Mary would receive an annual cash flow at the end of each year that will continue indefinitely. The cashflow at the end of year 1 is $48,157. This cash flow is expected to grow at 4% per annum. Calculate the present value of these cash flows if the effective annual interest rate is 13%.

Round your answer to two decimal places.

Question

Mary would receive an annual cash flow at the end of each year that will continue indefinitely. The cashflow at the end of year 1 is $48,157. This cash flow is expected to grow at 4% per annum. Calculate the present value of these cash flows if the effective annual interest rate is 13%.

Round your answer to two decimal places.

BlackTom AI · Accepted Answer

For a perpetuity that begins one year from now and grows at a constant rate, the present value at time 0 is given by the growing-perpetuity formula: PV = C1 / (r − g), where C1 is the cash flow at the end of year 1, r is the effective annual discount rate, and g is the annual growth rate (with r > g).

Derivation and reasoning:
1. The cash flows are C1 at t=1, C1(1+g) at t=2, C1(1+g)^2 at t=3, and so on indefinitely.
2. Present value = sum_{t=1 to ∞} C1(1+g)^{t-1} / (1+r)^t.
3. Factor out C1/(1+r): PV = (C1/(1+r)) * sum_{t=1 to ∞} ((1+g)/(1+r))^{t-1}.
4. The infinite geometric series sum is 1 / (1 − (1+g)/(1+r)) = (1+r)/(r − g).
5. Multiplying through gives PV = C1/(r − g).

Apply values:
- C1 = 48,157
- r = 13% = 0.13
- g = 4% = 0.04
- r − g = 0.13 − 0.04 = 0.09

Compute PV:
PV = 48,157 / 0.09
Intermediate multiplication: 1 / 0.09 = 11.111111...
So PV = 48,157 × 11.111111... = 535,077.777...

Rounding to two decimals gives PV = 535,077.78.

Why this is correct and not other formulas:
- Using C0/(r − g) would be wrong because C1 is the first cash flow one period from now; C0 would be the cash flow at time 0 (not given).
- Using PV = C1/r (a level perpetuity formula) ignores growth and would produce a lower value.
- Using r + g or other denominators misconstrues the discount-growth relationship; the correct denominator is r − g and requires r > g (which holds here).

Conclusion: The present value is 535,077.78, 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%.

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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