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米其林 · 2023年05月13日

每年的PMT为什么都是700

  1. CFA I
  2. Quantitative

NO.PZ2017092702000019

问题如下：

A client invests €20,000 in a four-year certificate of deposit (CD) that annually pays interest of 3.5%. The annual CD interest payments are automatically reinvested in a separate savings account at a stated annual interest rate of 2% compounded monthly. At maturity, the value of the combined asset is closest to:

选项：

A.

€21,670.

B.

€22,890.

C.

€22,950.

解释：

B is correct,

as the following cash flows show:

The four annual interest payments are based on the CD’s 3.5% annual rate. The first payment grows at 2.0% compounded monthly for three years (where FV is future value): FVN = €700(1 +0.02/12 )3×12 FVN = 743.25 The second payment grows at 2.0% compounded monthly for two years: FVN = €700(1 +0.02/12 )2×12 FVN = 728.54 The third payment grows at 2.0% compounded monthly for one year: FVN = €700(1 +0.02/12 )1×12 FVN=714.13

The fourth payment is paid at the end of Year 4. Its future value is €700. The sum of all future value payments is as follows:

根据EAR的公式：（1+2%/12）^12=1+EAR，得到EAR=2.0184%。

然后：

1. 首先按照年金的方式来计算利息再投资的终值：N=4，I/Y=2.0184，PV=0，PMT=700，CPT FV=-2885.9208；其中700是20,000每年按照3.5%产生的利息，I/Y是转化的EAR，因为利息（PMT）的频率也是一年一付的。

2. 然后把期初的这20,000再加回去，得到22,885.92。（和选项B有写出入是因为四舍五入的问题，但不影响选出答案）

这道题第一年的利息是700， 第二年的不应该是20700*0.035 = 724.5吗？


这样的话，每一年的PMT就不是固定的700了。这个地方没太搞懂，希望老师解析一下，谢谢

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星星_品职助教 · 2023年05月15日

同学你好，

根据题干中的“The annual CD interest payments are automatically reinvested in a separate savings account ....”，可知第一个account中的利息700在产生的同时，就被自动移到第二个account中进行投资了。不再参与Account 1中的复利过程。

所以Account 1中一直就是本金20,000，只要产生了利息就会被移走，20,000的金额不会产生增加。

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