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rmhamei · 2023年11月29日

能否把题干翻译一下？

  1. CFA I
  2. Quantitative

NO.PZ2015120601000010

问题如下：

At the end of the current year, an investor wants to make a donation of $20,000 to charity but does not want the year-end market value of her portfolio to fall below $600,000. If the shortfall level is equal to the risk-free rate of return and returns from all portfolios considered are normally distributed, will the portfolio that minimizes the probability of failing to achieve the investor’s objective most likely have the:

选项：

highest safety-ratio?
highest Sharpe ratio?

A.

No
Yes

B.

Yes
No

C.

Yes
Yes

解释：

C is correct.

The portfolio with the highest safety-first ratio minimizes the probability that the portfolio return will be less than the shortfall level (given normality). In this problem, the shortfall level is equal to the risk-free rate of return and thus the highest safety-first ratio portfolio will be the same as the highest Sharpe ratio portfolio.

这道题仍然是要从公式入手，SFR公式为 SFR=[E(Rp) – RL ] / σp 。所以可以看出，当threshold / minimum return 也就是RL被设定为 risk free return Rf时，这个公式恰好就等于Sharpe Ratio（SR）。所以SFR等于SR是有前提的。

Shortfall risk是SFR里Rp低于RL的风险(就是投资的收益率低于了最低要求的门槛收益率的风险)。所以要让shortfall risk最小，相当于让SFR最大，也就是让E(Rp)尽量的大于RL。由于这道题RL=Rf，所以也相当于让SR最大。

知识点都懂，不知道题干在问什么？

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1 个答案

品职助教_七七 · 2023年11月29日

嗨，从没放弃的小努力你好：


这道题的第一句话对于本题完全没有用处，给的信息也不全。可以直接忽略。

第二句话比较关键，说的是shortfall level 要和 the risk-free rate of return 相等（且return服从正态分布）。

最后一句话里说的.....that minimizes the probability of failing to achieve the investor’s objective，是要最小化没有达到投资者目标的概率。这就相当于要最小化shortfall risk，等价于要最大化SFR。所以要从选项中的highest safety-ratio这一列中选择为“Yes”的，排除A选项。

再看选项，除了SFR以外，还提到了Sharpe ratio。

由于SFR=[E(Rp) – shortfall level ] / σp，Sharpe ratio=[E(Rp) – risk-free rate of return ] / σp。可以通过公式看出，在shortfall level=risk-free rate of return这个条件下，SFR就等于SR。highest safety-ratio就相当于highest Sharpe ratio。故在B和C中选择C选项。


总之，题目就是在问当shortfall level=risk-free rate of return时，如果要最小化达不到投资者投资目标的概率，就必须要最大化该投资者资产组合中的SFR和SR。


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