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如此_AnnieCcc · 2021年02月02日

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  1. CFA I
  2. Quantitative

NO.PZ2017092702000006

问题如下：

For a lump sum investment of ¥250,000 invested at a stated annual rate of 3% compounded daily, the number of months needed to grow the sum to ¥1,000,000 is closest to:

选项：

A.

555.

B.

563.

C.

576.

解释：

A is correct.

The effective annual rate (EAR) is calculated as follows:

EAR = (1 + Periodic interest rate)m – 1   EAR = (1 + 0.03/365)365 – 1   EAR= (1.03045) – 1 = 0.030453 ≈ 3.0453%. Solving for N on a financial calculator results in (where FV is future value and PV is present value): (1 + 0,030453)N = FVN/PV = ¥1,000,000/¥250,000）So,N = 46.21 years, which multiplied by 12 to convert to months results in 554.5, or ≈ 555 months.

这道题我的答案是我自己豆出来的：pv=-250k，fv=1000k，pmt=0，i/y=3%/365=0.008219算出来n=16867～然后16867除以365乘以12就等于A～～～～～我这里不知道为什么要除以365再乘以12～～怎么去理解？n要和i/y保持一致对吧～所以乘以365？然后按月就除以12～～～～我知道这么算怎么去理解？

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2 个答案

星星_品职助教 · 2021年02月21日

@Bonnie8916

同学你好，

用EAR/12得到的“月利率”会体现不出按月复利的效果。

EAR是逐月“复利”后的结果，每个月都会复利一次，一共复利12次。而直接除以12是一种“单利”去年化的方式。

以本题为例，EAR=3.0453%。如果假设月利率等于3.0453%/12=0.2538%的话。那我们其实又可以计算一个新的“EAR”，即“新EAR”=（1+0.2538%）^12 -1=3.0882%。和之前的EAR=3.0453%就不一样了。“新EAR”一定更大，因为这里体现出了复利的效果。

------------------------

可以用过如下方式从EAR得到月利率。

（1+月利率）^12=1+EAR=1+3.0453%。可得月利率等于0.2503%

此时新的计算器按键为：PV=-250000， I/Y=0.2503，PMT=0，FV=1000000， CPT N=554.54（月）.

同样可以得到答案。但不如直接计算出46.21年后再乘以12计算简便。

-----------------------

所以我们不会用EAR/12这种复利和单利混在一起的方式，只有名义利率（stated annual rate）才可以用stated annual rate/12=实际月利率。

可以这么做的原因是所谓的“stated annual rate”的来源就是通过实际月利率×12得到的，所以才可以反算。

星星_品职助教 · 2021年02月02日

同学你好，

N，I/Y和PMT三者要保持一致，所以你计算的“pv=-250k，fv=1000k，pmt=0，i/y=3%/365=0.008219算出来n=16867”这个过程是一致的没问题的。由于这里I/Y用的是日利率，所以计算出来的也是N=16867（天）。

但答案要求的是多少个“月”，所以“除以365再乘以12”就是把天数转化为月数的过程。

要注意这个时候不能直接用16867去除以30，因为把月设定为都是30天是不合理的，所以需要先转成年过渡一下，再转成月。

Bonnie8916 · 2021年02月21日

请问那为什么不可以直接用EAR/12作为I/Y呢

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NO.PZ2017092702000006 问题如下 For a lump sum investment of ¥250,000 investea stateannurate of 3% compounily, the number of months neeto grow the sum to ¥1,000,000 is closest to: A.555. B.563. C.576. A is correct. The effective annurate (EAR) is calculatefollows: E= (1 + Perioc interest rate)m – 1   E= (1 + 0.03/365)365 – 1   EAR= (1.03045) – 1 = 0.030453 ≈ 3.0453%. Solving for N on a financicalculator results in (where FV is future value anPV is present value): (1 + 0,030453）N = FVN/PV = (¥1,000,000/¥250,000）So,N = 46.21 years, whimultiplie12 to convert to months results in 554.5, or ≈ 555 months. 这题我EAR已经算出来是3.045，带入计算器知四求一不知道为什么算出来是46.21.

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