Question

please explain the ubderlined step and include the formula used to get there

3. A perpetuity-immediate pays 100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. The annual effective rate of interest is 8%. Calculate X. (A) 54 (B) 64 (C) 74 (D) 84 (E) 94 PV = 100 w =1250 - the present value of the original perpetuity immediately after the 5" 0.08 108 1. payment. The present value of the new 25 year annuity is X- 1.08 1.08 1.089 1.08 Equation of values: X 25 =1250, 1.08 X = 54 Answer: A Wh

Answer 1

Since it is perpetuity with constant $ 100 payment per year,the present Value of perpetuity at interest rate =r:

(100/(1+r))+(100/((1+r)^2))+……….up to infinity=100/r

In this case interest rate of perpetuity is =8%=0.08

Value of the perpetuity at the end of year5=100/0.08=$1250

This amount is exchanged for another perpetuity paying X in the end of 6^th year.

Payment at end of 7^th year =8% greater than X=1.08*X

Payment at end of 8^th year =8% more than the payment in 7^th year =1.08*1.08*X=(1.08^2)*X

Payment at the end of 10^th year =(1.08^3)*X

The 25 year annuity will have last payment =(1.08^24)*X (The first payment was X)

Present Value of Cash Flow =(Cash Flow)/((1+r)^N)

r=interest rate =8%=0.08

N=Year of Cash Flow

We calculate the Present Value at end of year5 of the new annuity (with payment of X in the first year)and lasting for another 25 years

Since this is exchanged with the original constant payment perpetuity of $100 per year (Having a Present Value of $1250 at end of year5),

The Present Value of this new annuity should be =$1250

Present Value of new annuity at end of year 5=(X/1.08)+(((X*1.08)/(1.08^2))+(((X*(1.08^2))/(1.08^3))+ ……..+(((X*(1.08^24))/(1.08^25)=$1250

(X/1.08)+(X/1.08)+(X/1.08)+……………..+(X/1.08) ….(25 times)=$1250

25X/1.08=1250

X=(1250*1.08)/25=$54

ANSWER:(A)54