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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BlackTom AI · Accepted Answer

先把题目信息整理清楚：两项投资都是增长型永续年金，初始现金流都是 200 美元，现值分别为 4000 美元（投资 X）和 5000 美元（投资 Y），且两者的折现率 i 与现金流增长率 g 不同，但对两项投资的 i 相同。

对增长型永续年金的一般公式是：PV = C1 / (i - g)，其中 C1 是第一期的现金流，i 是贴现率，g 是增长率。已知 C1 = 200，因此对两项投资可以写成：4000 = 200 / (i - g_X) 和 5000 = 200 / (i - g_Y)。

接下来通过等式比较来推断各自的 (i - g) 值：
- i - g_X = 200 / 4000 = 0.05
- i - g_Y = 200 / 5000 = 0.04
因此 g_X = i - 0.05，g_Y = i - 0.04。

为了比较增长率的大小，可以把这两个表达式直接比较：g_Y 与 g_X 的差值为 g_Y - g_X = (i - 0.04) - (i - 0.05) = 0.01，即 g_Y 比 g_X 高 0.01。

也就是说，投资 Y 的增长率大于投资 X 的增长率，反过来可以说投资 X 的增长率低于投资 Y 的增长率。

如果题干还给出其他选项来判断，它们通常会涉及对 i、g 的关系的误解，比如将现值直接与增长率正相关等错误观点。具体来说：
- 错误理解之一可能是以为现值越大，增长率越高，但在上述公式中，现值与 (i - g) 成反比关系，且 i 相同，因此更高的现值对应更小的 i - g 值，并不直接等同于更高的增长率。
- 另一个常见误解是忽略增速对分母的影响，只看分子 200 的恒定，忽略了 i 与 g 的配对关系。

综上，给定的现值对比清晰地表明 g_Y > g_X，因此结论是投资 X 的增长率低于投资 Y。

MCD2170 金融基础 - 第二学期 - 2025年

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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