# Active Portfolio Management: What is the logic behind this equation? [closed]

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?

## closed as unclear what you're asking by phdstudent, vonjd, amdopt, LocalVolatility, HelinApr 18 '18 at 23:34

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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 $SR_p^2 = SR_b^2 + IR^2$.
• Sorry, should have been clearer. The “risk of the portfolio” is simply the standard deviation of the excess returns for the active manager $\sigma_p = std(r_p)$. If the standard deviation of the active portfolio’s returns is $\sigma_a$, then the standard deviation of the leveraged active portfolio returns is $c\sigma_a$, where $c$ is the amount by which we have scaled up the active portfolio. Given that the risk of the benchmark is $\sigma_b$ and the active returns are not correlated with the benchmark, then $\sigma_p^2 = \sigma_b^2 + c^2\sigma_a^2$. – Tim Wilding Apr 20 '18 at 17:15
• Perhaps one way to look at it is that the portfolio consists of a passive part which replicates the benchmark and leveraged active bets which are assumed to be orthogonal to the benchmark (hence finding $\sigma_p$ by adding the variances and taking the square root is valid). – Alex C Apr 20 '18 at 23:45