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牛魔王 · 2020年02月20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

老师，解析里那个total excess return 6%是哪里来的啊

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星星_品职助教 · 2020年02月21日

同学你好，

这个地方的描述比较晦涩。6%是benchmark return的9%-risk free rate 3%得到的。最后Sharpe ratio分子的7.217%的实际过程应该为=Rp-Rf==Rb+RA-Rf=9%+1.217%-3%。

但这个地方不是考点，而是验证Sharpe ratio确实等于0.365的过程，所以看一下就行。

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ivyisabelle · 2020年02月25日

老师，Indigo Fund的比例是1.014，benchmark 的比例是-0.014。那计算Rp时，为什么不是-0.014*9%+1.014*1.2%呢？您这里直接是9%+1.014*1.2%

星星_品职助教 · 2020年02月25日

算Rp的方法不是加权平均，而是用的excess return。这块不用去看，不是考点，考点是计算那两个系数。

李艳林 · 2020年03月20日

应该是benchemark的sr*标准差得到的吧。。。

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