Jennifer has a deposit of $5,000,000 at St. George bank. Effective annual interest rate is 15%. Jennifer wants to withdraw a series of consecutive payments at the end of each year that continue indefinitely, and each payment grows at 3%. Calculate the value of the first withdrawal at the end of year 1.

Question

BlackTom AI · Accepted Answer

We start by identifying the structure of the cash flow and the relevant formulas.
The deposit is the present value of an infinite series of end-of-year withdrawals that grow by a fixed rate each year. For a growing perpetuity with initial payment X at the end of year 1 and growth rate g, discounted at the effective annual rate i, the present value at time 0 is X / (i - g) provided i > g.
Here, i = 0.15 (15%), g = 0.03 (3%), and the present value (the deposit) is 5,000,000. 
Set up the equation: 5,000,000 = X / (0.15 - 0.03) = X / 0.12.
Solve for X: X = 5,000,000 × 0.12 = 600,000.
Thus, the value of the first withdrawal at the end of year 1 should be 600,000.

Now evaluate the given options:
Option 1: $750,000.00
If the first payment were 750,000, the present value would be 750,000 / 0.12 = 6,250,000, which exceeds the 5,000,000 deposit. Therefore this cannot be correct.

Option 2: $600,000.00
This yields a present value of 600,000 / 0.12 = 5,000,000, which matches the deposit exactly. This is consistent with the growth rate and interest rate given.

Option 3: $743,552.40
PV would be 743,552.40 / 0.12 ≈ 6,196,270, far above the 5,000,000 deposit. This option overstates the initial withdrawal.

Option 4: $855,085.26
PV would be 855,085.26 / 0.12 ≈ 7,125,711, again well above 5,000,000. This also overstates the initial payment.

In summary, the correct approach is to equate the present value of the growing perpetuity to the deposit and solve for the first payment. The only option whose PV matches 5,000,000 is $600,000.00, while the other options imply larger PVs and are therefore incorrect.

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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Amanda would receive an annual cash flow of $100,000 at the end of each year that will continue indefinitely. This cash flow is expected to grow at 5% per annum. Calculate the present value of these cash flows if the effective annual interest rate is 10%.

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?

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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