A wealthy donor invests $500,000 into an endowment fund that will pay a growing annual scholarship in perpetuity, starting one year from today. The fund earns an annual return of 6%, and the scholarship amount will grow by 2% each year to keep up with inflation. What will be the scholarship payment paid in year 3?[Fill in the blank]单项选择题
A
a. $20,808
B
b. $20,000
C
c. $20,400
D
d. $22,472

Question

A wealthy donor invests $500,000 into an endowment fund that will pay a growing annual scholarship in perpetuity, starting one year from today. The fund earns an annual return of 6%, and the scholarship amount will grow by 2% each year to keep up with inflation. What will be the scholarship payment paid in year 3?[Fill in the blank]单项选择题

A

a. $20,808

B

b. $20,000

C

c. $20,400

D

d. $22,472

BlackTom AI · Accepted Answer

We start by identifying the key financial relationships in this problem. The endowment funds a perpetuity that grows each year by a fixed rate, while the fund earns a constant return that sets the present value.
Option A describes the year-3 payment, so we need to determine the base level S1 (the year-1 payment) and then scale up by two compounding steps of the 2% growth rate to reach year 3.
First, recall the formula for the present value of a growing perpetuity: PV = P1 / (r − g), where P1 is the payment in year 1, r is the discount rate, and g is the growth rate of payments.
Here, r = 6% and g = 2%, so r − g = 0.06 − 0.02 = 0.04. The donor funds 500,000, so 500,000 = P1 / 0.04, which gives P1 = 500,000 × 0.04 = 20,000. This means the year-1 scholarship is $20,000.
Now proceed year by year with the 2% growth:
- Year 2 payment = Year 1 × (1 + g) = 20,000 × 1.02 = 20,400.
- Year 3 payment = Year 2 × (1 + g) = 20,400 × 1.02 = 20,808.
Let's evaluate the other options to see why they don't align with the calculations:
- Option B: $20,000 would be the year-1 payment, not year 3, and it ignores growth to year 3.
- Option C: $20,400 corresponds to the year-2 payment, not year 3, since it equals Year 1 grown once by 2%.
- Option D: $22,472 is higher and would imply a larger base P1 or a higher growth rate; applying 2% growth twice from 20,000 would not reach 22,472.
Therefore, the year-3 scholarship payment, computed as 20,400 × 1.02, equals $20,808, matching Option A.

类似问题

Investment X and Investment Y are both growing perpetuities with an initial cash flow of $200. Both investments have the same interest rate(i) and cash flows. The present value of Investment X is $4,000, while the present value of Investment Y is $5,000. Which of the following is true?

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投资 X 和投资 Y 均为增长型永续年金，初始现金流均为 200 美元。两项投资的利率 (i) 和现金流相同。投资 X 的现值为 4,000 美元，而投资 Y 的现值为 5,000 美元。以下哪项是正确的？

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