Renee buys a perpetuity paying 1000 every two years starting

Renee buys a perpetuity paying $1,000 every two years, starting immediately. She deposits the payments into a savings account earning interest at an effective annual interest rate of 6%. Ten years later, before receiving the sixth payment, Renee sells the perpetuity based on an effective annual interest rate of 6%. Using proceeds from the sale plus the money in the savings account, Renee purchases an annuity paying P at the end of every three years for thirty years at an annual effective interest rate of 3%. Find P .

Solution

Value of initial perpetuity immediately after the 5

th

payment (or any other time) = 100 (1/i) = 100/.08 = 1250.

Exchange for 25-year annuity-immediate paying X at the end

of the first year, with each subsequent payment

increasing by 8%, implies

1250 (value of the perpetuity) must =

X (v + 1.08 v

2

+ 1.08

2

v

3

+ .....1.08

24

v

25

) (value of 25-year annuity-immediate)

= X (1.08

-1

+ 1.08 (1.08)

-2

+ 1.08

2

(1.08)

-3

+ 1.08

24

(1.08)

-25

)

(because the annual effective rate of interest is 8%)

= X (1.08

-1

+1.08

-1

+..... 1.08

-1

) = X [25(1.08

-1

)].

So, 1250 (1.08) = 25 X or X = 54