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要紧雅冠 · 2023年05月06日

第二步就PV时为啥是^4呢

  1. CFA I
  2. Quantitative

NO.PZ2017092702000009

问题如下：

A perpetual preferred stock makes its first quarterly dividend payment of $2.00 in five quarters. If the required annual rate of return is 6% compounded quarterly, the stock’s present value is closest to:

选项：

A.

$31.

B.

$126.

C.

$133.

解释：

B is correct.

The value of the perpetuity one year from now is calculated as: PV = A/r, where PV is present value, A is annuity, and r is expressed as a quarterly required rate of return because the payments are quarterly.

PV = $2.00/(0.06/4) PV = $133.33.

The value today is (where FV is future value) PV = FV(1 + r)–N

PV = $133.33(1 + 0.015)–4

PV = $125.62 ≈ $126

</span>12\frac12

题目说makes its first quarterly divident payment of $2 in five quarters.

请问此处的5个Q该怎么理解呢？

我的理解是133/(1+0.015)^5

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2 个答案

YunPlanner · 2023年08月23日

题目中说的perpetual preferred stock makes its first quarterly dividend payment of $2.00 in five quarters,所以是在第5个季度末发放的dividend, 可以视为是ordinary annuity，而不是annuity due.


如果是Ordinary annuity, 通过perpetual annuity公式来计算时, PV=A/r, A是第5季度末的发放的dividend, r是一个季度的rate of return, 算出来的PV是第5季度的PV, 我们可以叫做PV(5), 同时也是第四季度末的FV值, 我们也可以称之为FV(4), 结果是133,那从0时刻到4时刻只需要折现4次, 所以计算时用的是PV(0)) = $133.33(1 + 0.015)^(–4), 用公式表示就是PV(0)(1+r)^(4)=FV(4), 也可以写成FV(n) = PV * (1 + r)^n. 其中，r是每期折现率，n是期数。这道题如果计算时折现5次就是把perpetual annuity公式计算的得出的PV(5)当初了FV(5), 实际上这道题里是ordinary annuity, 所以PV(5)是FV(4).


如果这道题变一下, 是季度初就发放, 那么这道题就成为了annuity due, 这时就要记着annuity due要比ordinary annuity需要多折现一期, 之前ordinary annuity 的公式FV(n) = PV * (1 + r)^n就变成了FV(n) = PV * (1 + r)^(n+1), 这个时候, 我们再计算0时刻的PV值就是PV = $133.33(1 + 0.015)^(–5)了.


所以做这类型的题, 首先要确定是ordinary annuity还是annuity due.

星星_品职助教 · 2023年05月07日

同学你好，

本题已回复，总结如下：

题干中“.....makes its first ...payment....in five quarters”指的是首笔PMT发生在5时点。所以首次折现需要折到4时点。即133.33是时间轴上4时点的值。上述逻辑同非递延的普通年金首笔现金流在1时点，折现需要折到0时点。

所以，133.33再折现到0时点就只需要再往前折4期即可。


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