Question

Jones purchased a perpetuity today for $7, 000. He will receive the first annual payment of $200 five years from now. The second annual payment will be $200 plus an amount C. Each subsequent payment will be the prior payment plus an additional constant amount C. If the effective annual interest rate is 4%, find C. I know the answer is 5.1 but I'm not sure how to get there.

Answer 1

The stream can be broken down into two streams:

First is constant payment of 200 every year starting from 5 years from now

Present value of perpetuity with first payment t years from now=Payment/rate*1/(1+rate)^(t-1)=200/4%*1/(1+4%)^4

Second is arithmetic gradient perpetuity of C starting 5 years from now

Present value of arithmetic gradient perpetuity starting t years from now=Payment/(rate^2)*1/(1+rate)^(t-1)=C/(4%^2)*1/(1+4%)^4

Given,

Present Value=7000

=>200/4%*1/(1+4%)^4+C/(4%^2)*1/(1+4%)^4=7000

=>C=(7000-200/4%*1/(1+4%)^4)*(1+4%)^4*(4%^2)

=>C=5.1024