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Deriving the Optimal Active Risk from Information and Benchmark Ratios

Article Quant Q&A · Author: Stannis John
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Summary

This item presents a portfolio-allocation claim: the active-risk level that maximizes a portfolio's Sharpe ratio is proportional to benchmark volatility, with the scale set by the information ratio divided by the benchmark Sharpe ratio. It defines active risk as the portfolio's active-risk measure, benchmark risk as the benchmark's standard deviation, and supplies names for the two ratios. The question asks for a proof or a reference but includes neither.

The formula points toward balancing expected active return against the risk added by deviating from the benchmark. However, the document does not state the assumptions needed for such a result, such as how active returns relate to benchmark returns or whether the ratios and risks use consistent units and horizons. It provides no derivation, evidence, or worked example, so the expression should be treated as a claim to verify rather than a universally applicable rule. Its usefulness lies in identifying the proposed relationship and the need to establish the optimization conditions.

Key ideas

  • The stated result links optimal active risk to the information ratio and benchmark Sharpe ratio.
  • The proposed active-risk scale is benchmark volatility multiplied by the ratio of those quantities.
  • The document requests a proof but provides no derivation or supporting reference.
  • Applying the claim requires assumptions about returns, risk measures, and consistent measurement periods.

Tags

Portfolio constructionRisk managementStatistics
Full text
# Optimal active risk


# Optimal active risk












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."

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.