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Hydefu · 2022年08月26日

如何使用计算器计算？

  1. CFA I
  2. Quantitative

NO.PZ2017092702000013

问题如下：

At a 5% interest rate per year compounded annually, the present value (PV) of a 10-year ordinary annuity with annual payments of $2,000 is $15,443.47. The PV of a 10-year annuity due with the same interest rate and payments is closest to:

选项：

A.

$14,708.

B.

$16,216.

C.

$17,443.

解释：

B is correct.

The present value of a 10-year annuity (A) due with payments of $2,000 at a 5% discount rate is calculated as follows: PV = $16,215.64.

PV=A[1−1(1+r)Nr]+2000PV=A{\lbrack\frac{1-\frac1{{(1+r)}^N}}r\rbrack}+2000

PV=2000[1−1(1+0.05)90.05]+2000PV=2000{\lbrack\frac{1-\frac1{{(1+0.05)}^9}}{0.05}\rbrack}+2000

PV = $16,215.64. Alternatively, the PV of a 10-year annuity due is simply the PV of the ordinary annuity multiplied by 1.05: PV = $15,443.47 × 1.05 PV = $16,215.64.

无论是求PV还是FV，Annuity due的值都相当于对应期数的Ordinary Annuity的值再往后复利一期。即可以先求出Ordinary Annuity的PV，在乘以1+r，就是对应的Annuity due的PV。

对于本题而言，Ordinary Annuity的PV直接给出，所以就用给出的15,443.47*（1+0.05）即可得到对应annuity due的PV。即答案B。

在计算ordinary pv的时候，计算器按N=10，I/Y=5，PV=15443.47，PMT=2000，算出来FV=50311.57037

然后再改成BGN模式，N=9,I/Y=5，PMT=2000，FV=50311.57037,为什么PV算出来的和答案不太一样？

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已采纳答案

星星_品职助教 · 2022年08月26日

同学你好，

1）这道题不需要计算FV。

2）本题只有年金的PMT现金流，期末没有单独的现金流，即FV=0。所以，PV=15443.47是根据FV=0计算出来的数值。即N=10，I/Y=5，FV=0，PMT=2000， CPT .PV= -15443.47。

不能把根据FV=0计算出来的PV=15443.47再代回去再求一次FV。



星星_品职助教 · 2022年09月01日

@Marissa

这道题计算FV的思路是错误的，不应该去计算FV，直接根据Ordinary Annuity的PV ×（1+r）=对应的Annuity due的PV 就可以了。

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本题中的年金是10笔期间现金流，在期初和期末都没有独立的现金流存在。所以，如果计算此年金的现值，则FV是0。题干中给出的后付年金的PV是15443.47就是这样计算出来的。即N=10，I/Y=5，FV=0，PMT=2000， CPT PV= -15443.47。

不能把根据FV=0计算出来的PV值再代回去再计算一次FV。

同理，如果要计算这笔现金的FV，则PV=0，此时后付年金的FV为：N=10，I/Y=5，PV=0，PMT=2000， CPT FV=-25,155.7851。

------

如果要计算本题先付年金的PV，首先要将计算器调成BGN模式，和上面计算普通年金的PV相比，按键是不变的，同样还是N=10，I/Y=5，FV=0，PMT=2000， 此时CPT PV=- 16,215.6434，即答案的B选项。注意此时FV仍然是0.



Marissa · 2022年09月01日

继续提问：老师我陷入的同样误区 先求普通年金FV 再求先付年金PV 结果算出来和A相似。可是还转过来 为啥不能这么算的原理。

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