Deriving the Sharpe Ratio of an Optimally Active Portfolio
Summary
The note explains a relationship between the Sharpe ratio of an actively managed portfolio, the benchmark’s Sharpe ratio, and the manager’s information ratio. It derives the result by scaling active bets and expressing portfolio return as benchmark return plus scaled active return, with risk calculated from the benchmark and active risks.
The derivation assumes active returns are uncorrelated with the benchmark and that the manager can adjust leverage to choose the optimal active risk. Under those conditions, optimizing the scale of active bets yields the stated squared-Sharpe relationship. The discussion is conceptual and supplies no empirical evidence; the result depends on its assumptions and should not be treated as a general identity for portfolios with correlated active and benchmark returns or constrained leverage.
Key ideas
- The active portfolio return is modeled as benchmark return plus scaled active return.
- Portfolio risk combines benchmark and scaled active risk under the assumption that they are uncorrelated.
- The optimal active-bet scale depends on benchmark risk and Sharpe ratio, and on active risk and information ratio.
- The squared-Sharpe relationship holds under the stated independence and leverage assumptions.
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# Active Portfolio Management: What is the logic behind this equation?
# Active Portfolio Management: What is the logic behind this equation?
In the CFA Curriculum Level II Readings (link) it is stated without further comment that:
$(SR_{p})^2 = (SR_{b})^2 + IR^2 $
where,
$(SR_{p})$ = Sharpe Ratio of an actively managed portfolio;
$(SR_{b})$ = Sharpe Ratio of benchmark;
IR = Information Ratio
What is the justification for this statement? More specifically, how is this equation derived?
## Answer by Tim Wilding (score 1, accepted)
https://quant.stackexchange.com/a/39332
This is related to the question of how much active risk an active manager should take. The assumptions are that the active portfolio is not correlated with the benchmark, and that the manager can leverage up the active bets so that the portfolio is optimal.
Given those two assumptions, we can calculate a scale factor $c$ to determine how much active risk we should take to get the highest Sharpe ratio for the portfolio ($=\frac{\mu_p}{\sigma_p}$).
If we scale the active bets by $c$ to take on the optimal level of active risk, then the return of the portfolio will be: $\mu_p = \mu_b + c\mu_a$, and the risk of the portfolio will be: $\sigma_p = \sqrt{\sigma_b^2 + c^2\sigma_a^2}$. This has the solution that $c = \frac{\sigma_b IR} { \sigma_A SR_b}$.
Given the optimal level of active risk, then the Sharpe ratio for the portfolio satisfies $SR_p^2 = SR_b^2 + IR^2$.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.