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?单项选择题
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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?单项选择题

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BlackTom AI · Accepted Answer

First, recall the formula for the present value of a growing perpetuity: PV = CF1 / (i - g), where CF1 is the next period cash flow, i is the discount rate, and g is the growth rate. Given both investments have the same initial cash flow of 200 and the same i, but different PVs (X = 4000 and Y = 5000), we can solve for the implied (i - g) for each: - For Investment X: 4000 = 200 / (i - g_X) => i - g_X = 200 / 4000 = 0.05 => g_X = i - 0.05. - For Investment Y: 5000 = 200 / (i - g_Y) => i - g_Y = 200 / 5000 = 0.04 => g_Y = i - 0.04. Now compare g_X and g_Y: - g_X = i - 0.05 - g_Y = i - 0.04 Since subtracting a larger number (0.05) from i yields a smaller growth rate than subtracting 0.04, it follows that g_X < g_Y. Thus, Investment X has a lower growth rate than Investment Y. This reasoning hinges on the inverse relationship between PV and (i - g) for a growing perpetuity with a fixed CF1: the higher the PV, the smaller (i - g) must be, which corresponds to a larger growth rate g when i is fixed. The option stating that Investment X has a lower growth rate than Investment Y aligns with these derived relationships. If you consider alternative interpretations or common pitfalls, remember that equal i and equal CF1 across both investments mean the difference in PV must come from the difference in (i - g). A larger PV implies a smaller (i - g), which translates to a larger g if i is fixed, supporting the conclusion that X’s growth rate is lower than Y’s as PV_X < PV_Y.

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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?

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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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