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?

Question

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.

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?单项选择题

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