Question

A perpetuity will make annual payments with the first payment coming 12 years from now. The first payment is for $4500, and each payment that follows is $150 dollars more than the previous one. If the effective rate of interest is 4.8%, what is the present value of the perpetuity?

Answer 1

This growing perpetuity has two parts; one is fixed payment $4500 every year and second is amount of $150 growing every year. We can calculate the value of this growing perpetuity at the time of its first payment (12 years from now) with the help of following formula

Value of perpetuity (after 12 years) = C/ r + A / (i^2)

Where,

First payment C = $4,500

Interest rate r = 4.8%

Perpetuity growth amount A = $150 / year

Therefore

Value of perpetuity (after 12 years) = $4,500/ 4.8% + $150/ (4.8%^2)

= $4,500/ 0.048 + $150/ 0.002304

= $93,750.00 +$65,104.17

= $158,854.17

But this value is after 12 years from now, to calculate the present value we have to discount it at the interest rate of 4.8% per year.

Present Value of perpetuity = Value of perpetuity (after 12 years) / (1+ interest rate) ^ time period

= $158,854.17 / (1+4.8%) ^12

=$90,503.05

Therefore Present Value of perpetuity is $90,503.05.