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shuer · 2022年03月30日

这道题可否用通俗一点的理解解释一下?

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[11(1+r)Nr]+2000PV=A{\lbrack\frac{1-\frac1{{(1+r)}^N}}r\rbrack}+2000

PV=2000[11(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。

比如按照年金的思路,后付年金的现值PV,折算出来相当于是后10年保险公司每年给我2000,按照利率5%每年,所以现在我要一次性存15443.47,那么如果同样的领取方式变成先付年金,由于保险公司在年初就付第一笔2000,按照货币的时间价值来说相当于我提前一年支付,所以目前我要用后付年金当年的现价乘上当年的利率折算出来为先付年金的现价?

就是怎么能去更好的在现实例子中理解先付年金现值等于后付年金现值✖️(1+r)?

1 个答案
已采纳答案

星星_品职助教 · 2022年03月31日

同学你好,

以一个三期的先付年金为例,可以看出,三笔PMT分别在0,1,2这三个时间点。

如果把这三笔现金流视作一个三期的后付年金,那么此时现值是求到0' 这个点的,即图中第①步计算得到的PV' 。

但先付年金的现值是需要求到0时点的,而PV'和0时点的PV0差了一期,所以就有了图中的第②步计算,PV0=PV'×(1+r)。转化成文字就是 先付年金的现值等于同期数的后付年金×(1+r)


上述内容只需简单了解即可。这个关系其实非常好记。

因为先付年金的所有现金流都要比后付年金早发生,先付年金的现值一定会比后付年金的大。所以,后付年金需要多乘以一个(1+r)才能得到先付年金的现值。

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