Five years from today, you hope to donate $10,000 to the alumni association. Thereafter, you intend for your annual contributions to grow at a rate of 3 percent per year, forever. If you can earn 7 percent per year on your investments, how much must you invest today to fund this donation?

Question

BlackTom AI · Accepted Answer

We start by parsing what the question is asking and identifying the key values: the person plans to donate 10,000 in five years, and after that the annual contributions will grow forever at 3% per year. The investment return (discount rate) is 7% per year.

First, determine the value, at time 4, of the perpetuity that begins with a payment of 10,000 at year 5 and grows at 3% thereafter. A growing perpetuity starting with payment P at year 5 has present value (one period before the first payment) equal to P / (r − g), where r is the discount rate and g is the growth rate. Here P = 10,000, r = 0.07, g = 0.03, so the value at time 4 is 10,000 / (0.07 − 0.03) = 10,000 / 0.04 = 250,000.

Next, discount that amount back to today (time 0). We discount four years at 7% per year, since the value at time 4 needs to be brought to present value today. The discount factor is (1 + r)^4 = 1.07^4. Compute 1.07^4 ≈ 1.310796. Therefore, the present value today is 250,000 / 1.310796 ≈ 190,740.

Now, evaluate each option in light of this calculation:
- Option "$190,724": This is very close to 190,740; small differences arise from rounding intermediate steps. It aligns with the precise calculation when using standard rounding conventions.
- Option "$142,857": This number looks like 1/7 of 1,000,000 or a common division used in simple perpetuity problems, but it does not reflect the 3% growth and 7% discount interplay here.
- Option "$222,122": This would correspond to a higher present value, which would occur if either the initial payment were larger, the growth rate were smaller, or the discount rate were larger than in the given scenario; none of these apply to the problem as stated.
- Option "$250,000": This is exactly the value of the perpetuity at time 4 (P/(r−g)) with no discounting to present. It ignores bringing the amount back four years, so it overstates the required today investment.
- Option "$215,652": This figure exceeds the computed present value, suggesting a smaller discounting effect or a larger initial payment than specified; it does not match the precise calculation from the problem setup.

In summary, the only option that corresponds to correctly discounting the time-4 value of the growing perpetuity back to today, given P = 10,000, r = 7%, and g = 3%, is the approximate value around 190,740. The remaining choices do not align with the required combination of a growing perpetuity starting in year 5 and the 7% discount rate.

Five years from today, you hope to donate $10,000 to the alumni association. Thereafter, you intend for your annual contributions to grow at a rate of 3 percent per year, forever. If you can earn 7 percent per year on your investments, how much must you invest today to fund this donation?

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