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蓝莓💭 · 2020年09月17日

问一道题：NO.PZ2017092702000012 [ CFA I ]

  1. CFA I
  2. Quantitative

问题如下：

A sweepstakes winner may select either a perpetuity of £2,000 a month beginning with the first payment in one month or an immediate lump sum payment of £350,000. If the annual discount rate is 6% compounded monthly, the present value of the perpetuity is:

选项：

A.

less than the lump sum.

B.

equal to the lump sum.

C.

greater than the lump sum.

解释：

C is correct.

As shown below, the present value (PV) of a £2,000 per month perpetuity is worth approximately £400,000 at a 6% annual rate compounded monthly. Thus, the present value of the annuity (A) is worth more than the lump sum offer. A = £2,000 r = (6%/12) = 0.005 PV = (A/r) PV = (£2,000/0.005) PV = £400,000

请问这题里面的6%为什么不先计算出EAR，再除以12得到实际每月的利率呢

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3 个答案

星星_品职助教 · 2021年11月13日

@Sylvia💝

1) annual discount rate在这里就是 stated annual rate，或者说用什么单词描述不重要，重要的是题干描述了一个名义上的年利率后面跟着一个计息频率，即这里的“compounded monthly”，这种情况下那个名义的年利率（无论用哪个单词形容）都必须转化；

2）可以转化为年利率EAR，但重复一遍此前的回复“EAR本身就是实际利率的概念，不需要再除以12得到实际利率了”。EAR已经是转化后的结果，可以直接（拿掉百分号后）代入I/Y，这是以一年为一期时的转化方法，通过EAR计算出来的N的数字对应的就是多少“年”，如果是PMT就是“年度”PMT；

3）可以转化为期间利率，例如本题就是用6%/12，最终的I/Y就是0.5。此时I/Y是月利率的概念，如果是算N，出来后的数字就是多少个月；如果算PMT，出来的就是月度PMT；

4）回到本题，最终要求的是“the present value”，所以以上两种转化形式都是可以的。但如果有题目要求求N是多少年，这时候就要转化为EAR输入计算器；如果需要求“月度PMT”是多少，那这时候I/Y就要用期间月度利率，不用去计算EAR。

星星_品职助教 · 2020年10月06日

@Jm

对的，EAR是effective “annual”rate，如果用EAR做I/Y，那么对应的N就是多少“年”，PMT也是每年付多少钱

星星_品职助教 · 2020年09月17日

同学你好，

EAR本身就是实际利率的概念，不需要再除以12得到实际利率了。此外，EAR是个实际“年”利率，并不是实际月利率。

需要除以12得到实际月利率的是stated annual rate。因为stated annual rate本身就是个“假的”非实际利率，是通过实际月利率×12得到的，所以将stated annual rate除以12就可以反求出实际月利率。

在本题里，6%就是stated annual rate的概念，所以直接除以12就可以得到实际月利率。不需要特意去转化EAR了

Jm · 2020年10月03日

换句话说，用得上EAR的情况一定是以”年”为一期来算 这样说对吗？

Sylvia💝 · 2021年11月13日

老师我还是不懂，这里“If the annual discount rate is 6% compounded monthly“ 如果把annual discount rate换成anual stated rate，那我就需要先处于12，算出EAR，再除以12来得到实际r来算吗？

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