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吴超 Cassie · 2022年06月18日

关于方差到底用哪个公式求的问题

  1. CFA I
  2. Quantitative

NO.PZ2021062201000003

问题如下：

A two-stock portfolio includes stocks with the following characteristics:


What is the standard deviation of portfolio returns？

选项：

A.

14.91%

B.

18.56%

C.

21.10%

解释：

B is correct. The covariance between the returns for the two stocks is

Cov (R1,R2) = ρ (R1,R2) σ (R1) σ(R2) = 0.20 (12) (25) = 60.

The portfolio variance is：

σ2Rp=w12σ2(R1)+w22σ2(R2)+2w1w2Cov(R1,R2){\sigma ^2}{R_p} = w_1^2{\sigma ^2}({R_1}) + w_2^2{\sigma ^2}({R_2}) + 2{w_1}{w_2}Cov({R_1},{R_2})

=(0.30)2(12)2+(0.7)2(25)2+2(0.30)(0.70)(60)

=12.96 +306.25 +25.2

=344.41

The portfolio standard deviation is：

σ2(RP)=344.411/2=18.56%{\sigma ^2}({R_P}) = {344.41^{1/2}} = 18.56\%

知识点：Probability Concepts

现在碰到好几种求方差的题型：最常规的就是基于历史数据直接求得算数平均，也可以根据概率求加权平均，或者像本题一样用公式计算。请问这道题为什么不能用概率求加权平均呢，然后怎么样根据题目选择合适的方法计算方差，老师可不可以分别举一下例子，谢谢！

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1 个答案

星星_品职助教 · 2022年06月18日

同学你好，

本题要求的是组合的标准差，并非单个资产的标准差，所以需要代入的是两资产组合的公式来求。

题干明确给出概率，求单个资产标准差时，用加权平均来计算，概率为权重。

如果没有给出概率，只给了一堆离散的数字，按照算术平均来求。

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