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fengsj · 2020年01月28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

Sharp ratio

portfolio 与 risk free asset 组合，VAR(Rp-Rf)=VAR(Rp)，Trecking error = standard deviation of portfolio

Infomation ratio

portfolio 与 benchmark 组合，VAR(Rp-Rb)=VAR(Rp)+VAR(Rb) ,Trecking error = active risk of portfolio，不等于standard deviation of portfolio

这样理解正确吗？如果正确题目里的active risk数字不正确。

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星星_品职助教 · 2020年02月04日

同学你好，

这道题不建议这么去想，这种题目有标准做法，按照上课老师讲的思路就行（也就是解析里的思路）。

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