Question

A perpetuity pays $1 at the end of every year plus an additional $1 at the end of every second
year. The effective rate of interest is i = 5%. Find the present value of the perpetuity at time
0.

Answer 1

solution

At the end of every year, the perpetuity pays $1. On every second year, an additional $1 is paid. The payment series formed is;

$$(time,\ payment) =(1,\ 1), (2,\ 2),(3,\ 1),(4,\ 2),...$$

Note that at the end of times $time = 1,3,5,7,...$ , The perpetuity pays $1.

At the the end of times $time\ = 2,4,6,8,....$ , The perpetuity pays $2.

$$i= 0.05$$

Assume (series 1) is the series of $1 payments and (series 2) be the series of$2 payments. Series 1 and 2 are level annuities. The present value can be obtained separately for each series and the solutions added together to get the present value of the perpetuity

series 1: (time = 1,3,5,7,...)

$payments,\ p=1$ start at $time,\ t=1$ and occur every $2\ years.$

The present value:

$$present\ value =p* \frac{1+i}{(1+i)^2-1}$$

$$=1* \frac{1.05}{(1.05)^2-1}=10.2439$$

series 2: (time\ = 2,4,6,8,....)

$payments,\ p= 2$ start at $time,\ t= 2$ and occur every $2 \ years$

The present value:

$$present\ value=p* \frac{1}{(1+i)^2-1}$$ $$=2* \frac{1}{(1.05)^2-1}=19.5122$$

answer:

The present value of the perpetuity is the sum of the 2 series of payments

$$=10.2439+19.5122$$ $$=29.7561$$