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lcrcp3 · 2022年02月02日

我的思考过程哪步出问题了？

  1. CFA I
  2. Quantitative

NO.PZ2017092702000011

问题如下：

An investment of €500,000 today that grows to €800,000 after six years has a stated annual interest rate closest to:

选项：

A.

7.5% compounded continuously.

B.

7.7% compounded daily.

C.

8.0% compounded semiannually.

解释：

C is correct,

as shown in the following (where FV is future value and PV is present value): If: FVN = PV(1 + rs/m)mN, Then: (800,000500,000)12×6−1=rs2{(\frac{800,000}{500,000})}^\frac1{2\times6}-1=\frac{r_s}2    rs = 0.07988 (rounded to 8.0%)

这题应该带入选项试吧？带入中间B选项，PV=-500000，FV=800000，PMT=0，N=2190，CPT I/Y=0.021464 EAR=0.021464*365=7.83。也就是每日复利的EAR是7.83，计息次数越大EAR才应该越大，如果这么考虑不断复利的选项如果是8我就直接选了，为什么半年计息一次EAR却是8呢？

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星星_品职助教 · 2022年02月02日

同学你好，

对于stated annual rate和EAR的理解和计算公式有误。

PV=-500000，FV=800000，PMT=0，N=2190，CPT I/Y=0.021464这个列式里，计算出来的是题干隐含的每日利率为2.1464%，所以题干对应的

stated annual rate（非EAR）应为0.021464×365=7.8342%。

而题干中隐含的每日复利的EAR列式应为（1+0.021464）^365 -1 =8.15%。或者通过题干直接算出PV=-5，FV=8，PMT=0，N=6，CPT I/Y=8.15得到。

这道题需要通过比较题干中的EAR和选项中转化的EAR的方法得到答案：

1）B选项为每日计息一次的stated annual rate（非EAR）为7.7%，对应EAR计算为（1+7.7%/365）^365 -1 =8%，和题干中隐含的8.15%差距较大

2）C选项为半年计息一次的stated annual rate（非EAR）为8%，对应EAR计算为（1+0.08/2）^2 -1=8.16%。和题干中隐含的8.15%相差无几。

3）A选项转化为EAR后为 e^0.075 -1 =7.79%,同样和题干中隐含的8.15%差距较大

综上，应该选择B选项。

ps，计息频率大，对应EAR大前提是stated annual rate相同，例如B选项中7.7% compounded daily对应的EAR就要比7.7% compounded semiannually对应的EAR大。这个规律本题中用不上，因为三个选项中的stated annual rate都不同。



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