
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Chapter 2
Relevant Aspects of Analytical
Geometry
This subject will be treated in a fairly cursory manner since the subject isvast and much of
it is of little concern in economics. This chapter will concentrate on aspects which are
relevant to economists.
The Straight Line
Figure 2.1
A Y
T ,
‘I
1‘
O
_ i, x
0 R b S l
—: D
ta. 9‘?
‘L 3 ti
B-'7‘ ‘bf
Consider the straight line AD in relation to the x and y axes cutting at origin O. The point
P whose coordinates are (x, y) represents any point on the line. PO is equal to x and is '
parallel to OS while RR is equal to y and parallel to TO.
Since As TQP and PRS are similar. it follows that:
PO TQ_ x c-- )4 '
§'ti'=‘1¥""(E>'-Tt)=_';T"
where c and b are respectively the intercepts cut off by the straight line on the y and x axes.
13 |
KONSTANTINOS
Rectangle
Cross-multiplying,
xy=cb—cx—by+xy
or cx + by - cb = 0 which represents the equation of the straight line.
Dividing each term by b and rearranging,
-c+c11- ht
The gradient of the line is given by Tan ¢ which is equal to -5: if m now replaces — E, the
equation can be expressed in the usual form:
y = mx + c where c is the intercept on the y axis.
Equation of straight line joining two given points
Given two points (xfyl) and (x;y;), to find the equation of the straight line passing through
them.
Figure 2.2
Y
A X1y1
__a|___.._=1.__.. .__..___'__..._.._
X
_: _ _ Cxv
_____ ____ ______ ‘E xzyz
In Figure 2.2, (x, y) is any point on the straight line joining (x|y,), (x;y;). Since As ABC
and ADE are similar
E_BC
AD_AB
. 32-31 3'31lei-=———
Y1"Y2 Y1“?
X " X1 _ Y "' Y1 . .or - --L s- which is a standard "result.
K2 ' X1 Y2 *' Y1
l
{F—---qr‘-r.¢-s.s.%,,-n-,1...-t--tn,-as--t.-“:1
Q.-...- *-----.1....s----—-—-—---—--.--'---1-r
.'-
t
Y!
1
‘,"__hI___‘U,-k,_“;_,_,?.‘M-},_,,.,.,_-.d'__.___.-.,,....---u---.-_-._.
!I
_--st.‘xS¢ 1ji--..-or:fh.Irn.4
--sew-"-2.1",-—-ZI"—
1‘iv
l
l
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Second Order Equations
A first order equation of the general form ax + by + c = 0 is a straight line and passes
through the origin only if c = 0.
A second order equation implies that either x or y (or both) is raised to the power of two.
Such equations are always curves, although parts may approximate to straight lines. e.g.:
y = 4x2 is a parabola passing through the origin. (Fig. 2.3)
Figure 2.3
Y
X
O
Y
y=—4x*
‘ X
O
KONSTANTINOS
Rectangle
The shape of the parabola is changed by varying the coefficient 0f~x2, i.e. the ‘parameter. so
that if y = ax: the parabola is of flatter form if a is reduced and vice-versa. Its position with
respect to the origin can be changed by introducing constants b and c such that
y = a(x — b): + c
A positive value of b moves the parabola bodily to the right while a positive value of c
raises it. (Fig. 2.4) - -
II! IIIIi -1(II ['0 -Ii
Y
y=2(x— 10)2+6
= ZXZ — 40X + 206
..-.....o.l'....,._______,_____,______x
It should be noted that an equation of the general form y = ax: + bx + c is a parabola and
always has a maximum or minimum value of y. This value can be positive or negative.
Figure 2.5
Y
XO
l
it
F
J
i
-r_|-t:|---
I
it
______1___‘___'_________,_Q_____,‘,_,_,,_gn¢.,,_____,,...._-‘to--H1,---cunt-unit-u-1--‘-_-v
O-
|
I
1*
'1
‘v
__---...a--‘it
st‘)
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The Rectangular Hyperbola
y = 3- where k is constant. (Fig. 2.5)x
As a second-order equation, this is of particular interest since the product of x and y is
constant. A demand curve represented by such a function would indicate constant total
revenue with demand having at all points an elasticity of unity (t'.e. Ed = -1). The curve is
asymptotic to both the x and y axes; tie. it meets them respectively at infinity.
Cubic Functions
These include third-order values of x or y and are of the general form:
y = ax’ + bx’ + cx + d where a, b, c, d are constants. (Fig. 2.6(i))
In general a cubic equation unlike a quadratic function has both a maximum and minimum
value of y. An exception occurs in an equation such as
y=x3—9x2+27x-27
which factorizes to y = (x — 3):‘ and in this case the maximum and minimum points coincide
giving a point of inflexion. (Fig. 2.6(ii))
Adding Functions
Two or more functions may be combined to yield a third. Consider the two linear functions
y=5x+4andy=2x+7.
These may be combined in either of two ways according to the purpose intended.
(i) y = 5x + 4 (A) adding vertically
y=2x+ 7 (B)
gives y = 7x + ll . . . (l) (Fig. 2.7(i))
(ii) Rearranging.
_z_fi
" ' 5 5 (A)
and adding horizontally
y 7X = 5 -5 (B)
- _fi_£g1VCS X-—
or 7y = 10x + 43
i.e. y = lg-’-5 + 553- . . . (2) (Fig. 2.-1(a))
KONSTANTINOS
Rectangle
Figure 2.6
X
y = (x + 6)(x +'2)(x -1)
=x3+7x3+4x-12
li)
X
y = (X ~ 3)’ '
Note point of inflexion
(ii)
2.‘)
i
l
r
i
q~_.-._q_.._._-__.
ut
I
i
it
1
i
~nu-urqI--.---fa-»--;.-_-.....
I
Q
i
t.
I4i
"4-§"—'_san-nntr-o-,_._*t|p.-.4...»
1
l
1:
I
“*9ii.1,“
l I
-.‘
E
7
we
4-s,_,,,_..nL
l
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
B.
, tin
Figure 2.7
(i)
Y x
(ii)
1 l
I .1 o
/ |
H
KONSTANTINOS
Rectangle
Combining Cost Curves
Example I A manufacturer produces 100,000 units of goods per annum which he ships
regularly to an overseas country. His costs (excluding production) comprise freight, storage
prior to shipment and ‘inventory’ or interest on capital tied up in goods during storage. He
wishes to minimise these costs per unit. Given the following information, calculate the
optimum size of shipment and the corresponding cost per unit:
-t
l
Value of goods: $150 per unit
Rate of interest: 10%
Cost of storage per unit per annum: $10
Freight per unit: according to the function (-400.000/x + 10) where =
x is the size of shipment.
i. x . 150 . (10/100)The inventory cost/unit if o A o where x/2 is the average number of units in
storage at any time.
, _ x. 10The storage COSI/LIIIII -
Inventory plus storage costs/unit
__ x (75+10 _17.5x
‘10u.000 ' J" 100.000
This is clearly a straight line passing through the origin. Now. adding vertically the two cost
functions
c = 400.000/it + 10 . . . (1)
c = 17.s>t/'100.000 . . . (ii)
>
gives C = 400.0001?‘ + l7.5x/100.000 + 10 . . . (iii)
These functions are plotted in Figure 2.8 below.
Graphical methods indicate that the optimum size of shipment is about 50.000 units giving a
minimums cost of $24 per unit.
Thisproblem, given the function, can however be solved more readily using the differential
calculus. for
C = 400.000x"‘ + 17.5x/100.000 + 10
dC ., _ _
= ——400.000x'~ + 17.5/100.000 = 0 for a minimum
».
400.000 .. e e 115 ts“'°“°e X2 'i00,000
d 2X105—47s09 'Elf! X-— — , UIIIIS
whence C = $26.73 per unit. _,.
--.o@;--i-|-s-uiq-
N
r’
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Figure 2.8
$24 - ---- 
50.000 units
Example 2 A country Y is the sole producer of a commodity for export to three areas A. B
and C. The demand functions are given by:
‘Q1128 ‘,2 Z9558
Q 3 A an
= -—-i Qt; 1%(A)P 500-+-100 0 first
3Q
(B) P = “W-F 150
Q
(C) P ' _ 2.000 + 75
where P is the price and Q the quantity that would be bought at that price per unit of time.
These functions are plotted in Figure 2.9.
In principle these functions must be aggregated or combined horizontally in order to
determine the total demand function. However, since a negative demand is not meaningful.
only the positive values of these functions can be considered. A lit_tle thought will show that
fromoa. pr.i.c.e..o.£.li0. .u_t_1i.t§__d_own__t_9___1,Q.Q_th§ demand will arise from country B only: and from
100 units down to__ 75 the demand will comprise an aggregate of B and A. At prices below 75
units the demand will arise from all three countries. The total demand function will not
therefore be a continuous function, being kinked at prices of 100 and 75 units.
The top part of thedemand function will therefore be:
P = -30/2.000 + 150
KONSTANTINOS
Rectangle
P
150
100
75
Figure 2.9
.150 Ql0O0s)
For the next part it is necessary to combine functions for countries A and B.
(i) Q = -500 P + 50.000
-2,000 P
(ii) Q = -—3,——- + 100,000
— 1,5whence Q = (
P._ —3o+900
°'. '3,s00 7
P
150
‘I00
003+ L000) P + 150,000
_ 71- 7
Figure 2.10
75
QiO0Osl
l‘
I
l
s
‘D
I
i
l
-|___,,=,_,_,¢|,_,_,_.-_.,_@v,..s-,.._.,-.1-u-so-I\tr-°--v--'-""-"-"““
't
_\
!
..--.-----Q»-—
}
. ii
4‘)
1-l------
I
I
I.
i22
1
KONSTANTINOS
Rectangle
For prices below 75 units this function must be combined with that of country C. Thus
(i) Q = —2,000 P + 150,000
-3,500 P(ii) Q = 13- + 150,000
—9,500 Pwhence Q = -—T- + 300,000
.,.-fi.,m
" '9,s00 95
The aggregate demand will appear as shown in Figure 2.10.
KONSTANTINOS
Rectangle
Chapter 3
Progressions and Series
An arithmetic progression in the general form. may be expressed as
a.a+d.a+2d.a+3d.....a+(n—l)d
where a is the first term and d is the common difference between successive terms of the
series. The rth term is denoted by a + (r - l)d and the sum (S) of the first n terms is given
bv
S=a+(a+d)+(a+2d)+(a+3d)+...+(a+(n—l)d)
which can be re-written as
S=(a+(n- l)d)+(a+(n—2)d)+._..+(a+d)+a
Adding these two expressions gives
2S=(2a+(n — l)d)+(2a+(n — l)d)+ . . . +(2a+(n- l)d)
= n(.?.a + (n — l)d)
HenceS=%(2a+(n— l)d)=%(a+L)
where L is the last term.
A geometric progression has the general form.
a. ar. ari. ar-‘. . . . ar""
where a is the first term and r the common ratio between successive terms of the series. The
pth term is denoted by arl’ " ' and the sum (S) of the first n terms is givenby
S=a+ar+ari+ar-‘+...+ar"'""...(i)
whence. multiplying throughout by r.
rS=ar+ar3+ar-‘+ar‘...+ar"...(ii)
and subtracting (ii) from (i)
S — rS = a — ar“
.'.S(l —r)=a(l —r") 4;,
and S=—-——-an_rn)l—r
If r < 1. as n becomes larger r" becomes smaller: so that in the limit as n—> ==. r"—> 0.
Hence. the sum of an infinite geometric progression is given by
S=iwhenr<1l—r
t‘
I’
i
--s--—uIe-'-——
-t-wtqv..IIn.1'-qvqr-H-I-.¢~-—-
i‘
t
l
---\§'Q,___I-is-4'.’-=Zi¢qr.;1q--vtF'r-----trim-IQ;-in‘-‘:11'»-cu-I-C--aw
1i
._;-s—;_-__=111:"-._-.=.-t.Mtg-waif‘:4412;
il
...-.._
..-...-....‘.--.---\iqn._
I
l
KONSTANTINOS
Rectangle
A different series may be obtained by combining the arithmetic and geometric progressions
given above.
The series may be written as
a. (a + d)r. (a + 2d)r3. (a + 3d)r~‘. . . . (a + (n — 1)d)r""
The sum S of n terms of this series is given by
S=a+(a+d)r+(a+2d)r3+...+(a+(n—l)d)r““
.'.rS = ar + (a + d)r3 + (a + 2d)r-‘ + . . . (a + (n — 2)d)r“" + (a + (n — 1)d)r"
Subtracting.
S(1—r)=a+[dr+dr3 +...+dr"“]-(a+(n— l)d)r“
= a + dr(l—:i——l) — (a + (n — l)d)r"
1- r
since dr + dri + dr-‘ + . . . + dr““
‘_ n-l
=dr(l+r+r*+...+r"‘2)=dr(%;-)
_S jg ax, + dr(lg— rt‘) _ (a + (n
"T1—r (l—r): l—r
Alternatively it may be shown that
a(1-r“) *dr(l-nr""') dr(n &_g1)r"
S: 1-t + (1-t)= + (1-r)=
Permutations and Combinations
The product of successive positive integers occurs often enough in mathematics. especially in
probability theory. to warrant a special notation. Thus the number n! (to be read as n
factorial) has the following meaning:
n!=n><(n—l)><(n—2)><(n—3)><(n—4)><...><3><2><l
and5!=5><-l><3><2><l=l20
and7!=7><6><5><-l><3><2><l=50-10
1t should be noted that 0! is defined as unity (0! = 1). Occasionally an alternative symbol L
is used to denote a factorial quantity: thus |§ E 5! and Q E n!.
An arrangement of a set of n objects in a given order is called a permutation of the objects.
An arrangement of any r Q n of these objects in a given order is called a permutation of the
n. objects taken r at a time.
The number of permutations of n objects taken r at a time is denoted by
n!,,P,=E-r?-_;T)?=n(n—l)(n-2)...(n-r+l)
e.g. Consider the set of letters w. x. y and z. How many permutations of these -l letters can
be made taking two at a time?
4! 4.3.2.1The formula tells us 4P; - - - 12
KONSTANTINOS
Rectangle
and these can be written down as
wx. wy, wz. xy. xz, yz. xw. yw, zw. yx, zx. zy, from which it can be seen that the order in
which the letters occur within each pair is important in this context because wx and xw are
considered as different permutations. It must be noted however that they would be classed
as the same combination because in combinations order does not count. A combination of n
objects taken r at a time may be denoted by ,,C, or (i‘) and is defined.
n! n(n—1)...(n-r+l)
nCr=i=r!(n - r)! r!
S! 5.4.12.1“-8- =m=sIi.'.t.—t= 1°
The number of permutations of n objects taken. r at a time always exceeds the corresponding
number of combinations of n objects taken r at a time.
i.e. ,,P, > "C,
€'.g. 5P3 > 5C3 SIHCB 5P3 = 60 afld 5C3 =
In fact ,,P, = r!,,C,
Consider again the set of letters w. x. y and z. How many combinations of these 4 letters
can be made taking 2 at a time?
4: 4.3.2.1
‘C’ I 2:2! Z 2.15.1 I 6
and these can be written down as
wx. wy. wz. xy. xz. yz
It may be noted that ,,C, = ,,C,, _,. This identity is sometimes useful in facilitating
computations of these combinations.
The Binomial series
(l + x)“ expands to give
where the (k + l)"‘ term is given by
n(n—l)(n-2)(n—3)...(n—k—l)x"
7 7' '1 _ _;_ 7' _7 ‘iii 1
It!
Adopting the notation used for combinations from the previous section. the binomial
expansion becomes. if n is a positive integer.
l+nC|nX+nC1cX:+nC3sX3+...+xn
which may be rewritten as 2 ,,C, . x’
r=tI
11
In general (a + -b)" = 2 ,,C, . a""b'
r=fl
=a“+na" b+—(-iT)a"'=b3 +...+nab""' +b“_, nn—l T’
is
I
\
..
‘I
I
I
I
_.__.._..__.u_.....-i-
I
______.___.___.._..-.--.-._---—--
_‘_.__-.-
-nu-——-'.1-I-17‘‘Avs
»
l‘ I
it
l
14*-.-.g=p-Ii-—~s.=r".
I
I
II
1'-.-_,:fi;:»,-
.---~t‘;.---.---r-:1-I“
|I
I
."_.
F
I
I
I i
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
e.g. (a + b)5 = as + 5a‘b + 10a3b2 + 10a2b3 + Sab‘ + b5
The following properties of the expansion (a + b)“ should be noted:
(i) there are n + 1 terms
(ii) the exponents of a and b in each term sum to n
(iii) the coefficient of any term is "Ck where k is the exponent of either a or b.
The coefficients of the terms of successive powers of a and b can be arranged in a triangle
known as Pascal’s triangle, from which the coefficients of the terms of the next power's
expansion can be derived. Pascal’s triangle is usually written as
(a+b)°= 1 1
(a+b)‘= a+b 11
(a+b)2= a3+2ab+b3 1 2 1
.,@“ -...»@
Sis
(a + b)3 = a3 + 3a3b + 3ab3 + bi
(a +‘b)“ = a" + 4a3b + 6a2b3 + 4ab3 + b‘1
(a + b)5 = as + 5a"b + 10a3b3 + 10a3b3 + 5ab" + bi 1 5 10 5 1 I
The coefficients of the terms in the expansion of (a + b)° can easily be derived from the next
line of Pascal‘s triangle
1 @ ® 10 5 1
1. \ 1
1 o ® 20 15 6 1
so that (a + b)“ = a“ + 6a-‘b + 15a"b2 + 20a~‘b-‘ + 15a3b‘ + 6ab5 + b“
A useful expansion for probability theory is that of (q + p)“ which is
(q + P)“ = ..Cs1>°q" + ..CtPq"" + ..¢=1>’q"" + ..CtP’q“" + - - -
The Exponential Series
. 1 " . . .The series (1 + expanded binomially gives
—1+1+n2_n+n3"3n:+2“+_ 2n= an‘
—1+1+1 1+1 1+1+" 2 2n 6 2n ant
and. in the limit as n—> ==
111 jIn
(1%-H) -1-I-1+5-I-6+iz+...
= 2.7182 . . . which is denoted by the letter e.
It is the base for Napierian or natural logarithms (ln).
KONSTANTINOS
Rectangle
An exponential series may be written as
x x xi X3
6-1-I-F-I-2|+3'+...
writing cx for x gives
cx cx cixz c3x3
B -1.-I-—'-I"i'+i'-I-...
1. 2. 3.
and letting e° = a (so that c = loge a) and substituting for c gives
:1 2 3] :1
a"=1+x1og,a+x (igfa) +x(c;g!°a) +...
Appendix A contains a note on Maclaurin's Series which is particularly useful in deriving the
expansions of a number of the more common functions.
The Logarithmic Series
It is not possible to obtain directly a series for loge x but it can be shown that:
I 1+)~ x:+x3 x4+ f 1< <1og,,( x x 2 3 4 ...or x~..
and similarly
1 1 )—— x~:—x3~x4 t 1-== <1og,,( x— x 2 3 4 ...or -.x
By subtraction.
fr
I
I
I
mew’:-nu-v-<-nmpnn-run-m¢g,_':;-amen-—=q-1.11”-n.:-=I-I-I-1=-311--II-Q'|t=ei:_“‘rI-I"==""'-'-=-
I
I
1+x 7 x3 x5
IOgg[""-'1_x:|=...[X+"3-+?+...] U
_ 1+x n-l
and letting -i = n so that x =i then1-x n+1
1 —2[(i)+-1-'“'1)3+-1-(“'1)5+ jt >0og°n_ n+1 3(n+1 5n+1 or“ '
This series does not converge very rapidly and is therefore of limited value.
ft‘-'L:-I;\1
--—-I_ fi“_;-1-st_-t._
fr
9
l
J
I
I
I
KONSTANTINOS
Rectangle
Chapter 4
Probability Theory and
Distributions
The probability of something happening (an event occurring) is expressed as a number
between 0 and 1. If it is impossible for an event to occur, its probability of occurrence will
be zero; but if the event is certain to occur the probability of occurrence will be unity. There
is a classical definition of probability which states that if an experiment can result in any one
of exactly N mutually exclusive and equally likely outcomes of which n involve the
occurrence of the event E, then the probability of the event E occurring is given by:
Pr(E) = g
or, to express the same idea in a different way, the probability of an event E occurring is
equal to the number of ways that E can occur divided by the total number of possible
outcomes. For example when a coin is tossed it can result in any one of two possible
outcomes; hence the probability of obtaining a ‘head’ is equal to the number of ways that a
head can occur (only 1) divided by the total number of possible outcomes (which is 2).
Hence Pr(head) = i. a well known result. Similarly the probability of obtaining a six (or any
number from one to six) when rolling a fair die is i .
The above definition is sometimes criticised for being circular since “equally likely" implies
equally probable so that probability has been defined in terms of itself. Mathematically too
the definition poses problems because theoretically there could be an infinite number of
cases. For these reasons the definition of probability is sometimes given as follows:
if in a sequence of N experiments performed under similar conditions the event E has been
observed to occur n times the probability of E is
_ I1we = in st
The current treatment of probability is such that the probabilities of events must satisfy
certain axioms.
Briefly these are:
(i) Pr(E) 2 0
(iii) Pr(E) =5 1
(iii) if two events, A and B, are mutually exclusive (i.e. they cannot occur simultaneously)
then Pr(A or B) = Pr(A) + Pr(B).
Axiom (iii) can be generalised for any events (not necessarily mutually exclusive) so that
Pr(A or B) = Pr(A) + Pr(B) - Pr(A and B)
where Pr(A and B) = 0 for mutually exclusive events.
ll
KONSTANTINOS
Rectangle
Example I Consider an ordinary pack of 52 playing cards. What is the probability of
drawing a spade or a king?
Let A = the event ‘drawing a spade’
B = the event ‘drawing a king’
Pr(A or B) = Pr(A) + Pr(B) — Pr(A and B)
13* I4 f lg _ lo *4
‘ls2+s2 s2 s2°'13'
This follows because there are 13 spades in a standard pack of cards, there are 4 kings, and
there is also one card (the king of spades) which is both a king and a spade. Hence the
events are not mutually exclusive and the generalised formula applies.
Cona'z'tz'0nal Probability
If A and B are two events, the probability that B will occur given that A has already
occurred is denoted by Pr(B,/A) or_Pr(B given A) and is called the conditional probability of
B given that A has occurred.
If the occurrence or non-occurrence of A (or B) does not affect the probability of
occurrence of B (or A) then
Pr(B,"A) = Pr(B) and Pr(A/B) = Pr(A)
and the two events are independent of each other. Letting AB denote the event that "both
A and B occur" then
Pr(AB) = Pr(A) . Pr(B/A)
which, if A and B are independent events, becomes
Pr(AB) = Pr(A) . Pr(B)
For three events A, B. and C this can be written as
Pr(ABC) = Pr(A) . Pr(B/A) . Pr(C/AB)
which, if A, B and C are independent, becomes
Pr(ABC) = Pr(A) . Pr(B) . Pr(C) '
This result may be generalised for any number of independent events.
Example 2 Calculate the probability of obtaining a head on both the first and second toss of
the same coin.
These two events are obviously independent of each other.
Let A = head on first toss, Pr(A) = %
B = head on second toss, Pr(B) = l
Pr(AB) = Pr(A) . Pr(B) = i X % = i
Example 3 Five vessels are due to arrive at a port on a given day. Their times of arrival are
unknown, but on arrival they will be allocated to the same berth on the basis first come, first
served. Three of these ships are general cargo vessels and the other two are semi-container
ships. What is the probability that the first-two vessels to be handled are general cargo
ships?
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
‘_-
Let A = first vessel a general cargo ship
B = second vessel a general cargo ship
Then Pr(AB) = Pr(A) . Pr(B/A)
3 2 3
= 3'1 = E
The explanation is straightforward. The first vessel can be chosen in any one of three ways
out of a total of 5 possible ways. Given that the first vessel was a general cargo ship. this
leaves four vessels of which only two are general cargo vessels. Hence. given that the first
has already been chosen, the second general cargo vessel can be selected in two out of the
four ways.
Bayes Theorem
It has been shown earlier that
_ Pr(AB) _ Pr(BA)
Pr(A/B) - -——-—P1_(B) and Pr(B/A) - -—F(A)
Now. since Pr(AB) = Pr(BA)
Pr(AB) = Pr(B) Pr(A/B) = Pr(BA)
P B P A B
Hence. Pr(B/A) =L( which is Bayes Formula.
In general. Pr(A,/B) e nPr(B/Ai)Pr(Ai)
2 Pr<B/A.>Pr(A.)
i= I
where n is the number of possible sub-sets.
This formula is useful for decision-making in evaluating additional information and arriving
at posterior (as distinct from prior) probabilities. A practical example will help to explain its
usefulness.
Example 4 Three berths in a container port are allocated to vessels involved in the carriage
of a particular commodity. The joint probability table of loading times against individual
berths is as follows: tn 1,
. §1l% F IA_;1_T_.€_..,‘_ Loading time (X) 3'. .. _
less than 6 hrs 6 hrs to 9 hrs more than 9 hrs
Berth I 0.09 . 0.13 0.11
(Y) II 0.08 0.15 0.09
III 0.06 0.24 0.05
P}. ’Ll J
Determine (i) whether a loading time is indep§ndent£f*_th hich is allocated to the
vessel. (ii) the respective probabilities of are berth used given that a vessel was loaded in 7
hours.
(i) Independence will be established if it can be shown that Pr(XY) = Pr(X) x Pr(Y) for
every X, Y cell in the table: so is Pr(loading time < 6 hrs and Berth = I) =
Pr(loading time < 6 hrs) X Pr(Berth = I)?
is 0.09= 0.23 x 0.33‘?
i.e. is 0.09 = 0.0759?
Obviously the answer is 'no‘ and hence the loading time depends on which berth is
allocated.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
(ii) What is Pr(berth = I,/loading time = 7 hrs)?
Z Pr(berth-_= land loading time = 6-9 hrs)
T Pr(loading time = 6-9 his)"" C
0.13
=F0.2s
Similarly.
Pr(berth = II/loading time = 7 hrs) = = 0.29
and Pr(berth = III/loading time = 7 hrs) = = 0.46
Notice that the sum of these three probabilities will equal one. This solution can be
presented explicitly in terms of Bayes Theorem which states that:
_ _ Pr(loading time = 6-9 hrs/berth = l)
Pr(berth = I loading time = 6-9 hrs) Q, e~ 3 *0 * 0 ~ rs 0
E Pr(loading time = 6-9 hrs/berth = j)
j=l
Z3 0113 f tbl)"0.i3+0.1s+0.2-i ('°‘“ a °
0.13 _ -
= E5E=0.2D
The probabilities of berths and loading times can be found by summing across the
rows and down the columns. Hence
____ 777 __ _ " _ " ' _ _ —— ___ —|7—— — _ __ ————
Loading time
Probability
Berth <6 hrs 6-9 hrs >9 hrs of berth
I 0.09 0.13 0.11‘ 0.33
II 0.08 0.15 0.09 0.32
III 0.00 0.24 0.05 0.35
Probability 0.23 0.52 0.25 1.00 ’
of loading
time I '7
_ ,__ it-, 3
which can be expressed in general terms as *1.' l.
/
j fife’ 7" 2 i. i v 4|
Loading time
Probability
Berth B1 B2 B3 of berth
A1 PT(Bi/A1) PT(B2/A1) P1'(B3/A1) P1'(Ai)
A: PT(B 1/A2) Pr(B2/A2) Pr(B:/A2) P1'(Az)
A3 W g PT(Bi/A3) H PT(B2/A3) PT(B3/A3) g P1'(A3)
Probability Pr(B1) Pr(B2) Pr(B 3) 1.00
of loading
time
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
P B A 0.13 0.13
Hence Pr(A1/B3) = 3 r( 1) P Prams ) I S2e 0.25 as given above.
2 -Z1 Pr(B;/Ai)
J=
Probability Distributions
Probability distributions are broadly classified into two types: discrete and continuous. A
discrete probability distribution is one where the variable involved can assume only discrete
values, e.g. the number of children in a family, or the number of vessels queueing for a
berth. A continuous probability distribution (sometimes called a density function) is one
where the variable can assume any value within a given range, e.g. the current in a
conductor, or the annual rainfall in a particular location. Certain key probability
distributions have been identified: these show that under certain specified conditions the
probability of events occurring always conforms to the same pattern and hence a standard
formula can be employed to calculate these probabilities. Three of these standard
probability distributions will be considered; the Normal, the Binomial and the Poisson
distributions.
(a) The Normal distribution is a continuous probability distribution of the form
_ 2
Pr(X) = ET/1—2Ee_§(£F-E)
which looks more complicated to use than it actually is. A standardised Normal (or
Gaussian) distribution has been tabulated to give the required probabilities. The distribution
is symmetrical about a mean value of u with a standard deviation of o (where rt and e have
the usual meanings and approximate values of 3.14159 and 2.71828 respectively). The total
area bounded by this curve and the x-axis is one, and the area under the curve between any
two values represents the probability that the random variable lies between those two
values.
The shaded area in Figure 4.1 represents the Pr(\/1 < x < V2) where x is the value of the
random variable X.
Figure 4.1
i‘ -._l
0
Q l
KONSTANTINOS
Rectangle
Values have been tabulated (see Table 4.1) for standardised random variable Z = ET;-E
1 .and the standardised form of the normal equation becomes Pr(x) = --— e"“'3 where Z
\/Zr:is Normally distributed about a mean value of zero and has a standard deviation of one.
Figure 4.2
' ‘ | ll
l i I . ‘
l
--5--_-J----~ —,_ l — ~___
-30 -20 -10 X +10 +20 +30
As well as being able to calculate probabilities from this distribution, it is also possible to
derive confidence intervals within which one can be cx% confident that a variable will lie. It
is usual to express these intervals as confidence limits of 95%, 98% or 99% probability
(though any level of probability may be used and the corresponding value calculated from
the table) and the limits are given by the mean value plus and minus a certain number of
standard deviations. (See Figure 4.2.)
. . . X "' . . .It is known that the standardised variable Z = -FE is Normally distributed about a mean of
zero and further that the distribution is symmetrical.
Figure 4.3
I
t
4.._,,.._
0
i.
!
3:"--. \
-0’,-_____.
__.
l
v
8.34 *1
‘l
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The Normal table (see Table 4.1) gives the area to the left of the Z-value, i.e. it shows the
probability that x < Z as depicted in Figure 4.3. If Z = 0 then it follows, because the curve
is symmetrical, that Pr(x < 0) = 0.5.
When confidence limits are calculated, a certain percentage (say 95%) of the variable values
are expected to lie in a particular range. The range is given by the mean 1 Z standard
deviations. How is the value of Z to be determined? Figure 4.4 shows the required
Table 4.1 The Normal Distribution
Z 0 1 2 3 4 5 6 7 8 9
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2L9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9773
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
* 1.0000
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0. 7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7703
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0. 8943
0.91 15
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7793
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.99930.9995
0.9997
0.9998
0.9998
0.9999
0.9999
0.9999
1.0000
,.
l
in
KONSTANTINOS
Rectangle
Figure 4.4
, 95%
2.5% 1 2-5%
i 1 '1 i i 3 I i I 1' 7 I "I i 1 i i
interval containing 95% of the values. This means that 5% must be excluded from the
interval and, given the symmetry of the curve. this implies that the 5% must be divided
equally between the two tails (see Figure 4.4). The appropriate Z value is found from the
Normal table by finding a value which has a 97.5% probability of not being exceeded or
equalled. Thus the area to the left of Z must be 0.975 and the value of Z corresponding to
this probability is 1.96. Similarly, for 98% and 99% confidence intervals, Z = 2.33 and 2.576
respectively. The figures only apply in two-tailed tests and readers should verify these results
from Table 4.1.
These Z values can now be used to determine the confidence intervals. Since
X " ll . . -Z = -T it follows that (11 — 0Z) < x <1 (ii + oZ) where it is the mean. 0 the standard
deviation. and Z the positive or negative value from the Normal tables corresponding to a
particular probability.
Roughly, the meani 2/3 of a standard deviation will contain 50% of the values
the mean 1 one standard deviation will contain 68°/,, of the values
the mean i two standard deviations will contain 95°/,, of the values
and the mean i three standard deviations will contain 99% of the values
Example 5 The annual rainfall figures (in cm) recorded at a particular weather centre and
referring to the 10-year period 1975-1984 are as follows:
63.1, 63.5. 66.0. 72.1, 73.1, 76.2, 68.8, 70.2, 77.4, 69.6.
A ‘wet’ year is defined as one during which more than 78 cm of rain falls while a ‘dry’ year
is one when less than 60 cm of rainfall is recorded. Assuming that the distribution of rainfall
approximates to a Normal pattern, find
(i) the probability that any particular year, say 1985, was a ‘wet’ year
(ii) the probability that 1985 was not a ‘dry’ year
(iii) the probability that 1985 was ‘dry’ and 1986 was not ‘wet’. assuming statistical
independence.
The first step in the solution is to calculate the mean (ii) and standard deviation (S) of the
data. Most calculators will do this automatically but alternatively the result is easily found by
using the formulae
E X. Z Xi Z Xi 2 .,'=1 i=1 i=
8 = T and $2 = T e --%l— In this example 2 xi = 700 giving a mean
i=1
KONSTANTINOS
Rectangle
3
Figure 4.5
1 1 Q I
1
-3 s.d. -2 s.d. -1 s.d. X +1 s.d. +2 s.d. +3 s.d.
l 8 68% ——-
- 06%
- 09% 4 .4 2 W-»-
H
it = 70 and 2 xf = 49214.72 leading to a variance S3 = 21.472 and hence the standard
i=1 .
deviation S = 4.6338.
l (i)
‘?
Pr (Wet) = Pr (rainfall > 78)
The data are continuous so there is no need to make any adjustment to the numbers
(see example 7).
x — |.l 78 — 70
I 5 ~55 7 2 _ 2
| Z 0 4.6338 L776
C)
¢I'II"\'Ih-\-p—uq,--..
i.
I
-=-.1"}-—-
-vg .
..._,_,,.’-_,,______,,_._(
I
“' 37
6
4.
Figure 4.6
i Dry , ‘- Wet
. .. "-
.. ,_ '__
. I
3) ' _ ._-:-::_-'-:: :_
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
From Normal tables the probability corresponding to Z = 1.7262 is 0.9579 but this is
the area to the left of 78 cm hence
Pr(wet) = Pr(rainfall > 78 cm) = 1 - Pr(rainfall < 78 cm)
= 1 — 0.9579 = 0.0421
(ii) Pr (1985 was not dry) = Pr(rainfall > 60 cm in 1985)
_x—|i*60—70
UseZ* O -4.6338e— 2.16
Find the area to the left of Z = +2.16 in the Normal tables and subtract from one to
get the value corresponding to Z = -2.16
Z = +2.16 corresponds to 0.9846
Z = -2.16 corresponds to 0.0154 but this is the area to the left of
rainfall = 60cm.
The required area is the one to the right i. e. the Pr(rainfall > 60 cm) which is
1 — 0.0154 = 0.9846.
theprobability that 1985 was not a dry year = 0.9846.
(iii) Pr(1985 was dry) = 1 - Pr(1985 not dry) = 0.0154
Pr(1986 was not wet) = 1 — Pr(1986 was wet) = 0.9579
Assuming statistical independence
Pr(1985 was dry and 1986 was not wet) = 0.0154 X 0.9579
= 0.01475
(b) The Binomial (or Bernouilli) Distribution
The Binomial distribution is a discrete probability distribution that arises when there are n
independent repeated trials each with a constant probability of success (p) and a constant
probability of failure (q) where q = 1 — p. The probability of there being exactly r successes
out of n such trials is given by
Pr(x = r) = ,,C,p'q“" for r = 0. 1, 2 . . .n
and is obtained from the Binomial expansion of (p + q)“ as explained in Chapter 3.
The mean of the Binomial distribution is given by np and the variance by npq where
q = 1 L p. It is possible to relate the Binomial and Normal distributions. If n is large and if
neither p nor q is close to zero the Binomial distribution closely approximates a Normal
distribution with standardised variable given by Z = This approximation improves as
n increases in magnitude and, in practice, the approximation is good if both np and nq are
greater than 5. Allowance must be made for mixing discrete and continuous variables; see
the second example which follows.
Example 6 In the selection of graduates for a management training scheme in a large
shipping company it is expected that on average only 70% will complete their period of
training. What is the probability that out of a group of 5 new recruits 3 or more will fail to
finish their training?
4
L
)-e___.._3___..-1._e_4
I
I
1
|
I
1
4.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
F
__-‘._-.2._.._._.
l
l
r
.~.-..._..-__--1-.
)
>
Pr( failure to complete training) = 0.3 = p .'.q = 0.7 and n = 5
Pr(3 or more will not complete)
Pr(3) = 5C3(0.3)3(0.7)2 = 0.1323
Pr(4) = 5C4(0.3)“(0.7)‘ = 0.0284
Pr(5) = 5C5(0.3)5(0.7)° = 0.0024
.'.Pr(3 or more fail to complete)
Example 7 A major ship chandler plans his orders on the basis that his expected share of
sales will average 21% of the total market sales per month. Assuming normality with a
= Pr(3) + Pr(4) + Pr(5)
= 0.1323 + 0.0284 + 0.0024 = 0 1631
standard deviation of 1% determine
(i) the probabilities that his market share will exceed 22.5% at least twice in four
successive months
(ii) the probability that his market share will lie between 20 6% and 21 8% not less than
20 nor more than 30 times in 50 successive months.
Figure 4.7
Qi-iDI = £112
 
21% 22.5
gx-it 22.5-21.0
Z 6 if 10 41.5
Pr(Z < 1.5) = 0.9332 from Normal Tables
Pr(x Q 22.5) = 0.9332
and Pr(x > 22.5) = 1 — 0.9332 = 0.0668
Hence p = 0.0668 and q =1 - p = 0.
For (i) use the Binomial distribution “C, p’ q“" with n = 4, r = 0 1, and 2 and
p = 0.0668. .
Pr(sales > 22.5% at least twice) = 1 : [Pr(sales > 22.5% never)
+ Pr(sales > 22.5% once)]
Pr(sales > 22.5% never) = 4C0(0.0668)_° (0.9332)‘ = 0.7584
9332
Pr(sales > 22.5% once) = 4C1(0.0668)‘ (0.9332)3 = 0.2172
Pr(sales > 22.5% at least twice) = 1 — 0.9756
= 0.0244
0.9756
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
nap...- unA_4u.>.4.
lp-..--1---1-vvn-vnwn M
q_ ,5 2 Figure 4.8 '
20.6 -— 21.0 21.8 - 21.0
Z; = -—T-0-—and Z2 =T
(ii)
.'.Z| = and Z2 =
From Normal tables the areas are
area to -left of Z2 = 0.7881
area to left of Z1 = 0.3446
by subtraction the required area between the two Z values = 0.4435.
' Figure 4.9
i
1 i
¢-ii
i! = >(—np
‘JFIDQ
,___._.___..___.... [_________
-26- 22
19.5 lnle$III 1'.‘ HI II
Use p = 0.44, q = 0.56, n = 50 and the Normal approximation where np = 22
npq = 12.32 \/npq -'= 3.51
NB The discrete x > 20 is equivalent to x > 19.5 on the continuous scale.
Z; = (30.5 - 22)/3.51 = +2.42
Z, = (19.5 — 22)/3.51'= -0.71
Area to left of Z, = 0.99224. Area to left of Z, = 0.2389. Hence the required probability is
Pr(20 < x < 30) = 0.9922 — 0.2389 = 0.7533
Figure 4.10
Frequency
X
31
.-_._-_._-.-.
1
.,-..-..-.0...--..._-.-....__.V
Q.
l
-:,_~
-_-'---1.-_._.-_.
4'=
.-124222.-.22{‘~"-»_--J_--3:61“-F-_--.4»-4
1
11
4‘)1
1
.4,‘-.
I
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
(c) The Poisson Distribution
The negative exponential probability distribution depicted in Figure 4.10 is a continuous
distribution which can represent theprobability pattern of the time between consecutive
occurrences of a particular event. The total area under the curve is one and the area under the
curve between t = 0 and t= T represents the probability of there being a time less than T
between consecutive events.
The related discrete probability function giving the number of occurrences during a time
interval t has the form
e"“ lit ’Pr(r events in time t) = --£4 for r = 0,1,2...
and is known as the Poisson distribution where lit is the average number of occurrences in
time t.
The Poisson distribution is sometimes explained as the one which deals with rare events. i. e.
with events whose probability of occurrence is very low. The classic example concerns the
number of cavalrymen being killed by a horsekick in the course of a year. The important
feature is that the number of times a particular event occurs is very low compared with the
number of times the same event does not occur. It is impossible to state how many times
something does not happen unless a limit on the total is given. and this demonstrates the
essential difference between the Poisson and Binomial distributions. The latter has a clearly
defined number of trials (n) and the event will occur r times and not occur (n - r) times: no
such precision concerning non-occurrence of an event is possible under the Poisson
distribution.
The mean of the Poisson distribution is denoted by lit which is also equal to the variance. It
is possible to relate the Poisson to the Binomial and the Normal distributions with the
. . . . - 71. . . . . . .standardised variable being given by Z = This approximation is valid provided p (the
probability of success) is small and n is large. Under these conditions q will tend to unity and
hence the mean (np) will approximate to npq. Conventionally, small samples are those where
n < 30, so for n to be large it must exceed 30. Similarly a small value of p would be when
p < 0.1 implying a value of q > 0.9.
Example 8 A ship is discharging bulk phosphate directly into a fleet of lorries which arrive at
the dock. on average. once every twelve minutes. Assuming a Poisson arrival pattern find
(i) the probability that there will be no arrivals in the next hour
(ii) the probability that more than one lorry will arrive in the next hour
(iii) the probability that exactly three lorries will arrive in the next two-hour period
= 65*‘ . (/11)’
Pr(r. t) H
1 arrival every 12 minutes is the equivalent of 5 arrivals every hour so it = 5
For(i)r=0,t=1,andltt=5
e“5 . 5" _Pr(0, 1) = —-5-_-— = 6 5 = 0.0067
—=.=--nr——
KONSTANTINOS
Rectangle
__ e“5 .5‘ _For (11) Pr(1, 1) = —i,—- = 5 . 6 5 = 0.0335
Pr(0. 1) + Pr(1, 1) = 0.0067 + 0.0335 = 0.0402
Pr(morethan 1 in the hour) = 1 — [Pr(0, 1) + Pr(1, 1)]
= 1 - 0.0402
= 0.9598
For(iii)r=3,t=2,A=5
-111.103 510.1Pr(3,2)e 3;. 1° 6000-- 0.00757
KONSTANTINOS
Rectangle
~
I8
l
F ___
I.
I
I
l"U .
B
-_{._.44-’.-q-1--4--4-_--v-.,_1-——-—4--.4-_
F
l
-“.14
1
l
i;H
l
1‘
I)
Chapter 5
Basic Economic Relationships
Demand and Elasticity
It will be assumed that readers are familiar with the basic principles of demand and supply
and with the derivation of the general shape of such curves as well as with exceptions to the
general case. Demand (average revenue) curves normally slope downwards from left to right
and reflect an inverse relationship between price and quantity demanded. Supply curves
slope upwards from left to right and reflect a direct relationship between price and quantity
supplied. In other words. if the price of goods or services rises. less will be bought and more
will be supplied.
Figure 5.1
/1 .-.7r:">-
P2‘ 4-» — 44 —-4—-—-P».i ’+
- H - — 4 ;lP144Pr'cG
‘U Ii
D=AR
OD: O01 8 8 5 8 8 061 O52 2*“
Quflfllflv I901-Ighl Quantity supplied
Clearly the demand for and the supply of particular goods or services are not divorced from
each other: they simply represent different -sides of the same theory. The two components
must be taken together to arrive at an equilibrium price and quantity which satisfy both the
consumers and the producers of the goods or service. (See Figure 5.2)
The equilibrium position can be disturbed by a change on either the demand or the supply
side (or both). and the “state of the market” or the “market conditions” are determined by
the relative strengths of demand and supply.
Demand conditions may change due to a change in income while supply conditions could be
altered through a change in technology, as for example. with the introduction of
containerisation. Such changes will alter the physical position of the demand and/or the
supply curves within the axes.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Figure 5.2
S
.3
if
Pe
l
l
J
'1
I1
i
D l
Qe
Quantity bought and supplied
.5
In Figure 5.3 there has been a shift in the demand curve from D, to D; (possibly caused by
an increase in consumers’ income) and also a shift in the supply curve from S, to S2. The
latter represents an increase in supply (i.e. the producers are prepared to supply more at any
given price) which could have been caused by, for example, a reduction in the costs of
production. The combined result of these two changes considered together is to give a much
larger output at a slightly higher price. It is a simple matter to consider the changes
separately as shown in Figure 5.4.
This type of analysis leads in two directions. First it leads to a discussion of the revenue
implications of different levels of demand and supply and. second. to a consideration of the
rates of change of price and quantity in response to market forces. The revenue aspect is
straightforward and revolves around three definitions, namely those of Total Revenue (TR).
8 1Figure 5.3
S1
S2
‘L -P2 __ __ __ __ i ._.
kg P1 _ -.
v ix D2
D1
Q1 Q2
Quantity demanded and supplied
l
_._._»._.______4n_._A_Z-imA
1
!
‘I.
1"‘
.';.u._Z.44.4
1
'1.
1
14
4
i
1
ii
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
.._i-.__..,fl’._..._-.-_-_.
i- s ‘
1
8
l.1
u-vg-.-_.gnu.-—-—-
l
4,.__+_____
1
1
l
l
1
i
+J44—-vfiw
llll
s
iI.
tr’
I
I»ii
‘_":§>"4
r
1.
l
?'
It
i
_,.
?
l
I
I
Figure 5.4
l, S1
P2 \ S2
I , Pi
8 P1 / M til"
D1 0‘ D
L
01 Oz 01 02
Quantity Ouamitv
Marginal Reinue (MR) and Average Revenée (AR). Total revenue is defined as the
revenue derived from all sales of particular goods or services and is the product of the
number of units sold and the price per unit. Marginal revenue is the change in total revenue
brought about by selling one unit more or one giiitzglmegss -— this unit is known as the marginal
unit. Average revenue is El€fiiiéTl“§§ tneaxii Pévenué resulting from all sales of goods or
services divided by the number of units sold. In other words
TR = P X Q where P = price and Q = quantity
MR =- = TR“ 1, — TR“ where n signifies the nth unit of the good.
TR P.Q
AR-Q- Q -P
The second direction leads to a discussion of the concept of elasticity which may refer to
either demand or supply. Three types of elasticity of demand are considered in the literature
and are known respectively as price. income and cross elasticity. of which the mostcommon
is plice-elasticity of demand.
The usual definitions are
(1) Price elasticity of demand (Ed) is the responsiveness of demand to a change in price
proportionate change in quantity demanded
Price Ed proportionate change in price
50
Q 60 P
Hence Price Ed = § = -6 .
P
P50
= 6'5
(2) Income elasticity of demand is the responsiveness of demand to a change in income and
is measured in the same way as price elasticity of demand but substituting income for
price in the formula. It will be positive for normal goods.
""--_._...
(3) Cross elasticity of demand (Ex!) is defined as the proportionate change in the quantity
demanded of a particular commodity x in response to a proportionate change in the
price of a different commodity y. '
7
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Hence
293.5
E _ Qx __6Qx l’_y_
*>"@'ox‘6Py
Pr
_@»Qx 21.
*61>y'ox-
The cross elasticity of demand will be negative if x and y are complementary goods. and
positive if they are substitutes for each other.
Reverting now to price elasticity. which is usually what is meant by the phrase "elasticity of
demand”. several points need to be made. First. there is a distinction between arc- and
point-elasticity. By definition. the price elasticity of demand is a measure of the
responsiveness of demand to changes in the commodity‘s own price. If the price changes are
very small. so that the points are very close together on the demand curve. the
point-elasticity of demand is being measured. If the changes of price are not small then
elasticity is being measured along an arc of the demand curve. and problems may arise in
calculating a value for .the price-elasticity of demand. For example. consider. in Figure 5.5. a
Figure 5.5
Price
P1 =12 (11:20
P2=1O Q2=25
12 ~ A
110 4 B
I
D=AR
Q 20 25 Quantity
_""' demanded
fall in price from £12 to £10 per unit which results in an increase in quantity demanded from
20 to 25 units.
6Q +5
Q 20The formula states that Ed = -5? = -:5
? T2’
5 12 60
5 x 664 : 4
J
1
"I
I
l
‘ 1
1
l
1
I
1
1
U
1
1
u1,-4-.---i"‘-+--.--»=-=Q‘%t;
i1
1 520 -2 -40 ' l
Hence the elasticity of demand measured along the arc from A to B has a value of -1.5. l
Will this be the same value as that of the elasticity of demand measured along the arc from
J
-4
d .
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
. 7 I 7 _ __ _ 7_7_ 7 7
l
i l
’ Unity 1 Ed = 1
. l
i‘ Inelastic ‘ 0 < Ed < 1 1
B to A? Unfortunately not; the arc-elasticity of demand from B to A has a value of -1. The
reason for this ambiguity is that A and B are not sufficiently close together to measure
point-elasticity of demand and the proportionate changes in price and the quantity are
significantly different when taken as proportions of the original and final figures. In other
words the direction of change has become important. To measure arc-elasticity in an
unambiguous manner the proportionate changes in price and quantity must be expressed as
proportions of the average price and average quantity respectively. Hence the correct
formula for measuring the value of arc-elasticity is
Ed = <0. + on/2 t t>o_ ‘(rt + P2)
6P 5P (Q1 + Q2)
(P1 + P2)/2
Secondly. it is conventional to multiply formulae for price elasticity of demand by the
number (-1). This has the effect of nullifying the inverse relationship between price and
quantity and means that the higher thedgnurnverical !;1lue_ of the price elasticity of demand the
more elastic is the demand. ' ' i '
Demand is usually ciassified according to whether it is elastic, inelastic or has an elasticity
which equals unity. It is the value of unity which is critical in this respect and the
appropriate terminology may be summarised as:
; Classification p Value of elasticity
l Perfectly elastic l - Ed = 1
Elastic 1 < Ed < ac l
1"’ _ _ _ _ ______ ——————— ~ — —
__ __ _ ___ __7_ ——7* *'— 7 ' 7 _____ *1l
A Perfectly inelastic * Ed = 0
With three exceptions the elasticity of demand varies at every point on the demand curve.
The exceptions are shown below in Figure 5.6.
Figure 5.6
. 0 9
l
.§ ._ .0.
l ‘v n D
in at pgflgcfly umu; dgmgnd mm; 5., = Q {ii} A Dcfioetly inelastic demand curve fie =0 um A rectangular hyperbole flu =1
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
To obtain an indication of the elasticity of a demand curve at a particular point it is
sometimes useful to consider the effect" on. total revenue of varying price. If a small
reduction in price leads to (i) a larger proportionate increase in quantity demanded so as to
increase total revenue then the demand will be elastic (Ed > I), (ii) a smaller proportionate
increase in quantity demanded so as to reduce total revenue then the demand will be
inelastic (Ed < 1), (iii) a corresponding increase in quantity demanded so as to leave total
revenue unchanged then the elasticity of demand will be unity (Ed == 1).
Consider again the previous example:
Figure 5.7
Price
P1 :12 01:20
P2=10 02:25
<5P=-2 6o=+s
112 “'" ”
10 ~—. e i B
l
( D=AR
Q 20 25 Quantity
_"' demanded
6 P PThe arc elasticitv of demand :< Q I + 2) (>1)' i>P'(Qi+Q;J'
_(W_1_s (1z+10)
' J-2'(20+2s)
-5 22-12222'2'4s"
Original total revenue = P, X Q1 = 12 X 20 = 240
Final total revenue = P; X Q3 = 10 >< 25 = 250
i'.e. the increase in total revenue suggests that the demand will be elastic and that the price-
elasticity of demand will exceed one. '
It follows that when the total revenue has increased (which implies that the demand is elastic)
the marginal revenue will be positive, while MR will be negative where the demand is
inelastic and a fall in price produces a decrease in total revenue. Revenue will be maximised
when the price elasticity of demand is equal to "unity or in other words when MR is zero.
48
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
This relationship can be expressed graphically as
Figure 5.8
Price
(Pl
€d>1
§d=‘
gdci
O
MR
or mathematically as:
P = f(Q)
.'.Total revenue (TR) = P . Q = f(Q) . Q
. d(TR) .Marginal revenue = TQ- = f(Q) . 1 + Qf (Q)
.-.MR = f(Q) + Qf'(Q)
Price elasticity of demand is defined as
Ed rd _ f(Q)
Q -<11’/dQ Qf'(Q)
. Q.Ed 1 - f(Q)hencei = -—- and inverting. f '(Q) =if(Q) f'(Q) Q - Ea
Substituting in the expression for MR
ii/iR=i(o)+Q1fi)l=(i>+El;)Q . Ed
. _ _l_ where the value of E is conventionally
Le‘ MR _ P0 + Ed) expressed as a negatidve number.
A demand or other function is often expressed in the form:
P=Q"
sothat LogP=xLogQ
. rI
dP _ dQand P - x Q
dP Px
°' 5:15"
AR = Demand
Quantity (Q)
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Now the elasticity of P with respect to Q
_ £91 - iii._ Q . P
d P d P l
5* dP IP51?Q1115 Q6"
Hence, if P represents price and Q is the quantity of goods demanded, the price elasticity of
demand is given by:
1
Ed = I
This relationship is often useful for empirical analysis using multiple regression techniques.
Derived Demand
It may be useful to comment here on the idea of a derived demand. A derived demand
exists when goods or service are demanded not for themselves but for their usefulness in
producing other goods or services. The demand for any factor of production is a derived
demand as is the demand for sea transport. A fuller discussion of this topic is contained in a
subsequent chapter but briefly it can be stated that the elasticity of demand for a factor of
production. or any goods or service the demand for which is derived. will depend on the
elasticity of demand for the goods being produced or transported. on the ease with which a
substitute can be used. and on the proportion of total costs accounted for by payments to
that factor.
Supply, Costs and Elasticity
This chapter has so far dealt with the theory of demand and the revenue implications of
changes in price but it is equally important to consider the supply side and the theory of
costs. The basic definitions of cost involve the ideas of total. marginal and average cost.
Total cost (TC) is defined as the total cost of all factors of production involved in the
production of a certain level of output. It is divided into two components. viz. total fixed
cost (TFC) and total variable cost (TVC). '
i'.e. TC = TFC + TVC.
Fixed (or indirect) costs are those which do not vary with output while variable (or direct)
costs are those which do vary as output changes. A distinction must be made between the
long and short-run time periods which refer, not to any length of calendar time. but to the
ability to alter the quantities of factors of production being used. In the long run all factors
of production and hence all costs are variable. The short run is defined as the period of time
over which the inputs of some factors cannot be varied.
Marginal Cost (MC) is the change in total cost brought about by producing one extra unit
and it is usual to distinguish between short-run (SRMC) and long-run marginal cost
(LRMC). Average cost is the totalcost of production divided by the number of units
produced and will have two components. viz. average fixed cost (AFC) and average variable
cost (AVC).
KONSTANTINOS
Rectangle
These definitions can be summarised as:
TC = TFC + TVC
ATC = -F-lg where ATC is Average Total Cost
TFC+TVC=-—-Q——=Ai=c+Avc
_ d(TC)
MC ' dQ
or MC“ = TC“ — TCd__, where n is the nth unit.
The supply curve will reflect the costs of production and as these costs change. due perhaps
to technological progress. so will the position of the supply curve between the axes.
Figure 5.9
51 52
 -
Bsp- . -
Quantity supplied
Figure 5.9 illustrates an increase in supply in response to changes in the conditions of
supply. In this case a reduction of costs permits more to be supplied at the same price and
the supply curve moves to the right. A great deal of what was said above concerning
demand theory can also be applied to the theory of supply; hence ‘movements along’ must
be distinguished from ‘shifts in’ supply curves and so the responsiveness of supply to changes
in price can be measured by the price elasticity of supply. Again the delineating value of
elasticity is one: if the elasticity of supply exceeds one then supply is elastic; if the elasticity
of supply is less than one then the supply is inelastic; and if the elasticity of supply equals
one then the supply curve is said to have unit elasticity.
An easy way to determine whether supply is elastic or inelastic at a particular point on the
supply curve is to draw a tangent to the supply curve-at that point. If the tangent when
extended cuts the price axis the supply will be elastic (E, > 1); if it cuts the quantity axis the
supply-twill be inelastic (E, < 1); -and if it passes through the origin the elasticity of supply
will be unity.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Figure 5.10
S
Pa
(Pl
../ '
l
Pr'ce'_ P1 ..
Quantity (Q)
Any linear supply curve which passes through the origin will have constant elasticity
throughout its length (E, = 1) as will perfectly horizontal (E, = ==) and perfectly vertical
(E, = 0) supply curves. Apart from these three cases the elasticity of supply (E,) will vary at
every point on a supply curve.
Figure 5.11
Pi-in Prieq Price I
. s ‘
8
l
‘ l
i 8 I’
T" "'65 ' "' " "' " E as
lil 5, = o liil§s= ii» liiil Es = 1
Relationships Between Cost Curves
Basic economic theory suggests that cost curves could be ‘U’-shaped in both the short and
long run. This shape stems from the law of diminishing returns and from the long-run
average cost curve being an envelope curve of the ‘U’-shapedshort-run average cost curves.
More advanced economic theory suggests that the cost curves might be ‘L’-shaped or
I
rI
l
—_-p1,-
l
I
I
I
?_IE1
___________._E
ll
;\beAif4*-7
_..T,.e:h__..._-L4
'.F
--_.-.--.-.---,_“'
in
I
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Figure 5.12
AC
MC
Cost
Output
saucer-shaped and incorporates the idea of reserve capacity. For the purposes of this book
cost curves will be treated as ‘U’-shaped and certain relationships will then follow.
From the diagram above it can be seen that the average cost (AC) continues to fall so long as
MC lies below it, and rises when MC is above it; and that the MC curve cuts the AC curve
from below at the lowest point of the latter.
Proof
Let AC = f(Q)
Then TC = Q . f(Q)
Cl TC
and MC = la?) =5 Qf’(Q) + f(Q)
When AC ..= MC.
f(Q) = Qf'(Q) + f(Q)
i.e. Qf'(Q) = 0
or f'(Q) = 0 - _
but since AC = f(Q)
it follows that AC is at a minimum when MC = AC
The AC curve depicted above is the average total cost curve which as defined earlier
contains two components. average fixed cost and average variable cost. Figure 5.13 explains
the relationships between these curves.
It was stated above that supply curves reflect the costs of production but in fact the
relationship may be stronger than that suggested by this statement. Under conditions of
perfect competition the firm's marginal cost curve above AVC has the identical shape of the
firm‘s supply curve. and it follows that the supply curve for a competitive industry is the
I
l
_._.-11-_a:rm-_-
KONSTANTINOS
Rectangle
Figure 5.13
0.4
0.3 MC
EQ °~1 ATC
AVC0.1
AFC
2 4 6 8 10 1 2
Output
ATC :- Average Total Cost
AVG :- Average Variable Cost
AFC :- Average Fixed Cost
MC :- Marginal Cost
horizontal sum of the marginal cost curves of all the individual firms in the industry. To put
this into a shipping context. simply consider the firm as an individual ship and the industry
as a fleet: the supply curve for a fleet is the horizontal sum of the marginal cost curves of all
the individual vessels in the fleet. (See also chapter 6)
Relationships Between Revenue Curves
For any downward sloping demand curve (or average revenue curve) there is a marginal
revenue curve which is also downward sloping but is‘ twice as steep as the average revenue
curve. Figure 5.14. using a straight line demand curve. depicts this relationship and point R
represents the point of maximum revenue.
Figure 5.14
Price
D = AR
0 MR Quantity
‘ l
Il
l
l
it
l
l
t‘
l
l
xy.-_.__.....___»----F»-----
i
1
I
‘u
Ii
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
If the demand curve is a straight line with the equation P = mQ + C then total revenue,
given by P X Q. will be mQ2 + CQ and marginal revenue will be 2mQ + C which is the
derivative of total revenue with respect to output.
The MR curve is coterminous with the AR curve on the price axis, falls twice as steeply and
consequently bisects the base of the triangle formed by the demand curve and the axes. This
relationship is true for all downward sloping demand curves (except under conditions of
perfect price discrimination).
However. under perfect competition. a market structure often applied to tramp shipping.
there ceases to be a distinction between average revenue and marginal revenue. The
perfectly horizontal demand curve facing the firm makes it impossible to draw a separate
marginal revenue curve. This follows because the market price is assumed to be unaffected
by variations in output. and so the marginal revenue arising from the sale of one more unit
is constant and equal to price.
Figure 5.15
Price
D= MR=AR
Output
If profit maximisation is the objective of the shipping company then it should expand output
up to -the point where MC = MR. The industry as a whole will be in equilibrium (no
tendency for new firms to enter or for existing firms to leave it or for firms to alter their
level of output) when AC = AR.
Figure 5.16 Short-run equilibrium of firm
lil liil
' ""°' '"°' AC
AC
oi-undaQui~ MC orndlou
mi'Irlllprdin
F ~ ' = P 1.2" ' ' '-:"i#-'- " iii _: F. I.‘ “ifiii;-2-I-:\-.1; =
o Q Quantity 0 0 Oiiantity
-_--_.__....__..-:?-_ic~_-aw--.
KONSTANTINOS
Rectangle
Figure 5.17 Long-run equilibrium of firm and industry
M M
Price ‘ Pill-79
AC ‘ S
MC
P AH=MR
~ l D=AH
l
" ' ti T Tisn'iTi; A I “Q-am-'-5
Consumers’ Surplus, Producers’ Surplus, and Price Discrimination
The interaction of demand and supply as shown in Figure 5.18 determines not only the
equilibrium price and quantity but also the values of consumers’ and producers’ surpluses.
Consumers‘ surplus (CS) is represented by the area below the demand curve but above the
price and is a measure of indirect benefit to the consumer. It represents the difl'erence
between the money that consumers in total are preapred to pay (PP) for goods or services and
the money actually paid (PA), i.e. CS = P, — PA; this area of consumers’ surplus is often the
focus of price discriminatory actions. Producers’ surplus is the area above the supply curve
but below price and represents the diflerence between the amount at which a producer is
prepared to supply a certain level of output and the amount at which he actually does supply
that level of output; alternatively, it may be regarded as the difference between the money
obtained for goods and the direct costs of producing them.
Figure 5.18
Price
Consumers’
surplus S
Pe
Producers’
surplusD
0 A Quantity
l
_l-
.1.
F
l
ll.
l
l_l
ll
l
l
l
ll
l.
5
l
‘i
_L.,..._*_-4
l
l;
l
'
l
l
.v-v-4
I
l
l
l‘
ll
L
1‘ \-
__T_,,.___v___
wqw__4m‘E;m
.l)
l
ii
l
I
l
KONSTANTINOS
Rectangle
Price Figure 5.19
A
19, E1
P, E2
0 Quantity
If price falls the area of consumers‘ surplus will increase.
The diagram above clearly shows that when the price falls from P, to P2 the area of
consumers’ surplus increases from AP,E, to AP,E,. The net change is given by the area
P,P,E,E,. -
Price discrimination is defined as the practice of selling identical goods or services to
different buyers at different prices or the selling of different units of identical goods or
services to the same buyer at different prices. Good examples arise in service industries
where prices differ as between individuals and firms, and also in passenger transport where
peak and off-peak charges differ for the same journey. Liner shipping is often accused of
practising price discrimination.
Three conditions are necessary for the successful application of price discrimination. First
there must be some degree of monopoly power on the part of the supplier so that some
control can be exercised over the level of output or frequency of service being offered.
Second it must be possible to split the market into separate submarkets where the different
prices can be charged without any fear of the lower-priced goods finding their way into the
higher-priced market. Finally the price elasticity of demand must be different in each
submarket to enable the higher price(s) to be charged. The determination of equilibrium
positions under the assumption of profit maximisation depends on equating marginal cost and
marginal revenue for the total market. This will determine the total output which can then be
divided between the submarkets in accordance with the marginalist rules and sold at prices
determined by the demand curve in each market.
Figure 5.20
Prim Price l Price
_ lil Market A , iii) Market B [iiil Totil MOI‘!!! me
"' 1 s
11 x 1 1x Q 1x @:$11 :¢ — 11--‘h-ii-— i1— q--—--nu-1-11111—L—-n-n-111-inn--.1-n-u
F1__
QI I
D
dI ZI DI
-III--"'-—"—
..._._ ._ . ._ l _._. .LL‘_
0 g 0 B O Output
KONSTANTINOS
Rectangle
Figure 5.20 shows how the different prices in each market are determined and how the total
output OT is divided between the two markets (OA + OB = OT). Under perfect price
discrimination the producer aims to appropriate the entire area of consumer surplus by
charging different prices for each unit sold. In practice this is extremely difficult and it
rarely, if ever, occurs; the producer would usually settle for second or third-degree
discrimination with different prices being charged for different blocks of output.
It is interesting to note that, under perfect price discrimination, price is equal to the
marginal revenue as each unit is sold for a different price. The demand and MR curves
coincide even though the demand curve is downward-sloping. Laingm provides the following
diagram (Figure 5.21) showing how the cargo volume (X) may be determined under a
perfectly discriminating price structure. The point of this diagram is that the MR curve
becomes the demand curve when perfect price discrimination occurs and no single price
prevails in the market. If profit maximisation was the aim then the volume of cargo carried
would be OW but this would result in an area of supernormal profits (TUV) which could
encourage new entrants to the trade. In order to deter such potential entrants the operator
may decide to expand the volume of cargo carried to OX. At this point AC = AR and
normal profits are being earned. The quantity WX is carried at rates below the average
cost, i.e. these cargoes cost more to carry than they earn in revenue, and the super normal
profits from other cargoes are used to compensate for these losses. Under this approach
normal profits are being eamed over maximum volumes of cargo.
_ Figure 5.21
Freight
rate ‘|'
'> -' -
‘n
. .. zu .. _ I O * *AC =MC
O W X Volume of cargo
In general if P = f(Q)
then under perfect price discrimination
TR=rf(Q)dQ
U
AR=%rf(Q)dQ
0
Hence if P = mQ + C
2
TR=%+CQ
.-.AR=%Q-+c
[_.-
5
l
i
I
S
>
i.
l
P
l
l
4.
0
l
)
_.¢:n‘?.;.-_.,-._i._.-
V
.l.i
__‘4“_7"7*1"‘i1"
-i.u_;.-.,,":;-3
fig=-<v=e=e'-be-*';_ij-r
ll
ll‘l
l*
l
ii’
....._P_..
KONSTANTINOS
Rectangle
so that the AR curve has half the slope of the MR curve where the latter is in fact the
demand curve reflecting a non-uniform price.
Opportunity Cost
In addition to the basic definitions of cost given above there is the concept of opportunity
cost which measures the cost of an item in terms of foregone alternatives. It is a measure of
what is given up (unable to be attained) by using resources in one way rather than another:
for example the opportunity cost of ship's time is the profit foregone when the ship is
delayed.
If R = the revenue for the voyage
C = running costs per day
F = fuel costs for voyage
D = port disbursements
N = number of days on voyage
then the profit per day
R — CN — F — D
N "3;
and if one day is lost in port ii
R - C N 1 — F - D
profit/day : ( N++ )1 "
the opportunity cost
_ N+1 {R1—CN—F-D R—C(N+1)-F-D}
'( ) 5 N N+1
_N+1[R R F+D+F+D}
'( )NN+1N N+1
_(N+1)[RN+R-RN_(F+D)N+(F+D)-(F+D)N
' N(N+1) N(N+1) l
R F+D
N N -
_R—G+D)
" N
When one speaks of the cost of ship’s time, it is the opportunity cost to which one should
refer. The equation indicates that the cost of ship's time depends upon the revenue on the one
hand and voyage costs on the other. When revenue is equal to voyage costs (F + D), the
opportunity cost is nil and the shipowner should be indifferent as to whether the voyage is
delayed or not. In fact, this could only happen in the very short term since, as a general rule,
when the revenue falls below (C + F + D) the nmning costs (C) may be largely dispensed
KONSTANTINOS
Rectangle
with by laying up the vessel. Thus, the relationship holds only so long as the ship remains in
service. It will be seen (chapter 8) that the time charter equivalent (TCE) of the voyage
charter freight rate is equal to {R — (F + D)}/N and so the market time charter rate may be
regarded as the opportunity cost of ship's time. The opportunity cost of time lost at sea due to
breakdown is exactly the same as a day lost in port. In both cases there exists some scope for
mitigating the cost of delay by slowing down the speed of the vessel subsequent to the delay:
in other words, the optimum speed would be reduced. (See chapter 7.) However, the benefits
of slowing down would normally be minimal.
With liners that try to maintain fixed schedules, the opportunity cost of a day’s delay due to
rain, labour or similar problems affecting cargo handling in port, can be different in different
circumstances. For example, if cargo is short, a delay of one day may have no effect at all; in
other cases, cargo may be shut out with consequent loss of revenue. This, however, is
unlikely. An accumulation of delays can result in the ship being forced to miss one or more
ports of call at a future date, cargo being feedered from such ports to ones at which calls are
made. It is generally the case that it is cheaper to increase speed at sea in order to make up
lost time than to lose revenue by shutting out cargo. An 18 knot ship burning 80 tonnes of
fuel oil per day could make up one day by steaming at 19 knots for 18 days at an additional
fuel cost of about $17,350 if the price of fuel were $100 per tonne. Clearly there is a limit to
the amount of time that can be made up by increasing speed, but equally clearly, it does not
take many containers at say $1,500 a box to compensate for increased fuel costs of this
amount.
*1
‘IF
I
iv
li'
i*";'I-.
I
l
l
'i
i l
lJ L
F
-.'-l
_""“‘_‘Ii"-magi:
a
ll
)
ll
l
l
ii
l1] r
'l-
I
E3;-_"-_5".—:—-ii.-—-1»..i-—~'_""1L-
KONSTANTINOS
Rectangle
Chapter 6
The Demand and Supply of
Sea Transport
The demand for sea transport is complex and variable. particularlyin cases where trade in
goods or in a commodity takes place between two countries both of which are producers of
the goods or commodity. Hence. the demand is a measure of the comparative advantage
enjoyed by one country over the other. In many cases. however. trade takes place between
countries where only one of the trading partners can produce a particular commodity or
where the importing country itself produces the commodity in insignificant quantities. or
only at high cost.
In the first instance it will be helpful to consider the general case where a producing country
with a given supply function is faced by a given demand in a single importing country:
although it is not necessary in all circumstances. the assumption of linear supply and demand
functions will help to clarify the general principles involved.
Figure 6.1
D
3111/
/
/
/
//'
/ S1
/
A//’
Pd “ -7
/ l
// A//
//
/P . - - z8 ’//
511
s D.
Q
KONSTANTINOS
Rectangle
The Elasticity of Demand for Sea Transport
Figure 6.1 illustrates the respective supply (SS') and demand (DD‘) functions of the
exporting and importing countries. When sea transport costs are included. the supply curve
is raised bodily to S“S“' by an amount equal to those costs. Pd is the price obtained by the
importer while P, is the price obtained by the exporter. Thus Pd - P, = f where f is the
freight per unit transported. '
' Figure 6.2
D
5111
l
l
6Pal
I
Pa —————— ——-l--Q2 A
e5P,l
S11 A
D1
Q
Figure 6.2 is part of the same diagram enlarged at A.
If at represents a small (+ve) change in f then it is clear that Pd increases by 6Pd, P, is
reduced by 6P, and the quantity (Q) of the goods transported is reduced by 6Q.
Thus at = 6Pd — 6P, . . . (i) (negative sign appears because 6Pd and 5P, move in opposition
to each other).
Now the elasticity of demand for the good is defined as:
P 50 ..Ed=6d.5*i,:...(ll)
The elasticity of supply of the good is defined as:
P5 50E, -6.6% . ..(1li)
ii
I-ii
i>
I
—I---V.‘Q-ilr===I""'i\'i
>
l
‘I
I
ll
ti-l
l
-'-'--i2-1!"I4|—-ii;fl—|-l--|_—
il;-i_i—-._-n.__|—-—h ni—_n.-_.
-
' F
"ll.
ll
.i§“
Q
3
Ii-1'
‘ .
-
I .
KONSTANTINOS
Rectangle
and the elasticity of demand for sea transport is:
f 60 .
Ed, -6. at ...(iv)
From (ii). (iii) and (iv) respectively
Pa 6Q P, 5Q f 6Q
6P =—-.—; 5P,=-—.—; 6f=—.-—“ E. Q E. y Q E... Q
and substituting these values in (i)
dry so ;Pd egg P, qg
Ed‘t3 E.'<) E,'<2
- L_1°a_Et.e.Eds —Ed ES
E fl *E"E‘ ]
or ds_ Espd ‘Edi
15,15,
— f [Espd _ Since Pd '“ P5 =f
and if p is the proportion of freight rate to c & f price (Pd)
dsi p El.“ E-all
Thus, the elasticity of demand for sea transport is dependent upon the elasticity of demand
for the goods in the importing country. the elasticity of supply in the exporting country and
the proportion of the freight to the price (Pd) in the importing country. Allthings being
equal therefore, the lower the freight rate the less elastic is the demand for sea transport.
Taking one or two extreme cases:
(i) if E, = 0 then Ed, = 0
(ii) if E,= oo then Ed, = pEd so that in this case the elasticity of demand for sea transport
is always less than the elasticity of demand for the goods.
(iii) if Ed = 0 so will Ed, = 0; and
E.
(iv) if Ed = - oo then Edd = p and Ed, can vary from 0 to —- oo as p varies from
0 to 1.
It should be appreciated that, as a general rule, if one variable changes in value all other
variables will change in value also. The exception to this occurs if the supply function is
linear and passes through the origin when the value of E, is constant and equal to l. It may
be -noted however that since 55 = K (See later in chapter), Ed, =
PdE' '-
The Demand for Sea Transport
It can be deduced from Figure 6.1 that the quantity of goods (Q) moving by sea is
dependent upon the freight rate (f) which is equal to the vertical separation between the
supply and demand curves SS‘ and DD‘. Given that the demand function is:
Pd=m1Q+C]
KONSTANTINOS
Rectangle
and the supply function is
P5 = H120 "I" C2
then (Pd — P,) = f = (ml — m;)Q + (C, — Cd) which is the demand function for sea
transport.
When f = 0.
C1 "' C -
Q = -nfi which represents the value of Q at the intersection of the supply and demand
l " 2
curves.
When Q = 0.
f = C, — C3. i.e. the separation between the supply and demand curves on the vertical axis.
Hence the demand function can be drawn as shown in Figure 6.3
Figure 6.3
l ' D
P I
l S1,.
C1-C2 Demand for
-- Sea Transport
W
C1-C
l , D‘l l
Q __...
In general D(Q) — S(Q) = D,(Q) where D(Q). S(Q) and D,(Q) are the respective functions
representing the demand for the goods. the supply of the goods and the demand for sea
transport of the goods.
Example A shipping company has a monopoly of sea transport between country A and
country B. The demand for sea transport is linear and at a freight rate of £100 the elasticity
is -2. It is found that when a certain quantity of goods is carried at rate P, the total
revenue is £50.000 while at rate P3 twice the quantity of goods moves but again yields a total
revenue of £50.000.
_..-_..i-
\
ii-:>
§"'T"'"'\--'-£1
~_41‘;4.1_--*4
l
l
til
4. lf
I
VT444
‘l llll lAl
I-l
rII\-.
ll
m-=e.<~
I
ii
,l
Q
r
-I
L
ll
<9
l
=l-
I
: _ rip-
<.~>
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Determine the sea transport demand function and. assuming that the marginal cost of
carriage is £10 per ton at all levels of output, calculate the freight rate to maximise profits
Let the demand function be in the form:
P=mQ+C
At a certain point (Qd, Pd), Ed, = -2
Pg 100
..-2-6;‘-1—$..Qgm— -'50
Hence 100 = -50 + C
and C = 150
Now there are two points on the demand schedule that yield revenues of £50,000; Thus
P101 = P202 =
and Q; = 2Q,
-'-P1Qi = 2P2Q1
P1 = 2P2
These two points both satisfy the equation for the demand function. Hence
P1== mQ1 + 150. . . (i) and
P2 =mQ2 + 150. . . (ii)
QNow 2P, = % + 150 (Substituting 21>, = P, and Q, = io, in (i))
or 4P2 = mQd + 300 . . . (iii)
subtracting (ii) from (iii) gives
3P2 = 150
.'.P2 = 50 and P, = 100
so that Q; = 1000 and Q1 = 500
substituting P, = 100 and Q1 = 500 in (i) gives 100 = m500 + 50
.'.500m = -50
. .. -_5@ - _ .1.3" "‘ " 500 " 10
Thedemand function is given by
QP - — -i-6 + 150
In order to maximise profits MC = MR
and MR = — % + 150 (See chapter 5)
.'.10= — 9-+1505
.'.Q = 700 and substituting this value in the demand function gives P = -70 -lr 150 = £80
KONSTANTINOS
Rectangle
A Formula to Find the Freight Rate for Maximising Profits“
Let the demand function for sea transport be represented by the straight line:
f=mQ+C...(i)
and MR=2mQ+C. ..(ii)
‘\l E ~l°- ii‘ Q‘tow d,-0.51,-Qmorm -Eds
From (i) and (ii)
MR = mQ + f
t 1..MREds+f t(Eds+1)
. FE E. . . .Since Ed, i PdEs; E: (Pd _ 0 (shown earlier in this chapter)
Pd-1E5 "" Ed(Pd " ‘I’ fEdE§
M — _ , 1-. ~ _,
R fI fEdE, I
_ f{PdE5 _ Edpd ‘I’ Edf ‘I’ fEdE5}
1 I fEilEisTii I
7 PdE,_ Edpd ’I’ Edf ‘I’ I-EdE,5
EdEs
i Pd Pd f
-Ed Eq+Eq+f
Pd Ps f . .I Ed — EA; -- E‘. E; +f (since P, + f = Pd)
-MR—P" P"‘+ton Eb
For profit-maximisation MR = MC
- f — P“ P“ + MC" (max) _ E‘ Ed
where f is now the profit-maximising freight rate.
Again with the exception of MC. the variables P,. Pd. E, and Ed are dependent on each
other.
An Alternative Approach to Profit-Maximisation
Suppose there are. in the first instance. two cargoes A and B with different demand
functions and it is required to set freight rates in such a way that profits are maximised.
ForA f,=m,q,+C,... (i)
For B f2 = Hlzqz ‘I’ C2 . . .
where f 1. fd are the respective freight rates and q,. q; the respective quantities of cargoes A
and B.
KONSTANTINOS
Rectangle
Total revenue from A = mlqi + C1q,
and from B = mzqi + Czqz
.'.Total revenue (A + B) = mtqi + m2q§ + Clql + Czqz
If the capacity of the ship is qu
Total revenue (A + B) = mlqi + m;(q0 — q,)2 + Clq, + C;(q0 — ql)
Differentiating with respect to ql
d TR-gal = 2m,q1+ 2m;(q0 -— q,)(-1) + C1— C; =0 formaximum
1
.'.2ITl|q1_' Zmzqg + 2m3q1+ C1“ C3 = 0
.'.2m1q1+2m;q1— 2m2(q, + qz) + C, - C; = 0
.'.2m,q, + 2lTl2q1"" 2m;q; — 2m2q2 + C, -— C; = 0
.'.2m1q1+ C1= Zmzqz + C;
i.e. MR (A) = MR(B)
It may be deduced that in order to fill a vessel with a number of cargoes and to maximise
profits the marginal revenues from each cargo should be equal. The marginal revenues
should however be positive (and >MC). otherwise profits could be increased by reducing
the volume of each cargo until the respective MR5 were equal to their MCs. The vessel
would not however be full.
Example A container ship has a capacity of 1.500 twenty-foot containers (TEUs).
Commodities in containers are available as indicated by their demand functions:
Rate Number (TEUs) Ed,
$300 S00 —
$400 600 —
$500 400 —
$600 300 -U01?-13> u-la-rt--no-
It is required to find the number of containers of each commodity and the rate to
charge in order to maximise profits. It is assumed that the marginal cost of handling the
containers is negligible.
If ql, C12, q3, q4 are the respective numbers of containers of commodities A. B, C. and D
then
¢l1+q2 ‘H-ls +q4=1=500---(1)
and the equations corresponding to the marginal revenues of each, assuming linearity. can
be deduced:
600MR1=-%q,+600...(ii)
390MR2‘ = — W2 + 800 . . . (111)
KONSTANTINOS
Rectangle
1.000MR, = --456-q, + 1.000.. .(iv)
1,200
MR4 = _""""iq4+1,200...(V)
Solving equations (i)-(v) for values of q give (to the nearest integer)
qi=
Q2:
qs=
Q4:
300
Number Freight Rate Revenue
388
500
346
266
fl =
fz =
n=5m
f4 = £668
Total
MR1: MR2 = MR3 = MR4 =
Assuming that MC is negligible, the usual way of maximising profits (MC = MR) yields a
total revenue of £680,000 and 300 TEUs being shutout. If the ship s capacity is l 800 TEUs,
both methods yield the same result.
Pd
P
P,
D
I b
£142,396
£233 ,500
£196,528
£177,688
-nip-pi
£750,112
Incidence of Costs of Sea Transport
Figure 6.4
Figure 6.4 shows the supply curve (SS1) of goods from an exporting country and the demand
curve (DDI) of an importing country. If there were no transport costs, the price paid and
received for the goods would be P; but when the freight element is equal to AB for
I
Q
F‘
41----.
example. the importing country suffers an increase in price equal to PP, while the exporting
country sustains a fall in price of PP,. Clearly PP, + PP, = f where f is the freight rate per
unit of goods transported. The relative magnitudes of PP, and PP, depend upon the slopes
of the supply and demand functions in the area under consideration.
Now the proportion of freight borne by the importer = a/f and the proportion of freight
borne by the exporter = b/f
DO Pd Pd C Pd C
Tl'ltlS,EdQ.6Pd"Q.5
5Q P P _vi
0'0
andE,=6.§,-S-=-6
.E__F!P
E, P, a
D
O
_-<-_i_---Q
\
it
4
1|
\
4
k
I
H»
U
.50 P11 ll. is, "E, P.'<r-bi
_f—b_ PdE,
" b P,Ed
_ 3 1 _ Pie. P.E.. — P.E.
7 P5Ed '7 PsEd
_ E: PsEd
"f 7 P,Ed - PdE,
and similarly it can be shown that
3. PdE5_
¥=Fill
The ratio of the freight borne by the importer to that borne by the exporter is a/b
PdE,
= 7 iii
and since Pd is always greater than P, the ratio a/b is determined largely by the relative
magnitudes of E, and Ed.
It is worth noting that in the case of linear supply and demand curves a/b is constant and it
PdE,
follows that the expression - F-15- is a constant also.
s d
Consumers’ Surplus
Figure 6.5'illustrates the supply (SS‘) and demand (DD‘) functions in an exporting and
importing country respectively. The demand function for sea transport is shown as FF‘
which. as has been shown earlier, bears a specific relationship to SS‘ and DD‘. In particular
FO = DS.
At a particular freight rate XQ = TW and the consumers’ surplus for sea transport is
represented by the area of AFPfX. _
In the importing country the consumers‘ surplus is represented by the area of ADPdT. while
the producers‘ surplus in the exporting country is given by the area of AP,SW.
KONSTANTINOS
Rectangle
Figure 6.5
D
S1
F Tpd **_*,_
'<
I
Ps|- .
I
S
Pf x \
o is D‘
o F1
Now Area ADSY = Area AFOF' (equal bases and heights)
and since TW = XQ, it follows that
Area ATWY = Area AXQF'
Hence by subtraction,
Area DTWS = Area FXQO
but Area of rectangle PdP,WT = Area of rectangle P;0QX,
Hence Area ADPdT + Area P,WS = Area FPfX.
i.e. Consumers‘ Surplus + Producers‘ Surplus = Consumers’ Surplus from sea transport.
The effect of a change in freight on the value of imports _
Figure (6.6) illustrates the supply and demand functions in an importing country. A small
change in freight (6f) produces an increase 6P, in import prices and a reduction in imports
6Q
The change in the value of imports (VI) is given by
50/I) = (Pd + 5Pd)(Q + 60) - P00
= PdQ + Pd6Q + Q6P,, + 6Pd6Q - P,,Q, and, as 5Q—> 0
d(VI) = + Pd(dQ) + Q(dPa)
w 5 ‘
d(VI) dQ e 01>, ‘
+Pd.¥-l-Q.‘-(F
KONSTANTINOS
Rectangle
, I
D Figure 6.6
sll
em,
______._+__
6Ps
‘-2
I
s I
50.
Now df = dP,, — dP, . . . (i) (negative sign appears because dP,, and dP, move with opposite
signs)
Pd dQ P, dQ
and Ed I b . dmt E, E . dps (by definition)
.,,,Pd d<>_ to Ps
. l 8 Ed . Q Q . E5
_ _ _ EsEd _) ‘ ..
" at T Q (E,P., 4 P,E,, "'(“)
P d d dP -but dP,, = E-3 . E0 = Yd . Ed and substituting in (ii) gives
dP,, E, E,Ed K
jdf ‘P; E,i=7,, -' P,E,
. dP.i _ PclE$
"8' Jr ' EIP,'- i>,i§,,
_Q(_.§.a_) .,,(_na._)
at E,P,, - P,Ed E,Pd - P,Ed
(Ea + 1) -
= Pd . Q . Es [(EsPd - PsEd)]
Q(Ed + 1) P,Ed= —-1-Y— where K = P—-E- which, as shown earlier is a constant for linear supply
- Cl
and demand functions. - S
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
d(VI)
“Ed = "1 IhCflT = O
Similarly it may be shown that the change in the value of exports (VE) with a change in
freight is given by:
d(VE) : QP,E,,(l + E,)
df T P,,E, — P,Ed
= 9§l._+L~l,,,h,,e K = E153
(1/K_ PdEs
_ d(VE) Qi<(i + 13,)
" at (l—i()
Since K is always negative it follows that an increase in freight rates will have an adverse
effect on the export earnings while the effect on imports depends upon the value of Ed.
Figure 6.7
I
I o
l\
I
r
I
\ \
\ \\ \ 5n
\ \
\\ \\ sin
\\ \
I \ \\
Pr I
| '_ \ .
I \ II \ .
\ I(D
0-""'
_ .
O Q1 O2 O
\ .
\\\ SI
\\
\\
l \ \ 5
\
\ I ~‘
\ 1
gm
KONSTANTINOS
Rectangle
. I
I J
It I
Common demand with more than one source ofsupply
So far, the only situation that has been considered is that of a single country exporting to a
single destination. It will be of interest now to examine the effect of two sources of supply
and a common destination.
Case 1
Figure 6.7 shows demand DD‘ with sources of supply SS’ and SS". These can be-combined
to yield an aggregated supply function SS”'. It is evident that the amount of goods supplied
will be Q at a common price P; also that one country will supply OQ, and the other OQ;
where OQ1 + OQ2 = OQ.
It can be deduced that, although exporting to the same destination, a separate demand
function faces each country, viz. DD" and D’” which combine to give demand function
Figure 6.8
D
I
on --- DIV
-"'“-I
aI""”-"-
-1"-if
-1"’---‘.—-.
-II
X’
ff
?f
If
.ffI
’f
I
(D
m V
(D‘U
.-_-r
CD-—'-_J+uI-"3';-—-., 44.
u-I"""'-H.
___._til’—_..-_--
--""""-
I
0’,
i’
I’?
I
/I’
I
<
I \|)vI
I W pm W ;
> I I I I DI
0 0| 0.2 Q
73
¥_ _ ___ _ *1 _, _ |
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
DD’. These will of course only remain fixed provided the total demand and other supply
functions also remain fixed. When there is a common starting point on the vertical axis as at
S. it can also be readily deduced that the elasticity of demand facing each exporting country
is the same. The conclusions for the particular case will hold good for any number of
sources of supply.
Case 2
Figure 6.8 shows a case of two exporting countries facing a common demand DD’.
However, since the two supply functions are not coterminous the combined or aggregate
supply curve S“'SS" is kinked at S.
At a common price P the country with the greater production costs (S'S") exports OQ,. the
second exports OQ; withthe aggregate being represented by OQ.
Figure 6.9
D
I
S1
S 11
S111
\
Q 01 Q2 D1 Q D11 D111
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
If the aggregate supply curve is raised bodily. the point S meets the demand curve DD’ at
Di‘ and this represents the price where the demand‘ from country S'S” vanishes and the
remaining demand DD“ is met by country S"' S“. Thus the demand facing the first country
is D"'D'" and the second country has an kinked demand function DD*“D". At the common
price P the elasticity of demand is greater for S'S” than for the lower cost supplier S'" S“.
Two sources of supply and two importing countries
In order to avoid complicating the picture unnecessarily it will be assumed that the supply
and demand curves are coterminous on the vertical axis. Figure 6.9 shows countries with
demands DD’ and DD" aggregated to give a total demand DD”'. The suppliers SS’ and SS"
have an aggregate supply of SS"’ with the result that price is set at P and the respective
outputs of the two countries are O01 and OQE. The demand from each of the two countries-
does not match (except in aggregate) the output from either supplying country and so if
market conditions are to be satisfied at least one of the importing countries must have two
suppliers. The price P will be the same in each importing country.
The Supply of Sea Transport
Introduction
In this section, only ships employed in the charter markets are considered, and, in particular,
only those employed in a particular sub-market rather than the bulk market taken as a whole.
The reason for this is that a perfect market is involved where, in the short term, at least, all
vessels will be paid the same freight rate regardless of size and speed considerations. This
means that a 10,000 dwt vessel will not be competing in the same market as one of say
150,000 dwt, but that it is quite possible, and indeed likely, that vessels from 50,000—70,000
dwt, for example, may do so.
In the first instance the effect of time spent in port will be ignored. In other words, for
simplicity, it will be assumed that ships do not spend time in port and consequently incur no
costs therein. The output of ships (expressed in ton-miles) only takes place when vessels are at
sea. Given that there is a fixed proportion of ships at sea and consequently a complementary
proportion in port at any given time, the supply function may be considered to apply only to
those vessels that are actually carrying cargo at sea or in ballast, thus forming part of a cargo
laden voyage at that time. If it is assumed that the time in port for a given voyage is fixed,
the only way of increasing the output of ships in a given (short) time period is by increasing
the speed of the vessels on both the loaded and ballast legs. Also, ships may be
re-commissioned from lay-up; but this aspect is diflicult to quantify and will be dealt with
later.
The supply function
The supply function for an individual ship is derived from its marginal cost curve. Let us
consider a bulk carrier on voyage charter where the objective of the shipowner is to maximise
its daily gross profit. In order to achieve this objective the vessel will be required to steam at
optimum speed. The optimum speed will be different from one voyage to the next according
to the freight rate and the distance steamed both laden and in ballast, and will also change
over time according to the price of fuel. Hence, the shipowner will order the speed to be such
that marginal cost is equal to freight rate (after deducting costs incurred in port and canal
charges, if any).
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Marginal Cost
Marginal cost may be defined as the cost of producing an additional unit of a product in a
given time. In the case of sea transport, the only way of increasing the output of a loaded ship
in a given time is by increasing the speed. The unit of output is taken here as the ship-mile. It
is assumed that fuel consumption per day varies as the cube of the speed so that:
Consumption/day = k . s3, where k is a constant for the ship.
Let Cr = running costs/day;
n = constant time period under consideration, in days;
= speed in miles/day;
= price of fuel oil/tonne;
= ship fuel consumption constant;
= distance covered in n days; then
Total Variable Cost (TVC) = Cr . d/s + pks-" . d/s.
But d/s = n
Therefore TVC = CR . n + pkd . dz/nz
The marginal cost is the rate of change of TVC with respect to distance d:
i.e. MC = d(TVC)/d(d) = 3pkd2/nz
or MC = 3pks2 . . . (1) (since n = d/s)
i.e. MC is the cost of producing an additional ship-mile and varies as speed squared. It should
be noted that the period ‘n’ in which the additional ship-mile is produced does not appear in
the expression, implying that the cost of producing an additional ship-mile is the same
whether a day, week, month, or any other (short) period of time is considered.
O-P¢"'Om
The supply function for a given ship is that part of the MC curve lying above the AVC curve.
If we take the ship-mile as the unit of output then for one day:
AVC = Cr/s + pks-"/s
= Cr/s + pksz . . . (2)
i.e. running costs/mile + fuel costs/mile. '
It is assumed that the output includes miles steamed in ballast, for this must be paid for and
included in the freight rate just as surely as if the ships were carrying cargo. One may argue,
of course, and with some justification, that two vessels of equal size may be paid the same
freight rate for a voyage involving the same commodity and ports of loading and discharge
when the ballast passages involved in steaming to the loading port have been quite difl'erent.
This difference will result in different optimum speeds, different marginal costs and therefore
apparently different freight rates — in terms of dollars per ship-mile — for what is essentially
the same voyage, occurring at the same time. Such a situation would appear to make
nonsense of the theory. However the problem can be resolved by the process of averaging. In
other words, for the purposes of calculating optimum speed it is necessary to take into
consideration not just a single voyage, but, as far as possible, several voyages that might be
foreseen. In this way, for a number of ships the average length of time spent in ballast should
be equalised and the optimum speed of all of them become the same. In parenthesis it should
be emphasised that for this -reason, profit maximisation should be calculated over several
voyages where possible rather than for each individual voyage. This statement is not relevant
of course in the case ofconsecutive voyages or where a bulk carrier is engaged in shuttle
services over a period of time.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
A Numerical Example
Taking a simple case, let us suppose that a bulk carrier of some 70,000 tons dwt has running
costs of $5,000 per day and a fuel consumption of 50 tonnes/day at a speed of 15 kts. The
price of fuel oil is $100/tonne. Since the fuel consumption/day = ks?’ we can deduce that
k = 50/3603 where s is in miles/day.
The marginal cost and AVC curves are plotted over a range of speeds from 0-360 miles/day
in figure 6.10. From this figure it can be seen that MC and AVC cross at an output
equivalent to 285 (285.7 more exactly) ship-mileslday or the ship steaming at a speed of 11.9
kts.
It can readily be shown by combining equations (1) and (2) that the speed at which
MC = AVC is given by:
32 1S = (Crl2pk)”’ and MC = s(C—"4‘l'-‘-)'
and of course this is the lay-up point which depends upon Cr and also the price of fuel (p).
Figure 6.10
$
60
50
MC40
30 AVC
20
IO
60 I20 I80 240 300 360
Miles
The supply curve for the whole bulk carrier fleet may be found by the horizontal summation
of all MC curves. This appears to make the supply curve flatter, but for a given speed the
marginal cost will remain the same regardless of the number of vessels in service. '
It is worth noting that within the same market or sub-market the marginalcosts of all vessels
will be the same. This means that:
2 _. 2
klSl — k2S2
where k, and kz are the respective fuel constants for ships A, and A, and s, and s; their
respective optimum speeds. It follows that modern fuel efiicient ships will be steaming at a
greater optimum speed than the older uneconomical (especially steam turbine) vessels.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Elasticity of Supply
Let us consider the supply, comprising the horizontal summation of the MC functions of
n vessels of respective deadweights W,, W2, W3, W4 . . . Wu, with fuel constants
k,,k,,k,,k,,...k,,(l ...i...n).
Then :
3 - 2MC (Supply) for vessel i= i::f- expressed in USS per tonne mile
Let output in tonne miles per day be Q where Q = qsW, and q is a constant representing the
ratio of loaded to total tonne-miles.
Thus :
3 k, 1
MC=-5%-. whence
MC _ witqz ll2
° = (“Til
Horizontal summation for n vessels gives
MC 2 U2 " "1Q+—q»> 0 <~>3p l=l ki
Since MC now represents the aggregate supply price, it is appropriate to substitute P (price)
for MC.
Thus, squaring and rearranging:
3 zP = PQ(s>":>i
The Elasticity of supply is given by:
E, = (dQ/Q)(P/<1?)
= P/Q(dP/dQ)
dP/dQ is found by differentiating P with respect to Q.
Thus, for any given values of P and Q we have:
2 2 ki ll 2E21 ,3pQ g qO2(,/1’) 05
' Q q’(E(W?/1<i)"")* 6PQ I '
Hence E, is constant and equal to one half for all values of s.‘”
The general conclusion to be drawn is that provided there are relatively few vessels laid up,
an increase in trade of 10% should result in an increase in freight rates of 20%. If the increase
in trade occurs in only one sub-market however there is every possibility that an increase in
demand will cause marginal tonnage from other sub-markets to enter that market, with the
result that the elasticity of supply will be somewhat greater than 0.5. Finally, it should be
noted that the assumption underlying this result, viz. that daily fuel consumption is
proportional to speed cubed only holds good over a limited range of speeds and, in particular
it cannot be extended far beyond vessels‘ design speeds. At this point the supply must become
completely -inelastic.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Chapter 7
Optimum Speed of Ships
As a general rule the speed of a ship is optimised at the design stage. That is to say, the
design speed is such as to minimise the required freight rate (RFR) when the ship is
employed in the way intended. Liners however may be provided with a design speed that is
not strictly optimal but one that is necessary to maintain a given schedule. It is also likely
that a small reserve of speed will be available for contingencies. Very high speeds are costly.
not only because of the greater fuel consumption, but also because of increased engine size
and weight, increased scantlin/gs and length-to-breadth ratio of the vessel.
When a ship is brought into service there are three main factors that determine her optimum
speed. These are: price of fuel, freight rates dictated by the market, and voyage distance.
In these circumstances the shipowner aims to maximise his profits per unit of time; it should
be appreciated that simply minimising costs associated with a voyage does not have the
effect of maximising daily profits.
Taking a simplistic view to begin, time in port is ignored and fuel consumption per day is
assumed to vary directly as the cube of the speed, so that the daily gross profit is given by
GS—Rw-C - k3_d/s R PS
where GS = Gross profit or surplus/day
R = Freight rate per ton of cargo
W = Deadweight available for cargo
CR = Running costs/day
= price of bunker fuel per ton
= distance steamed. including ballast passage if applicable
= speed in nautical miles/day
= constant of proportionalityWmfl-‘U
Since only the short run is considered, capital becomes a sunk cost and does not enter into
the argument.
For a given freight rate and voyage the only variable is the speed s. To find the optimum
speed it is simply necessary to differentiate the function with respect to s and equate to 0.
Thus:
d RWE; (GS) = T — 3pks2 = 0 for maximum
RW
whence 3pks2 = T
d _ (E)an S 7 3pkd
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
In order to find k it is necessary to know the fuel consumption per day at design speed so, so
that :
Consumption/day = ksfl
consumption/dayorkl A -Si
It should be noted that maximising profits/day is the same as minimising losses/day
The question of adjusting speed according to the market conditions has become of much
greater importance since the large increase in oil prices post-1974 Many shipowners now use
computers to calculate the optimum speed. Instead of using the ‘cube law they have
data held on file that may have been determined empirically or supplied by the builders
Example I A VLCC of 200,000 dwt has a fuel consumption of 120 tons/day at 15 kts The
voyage charter rate from Kuwait to Rotterdam is W30. If the round voyage distance is
21,000 miles and the price of fuel is $180/ton calculate the optimum speed and the
corresponding profit/day. (Running costs are $7,000 per day.)
30
Taking W100 = $25 for this voyage. the freight rate is 25 X W = $7 5 per ton of cargo
Thuss = v‘ 
F 1.5 >< 200.000= \/__._._-----—-— = 3363
3.180.. ,.21000
whence speed in knots = 226.8/24 = 9.45.
120
(15.24)-
The corresponding gross profits/day for a range of speeds are:
Speed (kts) Gross Profit/Day ($) Speed (kts) Gross Profit/Day ($)
8
8.5
9
9.5
10
Hence at 9.5 kts the gross profit/day is $3.798 compared with a daily loss of $2 886 at
15 kts. The above table shows that the profit is very sensitive to changes in speed except
near the optimum where there is no great difference in the 9-10 kt range of speeds
In practice the Cube law does not hold good over an extended range of speeds so that while
the Cube law is operative near design speeds, at lower speeds a square law may be more
appropriate.
One sourcel“ suggests that the fuel consumption per day obeys a law of the form
3.437
3.641
3.763
3.798
3.743
Consumption = CD x J
where CD is the consumption at design speed and J is dependent on s such that
J=As3+Bs1+Cs
3.339
2,512
1,225
-562
2
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
If s is given in knots then coefficients A, B and C are of the general order 0.000167
-0.00098 and 0.0363 respectively
i.e. J = 0.0001675?’ — 0.0009852 + 0.03635
Given suficient data relating to speed and fuel consumption the coefficients and hence the
curve can be determined by the method of Ordinary Least Squares as shown in Appendix B.
However, using fuel consumption/day = CD(As3 + Bsz + Cs) in example 1 and remembering
that the unit of s in this expression is the knot, the equation for profit/day becomes
RW
GS = T - CR - pCD(As3 + B52 + Cs)
24s
and differentiating with respect to s
d GS 24 RW{as ) 1 de e pCD(3As3 + 2Bs + C) = Ofor a maximum
-3A1+2B +C—g€-liv-.. S S —dpCD
4 Rw C 2Bi-—— — — s
Pdcpwhich reduces to s =
It will be noticed that the variable s appears on both sides of the equation, but in the right-
hand expression it is not of great significance.
The equation can be solved quite easily by entering say 15 kts for s in the RHS and after
evaluating the expression to yield s on the LHS this value can be substituted in lieu of 15 kts
and the expression evaluated once more. This process of iteration is continued until there is
no significant difference between the values of s in each side of the equation.
Thus, using the values for constants already given, the following values are obtained.
(s) LHS (s) RI-IS
1st Iteration 11.97 kts 15 kts
2nd Iteration 11.52 kts 11.97 kts
3rd Iteration 11.46 kts 11.52 kts
Hence using the expression for fuel consumption/day the optimum speed is approximately
11i kts compared with 9i kts using the Cube law. The difference between the result depends
onthe values of A, B and C in the expression for J and upon the extent that the optimum
speed differs from the design speed.
For the remainder of the exposition it will, for simplicity, be assumed that the Cube law
holds good. '
When time in port and disbursements, cargo handling costs, canal dues etc are considered,
the daily gross surplus is given by
Rw pks3.§—D sGS-"zi—"°R-T-<—»~> (—~>s s
Q
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
where D are the disbursements in port and t is the time in port in days
Differentiating this expression with respect to s gives
d(GS) _ (RW — D) (la) pkszd _ (_—_d_) _ (Q )"
T d 2 .( 1) S2 if2( 1) S2 s+t 2pksd
Is‘) (:1)
= 0 for a maximum
RW — D pkszd 2pks-___-__i__--_=0" a 2 <1 1 <1s2(—+t) s2 (-+r) (-1:-t)
S S _S
d(RW-L D) - pksid - 2pks3 (E + t) = 0
.'.(RW — D) — pkszd — 2pks2d - 2pks3t = 0
.'.(RW — D) — 3pks2d — 2pks3t = 0
RW — D RW —~ D
"'81 7 sipkd + Zpkst 5 pk(3d + 2a)
_ BYԤ:D
°’ S pk(3d + 25:)
The effect of D is to reduce the optimum speed and, similarly, the greater the value of t the
lower will be the optimum speed.
Again. this equation must be solved by iteration, since s appears on both sides of the
equafion.
The Marginal Cost of Sea Transport
Liner shipping The concept of marginal cost with liners is not too difficult. particularly
where there is a fixed schedule so that virtually all costs other than those for cargo handling
are fixed. It is argued by some that additional cargo carried causes an increase in fuel
consumption — which is true — but this can hardly be measured with respect to a small
quantity of cargo. On the other hand. by not carrying a certain quantity of cargo, there is a
saving in port time that could result in fuel saving by slow steaming. In the case of container
ships the marginal cost is often negligible because the containers may still be loaded even if
empty.
Tramp shipping It was shown in chapter 6 that
Marginal Cost (MC) = 3pks2 where
p = price of fuel
k = ship’s fuel constant and
s = ship's speed in miles per day
MC is expressed in USS per ship mile provided the price of fuel (p) is also expressed in US$.
This expression does ignore time in port and strictly can only be applied to the time spent at
sea. It is instructive to look back to page 79 where the optimum speed s was given by
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
my
3pkd
or RW = 3pks2d
Thus RW is the freight paid for ‘d’ miles and the freight per ship (or cargo) mile = Bpks’.
This has been shown to be the marginal cost, so that in this case, the freight rate per ship
mile is equal to the marginal cost. This accords with the theory of a perfectly competitive
market where output is increased (or reduced) to the point where MC = MR (and
MR = price) in order to maximise profits per day.
Extending the concept a little in order to include port disbursements, D, and time in port, t,
we saw (page 82) that for profit maximisation optimum speed is given by
,. -2
pk(3d + 2st)
or RW —- D = pks*(3d + 2st)
ff 3pks2d 2pks3t Dsothat R W + W +wMC...(1)
The second term raises the MC curve by an amount that varies with speed and time in port
while the third term raises the curve by a fixed amount, i.e. it is independent of speed.
It was seen in chapter 6 that the lay-up point occurs where the MC curve crosses the average
variable cost (AVC) curve —- at its lowest point.
Now, allowing for port disbursements and time in port
D Cllt Clld/S pkS3d/SAVC W + W + W + W ( )
Difl'ere'ntiating with respect to s and equating to zero for a minimum
d(AVC) Clld Zpksd_ — + ~-0
ds s2W W
or CR = 2pks3 . . . (3)
C II3
i.e. s = which is the same as before.
It may be noted by substituting CR = 2pks3 in (1) that
3C,,d Cat DR _._ .
2sW + W + W
_;
2sW sW W W
_ prised/_s Clld/S Cltt DW + W + W +w _AVC
In other words marginal cost is equal to average variable cost at the minimum point of AVC.
Fuel Consumption when Steaming in Ballast. It may be argued quite correctly that the fuel
consumption per day is much lower when the ship is steaming in ballast than when steaming
fully,laden. It follows that the optimum speed in ballast is likely to be different and probably
higher than when laden.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Let k,s3 be fuel consumption/day in ballast and k2S3 be fuel consumption/day when laden.
Hence.
d d
(RW Pkisi —‘- + pkzsi -i)
<—* + 4) <4 + 4)5| 52 S1 $2
where d, and dz are the distances steamed in ballast and laden respectively. 51 and s; are the
corresponding speeds. Differentiating partially with respect to s1,
E _ (Rw*'D) dl (kisidi ‘I’ k25id2) dl 2Pki5idi.-- .-- =065- (11.,e)1si P gnu)’ Si (a,e.)
51 52 51 52 51 52
foramaximum.
d d
.'.(Rw ‘T’ '- p(IE;S%(11-l- kzsgdz) -' 2pk1S::' + = 0 . . .I 2
Similarly, differentiating partially with respect to s2
d d
(RW — D) - p(k,s§d, + k;s§d,) - Zpkzsi + = 0 . . . (2)
1 2
whence k,s{ = kzsg for profit (GS) maximisation. In other words, the fuel consumption per
day must be the same, whether the vessel is loaded or in ballast. Nevertheless, it should be
clear that the marginal cost of moving the ship loaded must be greater than that of moving
the ship in ballast. (MC = 3pks*)
3 k2 kl 1/3
Thus sl = s; or Ks; where K =
l i
From equation (1)
RW - D = pk,sid, + pkzsfidz + 2pk,sfd1 + 2pk,s}d;/s,
and substituting Ks, = s,
RW - D = pk,I(*s§d, + pkzsidz + 2pk,I(2s§d, + 2pk,K3s§d2
RW — D - 3pk,I(2s§d, + pkzsidz + Zpkzsid;
= 3pk,I(1s§d, + 3pk,s§d,
or S J iiw-D U I R.W—D g
2 3p(klK2d1 '1" kzdzl 3P(ki”kindi 'l' kzdzl
and as a first approximation
S /ngw-pg law-D
’ 3Pk2(di'l'd2) 31'-‘kid
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Similarly,
\/“I RW—D
S‘ 3pik.<1I+ krkitdii
and as a first approximation,
lRW — D
S1 = "'—*'i""""'3pk,d
Time in Port Included
If t is the time in port in days,
RW — D P0415191 'l' kzsidzlos=-i---c,,——--i--(eieiq (eiem)
$1 $2 $1 52
t7GS (RW - D)d 2 k Id P(k1S2d, + i<,s=<i,) d.__=_____gi_ P151 _ 1 2 -1
‘.651 d1 dz 2 d dz d dz 2(-—+—+t) sf (—‘+—+t) (-——-1—+——-—+t)si
51 52 51 52 51 52
= 0 for a maximum.
(RW — D) - p(k,sid, + k;s§d;) — Zpklsi + % + t) = 0
1 2
and substituting Ks; for s, (where K = (kl/k,)”’)
RW - D = r>(kiK=sidi + kisidi) + 2pk,K=s§d, + 2pi<,i<,=*i-.54, + 2pk,I(3s§t
= 3pk,K1s§d, + 3pk2s§d2 + 2pk,K3s§t
_ 3Pk25idi 2 3-T-+ 3pk2s,d, + Zpkzszt
= pkistisidi/K + dz) + 2s=I}
e J RW—D 0":or s2 F :T~—e———
+ dz) '1' 2521}
or, as a first approximation, since K =1
S I _ RW - D
2 . pk;(3d + 28,1)
and similarly,
s\/'1 RW-D if or I RW—D
1 + '1' 25111} '1" 251'”
Example A bulk carrier is fixed to carry ore from Brazil to Port Talbot at a rate of $5 per
tonne. The deadweight available for cargo is 100,000 tonnes. The vessel's ballast passage
is 3,000 miles and the loaded passage is 4,000 miles. Given: total time in port is 7 days;
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
port disbursements: $40,000; price of fuel: $125 per tonne; fuel consumption loaded:
80 tonnes/day, and in ballast 65 tonnes/day @ 15 kts; daily running costs: $6,000.
Find the optimum speeds at which the vessel should steam laden and in ballast. Also
find the corresponding gross surplus/day.
RW = $500,000; D = $40,000; p = $125/tonne.
N... J1i-?_i.irvi1b ‘2 pk;{3(d,/K+d;)+2s2t}
460000~ ~ ll K=l.07l66\/125.80/360-‘{3(3,000/K+4,000)+7.2. 360} W °'°
On lst iteration, s2 = 290 miles/day or 12.10 kts.
On 2nd iteration, s; = 296 miles/day or 12.34 kts and s1 = Ks; = 12.34 X 1.07166 = 13.22 kts.
460,000 {(k, x 3111 >< 3,000) + (i<, >< 4,000 x 2961)}cs - , , - 6,000 125 -A ~ ~-e A(3.000/an + 4,000/296 + 7) (3,000/317 + 4,000/296 + 1)
= $5,087 per day.
Approximate answers can be found more simply using the expressions:
S _ / (R_W - D)
‘ pk,(3d + 2s,t)
and s ea l (RW _ D):
2 pk2(3d + 2s;t)
where d = d, + dz
Using the above formulae,
s, = 13.41 kts
s; = 12.18 kts
and GS = $5.085.7 per day.
In this example, therefore. theresults obtained from the approximate formulae are very
close to those obtained by the full formulae.
Optimum Speed allowing for changing fuel requirements
Previously the equations for Gross Surplus/day have tacitly assiuned that the deadweight
available for cargo remains constant. However, a little thought should make it clear that such
quantity depends upon how much fuel is required for consumption on passage and where the
bunkering port is situated in relation to the port of loading. It also depends upon the speed;
for since fuel consumption per mile varies as speed squared, the lower the speed the less fuel
required and the more cargo that can be carried.
The following equation makes allowance for fuel consumption by reducing the deadweight
and also total revenue by the amount of fuel consumed over distance d, which is the distance
from bunkering port to discharging port or, loading port to discharging port whichever is the
greater.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
{W — ks3d,/s}R — D pks3d/s
Gross Surplus/day (d/S + t) CR (dis + O
Where W = deadweight available for cargo (without deducting fuel consumed on passage)
R = Freight Rate (after commission)
= speed in miles/day
= total disbursements in port, including canal dues, taxes, cargo handling etc.
= total voyage distance
= price of fuel in USS/tonne
= time in port and canal (if applicable)
= ship's fuel constant
CR = Daily running cost
d, = distance to steam after bunkering, as defined above.
er.-noO-Um
Differentiating with respect to s, we have,
d(GS)* 1 (WR j D — ks*d,R j ks*pd)d *(2ksd,R 4?-_2pksd) W 0 for a max.
ds (d/s + t)2s2 (d/s + t)
Whence,
(WR — D)d - ks*d,dR — kszpdz — s2(2ksd,R + 2pksd)(d/s + t) = 0
and
(WR — D)d - (Rd, + pd)(3d + 2st)ks2 = 0
WR — D d .
so that s2 * k(Rdf +pd)(3; + Est) or, if d, = d
(WR — D)h z _ -1‘ °“ S k(R + p)(3d + 2a)
For an example, see chapter 8 (Voyage Estimates).
Industrial Carriers
The case of the industrial carrier is somewhat different from that of the independent carrier
where the objective is to maximise profits/day. Industrial carriers form part of an integrated
chain involving production, transport, processing_and often many other activities. Although
notional profits may be earned by the organisation concerned with the sea transport of the
cargoes -— often raw materials — the main concern should be with minimising costs,
commensurate with the carriage of the required volume of cargo in a given time.
Two situations may be distinguished, viz: (a) where the industrial carriers are so specialised
that any increase in capacity can only be achieved by newbuildings or by ‘stretching’ existing
vessels; (b) where tonnage controlled by the industry can be increased by chartering from
independents on the open market.
(a) The objective is to minimise the cost of carriage per ton and because the longer term is
considered, all costs including Capital costs must enter into the calculations. Hence the
cost/ton is given by:
,(c¢+C,,) (<1 ) pks3 d D
C1-~ . 5+1 + W .s+w
KONSTANTINOS
Rectangle
Iii
i
I
l|,'
B
where CT = Cost/ton
CC = Capital Cost/day
CR = Running Costs/day
W = Deadweight available for cargo
d = Distance
= Speed in miles/day
= Constant for vessel
= Price of fuel/ton
= Port disbursements
= Time in port
Cost/ton is minimised when
d(Cr) -Cc CR)(—d) + Zpksd f 0
ds _ I W s3 W
S0 that 2pkS3 = Cc + CR
_ 3 (Cc ‘l' CR)ors - ————-——2pk
It will be noticed that this optimum speed is independent of distance but would increase
with increased running costs and fall with increased fuel costs. It is also independent of port
disbursements and time in port.
Allowing for different ballast and loaded speeds it can be shown that:
3 3_ (Cc + CR) (Cc + CR)
S1 -- ~3Hd S; = T2
(b) The ability of the owners or controllers of industrial carriers to reduce costs by slow
steaming was explored by Artzm in 1975.
Suppose that a major oil company has a fleet of in ships. Each has a carrying capacitv of W
tons and design speed of so miles/day. Consider one route where the round voyage distance
is d miles. Let the ships be in service for 350 days per annum.
At design speed, the oil carried per "annum
350Wm _ d _ _ _ _
= T—- , since + t) is the time taken for each round voyage. t days being spent in
( + i) °
50
‘_._
port.
If speed is reduced to s then the oil carried per annum
350Wm
(iv)
This results in a saving in fuel costs but a reduction in oil carried which can be made good
through chartering in on the voyage market.
Loss of oil carried as a result of slow steaming
S0 S
= 350WM (—-— - --—) tons
d '1' Sgt (1 ‘l’ St
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
If the ocean spot rate is R per ton then the additional cost due to chartering in
—3s0Rw (-S-‘L---s—-)_ m d+sot d+st
Now the time spent at sea at full speed
350 d
= . — days per annum
d Sq
+ t
50
350d
(d + so .t)
If co = fuel consumption/day at speed so and p = price of fuel/ton then the cost of fuel per
vessel per annum
_ pco350d
(d '1" Sgt)
. 350 d . 0At speed s, time spent at sea =Em - —; and fuel consumption = -Egg-_%6 . ks3
Hence the saving in the cost of fuel per annum for all ships
_ osod _£>.___“i_)
_mp d+sot d+st
The total saving = saving in fuel costs less the additional cost of chartering.
Hence, the net saving per annum
— s50<i( °° ks} )3s0Rw (-1---5-)—mp d+sot d+st m d+sot d+st
Differentiating with respect to s and equating to 0 for a maximum
d -ks3 . mp350d d —3SORWms9» > < A I >0ds d + st ds d + st
.£(..-_1’E’£) ,i(_1£""_S) _o
“ds d+st ds d+st
_ -pks3d*(—g1)t I _3pks2d + RWs (—gl)t + RW I 0
" (d + st)2 (d + st) (d + st)2 (d + st)
.'.pks3dt — 3pks¢d(d + st) - RWst + RW(d + st) = 0
—2pks3t - 3pks2d + RW = 0
.'.s2.pk(3d + 2st) = RW
. _._ _ g W
“S *5 pk(3d + 25:)
This represents the optimum speed for an industrial carrier where the owners can make
good the shortfall in cargo carried by chartering in at rate R.
Taking into account the savings in disbursements resulting from the industrial carriers
making fewer voyages per annum, it can readily be shown that
89
i 
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
5 : 1 I £1) where D are the disbursements per voyage.
It may be seen that this result is independent of days in service per annum and the number
of ships in the fleet. Of particular interest is the fact that the optimum speed is identical to
that of an independent owner's vessel of similar characteristics and employed on the same
voyage. In other words, it should be expected that when a charter market exists to
supplement the work of industrial carriersthen all carriers (of similar size, technology,
employment, etc) should be steaming at the same optimum speed regardless of ownership.
Were this not so, a fall in demand would bear much more heavily on the vessels engaged in
the voyage or spot charter market, to the detriment of their owners, and result in much
greater fluctuations in market rates.
Optimum Speedfor Vessels on Trip Charter
When vessels are on voyage charter, the objective is for owners to maximise their gross
profits per day. On trip charter, however, the owner accepts a daily rate and, since the
charterer controls the speed and hence the fuel consumption, his objective is to minimise the
total cost per ton of shipping the cargo.
It was shown earlier that allowing for cargo displaced by fuel, the optimum speed on voyage
charter is given by,
RW — D
S2 I rth + p)(3d +I2st)
or aw - o = 3(R + p)ks2d + 2(R +' p)ks3t . . . (1)
The Time Charter Equivalent (TCE)
l(S2(1*)W— D f pkszd
(d/s + t)
_ __ , z _ 2
F Rw (dfilrg pks de and substituting the value of RW —D from (1) above
TCE __ 3Rksid -l:3pks1d + 2Rks3t +*2pks3t*— Rkszd —gpks2d
(d/s + t)
2(R + p)ks2d +fi2(R + p)ks1ft
if T (ti/S391) 5’ 5
_, 2(R+p)ks1(d+st)s
T 5 (<1 + st)
= 2(R + p)ks3
Now if v is the speed at which the charterer orders the ship to proceed,
__ TCE(d/v + t) + D + pkvzdTotal Cost/tonne F e so ~(w _ kvzd) ~ e ~
Differentiating with respect to v and equating to 0 for cost minimisation,
d(TC)_ {TCE(d/v + t) + D + pkv2d}2kvd {—TCE . d/vi + Zpkvd}
dv D (W — kv2d)i + (W — livid) I T 0
{TCE(d/v + 1) + D + pkvzd} {TCE/vi - 2pkv}
°“°° (w - kvzd) 2kv
KONSTANTINOS
Rectangle
T _ 3
i.e. C = ELwhere C is the Total Cost/tonne
Therefore, 2Ckv-" = TCE — 2pkv3
or TCE = 2kv3(p + C)
i.e. 2ks3(p + R) = 2kv3(p + C)
Now since R = C, i.e. the freight rate that the charterer would be willing to pay must be
equal to his cost/tonne of shipping the cargo under trip charter, it follows that s must also be
equal to v. In other words, whether cargo is shipped under voyage or trip (time) charter,
provided the ship is hired for an equivalent time, the speed of the ship should be the same.
Time in Port
Throughout this chapter it has been assumed that the only variable that can be optimised is
speed. It may also be possible to vary time in port by employing labour and equipment at
overtime rates.
Assuming that the actual rate of working is fixed and that time in port is reduced by working
outside normal hours then time in port can not be optimised.
In fact, there will be a minimum and a maximum time depending upon the amount of
overtime worked.
There is a simple rule to employ, viz., that overtime rates should be paid if the additional cost
for every day saved is less than the opportunity cost of the ship‘s time, viz., the Gross Surplus
per day or the market daily Time Charter rate.
1‘
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Chapter 8
Ships’ Costs; Voyage Estimates;
Worldscale
Tramp ships, which include dry bulk carriers and tankers, operate in a market which is
generally recognised to be very close to the classic economic model of perfect competition.
In particular, there are many buyers and sellers (charterers and shipowners) none of whom,
individually, can influence the market; in addition there is a very good knowledge of prices
or freight rates prevailing in the market. Such knowledge arises through the medium of
shipbrokers whose function it is to bring together the two parties to a freight contract,
thereby assisting in matching cargo and ship. Negotiations for long term contracts or charter
parties are often conducted directly between shipowners and charterers, while short and
medium term fixtures are generally, although not invariably arranged through brokers
working in a shipping exchange. It is essential in an efficient freight market that shipowners
should be aware of potential cargoes available whilst charterers, for their part, should have"
the opportunity of selecting, for the carriage of their cargoes, ships most suitable in size
commanding lowest freight rates.
The Baltic Exchange, situated in St. Mary Axe, London, since 1903 is the most well known
and important of all shipping exchanges, partly because of its geographic location but also
because of its size and formal constitution; 750 companies under a wide umbrella of
international ownership are Members of the Exchange. Such membership may be suspended
or terminated on the grounds of malpractice or conduct of business not in accordance with
the Baltic’s Code of Practice. Agreements made verbally between brokers on behalf of their
principals are considered binding in accordance with the well-known motto of the Baltic,
‘Our Word, Our Bond‘. Many Members of the Exchange are Fellows or Members of the
Institute of Chartered Shipbrokers which seeks to maintain professional standards with its
work in education. Links are maintained with other exchanges, situated in important business
centres such as New York, Tokyo, Hong Kong, Piraeus. The freight market is the most
important activity of the Baltic, although aircraft chartering and the sale and purchase of
ships and futures markets are also conducted. Futures markets in grain, a number of other
commodities and freight, also, are all situated in the Baltic. Cargoes fixed in the international
freight market include grains, coal, ores, fertilisers, cement, timber and metal scrap. Brokers
gain information on cargoes and ships available in a number of ways, one of which is through
daily meetings on the ‘Floor’ of the Baltic.
Brokers may specialise by commodity or by geographic region. Brokers such as John I.
Jacobs, for example, are known as specialists in the tanker market, while. H. Clarkson is .
better known for its interest in the dry bulk freight market. Most of the well known firms of
shipbrokers produce market reports periodically for distribution to their clients. Fearnley's in
Oslo is probably the most famous firm of shipbrokers as a result of the comprehensive and up
to date statistics which they produce and disseminate world-wide and which form the basis
for statistical analysis by many major organisations including OECD and UNCTAD.
I
I
I
I
i-I._A
1.
_ _;r___-a;:¥‘-A-1111*‘
\
J_---fats;
If
L
I
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The world freight market, in reality, comprises a number of sub-markets according to vessel
size and, to some extent, by specialisation. Although the level of freight rates is generally
diflerent in the difl'erent sub-markets, such levels may be influenced by changes in supply and
demand conditions in adjoining markets since marginal vessels may move freely from one
sub-market to another. In particular, combination carriers may trade in dry bulk cargoes, oil
cargoes, or in both — although not at the same time.
Since different cargoes are carried in different quantities over diflerent distances and under
different cost conditions, e.g. the carriage may involve slow working in ports of loading
and/or discharge or transit of the Panama or Suez Canal etc., it should be expected that
freight rates will not be uniform. A perfect market, however, cannot operate unless the
parties are able to compare prevailing prices or rates in order to make rational decisions.
While comparison for charterers is fairly straightforward, especially if charterparty terms are
identical, shipowners need to reduce freight rates per ton (or tonne) to the same unit, viz.,
gross profit (or gross surplus) per day. Once this has been done, it is a fairly simple matter to
convert this figure to a Time Charter Equivalent (TCE). This is necessary since a great deal
of business is done through Time Charters, either for one voyage (Trip) or for a number of
round voyages or for a number of months.
The purpose of Voyage Estimates is to allow comparisons to be made and thereby to improve
the efliciency of the market. Such estimates will quickly demonstrate whether charterers‘ or
shipowners‘ ideas have moved out of line with the market, and if so, will cause rates to come
back — if the principals wish to do business. Thus in the tramp market, contrary to what is
often read in text books on the subject, shipowners do not set rates at such a level to allow
them a reasonable profit, but begin negotiations with charterers, taking into account
charterparty terms which can affect profit levels, starting at prevailing market rates as
indicated by those published in the shipping press or market circulars. An owner who feels
that the market is hardening can always hold out for higher rates, but he is then in danger of
losing the business. In a similar way, if a charterer persists in oflering rates that are too low,
he runs the risk, if he is the seller of the goods, of not getting his cargo shipped by the time
specified in his contract of sale.
Brief details of charterers’ requirements could appear in circulars in the following form:
ROSTOCK/l—2 USEC LIVERPOOL/VITOR,lA*(27l)
3,000 TS ROCK SALT 20,000 TS SCRAP (52[53’)
7/10 AUGUST 10/15 AUGUST
4,000/5,000 FIOT 5,000/1,000 FIO
GENCON C/P — 5% GENCON C/P — 5%
(l5T GEAR)
The format and terms used are of course readily understood by those working in the market.
The details here are given in the order: loading port; discharging port; quantity and type of
cargo; dates between which ship" should be ready to load; loading and discharging rates/day;
who pays for loading/discharging; type of charter party and amount of commissionpayable.
The GENCON standard charterparty, developed by BIMCO is intended to be used when
there is no special C/P form available.
The availability of a vessel at a port or area might appear as follows:
‘SD 14‘ LIB 1979 90T H/L Open Philadelphia 18/22 August.
Fixtures are usually reported in the following style:
NSW/JAPAN ‘MINERAL ANTWERPEN' 120,000 COAL $8.65 FIO 35,000 SHEX/38,000
SHINC 15/30 8 BHP.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Interpreted, this means that the vessel ‘Mineral Antwerpen’ has been fixed by the Broken
Hill Proprietary to carry coal from New South Wales to Japan beginning between 15 and 30
August at a rate of USS 8.65/tonne, loading and discharging costs free to shipowner; rates of
loading and discharging 35,000 and 38,000 tonnes/day respectively. Sundays and holidays are
excluded for loading but included for discharging.
The shipowner is in a market that is known for rate volatility so that he does not necessarily
expect to make profits every year. In practice, he takes the good years with the lean, hoping
that the long term will produce profits at least suflicient to cover the cost of the vessel or
vessels. Nevertheless, in the short term, the owner is concerned with the lay-up point which
occurs approximately when the net freight revenue is only just suflicient to cover voyage and
running costs, i.e. when the gross profit is zero. At such times the shipowner must consider
seriously whether to keep the vessel in service or to pay ofl‘ the crew and place the vessel in
lay-up. A number of factors must be considered, including market expectations. The owner is
also concerned with the point where the net profit becomes positive, i.e. where depreciation is
covered as well as voyage and running costs. The owner can then look forward to a new car
or yacht —— and the payment of dividends if the company has shareholders. The costs with
which the shipowner is concerned are shown below.
Ships’ Costs
Many readers will be familiar with the structure of ships‘ costs although definitions are not
always precise. For example, ‘operating costs’ sometimes includes capital costs and sometimes
they are synonymous with running costs, q.v.
Capital Cost This, in its simplest fonn, is the actual cost of the ship. It is a sunk cost and in
the short term‘, at least, it must be regarded as a fixed cost. A ship is however, as a mobile
asset, more easily sold in the intemational sale and purchase market than a car factory. Some
shipowners are in fact more interested in buying and selling ships for profit than in
employing them in the carriage of cargo. The capital cost can be reduced to an annuity by
dividing it by the appropriate annuity factor. The resulting annual capital cost (or charge)
may be considered to include depreciation and interest on capital. The term ‘capital cost‘ may
be modified by including the effects of loans, interest, tax and capital allowances (i.e.
depreciation for tax purposes). '
A capital cost may be turned into a variable cost by means of a short term ‘lease’ or bareboat
charter.
For the purposes of voyage estimating which is primarily an exercise in accounting, annual
capital cost may be considered as equivalent to ‘depreciation’. This is commonly calculated by
the ‘straight line’ method where annual depreciation is simply the capital cost of the ship
divided by the projected economic life. Depreciation does not include return on capital or
profit, so that Net Profit = Gross Profit (i.e. Revenue — Operating costs) — Depreciation.
Running Costs These comprise certain costs that must be incurred, provided the vessel is in
service. Essentially, they do not vary with the specific voyage and are time related. Although
these costs are in some sense fixed,‘ they are predominantly variable with output (measured in
tonne—miles). Thus, in port, the running costs are related to the time taken in loading and
discharging, while at sea they are related to the number of miles steamed. Running costs, in
broad terms, are considered to comprise:
Crews’ Salaries and leave allowances as well as other ancillary expenses of training,
pensions, travelling etc.
Insurance (Hull and Machinery) This covers total loss as well as damage to hull from
collision and stranding; it also covers certain third-party claims.
KONSTANTINOS
Rectangle
Protection and Indemnity Many ships are entered with P and I Clubs that indemnify owners
against losses arising from strikes, quarantine restrictions, breakdown, and ofl‘er protection
against claims, often involving negligence, that are not covered by the hull insurance
policy. The clubs are non-profit making so that cost varies according to the claims that are
settled.
Maintenance of hull and equipment including painting and cleaning; the overhaul of
machinery, firefighting and lifesaving appliances.
Stores The supply of consumable stores, ropes, paints, cleaning materials for deck, cabin,
galley and engine room.
Spares Mainly for machinery.
Victuals Food and beverages for crew.
Lubricating oil This is an expensive item for diesel powered ships. Although included as a
running cost it should more properly be considered as a voyage cost q.v.
Drydocking and Surveys Surveys and drydocking are required to comply with regulations
and/or to keep the ship in class with one of the classification societies such as Lloyd’s
Register or Bureau Veritas. Owners try to postpone drydocking as long as possible by using
underwater inspection and hull cleaning methods.
Administration The running of a ship requires support from shore superintendents,
personnel managers and accountants as well as the commercial department whose function
it is to obtain employment for the vessel.
Running costs are, apart from insurance (which is greatly reduced) and certain administrative
costs, largely avoidable when the ship is in lay-up.
Voyage Costs These relate to a particular voyage being undertaken and include:
Fuel for propulsion machinery.
Port Charges including tugs, pilotage, conservancy, side wharfage, agency fees and light
dues.
Cargo Handling These include loading, discharging, trimming, lashing and provision and
laying of dunnage where required. The costs of refrigeration and special preparation of
holds are related to the -voyage but are not specifically cargo handling costs. The hire of
cranes and other cargo handling equipment is also included under cargo handling costs.
Of these costs, fuel and running costs are variable as discussed in chapter 6. The marginal
cost is defined exclusively in terms of fuel costs. Cargo handling costs are generally related to
the quantity of cargo carried and only in this sense are they variable; port charges are mainly
fixed costs for the voyage though they may be considered variable to the extent that they
increase with the number of voyages performed each year.
The costs normally borne by the shipowner under difl'erent types of charterparty are shown in
Figure 8.1 below. Other costs are normally borne by the charterer.
Figure 8.1
I I 1
I Capital Bareboat Charter I
Time (or trip) Charter “ I
Running Costs Mtlloyage Charter \
I Voyage Costs I
J I
‘L _i _ _ _ _I
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The burden of cargo handling costs are however determined specifically by the terms of the
charter party, e.g.
FIO (Free In and Out) Cargo handling costs are for the account (a/c) of the charterer.
Free Discharge Loading costs are for the a/c of the owner; discharging costs are for
charterer’s a/c.
FOB (Free on Board) Loading costs are for charterer’s a/c; discharging costs are for
owner's a/c.
Crew Overtime Costs are normally paid by the owner unless the costs are incurred specifically
for the voyage and provision is made in the C/P for such costs to be reimbursed to the owner.
Fuel Costs are, under Time or Trip Charters, paid by the charterer. However, the
charterparty specifies a minimum speed and corresponding fuel consumption which, if not
maintained withinlimits, will require compensation to be paid, by deduction from hire
money, to the charterer.
Commissions Shipbrokers are normally paid commission on gross freight earnings, including
despatch at the rate of l;l;°/,, per broker involved. Thus it is common for brokerage to appear
as 215% or 32°/,, for example. Address Commission is paid to the charterer —— not to brokers.
It should be noted that all commissions are paid from gross freight and are therefore paid by
the owner rather than the charterer.
Voyage Estimates
A number of estimates must generally be worked in order to determine the most
advantageous fixture. Time is of the essence since ships cannot be allowed to remain idle in
the hope that a more advantageous offer of employment will be forthcoming. The opportunity
cost of a day's unemployment is generally the prevailing time charter daily rate —— less
commission. The owner must therefore perform estimates speedily and, within realistic limits,
accurately. The use of standard forms can be an advantage, although nowadays the majority
of shipowners probably use microcomputers for the task. The computer can also be
programmed easily to perform sensitivity analyses on the effects on gross surplus of changes
in freight rates, speed, and quantities of cargo.
The following are required for Voyage Estimates:
(a) Vessel's capacity (Cubic and deadweight) as well as the stowage factor of the cargo. It is
necessary to calculate how much cargo can be loaded by weight, since revenue is earned
normally by the ton(ne); but at the same time the estimator must ensure that the cubic
capacity of the ship is suflicient to hold that quantity.
(b) Vessel's speed and fuel consumption at different speeds (loaded and in ballast) since the
former determines the time at sea and the latter the amount of fuel that will be used.
(c) List ofbunkering ports andfuel prices The bunkering port used can have an important
influence on voyage profit. The voyage to be estimated begins from the port of discharge
on the previous voyage — usually when the pilot disembarks. The laden passage is
normally preceded by a passage in ballast and since its cost must be included in the cost
of the voyage its length may be crucial to the financial result. Bunkering may take place
at the last port of discharge; at the loading port; or at a specific bunkering port en route.
Balboa, at the southern end of the Panama Canal is commonly used for bunkering when
ships transit the Canal in the course of their voyage. Curacao, an island in the Caribbean
ofl' the coast of Venezuela is also noted for cheap fuel. It must be appreciated that if
bunkering takes place between the port of loading and the port of discharge, a smaller
quantity of fuel can be carried and more cargo may be lifted. It is sometimes profitable to
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
bunker at more than one port on the loaded passage, but the revenue from extra cargo
carried would be offset to some extent by the additional cost of calling at the bunkering
port. Again, if the cargo is volume- rather than weight-constrainedrthe deadweight taken
up by fuel may be of little consequence. The price of fuel at the several alternative
bunkering ports should also be taken into consideration. It is sometimes cheaper to carry
more fuel and shut out cargo if the fuel price differential is greater than the freight rate
for cargo. <
(d) Maritime Atlas such as Lloyd's and a Table of Distances as published by BP or Reed’s.
An atlas is necessary to display different possible routes and the location of cargo and
bunkering ports. A knowledge of ocean currents and weather systems may also be useful
in deciding on a particular route. '
A map of Load Line Zones For safety reasons the world's oceans are divided into a
number of different zones which indicate the load line to which a ship may be legally
loaded. The zones are basically winter, summer and tropical. Some are seasonal in that
they may be winter between specific dates in the winter, and summer for the rest of the
year. Some tropical zones, e.g. N. Indian Ocean, N. Atlantic (l0°N—20°N) and parts of
the N. and S. Pacific, may change to summer zones during hurricane (cyclone or typhoon)
seasons. Care must be taken in performing calculations for the prospective voyage that
the vessel is not permitted to be overloaded at any time. For example, the vessel cannot
be loaded to her tropical marks in a tropical zone if within a short time of leaving port
she will enter a summer zone. Relevant information from the ship’s loading scale must
also be available. In particular, draught and deadweight corresponding to winter, summer
and tropical marks must be known The difference in deadweight between summer and
winter marks varies with the size of ship, but in the case of a vessel of about 25,000 dwt
the difference is in excess of 700 tons which can make a significant difference to the
revenue earned. Draught limits in, ports and canals as well as _length or beam restrictions
must also be taken into consideration. The Panama Canal has a beam restriction due to
the locks which are 110 ft (33.6 M) wide. There is also a maximum draught in the canal
of some 36'—38’ which varies according to the amount of rain that has fallen: a
considerable quantity of rain is needed to maintain the level of water against losses
occurring each time ships pass through the system. Such restrictions give rise to the tenn
Panamax vessel which is the largest size of vessel that can transit the canal fully laden. In
the case of bulk carriers a Panamax vessel is about 65,000—70,000 dwt.
Port disbursements, canal dues, rates of loading and discharging Some of this information is
available from oflicial port publications and from guides. Organisations such as BIMCO
(Baltic and International Maritime Council) and INTERTANKO (International
Association of Independent Tanker Owners) also provide information on port charges.
Such charges are difficult to determine even from official publications: many provide
charges on application. The best way of determining port charges as well as loading and
discharging rates is by experience —— so that accurate infonnation should always be
retained — or requested from port agents. It is essential to have accurate information on
cargo handling rates and weekend working in order to estimate accurately the time in
port. Also, the charterparty will contain a clause concerning time allowed for cargo
handling, viz., laydays. Time spent in addition to laydays will give rise to demurrage
(payment to shipowner), whilst despatch money (normally at half the demurrage rate) is
payable by the shipowner to the charterer if cargo-handling is completed before
laydays expire. The owner may be able to estimate such payments and, if so, they will
form adjustments to the freight revenue. Shipowners should also be aware that in some
countries taxes are payable on revenue earned. Such taxes may not be recoverable if the
country of the flag of registry does not have a double taxation agreement.
Rates of exchange In order to perform voyage estimates, all revenues and costs must be
converted to a common currency. This is often US dollars since freight rates, fuel and
certain other costs may already be expressed in this currency.
KONSTANTINOS
Rectangle
Even though the freight market may be technically perfect, the estimates are unlikely to
produce identical results for all options considered unless such options are concerned with
shuttles where, due to the strength of demand, many vessels may remain on the same trade
for a period of time and the ballast passage is exactly the same as the loaded passage but in
the reverse direction. Particularly when demand is high on such routes, and rates are at a
correspondingly high level, owners may ignore opportunities to carry backhaul cargoes. Such
cargoes which are likely to be in relatively short supply yield very moderate returns: they are
often of interest to marginal vessels only. Here, revenues are only slightlygreater than
marginal costs which for backhaul cargoes are very small. Nevertheless, over a number of
voyages, average returns should be similar. The shipowner, with the aid of his broker should
be able to plan a vessel’s itinerary to take advantage of seasonal shipments such as sugar or
other crops and thereby do a little better than the market. Where options yield similar daily
gross surpluses, the shipowner is likely to fix the one of shorter duration in a rising market
and the one of longer duration in a falling market. Consideration must be given to the
positioning of ships; repatriation of crews; surveys and drydocking that need to be carried
out at certain intervals and at repair yards favoured by the shipowner. With increase in size
of vessel, the opportunities available for different cargoes and ports are reduced. Time spent
in ballast may approach 50% and shuttles or regular pattems become more common. This is
particularly true of tankers in the VLCC/ULCC category where loading ports are, in general,
restricted to the Middle East and discharging ports are limited mainly to Europe and Japan.
Example ofa Voyage Estimate
Voyage: New Orleans to Yokohama. 25,500 tonnes i 5% Scrap. Feb. $25.00 fio. 3,000
Load/4,500 discharge both SHEX. 3.75%.
Find the Gross surplus/day and the TCE (5%) given:
1. Universe Cardifi" in Rotterdam: Summer dw 26,000 tonnes. Draught 32'09".
Hold Capacity 1,120,000 cu.ft.
2; Speed 14.5 kts.
3. Fuel consumption 40 tonnes/day f.o. at sea. 1 tonne/day in port
1 tonne/day d.o. throughout
New Orleans
Yokohama
Panama
Canal Dues
Rotterdam-New Orleans
New Orleans—Panama
Panama—Yokohama
Rotterdam
Panama
Rotterdam
4. Port disbursements
5. Distances
6. Fuel prices
Remain on Board
7. Safety margin
8. Running costs
9. Load Line'Zones New Orleans — Summer;
Yokohama —- Summer
$25,000
$36,000
$7,000
$27,000
4,880 miles
1,430 miles
8,400 miles
Fuel oil $110/tonne. Diesel oil $180/tonne
Fuel oil $105/tonne. Diesel oil $180/tonne
Fuel oil 200 tonnes @ $120/tonne.
Diesel oil 50 tonnes @ $200/tonne
Fuel oil 150 tonnes. Diesel 50 tonnes
$4,000/day -
Balboa — Tropical;
Also find the optimum speed and the corresponding Gross Profit/day.
Preliminary It is first necessary to be certain whether 25,000 tonnes i 5% is at owner’s or
charterer‘s option. In this case it is at owner's option,’ in which case the vessel must Iload a
minimum of 24,225 tonnes up to a maximum available of 26,775 tonnes. The scrap stows at
about 35 cu.ft./tonne which means that the quantity loaded will be weight rather than volume
constrained.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Voyage Legs
Rotterdam
New Orleans
Panama
Panama
Yokohama
Total
Port Details
Rotterdam
New Orleans
Panama
Yokohama
Total
Dist. (n.m.)
4.880
1,430
8,400
14,710
Days
10.7
1
8.0
Fl
Port
Dag
14.02
4.l l
24.14
42.27
Costs
Misti
25,000
7,000
36,000
68,000
Fuel (mt) Diesel (mt)
561 14
165 4
966 24
1 ,692 42
Fuel (mt) Diesel (mt)
ll ll
1 1
8 8
5 E
Note Time in port is based upon an estimated cargo of 24,500 tonnes loaded at
3,000 tonnes/day; discharged at 4,000 tonnes/day; a Si day working week augmented by one
day to allow for notice. It is assumed that all lay time will be used.
Bunkering Plan
P22
ROB
Rotterdam
Balboa
Yokohama
ROB
Total
Fuel (mt)
200
688
974
150
Price ($1 D.O.
120
110
105
105
(milso
62
_-.-1
50
Price (§) Total C. ($1
200 34,000
180 86,840
—-— 102,270
180 (24,750)
198,360
Note Fuel remaining on board at the beginning of the voyage is charged at the price paid on
the previous voyage. Fuel remaining on board at the end of the voyage is credited at prices
paid during the current voyage.
Cargo Details
LIL9
Summer Deadweight 26,000
Stores etc.
Fuel oil
Diesel oil
Safety margin
Available for cargo
+41} days
Total Cargo
1,606
24,394
185
24,579
‘iii-1
LL19
400
974
32
200
1,606
IIQTI ' »
- ,_ 3 _ -_ ..".. =4 r ;' ._ ‘E. ‘J
' E; ,. d_. ' I; _- _ 1_-_-cl‘: _(:- vi‘ _- 5 __, ._?°‘. ‘ _ €_
,1 '-'..""i.-tr. -L.J.:§,i_.eiP.-iur....--—"" I -- -=' ‘F__ ' _-=-.-
“ § .’.'--‘~.-is .1,“‘L T“ 531'-?B§"fi “fig”-1'. ,, ...
ii? er -é .=.
(Allowance for Tropical Zone)
Note Since the vessel is bunkering at Balboa, the critical time for determining cargo weight is
at Balboa. An extra allowance of 185 tonnes is made since vessel leaves Balboa in a Tropical
Zone and has 4») days steaming until she reaches the Summer Zone.
Cargo carried complies with the C/P terms of 25,000 tonnes 1 5%.
99
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Expenses {$1
Fuel Oil/Diesel Oil 193,360
Port Expenses 68,000
Cargo handling —-
Canal Dues 27,000 Net Revenue
Taxes — Less Expenses
Sundries —
Warranties —-
24,579 tonnes @ $25
Less Comm. (3.75°/,,)
Voyage Surplus
293,360
Total Days on Voyage = 62
Hence, Daily Surplus = $4,808
Revenue {$1
614,475
23,043
591,432
293,360
298,072
Therefore Gross Profit/day = $4,808 — $4,000 (Running costs = $4,000/day) = $808
TCE = $4,808 plus allowance for Commission = $5,061
Optimum Speed (See Chapter 7)
This is given by s where
S, (WR - 1))
(Rd,/d + p)k(3d + 261)
and w = (26,000 - 400 - 32 - 200)
= 25,368
R = 25 >< 0.9625
= 24.0625
1) = 95,000 + (19.7 >< 180) (days in port)
= 98,546
= 14,110
= 105
= 19.7
= 40/3431
<1, = 6,334
S21 f f W Sgl li,8fi7l.5 X 14,710
40/3483 X l,708,993 >< 55,595.4
= 73,618
So that
s = 289 miles/day or 12.04 kts.
7'¢'u-r"UO-
Taking an optimum speed of 12 kts the corresponding fuel
consumption/day = 40(288/348)’ = 22.67 tonnes.
The Estimate is now reworked with the new speed and fuel consumption with the result that
the Daily Surplus = $5,074 and the TCE = $5,074/0.95 = $5,340.
The result of steaming at optimum speed is therefore an improvement for the shipowner of
$5,074 — $4,808 = $266/day.
It may be shown (Chapter 7) that TCE at optimum speed
= 2ks3{p + (d,/d)R} = 2 X 40 x (288/348)3{l05 + 24.0625(6,834/ l4,7l0)}
= $5,268
This is slightly higher than the previously calculated figure of $5,074 because that calculation
allowed for a diesel oil expense of $180 (i.e. 1 tonne) per day which should more correctly
have been included with running costs. Thus the revised figure = $5,268 — $180 = $5,088
which is close to that calculated by the more laborious method.
KONSTANTINOS
Rectangle
Tanker Voyage Estimates
Whereas in the dry bulk freight market, voyage charterparties are usually fixed in terms of
US dollars per tonne, with tankers they are normally fixed by reference to Worldscale, or,
since lst January 1989 to New Worldscale. Worldscale is the code name for the Worldwide
Tanker Nominal Freight Scale, a volume comprising standardrates and diiferentials for the
majority of tanker terminals in the world as well as for combinations of ports and trans-
shipment areas. Worldscale is jointly sponsored by the Worldscale Association (London) Ltd.,
and the Worldscale Association (NYC) Inc. The area of responsibility for New York is the
western area comprising North, Central and South America including the Caribbean Islands,
Greenland and Hawaii. London is responsible for the eastern area covering the remainder of
the world. Since lst January 1972, all Worldscale rates have been quoted in terms of US
dollars.
There are two main reasons why Worldscale is used for negotiating fixtures in the tanker
market. First, it is a standard o_f reference; this means that at a given Worldscale rate —-— the
standard is Worldscale 100 (now New Worldscale 100), all other rates being expressed as
percentages of this figure —- a tanker owner should earn approximately the same gross
profit/day regardless of the voyage on which the tanker is engaged. The owner will have
calculated for every tanker in his fleet, inter alia, the (New) Worldscale rate required to cover
all costs and also the rate at which the tanker would be on the point of lay-up. Such rates
depend upon the size of the tanker, flag, age and type of propulsion. Suppose, for example
that the New Worldscale rate for a particular voyageis shown in the Worldscale Schedule as
$15.90. An owner of a new 200,000 dwt tanker may find that a rate of $12.20 per tonne is
suflicient to cover the total costs of the vessel for that voyage. He therefore knows that for all
voyages the New Worldscale rate required to break even is approximately
{l2.2/15.90} x 100 = NWS 77. In general, the larger the tanker the lower is the NWS rate
required to break even and vice versa. If may also be noted that tankers bought second hand
in a depressed freight market at low prices also require lower Worldscale rates to break even
than do their newly delivered counterparts.
The second reason for using Worldscale is that it saves a great deal of effort in negotiating
fixtures that involve a number of optional ports.
Calculation of rates for New Worldscale Schedule
All rates are calculated in US$ per tonne (1,000 Kg or 0.98421 long tons) for a full cargo
based on a round voyage from loading port or ports to discharging port or ports and back to
the first loading port. For the purposes of calculation the following factors are used. These are
purely nominal and are for rate calculation purposes only.
(a) Standard Vessel 75,000 tonnes (Total capacity)
Average Speed 14.5 kts
Fuel consumption Steaming S5 tonnes/day
Other 100 tonnes/round voyage
Port 5 tonnes for each port involved
Grade of fuel 380 cst
(b) Port Time 4 days (One loading port/one discharging port)
+1} day for each extra port involved
(c) Fixed Hire Element US$ 12,000 per day (Notional running plus capital costs)
(d) Bunker Fuel Price (1989) US$ 74.5/tonne
(e) Port Costs Assessed by the Worldscale Association from information
available up to the end of September of the previous year
(f) Canal Transit Time 30 hours for each transit of Suez Canal
24 hours for each transit of Panama Canal
(Mileage not taken into account)
_ _ _, , ,=,n,,_._ _
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Every lst January a revised edition of Worldscale will be issued, taking into account
updated port charges and the average price of 380 cst bunker fuel prevailing during the
month of September of the preceding year, as assessed by Cockett Marine Oil Ltd.
From lst January 1990 the term New Worldscale will disappear and, in line with industry
practice, the Schedule will be known as Worldscale once more.
The following is a hypothetical example of a rate calculation involving a round voyage from
Mina al Ahmadi to Milford Haven on the basis of a ballast passage via Suez and loaded via
the Cape of Good Hope. Other rates are calculated for voyages involving the Cape both ways
an_d also for Suez both ways. Rates are designated C, CS or S accordingly.
In practice, rates are calculated from Quoin 1., at the entrance to the Persian Gulf, to a range
of discharging ports and, in addition, amounts by which these must be augmented to allow
for port charges and additional distances from the many different loading terminals in the
Gulf to Quoin I.
Distances Milford —-— Mina via Suez 6,257 miles
Mina — Milford via The Cape 11,115 miles
Fuel Consumption
Days
Expenses
Cargo
Total
Sea
Port
Other
Total
Steaming
Port
Canal
Total
Fuel
Hire
Port
Total
Deadweight
Stores plus Fuel safety margin
Fuel for steaming
Other
Cargo
17,372 miles
2,746 tonnes
10 tonnes
100 tonnes
2,856 tonnes
50
4
It
552%
$212,772 (2,856 x $74.5)
$663,000 (12,000 x 55%)
$49,000
$924,772
75,000 tonnes
2,312 tonnes
73,688 tonnes
500
1,757
55
2,312
From these calculations the Schedule Rate (NWS 100) = 924,772/73,688 = $12.55 per tonne.
It must be emphasised that this does not actually represent the cost of carriage for a tanker of
such specifications, essentially because of the notional nature of the ‘Hire Element’ used.
Canal dues for Suez transit do not enter the calculations since they form a fiked additional
which are specified in the Schedule and paid ‘up front’ to the shipowner together with the
agreedcharter money. This differential, which is not subject to market variations, is paid by
the charterer even though the tanker may have started her voyage in Japan and is not
therefore required to transit the Suez Canal in ballast as part of her voyage.
KONSTANTINOS
Rectangle
1
1»
i
i
li
l
'\
__?_.--....w,___+"l
-l
The actual charges for transiting Suez by a crude oil tanker are shown in the following table.
TABLE 8.1
Suez Transit Dues for Crude Oil Tankers 1.1.1989
Laden Ballast
SDR per ton SDR per ton
First 5,000 tons
Next 5,000 tons
Next 10,000 tons
Next 20,000 tons
Next 45,000 tons
Rest of tanker’s tonnage
5.50
3.10
2.90
1.30
1.30
1.10
4.40
2.48
2.32
1.04
1.04
0.88
The fixed differential as specified in the Schedule comprises an amount designed to
compensate the shipowner for canal transit both laden and in ballast. The amount consists of
a rate (in US$) per SCNT (Suez Canal Net Tonnage) together with a lumpsum payment.
For example, the fixed differential to be added to the freight in the case of a tanker of
145,000 dwt with SCNT 72,000 carrying crude oil on the laden leg, would be calculated as
follows:
$3.28 x 72,000
Lumpsum
L.S. increase
= $236,160
= $116,500
= $7,095
Total = $359,755
(To cover cost of escort tugs for vessels of this size)
From the above table, the actual charges would amount to 251,280 SDRs which is equivalent
to $313,095 @ $1.246 to the SDR. To this must be added an allowance for light dues, harbour
dues, escort tugs, agency fees and sundries which are incorporated in the figures calculated
using the fixed difierential.
The Rate Schedule indicates clearly which costs are for charterer’s and which for owner’s
account as well as providing a table of demurrage rates that are subject to market variations
of New Worldscale. For example, if a vessel of 180,000 tonnes dw is chartered at NWS 60
and in the course of the voyage the owner is to be paid three days demurrage, the charterer
will pay the shipowner 3 X 0.6 X $30,000 = $54,000. The figure of $30,000 is taken from the
Table of Demurrage rates for a tanker of that size. The above assumes that the same
percentage variation is used for demurrage as is used for freight, but essentially it is a matter
for negotiation.
A New Worldscale demurrage rate represents the daily net revenue at a freight level of New
Worldscale 100 for a typical vessel within the size range concemed (on the basis of a voyage
that is representative of the average length of round voyage performed by tankers) plus an
allowance for one day's consumption of bunkers in port with engines at stand-by.
No allowance is made in the demurrage rates for port costs incurred when the vessel is on
demurrage, nor is any allowance made for the cost of cargo heating when on demurrage.
Owners and charterers are not compelled to use New Worldscale; they may simply negotiate
rates in US$ or any other currency. Nevertheless, New Worldscale makes a useful basis to the
contract or charterparty.
KONSTANTINOS
Rectangle
The Origins of Worldscale
Tanker Rate Schedules have their origins in World War II. Before the war, rates of freight in
respect of tanker voyage charters were negotiated and agreed in terms of a monetary unit per
ton. This meant that a rate of freight had to be established for every voyage that was
contemplated under the charter and for certain trades it was necessary to agree several pages
of rates.
During the 1939-45 war, first the British Government and later the US Government
requisitioned tankers, and owners were remunerated on what was essentially a Time Charter
basis. From time to time these tankers were re-let to oil companies on a voyage basis and
consequently schedules of rates were devised with the underlying principle that irrespective of
the voyage performed, the Government should receive from the oil companies the same net
retum per day. This retum could be compared with a Time Charter hire rate.
By the time Government control of shipping was relinquished in 1948 the tanker trade had
cometo recognise the advantages of having schedules of freight rates and therefore in the free
market the system evolved of quoting on the basis of MOT (Ministry of War Transport) or
USMC (US Maritime Commission) plus or minus a percentage as dictated by the
supply/demand position in the market.
Between 1952 and 1962 a number of different rate schedules were issued: Scale Nos 1, 2, 3
and Intascale in London; ATRS (American Tanker Rate Scale) in New York. Then in 1969
the joint London/New York production Worldscale was issued to replace Intascale and
ATRS. A number of changes were introduced as one scale replaced its predecessor but until
1965 the daily hire element was always such as to yield the historic rate of 32/6d (£l.625) for a
round voyage between Curacao and London.
The first edition of New Worldscale contains over 60,000 rates and over 1,000 ports are
mentioned in the Schedule. The Schedule includes a number of standard conditions, the most
important of which are probably those specifying which items of port costs are to be paid by
the respective parties. It is customary to incorporate these standard conditions, including the
specified laytime, into a charterparty by a simple reference to New Worldscale terms and
conditions.
The main change in the construction of New Worldscale vis 6 vis Worldscale is the increase in
nominal size of tanker from 19,500 tons dw to 75,000 tonnes dw with associated changes in
speed, fuel consumption and fixed hire element. It was felt that a scale based on a larger
vessel would produce less distortion in voyage results when different percentages of New
Worldscale were applied across a range of different voyages.
As a result of a lawsuit filed by the US Department of Justice against Worldscale Association
(NYC) Inc., in which it was alleged that the inclusion of a brokerage element in the
calculation of Worldscale rates was a violation of the anti-trust laws, it was agreed to amend
the method of calculating Worldscale rates by excluding this brokerage element beginning
with the Edition of lst January 1982.
The bodies responsible for Worldscale see their role as limited to providing a set of reference
rates calculated on a common basis in accordance with the published formula and revising
those rates in line with the announced revision policy.
Standard Voyage Estimates
Most tramp voyages are associated with a ballast passage and this is included as part of the
voyage. The gross surplus per day therefore depends upon the relative lengths of the loaded
and ballast passages. In order to standardise the expected returns a particular voyage may
be selected involving a loaded and ballast passage of equal lengths and the gross surplus per
1
I.-1.
I1‘
1
r
__.,_,"\1-
"H
l
1‘
____..__.-.‘L_--.4...‘-_-._.-;_._.__._._._.
1
1
u
1
1
'3"
1
,1.
1
1
1 11
1
1
\
1
A44-‘I-=4
1
11-1
|1
!
1
.1
P
1
1
Q.
1
f,\
day calculated for a given freight rate. The prevailing market rate can then be related directly
to the returns associated with the selected rate and an indication of expected profits found
The Standard Estimate, which is probably of greater use in the tanker markets, can be used
to predict profits from operating on any trade route since, on average, the lengths of the
ballast and loaded legs are equal.
The following is an example of a Standard.Voyage Estimate for a tanker of 102,500 tonnes
dw and a loaded voyage from Sidi Kerir terminal to Tarragona using New WS 100* as the
rate reference.
Standard Yoyage Estimate @ NW_S 100
11;. T. Myfanwy 102,500 tonrlesggdw
Voyage Sidi Kerir—Tarragona
Tarragona~—Sidi Kerir
Total
Fuel oil requirements
Bunker Stems
Miles per day
Days at sea
Days in port
Fuel oil per day
F.O. tonnes at sea
F.0. tonnes in port
F.O. tonnes pumps
Total
1,500 miles
1,500 miles
3,000 miles
360
8.33
3.5
65 tonnes
542
30
80
652
Price Cost
R.O.B. Sidi Kerir 500 mt $70 $35,000
Tarragona 652 mt $70 $45,640
Sidi Kerir —— -— —
Remain. Sidi Kerir 500 mt $70 ($35,000)
Total cost
Summer Deadweight 102,500 tonnes
Stores etc.
Fuel safety margin
Fuel steaming
Fuel oil
Total expenses
Freight: 101,529 tonnes @ $3.25
Commission: (21%)
Net freight
971 tonnes
M--ii
Available for cargo 101,529 tonnes
Running expenses 11.83 days @ $8,150/day
Disbursements Tarragona
Disbursements Sidi Kerir
Estimated gross profit
gross profit/day
$45,640
500 tonnes
200 tonnes
271 tonnes
971 tonnes
$96,415
$25,000
$18,000
$45,640
$185,055
$329,969
$8,249
$321,720
$136,665
$1 1 ,552
KONSTANTINOS
Rectangle
_-1.
@ New Worldscale 110 Gross freight: 101,529 X $3.25 X 1.1
Commission (211-7,)
Expenses
Gross profit (voyage)
Difference
rm
(day)
@ NWS 100
Hence each 10 points of NWS is equivalent to $2,719 profit per day.
$362,966
$9,074
$353,892
$185,055
$168,837
$14,271
$11,552
$2,719
“‘ The rate used in the example is estimated. It is not taken from the NWS Rate Schedule.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
I‘
I
F"
1
1
is
b
1
1
-.-“ii____“,__
1
I1-
1-
'1
44;4~1-:44
1
11
1
1
1
0
1
t
1
1
r
1
1
I
1’
Chapter 9
Freight Futures
Market fluctuations, whether in currency, commodities, or shipping, generally result in
exposure to risk by buyers and sellers involved in their transactions. Such market fluctuations
arise because demand may vary unpredictably or more often because of uncertainty of supply.
Certain crops whose harvest depends very much on the weather are, in particular, subject to
alternate glut and scarcity and consequently, the market prices at harvest time will reflect the
particular market situation. For the grower there is the risk that a bumper crop will result in
low prices and even the possibility of some of it being deliberately destroyed, while for the
buyer who may already have made arrangements to dispose of his purchases there is a risk
that a scarcity of supply may exist with prices at unattractively high levels.
Whereas some producers and buyers are prepared to accept such risks together with the
accompanying profits or losses, others may not be so inclined. The former are termed risk
prone while the latter are risk averse. A risk of prices moving against the buyer or seller can
be managed in various ways. Buying currency forward at a known rate can be used to
manage risks due to fluctuations in exchange rates. Such hedging, -as it is called, can also be
used in the trading of a relatively large range of commodities including potatoes, grain, rice,
cotton, animals, gold, copper, tin and oil, but such methods lack flexibility with respect to
quantity, delivery and time. A more satisfactory method of hedging is available however in
the case ofmany commodities through the medium of the futures markets. The main
advantages of futures markets lie in their flexibility and the fact that the contracts may be
settled by cash payments in lieu of physical delivery of the goods.
Observers in the shipping markets will note that the majority of shippers and shipowners
appear to be risk averse. In other words, rate stability appears to be of major importance to
the smooth conduct of international trade. In liner shipping, rate stability is achieved largely
through the conference system where the general practice is for conferences to give a
minimum of three months notice (the UN Code of Conduct for Liner Conferences, where it
applies, requires 150 days) of proposed rate increases. In the tramp market some of the minor
bulks, e.g. timber, move under ‘liner’ terms and a considerable number of the smaller ships
are employed under contracts that guarantee owners a fixed rate. Many, however, are fully
exposed to the vagaries of the freight market where profits can at times be extremely high
whilst at other times losses can be prolonged and heavy. At the same time shippers can be
exposed to great uncertainty with respect to freight costs. Such exposure can be attractive to
shipowners who ‘play’ the markets, buying and selling second hand ships at appropriate times
and taking full advantage of booming freightmarkets. These shipowners, however, are in
reality, speculators who use their knowledge and skills in an attempt to turn market
fluctuations to their advantage.
In the majorbulk trades that include oil, ore, coal, grain, bauxite and fertilisers, both
shipowners and charterers may take steps to limit their exposure to the market. They may
accomplish this through the use of integrated carriers, particularly in the ore, coal and oil
trades where flexibility of employment is limited. Integration implies that major firms such as
oil and steel producers establish subsidiary shipping companies for the purpose of carrying
_ _ fl-—$ . . __
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
F_
it
It
._-4..cw ,.._f_
raw materials that they have produced or purchased in overseas countries. Such subsidiary
companies own vessels and also charter-in considerable numbers, on a long term basis,
from independent owners, thereby guaranteeing a steady income to the owner and stable costs
of carriage for the charterer. Very long term charters are rarely found in the grain trades
although some of the larger and more important grain houses such as Louis Dreyfus do have
shipping interests.
In the 1960s, Contracts of Affreightment (CoAs) became very popular. Bulk shipping pools
and consortia were establishedto negotiate and service these contracts that involved the
shipment of some hundreds of thousands or even millions of tons of coal, oil, etc., over a
period of several years. With such CoAs freight rates are fixed but possibly subject to annual
review. For many charterers, tight supply and rising freight rates provide the signal to hedge
by time chartering bulk carriers or tankers for say 6-24 months. This type of hedge does not
necessarily imply that the charterer will employ the vessel in the carriage of his own cargoes,
but that any general increase in freight rates resulting in higher rates paid by the charterer for
shipment of his cargoes through the spot market are offset by higher earnings from the
employment of the time chartered vessel.
Interest in a- futures market for tramp shipping has existed for many years, particularly for
the grain trades, but initial problems of standardisation of the commodity (shipping space)
and physical delivery needed to be solved. Whereas a contract to deliver potatoes of a
standard type and quality at a given future date can normally be fulfilled without difficulty, a
shipping contract involving the supply of a certain quantity of shipping space at a given port
and time, given the nature of the tramp market, would be totally impracticable. It is of course
essential that a futures market is based upon an underlying physical commodity whose price
at any time may influence the price of futures transactions but, more importantly,-governs the
cash settlement price of a futures contract that has not resulted in physical delivery of the
commodity.
A freight futures market called BIFFEX, or the Baltic International Freight Futures
Exchange, to give it its full title, opened in the Baltic Exchange in London in May 1985. The
value of the underlying commodity, viz., the transport of dry bulk cargoes by sea, is based
upon a- weighted index of freight rates on 12 trade routes as illustrated in Table 9.1. The
trades are heavily Panamax weighted and coal and grain comprise two thirds of the total
number with a combined weighting of 82.5%
TABLE 9.1
Present Day Components of the BFI
Route Tonnage Commodity Weighting (°/0)
1. US Gulf/Cont.
2. US Gulf/Japan
3. N. Pacific/Japan
4. US Gulf/Venezuela
5. Antwerp/Jeddah
6. H. Roads, R. Bay/Japan
7:‘ H. Roads/Cont.
,8. Queensland!Rotterdam
9. USWC/Continent
10. Monrovia/Rotterdam
11. Hamburg/WC India
12. Aqaba/WC India
55,000
52,000
52,000
21,000
35,000
120,000
65,000
1 10,000
55,000
90,000
15/25,000
14,000
HSS
HSS 0 1 20
20
HSS 15
HSS
Barley
Coal
Coal
Coal
Petcoke
Iron ore
Potash
Phosphate Rock
5
5
7.5
'U'|!\)LhLh\l\LlI
FJI
E
108
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
1
IT
7
-ini-I—J_‘:_‘
1
‘J
ll
11
a
1,1
.1
11*
1;
1
1
i
l1
111
-4-‘.ij__
-.4
1.
1
1
1-
1
1
lr
if
‘I
The index, called the Baltic Freight Index (BFI) is compiled daily from information supplied
by a panel of brokers: estimates are made of rates on routes for which no fixtures have been
reported and safeguards are employed to ensure that the BFI cannot be influenced by
malpractice, The index was started at a base level of 1,000 in January 1985 and for the
purposes of the market each index point has a value of $10 (originally $5 for each half point).
Thus, the value of a Contract or ‘Lot’ is, the product of the BFI (reported to the nearest whole
number) and $10, i.e. $10,000 for an index level of 1,000. Although the unit of the contract is
given in terms of BFI, the level at which futures contracts are bought and sold is determined
by supply and demand. In other words, the greater the number of contracts offered for sale in
relation to the number for which there are buyers, the lower will be the price expressed in
terms of the Index. In the freight futures market there is no physical delivery of the
underlying commodity so that all contracts are settled by cash payment.
Simplistically, the prevailing futures price for a given settlement date represents the rate level
at which the owner can effectively obtain employment for his vessel or vessels on the spot
market at that future date, or conversely, the rate at which a charterer could obtain sea
transport for a shipment of cargo.
If the owner finds the rate unattractively low, there is no point in his hedging; if, on the other
hand, he feels that the futures price is a good one and he is afraid that the rates in the freight
market may fall below this level he can ‘lock in’ this rate by selling sufficient futures
contracts to cover the revenue at risk. A quotation taken from the ICC publication Futures
Trading in Commodity Markets states, ‘A naive hedger is like a man who goes out and buys
life insurance. The merchant, however is like a man who already suspects he is ill but is still
able to take out life insurance’.
When the owner actually fixes his ship, the rate he obtains may of course be above or below
the level of the-rate locked in or hedged. At settlement day, rates in the freight market.
through the BFI, govern the level of the contract price, at which time BFI and settlement
price are notionally equal. In other words, the settlement price of all futures contracts is based
upon the BFI prevailing at the time. Figure 9.1 below illustrates the changes in futures prices
Figure 9.1
Development of October 1989 and January 1990 Freight Futures Market DUHTIQ September and October 1989
-_-____’4\\
“ I‘
3 F Jl
ll,
JANUARY 1900 1,____\ / -_./“'-
/ \./
4--—’ ii-
1 ,\\ /_____/
/.
/ ,. i
160044 44 4 4 4 -4,-’ 4444 4 _ ~ ~ 4 A A
l.1 f /"—--
l... I "7’ \\ '
\ I \ '\ 1
\.-J \ rl \\ KJ .
\ I -— /
,1“ J./\-..____,,\ \.__,' ‘--~.,/’ OCTOBEFl/1_9_99/'/
1 \\ _/A/44-4./I
1500: ~44 \ //1. as ~ /_ <4 — ~ — 1
" - - H
P /_ BF1
1400 
SEPTEMBER 1989 OCTOBER 1989
KONSTANTINOS
Rectangle
l
relative to BFI over a period of two months, viz., September and October 1989. In practice,
the settlement price is equal to the average value of the BFI over the last five trading days of
the delivery month.
It will be noted that futures prices and BFI move independently but always converge at
settlement day.
If the freight market has improved compared with the level hedged by the shipowner, gains
will be made, but the consequent improvement in the level of BFI means that at settlement in
the futures market the owner must pay the difference between his original selling price and
the improved settlement price. The result is a gain in the physical market offset by a
corresponding loss in the futures market. This is a perfect hedge. The astute observer will
note that the shipowner would have been better off not hedging at all, but this is just being
‘wise after the event’. The point is that had the freight marketfallen, the ‘loss’ in revenue
sustained would have been compensated by a profit in the futures market. The risk averse
shipowner can be certain of his revenue as locked in by his hedge regardless of subsequent
movements in the freight market. He cannot however take advantage of a booming freight
market; but neither would he have been able to do so had he hedged by time chartering.
Nevertheless a shipowner who has hedged against a falling freight market by selling futures
contracts can, if he perceives, in the meantime, a hardening of the freight market, close out
his position by buying back the same number of contracts as he originally sold. In this,-way he
may still make a loss in the futures market but make a bigger gain in the physical (i.e. freight)
market. The charterer can hedge in exactly the same way as the shipowner except that he
buys contracts rather than sells them. In futures parlance, selling is known as going short (or
a short hedge) while buying is known as going long (or a long hedge).
Example of a ship0wner's hedge
_[_)_aE Cargo size 50,000 mt
Commodity Heavy grain
Prospective employment US Gulf — Japan
Fixture date End July 1989
Date of Hedge 9 May 1989
Current freight rate $28.87/tonne
BFI 1,699
July futures price _1,467
Owner's view rates likely to fall
Freight at current rates = $28.87 X 50,000= $1,443,500
Implied futures rate to be ‘locked in’ = (1,467/1,699) >< $28.87 = $24.93
Number ofcontracts required = 1,443,500} 16,990 = 85
Owner therefore goes short for 85 ‘lots’ @ 1,467 >< $10 = $1,246,950
In the meantime freight rates have fallen to $20.67 and BFI to 1,376 on 22 July when the ship
is fixed.
Owner then earns $20.67 x 50,000 = $1,033,500
and he closes out his position in the futures market by buying 85 contracts @ 1,376 X $10
= $1,169,600
Hence, profit in futures market = $1,246,950 — $1,169,600
= $77,350
Loss in the freight market = ($24.93 —-— $20.67) >< 50,000
= $213,000
Hence, net loss = $213,000 — $77,350
= $135,650
KONSTANTINOS
Rectangle
Note A perfect hedge would have resulted in the shipowner‘s profit in the futures market
being equal to his loss in the freight (physical) market. That this did not happen is a
consequence of imperfect correlation between the owner’s route and BFI. Nevertheless the
shipowner was better off by hedging because of gains made in the futures market. This does
not take into account savings from slow steaming.
One of the important advantages of BIFFEX is that all contracts are fixed by open outcry
around a ring at which 30 members are permitted to deal. Prevailing market prices are
therefore known throughout the trading period day by day. Informed opinion of shipowners,
charterers and speculators are reflected in the futures prices and to some extent it might be
argued that if futures prices were self-fulfilling there would be no point in hedging at all.
Although there appears to be a strong correlation between futures prices and their
corresponding spot prices in the near positions, this correlation generally becomes worse the
further away the futures date lies. The greatest volume of trading does in fact take place with
respect to the nearby positions but there are many factors that can unexpectedly affect the
spot rates in the freight market. These include, political events, crop failures, congestion in
ports, changes in energy prices and natural or other disasters.
Trading futures requires certain features to be present. These are:
market volatility
standardisation of the commodity
attraction to speculators
large ‘physical’ market in the underlying product
the product should normally be storable.
While the fundamental function of the futures market is to allow traders (shipowners and
charterers) to hedge, the presence of speculators is needed to give liquidity to the market.
Without speculators a shipowner who wished to make a short hedge would need to find a
charterer or charterers wanting to make a similar long hedge. Speculators (who may in fact be
shipowners or charterers) go short or long, after taking a view of the market, in an attempt to
profit from the futures market itself. They frequently close out their contracts shortly after
buying or selling in order to secure their profits or minimise their losses. Since the futures
market is a ‘zero sum’ game, all profits made must be balanced by an equal amountof losses.
Even though BIFFEX appears to function quite efficiently, one of the criticisms is that the
daily volume of contracts traded may be insufiicient to permit hedgers to dispose of a
relatively large number of contracts without unduly affecting the price. The Baltic Tanker
Index (BTI) which was established for trading in tanker freight futures was discontinued at
the end of 1986 due to the low volume of business conducted. INTEX, established in
Bermuda operated until the end of 1988 a freight futures market in both dry bulk and
tankers, prices being determined through a computer network rather than open market
outcry.
At the Baltic it is possible to buy freight futures for the present month, the following month
and for the months of January, April, July and October up to two years forward. These are
called Delivery Months. To avoid confusion, trading on the floor only takes place for one
particular month at any one time.
International Commodities Clearing House (ICCH)
All contracts agreed between Members on behalf of their clients are registered with the
ICCH after which the two parties to the contract are no longer bound to each other but each
has a contract with the ICCH.
Members are required to maintain funds in the ICCH sufficient at all times to enable the
Clearing House to settle any potential debts. This is done by means of an initial deposit,
KONSTANTINOS
Rectangle
2
called the Original Margin (about 5% of the contract price), and maintenance or variable
margins on a day to day basis if the value of the futures contract moves against the trader.
If a charterer, for example, bought 50 futures contracts for 50 X 1,600 X $10 = $800,000
yesterday and today the price has moved down to 1,590 index points, the charterer is
potentially in debt to the extent of $(50 X 1,600 X l0) —- $(50 X 1,590 X 10) = $5,000. It must
be made clear that when contracts are bought or sold the traders do not actually pay the value
of the contract at the time. Nevertheless the owner or speculator, in the above example, who
has sold the 50 contracts @ 1,600 might wish to buy them back on the following day. In that
case he would make a profit of $5,000 to be paid by the Clearing House —— not the charterer
who originally bought the contracts. It is for this reason that the ICCH must be kept in funds
by means of maintenance margins. Clients do not benefit on a day-to-day basis when futures
prices move in their favour above or below the original contract price. Floor members usually
charge $30-—$40 per contract bought and sold according to the status of their client and the
volume of business transacted. In their accounts with the ICCH they can, unlike their clients,
offset deposits due to unfavourable movements with credits arising from favourable ones. It
appears therefore that they are in the most favourable financial situation when they have a
portfolio of short and long hedges on behalf of their different clients.
If contracts are not closed out by the sale or purchase of opposite contracts, Members must
settle accounts with the Clearing House on Settlement Day which is the first business day
after the last Trading Day. The Settlement Price is, as stated previously, the BFI averaged
over the last five trading days of the Delivery Month X $10 per point.
The Basis In the case of commodities that are completely standardised and for which there is
an efficient market there should be no problem in establishing a perfect hedge. Where,
however there is a difference in the quality of the goods or circumstances surrounding the
delivery of those goods then there exists a Basis risk which is the difference in price betweenwhat is obtained for the goods and the market price for the standardised product. In._these
circumstances a perfect hedge may not be possible. In the freight futures market the
underlying commodity is a basket of freight rates as represented statistically by the BFI. The
freight market comprises a number of sub-markets which are not totally differentiated and so
the rates tend to move in the same way. Nevertheless, because the sub-markets, which are not
easily defined, may relate to different trades, different commodities and different ship sizes,
the correlation between movements in freight rates is unlikely to be perfect and lags may
exist between such movements. Relative movements in rates for three of the routes
(1, 2, and 7) comprising BFI as well as BFI itself are shown in Figure 9.2
These show a generally good correlation between the three routes and, not surprisingly, with
BFI.
Nevertheless it is essential for any charterer or shipowner who wishes to hedge using freight
futures first to establish the degree of correlation between the BFI and the freight rates on the
route of interest. It should be noted that a poor correlation between rates on a trade route and
BFI could result in a windfall loss or gain, especially when the BFI fluctuates against a more
static level of rates in the physical market.
An owner who has a fleet of ships employed on trade routes world wide should nevertheless
have little hesitation in going short in the market to cover the whole fleet in a ‘blanket’
hedge. The fact that a correlation is not perfect does not mean that hedging cannot be
undertaken.
Rising Fuel Prices
One of the shipowner‘s risks that appears to present some difficulty concerns the possibility
that costs may rise, leading to higher freight rates, a raised BFI, subsequent losses in the
futures market but no compensating gains in the freight market.
Figure 9.2
Development of Selected Floute Index Numbers and BFI Jan - Oct 1989
{'1
,1 \-2000 A ~ e ,1 1_ e — e W"
l \.. - I l1900 ,..*-1. _r'-' '\-(=3...-"=,_,_.-" \_
I L.‘ 3-' -l i_ J
180° is 3 ! 1_ r
\.,__,,
fl—‘
\I .3
I.
r-""3
_._.-I
1"’,-
§__
—L.-.,
I I I
Q ‘s
\-__‘
IHi-
Q
-1
nil.
I?‘
)
F F
__'.'-- ,0.-f""": 3"”.”°° - -"’ ROHUTE2‘-_ !
F ',J
J11’
.¢-#4 '1.-
_..,
B. -xii -'
1600 ,- '
Q " AP .
1.
1.. \-_
‘L
§§-
H‘*;-—-
J""\
__,J
.3r._.
QM‘
I;-J
L11
“L
UJI
\
0--\J'\ ,_J~% r‘ ___ ‘ lg _ _1500 ———J w-v,,._P'\--I x
/_._,__- "--..
‘-1"'h-,
'-“_"-='!a-::'='--
I..
\‘(
Q‘-.
}_.r.
fi"T'--L‘
\I
-._____
1400 "'
1so ,_ --
O "" \\_ ROUTE__ \ .______ __ ~J__/
_\.,_/\v_ ":7/:~\,‘_/-—-\;-d
ROUTE 7
lllll'I""llTl'1lllIl"1'1""'T”'l"'1llll'T-"‘l
JANUARY 1989 OCTOBER 1989
WEEKS
3'\
1100
average market, increases in fuel prices will, as a first approximation, cause freight
rates to rise proportionally (see Chapter 7). The actual amount will depend upon the elasticity
of demand and the proportion of fixed voyage costs, viz., expenses in port, canal dues, etc
comprising the freight rate. Increased freight rates will be offset to some extent by higher fuel
prices, but nevertheless, increased profits will follow.
In an
Example
gag Bulk carrier 50,000 tonnes capacity
Speed . l5 kts
Fuel consumption 60 tonnes/day
Distance 14,000 miles round voyage
Port disbursements $125,000
Time in port 14 days
Running costs $5,450 per day
Price of fuel oil $100/tonne
Freight rate $20/tonne
Using the methods of voyage estimating (Chapter 8)
Gross profit/day = $6,682
1~1<>’.'t‘, if the price of fuel rises by 50°/,, to $150/tonne and the freight rate (exclusive of costs in
port) by the same percentage to $28.00,
Gross profit/day _ = $12,040
This does assume that demand is tot-ally inelastic and that supply (speed) does not contract.
The assumption is not wholly unrealistic and therefore gross profits will rise as a result of
increased fuel prices.
KONSTANTINOS
Rectangle
The improved profit for the voyage is therefore ($12,040 -— $6,682) X 52.9 = $283,379
where 52.9 represents the time for the round voyage. I-lad the shipowner hedged in the
freight futures" market and rates risen solely because of increased fuel prices and not as a
result of changes in the supply/demand balance the shipowner would have lost
400 X $10 X 100 = $400,000 where 400 represents the change in the index and 100 the number
of contracts sold. The example assumes that the contract price index was originally 1,000.
Had it been any other price the outcome would have been exactly the same. Thus the net loss
is $400,000 —- $283,379 = $116,621 rather than the full $400,000.
An alternative scenario is that the demand is proportionately elastic (Eds = -1). In such
a case the freight rate would rise to $22 and the shipowner would make the same daily
profit as if the price of fuel had not risen. Slow steaming would result in the voyage
being longer by about 5.4 days. However losses in the futures market would be lower at
100 x $10 x 100 = $100,000 as a result of the lower increase in freight rates. Thus the
shipowner would appear to be out of pocket by an amount ranging from $100,000 to $116,621
depending upon the elasticity of demand. While the above problem does not affect charterers
who contract for the carriage of cargo on voyage charter, such is not the case with trip
charters when the charterer becomes responsible for voyage costs including fuel.
Trip Charters
There has, it appears, been a considerable amount of discussion with regard to trip charters.
The earliest proposed routes that would determine the BFI included three based upon trip
charters. However, although some experts thought that trip charters should be included,
others considered that there could be substantial variations in trip charter rates as a result of
differences in age and performance of the vessels involved. This may indeed be so, but it does
not matter greatly - any more than the fact that rates for the BFI routes are all different.
Figure 9.3
GCBS TRIP CHARTER AND BFI FREIGHT INDEXES: 1985- 1989
l
,, GCBS TRIP/C x 5
\
l‘/I \\ I
I \ 1'
I \ 1
1 \ I
/ \ /1
; \ I
/I \ I
/ ‘.-’/
I, BFI >< 0.5
["11
\\
it ’)
,4’
I1 *4
i \ __ ____V_i_ L iiiiri
_"' 10071000
-' /
\ I
\\ If \\ ’~‘ If’
\v \\ I \‘l
\\ I
\\ I
1 \,J
l
1
1111lF‘Fr"lI"rt‘T1l"l"lF‘1 ffthtrl It llltlfdl t'1'I1t1'f'TfTl viii F1
1985 1986 1987 1988 1989
What does matter is that the trip charter rates for individual vessels should move in sympathy
with the BFI. Indeed, Figure 9.3 illustrates clearly the high degree of correlation between the
GCBS monthly index of trip charter rates and the BFI monthly average. Taken over four
years from 1985 to 1988 inclusive the correlationcoeflicient was found to be 0.966.
The GCBS trip charter index is a Paasche-type indicating the development of weighted
average rates for vessels ranging from 12,000 dwt upwards. Market changes in rates may not
therefore exhibit identical variations for each vessel size. Nevertheless, it should be possible
by comparing changes in trip charter rates for a particular vessel with changes in BFI to
evolve a simple rule or scale that will enable owners to hedge trip charter rates satisfactorily.
KONSTANTINOS
Rectangle
Chapter 10
The Optimum Size of Ships
The casual observer will note that ships engaged on long international voyages tend to be
large while those engaged on short sea passages are usually very much smaller. For the
purposes of this chapter. optimum size of vessel is defined as that which can carry_ a specific
cargo over a given distance at the lowest cost/ton. In particular it should be noted that °
constraints on the demand side are ignored (i.e. the availability of cargo to fill the vessel) as
are physical constraints associated with rivers. canals, locks. berths etc.
The methodology to be used broadly follows that developed by Thorburnl“ in his excellent
work, "The Supply and Demand of Water Transport”. Here however. no attempt is made to
justify assumptions made. other than by a fairly casual look at published data and by a
prion‘ hypotheses.In order to formulate a general model that can be used for all sizes of vessel and all
distances it is necessary to assume that the vessels under consideration are all similar in
type. have similar main propulsion units and similar length/breadth/depth ratios.
Let C0 = capital cost of vessel
Co = daily capital cost ; C9 Xqapnal gzszcovery act?!
C)
Q-"1€"UI-um”
= daily running costs
= speed of vessel in miles/day
= fuel consumption/day
= price of fuel/ton
= deadweight
= rate of cargo handling/day
= distance steamed (loaded and ballast)
It is assumed for the purposes of this model that the cargo handling and port costs/ton of
cargo are the same for all sizes of vessel and therefore do not influence the optimum size of
vessel for a given distance.
W 2
Time in port = ——:— (since cargo must be loaded and discharged)
.'.Cost of time in port
. 2W
= T‘ (Co + CR)
and cost of time at sea
d d
+ CR) ‘f’;-pf.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Hence, total cost for distance d
2W d d
and total cost/ton (excluding cargo handling and port costs)
2 d
=;(Co +CR) 'i'§(Co + CR '*Pf)
Thorburn tested the hypothesis that the cost of a vessel is a function of the area rather than
the volume, i. e. that
C1 (W,)2/3
C2 T W,
where C, and C2 are the capital costs of vessels of size W, and W2 respectively. Thorburn
used NRT rather than deadweight, but, since he assumed that each vessel (initially) carried
1.6 tons of cargo per nrt the use of deadweight rather than nrt is of no consequence.
Some data on capital cost of bulk carriers ca 1980 indicate that the hypothesis is at least
reasonable.
(1) Dwt (2) Cost (average) ($) (3) Calculated cost (4) Error %
15,000
26,600
64,500
120,000
167,500
14m
21m
31m
44m
56m
l1.7m
17.2m
31.0m
46.9m
58.6m
-16%
-18%
0%
+7%-
+5%
Column (3) is calculated starting from the mid-range vessel of 64,500 dwt.
In this model it might be assumed that the 26,600 dwt bulk carrier has a speed of 14.5 kts
and that the speeds of the other vessels are a function of the (length)‘ '1. Since the length is
a function of (DWT)""3‘ the speed would be given by:
S1 (W1 1/6 (w2)1/6
S2— W2) OTS2"-S] W1
This implies that a doubling of the size of a vessel increases its speed by 12%. in line with
the assumption made by Thorburn.
However, this relationship produces the following results:
Size (dwt) Calculated speed (kts) Actual speed (kts)
15,000 13.2 15.0
26,600 14.5 15.1
64,500 16.8 15.2
120,000 18.6 15.5
167,500 19.7 15.0
It is evident that the theoretical speeds (for bulk carriers) are not consistent with the actual
design speeds, and it is not altogether clear why this should be so other than that the block
coefficients of vessels tend to become greater and the length/beam ratio smaller with
increased size.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
For the model, therefore, it is assumed that the speed for all vessels is 15 kts or
360 miles/day.
Fuel Consumption. Clarkson’s Bulk Carrier Register“) indicates fuel consumptions for motor
ships of the sizes under consideration as follows:
(1) Size (dwt) (2) Fuel consumption (tons/day) (3) Theoretical consumption
15,000 18 16
26,600 37 25
64,500 49 49
120,000 80 78
167,000 105 100
While Thorburn assumes that fuel costs are in direct proportion to size, it is clear from the
above figures that considerable economies of scale exist in fuel consumption albeit at speeds
assumed constant at 15 kts.
The theoretical consumption shown in column (3) is based on the relationship
Cl (W1 ) 0.75
C2 T w,
where C1 and C2 are the fuel consumptions/day for bulk carriers of W1 and W; dwt
respectively. This relationship appears to give reasonably good results and may be used for
the purposes of the model.
The price of fuel is taken as $180 per ton.
The rate of cargo handling is an input to the model and is assumed constant for vessels of all
sizes. This assumption is a ‘brave’ one. but may not in fact be altogether unreasonable since
most berths have a fixed number of installations for the purpose of handling bulk cargoes.
Thorburn himself assumed that the rate of cargo handling was directly proportional to the
length of vessel.
Running costs These include wages, insurance, maintenance and repair, stores, drydocking
and administration. There are wide variations to be found in these costs but the following
data give some indication of such costs for a UK shipowner.
(1) Size (dwt) (2) Running costs ($) (3) Theoretical running costs ($)
15,000
26,600
64.500
120,000
167,500
1,200,000
1,600,000
1,970,000
2,350,000
2,600,000
1,271,000
1,510,000
1.970.000
2,373,000
2,623,000
Again there are clear economies of scale to be found in running costs
The theoretical running costs, based on actual data for the vessel of 64.5000 dwt, are shown
in column (3).
These are based on the relationship:
CR“) - (Y1-)0.3
Cam W2
where Cam and Cm, are the running costs for vessels of W, and W; dwt respectively.
KONSTANTINOS
Rectangle
Daily Capital Costs are based upon a vessel life of 18 years, an opportunity cost of capital of
5% in real terms and 350 days in service per annum. Taxes are ignored.
Ballast passages Every loaded passage is assumed to be associated with a ballast passage of
equal length.
On the basis of the above assumptions, Figure 10.1 demonstrates the cost curves for vessels
of 20,000—160,000 dwt for a rate of cargo handling of 5,000 tons/day. The co-ordinates of
the points of intersection of each cost curve with the one in the next size category above are
tabulated. Thus at a round voyage distance of 3,780 miles the 20,000 dwt vessel has the
same cost ($6.9) as the vessel of 40,000 dwt. It may be noted that, for short distances, the
smaller vessels have the lower unit costs whereas at extreme distances the larger vessels are
less costly. At intermediate distances the vessel of optimum size is neither the largest nor
the smallest but one of intermediate size. At a distance of 8,000 miles, for example, the
vessel with the lowest unit cost is one of 60,000 tons.
Figure 10.1
 Hate or Cargo Handlung lTons Davl 5000
I I DrstancelMl I Cost l$l
1o arses as\\\\\\ 20 I045? 93
30 104244 113
40 13904? 131
50 I7‘-4692 143
60 211045 163
to 248004 I78
Distance l M l
Figures 10.2, 10.3, 10.4 show the same cost curves when the rates of cargo handling are
respectively increased to 10,000, 15,000 and 20,000 tons per day. It can be seen that the
effect of an increase in the rate of cargo handling is to reduce significantly the optimum
distance of a given size of vessel which implies, as a corollary, that the optimum size at a
given distance is increased.
It can be observed from the figures that at very long distances far greater benefits accrue
from using 40,000 dwt vessels instead of 20,000 dwt vessels than by substitution in the larger
size range e.g. from a 140,000 dwt vessel to one of 160,000 dwt. In other words the main
economies of scale are seen in the lower size ranges of vessel.
KONSTANTINOS
Rectangle
hi
.--._,..._:
.-g
-n
Cost$
CostIS
\\\
If
Figure 10.2
Rate of Cargo Handling [Tonal Day} 10000
IO
Z0
30
40
50
60
JO
Distance lMl Cost [$1
18902 35
35225 46
52122 57'
E9524 G6
87346 1'4
I05523 B2
124-$2 B9
Dlstance lMl
Figure 10.3
Rate of Cargo Handling {Toast Day] 15000
1.0
20
30
4.0
5.0
BO
70
Distance {Ml Cost [$1
1 250.2 2.3
za4a.e a 1
3474.! 3.9
4634.9 I 4.4
5523.1 4.9
roan. 9 5 4
Distance [MI
C061IS}
CostIS
f
\\
Figure 10.4
Rate of Cargo Handling (Tons! Day) 20000
Distance (M) Cost [$1
10
20
30
40
50
60
70
945 1
I761 4
2508 1
3476 2
4867 3
5276 ‘l
82@1
Distance IMl
Figure 10.5
/
Rate of Cargo Handling lTons.1'Davl 10000
Hate of Interest :10%
‘_/
10
20
JO
4.0
50
80
70
Distance tMi Cost t$l
21536
39984
5897 3
78450
9832 8
11854 1
13903 7
45 -
61
15
BB
10.0
110
I21
Distance IM]
KONSTANTINOS
Rectangle
Cos:S
‘—-
n-v
1-
C051{S
/
f
 \
Figure 10.6
Rate of Cargo Handiing (Tons! Day} 10000
Fuel Cost: $200fTon
10
20
30
40
50
60
I0
Dnstance 1M] Cost IS]
IBBEH 35
3520‘ 4?
52081 5 1'
5946555
81268 I4
10541 3 ' S2
123881 B9
Dustance IMI
Figure 10.7
Rate of Cargo Handlmg {Tons£ Oay) 10000
Speed: 14.5 Knots
10
Z0
30
40
50
BO
TO
D:stanca (M1 7 Cost {$1
18212 35
34054 46
50305 5?
EH06‘ 66
B4435 T4
102005 B2
119869 B9
Distance IM}
\..
Figure 10.8
Cost$
4-
-|_
' Rate of Cargo Handling (Tons; Day) 10000
Running Costs. +10%
Distance {Mi Cost {$1
18253
20 34035 48
30 50303 I‘\\\\\\ 58
40 67230 6?
50 B4516 15
60 102150 83
T0 120039 91
Dastanoe-{Mi
Figures 10.5, 10.6, 10.7 and 10.8 explore the effects of varying the rate of interest on capital,
the fuel cost, the speed of running costs respectively, when the rate of cargo handling is kept
constant at 10,000 tons/day. An increase in the rate of interest, implying an increase in
capital cost of the vessels, causes a significant increase in costs as might be expected and also
an increase in optimum distance. An increase in fuel costs from $180-—$200/ton has little
effect at short distances but reduces the optimum distance to a small extent. A reduction in
speed to 14$ knots is also of least importance at short distances but has the general effect of
increasing the optimum size of vessel.
Finally. an increase in running costs of 10% has very little effect on optimum size but
naturally results in an increase in unit costs.
' .
123
KONSTANTINOS
Rectangle
Chapter 11
Liner Freight Rates
The determination of liner freight rates has caused many problems over the years. This is
because ships have capacities in two ways. viz. by weight and by volume. There would be
no great difficulty if the cargoes carried were homogeneous, for then, as in the tramp freight
market, rates could be negotiated that would automatically take account of the relationship
between weight and volume.
If all cargoes had stowage factors less than that of the ship, i.e. less than (M/W) where M is
the total cubic capacity of the ship and W the deadweight available for the cargo. then it is a
relatively simple matter to charge according to weight. Conversely if all cargoes had stowage
factors greater than (M/W) then they could be charged according to volume occupied. The
main difficulty arises because some cargoes are dense and others relatively light.
Consider a ship whose cargo capacity is 600,000 cu ft and whose deadweight available for
cargo is 12.000 tons. This implies that. if the ship were loaded with a cargo having a stowage
factor of S0 cu ft/ton, the ship could be down to her marks and full by volume.
In the first instance, suppose that the ship is full of a Cargo A of stowage factor 60 so that
the total cargo is 10,000 tons occupying the full space of 600.000 cu ft. The marginal
opportunity cost of carrying Cargo A would be equal to the revenue from 3 tons of a Cargo
B having a S.F. of 20 cu ft/ton. Thus, provided the freight rate of Cargo A is not more that
3 times that of B per ton it will pay to release space occupied by A to accommodate Cargo
B until the full 12,000 tons has been loaded.
Thus if X tons of A are released and 3X tons of B loaded
10.000 — X + BX = 12,000
X = 1.000
and 3X = 3.000 tons.
Hence maximum revenue is obtained, subject to the proviso stated above, when the ship is
full and down with 9,000 tons of A and 3.000 tons of B.
The result could equally have been obtained by the solution of the two simultaneous
equafions
A + B = 12,000
60A + 20B = 600,000
which gives
40B = 120,000
B = 3,000
and A = 9,000
In general, if two cargoes X1 and X2 having stowage factors S, and S; are loaded to obtain
maximum revenue, then
X _ W (52 - S) T
‘ ts. - so
and 1* where W is the deadweight and S the
j ship‘s mean stowage factor.
(51"S) Im=w—--
b (S1_ S2) ,1
If for the two cargoes A and B the freight rates are $50 per ton then the total revenue
= 12.000 >< 50 = $600,000
On the other hand, if rates are charged by volume. then to obtain the same revenue it
would be necessary to charge $40 per measurement ton of 40 cu ft which is equivalent to $60
per ton for A and $20 per ton for B:
thus 60 X 9,000 + 20 >< 3,000 = $600,000
There is no general principle that can be used to decide whether cargoes should be charged
by weight or volume: this is particularly irksome because each method leads to a vastly
different result. See Table 11.1.
Table 11.1
Cargo S.F. Wt. F.R. (W) F.R. (M) Equivalent per ton (W)
A 60 9,000 50 40 60
B 20 3,000 50 40 20
Cargo Selection to Maximise Revenue
It is not always the best policy to select the cargoes that yield the highest freight rates since
the cost of handling. stowage, time in port, damage. risk etc may mean that greater profits
arise from lower-rated cargoes.
For the purpose of this section however it will be assumed that the objective is to earn maxi-
mum revenue from the cargoes available.
Consider four cargoes A, B, C, D whose stowage factors and freight rates are given in
Table 11.2.
nmuz
mi Rnmmm
80 100
60 50
30 60
20 40oow>
It is required to determine how many tons of each. assuming they are available in unlimited
quantities, would maximise the revenue for a ship of 12,000 tons and cubic capacity
600,000 cu ft. If the ship is to be ‘full and down’. clearly, there will be two cargoes whose
stowage factors straddle that of the ship, viz. 50, that will yield maximum revenue. The
KONSTANTINOS
Rectangle
i<-—w-_
possibilities are: A and C; A and D; B and C, or B and D It is not immediately obvious
which two should be selected. There are however some general pnnciples that can be used
to advantage:
Q?
(a) Since the
mean for
(b) If the deadweight of the ship is W, its mean stowage factor S, and the weights and
stowage factors of the two cargoes are respectively W, S, and W, S2 then the
average S.F. must be 50 it follows that the closer the S F of the cargo is to the
the ship, the greater the proportion of that cargo that will be carried
fundamental conditions are given by
W1+W;=W . . .
and W181-'r W282 = . . .
These two equations yield the relationships
c_ (s—S.)_ = <s.—s)
‘Y‘“w<s.—s.>‘w’ w(s.—S.>
and dividing,
W1 S '_
W:-=S1_'S
sothat W1=0and W2=W
and ifS,=S, W2=0 and W,=W
Returning to Table 11.2 and the decision between cargoes A or B it follows that, if Cargo B
is selected, more, will be loaded (due to its lower S F ) than if A were selected At first sight
it might appear that A would be the better choice because of its relatively high freight rate
but this could be more than offset by the lower quantity carried
Datzm suggests that the problem can be resolved in the following way The cargoes are set
out in descending order of freight rates and underneath each 15 placed its stowage factor ant
also its freight rate per cu ft.
F.R./ton
S.F. - 80 30 60
F.R./cu ft
Starting from the left and working to the right, each FR/cu ft 1S underlined if it is greater
than or equal to the preceding one. The two cargoes that will maximise revenue are those
closest together whose data are underlined and which straddle the ship s S F (50) A and 1
are the two cargoes that satisfy the conditions and calculations show that the revenues from
A C B D
100 so so 40
20
1.25 2.0 0.83 2.0
the four possible selections are:
Total Revenue
iii- 
AandC
AandD
BandC
BandD
$912,000
$840,000
$640,000
$570,000
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
This method, unfortunately, does not always work, as the following example will show:
A B C D
F.R./ton 100" so 60 45
S.F. 120 as 40 2s
F.R./cu ft 0.83 0.94 1.5 1.6
According to the method employed by .Datz the selections for maximising revenue should be
B and C.
Calculations produce the following results:
A and D $697,826
B and D $702,105
A and C $780,000
B and C $773,333
indicating that A and C should in fact be chosen.
To understand the problem, further analysis is necessary.
Total Revenue (TR) is given by
TR W rs—s.>+w ts.-S)= r--—-— r———-—-‘rsi — S.) 2 (st — s.)
where r, and 1'2 are the respective rates corresponding to cargoes of stowage factors S, and
S’).
Differentiating TR partially with respect to S1 gives
5TR (S " S2) {(31 "' S2)" (S1 - 5)} (S _ S2)(1'2 _ 1'1) .i = —W ——-i W W. W ~ if ...as. " ts. - s.>= + " ts. — s.)= is. — s.>= (Q
and assuming that r, is greater than r; this function indicates that an increase in S1 will
result in a decrease in TR and vice-versa.
Differentiating TR partially with respect to S2 gives. in a similar way:
S — S —3TR E wi 1 )(1'2 21'1)_H(ii)
552 (Si " 52)
which will normally be negative. implying that an increase in S; causes a reduction in total
revenue.
Differentiating TR partially with respect to r, gives
3TR (5 - 52) ..at W ;_eS*2)...(11)
C 5T1 (S1
which is always positive. and substituting this in equation (i) gives
are art} (r,~i,) (_)
BS] 61'] .(S1'_S2)... DIV
and similarly
E-JTR 6TR (r2 — r1) ( )
6S2 61'; (S1 _ S2)
which is the same as (iv).
KONSTANTINOS
Rectangle
Examining equation (iv) it will become clear that
BTR _ 6TR _ _-5-5- will be > -E?-— if (r; - r1) is >(S1 - S2) and conversely
1 1
BTR _ 6TR _ _ _ _ _
K will be < 6Tif (r2 —-— rl) is <(S, — S2), no account being taken of the different signs
1 1
in either case.
The method used by Datz, as outlined above, for cargo selection does in fact work only
when (r3 — rl) is greater than (S, — S3) when change in S.F. has a greater influence on total
revenue than change in rate. In the last example where it was expected (after Datz) that B
and C should be selected (r; — r1) = 35 and (S1 — S2) = 57.
Consider then, Table 11.3:
A B C D E
F. R. 100 90 80 70 60
S.F. 100 85 70 55 40
F.R._/cuft 1.0 1.06 1.14 1:27 1.5
Cargo E should be selected (SF < 50) with either A. B. C or D and again, at first sight it
might be thought that D would yield the highest revenue. In fact all yield the same.
r — r r — r
viz. $800,000. The general rule is that if W3 B 3 E , then the total revenues from
SA _ SB SA '— SE
A and E will be the same as from B and E (for proof, see Appendix C). If, on the other
FA ‘ TB TA “ TE .hand. < then B and E will produce more revenue than A and E.
5s - Se 3s " 5|;
Changes in the Value of S
Normally the value of S is fixed for a particular ship. An increase in S would imply an
increase in volume or a reduction in deadweight. Clearly if space could be increased this
would produce an increase in revenue given the same cargo availability. while a reduction in
deadweight would cause a loss of revenue. An effective change in the value of S may be
accomplished by loading a cargo of which there is a fixed quantity available. provided the
stowage factor is different from S.
Exmnple I Three cargoes are available. having particulars as follows:
A B C
F.R. 80 70 40
S.F. 60 54 40
F.R.,"cu ft 1.33 1.3 1.0
Cargo B is available to the extent of only 1,000 tons
Deadweight of Ship: 12,000 tons; S.F.: 50
To maximise revenue, given unlimited quantities of these cargoes, C must be loaded with A
or B. Using the test shown previously, B is preferred to A if
:<u—70<8()—40_ _f10<4 h_h__
so-154 so-4t)""" 6 zwlc "'5'
The problem can be approached in two different ways
(i) Assume that 1,000 tons of B are loaded, giving a revenue of 1 000 X 70 = $70 000
S for the ship now becomes ’ -1 es’ W — —--4 1 = 49 64
A and C are now loaded into the remaining space so as to maximise revenue
600 000 — 54 000 546 000
11,000 11,000
_ (49.64-40) _w,-11,000 (60_4O) -5,300
(60 — 49.64)
W3 =11,000"'i26-i = 5,700
.'.Total revenue = 70,000 + (5,300 x 80) + (5,700 >< 40)
(ii) Alternatively, assume that C and B are loaded in such a way that the ship s stowage
70,000 + 424,000 + 228,000
$722,000
ft S in‘ ttt50 S‘ W‘ 8-82r . -— =—-—-ac 0 re 81115 COIIS an a 11106 W2 S1 _ S
(50 — 54) 4then WC = 1,000 -—-—-— = 1.000 x —- = 400 tons.
Revenue =
The remaining capacity of 10.600 tons is loaded with A and C to bring the ship full
and down.
(40 - s0)_ 10
(1.000 >< 70) + (400 >< 40)
70.000 + 16,000
sss .000
_ (50 - 40)_ _ (60 - 50)w,, -10.600(60 _ 40)_wC -10,600(60 _ 40)
= 5,300 tons l = 5.300 tons.
.'.Total revenue = 86,000 + (5,300 X 80) + (5,300 X 40)
Example 2 Four cargoes A, B, C, D have characteristics and availability as follows
= 86,000 + 424,000 + 212,000
= $722,000 as before.
A B C D
ft .
F.R. 90 70 50 30
S F 70 65 40 30
129 108 125 100
Availability (3,000t) (6,000t) (5,000t) (6,000t)
It is required to load a ship of deadweight 12,000 tons and S.F. 50 so as to maximise
ICVCIIIIC .
Using the method previously explained it can be seen that a combination of A and C are
preferred cargoes but they are not available in sufficient quantity to fill the ship Thev
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
should be stowed in the ratio:
WA : S--—S¢ 50-40 *1
WC T5,, —s ‘70-50 *2
Hence, 2,500 tons of A should be combined with 5,000 tons of C.
This exhausts the supply of C.
Now the remaining 500 tons of A should be combined with D in the ratio
WA i S _ SD "
w,, Ts, -is 10-5071
500tons of Agis loaded with 500 tons of D.
Deadweight remaining = 12,000 — 8,500 = 3,500 tons.
This must be taken by cargoes B and D
(50 - 30) (es - so)w = 3,s00—-; w = .500-—“ (es - 30) ° 3 (es - 30)
20 15w,,=3,s00><§§f w,,=3,s00><?5-
W, = 2,000; WD = 1,500
Cargoes loaded to maximise revenue are:
A 3.000 >< 90 = $270000
B 2.000 >< 10 = $140,000
c 5,000 >< 50 =‘s2s0.000
0 2.000 >< 30 = s 60,000
12,000 Total = $720.000
KONSTANTINOS
Rectangle
Chapter 12
Linear Programming and
Transportation
1. Introduction
Economics is sometimes described as the study of the allocation of scarce resources. The
problem of how to utilise limited resources in the best way occurs often enough to have
justified the development of a set of techniques. The generic term for these techniques is
mathematical programming and the simplest of these is linear programming. Linear
programming problems may be solved graphically, by constructing a feasible region bounded
by linear constraints and containing all possible solutions; or by using the Simplex method
which involves the construction and manipulation of a tableau. This latter approach, in its
dual form, is often used to calculate shadow prices.
Linear programming is arguably the most widely used technique in operational research and
aims at allocating limited (i.e. constrained) resources in an optimum way. Linear
programming problems typically appear to contain a great deal of information; they have a
clearly defined objective which can be stated quantitatively in the form of an equation (the
objective function); they have a set of constraints which must be satisfied simultaneously and
have linear relationships. If they are not linear then some other version of programming must
be used, e.g. dynamic or quadratic programming.
2. The Graphical Solution
The graphical solution involves expressing the constraints as straight lines on a graph and
gradually, as more constraints are included, reducing the area bounded by the axes and the
straight lines, to a feasible region which will contain all possible (feasible) solutions. In other
words, any point within this region will satisfy the most limiting constraints and hence will
also satisfy the less rigorous constraints. The procedure is illustrated by the following simple
example.
A ship supplier produces two difl'erent types of rope. Type I is made entirely from sisal while
Type II is threaded with wire. Sisal rope requires 0.5 man-hours per coil whereas wire rope
requires 0.4 man-hours of wire workers and 0.6 man-hours of sisal workers per coil.
Packaging of sisal rope takes 0.2 man-hours and wire rope needs 0.3 man-hours. The supplier
makes £10 profit on each coil of sisal rope and £15 profit on each coil of wire rope. I-low
many coils of each should be produced each day in order to maximise profits if there are
112 man-hours available for sisal and packing and 70 man-hours available for working wire?
The usual approach to such a problem is to let the unknown quantity be x and then to sift the
information. Clearly, there are two unknowns (x and y) in this question: the quantities of
each type of rope to be produced per day. The supplier is attempting to maximisehis profit
and the obvious first impression is to produce only the type of rope which yields the greater
profit. Therefore, he should produce only type II, but in what quantity‘? He has at his disposal
70 man-hours of labour for working wire and each coil of type II rope requires
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
0.4 man-hours of wire workers implying that 175 coils of such rope could be produced each
day. However, I75 coils of wire rope require 105 man-hours of sisal workers’ time which
leaves only 7 man-hours for packaging. Clearly this is not suflicient and it shows that the
interaction of the constraints must be considered when determining a feasible solution. Profit
will be maximised by producing a certain number of each type of rope and considering all the
interacting constraints. One approach is as follows:
Let x = number of coils of Type I rope to be produced per day;
y = number of coils of Type II rope to be produced per day;
Then daily profit P = 10x + 15y
This is known as the Objective Function and the solution seeks to maximise this, subject to all
the relevant constraints. What then are the constraints?
(i) The number of man-hours for sisal and packing is not to exceed 112:
0.5x + 0.2x $112
(ii) The number of man-hours for wire working is not to exceed 70:
0.4y $.. 70
(iii) the number of man-hours for total production and packing is not to exceed 182:
0.5x + 0.4y + 0.6y + 0.2x + 0.3y Q 182
i.e. 0.7x+l.3y-$182
Formally, this probleni can be expressed as follows:
Maximise P = 10x + lSy
subject to 0.7x Q 112
0.4y Q 70
0.7x + l'.3y Q l82
and x, y 7; 0
Figure 12.1
180- 0.4y_-570 'v» ~.— r ~
. ‘°.?l,§1'|2
160—* I
140 ‘A’ “
l
\\\\
120
0.)‘, ‘ 4
7.
100 er“ '0;
80 1
so | B
‘ FEASIBLE REGION
40
20
20 40 60 80 1 00 1 20 140 160 180 200 220 240 26
\0 -F 4 .
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The last (non-negative) constraint is included because negative amounts of rope cannot be
produced. Each of these constraints can now be represented as straight lines on a graph.
Given the non-negative condition, only the positive quadrant of the axes is relevant to the
solution.
The shaded area in Figure 12.1 is the feasible region and the answer will be a point contained
within its boundaries. In fact, it will be seen that the solution will correspond to a point on
the boundary and at an apex.
If any point within the feasible region is a possible solution then x = 0, y = 140 is acceptable.
At this point the profit would be 140 X 15 = 2,100. However, the point y = 0, x = 160 is also
acceptable; though here the profit is only 1,600. By moving vertically upwards from this point
it is possible to increase output of type II rope while still producing 160 of type I and remain
within the constraints. Such a movement will increase total profit; and the maximum number
of coils of type II rope which can be produced before resources are diverted from the
production of type I rope is approximately 54. The total profit will then have risen to
(160 x 10) + (54 x 15) = 1,600 + 810 = 2,410.
Profit will increase by moving as far as possible along any side of the feasible region so that
the points of interest are the apexes.
fit Combination _l:_r_ofil
A Ox, l40y 2,100
B 160x, 54y 2,410
C 160x, 0y 1,600
The solution given by point B would involve:
- 160 X 0.7 = ll2 man-hours for sisal and packing
54 X 0.4 = 21.6 man-hours for wire working
i.e. (0.7 X 160) + (1.3 X 54) = 182.2 man-hours in total.
This last requirement is 0.2 hours too large and arises because the correct output of type II
rope is only approximately 54. The correct quantity is in fact slightly less than this. This
example satisfies the constraints and further shows that the wire working constraint is not
effective as it is dominated by the more rigorous constraints. This is shown on the graph by
the fact that this constraint does not form any part of the boundary of the feasible region.
One obvious weakness of this approach is that it is, at best, only three dimensional. Anything
more complicated becomes difficult to visualise and impossible to draw. ‘It does, however,
afford a clue to a systematic method of solution of linear programming problems, namely:
(i) Locate an extreme point of the feasible region.
(ii) Examine each boundary edge intersecting at this point to see whether movement along
any edge increases the value of the objective function.
(iii) If this is so, move along this edge to the adjacent extreme point.
(iv) Repeat (ii) and (iii) until movement along an edge no longer increases the value of the
objective function‘.‘"
More diflicult (realistic) problems will require a more sophisticated method of solution; this is
provided by the Simplex Algorithm.
KONSTANTINOS
Rectangle
134
3. The Simplex Method
The simplex method is an iterative procedure involving a series of calculations starting from
the origin and moving through a sequence of vertices of the feasible region?’ At each stage
the coordinates of the vertex are evaluated, along with the current value of the objective
function: at the same time, an indication is obtained of whether further improvement is
possible. If improvement is possible then the focus shifts to the next vertex in the sequence
and the calculations are repeated. When no further improvement is possible, the process is
terminated. Note that these calculations are all algebraic manipulations and no actual graphs
are drawn.
Any inequality can be converted to an equality by the use of a slack variable. In the example
above the first constraint was expressed as:
0.5x + 0.2x Q 112
which may be written as:
0.7K + S1 = 1
where s, is a slack variable used to take up the shortfall between 0.7x and 112. The non-
negative constraints x 2 0 and y 2 0 can similarly be written as x — s2 = 0 and y —- s3 = 0
respectively where, again, s; and s, are slack variables. Proceeding throughout in this
manner results in a set of simultaneous equations which can then be expressed in tabular
form. This provides the simplex tableau which is the basis for the calculations.
The original formulation was:
Maximise P = 10x + 15y
subject to 0.7x Q ll2
0.4y -4. 70
0.7x + l.3y $._ 182
which may be rewritten as:
Maximise P = 10x + 15y
subject to 0.7x + s, = 112
0.4y + S2 =
0.7x + 1.3y + s3 = 182
and presented in the form of a tableau as:
x y s, S2 s3 P C R
s, 0.7 1 ll2
S2 0.4 l 70
s3 0.7 l.3 l 182
P -10 — l5 1 0
The bottom row signifies the objective function and the columns C and R depict constants
and ratios (not yet calculated). The variables in the left hand column are the active variables
and the Simplex technique involves starting with the slack variables and gradually replacing
them by more relevant variables (in this case x and y). The iterative process by which this is
accomplished will cause all the coefficients in the table to be changed and eventually lead to
the level of profit (currently zero) being maximised. The technique involves a pivot and a_
series of algebraic manipulations. The pivot is selected by choosing the largest negative item
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
in the bottom row and then dividing the values in that column into their respective row
constants. This creates the numbers in the R column as shown below.
x y s, s2 s3 P C R
s1 0.7 0 1 0 0 0 112 oo
S; 0 0.4 0 1 0 0 10 175
S3 0.7 @ 0 0 1 0 1s2 140 +-
P -10 -15 0 0 0 1 0 -
1
The pivotal element is the one at the intersection of the column containing the largest
negative item in the objective function and the row containing the smallest non-negative R
value. This pivot shows that y enters the tableau and the slack variable s3 leaves. The
coefficient of y must be unity so dividing the row by 1.3 gives the correct formulation for the
next iteration. Multiples of this new line are then added to or subtracted from the other rows
of the tableau so that the new variable only appears in one row of the matrix. Hence the
second formulation is:
x y s, S2 s3 P C R
S, 0 1 0 0 0 1121s0.o0<~
S2 —-— 0.2154 0 0 l --0.3077 0 14 — 64.99
y 0.5385 1 0 0 0.7692 0 140 259.98
P — 1.9225 0 0 0 11.538 1 2,100 — 1,092.33
1
The pivot shows that x enters while s, leaves to give
x y s, s; s3 P C R
x 1 0 0.7 0 0 0 160
s2 0 0 0.1508 1 — 0.3077 0 48.464
‘y 0 1 —0.3769 0 0.7692 0 53.84
P 0 0 1.3458 0 11.538 1 2,407.6
Now there are no negative items in the final row so that the position cannot be improved by
replacing them by another variable. The iterations are thus complete and the above
represents the final tableau. The result_is found by reading across the rows and matching the
left hand variables with the values in the C column. Hence the maximum profit P is 2,407.6
when x = 160 and y = 53.84 (approximately 54). Compare this result with the graph in
Figure 12.1 and note further how this method also calculates the amount of slack (i.e. unused
resources) in the solution. The slack variables s, and s3 have disappeared from the tableau
signifying that their values are zero while the value of s2 can be read as 48.464.
The Transportation Method
The ‘Transportation Problem’ is a classic problem in operations research and usually involves
minimising the cost or the distance involved in transporting goods from a number of origins
to a number of destinations. If there are m origins (e.g. factories) and n destinations (e.g.
ports) then the optimum solution will involve m.+ n -1 routes of the mn available
possibilities. The problem may be formulated as a linear programme and solved using the
simplex method but the size of most realistic problems will be such as to render manual
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
solutions very difiicult, if not impossible. Computer packages exist to solve such problems
more easily. A simpler procedure than the simplex method is the transportation -method and
this enables manual calculations to be carried out. The following example will serve to
illustrate the technique.
A major oil company has to arrange the transportation of several small petrol consignments
from two coastal refineries to four inland storage terminals using barges. The outputs of the
two refineries are 1,000 and 1,600 barrels per week to be shipped to the four inland storage
terminals which require 800, 600, 700 and 500 barrels per week respectively. If the cost per
barrel of shipment is as shown in the matrix how should the shipping oflice move the
consignments so as to minimise the costs?
TO TERMINALS
FROM I II III IV OUTPUT
Refineries A 10 13 12 13 1,000
. B 1 1 1 l 14 13 1,600
Requirements 800 600 700 500 2,600
The problem is one of minimising total costs subject to various constraints; for example the
quantity delivered to terminal I must not exceed 800 barrels and the total taken from refinery
A must not exceed 1,000 barrels. This problem is the most straightforward case as the total
requirement is exactly matched by the total available output. As the objective is to minimise
costs. a sensible policy is to load each route according to cost; the route with the lowest cost
carries as much as is permitted by the constraints. Clearly the route from refinery A to
terminal 1 is the cheapest and could be allocated 800 barrels which would imply that no oil
would be supplied from refinery B to terminal I.
I II III IV
A 80010 013 2001; 0,3 1,000
B 011 60011 50014 500,3 1,600
800 600 700 500 2,600
Proceeding in this way gives the matrix above corresponding to an initial feasible solution.
Note that the number of routes used is m + n —- 1 where m = 2, n = 4 and hence 5 routes are
used. The total cost is found by multiplying the amount shipped on each route by its unit cost
of shipment. Hence the total cost {is
(800 x 10) + (600 x l1)+ (200 x l2)+(500 x l4)+ (500 X13)
= 8,000 + 6,600 + 2,400 + 7,000 + 6,500
= £30,500
But is this optimal?
The reason why this allocation might not be optimal even though it is based on transporting
as much as possible on the lowest cost route available is that the allocation to the lowest cost
route might prevent the next lowest cost route being used at all and thus causing the balance
to be shipped on a route with a higher cost. In this example placing the full allocation of 800
barrels in route AI prevented any allocations at all in route B1 (where the transport cost per
barrel was £11) causing the output of refinery B to be placed in routes BII, BIII and BIV with
respective costs of £11, £14 and £13.
This allocation will be optimal if it provides the lowest cost solution. If however, the
allocations can be altered to reduce the costs then the new allocation might be the optimum.
How do we know? The basic feasible solution rarely provides the optimum allocation which
is usually only reached after several iterations and comparisons of costs on used and unused
routes. In this example the initial solution does not use the route from refinery B to terminal
KONSTANTINOS
Rectangle
F
I, yet this is cheaper than some of the routes actually used so would a reallocation to this
route be beneficial?
The procedure for determining whether or not to reallocate traffic is to calculate the shadow
costs associated with each occupied cell. If the actual cost of an unused cell exceeds its
shadow costs then no allocation to this cell is appropriate but if the actual cost is less than the
shadow cost then the total cost will be reduced by allocating as much trafiic as possible to this
cell. Such an allocation automatically involves readjusting the figures in some of the other
cells; this is achieved by creating a closed loop so that the row and column total remain
unalfected.
The idea of a shadow (sometimes called artificial, fictitious or partial) cost of any route may
be artificially divided into a cost of sending (x) and a cost of receiving (y). Hence the actual
cost in each cell (Cmn) is comprised of xm plus y,,. It must be stressed that these costs are
purely artificial and bear no relationship to any real costs; they are a device used only for the
purposes of calculating an optimum solution. Furthermore only differences in costs between
routes are relevant to the allocation so that one of the row or column costs can be arbitrarily
chosen as a basis for calculating the rest. If the cost of sending from refinery A is set to zero
then the shadow costs corresponding to the original feasible solution are shown in brackets
below.
y I II III IV
X (10) (9) (12) (ll)
A (0) 800 0 200 0 1,000
B (2) 0 600 500 500 1,600
800 600 700 500
These shadow costs follow because
X1+Y1=c11
0+ l0= 10 hence ifx, =0, y1= 10
similarly x, + y, ='c,;,
0+ l2= 12 hence y3 =12
if ya = 12 then since
x2 + y; = 14 x; must equal 2
and ifxz =2 then y4=l1 since X2 +y4= 13
and y2=9 since x;+y2=1l
Note that the shadow costs are calculated using only the occupied cells. The actual transport
costs in the unoccupied cells are now compared with the shadow costs of these cells. Any
savings will be obvious by subtracting actual costs from shadow costs for each unoccupied
cell. Hence
forcellA 11 (x,+y=) ~13— 9 .13 ~~4
force1lAIV(x,-I-Y4) l3~ 11*; 134 A 2
forcellB I(x2+y,)—11=12—l1= l
The only route which will lead to a reduction in overall costs is that from refinery B to
terminal I. Suppose we reallocate an amount 0 to this route from other routes then it must be
diverted from routes B11, BIII or BIV. Shifting any amount from BI1 or BIV would cause a
balancing readjustment in the cells AII and AIV which would then imply that too many
routes (greater than m + n — 1) would be used.
1 11 111 IV
A soo - e 0 200 + e 0 1,000
B 0 + 9 600 500 - 0 s00 1,600
soo 600 300 500
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The maximum value of 9 will be 500 units since negative quantities are not allowed. Cell B11
determines this value of 9 and reallocating the 500 units gives
I II III IV
A 30010 013 7001; 013 1,000
B 50011 60011 01,, 500,3 1,600
800 600 700 500
The number of routes used is still m + n — 1 = 5 and total costs have fallen to
(300 x 10) + (700 x 12) + (500 x ll) +(600 x ll) + (500x 13) = £30,000
The process of checking whether any improvements are possible is repeated after each
iteration by calculating the fictitious shadow costs.
Setting x, = 0 gives
I II III IV
(10) (10) (12) (12)
A (0) 300 0 700 0
B (l)" 500 600 0 500
where the fictitious costs are shown in parentheses.
In no case does the shadow cost exceed the actual cost for any of the occupied cells so that
any reallocation would cause transport costs to rise. Hence the optimum solution is found
when no reallocation is possible.
Extensions to the Transportation Problem
So far the method used to obtain the basic feasible solution has involved allocating as much
as possible to the lowest cost route. An alternative method to obtain an initial solution is to
adopt the ‘North West Corner’ rule which involves loading cells in the top left hand corner 2
heavily as possible according to the constraints. Furthermore, it is possible that the problem
does not call for the minimisation of anything (cost, distance, etc.) but instead might require
the maximisation of an objective function (e.g. profit) subject to certain constraints. Both
these variations are straightforward to incorporate and largely follow the method described
earlier.
The problems so far have been ‘balanced’, i.e., the total requirements have exactly matched
the total available, though this is unlikely always to be the case. When such a balance does
not exist it is necessary to introduce a dummy variable to take up the slack as shown by the
following example.
A freight forwarder has to arrange the transfer of 140 empty containers from three UK port:
(I, II and III) to three container depots (A, B and C). The transportation costs per unit are:
H To A B C
From I 230 210 215
II 190 190 180
III 220 210 200
with availability and demand constraints as follows:
' Port Location Depot Requirements
I 50 A 40
II 49 B 50
III 47 C 50
Totals 146 140
Clearly there is an imbalance between the supply and demand for containers and so a dumm‘
container depot is used to aid the solution by restoring the balance.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
I To A B C D
From I 40 10 —-
II — 40 9 49
III — — 41 6 47
40 50 50 6 146
— 50
Adopting the North West Corner rule gives the initial solution above. Note that the inclusion
of the dummy variable mean that 6 (m + n - l = 3 + 4 — 1) routes must be chosen for an
optimum solution. As the dummy variable does not really exist the cost of transport to it may
be set to zero or to an extremely high value (say 1,000) so as to make its choice irrational. The
initial cost of the solution above is £28,720 and so, in order to determine whether this is
optimal, fictitious/shadow costs are calculated as explained earlier.
3* w O
§U
I (0)
II (-- 20)
(230) (210) (200)
230 210 21 5
190 190 180
III (0) 220 210 200 COG
For route IIA (230 —- 20) — 190 = 20
IIIA (230 + 0) — 220 = 10
IIIB (2l0+0)—2l0 =0
IC (200+0)—2l5 =-15
ID (0+0)-0 =0
IID (0-20)—0 =—20
Savings may be obtained by using the routes from II to A or from III to A, but the former is
chosen as the savings are greater. Allocating 0 to this route and reallocating gives either:
(a) A B C D or (b) A B C D
I 34 10 — 6 I — 50 — —
II 6 40 3 - II 40 — 9 —
III — ~— 47 --- III -— — 41 6
Note that (a) uses 7 routes and (b) uses 5 routes so that degeneracy has occurred. The method
has broken down because, in the case of (b), there is now insufficient information to
determine row and column costs. Degeneracy can arise in an initial solution if the
‘availability of a source and the requirements of a destination are simultaneously satisfied‘ or
during reallocation when a new route may be introduced but more than one existing route
may drop out. Degeneracy may only be temporary; in which case it could disappear during
the next iteration -— this can be achieved by assigning a ‘positive zero’ allocation to one of
the routesf”
Tackling the same problem by finding an initial feasible solution using the least cost
allocation method gives:
A B C D
(230) (210) (200) (0)
I (0) 40 — 0 4 + 9 —-— 6 50
II(—20) +0 — 49-6 — 49
III (0) ' — ' 46 -— 9 l + 9 -— 47
40 50 50 6
Six (3 + 4 — 1) routes have been used so this is an acceptable allocation.
Calculation of the fictitious costs shows that there should be a reallocation of units to route
IIA. Inspection will show that it is possible to move 40 units round a closed loop to form the
following allocation:
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
,i_
I.
l i it1:1
llu
A B C
(210) (210) (200
I (0) — 44 —
II (— 20) 40 — 9
III (0) — 6 41
40 50 50
The calculation of fictitious costs on the m + n — 1 routes enables routes to be compared to
see if further savings are possible.
Route I to A 210 — 230 = -20
III toA 210--220=—l0
II toB l90—l90=0
ItoC 200—2l5=—-15
No improvement is possible so the optimum allocation is shown at a total cost of £27 920 By
allocating tralfic to the route from port II to depot B the total cost will remain unaltered
There are thus two least cost solutions to this particular problem, the altematrve solution
being :
A B C
(210) (210) (200)
I (0) — 44 —
II (-20) 40 6 3
III (0) —— — 47
40 S0 50
and the total cost is again £27,920.
-"Readers should be reassured that it is not the use of the North West Corner rule per se which
caused the problem to become degenerate; rather it was the particular numbers of containers
to be redistributed in each route. Suspicions may be allayed by confirming the solution to the
following problem, using the North West Corner rule to obtain an initial feasible solution
Example
A freight forwarder has to arrange the transfer of 200 empty containers from three UK ports
D
(0)
6
6
D
(0)
6
6
to three container depots. The transport costs per unit.are
To A B C
From I 50 60 70
II 40 40 60
III 70 80 50
with availability and demand constraints:
PORT AVAILABILITY DEPOT REQUIREMENT
I 36 A 50
II 60 B 80
III 104 C 70
An initial solution using the North West Corner nile would be
To A B C
From (50) (50) (20)
I (0) 36 — —
II (- 10) 14 46 —
36
60
III (30) -— 34 70" 104
50 80 70 200
Total cost = £10,420.
KONSTANTINOS
Rectangle
"2
The method of fictitious costs shows that an improvement would ensue by using the route
from port III to depot A. Allocating 14 units to this route gives:
To A B C
From (50) (60) (30)
I (0) 36 — — 36
II (-20) —-— 60 —~— 60
III (20) 14 20 70 104
50 80 70 200
Total cost = £10,280
which appears to be an optimum allocation — though it may not be unique.
In reality such problems may involve several origins and destinations, fixed costs and variable
costs, trans-shipment of the product and other complications. A manual solution is not
advised to such complicated problems: it is far better to use a computer package based on
linear programming techniques.
KONSTANTINOS
Rectangle
-_
!l‘l
1"1. ..
Chapter 13
Regression and Correlation
Relationships often exist or are suspected to exist between two or more variables and it is
useful to express such relationships in the form of mathematical functions or equations.
These equations can be polynomials of any degree and do not have to be linear; they may
involve either the variables themselves or some transformation of the variables. The purpose
of the equations is to explain how one variable (the dependent one) is influenced by or
related to the others (the independent variables). The standard technique for solving such
equations is that of ordinary least squares.
The method of Ordinary Least Squares (OLS)
A regression equation is an attempt to explain one variable in terms of one or several other
variables. When the regression equation has been calculated. it is possible. given a value of
the independent variable(s). to predict the value of the dependent variable. The simplest
case of regression (bivariate analysis) concerns two variables X, the independent one, and
Y. the dependent one. with the regression equation being most simply expressed as the
straight -line Y = a + bX where a and b areconstants. The purpose of regression analysis is
to calculate the coefficients a and b and to find an expression that measures the degree of
correlation or association between the dependent and independent variables. The OLS
technique enables the equation of the line which best fits the data to be found: but this begs
the question of what is meant by ‘best fits’ and what ensures that the equation derived is the
best‘? The criterion used in fittingmthe required line is that the sum of the squares of the
distances of each point from the line is minimised. A scatter diagram will help to explain
this.
Figure 13.1
Y
X X Y=a+bx
X X X X ><
Y1‘- -X_ _\\ X
* x
gT---- X
l3 1
A
I_J 
*1 X
KONSTANTINOS
Rectangle
,_
A consideration of Figure 13.1 shows that some points are above the regression line while
others lie below it. The linear regression equation Y = a + bX gives the predicted or
estimated Y values (sometimes denoted in the literature by Y) for given values of the
independent variable X. It follows that when X = x1 the estimated Y value is 9, whereas the
actual Y value is in fact equal to y1. The difference between the actual Y value and the
predicted Y value is thus y, — 9,. This distance could be positive or negative depending on
whether the point was above or below the line. The positive or negative bias is removed by
squaring the distance and the equation of the line of best fit is found when the sum of the
squares of the distances of all the points from the line is minimised. More formally:
Y = a + bX (where a and b are constants) represents the equation of the least squares
regression line of Y on X for the n points (x1, yl), (X3, y,), . . ., (x,,, y,,).
The square of thedistance between any point (xi, yi) and the line is given by
(Y1 - (a + bxi))2 '
Summation for all the points (xi, yi) gives
I1
S = 2 (yi — (a + bx,))2 where S is the sum.
i=l
The problem is to choose a and b so that S is a minimum. Expanding the right-hand side of
the above equation gives:
S =(y1 — (a + bx1))3 + (y; -— (a+ bx2))2 + . . . . . + (yn — (a +bx,,))3
Differentiating S partially with respect to ‘a’ gives
6S
£=2(y, —a—bx1)(—1)+.....+2(y,,-a—bx,,)(-1)
= —2(y1-—a—bx,)+.....-—2(y,,—a—bx,,)
= "2(2 Yi'1‘1a“b2X1)i=1 i=1 -
Dillerentiating partially with respect to ‘b‘ gives
6S
53 = 2(y, — a -— bx1)(——x1)+.....+ 2(y,, — a — bx,,)(—x,,)
1'1 1'1 I1
= -—2(i2lx,y,-ai2lx,—bi2lx,2)
For a minimum value it is required that
as as azs azs
ga'—0.-é'5—0and5;i-5>0,5B-3>0
The second order derivatives will not be investigated here; keen students may like to prove
for themselves that the conditions are fulfilled while others may consult any standard text.
some of which are given in the bibliography relating to this chapter.
6S 6S
Setting -3; = 0 and 5 = 0 gives
I1 ll
_2y,—na—b_2x,=0...(1)
1=l 1=l
n n n
Zlxiyi-—a;1xi--b;1x§=0...(2)
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Equations (1) and (2) are known as the NORMAL EQUATIONS of regression analysis
This system of simultaneous equations (two equations 111 two unknowns) can now be solved
to find values of a and b. Abbreviating the symbol to 2 for clarity and multiplying
( 1) by Ex, gives Zxiiyi — naEx, — b(Zx,)2 = 0 (3) and
(2) by n gives nEx,y, - naZx, —- nbixf = 0 (4)
Subtracting (4) from (3)
Exiiyi — nixiyi — b(Exi)2 + nbEx,2 = 0, and rearranging
P012312 " (2302) = nzxiyi _ 2312}?
h be 2 A ,2 ~sot at nixf — (2x,)2
and by substitution
T123130 ‘“ 2741235
I Eyizxi — Exiixiyi
a : nixi — (E14,):
Example The total distance steamed and time taken by a vessel on each of SIX voyages were
recorded and used to find an average speed (X) The fuel consumption for each voyage was
also recorded an.d calculated as consumption per day (Y) corresponding to each voyage and
hence to each average speed. Examine the relationship between the two sets of results
Average speed (knots)
Solution Assume Y = a + bX
X Y XY X2
15
18
14
16
l6
13
29
48
22
32
34
17
435
864
308
512
544
221
225
324
196
256
256
169
92 182 2.884 1,426
The normal equations are
ZY — na — bEX = I
ZXY — aZX — bEX2 = I
sol82—6a—92b =I
2884 — 92a — 1426b = |
16744 — 552a -— 8464b = 0
17304 — 552a — 8556b = 1
15 18 14 16 16 13
Consumption per day (tonnes) 29 48
(1)
(2)
(1) x 92
(2) X 6
Eq (2) — (1) gives 560 — 92b = 0 and b = 6.087
from (1) -378 — 6a = C so a = -63
Hence Y = -63.0 +-6.087X
This result can be verified by direct use of the formulae given for a and b above.
Standard Error of Estimates
The equation Y = a + bX will provide values of Y for given values of X and is derived from
pair-wise data of n points (x, y). Not every point will lie on this line —- in fact most points
will not lie on it — and for every given value of X the equation will estimate the
corresponding value of Y which is denoted by 9 oryest. A measure of the scatter of the
points about the regression line of Y on X is called the standard error of estimate of Y on X
and is given by
EU’-9)’s,,, = , /f
The standard error is similar to the more familiar standard deviation and is useful in the
construction of confidence intervals. If lines are drawn parallel to the regression line at
respective vertical distances S“, 2S,,,' and 3S’, from it, then, if n is sufiiciently large, there
should be included between these lines about 68%. 95% and 99% respectively of the sample
points.
Correlation Coefficients
The total variation of Y is given by Z(y — 9): which is clearly the sum of the squares
of the deviations of the values of Y from the mean 9. Spiegell“ expresses this as
Z(y — 501 = 2(y — 9)-1 + z(5> — 9)1 where Z(y — y)= is the unexplained variation so-called
because the deviations behave in a random manner.
z(y — y)1 is the explained variation so called because the deviations have a definite pattern.
and Z(y -— y)= is the total variation.
The ratio of the explained variation to the total variation is called the coefficient of
determination and is denote by r2, which lies in the range zero to one. Hence
2 explained variation 2
r an ~ and0sr $1
total variation
It follows that r. the coefficient of correlation. must lie between -1 and +1.
I : + explained variation _ _._
_ total variation _ E(y — 9)2
If two variables are highly positively linearly correlated, i.e. one exerts a strong direct
influence on the other, the value of r would be towards the top end of the range. say +0.72.
If the variables were highly negatively linearly correlated, where one exerts a strong inverse
influence on the other, the value of r would tend towards -1. If r tended towards zero it
would mean an absence of linear correlation though this does not imply a total absence of
any correlation as there could still be a high non-linear correlation between the variables.
Scatter diagrams are useful in deciding the type and likely extent of correlation between two
or at most three variables.
KONSTANTINOS
Rectangle
Example From the data given earlier (p. 144)
x 151814161
Y
AK
X
X
X
X
X X
*5)-tx
Figure 13.2
Y Y
x
x xx XXX
xx xx
xx ’$<>e< xxxx
XX
Xx
*x
xx
x
.xx
‘X
x
‘I
\
In] Positive Linear Correlation X [bl Negative Linear Correlation
X .\ \
* x,‘ \
x . \
x
V‘ Y
fit
lo] Parabolic Correlation I Id] Absence of Correlation
6 13'
29 48 22 32 34 17
it was found that Y = -63.0 + 6.087X
Hence
X
15
18
14
16
-16
13
__Zy_l82_
y_ n _ 6 _
'\
Y Y
ii
48
22
32
34
17
iii}
28.305
46.566
22.218
34.392
34.392
16.131
30.33
QY-Y (y-9)’
0.695
1.434
-0.218
-2.392
—0.392
0.869
0.483
2.056
0.048
5.722
0.154
0.755
(9 ~ ii:-1
4.10
263.608
65.805
16.500
16.500
201.612
 
-1-i--ii-,
9.218 568.126
Hence the standard error of the estimate of Y on X is
_ 'E(y — 9)’ 9.218 f
S" \/ i{ 7 657 ‘
When the sample size is small (less than 30) the denominator should be n -— 2 instead of n
1.2395
and hence the adjusted Sy, value becomes 1.5
. . . Z0 2The coefficient of determination r3 — Y :
181.
- 9) 568.126
Z(y — 501 577.334
(Y - 9):»
_ 
1.769
312.229
69.389
2.789
13.469
177.689
-ii
577.334
-iii
.'.r2= 0.9841 which means that 98.41% of the variation in Y is explained by the regression
line i.e. by variations in X.
The correlation coefficient r = 0.9920 which suggests a very high positive linear correlation
between the X and Y variables.
Transformations
The examples so far in this chapter have dealt only with linear regression but there is no
reason why the data could not be transformed or other types of regression lines/curves
fitted. It is however often easier to transform the data to fit a linear model than it is to
attempt to fit a polynomial to data which are non-linear. The use of computer packages
makes it easy to try different transformations to see which gives the best linear fit. This
process is not as haphazard as might appear and an ordered scale of transformations
(\/y, log y, -1/y) exists (See Ryan, Joiner, Ryanm). A common transformation is to
regress log y against log x.
Multiple Regression
Multiple regression is only an extension of bivariate analysis and in its simplest form.
involving three variables. the regression equation is that of a plane and no longer of a line.
In this case perfect correlation indicates that all points lie in the same plane.
Figure 13.3
Y Z
8 ><X ><
i " X ><
X x
X
X X X
XX . X
X
There is no mathematical limit to the number of variables being considered and matrix
algebra will permit solutions to quite complicated problems. It is not the purpose of this
book to delve into the intricacies of this topic but the reader can learn more about it from
any text on advanced statistics or econometrics. some of which are listed in the bibliography
However, it is important to explain certain points. Multiple regression is best handled by
computer packages (e. g. Minitab or the Statistical Package for the Social Sciences) which
KONSTANTINOS
Rectangle
can perform quickly and accurately the large number of calculations necessary for the
successful determination of the parameter coefficients and for the assessment of the
equation‘s forecasting value. Nevertheless it is essential to have an appreciation of what is
involved in such computations in order to interpret the results which emerge.
The purpose of multiple regression is to explain variations in a dependent variable on the
left-hand side of an equation in terms of the independent variables on the right-hand side.
The degree of explanation is again measured in terms of the r2 value which is now the
coefficient of multiple determination (a number between 0 and 1) and r is the coefficient of
multiple correlation (a number between -1 and +1). It is mathematically quite easy to
obtain a high r3 value but the result is spurious unless the independent variables have been
included in the equation only because there is good reason to believe that they affect the
dependent variable in a consistent way. Theoretical considerations will help to determine
which factors should be included as independent predictors. The results can be tested using
standard error or t-tests to see whether the variables are significant.
The example which follows later will help to explain these points.
The relevant formulae for the solution of multiple regression problems can be derived in a
similar way to those of simple regression. In the case of three variables the simplest
regression equation of Y on X and Z has the form Y = a + bX + cZ where a. b. and c are
constants. and this represents the equation of the least-squares regression plane of Y on X
and Z. The constants a. b and c are determined by solving simultaneously the normal
equafions
Zy = an + bZx + cZz
Zxy = aZ'.x + bZx3 + cZxz
Zyz = aZz + bZxz + c223
These equations are obtained in the same way as those for simple regression. as follows:
If Y = a + bX + cZ
theny=a+bx+cz
The intention is to minimize the sum of the squares of the distances of each point from the
plane. For any point. its distance from the plane is given by (y — a — bx - cz). The sum (S)
of the squares of the distances of all points from the plane is
S=Z(y—-a—bx—cz)2
and the problem in multiple regression is to find values of a, b and c so that S is minimised
Hence it is required that _
6S 6S 6S
6a_0'6b_0‘6c 0'
ThusZ2(y — a - bx ~— cz) (-1) = 0
Z2(y — a — bx — cz)(—x) =0
E2(y — a — bx — cz) (-2) = 0'
whence
Zy = an + bZx + cZz
Zxy = aZx + bEx2 + cZxz The normal equations
Zyz = aEz + b2xz + c222
KONSTANTINOS
Rectangle
The standard error of the estimate of Y on X and Z is given by
Sm = /EU II $02
where 9 indicates the estimated value of Y as calculated from the regression equation. For
small samples the denominator is usually taken to be n — 3 instead of n.
In the case of two independent variables the coefficient of multiple determination is given by
2
s,.
where S§, is the variance of the variable y. The coefficient of multiple correlation is given by
l Snxz
ryxz = 1 _ FY32!-
Example It is often suggested that shipping freight rates are determined by the value and
stowage factor of the cargo carried. Given the data below. investigate this relationship by
performing a regression analysis and calculating the coefficient of multiple correlation. Use
the regression equation to estimate the freight rate of a commodity having a value of 60 and
a stowage factor of 29.
Data Freight rate Value of cargo Stowage factor
190 175 45
215 88
135 132
110 109
200 102
275 103
250 92
255 95
255 163
192 229
x y z xy
190 175 45 33 .250
215 88 95 18.920
135 132 28 17.820
1 10 109 22 1 1 .990
200 102 80 20,400
275 103 133 28,325
250 92 120 23 .000
255 95 120 24.225
255 163 99 41.565
192 229 34 43 ,968
2 .077 1 .288 776 263 .463
95
28
22
80
133
120
120
99
34
xz
8.550
20,425
3 .780
2.420
16.000
36,575
30.000
30.600
25 .245
6.528
180.123
yz
7.875
8.360
3.696
2.398
8,160
13.699
11.040
11.400
16.137
7.786
90.551
ya
30.625
7.744
17.424
11.881
10.404
10,609
8.464
9.025
26.569
52,441
185,186
Z:
2.025
9.025
784
484
6.400
17.689
14.400
14.400
9.801
1.156
76.164
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The normal equations are:
Ex = an + bZy + cZz
Zxy = aEy + bZy2 + cZyz
Zxz = aZz + bEyz + cEz2
Hence 2,077 = 10a + 1,288b + 776c . . . (1)
263,463 = 1,288a + 185,186b + 90,551c . (2)
180,123 = 776a + 90,551b + 76,164c. . . (3)
Multiplying equation (1) by 1288 and equation (2) by 10 gives
2,675,176 = 12.880a + 1.658,944b + 999.488c . . . (4)
2,634,630 = 12,880a + 1,851,860b + 905,510c . . . (5)
Subtracting to eliminate ‘a’ we have
=—40,546 = l92,9l6b — 93,978c . . . (6)
Multiplying equation (2) by 776 and equation (3) by 1,288 give
204.447.288 = 999.488a + 143,704,336b + 70,267,576c . . . (7)
231,998,424 = 999.488a + 116.629,688b + 98.099,232c . . . (8)
Subtracting to eliminate ‘a’ we have ,
- 27,551,136 = 27.074.648b -— 27,83l,656c . . . (9)
Taking equations (6) and (9) together and multiplying equation (6) by 296.1508 gives
—-12.007,728.66 = 57.132.219.776 — 27,831,656c . . . (10)
—27.551.136= 27.074,648b — 27.831,665c. . . (9)
Subtracting equation (9) from (10), gives
15,543,407 = 30,057.57l.77b
whence b = 0.5171
From equation (6) c = 1.493 after su,bstituting.for b and from equation ( 1) a = 25.2407 after
substituting for b and c.
The regression equation becomes
Freight Rate = 25.24 + 0.517 Value + 1.493 Stowage Factor from which it can be seen that
the freight rate will have a value of 99.56 when the stowage factor is 29 and the value of
cargo is 60. '
The coefficient of multiple correlation is given by
l Sip
z = 1 _ i-rxy
E(x — x)2
where S“: = - T?
and s1 = z—-—(x_ Y2
" n
KONSTANTINOS
Rectangle
T
The regression equation determined above is now used to calculate the estimated values of
freight rate (ii) from given data of values and stowage factors. The following results are
obtained:
x
190
215
135
110
200
275
250
255
255
192
2.077
_ EX 2,077
x—"lT—T-—207.7
it
182.92
212.57
135.30
114.45
197.42
277.07
251.97
253.52
257.34
194.42
x - it
7.08
2.43
-0.30
-4.45
2.58
-2.07
-1.97
1.48
-2.34
-2.42
I 104.27 77
S0 rxyz = 1 — £5-a5—gT' = V 0.9943
= 0.9972(X - 1)’50.13
5.90
0.09
19.80
6.66
4.28
3.88
2.19
5.48
5.86
104.27
(X - i)-17.7
7.3
-72.7
-97.7
-7.7
67.3
42.3
47.3
47.3
-15.7
(X - if313.29
53.29
5,285.29
9,545.29
59.29
4,529.29
1.78929
2,237.29
2,237.29
246.49
26.296.10
 -
which suggests a strong direct relationship between freight rate and the two other variables.
In real life of course examples tend to be much more complicated than this one and such
problems are best solved using computer packages. The example which follows is an
extended version of the one above where 30 observations of each of the three variables
(freight rate, stowage factor and value of cargo) are considered. The results are given in the
form of the abbreviated computer printout from the Minitab statistical package.
A.
9
10
11
12
13
14
15
16
17
W
oo-.io~ui4=-oars»-2
190
215
135
110
200
275
250
255
255
192
225
210
130
162
250
165
190
175 45
95
28
22
80
133
109
102
163
229
113
225
164
Notes
Abbreviated output from a Multiple Regression Computer Program
C7 C8 C9
A. C7, C8 and C9 are three columns containing the data
on freight rate, value of cargo, and stowage factor
respectively.
B. The regression command instructs the computer to
carry out the regression of freight rates on two predic-
g tors, the value of cargo and the stowage factor.
120 C. The results are commented on fully in the text.
120
99
34
98
109
49
52
81
65
57
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Abbreviated output from a Multiple Regression Computer Program
A. Row C7 C8 C9
18 240 61 127
19 230 62 118
20 237 241 61
21 266 95 129
22 300 150 140
23 170 226 . 24
24 215 153 78
25 93 60 29
26 278 163 109
27 225 160 77
28 310 _ 182 127
29 280 191 1 1 1
30 166 213 24
B. MTB 2 REGRESS C7 2 C8 C9
C. THE REGRESSION EQUATION IS
C7 = 26.5 + 0.496 C8 + 1.48 C9
ST. DEV.‘
COLUMN COEFFICIENT OF COEF. T-RATIO = COEF/S.D.
26.490 3.472 7.63
C8 - 0.49576 0.01616 30.68
C9 1.48359 0.02434 60.95
S = 4.779
R-SQUARED = 99.3 PERCENT
R-SQUARED = 99.3 PERCENT. ADJUSTED FOR D.F.
“'(more usually called the standard error)
The regression equation, from the printout, is
FR = 26.5 + 0.496.V + 1.48SF .
where FR = freight rate, V = value of cargo, and SF = stowage factor.
It is usual to summarise such output in the form
FR = 26.5 +0.496V+1.48SF
(5.6.) (3.472) (0.016) (0.024)
degrees of freedom = 29 rz = 0.993
The degrees of freedom relate to the number of observations less one. The r2 value has the
usual meaning though it may have been adjusted for the degrees of freedom; this will
produce a different rz value when the sample size is small. The standard errors (s.e.) are
given in brackets below each coefficient and are extremely useful as they enable significance
tests with respect to the variables to be performed. The tests show whether the estimates are
significantly different from zero; in other words whether the variable to which the estimate
relates is important in the regression. The standard error test may be stated as follows:
if the standard error is smaller than half the numerical value of the parameter estimate then
the estimate is statistically significant.
From the above results a comparison of the estimates of the coefficients of value and
stowage factor with their respective standard errors shows that both these variables are
KONSTANTINOS
Rectangle
highly significant. Numerically the ratios are
Value Stowage factor
estimate 0.496 1.480
0.5 X estimate 0.248 0.740
s.e. 0.016 0.024
and clearly the standard error is much less than half the value of the estimate in both cases.
This test implies a two-tail test at the 5% level of significance and is a useful rule of thumb
when considering regression equations. It can be shown (See (3) p. 90) that this standard
error test is formally equivalent to the student's t-test. The t-statistic is found by dividing the
estimated value of the coefficient by its standard error -- hence 0.49576/0.01616 and
1.48359/0.02434 give 30.68 and 60.96 as shown in the printout. These t-statistics should be
compared with the value in the probability table of the t-distribution. For a sample size of
30 the value from the table for a 5% level of significance is 2.04. If the calculated t-statistic
exceeds the tabulated t-value then the variables are significant; clearly this is so in this case.
A third test of significance, that of the F-test and based on the F-distribution, is also
possible and the Minitab package provides the necessary information to perform this test
though it is not presented here.
Modelling
Multiple regression techniques play an important role in mathematical modelling and
forecasting. A model is simply a set of simultaneous structural equations containing
variables. These variables may be exogenous (i.e. determined from outside the system and
inserted as data) or endogenous in which case they are to be determined from inside the
system of equations as the solutions to those equations. A model will contain more variables
than equations but will be solvable provided there are at least as many equations as there
are endogenous variables.
The problem of identification (see 19) occurs with systems of simultaneous equations. An
equation is only identified if it is in a unique statistical form so that unambiguous estimates
of its coefficients can be found. The problem of identification has long occupied
econometricians because it has important implications for the choice of the appropriate
regression technique for solving the equations. Some authors (e.g. Koutsoyiannis) refer to
the paradox of identification whereby an equation is identified by variables it does not
include; these variables must be excluded from the equation concerned but included
elsewhere in the system of equations. The problem is essentially one of specification in that
the structure of the model and the extent of available information prevents the complete
estimation of the coefficients. The choice of regression technique depends on whether the
equation is exactly identified, when Indirect Least Squares would be appropriate;
over-identified, when Two Stage Least Squares would be appropriate; or under-identified in
which case the equation cannot be solved. The identification status of an equation may be
found by the application of two formal rules known as the Rank and Order Conditions
whereby certain conditions have to be met. The rules, which form a two-part test, determine
first whether the equation is able to be identified and then the state of identification which
leads ultimately to the correct choice» of regression technique.
Further problems may be caused by multicollinearity and autocorrelation which can
invalidate regression results, but a good statistical computer package will test for the former
and provide the Durbin-Watson statistic so that the latter can be investigated.
Multi-collinearity occurs when one of the independent variables is highly correlated with one
or more of the others; this reduces the number of independent equations in the system and
KONSTANTINOS
Rectangle
prevents its complete estimation. It is usual to specify inexact linear relationships between
variables e. g.
Y=a+bX+u
where u is a disturbance term and refers to a disturbance of the relationship between Y and
X. Autocorrelation occurs when successive values of the disturbance term are not
independent of each other, i.e. the value of u. depends on u,_1 which in turn depends on
u._; and so on. The existence of autocorrelation invalidates one of the assumptions upon
which the regression technique is based and hence the technique is no longer valid.
These problems and difficulties have been presented in this book because it is important for
students to realise that, powerful though regression analysis is. it is not a panacea; it is only
applicable under certain specified circumstances. It should be used wisely and cautiously.
and indeed many of the problems will be overcome by a correct specification of the model in
the first instance. As Achenlsl writes . . if the researcher setsup the problem correctly.
regression will tend to the right answer under any reasonable practical circumstances . .
The results of the regression package should be examined in the light of theoretical,
statistical and econometric criteria and only if all these are satisfied should any reliability be
placed on the output.
It does not follow that a model needs to contain many equations or even more variables. As
the number of equations increases so does the risk of interrelationships between them and
between the variables involved; further, the cost in computer time increases as more data,
variables and equations are added. The final example in this chapter presents a simple
demand and supply model of the dry bulk shipping market, and, while the data on which
the results are based are somewhat dated, the principles remain valid.
Example This model (see (6)) constitutes an attempt to derive the demand and supply curves
for dry bulk shipping services and these equations are taken as linear functions of certain
relevant variables. The structural equations define the demand and supply curves and show
the relationship which exists between quantities demanded and supplied and the other
variables of the system.-..The variables used are: voyage charter freight rate indices’ deflated
by a wholesale price index, gross national product of the developed (OECD) countries. the
freight rate index lagged by three periods, the combined ton-mileage of the five major dry
bulk cargoes each year, the ton-mileage figure lagged by both one and three years, and an
efficiency index. Data for each year of these variables were collected or generated”) for the
period 1953-73 and these 21 observations form the basis for the application of the model.
The approach used is that of Two Stage Least Squares (2SLS), a multiple regression analysis
for an over-identified system of equations. and the ‘period’ in question is one year.
Data Base Annual observations 1953-73
Number of observations = 21
Variables
f, = voyage charter freight rate index deflated by an index of wholesale prices
and with 1963 = 100.
GNP = gross national product of developed (OECD) countries in billions of US
dollars at 1963 prices.
f,_3 = f, lagged by three years.
q, = total shipments of the five major dry bulk cargoes per annum measured in
thousand million ton-miles.
q,_1, q,_3 = q, lagged by one year, three years.
KONSTANTINOS
Rectangle
EFF = an index to reflect the efficiency of the developed countries in converting
raw materials into a given tonne of output (1963 = 100). This index was
derived from the Annual Steel Statistics published by the Iron and Steel
Statistics Board and measures the ability to convert raw materials into pig
iron.
Model
The system of structural equations comprises a demand equation, a supply equation and a
market clearing condition that quantity demanded must equal quantity supplied.
In a general form there are three equations:
(1) qg = an + alf, + a;GNP + a3EFF
(2) qr = 90 '1‘ blft '1' b4ft-3 '1' 9591-1 '1‘ 969:-3
(3) <16 = qt
and the a priori economic constraints a1 < 0, a, > 0 and bl > 0, must be satisfied. Applying
multiple regression analysis provides the following demand and supply equations for dry
bulk shipping services:
(4) qr -= -6,603.68 - 224?. + 3.28GNP + 39.84EFF
(1.36) (0.21) (7.38) 5
(5) q, = -79.87 + 1.6‘/I. - 0.83:.-. + 0.ssq.-, + 0.69q.-.
(0.89) (0.58) (0.17) (0.19)
and the most significant variables are GNP, EFF, q.-1. and q,_3 as shown by the standard
error (in parentheses) of the coefficients. The exogenous variables in the demand equation
(4) are GNP and EFF and substituting for these yields a simple price-quantity equation
capable of representation on a conventional two-dimensional graph.
KONSTANTINOS
Rectangle
Chapter 14
Decision Theory
In all industries (and shipping is no exception) decisions have to be taken under conditions
of uncertainty where the outcome of a particular action (or decision) is influenced by what
subsequently happens in the market. For instance a shipping company might decide to
purchase a new VLCC because the expected rate of return and the investment appraisal
calculations lead the company to expect-a profitable investment. Such calculations are based
on expectations about future cash-flows and future trading conditions as well as projected
rates of inflation. It is easy to see that such determinants are uncertain in the real world;
trading conditions could be totally altered by say the closure of the Suez Canal or by a war
in the Middle East; rates of inflation and interest are determined by forces outside the .
control of the shipping company and could easily affect the value of a vessel‘s foreign
exchange earnings. Decision theory is an attempt to formalise the decision-making process
by taking account of the expected circumstances and by involving probabilities where they
are quantifiable. either from previous experience or from market research. This chapter will
consider both non-probabilistic and probabilistic criteria after first introducing the
appropriate terminology.
Definitions '
It is usual to distinguish actions and states .of the world. An action clearly results from a
decision to do something, e.g. purchase a new vessel. and the state of the world defines the
conditions existing in the world. e.g. the market conditions under which the vessel must
operate. The consequence of any decision is expressed in terms of a payoff or a loss and is
the result of the interaction of the action taken and the actual state of the world which
prevails.
A payoff is defined as the net change in total wealth resulting from such interaction. It may
be positive or negative in value. A loss can only be positive or zero and is defined as the
positive difference between any given payoff and the highest possible payoff under any
specified state of the world. It follows that if there are m actions and n states of the world
then there will be mn payoffs (and correspondingly mn losses). The results of the
interactions are expressed in the form of payoff or loss tables.
If a payoff table is given as
States of the World
Action ‘ 01 02 03 1
555"5 ' 5' " 1 '5 ' 5 5 ' 5 " 5' T 5 5T*5 .
A1 1 15 12 5 1
A2 i 8 10 11 y
KONSTANTINOS
Rectangle
then the corresponding loss table will be
_ if 7 i _ _ ___
States of the World l
, __ , ~— 4 — _ l
ACIIOH ll 61 62 63
A1 0 0 6 1
A2 ' 7 2 0 l
It will sometimes be the case that one action will dominate another. This occurs when, for
all states of the world, the first action leads to at least as high a payoff (or as small a loss) as
the second action and, for at least one state of the world, the payoff from the first action
exceeds that of the second. In other words it is never rational to choose the second action as
the outcome from so doing is worse than would have resulted from choosing the first action.
Hence the first action dominates the second one which is then said to be inadmissible.
Example A payoff table is denoted by
-77 _ 7. _ ' _
W J 91 ,1 _ 93 #1
A, . ~50 80 20 1
' 5 5715' ‘ ” l
1‘ A2 30 . 40 70 .
.2 , -1, 4 4 L _ .
.71, 10 y 30 -30
,-
. g _ . , l_ , I , ,, - 3i
(A4 ~ -10 -50 75 ‘
and it is clear that action A3 is dominated by action A3. Hence action A3 is inadmissible and
should play no part in the consideration. Thus, the revised payoff table becomes
g y‘ 91 92 H. 93 1
A1 . 3-50 1 80 20
,, _\_ 1?
A2 Li 30 l‘ 40 70
A 4- 3. as . A. - - 1
A4 ~ -10 " -50 J 75 ~
1~ ,2 _. _ l
and the decision will be based on this according to one of the criteria outlined below.
Non-probabilistic criteria for decision making
Three non-probabilistic criteria may be distinguished. namely the maximin. the maximax and
the minimax loss (or regret) rules.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Z-
The maximin rule is based on finding the smallest payoff for each action and then choosing
the action for which this is largest. From the revised payoff table above:
Action Smallest Payoff
A1 "5030 of which the largest is 30and
:3 so leads to the choice of action A;4 _
The maximax rule finds the largest possible payoff for each action and chooses the action for
which this is largest. Hence
A 70 of which_the largest is 80 and
Action Largest Payoff
A1 80 }
A4 75 leads to the choice of action A1
The minimax loss rule finds the maximum loss for each action and then chooses the action
where this is smallest. The loss table corresponding to the revised payoff table is
0, 02 0..
A1 80. 0 1‘ 55
A; H 0 1 40 ‘ 5
|.4. 40 N, 130 0.
from which the maximum loss corresponding to each action is
Action Maximum Loss
2' of which the smallest is 40 and
A3 130 ' leads to the choice of action A;
4
These three different criteria reflect the decision makers‘ attitude to risk. The maximin rule
reflects a very conservative or pessimistic attitude while the maximax rule reflects an
optimistic or gamblers‘ attitude to risk and ignores losses. An example of the use of the
minimax regret rule is contained in Meredith and Wordsworth“) and concerns the depth of
water that should be provided to accommodate large vessels in a new harbour.
Suppose a harbour is built to accommodate ships with a maximum draught of 41 feet; then.
if this proved to be the optimum. profits would be maximised. If, however, it proved not to-
be optimum in relation to the size of ships, profits would be reduced and the extent of this
reduction would vary with the actual value that the best harbour size turned out to be. This
reduction in profits measures the "regret" in taking the decision to build at 41 feet.
Adopting the minimax loss rule involves minimising the maximum regret and ucha policy
would be consistent with choosing a harbour depth so that, if events proved the choice to be
wrong, the greatest loss of profit incurred would be less than if some other size had been
chosen and‘ subsequent events had proved it to be wrong. Meredith and Wordsworth explain
their solution graphically by means of a curve of maximum regret which has its minimum
value at about 42 feet. This maximum regret is close to zero and suggests that the minimax
loss criterion is most appropriate in this case.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Example A shipping company running a weekly container operation is uncertain whether to
provide vessels of 1,000, 2,000 or 3,000 TEU capacity on the route. Economies of scale
enable the company to provide 1,000 TEU vessels at a cost per TEU of £1.000. 2.000 TEU
vessels at a cost per TEU of £900, and 3,000 TEU vessels at a cost per TEU of £700: they
expect to be able to sell this shipping space at a rate of £1,500 per TEU. The problem of
loss of goodwill associated with shutting out cargo will be ignored. Previous experience of a
different route leads the company to believe that the demand is likely to be either 1.500.
2,000 or 2,500 TEU per week. How much capacity should the company provide?
The problem can be formulated as:
A1: company provides 1,000 TEU capacity
A2: company provides 2.000 TEU capacity
A3: company provides 3.000 TEU capacity
01: level of demand is 1,500 TEUs
0;: level of demand is 2,000 TEUs
63: level of demand is 2.500 TEUs
and the payoff table may be derived as:
. *7. 'T ___
Z?.”§535Z§"" . .91 - 9 9 l
A1 1 500 4 500) 500
' _ ’7"_ *1 __ 37 *__ -l F 1A, ‘ 450 1.200 1.200 ?
~ ; _ 447 _ f _ 2 4 __ ~ .
i l
A3 150 ‘ 900 1.650 ;
l__ __ _1 _ l
It can be easily seen that no action dominates another hence the corresponding loss table
becomes:
Loss Table (5 ' 557' " ‘ 5 5 l
(in £0005) ‘ 91 9: 93
__ ___ ——l[ _ — __ F _ ,——
A, 5 0 1 700 "1.150.
A; I 50 1 0 450
_._ 1 _ _ — ‘
1 til
A3~ i 350 1 300 ‘ 0
The maximin rule would suggest_ that action A1 should be chosen since
Action Smallest Payoff
A, 500
A; 450
A 3 150
The maximax rule would suggest that action A3 should be chosen since
Action Largest Payoff
A, 500
A2 1,200
A3 1.650
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The minimax loss (regret) rule would also suggest that action A3 should be chosen since
Action Maximum Loss
A1 1 .150
A; 450
A3 350
Clearly such non-probabilistic criteria do not lead to an unambiguous answer. The decision
taken will reflect the decision-maker’s subjective assessment of risk which might not be the
best basis for an investment decision involving the expenditure of large sums of money. The
approach can be made more scientific and less subjective by adopting one of the
probabilistic criteria.
Probabilistic criteria for decision-making
There are just two such criteria; they are, in fact, the same although taken from two
different points of view and will lead to the same decision. Hence an unambiguous answer
based on expected probabilities of the different states of the world will be reached. The two
approaches involve calculating either the expected payoff (EP) or the expected loss (EL)
based on weighted average values of the payoffs or losses using their respective probabilities
of occurrence as weights. Formally the criteria can be stated as: choose the action with the
highest expected payoff (or the smallest expected loss). The probabilities may be derived
from previous experience or from probability distributions if it is known that events follow a
particular pattern.
Consider again the previous example where the payoff table. expressed in thousands of
pounds sterling, was:
i Bl ~ 92 l 93 ,
.2 _i.,_ I _, l ___,2. 1 m ____ _ ,
A. 1 500 500 i 500
E —~ Woe. ~~ T 3.1: ____ 44;
I A; 1 450 * 1.200 1,200 f
A. y 150 900 ‘ 1.650 ,
_ — - I _____ __ — __
l
‘p(Hi) 1 0.3 0.4 A 0.3
and the probabilities of each of the states of the world occurring have been added as an
I1
extra row. Note that 2 p(6j) = 1 where n is the number of states of the world. The
i=1
expected payoff (EP), in £000, of action A, is given by
(500 >< 0.3) + (500 >< 0.4) + (500 x 0.3) = 500.
Similarly EP(A2) = (450 >< 0.3) + (1.200 >< 0.4) + (1,200 >< 0.3) = 975
and -EP(A3) = (150 >< 0.3)+ (900 x 0.4) + (1650 >< 0.3) = 900
The action with the largest expected payoff is action A2.
The argument can be reversed by using the loss table instead of the payoff table and
choosing the action with the smallest expected loss.
The loss table with associated probabilities is given below where losses are in thousands of
pounds sterling.
KONSTANTINOS
Rectangle
1"
1.
I
|
i
l
I
ii
I
i
iijjii-?_ii—.1i
l
1
i 61 92 .; 63, , , __ _ ,
l
ll A1 1. 9
L1 —— 4 A mfl —— W _.>__
“ l
l
--I8 I-5 '1-1 ui ca
A2 ~ 50 0 450 5
_ 7'7" ' _ 71
. A, 350 5. 300 0 ;
_ ___;-; —— H __
l
p(9j) 0.3 0.4 0.3
The expected loss (EL), in thousands of £ sterling, for each action is:
EL (A1) = (0 >< 0.3) + (700 >< 0.4) + (1,150 >< 0.3) = 625
EL (A2) = (50 >< 0.3) + (0 >< 0.4) + (450 >< 0.3) = 150
EL (A3) = (350 >< 0.3) + (300 >< 0.4) + (0 >< 0.3) = 225
The action with the smallest expected loss is action A2.
It must always be the case that the action with the smallest expected loss is also the action
with the largest expected payoff.
All decision-making problems contain similar features
(a) a choice or sequence of choices
(b) some available information (if only subjective)
(c) the possibility of obtaining further information at a cost.
The problem is to determine the best course of action to take to reach t-he correct decision.
Should more information be obtained or might the cost of it outweigh its usefulness?
One way to decide whether or not information is worth purchasing is to determine the value
of perfect information which will set a maximum value on the price the decision-maker is
prepared to pay for additional information. Consider the following payoff table:
9 0, 1 0, Expected .
y. Payoff .
___ _.__ 1, 1-_ 1 ,.___f _ _ __‘
, l
A, y 45 \ -25 4 17
A2 -10 60 18
___ F _ ,3” , __l ,_
p(0j) fi70.6 0.4 "H l
from which the corresponding loss table is:
\ 0, 0; 1 Expected
1. Loss
_ _ #__. __.__._....
l l
A, 0 85 34
_, ,_ __ WE, ~~ Al . __ 2
A2‘55 0.33‘
p(0,) 0.6 l 0.4
KONSTANTINOS
Rectangle
In the absence of perfect information it follows that the optimum action isA2 which has the
highest expected payoff (and of course the smallest expected loss). If an action was to be
taken knowing the state of the world then the decision-maker would choose the action which
gave the highest payoff under that state of the world; action A1 under state 9, or action A2
under state 9;. These states of the world have different probabilities of occurrence so that
the current expected payoff of such action will be
(45 >< 0.6) + (60 >< 0.4) = 27 + 24 = 51.
The expected value of perfect information is the difference between this payoff and the.one
which would have been obtained in any case. i.e. 51 - 18 = 33. It is always the case that the
expected value of perfect information equals the minimum expected loss. If the
decision-maker was offered additional information at a cost of 35 units it would obviously
not be worth his while to accept it since the maximum possible benefit he could obtain is
only 33 units.
More complicated problems are best considered using tree diagrams to formulate a
comprehensive approach.
Tree diagrams
A tree diagram grows from left to right as a logical sequence of events unfolds and its
purpose is to enable a path to be traced back to the origin to represent the best sequence of
decisions. The diagrams comprise decision forks and chance forks. The probabilities and
expected payoffs at each chance fork are computed and at each decision fork the action with
the largest expected payoff is chosen.
The tree diagram for the example of the shipping company operating a container service is
given below.
Figure 14.1
EP= 5000 91 (5000) Pr = 0.3
— Z 4- 9z(5000| Pr = 0.4
/ 9a(5000) Pr = 0.3
. A‘
9750 EP = 9750 91 (4500) Pr = 0.3
Abe es‘ ~ e — "9z(12000)Pr=0.4
9=(1200o; Pr= 0.:M
91(1500) Pr = 0.3
~ +9z(9000) Pr = 0.4
EP= 9000 93(1650O) Pr = 0.3
The correct decision for the company to take is still action A2. A tree diagram is simply the
visual represen-tation of the probabilistic criteria and enables quite complicated problems to.
be tackled with the branches of the tree increasing to incorporate extra information or
courses of action. Bayesian theory dealing with conditional probabilities will be invoked to
KONSTANTINOS
Rectangle
calculate probabilities as subsequent events become conditional upon other events already
having occurred.
Example A shipping company is considering entering a new trade which it intends to serve
by purchasing one vessel. The likely cargo flows are uncertain though an international
agency puts the probability that they are good at 60%. The company is considering three
alternative courses of action:
A1: the purchase of a new vessel at a cost of £18 million
A2: the purchase of a 4-year-old vessel from a reputable shipping company for £14 million
A3: the purchase of a 3-year-old vessel from a recently established FOC-based company for
£15 million.
The company is unsure of how well the second-hand vessels have been maintained and
operated and is considering whether to commission an independent surveyor’s report at a
cost of £250,000 for each vessel.
The company would use the report to assess the probabilities of future maintenance
expenditure being heavier or lighter than usual.
The data may be summarised as
Age of vessel Maintenance costs Trade prospects Payoff (£m)
25New
New
4 yrs
4 yrs
4 yrs
4 yrs
3 yrs
3 yrs
3 yrs
3 yrs
Normal
Normal
Heavy
Heavy
Light
Light
Heavy
Heavy
Light
Light
Good
Poor 15
Good 28
Poor 18
Good 30
Poor 20
Good 30
Poor 20
Good 33
Poor 21
and the appropriate probabilities might be as shown in the table below.
_ 7 L7 __ 7________ _ ___ 7 T I
l
‘ Survey No Survey
Age of vessel l _ _
‘ P, (Maintenance costs ‘ P, (Maintenance costs
\ Heavy Light) Heavy Light)
4 ~ . , ________ __.... - —-—~—--~- _ . cw ~~ "17"" *—* —* W e e
4 yrs ' 0.6 0.4 0.5 0.5
3 yrs 0.75 0.25 0.5 0.5
A decision tree may be used to present all this information and enable the shipping
company to make a decision. Each chance fork is denoted by a circle and each decision fork
by a square. Combining the probabilities and the payoffs to calculate expected payoffs yields
answers of £3m, £11m, and £12.1m respectively for each of the three actions. It is clear from
the tree that the company's best course of action is to purchase the three-year-old ship and
to dispense with the survey. (See Figure 14.2)
Example A company producing radar systems for use on board ship knows that any newly
produced system may be defective. in the sense that it may not immediately work properly.
afterinstallation. A thorough process of inspection prior to installation can ensure that it
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
will function properly but this testing procedure will cost £750. The alternative is not to test
the system but. if it fails to function correctly. pay £1,200 to have it removed plus £800 to
repair it. From previous experience the company knows that 70% of its systems are sound.
Construct a payoff table for the company. A compromise alternative is possible whereby the
system is subjected to a quick examination that costs only £200. but this is not completely
Figure 14.2 3"“ “_M.,.,ato-Bi aw
3M ‘rd
I Buy 2nd mm: '0-4)
14 years oldll-14M] 34;; om
1 1“ 0° . 0 5‘ @3110 Good‘ 28“
cm‘
t>~"“°“
o
fa
3
\
\~\“’\
\
zass
‘[0
26.: '\.<>T""
9,<~I‘*
"0 so 27.127.1 '1',’
 15M
¢»°‘\-‘“m tom M“ t.sues°°°‘°'°‘ 13"W we 24.8 W “gnaw 1m
24.55 \_a1"‘n ""'===a= "'9hrro.4, “M t»¢=e°°°‘°'°‘ 3°"
Haw
c°*£s :59," to tw-1° 9°“
.5) t 
I
N. C0515 lradfi poor
M.°°3ls!5gh 28.2” trade s°°‘“°‘6\ 33M
 
F5119 poor M
zenn Md, 00¢ °-5‘ 30M
W cQ5\5"\Ba‘~ ffadi p0Or 20M
M'°°=r=:i h 28.2" trade ‘om 33'“
9 '(0.5;
fr“E =>o@.,o__,) am
reliable. When the test was applied to systems which were known to be good it declared
20% of them to be faulty. When applied to systems known to be faulty the test declared
10% of them to be sound. The company is prepared to accept the results of this test and will
not install any system which fails the test nor inspect any which passes.
Use a decision tree to find the correct action for the company under these circumstances
Let A, = inspect
A3 = not inspect
9] = Sotmd
92 = faulty
then the simple payoff table is given by:
B1 8;
750 750A1
A: 0 2,000
PW . 0.7 0.3
"J
KONSTANTINOS
Rectangle
where the payoffs are given in terms of money costs and the probabilities of each outcome
(i.e. state of the world) are known from previous experience. In the absence of any further
information the company would adopt the probabilistic criterion of minimising the expected
loss (or maximising the expected payoff). From the table it can be seen that
for A1 the expected payoff = 750 and
for A; the expected payoff = 600
As these payoffs are in terms of costs it is clear that the company should take action A; and
not inspect the system. However, more information is available. at a cost. and the company
could make use of this to arrive at a posteriori probabilities which may lead to a change of
decision. A decision tree will help to clarify the decision making process. (See Figure 14.3)
The conditional probabilities contained in the tree are found as follows:
P,(system passes test)
= P,(system passes test/system sound) + P,(system passes test/system faulty)
= P,(system passes test/system sound) X P,(system sound)
+ P,(system passes test/system faulty) X P,(system faulty)
= (0.8 X 0.7) + (0.1 >< 0.3)
= 0.56 + 0.03 = 0.59
Similarly
P,(system fails test)
= P,(system fails test/system faulty) + P,(system fails test/system sound)
= (0.9 >< 0.3) + (0.2. >< 0.7)
= 0.27 + 0.14 = 0.41
Using Bayes Theorem
P,(system is sound/passed test) =
g P,(passed test/sound) X P,(sound) i
P,( passed test/sound) >< P, (sound) + P, ( passed test,/faulty) X P, ( faulty)
_ _0.56 _ :0.ss O95
T 0.56 + 0.03 0.59 _ '
and similarly
0.03
P,(system is faulty/passed test) = 0 -jg = 0.05
The probabilities are found using Bayes Theorem as explained above (see also Chapter 4)
and the payoffs are found by calculating the total costs involved in each courseof action.
For example, if a system is found to be faulty after it has been installed following its passing
the quick test then the company will have incurred costs of
£200 for the quick test
£1,200 to remove the installed system
and £800 to effect repairs
making £2,200 in total as shown in the tree diagram.
It now follows from the diagram that the company's best course of action, i.e. the one which
involves the lowest costs. is to undertake the quick test at a cost of £200 and then to install
KONSTANTINOS
Rectangle
Figure 14.3
£750/pf
£200
\ t0 95) wpolflflill (£300: '59)
0°” kc '0.050 ck mt l£259l I"' ‘M, ezzoo
fail (0 q ’
' £200'4:
%‘%<%%
J ‘O
row I.”
"" czooo
_1\ WM 0
any system which passes this test. The expected cost of this action is £259 compared with
£600 or £750.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Chapter 15
Queueing Theory
A queue is defined as a line of persons or vehicles awaiting their turn; such a line will form
whenever there are more arrivals in a period than there are service points. Queueing theory
is concerned with the general behaviour of any service system which can involve a build-up
of customers waiting in queues for their service to begin; for example aircraft being held in a
stack while waiting for permission to land or vessels queueing for passage through a canal or
lock or awaiting a berth.
The mathematical theory of queues is based on probability distributions (often the Poisson
distribution) and was originally developed in the context of telephone exchanges. Queueing
theory is an attempt to describe various types of service systems by models that enable an
assessment to be made of how the system would cope with different demands. In any system
five basic factors need to be considered:
(i) the arrival pattern of customers
(ii) the service-time pattern
(iii) the layout of the servicing system
(iv) the capacity of the system
and (v) the disciplineof the queue.
The arrival pattern of customers is usually taken to be a random one described by some type
of statistical distribution. A negative exponential distribution of inter-arrival times (and
hence a Poisson arrival rate) is the most popular type, though other types are possible. The
Flgure 16.1
Numberofservices
Service time
KONSTANTINOS
Rectangle
mean arrival rate is given as A per unit time. The assumption in most queueing systems is
that arrivals occur singly and only the probability of 0 to 1 arrivals in a small time period,
bt, need be considered. It is not possible for an arrival to be serviced and leave the system
during a period of 6t, i.e. the increment of time 6t is too small to permit this.
The pattern of service times is normally taken as a negative exponential distribution as
depicted in Figure 15.1. This assumes that a large number of services take a short time and
a much smaller number take a long time. The mean service rate is denoted by p..
The layout of the service system is very important as it may allow customers to be served in
parallel or in series through multiple service channels and the behaviour of queueing systems
is influenced not only by the number of channels (or service points) and the way in which
they are arranged but also by their reliability. Figure 15.2 presents a possible layout of a
service system involving multiple channels arranged in parallel and in series to produce a
two-phase system.
Figure 15.2
3 service channels .. 2 S8l'VlC8 channelsln parallel .
II1 parallel
/’l:l\\ 4:_ _L.§‘i"§.
/ \ /
/' \ /Arrivals ———/ \ 1- /———————-Queue q ————— I |— —— ——} Queue C
-ii \ I it ""--l.\_
\ / \
""-. "‘--.
\
__LE@’°
2 phase system
{-1-I I 7 _ — -—' ' —’ — ~ * $-
The capacity of the system is an obvious factor in determining whether queues will be
formed and what their lengths will be. If the queueing area is sufficiently large the system
may be regarded as unrestricted. The capacity of the system must include those customers
actually being served as well as those queueing.
Queue discipline refers to what happens between a customer arriving in the queue and being
served. The simplest discipline is for all customers to form a single queue and then be
served in arrival order (S.I.A.O.); in the case of a single-service channel this is equivalent to
the F.I.F.O. (first in first out) system and allows the customer who has been queueing
longest to be served first. Other disciplines, e.g.~" S.I.R.O. = service’ in random order or
L.I.F.O. = last in first out, are also possible. The latter is often used in warehouses when
dealing with stack-s of inanimate objects piled on top of each other where it is easier to use
the top one than to turn the stack upside down to reach the one at the bottom. This
discipline is unlikely to be relevant in shipping. Reneging (leaving the queue before being
served) or jockeying (switching from one queue to another) are not allowed.
For simplicity of description a simple queueing system is described in the form:
M/M/N : L/SIAO
which means ,
Poisson arrival ' negative number of : space queue
pattern exponential channels limit discipline
I service
pattern
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Consider the simplest theoretical queueing model consisting of a single service point.
unlimited space for customers to queue, service given in arrival order. a Poisson arrival
pattern of customers with, on average, A arrivals per unit time, and service times which have
a negative exponential distribution. It follows that in an increment of time at
—7t5t r
P,(r arrivals in time 6t) = PZE-I-9-
1>,(0, at) = e‘“"
_ _ (-my (-tar)?_1+( u>t)+ + 3! +...
As 8t is small (8t)2 is very small so that
P,(0, 6t) = 1 — két and
P,(1, at) = e*’“‘“>.6t
= lt6t(1— ltét . . .)
= ltfit
Since P,(0, fit) + P,(1, 5t) = 1 — ltfit + ).6t = 1.
there will be either no arrival or one arrival during the period 6t.
It is usual to consider queues only after the system has settled down. and various formulae
have been derived to describe some of the standard average characteristics of such stable
systems. These formulae will differ depending on variations in the type of system. and their
use is restricted to the type of system specified.
If the mean arrival rate is denoted by it and the mean service rate by u then the traffic
intensity. given by p, is defined as
_ it meangarrival rate
p T u mean service rate
A necessary condition for the system to have settled down is that p < 1 or the mean arrival
time is less than the mean service rate.
It follows that the average service time will be 1/u and. further. that the average time in the
system will be equal to the average time in the queue plus the average service time. The
utilisation factor of the service channel is given by:
the average number entering per unittime g_
the average number able to be served per unit time
A queue may form whenever the number of arrivals exceeds the number of available service
channels (c). In a single channel system the probability of no queue in a given time period is
given by P,(0 arrivals) + P,(1 arrival); if either of those two events occurs no queue will be
formed as the number of service channels is sufficient to serve the number of arrivals but a
queue could form if there were two or more arrivals in a particular time period. It follows
that
P,(a queue)= 1 — P,(no queue)
= 1 — P,(0 arrivals) — P,(1 arrival)
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The following standard results apply.
Average number of customers in the system
Average number of customers in the queue
Average waiting time in the queue
Average waiting time in the system
Probability of no units in the system = P,(0)
Probability of n units in the system = P,(n)
Single channel;
infinite capacity;
FIFO system
L.
1 - 0
.2’_
1 - 0
__P_ =L
Hq“P) Mu-M
|.t - ll.
1-P
l>"(1 - P)
The following table is useful when verifying some of the above expressions.
Number in Number in Number in
system(x) service (y) queue (q)
0 0 0
1 1
hula I-0!--4' I-Jr-Cl
n 1 n;1
2 . PThe average number in the system == (3%!
Eu-P) =0+o(1-o)+2p=(1-o)+3P’(1-l>)+- - -+np“(1-P)+
=p-p1+2p_3-2p1"+3p3—3p‘+. ..
I p+p3i-l-p3+p"+. ..
I YET from the formula for the sum of an infinite geometric progression given 111
Chapter3.
2P'(1‘P)'*P(1"P)+P2(1"P)+---
=(1-p)l1+n+|=>’+p’+---1-<1-»><l-:-.>-1
Hence the average number in the system I 1-E—p
Probability
(P)
1- P
l-‘$1 - 0)
P"(1" 0)
P"(1- P)
t>"(1 - 0)
KONSTANTINOS
Rectangle
Z -. PSimilarly the average number in service = ~€%) == E(y . P). Since ZP = 1
i.e. p(1-p)+p2(1—p)+p3(1—p)+...
=p-0’+p’-p’+0’-p‘+---
= P
The average number in the queue = average number in the system - average number in
service
0 , _p-i>(1—o)
1-P pi T1“P K
P-_9j¢i>’ W P’
D 1—p T1—p
A customer who has just arrived will on average find 1%-6 customers in the system, each
. . . . 1 . . 1requiring an average service time of —. Therefore the new arrival can expect to wait ll?M _
time for his service to begin.
1The average time in the queue = -—p— . —
1 - P 11
I 1/ll 1
(1- l~/l1)- ll u(v — K)
The average time in the system = average time in the queue + average service time
it 1
=-—i-—- +-
u(u—7\) ll
_ 1 + Pf l~ _- _
u(v-1) u(v-llwu-1
For several channels in parallel the formulae become:
c channels in parallel;
infinite capacity; SIAO system
itTraffic intensity p = E
. . . . ¢!(1 - P)Probability of no units in the system = P,(0) -—--+1
(P<=)‘ + ¢!(1- P) Z10 5-, (p<=)"
Average number in system E2--3 . P,(0) + pc¢!(1- P)
Average number in queue lg-CL: . P,(0)
<=!(1 - P)
. . (P<=)°Avera e time in ueue -‘—-— . P, 0g q c!(1—p)2.cp. ()
. . (pc) 1Avera etimeins st —-—-+-.P, 0 + -g y em °!(1“P)“-C11 ( ) ll
KONSTANTINOS
Rectangle
l'l
Probability of n units in the system = P,(n) 92+) . P,(0) for n = 1, 2, . . ., c
or %;p".P,(0) forn=-'c+1,c+2,...,==
Clearly the value of P,(0) is essential if these formulae are to be evaluated. If c = 1 these
expressions reduce to those given earlier for a single channel system.
For a single channel and a limited capacity system the formulae become:
Single channel; '
limited capacity; FIFO system
Limited capacity of system L
. p_(L+1)pL+1+LpL+2
A b t N "cc 6 canveragenum erinsys em( ) (1_p)(1_pL*,)
1 _Probability of no units in system = P,(0) I-_TL9;;
Average number in queue N — 1 + P,(0)
N
Average waiting time in queue I
_ _ N + 1
Average time in the system —--—
)-lb
:1
/-x‘;Q1!
Probability of n units in the system --1-51+‘) for n =5 L
It is to be stressed that these formulae only apply to the systems described and under the
assumption of Poisson arrival patterns and a negative exponential service pattern. Problems
involving circumstances outside these assumptions are best solved by simulation techniques
and theuse of computers.
Examples
1. For a simple queue, what is the average time a customer is in-the queue if the mean
service rate is 12 per day and the traffic intensity is (a) 0.3 and (b) 0.9? What is the
average number of customers in the system corresponding to each traffic intensity?
_P_ Lxl
P 1-P 1-0 1-P ll
0.3 0.7 0.4286 0.86 hours
0.9 0.1 9.0000 18.00 hours
. 1 1where p. = 12 per day and E = E days or 2 hours.
The average number of customers in the system is given by 1—p—- and the average time a
- P
1customer is in the queue by iii; x E
2. A port has three berths available for the handling of general cargo, each offering similar
facilities and capable of providing, on average, a service time of two days. On average
KONSTANTINOS
Rectangle
one ship will arrive each day and there is unlimited room in the anchorage for vessels to
wait should a berth not be available. Service will be given according to arrival order.
(i) betermine the average time a vessel will spend in the port, and the average time
spent in queueing.
(ii) Find the probability of a queue and its average length.
Assuming a Poisson arrival pattern with a mean of 1 per day and service times which
have a negative exponential distribution with a mean service time of two days then this
queueing system can be summarised as M/M/3:==/SIAO
(i) The average time a vessel will spend in the port is given by
° 7t
 x wherep=_._...
<=!(1 - 0) -cu u cu
l 1-- .
and Pl_(0) =
(per + cm - 0) Q05; (per
it 1 1
Hence p - 6.1- 3&5) F -1-5 ~ 0.6667
3! (1 - 0.6667) Q
and P,(0) = 1
(0.6667 X 3)?’ + 3!(1 — 0.6667) 2 —' (0.6667 X 3)“
|'|"-:0 H.
_ 6 >< 0.3333
. 1
8 + 6(0.3333)[1+1.2‘ +5. 22]
- " 2 - 2 0 111108+2[1+2+2]S18" '
and the average time a vessel will spend in port becomes
(0.6661><3)* _i_
6x0.11ll x1.5 xmm + 0.5
= 0.8889 + 2 = 2.8889 days.
The average time the vessel spends in the queue will be the average time in the port less the
average service time
2.8889 — i = 0.8889 days.
(ii) The probability of a queue = 1 — P, (no queue)
There are 3 berths available so a queue will only form when all 3 are occupied. That is to
say, there will not be a queue if there are 0, 1, 2 or 3 vessels in the system.
Hence P, (no queue) = P,(0) + P,(1) + P,(2) + P,(3)
P,(0 vessels in the system) = 0.1111 from previous answer
P,(n vessels in the system) = £%)—- X P,(0) for n = 1,2, . . . c
0.6667 3 1
Thus P,(1 vessel in the system) = >< 0.1111 = 0.2222
0
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
0.6667 3 3P,(2 vessels in the system) = (ii.-Q X 0.1111 = 0.2222
0.6667 3 3P,(3 vessels in the system) = (——5-‘LL X 0.1111 = 0.1481
The probability of a queue = 1 — (0.11l1 + 0.2222 + 0.2222 + 0.1481)
= 1 — 0.7036
= 0.2964
The average length of the queue is given by
p(l><=)° _ (06661) - <2)’E:'(T_-6)? X - X ‘
5.3333
— 56?“; X
= 0.9 ships i.e. 1 ship in round figures.
State any assumptions which have to be made before considering a queueing system
having only one service channel when the system capacity is limited to a maximum of five
customers.
If in such a system an average of seven customers arrive each hour with each requiring an
average service of 6 minutes, determine
(i) the average number in the system at any given time
(ii) the average time that a customer can expect to be in the system and
(iii) the utilisation factor of the system.
Assumptions
Arrivals: negative exponential inter-arrival pattern
Services: negative exponential service pattern
Layout: single service channel
Discipline: service in arrival order
Capacity: limit of 5 customers
it arrivals on average per unit time
ii services on average per unit time
_ _ it 7 per ho'urTraffic intensity p - p - my-éwl-1;
This system can be described as M/M/1: 5/SIAO and it follows that P,(n in
queue) = P,(n + 1 in system) for n < 5
ll 1 _,
P,(n in system) = £;%5,+? for n s 5
and P,(n + 1) = p X P,(n)
s
It is knowii that 2 P,(n) = 1 hence
ii=0
P,(0) + P,(1) + P, (2) + P,.(3) + P,(4) + P,(5) = 1
.'.P,(0) + pP,(0) + p2P,(0) + p3P,(0) + p‘P,(0) + p5P,(0) = 1
Pi(0)[1+t>+p2+o’ +0‘ +i>’l=1
and when p = 0.7, P,(0) = 0.34
Number in system Number in queue
0 0
1 0
Probability
0.34
0.7 X 0.34
0.72 X 0.34
0.73 x 0.34
0.7‘ X 0.34
0.75 X 0.34U1-Pub-Il\J
l—l
Jib-ll-J
(i) Average number in system
= 0.34 [0 + (1 >< 0.7) + (2 >< 0.71) + (3 >< 0.73) + (4 >< 0.74) + (5 x 0.75)]
= 1.533
Average number in the queue
= 0.34 [0 + 0 +(1>< 0.71’) + (2 >< 0.73) + (3 >< 0.7‘) + (4 >< 0.75)]
= 0.873
1 1
(ii) Average time in the system I-I X Ave. no. in system + E
= 0.2533 hours = 15.2 minutes
(iii) Probability (system full) = p5 . P,(0) = 0.0571
Average number entering per unit time
Utilisation factor 7 is . ~ " -- FAverage number serviceable per unit time
7(1 - 0.0571) _*-2 10 A _ 0.66
4. A port imports on average 2,000,000 tons of ore per annum which involves using either
(a) 25,000 dwt vessels with an average discharging rate of 400 tons per hour
or (b) 200,000 dwt vessels with an average discharging rate of 800 tons per hour.
The vessels can only be unloaded at one particular berth but there is unlimited queueing
space in the estuary. Vessels will be serviced in arrival order and the queueing system
satisfies the necessaryconditions of a Poisson arrival pattern and negative exponentially
distributed service times. Thus the system can be described as M/M/1: =<=/SIAO or
equivalently in this case M/M/1: =0/FIFO.
From this information, assuming 360 working days per annum,
Mean service rate (ll) for (a) = 138.24 per year
(b) = 34.56 per year
Mean arrival rate (it) for (a) = 80 per year
(b) = 10 per year
so that
1
Average waiting time (including discharge) = E
1
for (a) = — 0.0172 yrs or 148.35 hours
1
for (b) = = 0.0407 yrs or 351.79 hours
Average waiting time in queue = -‘-IE‘-ff‘) 3
LIBRARY ,
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
80
for (a) = 0.0099 yrs or 85.85 hours
10
for (b) HEW = 0.0118 yrs or 101.79 hours
Clearly these results could be obtained by subtracting the average service time from the
average time spent in the system
for (a) 148.35 — 62.5 = 85.85 hours
for (b) 351.79 — 250.0 = 101.79 hours
Total waiting time per year
for (a) 85.85 X 80 = 6868 hours or 286.2 days
for (b) 101.79 X 10 = 1018 hours or 42.4 days
If the cost per day of each vessel were known then the costs of ships’ time awaiting berths
(i.e. queueing) could be calculated.
KONSTANTINOS
Rectangle
T
 _
Chapter 16
The Theory and Practice of
Index Numbers
Index numbers are used to measure how much something has changed from one period to
another, or to compare one thing with something else. Spiegelm has defined an index
number as “a statistical measure designed to show changes in a variable or group of related
variables with respect to time, geographic location or other characteristics such as income.
profession etc”. Index numbers were originally developed for prices but they have now been
extended to cover quantities (volumes) and values over a whole range of items such as
shipping freight rates, commodity prices, levels of industrial production, death rates, and
profits. In effect an index number is a number (say 106) which shows how mu.ch a variable
has changed by comparison with its base value (which always equals 1.00).. Allen”) provides
the classical definition of an index number as given originally by Edgeworth who stated that
an index number “shows by its variations the changes in a magnitude which is not
susceptible either of accurate measurement in itself or of direct valuation in practice”.
This definition provides a clue to some of the difficulties encountered with index numbers —
very often they are seeking to measure changes in some average value of a particular
variable (say ocean freight rates) over time but that particular variable may not be easy to
define or measure. There is no such thing as a standard ocean freight rate. "It is easy to
appreciate that a change in an index number from 100 to 120 represents a 20% increase in
the value of the variable measured by that index but if the variable was ‘freight rates’ this
does not mean that every freight rate has risen by 20%. It means rather that on. average the
change in those freight rates used to compile the index number has been 20%. The issue is
further complicated when comparing price index numbers over time as inflation is a process
which ensures that the value of the currency is itself changing over time (though on
occasions this may be what is being measured). In other words the yardstick is not constant.
L1
The simplest form of index number is the price relative which compares the price of a single
commodity in a particular period with its price in the base period. Such a price relative is
usually written as pg)“ = % X 100 where p,, is the price in the particular period and po is the
0
price in the base period. The purpose of index numbers is to focus attention on relative
changes and a price relative does exactly that; but other index numbers are not so
straightforward and the choice of a base year becomes an important step in the construction
of an index. Ideally a base year should be one containing a typical — i.e. normal — result
and should therefore be a past time period not too far removed from the present. If no one
year suggests itself as a base then it is possible to use an average of several years‘ results to
form a base figure of 100 against which all other results will be measured. This average
value could well not occur as actual data so that the index number series may not contain
KONSTANTINOS
Rectangle
the value of 100. For example
Year Data Index Number
1976 75 88
1977 81 95
1978 83 98
1979 79 93
1980 80 94
1981 81 95
1982 86 101
1983 88 104
1984 90 106
1985 91 107
If none of the individual annual observations are deemed suitable as base years it might still
be possible to construct an index number based on the average result for the period
1981-1983. The average value of data during this period is 85 and the index numbers can be
calculated using 85 as a base. It will be seen that the index number 100 does not appear in
the series. This choice of base year is very important because if too high a figure is chosen
to represent the base year then this would result in the whole index series being chronically
depressed. The importance of this is obvious in shipping when the impact of major events
such as closure of the Suez Canal or a world war or'a crop failure are considered.
Usually index numbers are not concerned with how the price (or quantity or value) of one
particular commodity has changed over a period; in practice they often relate to changes in
price (or quantity or value) of large groups of commodities. This introduces the second
problem of what amounts of each commodity to include in the index and how to assess the
quality of the goods being included. The phrase “a basket of goods" is often used in this
context and the essential point about the basket is that its contents should differ as little as
possible from period to period; (e.g. note the use of the basket of currencies when
comparing the purchasing power of the £ sterling in terms of other European currencies.)
The procedure is to weight the price of each commodity by a suitable factor, such as the
quantity of the commodity sold during a typical year, to indicate the importance of the
particular commodity.
The second problem, that of the changing quality of goods, is usually countered by including
only staple commodity prices in measurements of price levels over long periods of time,
provided such staple components can be identified, and by using the link-chain procedure to
coverthe changeover from one index series to another based on different baskets of goods.
The linking procedure involves constructing an index (the link index) with the immediately
preceding period as base. The link index numbers may be chained back to a common base
period by multiplication. The chaining procedure involves multiplying together numbers
which are related to each other.
The meaning of a chain index becomes less clear as time progresses and as more old
commodities are replaced by newer ones. The most valid comparisons are between index
numbers close to the fixed base.
\
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Two commonly used types of index are based on either the Laspeyres base-year method or
the Paasche current-year method. These index numbers can be summarised as:
Type of Index Price Index Quantity Index
Z .1 73 ..Laspeyres index -B-(19 x 100 --£2-q— x 100
EPBQO "zpoqo
2 I1 I1 ll I1Paasche index i x 100 ml x 100Z1160. Zi>..q6
The Laspeyres index is a weighted aggregate price (or quantity) index with base-year
quantity (or price) weights and gives the cost of maintaining the base-year rate of
consumption at prices for a given year compared with the base-year cost.
The Paasche index is a weighted aggregate price (or quantity) index with given year quantity
or rice wei hts and com ares the cost of consumin iven- ear uantities at iven- earP _ B _P 8 8 Y q 8 Y
prices with that of consuming the same goods at prices which prevailed in the base year.
Example
Price Quantity
Commodity 1980 1983 1985 19801983 1985
x 24 26 32 100 85 80
y 18 16 10 40 45 50
z 28 32 40 60 62 70
Letting 1980 be the base year
E
Laspeyres’ priceindex number for 1985 is given by- x 100
2pl980q I980
((32g>g< 100) + (10 >< 40) + (40 >< 60)) X 100
*5 ((24 >< i00)2f(1s”><“ 40) + (2s >< 60))
-Warm-125“4,s00 "
Z
Paasche's price index number for 1985 is given by -E29-3% X 100
zp1980q198S
I XW80) + (10 >< 50) + >_<_ 70)) X 100
((24 x 80) + (18 >< 50) + (28 X 70))
5,860= mfi x 100 = 122.56 or 123 to the nearest whole number.
Letting 1985 be the base year
Z
Laspeyres‘ price index for 1980 becomes §= 81.57
21) was q 1985
E
and Paasche’s price index for 1980 becomes ii-0-9-1?-82 = 80.00
EP 198$q1980
It is desirable that index numbers should meet certain technical requirements, the first of
which is that they should satisfy the Time Reversal Test. This states that if two periods are
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
interchanged the corresponding results should be reciprocals of each other. Letting 10,, be an
index number of year n based on year 0, and Ina be an index number for year 0 based on
year n, then the Time Reversal Test requires that 10,, X Inn = 1. It can be seen from the
example given above that neither the Laspeyres nor the Paasche index numbers satisfy this
test. The second consideration is whether the index number satisfies the Factor Reversal Test
which requires that the product of a price index and the quantity index should equal the
corresponding value index. Very few index numbers satisfy this test.
The Circular Test is an extension of the Time Reversal Test and requires that
10,, x Im x Imo = 1 or equivalently that 10,, x Im = 10",. Again neither the Laspeyres nor the
Paasche index numbers satisfy this test. The reader can show that in both cases
Isozss - Isa/ss - Iss,/so 7‘ 1
Various attempts have been made to design index numbers which would satisfy these tests
and the one which comes closest is the Fisher Ideal Index which satisfies the time reversal
and factor reversal tests but fails the circular test. This ideal index is the geometric mean of
the Laspeyres and Paasche index numbers but while it satisfies two out of the three
mathematical tests, its meaning is unclear.
However this Ideal Index may be defined as
7-; " ‘ 7 7
IF: X X
2P0q0 zp0qn
This brief introduction to index numbers has tried to explain the main methods of
calculation and desirable properties of a good index number. There are many problems with
real-life calculations of weighted aggregate index numbers which do not occur in the
calculation of simple aggregate index numbers or simple price relatives. The choice of base
period has already been discussed but, in addition, the person constructing the index would
need to have a knowledge of the data and use his or her judgement to determine which data
would form the basis of the calculation. There are problems too with baskets of goods not
being identical from year to year and also with the accuracy of the formula used to calculate
the index. The one most commonly used is probably the Laspeyres formula but a better
one, from the point of view of meeting the mathematical criteria, would be Fisher’s Ideal
Index formula. The snag here is that this is difficult to interpret as to its exact meaning and
it involves considerably more computation. Two further points which need consideration are
how to deflate an index series and how to change the base year.
In times of inflation the money value of items will rise, but a time comparison should be
based on the real value of goods and not the inflated money value (unless it is the rate of
inflation itself which is being measured); in other words the comparison must be based on
the value of the items assuming that the value of money has remained constant. This is
achieved by deflating (dividing) the index measured in current prices by an appropriate price
index. Suppose that an exporting company has sales valued at £3.7 million in 1985 and the
index for prices of the same company in that year is 80, based on 1980 = 100, then the value
. 3,700,000 . .
of sales for 1985 at constant prices = ——8T)—- X 100 = 4,625,000 which means that if the
selling prices in 1985 had been the same as those in 1980 the sales volume in 1985 would
have been £4.625 million.
The base of an index number series may need to be updated as the series grows over time.
This may be achieved by dividing each of the original index numbers by the index value for
the new base year selected. If it is desired to change the base year in the following example
from 1980 to 1985 the index numbers in the original series should be multiplied by
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Year Index (1980 = 100) Index (1985 = 100)
1980 100 87
1981 103 90
1982 106 92
1983 110 96
1984 112 97
1985 115 100
This same procedure may be used to splice two sets of index numbers.
Freight Index Numbers
Development of Dry Cargo and Tanker Voyage Index Numbers
NORWEGIAN SHIPPING NEWS DRY
CARGO VOYAGE INDEX: 1 965 -1966 =100
300 --
‘. ______ NORWEGIAN SHIPPING NEWS onv cmoo
7"‘ INDEX DEFLATED AT CONSTANT 1970 PRICES
. I 'l
26° -i __ . . TANKER VOYAGE INDEX EXPRESSED IN POINTS
4 IN I -—-—--—-— WOFILDSCALE. N.S.N TO 1974:GEFiMAN TANKER
280
246 I I l INDEX FF1OMi974
220- l‘ I l
I
200-) ll‘
. i l l
180-4 ' I l l. . 4I’ l _l/'0.‘ 7-, \/
I’ l14 _i - l . . - . ,oJ/ I L! \‘\\\
120 A \ l /-, \(\ \
‘°°‘7/‘$(l'l\_ ‘J: 5 \.\~\ ,--"l"-.\ 7
\\\\ \ ‘T I/Z l \\so — . - A """" .I x. .\,./\.\,'/“.J \\\
60 -5 \- ‘ ‘---4 -=- '\ ------/s
_,_-I-—--1..--.-_
\‘ _-'/,#n.~\._“/.A_\'/
40 E
l I l l _ l ""1 l ""1 l l l 1" l I I I l _l' ' 'l_
1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 ‘I981 1982 1983 1984 1935 1986 1987 1988
Source : OECD Maritime Transport Paris I989
Many different freight» index numbers have been compiled and published in trade journals
from time to time. Some of these have been explained in an UNCTAD publication, “Freight
Markets and the Level and Structure of Freight Rates”‘3l.
Index numbers relating to the liner market give rise to the greatest difficulties of calculation
as a result of the large numbers of cargoes carried under very diverse rate structures. It is
not possible to produce a rate index simply from information contained in the tariff since
this gives no indication of weights to be apportioned to the different cargoes, and. of course.
many of the rates are so-called ‘paper‘ rates relating to cargoes that may not be carried at
all. Consequently only one index for liner rates has been published regularly. This is the
German Liner Index that takes account only of liner cargoes moving through German ports.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
In the tramp market freight index numbers are published in respect of
(i) Voyage Charters
(ii) Trip Charters
(iii) Time Charters
(iv) Tanker Voyage and Time Charters expressed in points Worldscale.
The function and derivation of Worldscale is dealt with on page 101.
Chamber of Shipping of the United Kingdom Tramp Freight Index
The basis of calculation of tramp freight index numbers was reviewed by a special committee
of the Chamber of Shipping (COS) in 1952. They pointed out that the index number
provides a measure of changes in tramp shipping freights; it does not measure directly the
level of earnings. still less does it measure profit or loss“).
The need for periodic revision arises from changes not only in the relative importance of
tramp trades. (“trade" meaning the carriage of a specific commodity on a specific route) but
also in charter terms and in methods of loading and discharge which affect the freight rates
and invalidate a straightforward comparison of freight rates in one period with those in
another.
It follows that a great deal of specific information is required before a useful index can be
compiled. In 1958 the General Council of British Shipping(GCBS) carried out an inquiry
into the contribution made by the shipping industry to the UK balance of payments and the
information gathered afforded the opportunity for instigating a revised index.
“The criteria adopted in determining whether or not a particular trade should appear in the
calculations were (i) that the freight earned by United Kingdom tramp ships in that trade in
I958 was large enough to make the trade sufficiently important for inclusion and (ii) that
reported fixtures of ships in that trade appear with reasonable regularity.“(5l The use of
logarithmic instead of arithmetic averages was considered. but experiments showed that
because tramp freight rates in the various trades tended to move together. very little
difference in the results was discernible.
'\
The revised index number used the pattern of trade as shown by the invisible exports
inquiry and with the average freights in 1960 taken as 100. Because the level of freights was
considerably lower in 1960 than in 1952 an approximate comparison of the new series with
the old could be made by taking 74.2% of the figure on the new basis.
The index was based on the reported freights of seven main commodities, viz: coal, grain.
sugar, ore. fertilisers. timber and sulphur. Information supplied by British tramp owners
showed that more than 93 per cent of the total freight on voyage charter was earned in the
carriage of these commodities. Weights used were based upon total eamings which took into
account not only the total quantity of goods but distance carried as well. Table 16.1 shows
the commodities and their respective weights. indicating that under the new basis (1960) as
well as the old (1952), grain was the single most important commodity for freight earnings,
having a weight of approximately one-third of the total.
3
TABLE 16.1
Tramp Shipping Freights
Commodities and Weights
Commodity
Weight
Old basis New basis
Coal
Grain
Sugar
Ore
Fertilisers
Timber
Sulphur
Esparto
183
5 362
116
136
40
143
20
125
316
150
112
132
135
30
Total
T _ _._
l 1,000 1 ,000
For each commodity one or more representative routes for the main commodity movements
were selected according to their importance as revenue earners, appropriate weights being
allocated as shown in Table 16.2.
TABLE 16.2
Trade Routes and Weights
Main movement Representative route Weight
Coal
1. UK/Portugal, Spain,
Mediterranean
2. Poland/River Plate
3. USA Northern Range/Japan
4. USA Northern Range/NW
Europe (France-Norway)
UK (Bristol Channel)/West
Italy
Poland/River Plate
USA Hampton Roads/Japan
USA Hampton Roads/
Continent (Antwerp-
Rotterdam)
46
123
329
502
Total 1,000
Grain
1. Australia/UK, Eire, Continent
2. Eastern Canada/UK, Eire
3. North Pacific/UK
4. River Plate/UK, Eire Continent
5. River Plate/Mediterranean
6. USA Gulf/UK Continent
7. USA Gulf/India
West Australia/UK
St Lawrence/UK
North Pacific/UK
River Plate/Continent
(Antwerp—Hamburg)
River Plate/West Italy
us Gulf/UK
US Gulf/West India
123
185
73
371
29
147
72
Total 1 ,000
' 771
KONSTANTINOS
Rectangle
TABLE 16.2 (continued)
Trade Routes and Weights
_ Main movement Representative route Weight
Sugar
1. Queensland/UK
2. Mauritius, South Africa/UK
3. Cuba & San Domingo/Japan
4. Cuba/UK, Continent. Eire
Queensland/UK
Mauritius or Durban/UK
Cuba/Japan
Cuba/UK
204
332
295
169
Total 1.000
Ore
l. Brazil/Poland
2. Mormugao/Continent
3. Mormugaoflapan
4. Cyprus/Netherlands (Pyrites)
Brazil/Poland
Mormugao/Continent
(Antwerp—Hamburg)
Mormugao/Japan
Cyprus/Netherlands
461
461
29
49
Total 1,000
Fertilizers
1. Casablanca/Belgium.
Netherlands
2. Belgium. Netherlands. China
3. Italy/China
4. Casablanca/China
Casablanca/Belgium.
Netherlands
Antwerp-Hamburg/S China
East Italy/S China
Casablanca/S China
325
.478
96
101
Total 1.000
Timber
1. Canada and USA, N Pacific/UK North Pacific/UK
2. Finland and Sweden/UK
3. Russia/UK
Lower Zone/E Coast UK
Archangel/E Coast UK
839
139
22
Total 1,000
Sulphur
1. Gulf/UK Gulf/UK 1,000
Method of Calculation“)
(a) For each trade route, the arithmetic mean of the freight rates for fixtures reported in
each month is calculated. The arithmetic mean for the year 1960 of these monthly
averages forms the basis of comparison of freight rates in that trade route.
(b) To each commodity is attached an index number which is the weighted arithmetic mean
of the price-relatives for the trade routes in which fixtures were reported in the month
The weights used are set out in Table 16.2.
(c) The index number of tramp shipping freights is the weighted arithmetic mean of the
commodity indices for the month. The weights used are set out in Table 16.1
Index Number of Tramp Time Charter Rates (1960 = 100)
At the same time as the Voyage Index Number was revised, the COS also revised its Time
Charter Index to base-year 1960. This index was published monthly and was based on a
simple average of Time Charter rates compared with the average for 1960. The criteria for
including reported fixtures were:
(i) Motor ships (i.e. steam vessels excluded)
(ii) Dwt > 9,000
(iii) Round voyages or time charters < 9 months
(iv) Exceptionally fast vessels excluded (>15kts)
(v) Ports of delivery and re-delivery should not be widely separated.
In 1969, as a result of insufficient reported fixtures, the COS discontinued the Voyage Index
as being no longer representative of the market. In 1970 the time Charter Index was revised
to take account of different ship sizes so that separate calculations were made for ships:
(a) of 9,000—16,000 dwt
(b) of 20,000—40,000 dwt
(c) of 40,000 dwt and over
Fixtures for vessels of 16,000—20,000 dwt were not included on the grounds that they_did not
form a homogeneous group. At the same time the upper speed limit was raised to 16 kts and
the maximum length of charters for fixtures used in the calculations was raised to 24 months.
For the combined index number a system of weighting based on the number of reported
fixtures in each group was introduced. “Because of the differences in the average size of
ships in each of the three groups, it seemed appropriate to multiply the number of fixtures
in the 20-to-40,000 dwt group by 2 and of ships in the over 40,000 dwt group by 3 to arrive
at the actual weights used“.l"’)
Calculations for this index number (1968 = 100) were made quarterly rather than monthly as
they had been hitherto.
In 1975 the Chamber of Shipping of the UK and the British Shipping Federation merged to
become the General Council of British Shipping (GCBS) which took over the responsibility
of producing the Time Charter Freight Index Numbers.
In 1977 the GCBS introduced a further revision of the Time Charter Index and also
introduced a Trip Charter Index (Base Year 1976 = 100) to replace the Voyage Freight
Index discontinued in 1969.“) For various reasons, the Trip Charter (i.e. a Time Charter for
a voyage from A to B) has become very popular, and, while there may be some
disadvantages attached to this type of fixture with regard to the calculation of an index
number, there are, on the other hand, many advantages. These advantages are primarily
that details of fixtures reported in the Daily Freight Register provide homogeneous data in
the form of rate of hire per specified period of time: thus information on voyage and
commodity carried is unnecessary for the purpose of weights.
The method of compiling the Tramp Trip Charter Index Number and that of the Time
Charter is essentially the same.
(a) There are five size ranges:
12,000—19,999 dwt (small bulk carrier)
20,000—34,999 dwt (handy size)
35,000—49,999 dwt (medium bulk carrier)
50,000-84,999 dwt (Panamax size)
85,000'dwt and over (large bulk carrier)
KONSTANTINOS
Rectangle
(b) Specialized ships are excluded; also spot charters, charterers‘ taxes, charterers‘
lightening, ships trading on the Great Lakes and fixtures where the quoted price includes
overtime.
(c) Published information shows:
(i) Combined Index Numbers
(ii) Group Index Numbers
(iii) Weighted Average Price, $/dwt/month(iv) Number of Fixtures
(v) Average Size of Ship
(d) The Combined Index Number. One figure is shown each month for Trip Charters, and
quarterly for Time Charters in which the data in the size ranges are combined to give an
overall index. This index is a Paasche-type defined as:
E cm
21-—— x 100
Z am
i=I
where Ci = weighted average price ($/dwt/month) in the current period for size range i
Bi = weighted average price ($/dwt/month) in base year for size range i
Ti = Total tonnage in size range i in current month.
n = number of size ranges.
Essentially, the above formula provides an index which compares the current price of the
tonnage fixed in the current period with the price which the same quantity of shipping would
have cost at the average prices in the base year.
(e) Weighted average prices for each size range in the base year 1976 and used in the
calculations are as follows:
Trip Charter Tim
$7 .01
$4.76
$3 .38
12,000-19,999 dwt
20,000-34,999 dwt
35,000—49,999 dwt
50,000-84,999 dwt $2.47
85,000 dwt and over $1.50
e Cha
$6.03
$4.37
$3.00
$2.28
$1.15
TICK
Time Charters for periods exceeding 24 months are excluded and for the Time Charter
index, Trip Charters are excluded. The Time Charter Index continues to be published
quarterly while the Trip Charter Index is published monthly.‘
An example, using hypothetical figures, of a Time Charter Index Number for the 3rd
quarter of 1985 is given below. The procedure is that followed by the GCBS.
"' The GCBS has announced (March 1990) that it will no longer be publishing the time and trip charter
index numbers. It is hoped that these index numbers will however continue to be produced on the same
basis by a commercial organisation.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Calculations for Time Charter Index for 3rd Quarter 1985 m
accordance with GCBS procedure
Base Year 1976 = 100
Group (1) 12,000-19,999 dwt
(2) 20,000-34,999 dwt
(3) 35,000—49,999 dwt
(4) 50,000—84,999 dwt
(5) 85,000 dwt and over.
The following data are given for illustrative purposes only and do not represent actual
fixtures. The numbers of reported fixtures are also kept small for computatlonal expedlencv
Group I
A B
DWT Rate
. 12_0C $7.60
13,01 $7.00
. 18,0C $6.5I
16,5012 $8, 1|
. 19.51 $7.16
. 13, $8.41.
. 18.0ID£ $6.86
. 16.800 $6.5IIToo-.1o\u1_4=-w§u»-
U1 QQ=<3_
_¢_2r_1_LJc1f3
_a_1_c_1@
Z = 127.300
00 E A B
Wt. Av. Rate Group (1) = fiégo-B = $7.20 per dwt/month ( X )
1' = --Z =G 0 I d - 7'20 >< 100 119Up 1'] CX 6
Number of Fixtures = 8
Average Size of Ship = 15,913 dwt.
E = 916.300
1977 Basic Rate
AXB
91 ,200
91.000
117,000
133 .65!
138,451
113,401
122,400
109.200
1C41(41Q_
Ii — - _ i 9 —i 1 G 1 _
$
6.03
4.37
3.00
2.28
1.15
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Group 2
A B
DWI‘
31 ,000
28,000
. 22,001
. 23,001
. 25,5011
. 32,801.
. 29,50C
. 21 ,250
. 23,5613
. 24,0011‘&5@m-..|@ot4=.1.=;6N- _f___1_C*J(_1
Z = 260,610
1,224,959
Wt. Av. Rate Group 2 = - $4.70 per dwt/month
4.7
Rate
$4.20
$4.68
$5 .20
$5.30
$4.90
$5.15
$4.60
$4.50
$4.40
$4.20
A >_< B
130,200
128,800
114,400
121,900
124,950
168,920
135,700
95,625
103,664
100,800
Z = 1,224,959
0
Group Index = —:F X 100 = 1084.
Number of Fixtures = 10
Average Size of Ship = 26,061 dwt.
Group 3
A B
DWT
. 36,50
. 48,00
. 38,00
. 40,0011
.- 39,001
. 49,501
. 46,30
41 ,25'
. 40,351
38,5011
. 40,0011
47,601
2 = 505,000
§';-'E~;>m--.|o~u\4=-uor~..:»-- _,fi__¢i;c_1r_rrg_K:1FjC:1
Wt. Av. Rate Group 3 = —-—? = $3.41 per dwt month
K
Rate
$4.011
$3.011
$3.811
$3.511
$3.311
$3.00
$3.311
$3.50
$4.00
$3.60
$3.30
$3.00
A><B
146,000
144,000
144,400
140,001
128,700
148,500
152,790
144,375
161 ,40C
138,600
132,00C
142,801
Z=1 723,565
1,723,565
505,000
G '0 -3'41><100—114I'Ollp IH €X -- —
Number of fixtures = 12
Average Size of Ship = 42,083 dwt.
KONSTANTINOS
Rectangle
Group 4
Z = 531,000
A B
DWT
50,000
54,000
58,000
65,000
75,000
82,000
80,000
67,000
Rate
$3.20
$3.10
$3.00
$2.80
$2.60
$2.30
$2.35
$2.75
1.439.250Wt. Av. Rate Group 4 = —-5-lmob-0- = $2.71 per dwt/month
Group Index = X 100 = 119
Number of fixtures = 8
Average Size of Ship = 66.375 dwt.
Group 5
PP’!‘-""
A
DWT
120.000
. 90.000
98.000
105.000
E = 413,000
WtA Rt G 5—526‘150-$127 61/ m. v. ae __roup —4l3_000— ... per w mon
1.27
Gr0upIndex=mX100=110
-nu-_|-A-,.
B
Rate
$1.30
$1.20
$1.55
$1.05
Number of fixtures = 4
Average Size of Ship == 103,250 dwt.
Combined Index No
= [(127,300 X 7.2) + (260,6l0 X 4.7) + ($05,000 X 3 41)
+ (531,000 X 2.71) + (4l3,000 X 1 27)] X 100
+ [(121300 >< 6.03) + (260,610 >< 4.37) + (505 000 >< 3 00)
+ (531,000 >< 2.28) + (413 000 >< 1 15)]
- 5’812‘947 >< 100 - 1141 5,107,115 '
A><B
168 ,0‘
167 ,4‘
174,01
182 ,000
195,000
188,600
188,000
184.250
C-J
88¢
ii-_-i
Z = 1,439.250
AXB
156.000
108.000
151.900
110.250
Z = 526.150
 i
KONSTANTINOS
Rectangle
Great importance appears to be attached to the availability of Freight Index Numbers by
shipowners, brokers and charterers alike. It is not altogether clear, however, what specific
use can be made of them. Some believe that trends can be extrapolated, albeit with some
caution. Certainly it is easier to detect definite movements in tramp freight rates with the aid
of Index Numbers. From an historical perspective they help to establish market patterns and
aid in the economic analysis. Economists have used Index Numbers to establish supply and
demand relationships and in many other ways using Multiple Regression Techniques (see
Chapter 10).
A graphical presentation of Index Numbers should be viewed with the following in mind:
(i) Index Numbers are statistical averages and no indication is given of deviations from
the mean.
(ii) Index Numbers are based upon current dollar charter rates. In periods of inflation the
Index Numbers will tend to rise regardless of the market conditions. Changes in fuel
costs and port costs are likely to influence rates, but their effects may be masked by
changes in supply/demand conditions. If Index Numbers are to be used for statistical
purposes a suitable ‘deflator' must be used.
(iii) Falling Index Numbers may indicate increasing efiiciency such as economics of scale,
more economic propulsion units, faster turnround in port or lower manning costs
associated with ‘Open Registries’. '
(iv) An Index Number of 100 has no special significance with regard to profitability. If
possible, the base year should be one where rates are relatively stable and are neither
excessively high nor excessively low.
Tanker Voyage Index Numbers
All tanker indexes are based upon Worldscale (see Chapter 8). They sufi'er the same general
drawbacks as do dry cargo indexes in that they are based upon US dollars in current terms.
Nevertheless, it is difficult, over a period of time, to compare rates expressed in units of
currency with those based upon Worldscale, since Worldscale is not fixed in money or ‘real’
terms. The value of Worldscale 100 for any given trade, although fixed for a calendar year at
a time, changes from one year to the next according to changes that have occurred in fuel
prices and port costs. Thus, market rates expressed in Worldscale only partly reflect changes
in the value of the dollar (or inflation); but they do reflect changes in the balance between
supply and demand.
The construction of a tanker voyage index is a relatively easy matter since most tanker
fixtures are reported in Worldscale, a fonn that might be compared with the dry cargo trip
charter. Whether an index that purports to cover the total tanker market is truly
representative for all sub-markets is however open to question.
‘P
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Chapter 17
Currency, Bunker and Inflation
Differential Factors
Liner freight rates. though to some extent subject to market forces. have historically tended
to be stable over many months and sometimes even years. In recent times however. rising
costs have meant that. through general rateincreases (GRIs). liner conferences have
increased rates rather more frequently. One may note however that the UN Code of
Conduct for Liner Conferences. which entered into force at the beginning of October
19831“. states in Art. 14 that, unless otherwise agreed. the period between the date when
one GRI comes into effect and the date of notice for the next GRI shall be at least 10
months; and furthermore the period of notice of a GRI should be not less than 150 days.
Increases in costs may be the result of increased capital costs. running costs. fuel. port or
cargo-handling costs; or they can arise through congestion. changes in currency exchange
rates or other factors beyond the control of the shipowner. While general rate increases are
applied or negotiated as a measure to cover costs that tend to rise more slowly. surcharges
are applied more immediately to offset serious losses that could arise from such
contingencies as port congestion. increased fuel costs. exchange rate variations or from such
events as the closure of the Suez Canal. Such surcharges are intended to be of a temporary
nature: where. however. t-he condition that causes the surcharge to be imposed attains a '
more permanent status. then it is normal for that surcharge to be incorporated. in time. into
the tariff structure.
Currency Adjustment Factor
When currencies are stable. rates can be set and paid in terms of any currency. provided
they are freely convertible. The fact that the tariff specifies rates in US$ or £ stg does not
mean that the rates must be paid in either of those currencies: in fact they are usually
(though not always) paid in the local currency of the country where the goods are loaded or
discharged. Sudden devaluations in currencies have in the last decade or so been replaced by
a system whereby most of the world‘s hard currencies have been allowed to float against
each other, the exchange rates being determined primarily by market forces. Since many
liner conferences are international in character, with the Far Eastern Freight Conference for
example comprising 34 lines from 22 different states. changing currency exchange rates could
cause serious losses to some. or possibly windfall gains. It has been found convenient to
express the tariff freight rates in a single currency and preferably a neutral one with respect
to the member lines. The US$ is now the most commonly used tariff currency. but this does
not imply that currency problems are eliminated. The effect of a currency devaluation with
respect to conference lines and shippers is demonstrated by a simple case outlined below.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Currency Devaluation Example
In this example a conference operating between the United States and the UK comprises
equal numbers of US and UK lines only, the tariff currency being in US$.
Before devaluation of the US dollar, the lines incur costs on the following basis: for every
$100 of freight, US lines spend the equivalent of $25 in UK ports, $25 in US ports. $25 for
fuel and $25 is retained for depreciation, interest and profits. The UK lines incur costs of
$25 in US ports. $25 for fuel, the equivalent of $25 in UK ports and retain a similar amount
for depreciation etc. At an exchange rate of $1.40 to the £ stg, $25 is equivalent to £17.86,
and $100 E £71.43.
If the dollar is devalued by 20%, the rate of exchange becomes $1.40 = £0.80, i.e.
$1.75 = £1.00.
The US lines now retain from each $100. $18.75, which means they have suffered a net loss
of $25 — $18.75 = $6.25 or 6.25%.
It would therefore require an increase of 6.67% to the tariff to return them to their previous
position.
25 1
The UK lines now retain £10.71 from each $100 (100 — 50 -- 1—4 X 1.75) X l—_7§ and they
are worse off by £17.86 — £10.71 = £7.15; this represents a loss of 10% (£7.15 in £71.43),
requiring an increase of 11.1% to the tariff rates to return them to their previous situation.
This simplistic case illustrates the fact that a large devaluation may result in a relatively
small increase in freight rates.
A currency adjustment factor could be calculated by first assessing the proportion of costs in
£ stg. viz: 3/8: this would be multiplied by the change in the value of sterling, viz: +25%.
Note that although the dollar has suffered a devaluation of 20%. the £ stg. has effectively
been revalued by 35/140 = 0.25 or 25%. Thus the CAF required to return the total of the
lines‘ earnings to their previous level is 25 X 3/8 = 9.375%. a result that would place the UK
lines in a slightly less favourable position while slightly improving that of the US lines.
The commercial risks resulting from currency fluctuations ‘have been the subject of extensive
consultation between the Council of European and Japanese National Shipowners’
Associations (CENSA) and the European Shippers Council (ESC) and are covered in their
joint recommendation No. 11 to conferences.
The FEFC and allied conferences have, on the basis of this recommendation. developed a
formula to “achieve the object of a-return to parity of value of their income and
expenditure. in terms of the tariff currency, to the net level prevailing prior to the
Smithsonian Agreement“). The result is known as a Currency Adjustment Factor (CAF)
that is expressed as a percentage (+ or —-) to be applied to the tariff base freight rate. CAFs
are computed for each loading area, and take into account the various currencies in which
the lines incur their costs. The amounts disbursed in the different currencies are calculated
from the proportions of cargo carried by the respective conference members, their _
nationality, the origin and destination of the cargo and the various cost centres. These are
then computed in terms of 1971 (pre-Smithsonian) dollars and expressed as a percentage of
the total costs for the area of shipment. These are shown in line 1 in Tables 17.1 and 17.2
containing the data for the several loading areas (UK/Eire, West Germany. Benelux etc.).
Line 2 shows the re/devaluation percentage corresponding to the respective currencies and is
the percentage change of the average review exchange rate on the pre-Smithsonian parity
exchange rate. The CAF review total in line 3,15 the sum of the currency movements in line
2 weighted by the proportions in line 1. Since some of the lines’ costs, especially bunker
fuel. are incurred in US$ (the tariff currency) and are zero-rated, these do not appear in the
tables.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
-1-
|'
1
r
Table 17.1 Eastbound
UK-Eire
t stg.
1 35.01
2 -41.06
Yen DMark
10.58 7.22
43.80 18.94
I-[KS
4.81
-22.29 .
'l'W$ SlM$
4.58 8.06
0.45 31.20
FEIO
6.30
-53.70
SDR
2.91
-0.46
Others Total
80 .490.16
-67.43
3 -16.88 4.64 1.37 1.07 0.02 2.51 -3.38 -0.01 -0.11 -12.91
West Germany-Benelux
0.11-at Yen
1 19.30
2 18.94
11.45
43.80
D. GI.
10.75
4.45
Stg/BFIFF
6.96
Kroner SIMS
6.76 8.26
-32.10 -31.70 31.40
FElO
12.78
-31.50
SDR
2.91
-0.46
Others
1 .32
— 16.70
80.49
3 3.66 5.02 0.48 -2.23 -2.14 2.59 -4.02 -0.01 -76.5 3.13
France
Fl F I
1 30.49
2 -40.75
Yen
16.68
43.80
sq;1n1wo1.
5.15
-10.30
Kroner PH/TH SIMS
2.56 4.10 9.78
-31.30 -52.20 31.80
FEIO
8.66
-0.26
SDR Others
2.91 0.16 80.49
-0.46 -67.43
3 -12.43 7.31 -0.53 -0.80 -2.14 3.11 -2.24 0.01 0.11 7.84
Scandinavia
SJCR. D.KR.
1 27.34 9.95
2 -41.88 -32.07
N.KR.
4.61 9.93
-19.27
D.GL. Yen F.F.
6.65 4.63
_ 4.45 43.00 -40.75
SIMS
7.26
30.30
SDR Others
2.91 7.21
-0.46 -35.711
80.49
3 -11.45 -3.19 -0.89 0.44 2.91 -1.89 2.20 -0.01 -2.57 -14.45
N.B. Non US Dolla
Source: Far Eastern Freight Conference.
Table 17.2 Westbound
Japan
Yen
1 40.90
2 45.00
D. Mark £ Stg. S. Arabia Kroner SDR El.|rlO
6.06 4.17 3.75 4.51 2.53 7.74
18.94 -47.06 -23.97 -32.15 -0.46 -23.64
FEIO
2.23
2.24
Others
H.611
- 38.1111
r currencies 80.49%, U.S. Dollar 19.51% of which 17.09% bunkeirsiflif
Tnlttl
811.49
3 17.92 1.15 -1.96 0.90 -1.45 -0.01 -1.83 11.05 -3.32 11.45
Korea
Won
1 25.39
2 -60.25
D. Mark
10.85
18.94
£ Stg. Yen KronerSDR EurlO
8.17 9.22 6.63 2.53 10.20
-47.06 43.80 -32.45 -0.46 -21.76
FElO
2.10
+16.l9
Others
5.411
+2.22
811.49
3 -15.30 2.06 -3.85 4.04 -2.16 -0.01 -2.22 +0.34 -1-11.12 -16.98
Hong Kong
HKS
1 21.41
2 -22.29
D. Mark
11.97
18.94
I Stg.
13.88
-47.06
Yen j Kroner SDR Eur /O
4.90 6.76 2.53 9.39
43.80 -32.69 -0.46 -24.49
FE/O
6.23
-5.94
Others
3.42
- 2 . 63
811.49
3 -4.77 2.27 -6.53 2.15 -2.21 -0.01 -2.3 -0.37 -0.09 -11.86
Taiwan
rws
1 25.25
2 0.45
D. Mark
13.97
18.94
. 4 7.24 2.53 11.91
£ Stg. Yen Kroner SDR EurlO FEIO
8 23 5 2 3 36
-47.06 43.80 -32.87 -0.46 -26.20 -24.41
Others
2.76
-9.78
80.49
3 0.11 2.65 -3.87 2.30 — 2.38 -0.01 -3.12 -0.82 -0.27 5.41
KONSTANTINOS
Rectangle
Singaporelw Malaysia
S.-‘MS D. Mark £ Stg. D. GL Kroner SDR EurlE Eur.'O FEIO Others
1 26.36 0.21 1.65 4.12 3.60 2.53 0.44 8.13 6.19 4.06 110.40
2 30.04 10.04 -41.06 4.45 -31.19 -0.46 -58.29 -46.62 31.03 -34.56
3 8.31 1.56 -3.60 0.21 -1.17 -0.01 -4.92 -3.79 1.97 -1.41 -2.85
Philippines
P11. Peso D. Mark E Stg. D. Gl. Kroner SDR EurlO F1110 Others
1 22.50 17.82 9.91 5.57 9.79 2.53 8.80 1.67 1.90 80.49
2 -66.43 18.94 -47.06 4.45 -32.69 -0.46 -46.82 43.80 -52.63
3 --14.95 3.38 -4.66 0.25 -3.20 -0.01 -4.12 0.73 -1.00 -23.58
N.B. CAF currencies 80.49% plus 2.42% U.S. Dollars 17.09% equal 100% (dollars are zero
rated)
Source: Far Eastern Freight Conference.
Review Arrangements
The CAF formula is reviewed quarterly taking the average exchange rates to the US$
applicable during a set period of 10 banking days in the last month of that quarter. If the
resultant CAF shows a downward or upward swing of 1.5% or more. the CAF is revised.
effective from the first day of the month following a quarterly review period. Thus the first
quarterly review is effective on April 1st and is based on 10 banking days in March.
Should a radical movement in a currency or group of currencies be observed to produce a
swing in the CAF of 4% or more. based on the average rates of exchange to the US$ over a
three-day banking period from the commencement of the radical movement. the CAF is
then changed (up or down) effective from the 10th day subsequent to the commencement of
the radical movement.
A summary of the results for the FEFC for the second quarter of 1985 is shown in Table 17.3.
Results
Table 17.3
Loading Area Previous CAF Review CAF Result
UK,"Eire -13.00 - 12.91 No Change
W Germany/Benelux +3.0 +3.12 No Change
France -11.0! -7.86 -7.50
Scandinavia -14.I -14.45 No Change
Japan +ll.51 +11.-15 No Change
S Korea -20.01 - 16.97 - 16.50
Philippines - -23.58 No Change
Taiwan -8.01 -5.41 -5.00
Hong Kong -11.51 -11.86 No Change
Singapore/W Malaysia -1.50 -2.85 No Change
to Ln
JU1(2
n..J¢_1fi3r;Jr_1C.J~*_r-c_1
The above revised CAFs for France, S. Korea and Taiwan were applied to cargo as per the
tarifl", eflective from July 1, 1985. Since January 1, 1990, there are only three CAFs, i.e. from
Europe; from Japan; and from areas other than Japan. CAFs now cover only the sea
leg of the voyage and were incorporated into the Freight Tariff as set on January 1, 1990.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
The Bunker Adjustment Factor (BAF)
As a direct consequence of the success of OPEC in raising the price of crude oil by creating
an artificial shortage in 1973/74. oil prices have since been increased at frequent intervals by
agreement among the OPEC nations. In the decade 1970 to 1980 crude oil prices rose from
an average of just under US$2 to $32/34 and even $38 per bbl. Faced with escalating fuel
costs — changing almost daily — shipowners had no option but to reflect the increments as
a Bunker Adjustment Factor (BAF).
The BAF operates under the principle that BAF adjustments should reflect the current cost
of bunker fuel against the base level incorporated in the freight rates. By the 1st December
1980 the BAF had reached a level of 49.50%. and patently. freight rates no longer
realistically reflected the total charge to shippers. Accordingly. member lines of the FEFC
decided to consolidate part of the BAF into the Base Freight Rates. recognising the
near-certainty that even allowing for subsequent drifting down of oil prices. energy costs
would remain pitched at a high level and that this development was likely to be irreversible
in the long run.
Fuel prices incorporated into tariff rates with effect from 1st January 1981 were based on the
BAF ruling on 1st November 1980 (42.54% with an average bunker fuel price of $194.65).
Subsequent to the incorporation of this part of the BAF. an interim BAF was introduced to
reflect changes in fuel prices both above and below the incorporated fuel price of $194.65.
Rates were next revised generally on lst January 1984: the Interim BAF (-1.48???) was
incorporated and the procedure began again on the basis of a new bunker fuel price of
$183.15.
Mode of Calculation
To assess the Interim BAF requirement it is necessary to calculate the relationship of fuel
costs to tariff freights (Bunker Weighting). The current Bunker Weighting (since 1st January
1984) is 17.09%. Increases or decreases in fuel prices result in a positive or negative interim
BAF based on the weighted fuel price movement above or below $183.15 per ton.
Member lines lodge the actual Marine Fuel Oil and Diesel Oil prices they pay with
independent accountants. who produce a conference average price per ton for each area.
When weighted in accordance with ‘bunker offtake' in each area. an overall average price
per metric ton is found. Fuel prices are reviewed monthly for normal movements of 2?? or
more. and wee--kly for radical movements of 4% or more. The result of monthly reviews and
the calculation of changes in the BAF are advised to Shippers‘ Councils.
The current interim BAF (minus 2.55 percentage points as from 1.7.85) was calculated by
comparing the average bunker fuel price in the June 1985 review ($155.79) with the base
bunker fuel price ($183.15) and the decrease ($14.94) was applied to the Bunker Weighting
(17.09%) to give an Interim BAF of 2.55 percentage points.
European Area Inflation Differential (EAID)
Although the Currency Adjustment Factor serves to retrieve the status quo for the
Conference lines taken as a whole. following significant changes in currency exchange rates.
one of the problems that is likely to arise is that freight rates become relatively more
expensive for shippers in one loading area vis-a-vis the others: and insofar as the shippers in
neighbouring areas are competing with each other in the same overseas markets. there
appears to be no good reason why one group should benefit from cheaper freight rates at
KONSTANTINOS
Rectangle
the expense of the others. The Inflation Differential that has been computed and applied by
the FEFC following discussions with the European Shippers Council is an attempt to remove
the distortion in freight rates resulting from the application of CAFs.
The formula works by calculating cost changes in each of the four eastbound areas. From
there. an overall level of increase for the eastbound trade as a whole is calculated pro-rata
to trade flows from each area. The final stage is to adjust the overall increase calculated in
this manner to that of the General Rate Increase introduced for the trade as a whole. This
enables a relationship to be established between cost levels in each of the four eastbound
areas and a differential to be determined against the General Rate Increase. At the same
time total recoveries are as if freight levels overall were identical in each direction.
EAID factors in force as at 1.1.84 were as follows:
Percentage Points
UK/Eire +10.71
Germany/Benelux -3.28
France +3 .35
Scandinavia + 1.90
The UK/Eire EAID also reflects the disproportionately higher inflation levels, compared
with other European areas, prior to July 1977 (which had not been reflected in freight rates)
and for convenience has been added to the EAID. The EAID remains fixed between ‘
general rate adjustments and, so as to facilitatefreight calculation, is added to or subtracted
from the ruling CAF.
However, as from January 1, 1990, inflation surcharges/decreases in the FEFC have been
discontinued.
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
T
'|
j
I
I
l
l
l
l
||
T
l
|
1
I
I
.-tn-_.e__—___,r
Chapter 18
Investment Appraisal
in Shipping
Investment is the act of sacrificing current consumption in the expectation of some future
gain. The amount of the sacrifice is known accurately and its timing is certain: it takes place
now and is measured by the cost of the investment. The benefits of the investment are,
however, much less certain. That they are expected to materialise may be taken for granted,
since the investment would not otherwise be undertaken: but they will arise, if at all, only
over a period of time. Hence, the costs are incurred now, but the benefits, arising at some
time in the future, are likely to be represented by a time stream of profits (sometimes losses),
i.e. as annual cash surpluses (or occasionally deficits) over the life of the investment. A sum of
money arising at a future date is less desirable than the equivalent sum of money arising
today: hence there is a need to discount future sums of money to a common point in time.
Much of this chapter may be commonplace to those familiar with the techniques of
investment appraisal using discounted cash flow (DCF) methods. It is hoped, however, that
the ‘short cut’ formulae presented are both comprehensive and intelligible. Proofs of these
formulae are contained in Annexes and they should be particularly valuable for use with
pocket calculators or microcomputers.
Basic Definitions and Principles
Cash Flows
These are usually expressed in monetary units arising during 12 month intervals of time. In
most investment appraisal problems the net cash flows are expressed ‘after tax’, i.e. gross
profit less tax. It should be noted that depreciation — an imputed cost — does not appear as
part of the cash flow.
Discounting
It must be recognised that cash flows occurring in different years cannot be summed before
reducing them all to the same unit. £x receivable in n years time has a smaller value than £x
receivable now. The difference arises because of the opportunities available to invest and
obtain a rate of return on capital, so that if r is the annual rate of return, expressed as a
fraction, then in n years time £x would be worth:
£x(l + r)“
Conversely £x receivable in n years time would have a Present Value of:
£x(l + r)"‘
where r may now be regarded as the discount rate and (1 + r)"‘ the discount factor for year n.
The present value or PV is the unit for summation normally used. The discounting rate
used by individual firms depends upon many factors and its determination is complex.
For information on this aspect text books such as Merrett and Sykes’s ‘Finance and Analysis
of Capital Projects’ should be consulted.
KONSTANTINOS
Rectangle
Net Present Value
If A1, A2, A3, A4 . . . An are the cash flows occurring in years 1, 2, 3, . . . n, Co the
investment capital, and d the rate of discount then
NPV = A1(l + d)" + A;(l + d)‘: +...A,,(1+ d)"‘ - C0
or NPV = E A,(1 +<1)-i - co
i=-=1
and provided the NPV is positive the investment represents a return on capital equal to d
together with a sum equal to the NPV which is clearly in present value terms. More will be
said about NPV later.
Internal Rate of Return (IRR), DCF Rate of Return, or DCF Yield
Internal Rate of Return is a fully discounting method which is sometimes preferred to that of
NPV. The mechanics are similar but the criterion by which the assessment is made is
different.
The method sets out to find that rate of discount which will yield a NPV = 0
i.e. 0 = E A,(i + an-* - co
i=1
n
6; EA,(1+<1*)-‘=0, and IRR=d"'
i=1
The advantage of this method is that a discount rate need not be assumed initially: it remains
however to decide whether or not the IRR is satisfactory.
One disadvantage of the method is that iteration is required; another, that difliculties occur
in comparing mutually exclusive projects. An example may make this clear. The discount rate
for projects A and B is assumed to be 7%.
Project A
1 Year Cash Flow DF 7°/0 DCF DF 15°/0 DCF
1 I (10.000)
2.229 I
1 2,229
11 2,229 * 0.816 1,819
*i 2,229 0.163 1,200
I 2,229 1 0.113 1,589
2,229 1 0.666 1,484 i
1 V 2,229 . 0.623 1,388
2,229 l 0.582 A 1,291 1
(10,000)
2,081
1,946
@---JO\Ll'I-hbJI*~J'-'¢
1.000
0.934
0.873
1.000
0.870
0.756
0.658
0.572
0.497
0.432
0.376
0.327
. -—-—-_i
3.304 i
(10,000)
1,939
1,685
1,466
1,275
1 , 107
962
838
728
0
NPV = 3,304 @ 1% IRR = 15%
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Project B
Year j Cash Flow j DF 7% , DCF DF15% DCF
-__. U, _. jr’ , or . __ 1.
@\lO\L-II-§LaJl"~J'-‘Q
(10,000)
1,452
1,670
I 1,900
1 2,175
1 1 15°"1 1 2,890
y 1 3,320
W 1 3,800
1.000
0.934
0.873. 1
0.816
0.763
0.713
0.666
0.623
0.582
(10,000) (1.000) 3 (10,000)
1,356
1,458
1,550
1,659
1,782
1,925
2,068
2,212
0.870
0.756
0.658
0.572
0.497
0.432
0.376
0.327
1,263
1,263
1,250
1,244
1,242
1,248
1,248
1,242
4,010 1 j 0
L 1 _ , . ,. .. ,__ , , l__ __
1 NPV = 4,010 @ 7°/, IRR = 15°/0
u ——' — —~~ K
It is clear from this example that although the IRRs are equal at 15°/,, on a capital outlay of
10,000, at the required rate of return corresponding to 7°/,, the NPV of project A is 3,304
while that of project B is 4,010. The reason for this is that project B has an increasing cash
flow profile compared with A where it is uniform and at higher rates of discount the discount
factors bear more heavily on cash flows than at lower rates.
When discount rates are small the graph of discount factors against years shows a curve with
small slope and little change in gradient, while larger discount factors exhibit curves that
resemble rectangular hyperbolae. (See Figure 18.1)
It should be noted that IRR is the rate of return on capital remaining in the project at any given
time; this capital diminishes as amortisation progresses. Although arising continuously
throughout the year, for the purposes of discounting, it is normally assumed that cash flows “
actually occur at the end of the year in question. Year 0 is usually (but not necessarily) taken
to be the point in time where the asset is brought into use and starts earning revenue so that
year 1 is one year later; and so on. More will be said about cash flows later.
The Annuity Factor
The example below illustrates a case where the cash flow profile is uniform.
Year Capital Cash Flow or=12°/, ocr
_ _ 1 1
‘q 1,200
O\Lh-hLnl\J'-‘Q
1 1 . L- _ ,1
§§§§§§
1.000
0.893
0.797
0.712
0.636
0.567
0.507
(1,200)
357
319
285
254
227
203
NPV =445
_ _ , _.. ___ _1
4
1
KONSTANTINOS
Rectangle
Figure 18.1 Curves of discount factors against years
.9-
.8-
.7—-
D'scountFactor
6-.
5-1
' 4-1
.34
.2—1
l-1
(1.os)'"
(1.01)"‘
(1.10)'"
(1.1s)‘“
A FT" T771” If I
5 10 15 20 25
Years
It may be seen that the Present Value of the Cash Flows (excluding Capital Investment) is
given in general terms by:
PV = A(l.+ d)" + A(1+ d)": + . . . + A(1+ d)‘“"‘ + A(l + d)“
where A is the uniform annual cash flow, d the rate of discount and n the life of the project
It can be shown that this reduces (see Annex I(b)) to:
1 _._ -Tlwhere —~§1di is termed the Annuity Factor.
This gives a fast and easy method of calculating the PV of a uniform cash flow profile
In this example, PV = A >< 4.1114
= 400 x 4.1114
= 1,645
So that NPV = 1,645 — 1,200
= 445 as before.
L4
KONSTANTINOS
Rectangle
. 1 -- 1 "‘ . . .The annuity factor -431-L is extremely useful and its value may be found in tables or
calculated very quickly using a pocket calculator.
Even if there are some anomalies in the uniform cash flow it is often still possible to save
time by use of the annuity factor. For example, a cash flow series might take the form of :
1.0; 2. 0; 3. $3,000; 4. $3,000; 5. $3,000; 6. $3,000; 7. $1,000; 8. $3,000; 9. $3,000;
10. $3,000.
The anomalous years are 1,2 and 7. A method of finding the PV of these cash flows is to
discount all cash flows first of all to year 2 so that (ignoring initially the anomaly in year 7)
_ -8Pvm = W, (1..<1T+.<.1>_)
Now discount this back to year 0 by using the discount factor (1 + d)". Finally take account
of the shortfall in year 7 by subtracting 2,000(1 + d)"".
Thus,
_ -8
PV = 3,000 (1 + d)" — 2,000(1 +d)"
The annuity factor is sometimes known as the Series Present Worth Factor (SPW) and may
be written as:
(1 + d)“ - 1 K
(1 + d)"d
One of the many uses of the annuity factor is in finding the annual cash flow required to
amortise an investment. This annual cash flow is referred to as the Annual Capital Charge
or ACC.
Thus ACC =
(1— (ld+ d)"‘)
where C0 is the capital invested in year 0, d the rate of discount and n the life of the project.
Using the data given in the previous example
ACC = 1,200 _ 1,200
1-(1+0.12)-t) 4.1114
0.12
= 292
Thus 292 units are required per annum to amortise the investment and provide a Rate of
Return (DCF) of 12%. If the Annual Cash Flow were only 292 rather than the 400 units given
it should be evident then that the DCF rate of retum on the project would be 12°/,,.
This example may help further to illustrate the meaning of NPV. If the NPV is added to the
capital invested, i.e. 445 + 1,200 = 1,645, this now represents the maximum amount that the
firm would be willing to invest if the required rate of return is 12%. Otherwise, if the NPV is
divided by the annuity factor:
=i= 108l—(1+d)‘" 4.1114
d
it may be seen that this represents the annual surplus to the ACC over the life of the project.
It may also be noted that the reciprocal of the annuity factor is the Capital Recovery Factor
(CRF) so that:
ACC = c, >< car = co (141--1_ (1d+ d)_n)
KONSTANTINOS
Rectangle
KONSTANTINOS
Rectangle
Theoretical Value of an Asset
In accounting practice the book value of an asset such as a ship is often represented, by
subtracting from the original cost, depreciation calculated on a straight line basis. If,
therefore, a ship costing $10,000,000 has an expected life of 20 years, then after 10 years
(ignoring scrap value) its book value would be $5,000,000; after 11 years $4,500,000, and so
on (i.e. depreciation $500,000 p.a.).
Now the long term value of an asset should be represented by its projected eaming potential.
_ 1 —- 1 d *2" _
A vessel costing $10,000,000 would need to earn 10,000,000/ p.a. to provide a
return d and recover the capital invested. If d is 10°/,, or 0.1 then
ACC = $1,174,596
Assuming that this, in a competitive market, represents the constant long term earning
power of the vessel, then after 10 years the value of the ship can be represented by
l—(1+0.l)"° _
$1,174,596 A A A 0—~l~~~ $7,217,384, i.e. The PV (at the 10th year) of the cash flows
arising annually up to the end of the vessel‘s life. This should be compared with the value of
$5,000,000 obtained by the straight line method.
Sinking Fund Factor (SFF) An alternative approach to Annual Capital Charge is via the
. . . . 1 " — 1Sinking Fund Factor. This factor given by $——_%—— (See Annex 1(a)) where r is thegrate
of interest, and n the life of the investment, provides, when divided into the capital investment
C0, the amount of capital to be invested each year at rate r (compound) that will accumulate to
C0 at the end of n years.
This amount —§%’F is the Depreciation by Sinking Fund method. Using the same example, if
Co is 1,200, r is 12% and n is 6 then depreciation or annual sum required to amortise the
capital investment is:
1,200 = 1,200 =l48
(1 +0.l2)°—1 8.11519
0.12
The total annual cost is the sum of this depreciation plus the interest on capital. Thus if
interest is also 12°/,,, the total annual cost
12.1200=148 ——'—--+ 100
= 148 +144
=292
It will be noted that this is the same as that found by ACC. Thus the general connection
between ACC and capital cost by sinking fund method is:
CO ggwfgggw
Fi'i'1fi+'C° C°((1+r)-1-1+’)
I‘
__ r+r(1+r)"—r
C°( (l+r)"—l l
r _ 1._ Co i.e. C0 X CRF or (Co x Annuity Factor)
This relationship is valid only so long as the depreciation is invested at the same rate as the
required retum on capital, i.e. r = d.
Internal Rate of Return by Annuity Factor
Where annual cash flows are uniform, the IRR may be found by inspection of annuity tables.
Thus if a project costing £1,000,000 is expected to produce an annual cash flow of £80,000
over 18 years, the annuity factor is equal to:
1_@;m=,,,
80,000 '
An inspection of tables indicates that for a life of 18 years the annuity factor corresponding to
4% is 12.6593 and for 5%, 11.6896. By interpolation, if required, the IRR is found to be
approximately 4.2%.
Growth Factor
It has been shown that where a uniform cash flow profile exists, the PV can be found quickly
using the annuity factor. It may sometimes be assumed that the cash flow has a given value in
year 0 and grows at a given rate (compound) per annum. In the following example the cash
flow starts in year 0 at £500 and grows at 8%. The corresponding cash flows are then
discounted at 10%.
I _ 7 ' 47'“ J _ 1 _ ._ 1 .4 1 74' W4 _7 _ .4 ,
‘ 1 1
. Year W Cash Flow DF 10% DCF A
1] __ -1. .- _ . ..,_ 1 __.,.._
1 £540 0.909 £491
j 583 0.826 482
1 1 630 1 0.151 413
680 0.683 464
135 ~ 0.621 1 456
193 1 0.564 441
j 851 0.513 1 440
J 925 0.467 432%'--lO\'-It-FU-ll\J'—'
1
PV = 3,685
_ _ __i iii _ ‘ i 7 _ _
However, a faster approach is found using the growth factor (See Annex I(c)). The product of<1 "11 " TIEthis factor, I: (d 8) il (1 + g) and the cash flow in year 0 yields the PV directly.
l_(l.08)'
1.1h , PV= .T us £500l: 0.02 ](l +008)
= £500’x 7.37248
= £3,686 which compares with the answer previously found.
Should the cash flow start at A in year 1 rather than year 0 then the formula becomes:
_ l__“
Pv=Al1
KONSTANTINOS
Rectangle
Inflation
If all cash flows are expressed in money terms it is evident that over the ensuing years the
value of money will diminish in real terms. If k is the rate of inflation, then cash flows A,
arising in year i, will be worth, in real terms, A(l + it)“.
It follows that the PV of cash flows A arising in year i is given by:
PV = A(1 + d)“(l + k)" where d is the rate of discount in real terms.
Hence 'Pv = A[(l + d)(l + 1<)1-i
=A[l +k+d+kd]'*
= A[l + (lt + d)]" as a first approximation, since kd is small.
Thus, unless greater accuracy is required and provided both d and k are reasonably small the
discount rate of use may be found by summation of k and d.
Returning now to the example, using the growth factor, if the rate of inflation is 8% then the
discount rate in real terms becomes [(1.l)(l.08)—- 1] = 0.188.
I _( 1.08 T
Hence PV — £500 L188 (1 08)
. 0.108 '
= £500 x 5.3346
= £2,667
It should be evident that since all the cash flows have been increased by 8% compound and
then deflated by 8% the same result could have been achieved by assuming a uniform cash
flow profile of £500 and discounting using the annuity factor at 10%. E.g.
-PV = £500
= £500 x 5.3349
= £2,667 as before.
The Rate of Discount
All investment appraisal (i.e. net present value or internal rate of return) calculations, be they
public or private, involve discounted cash flows. Problems arise not only in the forecasting of
future cash flows but also in the choice of an appropriate rate of discount. The discount rate
is predetermined or exogenous and is used either to discount cash flows which can then be
aggregated to obtain the net present value or as the minimum rate of retum on projects which
companies require before they will make the decision to invest. The choice of the discount
rate is thus vital to the investment decision and there is an inverse relationship between the
rate of discount and the net present value. For an investment to be undertaken, the net
present value must exceed zero which implies that the intemal rate of return must exceed the
required rate of return on capital. The higher the discount rate the lower will the net present
valuenormally .be and the more difficult for the project’s internal rate of return to exceed the
required rate; more and more projects will then fail to meet the appropriate criterion and
companies will thus not make the decision to invest. At the margin, the company will
undertake projects provided the NPV is just greater than zero or provided the IRR just
exceeds the required rate of retu-rn.
KONSTANTINOS
Rectangle
The marginal efficiency of capital (or investment) curve is downward sloping, reflecting the
fact that the IRR falls as more investments are undertaken. Theory states that investments
should be undertaken, in the absence of capital rationing, provided that the marginal return
exceeds the marginal cost of the investment. In other words funds should continue to be
invested up to the point where the marginal cost of capital is just equal to the marginal
eficiency of capital (the internal rate of return). Figure 18.2 explains that investment should
be expanded up to point I, the point at which the marginal cost of capital is just equal to its
marginal efliciency, and the discount rate to be used (d) is determined by the cost of capital
at this point. In other words the choice of discount rate is determined by the firm’s marginal
cost of capital.
Figure 18.2
: Marginal cost of capital
etum% Average cost of capital
captaIrateofr
.-' /;
Costof
O _l_.._.._
Marginal efficiency
of capital schedule
Volume of investment (£)
The literature contains references to the social rate of discount and to the ideas of social time
preference rates (STPR) and the social opportunity cost (SOC) rates of discount. These are
simply two theories of how to derive the correct discount rate; the STPR theory argues that
the discount rate should reflect society’s preference for present benefits over future benefits
while the SOC theory considers that the discount rate for use in public projects should reflect
the rate of return forgone (the opportunity cost) on the displaced project. A third theory
provides for a ‘synthetic’ rate which is a weighted average of the SOC and STPR rates.
Regardless of which school of thought one follows, it is clear that, (i) in the interests of its
shareholders, a company should require a retum at least as great as the highest return
(allowing for differences in risk) obtainable from alternative projects and (ii) a company
should require a return at least as great as its marginal cost of finance. Once the costs of the
various sources of capital have been determined the overall cost of capital is the weighted
average of these individual costs, the weights being proportional to the relative importance of
each source. It is possible to interpret this average cost of capital as the cost of capital of the
marginal project which is to be used as the rate of discount.
Public investment appraisal often uses a test discount rate (TDR) of about 8°/,,-10%
depending on the level of risk. One way to allow for risk in investment appraisal is to
increase the discount rate by adding a risk premium to it, though this has the obvious
drawback of increasing the premium over time with associated consequences for the
discounted cash flow calculations.
KONSTANTINOS
Rectangle
Tax Positions
Shipping economists involved in calculating the net present value (NPV) of a shipping
project usually distinguish between the different tax positions in which a company might find
itself. This distinction is necessary because the NPV will vary according to the tax position of
the shipping company since the ability of the company to make use of capital allowances and
other fiscal incentives will be different for each tax position. Three key tax positions are
defined;
(i) The Fall Tax Position A company is in this tax position if it is earning suflicient
profits from other sources to take full and immediate advantage of any tax allowances
available on the current investment.
(ii) The No Tax Position A company will be in this tax position if it has accumulated tax
allowances from earlier years (previous investments) to the extent that it is now
unlikely to have any tax liability for the indefinite future and certainly for the life of
the proposed investment.
(iii) The New Entry Position A company will be in this tax position if, given the tax
allowances available on the capital cost involved in the investment and the level of
profit expected to be earned on it, it is neither earning sulficient profits from other
sources, nor has it accumulated sufiicient allowances from earlier years for either of
these factors to have any influence at all on the net present value of the investment
proposal. As neither profits from other sources nor accumulated tax allowances enter
the calculation of the NPV this tax position is equivalent to that in which a newly
established company would find itself on entering the industry; hence its name.
Tax allowances take many forms and some of the more common are listed below:
(a) Writing down allowances which permit an asset (a ship) to be depreciated for tax
purposes at specific rates. These allowances are set off against profits before tax liability
is assessed and are usually specified in terms of annual allowances according to either the
straight line or reducing balance methods of depreciation. The difference between these
two methods will be obvious from the following example.
Straight line Reducing balance
depreciation T; depreciation
Year allowance at 20% Residual ll allowance at 20% Residual
20% 80% 20% 80%
20% 60% 16% 64%
20% 40% 12.8% 51.2%
207 20%
0%til-ht:-JI'~J'—'
, 10.24% 40.96%
20°/, l s.19% 32.11"/,
The straight line method takes a constant percentage of the depreciable base each year
while the reducing or declining balance method takes a constant proportion of the
residual value each year.
Other methods, e.g. free depreciation and sum-of-the-years-digits, are also possible but are
less common.
(b) Accelerated depreciation allowances allow the vessel to be depreciated more quickly.
They may take the form of initial or first year allowances, which permit a higher than
nonnal depreciation rate during the first year of operation, or of advance depreciation
allowances provided in advance of the delivery of the vessel.
F.
i
1
l
s
I
1l
K.
l
l
|
1
__?..,_--1,-
l
|
I
s.
l
P
I
\
l
l-
|
I
I
I
E
'l.
1
i
5l
l
'\1\.
I
l
H
J
(c) Investment allowances permit the writing down of more than 100% of the cost of the
vessel, e.g. an investment allowance of 40% means that 140% of the cost of the asset may
be set against profits for tax purposes.
(d) Investment grants are cash payments of a certain percentage of the contract price of the
vessel and are paid direct to the shipowner. Strictly, the value of these allowances is not
dependent upon the company’s tax position though there is a tax aspect to such grants in
that the depreciable base of the asset may be reduced by the amount of the grant.
(e) Tax free reserves are monies set aside in a fund designated for some specified purpose,
e.g. the purchase of new vessels or the modernisation of existing ones, at a future date.
Such funds will not be taxed, provided their use satisfies certain requirements and
constraints as specified by the appropriate tax authorities.
(f) Favourable credit arrangements often exist for shipowners to purchase new vessels.
Typical of these arrangements is the OECD Understanding on Export Credit for Ships
which currently allows for loan arrangements no more favourable than 80% of the cost of
the vessel to be borrowed over a period of 8§ years at an annual rate of interest of 8%.
Domestic credit arrangements, provided by a country for its own shipowners, may be
more favourable. The interest payments associated with such loans are allowable against
corporate tax liability.
Briefly, the relevance of tax allowances to the tax positions defined earlier is that a company
in the full tax position will obtain the maximum benefit from such allowances by using them
as soon as they become