Course 2-Annuities

Prévia do material em texto

Financial and Actuarial Mathematics
2nd Course-Annuities
Prof. dr. Paula Curt
FSEGA, Statistics Forecasts and Mathematics Department
paula.curt@econ.ubbcluj.ro
Annuities
The Time Value of Money
Money has time value:
receiving $ 100 today is not the same as receiving $ 100 one year ago, nor
receiving $100 one year from now.
we can’t add, substract or compare payments which are made at different
moments of time. In order to compare such kind of payments we have to
evaluate them 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 2nd Course-Annuities
2 / 12
Annuities
Annuities
Annuity = Sequence of Periodic Payments
Annuity: {(rk , tk ), k = 1, n}; i is the interest rate/year; u = 1 + i , v = 11+i ;
rk−the kth payment; payment interval=time between successive payments
tk− the moment of kth payment; t1 < t2 < · · · < tn;
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
3 / 12
Annuities
Annuities
V (t)=Value of the Annuity at moment t; t=the evaluation moment
Let {(rk , tk ), k = 1, n} be an annuity; t1 < · · · < tk−1 < t < tk < · · · < tn.
V (t)=the Value of the Annuity at the moment t is the sum of the evaluated
values at moment t of all payments rk , k = 1, n.
r1 r2 . . . rk−1 V(t) rk . . . rn payments
——|——–|————|——–|——–|———–|——————>
t1 t2 . . . tk−1 t tk . . . tn time
r1, · · · , rk−1 are to be made before the evaluation moment t; their values at the
moment t are accumulated: r1ut−t1 = r1v t1−t , · · · rk−1ut−t1 = rk−1v t1−t
rk , · · · , rn are to be made after the evaluation moment t; their values at the
moment t are discounted: rkv
tk−t , · · · rnv tn−t
V (t) =
n∑
k=1
rk · v tk−t = r1 · v t1−t + r2 · v t2−t + ... + rn · v tn−t
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
4 / 12
Annuities
Ordinary Annual Constant Annuities; OACA
CA; Constant Annuities: equal payments at equal time intervals
ACA; Annual Constant Annuity: Constant annuity with one payment per year
OACA; Ordinary Annual Constant Annuity: ACA with payments made at the end of
each year
Ordinary Annual Constant Annuity; OACA; rk = r ; tk = k, k = 1, n
OACA: equal payments at the end of each year; {(r , k); k = 1, n}
r r . . . r r payments
———|——|——|————–|——|—————>
0 1 2 n − 1 n time
V (t) = r
1− vn
i
ut ; V (0) = r
1− vn
i
= PV ; V (n) = r
un − 1
i
= FV
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
5 / 12
Annuities
Ordinary Annual Constant Annuities; OACA; Example
Example1: Woud you prefer
a million today?
or 100000/year for the next 50 years of your life (with the first payment one year
from now)?
Suppose that the annual interest rate is 10% for the next 50 years.
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
6 / 12
Annuities
Annual Constant Annuities Due; ACAD
CA: Constant Annuity: equal payments at equal time intervals
ACA; Annual Constant Annuity: Constant annuity with one payment per year
ACAD; Annual Constant Annuity Due : ACA with payments made at the beginning
of each year
Annual Constant Annuity Due; ACAD; rk = r ; tk = k − 1, k = 1, n
ACAD: equal payments at the beginning of each year; {(r , k − 1); k = 1, n}
r r r . . . r payments
———|——|——|————–|——|—————>
0 1 2 n − 1 n time
V (t) = r
1− vn
i
ut+1; V (0) = r
1− vn
i
u = PV ; V (n) = r
un − 1
i
u = FV
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
7 / 12
Annuities
Annual Constant Annuities Due; ACAD; Example
Example2: Woud you prefer
a million today?
or 100000/year for the next 50 years of your life (with the first payment now)?
Suppose that the annual interest rate is 10% for the next 50 years.
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
8 / 12
Annuities
Fractional Constant Annuities; FCA; Constant annuities with m payments per year
Term of an Annuity=the time from the beginning of the first payment interval to the
end of the last payment interval
t-evaluation moment (in years); im-interest rate/period; um = 1 + im; vm =
1
1+im
Ordinary Fractional Constant Annuities; OFCA; m payments per year
n year term OFCA: nm equal payments at the end of each payment interval;
r r . . . r r payments
—–|——–|——–|—————–|——–|—————>
0 1
m
2
m
. . . mn−1
m
mn
m
= n time
V (t) = r
1− vnmm
im
utmm ;V (0) = r
1− vnmm
im
= PV ;V (n) = r
unmm − 1
im
= FV
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
9 / 12
Annuities
Ordinary Fractional Constant Annuities; OFCA; Example
Example3: A car costing $10000 is to be paid off by equal payments over 4 years at
12% annual interest rate. What is the value of each payment if the payments are
made at the end of each month?
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
10 / 12
Annuities
Fractional Constant Annuities; FCA; Constant annuities with m payments per year
Term of an Annuity=the time from the beginning of the first payment interval to the
end of the last payment interval
t-evaluation moment (in years); im-interest rate/period; um = 1 + im; vm =
1
1+im
Fractional Constant Annuities Due; FCAD; m payments per year
n year term FCAD: nm equal payments at the beginning of each payment interval
r r . . . r r payments
—–|——–|——–|—————–|——–|—————>
0 1
m
2
m
. . . mn−1
m
mn
m
= n time
V (t) = r
1− vnmm
im
utm+1m ;V (0) = r
1− vnmm
im
um = PV ;V (n) = r
unmm − 1
im
um = FV
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
11 / 12
Annuities
Fractional Constant Annuities Due; FCAD; Example
Example4: A car costing $10000 is to be paid off by equal payments over 4 years at
12% annual interest rate. What is the value of each payment if the payments are
made at the beginning of each month?
Prof. dr. Paula Curt
Financial and Actuarial Mathematics 2nd Course-Annuities
12 / 12
	Annuities