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%.

Question

BlackTom AI · Accepted Answer

For a perpetuity that grows at a constant rate, the present value is given by the growing-perpetuity formula: PV = C1 / (r − g), where C1 is the cash flow at the end of the first period, r is the effective discount rate, and g is the growth rate.

1) Relevant knowledge points and concepts
- Growing perpetuity: an infinite series of cash flows that start at C1 and grow by a constant rate g each period.
- Formula derivation uses an infinite geometric series and requires r > g for convergence.

2) Complete reasoning process (step by step)
- First cash flow at t = 1 is C1 = $100,000.
- Growth rate g = 5% = 0.05.
- Discount (effective annual) rate r = 10% = 0.10.
- Since r > g (0.10 > 0.05), the series converges and the formula applies.

3) Derivation / intermediate steps
PV = sum_{t=1}^∞ C1 * (1+g)^{t-1} / (1+r)^t
   = C1/(1+r) * sum_{t=0}^∞ [ (1+g)/(1+r) ]^t
The infinite geometric sum = 1 / [1 − (1+g)/(1+r)] = (1+r) / (r − g)
Thus PV = C1/(1+r) * (1+r)/(r − g) = C1/(r − g).

Plugging numbers:
PV = 100,000 / (0.10 − 0.05)
   = 100,000 / 0.05
   = 2,000,000
So PV = $2,000,000.00.

4) Why this answer is correct
The derivation shows the present value equals the first payment divided by the excess of the discount rate over the growth rate. With the given values, the algebra yields $2,000,000 exactly.

5) Why other potential answers would be incorrect
- Using the non-growing perpetuity formula C/r would give 100,000 / 0.10 = $1,000,000 (incorrect because it ignores growth).
- Using (g − r) in the denominator or forgetting r > g would produce a negative or divergent result (invalid here).
- Any other numerical mistake in substitution or arithmetic would not match the derived formula result.

6) Conclusion
The present value of the growing perpetuity is $2,000,000.00, which matches the provided correct answer.

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%.单项选择题

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