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Can someone help me prove the statement or share a link of the proof -

"The optimal amount of active risk is the level of active risk that maximizes the portfolio’s Sharpe ratio. This optimal amount of active risk is $\sigma_{A}^{*}=\frac{IR}{SR_{B}}\sigma_{B}$

where 1> $\sigma_{A}$ is the active risk of the portfolio 2> $\sigma_{B}$ is the standard deviation of the benchmark portfolio 3> $IR$ is the information ratio 4> $SR_{B}$ is the Sharpe ratio for the benchmark portfolio."

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  • $\begingroup$ Hi. Can you provide a link to where you've seen this, in order to get some more background information? :-) $\endgroup$
    – Pleb
    Aug 15, 2021 at 14:45
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    $\begingroup$ Grinold and Kahn book may have this (although I do not have a copy of the book with me at the moment). Another idea: If you plug in the definition of IR and SR, that may help you get started. $\endgroup$
    – nbbo2
    Aug 15, 2021 at 15:05
  • $\begingroup$ @Pleb I found it on the CFA level 2 material. They don't have the proof. Also, summary book (Schweser) don't have it too. In case you need the definition, here you go - 1> Sharpe ratio (SR) is calculated as excess return per unit of risk (standard deviation) $SR=\frac{(R_{P}-R_{F)}{\sigma_{P}$ 2> information ratio (IR) is the ratio of the active return to the standard deviation of active returns, which is known as active risk. You can use this link too - "breakingdownfinance.com/finance-topics/alternative-investments/…" $\endgroup$ Aug 15, 2021 at 18:12

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