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

Use the growing perpetuity formula: PV = C1 / (i − g), where PV is present value, C1 is the payment at the end of year 1, i is the effective annual interest rate, and g is the annual growth rate of the payments.

1. Relevant knowledge points and concepts
- A perpetuity that starts with payment C1 at the end of year 1 and whose payments grow at a constant rate g each year has present value PV = C1 / (i − g), provided i > g.
- This formula comes from summing an infinite geometric series: PV = Σ_{t=1 to ∞} C1(1+g)^{t−1} / (1+i)^t = C1/(i−g).

2. Step-by-step reasoning
- Given: PV = $5,000,000, i = 15% = 0.15, g = 3% = 0.03.
- Rearranged formula: C1 = PV × (i − g).

3. Calculations with intermediate steps
- Compute the spread: i − g = 0.15 − 0.03 = 0.12.
- Multiply by PV: C1 = 5,000,000 × 0.12 = 600,000.
- So the first withdrawal at the end of year 1 equals $600,000.00.

4. Why this is correct
- The formula directly equates the infinite sum of discounted growing payments to the known present value. Solving for C1 yields the exact first payment that makes the perpetuity’s present value equal to $5,000,000 when discounted at 15% with 3% growth.

5. Why other common answers would be incorrect
- Using a non-growing perpetuity formula (C1 = PV × i) would give $750,000, which ignores growth and thus overstates C1.
- Using i + g or reversing the subtraction would produce incorrect spreads and wrong results.
- Ignoring that i must exceed g would also be invalid (here i > g, so the formula applies).

Conclusion: The first withdrawal is $600,000.00, which matches the provided correct answer.

类似问题

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%.

A project has a required return of 12.6 percent, an initial cash outflow of $42,100, and cash inflows of $16,500 in Year 1, $11,700 in Year 2, and $10,400 in Year 4. What is the net present value?

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