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Lee13 · 2023年02月16日

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:

  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）

老师您好，这道题逻辑还是没弄清楚，烦请详细推导一下。还有可以直接用公式计算吗？公式怎么列呢？EAR在什么情况下需要计算，什么情况下可以直接用期利率，这类题型每回都不太懂怎么计算，谢谢！

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3 个答案

星星_品职助教 · 2023年02月17日

回复提问3：

使用什么利率取决于具体题目。如果期间利率可以直接计算，则不需要转化。如果题目的计算要求以年度为单位，则可以考虑转化为EAR。

以本题为例，中文解析中已说明，转化为EAR的原因是PMT=700是年度现金流的概念，这种情况下，I/Y就必须也得是年利率。

核心原则是N、PMT、I/Y这三者的期间必须一致。

星星_品职助教 · 2023年02月17日

回复提问2：

年金的公式上课没讲过，统一都用计算器来计算。但如果将账户2中每年收到的700看成单一现金流，则可用公式逐一计算FV后再加总（英文解析的思路）。

以第一年末收到的首笔700为例，这笔现金流到第四年末时复利了三年。

如果直接以0.02/12这个月利率来计算，则一共复利了3年×12个月=36个月。写成公式为：

也可以将0.02的stated annual interest rate转化为EAR来计算，此时期间为年度，也就是复利了3年。这种方式写成公式后，实际形式和上面是完全一样的。

由此可得到首笔700的现金流复利三年后的终值为743.25。

同理，第二笔700（第二年末收到的700）到了第四年末复利了两年，第三笔700复利了一年，第四笔700就在第四年末。四笔700分别复利到第四年末的FV值如下图。此后加总，再加上账户1中不变的20,000。即可得到和计算器同样的答案。

星星_品职助教 · 2023年02月17日

同学你好，

这几个问题在中英文解析中已经全部涉及到，此后提问请具体标明是哪里没弄清楚/解析哪里不懂。

-------

回复提问1：

本题为第一个账户里可以产生每年3.5%的利息，即每年可以产生€20,000×3.5%=€700。这个利息要直接拿出去在第二个账户中以“2% compounded monthly”的利率进行投资。要求算出采用上述投资方式，资产在四年后一共会变成多少钱。

由于第一个账户中的€20,000产生了利息后就会被拿走，所以这个数字是一直不变的。变化的是每年都会在第二个账户中新增的700。对于第二个账户而言，相当于第一年末打入首个700，第二年末打入第二个700……以此类推。单独来看，账户2的现金流如下：

可以看出，这是一笔四年期，PMT为700的年金。进而通过N=4，I/Y=2.0184（转化的EAR），PV=0，PMT=700，得到 FV=-2885.9208（现金流符号只代表方向）。

再加上第一个账户中不变的€20,000，得到答案22,885.9208。选择近似的B选项。



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