Question

Consider a perpetuity whose payments at the end of each year are R, R+p, R+2p,..., R+(n-1)p, R+np, R+np,... . The payments increase by a constant amount p until they reach R+np, after which they continue without change. Show that the discounted value A of such a perpetuity at rate i per annum is given by:

A = (R+pa(n)i) / i

Answer 1

The formula for determining A is given by:

A=$\frac{R}{(1+i)^1}$ +$\frac{R}{(1+i)^2}$ +$\frac{R}{(1+i)^3}$ ......+$\frac{R}{(1+i)^n}$ =$\sum^\infty_{n=1}$ $\frac{R}{(1+i)^n}$ =$\frac{R}{i}$ ................equation 1

Where R = the payment or receipt each period

i = the interest rate per payment or receipt period 

The discount at a perpetuity rate i per annum is calculated by;

$pa(n)^{i}$ ......................................................equation 2

Where pa= the amount

i = the interest rate per payment or receipt period

n=number of years

Inserting equation 2 into value of R in equation 1 to get the new value of R at a discounted rate, we get the discounted value A as:

A = $\frac{(R+pa(n)^{i})}{i}$