The annual payment of growing perpetuity starts at $2,000 in year 4 and grows at a rate of 4% per year thereafter. If the discount rate is 8%, what is the present value of the perpetuity at time 0?

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BlackTom AI · Accepted Answer

To approach this problem, I start by identifying the structure of a growing perpetuity: the present value at the moment the perpetuity begins paying is equal to the first payment divided by (r − g), where r is the discount rate and g is the growth rate, provided r > g.

Option 1: 50,000.00. This value would be the correct present value immediately at the moment just before the first payment if we were calculating the value of the perpetuity at t = 3 (the moment just before year 4 payments begin) using the formula PV at t=3 = Payment1 / (r − g) with Payment1 = 2000, r = 0.08, g = 0.04, which gives 2000 / (0.08 − 0.04) = 50,000. However, this is the value at t = 3, not at time 0, so it must be discounted back three years to time 0.

Option 2: 25,000.00. This value seems to reflect a miscalculation such as erroneously using a 50,000 value divided by 2, or misapplying a factor of (1 + r) or (1 + g) incorrectly. It does not align with the correct PV formula for a growing perpetuity or the correct discounting horizon.

Option 3: 36,751.49. This number appears to come from discounting a value near 50,000 by 1.08^2 or from misplacing the timing (e.g., discounting by only two years instead of three). It does not match the proper three-year discounting from t = 3 to t = 0.

Option 4: 39,691.61. This is the value our calculation yields when we correctly apply both steps: (i) compute PV at t = 3 as 2000 / (0.08 − 0.04) = 50,000, and (ii) discount back three years to t = 0 using 1.08^3, giving 50,000 / 1.08^3 ≈ 39,691.61.

Putting it together: the perpetuity has first payment 2000 in year 4 and grows at 4% per year, with a discount rate of 8%. The value at t = 3 is 50,000. Discounting three years to time 0 yields 50,000 / 1.08^3 ≈ 39,691.61. The option that matches this result is the last one. The other options arise from misplacing the timing, misapplying the formula, or miscomputing the discounting period.

25400 Financial Literacy - Spring 2025 🔴 Practice Questions 2 Deferred Perpetuity

The annual payment of growing perpetuity starts at $2,000 in year 4 and grows at a rate of 4% per year thereafter. If the discount rate is 8%, what is the present value of the perpetuity at time 0?

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