Question

Today is January 1st and you have just won Publisher Clearinghouse’s grand prize of four perpetuities. The first perpetuity will pay you $2,000 every year on April 1st, with the first payment occurring exactly three months from today. The second perpetuity will pay you $3,000 every year on July 1st, with the first payment occurring exactly six months from today. The third perpetuity will pay you $4,000 every year on October 1st, with the first payment occurring exactly nine months from today. The fourth perpetuity will pay you $5,000 every year on January 1st, with the first payment occurring exactly twelve months from today. If you can borrow and lend at an APR of 12%, compounded quarterly, what is the time t=0 present value of this grand prize?

Answer 1

The first payment is received 3 months from today on 1st April. Hence the date today is 1st January

Annual Interest Rate = 12%

Number of quarters in an year = 4

Hence, quarterly interest rate = r = 12/4 = 3% or 0.03

First Perpetuity
Amount received on 1st April each year = P1 = $2000
Present Value of 1st Perpetuity = P1/(1+r) + P1/(1+r)5 + P1/(1+r)9 + ...
= [P1/(1+r)]/(1 - 1/(1+r)4)
= [2000/(1+0.03)]/(1 - 1/(1+0.03)4)
= $17412.75

Second Perpetuity
Amount received on 1st July each year = P2 = $3000
Present Value of 2nd Perpetuity = P2/(1+r)2 + P2/(1+r)6 + P2/(1+r)10 + ...
= [P2/(1+r)2]/(1 - 1/(1+r)4)
= [3000/(1+0.03)2]/(1 - 1/(1+0.03)4)
= $25358.38

Third Perpetuity
Amount received on 1st October each year = P3 = $4000
​​​​​​​Present Value of 2nd Perpetuity = P3/(1+r)3 + P3/(1+r)7 + P3/(1+r)11 + ...
= [P3/(1+r)3]/(1 - 1/(1+r)4)
= [4000/(1+0.03)3]/(1 - 1/(1+0.03)4)
= $32826.38

Fourth Perpetuity
Amount received on 1st January each year = P4 = $5000
​​​​​​​Present Value of 2nd Perpetuity = P4/(1+r)4 + P4/(1+r)8 + P4/(1+r)12 + ...
= [P4/(1+r)4]/(1 - 1/(1+r)4)
= [5000/(1+0.03)4]/(1 - 1/(1+0.03)4)
= $39837.84

Sum Total of all the prize = 17412.75 + 25358.38 + 32826.38 + 39837.84 = $115435.35

Hence, the present value of the grand prize = $115,435.35