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孙璐 · 2021年02月06日

老师,支付时的4年,为什么不是用BGN先付年金来计算呢?

  1. CFA I
  2. Quantitative

NO.PZ2015120604000170

问题如下：

A.J. plans to pay his child's tuition fee for four years starting 15 years from now. He estimates the fees is $22,000 per year. If the saving account interest rate is 5.5% annually, and how much payment A.J. should make at the end of each year from now?

选项：

A.

$3,441.23

B.

$3,800.07

C.

$4,009.08

解释：

B is correct.

Step 1: N = 4, FV = 0, I/Y = 5.5, PMT = 22,000, CPT PV = -77,113.30

Step 2: PV =0, N = 14, I/Y = 5.5, FV = 77,113.30, CPT PMT =-$3,800.07

老师,支付时的4年,为什么不是用BGN先付年金来计算呢?

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星星_品职助教 · 2021年02月06日

同学你好，

这道题是一个协会出题的固定表达方式，“ ......for four years starting 15 years from now”这句话指的就是这四笔现金流发生的时间点分别是15,16,17,18.（至于学费是用什么方式支付的不需要去管，因为反正最后现金流就发生在这四个时间点）

所以当已知四笔现金流的发生时间后，常规的计算方法就是按照后付年金来算这四笔现金流的现值，这样这四笔现金流的PV是计算到14时点的。

这样计算的好处是：

①不用反复调计算器。

②最终要求的问题是“how much payment A.J. should make at the end of each year from now”，即求PMT。由于15时间点学费现金流就开始流出了，所以这个PMT是付到14时点（第14年末）的，不然15时点就又存又付了。

这样将4笔学费的年金现值算到14时间点，可以和最后一笔PMT正好连上，14时间点的4笔年金的PV正好是下一步的FV。便于下一步继续计算。

可以参照一下最终的步骤。可以发现step 1的PV正好就是step 2的FV（忽略只表示方向的符号）

Step 1: N = 4, FV = 0, I/Y = 5.5, PMT = 22,000, CPT PV = -77,113.30

Step 2: PV =0, N = 14, I/Y = 5.5, FV = 77,113.30, CPT PMT =-$3,800.07

具体的现金流可参照下图。

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