Investment X and Investment Y are both growing perpetuities with an initial cash flow of $200. Both investments have the same interest rate(i) and cash flows. The present value of Investment X is $4,000, while the present value of Investment Y is $5,000. Which of the following is true?

Question

Investment X and Investment Y are both growing perpetuities with an initial cash flow of​ $200. Both investments have the same interest rate​(i) and cash flows. The present value of Investment X is​ $4,000, while the present value of Investment Y is​ $5,000. Which of the following is​ true?

BlackTom AI · Accepted Answer

Let’s break down what the problem says and how the present value of a growing perpetuity is determined.

Option a: 'The answer cannot be determined without knowing the interest rate for both investments.' In this setup, the two investments share the same interest rate i and the same cash flow pattern, only the growth rate g differs. Since PV of a growing perpetuity is PV = CF1 / (i − g), if CF1 and i are the same for both, differences in PV must come from g. Therefore, you can determine the relationship between growth rates from the PV values provided, so this statement is not correct.

Option b: 'Investment X has a lower growth rate than Investment Y.' Here, note that the present values are 4000 for X and 5000 for Y, with the same CF1 and same i. Since PV grows as i − g in the denominator, a larger PV implies a smaller i − g, hence a larger g. Thus, the investment with the higher PV (Y) must have the higher growth rate, and correspondingly X has the lower growth rate. This reasoning supports option b as true.

Option c: 'Investment X has a higher growth rate than Investment Y.' This would imply a smaller PV for X than Y if all else were equal, which contradicts the given PVs (4000 vs 5000). If X had a higher g, i − g would be smaller for X, leading to a larger PV for X, not 4000. Therefore, this option is inconsistent with the data.

Option d: 'With the same initial cash flow and the same interest rate, Investment X and Investment Y should have the same present value.' The premise states their PVs are different (4000 vs 5000), so this statement cannot be true. The difference in PV directly points to different growth rates, given CF1 and i are the same.

To summarize the reasoning: since CF1 and i are identical for both investments, PV is inversely related to i − g. The higher PV corresponds to the smaller i − g, which means the higher growth rate. Because Y has the higher PV (5000), Y must have the higher growth rate, and X the lower one. Consequently, option b is the consistent choice with the given data, while options a, c, and d conflict with the PV relationship or the provided numbers.

类似问题

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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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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%.

Investment X and Investment Y are both growing perpetuities with an initial cash flow of $200. Both investments have the same interest rate(i) and cash flows. The present value of Investment X is $4,000, while the present value of Investment Y is $5,000. Which of the following is true?

投资 X 和投资 Y 均为增长型永续年金，初始现金流均为 200 美元。两项投资的利率 (i) 和现金流相同。投资 X 的现值为 4,000 美元，而投资 Y 的现值为 5,000 美元。以下哪项是正确的？

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