1. A perpetuity will pay X every year. The effective annual interest rate is 7.5
ID: 2383658 • Letter: 1
Question
1. A perpetuity will pay X every year. The effective annual interest rate is 7.5%. The
present value of this perpetuity 4 years before the first payment is 125,000. Find X.
2.Suppose you deposit $20,000 at the end of each of the next 30 years into a retirement
account. Immediately after your last deposit, you take the entire accumulated value in
your account and purchase a 20-year annuity, which will pay you X at the beginning of
each year for 20 years. The price of this 20-year annuity is equal to the present value (at
the time you purchase the annuity) of the 20 annual cash flows. The effective annual
interest rate throughout the entire 50-year period is 10%. Find X.
Explanation / Answer
1. Present value of perpetuity = amount/interest rate
amount = X. interest rate = 0.075
PV = X/0.075. This is the present value at time 0 (i.e start of year 1 when). The 1st payment is made at the end of year 1. 4 years befor end of year 1 is 1-4 = -3. Thus, we have to find the value of perpetuity 3 years before time point 0.
Amount at point 0 = X/0.075, time = 3 years, rate = 7.5%. discount rate = 1+7.5% = 1.075
PV at time point -3 = amount/(discount rate)^time = 125,000
(X/0.075)/(1.075^3) = 125,000
(X/0.075)/1.2422 = 125,000
(X/0.075) = 155,287
OR x = 155,287 * 0.075 = 11,646.53
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