Financial Mathematics for Actuaries
Financial Mathematics
for Actuaries
Chapter 2
Annuities
Learning Objectives
1. Annuity-immediate and annuity-due
2. Present and future values of annuities
3. Perpetuities and deferred annuities
4. Other accumulation methods
5. Payment periods and compounding periods
6. Varying annuities
2
2.1 Annuity-Immediate
? Consider an annuity with payments of 1 unit each, made at the end
of every year for n years.
? This kind of annuity is called an annuity-immediate (also called
an ordinary annuity or an annuity in arrears).
? The present value of an annuity is the sum of the present values
of each payment.
Example 2.1: Calculate the present value of an annuity-immediate of
amount $100 paid annually for 5 years at the rate of interest of 9%.
Solution: Table 2.1 summarizes the present values of the payments as
well as their total.
3
Table 2.1:
Year
1
2
3
4
5
Present value of annuity
Payment ($)
Present value ($)
100 (1.09)?1
100 (1.09)?2
100 (1.09)?3
100 (1.09)?4
100 (1.09)?5
100
100
100
100
100
Total
= 91.74
= 84.17
= 77.22
= 70.84
= 64.99
388.97
2
? We are interested in the value of the annuity at time 0, called the
present value, and the accumulated value of the annuity at time n,
called the future value.
4
? Suppose the rate of interest per period is i, and we assume the
compound-interest method applies.
? Let anei denote the present value of the annuity, which is sometimes
denoted as ane when the rate of interest is understood.
? As the present value of the jth payment is v j , where v = 1/(1 + i) is
the discount factor, the present value of the annuity is (see Appendix
A.5 for the sum of a geometric progression)
ane = v + v 2 + v 3 + ¡€ ¡€ ¡€ + v n
?
?
1 ? vn
= v¡Á
1?v
1 ? vn
=
i
1 ? (1 + i)?n
=
.
i
5
(2.1)
................
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