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Yuki119808 · 2023年03月20日

完全没有懂，可以仔细讲下步骤吗

  1. CFA I
  2. Quantitative

NO.PZ2021062201000011

问题如下：

A portfolio has an expected mean return of 8% and standard deviation of 14%. The probability that its return falls between 8% and 11% is closest to:

选项：

A.

8.5%

B.

14.8%

C.

58.3%

解释：

A is correct.

P(8% ≤ Portfolio return ≤ 11%) = N(Z corresponding to 11%) – N(Z corresponding to 8%).

For the first term, NORM.S.DIST(11% – 8%)/14% = 58.48%. To get the second term immediately, note that 8% is the mean, and for the normal distribution, 50% of the probability lies on either side of the mean.

Therefore, N(Z corresponding to 8%) must equal 50%, So, P(8% ≤ Portfolio return ≤ 11%) = 0.5848 - 0.50 = 0.0848, or approximately 8.5%.


基于下列总公式：

P(8% ≤ Portfolio return ≤ 11%) = N(Z corresponding to 11%) – N(Z corresponding to 8%).

本公式第一部分：利用excel中NORM.S.DIST这个公式，我们解得：58.48%

本公式第二部分，8%为均值，且为正态分布，则50%的概率落在均值两侧，所以，N(Z corresponding to 8%)=50%，

带入总公式：P(8% ≤ Portfolio return ≤ 11%) = 0.5848 - 0.50 = 0.0848 = 8.5%

考察的哪部分的知识点

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星星_品职助教 · 2023年03月20日

同学你好，

本题考察正态分布标准化。

题目要计算收益率落在8%和11%之间的概率。由于普通正态分布不能查表，所以需要将8%和11%这两个数字标准化后，通过查表来得到概率。

1）对11%的处理，代入标准化的公式 （X-μ）/σ，可得(11% – 8%)/14%=0.2143。根据0.2143查Z表。由于z表只能精确到两位小数，所以可以得到Z(0.2143)的值在0.5832（对应Z=0.21）和0.5871（对应Z=0.22）之间


2）对于8%的处理比较简单，由于8%正好是均值，所以处于分布的一半，即50%。

3）最后用11%对应的概率（约为58.32%），去减掉8%对应的概率50%，即可得到答案近似为8.32%，选择最靠近的8.5%即可。

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