Course 1-Interest rates (2)

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Financial and Actuarial Mathematics
1st Course-Interest Rates
Prof. dr. Paula Curt
FSEGA, Statistics Forecasts and Mathematics Department
paula.curt@econ.ubbcluj.ro
Interest Theory
Simple Interest
Definition : Interest
Is a payment made by the borrower (debtor) for using the lender’s capital. The amount
of money which is borrowed or invested is called the principal (the original principal).
Simple Interest (S.I.) is computed on the principal only; is not reinvested to earn
additional interest; is used for short term loans (less than a year)
Elements of Simple Interest
s - the principal; the present value, the initial value (m.u.)
t - time period (lenght of the loan or of the investment) (years)
I - the amount paid to the lender for the use of money (m.u.)
S - the acumulated value, the future value or the final value; S = s + I ; (m.u.)
i - interest rate/year = the interest earned over a year when 1 m.u. is invested
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
2 / 14
Interest Theory
Simple Interest
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
3 / 14
Interest Theory
Simple Interest; I = sit; S = s(1 + it)
Simple Interest; Basic Formulas; I = sit;
S = s(1 + it) accumulation relation;
s = S1+it present value (discount) relation;
Time value of money: money received now is worth more than in the future
u = 1 + i =accumulation factor; v = 1
1+i
=present value(discount)factor
u=value over 1 year of 1 m.u. of today;
v=present value of 1 m.u. a year from now
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
4 / 14
Interest Theory
Simple Interest; I = sit; S = s(1 + it)
Example1: Compute the interest if we invest $ 2,000 at a rate i = 6% for 3 months.
Remark: The interest rate and the time period must be consistent to each other. If
the time is given in something else than years then we must transform into years
I = sit = si ’no. of days’360(365) = si
’no. of months’
12
Example2: Bank 1 offers us a loan at an annual interest rate of 18%. Bank 2 offers us
a loan at an monthly interest rate of 1.5%. Which is more convenient for us?
Sometimes we use interest rates which correspond to fractional periods of time
such as months, weeks, days, etc.
We split the year in m equal periods of time (the length of each period is 1/m of
a year). Let im be the interest rate/period. If t is the time measured in years
then tm = t ·m is the time measured in no. of periods.
m=2 =⇒ the year is split in 2 semesters
m=4 =⇒ the year is split in 4 quarters or trimesters
m=12 =⇒ the year is split in 12 months
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
5 / 14
Interest Theory
Simple Interest; I = simtm; S = s(1 + imtm)
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
6 / 14
Interest Theory
Simple Interest; I = simtm; S = s(1 + imtm)
Equivalent interest rates
Two interest rates are said to be equivalent if they produce the same interest over the
same period of time starting from the same principal
Remark: i
S.I.∼ im ⇐⇒ im = im
Example2: Bank 1 offers us a loan at an annual interest rate of 18%. Bank 2 offers us
a loan at an monthly interest rate of 1.5%. Which is more convenient for us?
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
7 / 14
Interest Theory
Compound Interest
Compound Interest (C.I.)
In compound interest the interest due is added to the principal at the end of each
interest period and thereafter earns interest in the next period at the same rate. The
time between two successive interest computations is called compounding period or
conversion period. C.I. is used for loans whose terms are longer than one year.
Example3: Find the accumulated value of $1,000 after three years at a rate of interest
of 24% per year convertible annually.
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
8 / 14
Interest Theory
Compound Interest
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
9 / 14
Interest Theory
Compound Interest
C.I. Basic Formulas; conversion period=1 year; i=interest rate/year; t=term in years
S = s(1 + i)t = sut -accumulation relation; S=final value or future value
s = S 1
(1+i)t
= Sv t -present value (discount) relation; s=principal; present value
I = S − s = s(ut − 1); t =
ln S
s
ln(1 + i)
; i =
(
S
s
) 1
t
− 1
Example4: How long will it take to double a capital attracting annual interest of 6 %.
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
10 / 14
Interest Theory
Compound Interest
C.I. Basic Formulas; m is the number of compounding periods; conversion period=1/m
of a year; im=interest rate/period; t=term in years; tm = tm=term in no. of periods
S = s(1 + im)tm = su
tm
m ; um = 1 + im; s = S
1
(1+im)tm
= Sv tmm ; vm =
1
1+im
Example5: Determine the C.I. earned on $ 1000 for one year at 1%/day compounded
daily and compare it with S.I. earned on $ 1000 for one year at 1%/day.
Remark: i
C.I.∼ im ⇐⇒ im = m
√
1 + i − 1
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
11 / 14
Interest Theory
Nominal Interest Rate; Effective Interest Rate
Example6: We put some money in a saving account at an annual interest rate of 12
%, compounded monthly. Which is the accumulated value after 1 and a half years?
Nominal Interest Rate (N.I.R.)
An interest rate is called nominal if the period of time on which the interest rate
is announced (usually one year) is different to the compounding period.
Notation for N.I.R.: ρm; m is the number of compounding periods
im =
ρm
m
N.I.R. Basic Formulas; ρm-N.I.R/year; m compounding periods/year; im=interest
rate/period; t=term in years; tm = tm=term in no. of periods
S = sutmm = s(1 + im)
tm = s
(
1 +
ρm
m
)tm
Effective Interest Rate: ieff =the interest earned over a year when 1 m.u. is invested
ieff = i
Remark: i
C.I.∼ ρm ⇐⇒ i =
(
1 +
ρm
m
)m
− 1
Example7: We deposit $ 1000 for one year at an annual interest rate of 8%. What is
the final value if the conversion period is: one year; one semester; one month? What is
the greatest final value that can be obtained only by shortening the conversion period?
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
12 / 14
Interest Theory
Nominal Interest Rate; Effective Interest Rate
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
13 / 14
Interest Theory
The Time Value of Money
The Time Value of Money:
receiving $ 100 today is not the same as receiving $ 100 one year ago, nor
receiving $100 one year from now.
we can’t add, subtract or compare payments which are made at different
moments of time. In order to compare such kind of payments we have to
determine the value of such payments at the same moment of time (by
accumulating or discounting)
(r1, t1) (r1 due at the time t1) is equivalent with (r , t) at a given interest rate i ;
r = r1(1 + i)t−t1 =
r1
(1+i)t1−t
if t > t1, we move money forward in time i.e. we accumulate
r1
accumulation−→ r = r1(1 + i)t−t1 payments
——|—————————|———————–>
t1 t time
if t < t1, we move money backward in time i.e. we discount
r = r1
(1+i)t1−t
discount←− r1 payments
———-|—————————|———————–>
t t1 time
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 1st Course-Interest Rates
14 / 14
	Interest Theory