标准差计算

问题如下：

Question

A risk manager would like to calculate the coefficient of variation of a portfolio. The following table reports the annual returns of the portfolio and of the risk-free rate over the most recent five years:

The coefficient of variation of the portfolio is closest to:

选项：

A.1.00. B.0.74. C.0.90.

解释：

Solution

A is correct. First calculate the sample mean return as follows:

ˉX=(4.0%-1.0%+7.0%+11.0%+2.0%)/5=23.0%/5=4.6%$\begin{array}{c}\bar{X}=\frac{\left(4.0\%?1.0\%+7.0\%+11.0\%+2.0\%\right)}{5}\\ =\frac{23.0\%}{5}\\ =4.6\%\end{array}$

Then calculate the sample standard deviation with the following formula:

s =[(0.00004+0.00314+0.00058+0.00410+0.00068) /(5-1) ] ^1/2 = 4.62%

The coefficient of variation (CV) is calculated with the following formula:

CV = s/X=4.62%/4.6%$\frac{s}{\bar{X}}=\frac{4.62\%}{4.6\%}$ = 1.0

B is incorrect. It is the Sharpe ratio, not the coefficient of variation. First calculate the mean annual risk-free return over the five years:

ˉRF=2.0%+1.5%+1.0%+1.0%+0.5%5${\bar{R}}_F=\frac{2.0\%+1.5\%+1.0\%+1.0\%+0.5\%}{5}$ = 1.2%

Then calculate the Sharpe ratio with the following formula:

S_h = ˉRp-ˉRFsp=4.6%-1.2%4.62%$\frac{{\bar{R}}_p?{\bar{R}}_F}{s_p}=\frac{4.6\%?1.2\%}{4.62\%}$ = 0.74

C is incorrect. In the formula for the standard deviation, it uses n instead of “n - 1” in the denominator:

s = [(0.00004+0.00314+0.00058+0.00410+0.00068) /5 ] 1/2 = 4.13%

Then, calculate the CV:

CV = 4.13%/4.6%$\frac{4.13\%}{4.6\%}$ = 0.90