FM for Actuaries

Annuities 61
n is an integer and 0 <k<1. We note that
a n+k⌉
= 1 − vn+k
i
= (1 − vn )+ ( v n − v n+k)
i
[ (1 + i)
= a n⌉ + v n+k k ]
− 1
i
= a n⌉ + v n+k s k⌉
. (2.46)
Thus, a n+k⌉
is the sum of the present value of a n-period annuity-immediate with
unit amount and the present value of an amount s k⌉
paid at time n + k. Note that
s k⌉
should not be taken as the future value of an annuity-immediate as k is less than
1, the time of the first payment.
When the equation of value does not solve for an integer n, care must be taken
to specify how the last payment is to be calculated. The example below illustrates
this problem.
Example 2.15: A principal of $5,000 generates income of $500 at the end of
every year at an effective rate of interest of 4.5% for as long as possible. Calculate
the term of the annuity and discuss the possibilities of settling the last payment.
Solution: The equation of value
500 a n⌉ 0.045 = 5,000
implies a n⌉ 0.045
=10, which can be solved to obtain n =13.5820. Thus, the
investment may be paid off with 13 payments of $500, plus an additional amount
A at the end of year 13. Computing the future value at time 13, we have the equation
of value
500 s 13⌉ 0.045 + A = 5,000 (1.045)13 ,
which implies A = $281.02. We conclude that the last payment is 500 + 281.02 =
$781.02. Alternatively, if A is not paid at time 13 the last payment B may be made
at the end of year 14, which is given by
B = 281.02 × 1.045 = $293.67.
Finally, if we adopt the approach in (2.46), we let n =13and k =0.58 so that the
last payment C to be paid at time 13.58 years is given by
[ (1.045) 0.58 ]
− 1
C = 500s k⌉
= 500 ×
= $288.32.
0.045