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ditto · 2023年10月22日

怎么用数学公式推导证明正确选项

NO.PZ2022090804000010

问题如下:

Erin Chou is reviewing a profitable investment that has a conventional cash flow pattern. If the cash flows for the initial outlay and future after-tax cash flows all double, Chou would predict that the IRR would:

选项:

A.increase and the NPV would increase. B.stay the same and the NPV would increase. C.stay the same and the NPV would stay the same.

解释:

Solution

B is correct. The IRR would stay the same because both the initial outlay and the after-tax cash flows double, so the return on each dollar invested would remain the same. All the cash flows and their present values double. The difference between the total present value of the future cash flows and the initial outlay (the NPV) also doubles.

助教老师在其他问题下面的解答:https://class.pzacademy.com/qa/18056 ;提问:

  1. 0 = NPV 是什么意思, 题目中没有说NPV =0 呀,
  2. double老师写的是CF都乘以2, 这个时候NPV也乘以2是新的数值了, 为什么能得出IRR不变了。 (问题1中npv =0 那么乘以2就可以约去,可是NPV很少等于0)
  3. npv如何变化是从公式中怎么得出来的没有看懂


1 个答案
已采纳答案

王琛_品职助教 · 2023年10月23日

嗨,爱思考的PZer你好:


1)0 = NPV 是什么意思, 题目中没有说NPV =0 呀

NPV 等于 0 时的折现率,是 IRR

题目没有说 NPV = 0,但是问了 IRR 的变化情况

所以我们先写出 IRR 的定义,就是当折现率等于 IRR 时,NPV = 0

然后再考虑现金流的变化,对这个等式的影响

2)double老师写的是CF都乘以2, 这个时候NPV也乘以2是新的数值了, 为什么能得出IRR不变了。 (问题1中npv =0 那么乘以2就可以约去,可是NPV很少等于0)

NPV 是很少等于 0,或者说只有当折现率等于 IRR 时,NPV 才等于 0

或者我们换一个举例

期初现金流是 -100,项目只有一期现金流,CF1 = 150

那么,IRR 的计算方式是:0 = -100 + 150/(1+IRR)

得到 IRR = 50%

如果现金流均翻倍,即

期初现金流是 -200,项目只有一期现金流,CF1 = 300

那么 IRR 的计算方式是:0 = -200 + 300/(1+IRR)

公式变形,即 0 = 2 x [-100 + 150/(1+IRR)]

等号两边同时除以 2,得到

0 = -100 + 150/(1+IRR)

会发现,和现金流翻倍前的公式一样,所以 IRR 是不变的

也就是说,无论现金流翻几倍,IRR 都不变

3)npv如何变化是从公式中怎么得出来的没有看懂

假设还是之前的举例,折现率为 10%

现金流翻倍前,NPV1 = -100 + 150/(1+10%)

现金流翻倍后,NPV2= -200 + 300/(1+I0%)

公式变形,把 2 提出来,即 NPV2 = 2 x [-100 + 150/(1+10%)]

中括号里面的数据,刚好是 NPV 1

即:NPV2 = 2 x NPV1

所以 NPV 翻倍

----------------------------------------------
虽然现在很辛苦,但努力过的感觉真的很好,加油!

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