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kkyy · 2020年02月14日

问一道题：NO.PZ2015121810000013

问题如下：

Which of the following pairs of weights would be used to achieve the highest Sharpe ratio and optimal amount of active risk through combining the Indigo Fund and benchmark portfolio, respectively?

选项：

A.

1.014 on Indigo and –0.014 on the benchmark

B.

1.450 on Indigo and –0.450 on the benchmark

C.

1.500 on Indigo and –0.500 on the benchmark

解释：

A is correct.

The optimal amount of active risk is:

{$table2}

The weight on the active portfolio (Indigo) would be 8.11%/8.0% = 1.014 and the weight on the benchmark portfolio would be 1 – 1.014 = – 0.014.

We can demonstrate that these weights achieve the maximum Sharpe ratio (of 0.365). Note that 8.11% is the optimal level of active risk, and that Indigo has an expected active return of 1.014(1.2%) = 1.217% over the benchmark (and a total excess return of 6.0% + 1.217% = 7.217%. The portfolio total risk is

$STD{(R_P)}^2=STD{(R_B)}^2+STD{(R_A)}^2=18.0^2+8.111^2=389.788$

Taking the square root, $STD(R_P)$ = 19.743, and the optimal Sharpe ratio is indeed 7.217/19.743 = 0.365.

考点：Optimal amount of active risk

解析：Optimal amount of active risk

$\sigma_A=\frac{IR}{SR_B/\sigma_B}=\frac{0.15}{0.333/18\%}=8.11\%$

Indigo Fund现在的active risk是8%，为了使active risk达到最优水平，就将Indigo Fund与benchmark再做组合，形成active risk最优的combined fund。

假设Indigo Fund的权重为c, 那么

$\sigma_A=c\sigma_A^{fund},\;8.11\%=c8\%,\;c=1.014$

因此，benchmark的权重为1-1.014=-0.014

请问一下，portfolio的active risk 等于各个资产的active risk乘以权重加和这是为什么呢？

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星星_品职助教 · 2020年02月14日

同学你好，

你提问的这个公式是在哪里看到的？

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