# 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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## 1 Answer

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 $SR_p^2 = SR_b^2 + IR^2$.

• Thanks a lot, Tim. I'm unable to understand the logic behind the equation that you've stated here for the "risk of the portfolio". Will you be able to elaborate, please! – Anshul Modi Apr 20 '18 at 9:18
• 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
• This makes a lot of sense. I was absolutely confused as to how this makes sense algebraically. But now it makes sense. Here, the σa=c*std(ra) is squared, which is why 'c' is squared in the main equation. Thanks a lot, Tim & Alex! – Anshul Modi Apr 21 '18 at 16:28