Lucy has a deposit of $148,994 at Westpac bank. Effective annual interest rate is 15%. Lucy wants to withdraw a series of consecutive payments at the end of each year that continue indefinitely, and each payment grows at 2%. Calculate the value of the first withdrawal at the end of year 1.

Round your answer to two decimal places.

Question

Lucy has a deposit of $148,994 at Westpac bank. Effective annual interest rate is 15%. Lucy wants to withdraw a series of consecutive payments at the end of each year that continue indefinitely, and each payment grows at 2%. Calculate the value of the first withdrawal at the end of year 1.

Round your answer to two decimal places.

BlackTom AI · Accepted Answer

Present value of a growing perpetuity is PV = C1 / (r − g), so the first withdrawal C1 equals PV × (r − g).

1. Relevant concepts
- A perpetuity with payments that grow at constant rate g each period and are paid at the end of each period has the PV formula PV = C1 / (r − g), where C1 is the payment at the end of year 1, r is the effective interest rate, and the formula requires r > g for convergence.
- Here PV = 148,994, r = 0.15 (15%), g = 0.02 (2%), and payments are end-of-year, growing indefinitely.

2. Derivation (brief) and reasoning
- Start from PV = sum_{t=1 to ∞} C1(1+g)^{t-1} / (1+r)^t.
- Factor C1/(1+r) and recognize a geometric series with ratio (1+g)/(1+r).
- Sum = C1/(1+r) × 1 / [1 − (1+g)/(1+r)] = C1/(r − g). Thus C1 = PV × (r − g).

3. Step-by-step calculation (all intermediate steps)
- Compute r − g = 0.15 − 0.02 = 0.13.
- Multiply PV by this difference: C1 = 148,994 × 0.13.
- 148,994 × 0.10 = 14,899.40
- 148,994 × 0.03 = 4,469.82
- Sum = 14,899.40 + 4,469.82 = 19,369.22

4. Why this is correct
- The growing-perpetuity formula exactly matches the situation: infinite, end-of-year payments with constant growth and a constant effective interest rate. Using PV = C1/(r−g) directly yields C1 and produces the computed value.

5. Common incorrect approaches
- Using C1 = PV × r (i.e., treating payments as level perpetuity) would give 148,994 × 0.15 = 22,349.10, which ignores growth and overstates the first payment. Another mistake is failing the r > g requirement; here r (0.15) > g (0.02), so the formula applies.

Conclusion: The first withdrawal at the end of year 1 is $19,369.22, which matches the provided 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%.

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