Mr. Taylor wants to provide money in his will for an annual bequest to his oldest daughter Alison. That bequest will provide $5000 in the first year (t=1), and will grow by 4% per year, forever. If the interest rate is 6% per year (i.e., EAR = 6%), how much must Mr. Taylor provide today to fund this bequest?

Question

BlackTom AI · Accepted Answer

Present value of a growing perpetuity with first payment at t = 1 is C1 / (r − g).

Relevant concepts:
- A perpetuity is an infinite series of cash flows.
- When those cash flows grow at a constant rate g and the discount rate is r (with r > g), the PV at t = 0 of the first payment C1 occurring at t = 1 is PV = C1 / (r − g).
- This formula is derived from summing an infinite geometric series.

Step-by-step derivation and calculation:
1. Cash flow at time t is: C_t = C1 * (1 + g)^(t−1). Here C1 = $5,000 and g = 4% = 0.04.
2. Discount factor for time t is (1 + r)^t with r = 6% = 0.06.
3. PV = sum_{t=1 to ∞} [C1 *(1+g)^(t−1) / (1+r)^t].
   Factor out C1/(1+r): PV = C1/(1+r) * sum_{t=1 to ∞} [(1+g)^(t−1)/(1+r)^(t−1)].
   Let q = (1+g)/(1+r). Then PV = C1/(1+r) * sum_{k=0 to ∞} q^k = C1/(1+r) * 1/(1−q).
   Substitute q and simplify: PV = C1/(1+r) * 1/(1 − (1+g)/(1+r)) = C1/(r − g).
4. Plug in numbers: r − g = 0.06 − 0.04 = 0.02.
   PV = $5,000 / 0.02 = $5,000 * 50 = $250,000.

Why the correct answer is correct:
- The formula directly applies since payments start at t = 1 and grow forever at 4% while the discount rate is 6%; r > g so the series converges and yields $250,000.

Why the other options are incorrect:
- $200,000 would result if one incorrectly used r − g = 0.025 (or otherwise mis-specified rates).
- $160,000 and $220,000 correspond to other incorrect differences between r and g and do not follow from the correct formula with r = 6% and g = 4%.

Conclusion: The present value required today to fund the growing annual bequest is $250,000, which matches the provided answer.

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