# Proof that Sharpe ratio of the benchmark is related to the maximal information ratio and Sharpe ratio

I understand the economic logic behind it, that the active portfolio with the highest information ratio will also have the highest Sharpe ratio, but I can't see how $SR_B^2 = SR_P^2 - IR^2$

• It follows from "sum of independent variances" type reasoning. The portfolio return is equal to the benchmark return plus the tracking error. So $SR_P^2=SR_B^2+IR^2$ Commented Apr 26, 2017 at 8:40
• @noob2 that's the independence part I don't get, why wouldn't there be correlation between the bench and the active part? Commented Apr 26, 2017 at 8:42

This has been shown in Grinold & Kahn (1999), Active Portfolio Management (p. 137ff). First, write $$SR_P^2=SR_B^2+IR^2$$ as $$\left(\frac{f_Q}{\sigma_Q}\right)^2 = \left(\frac{f_B}{\sigma_B}\right)^2 + IR^2$$. Also note that the maximum information ratio is related to the portfolio's $$Q$$ Sharpe as follows: $$IR=\frac{\alpha_Q}{\omega_Q}=SR\cdot \frac{\omega_Q}{\sigma_Q}$$ where $$\omega_Q$$ is the residual risk. It is defined as $$\omega_Q=\sqrt{\sigma_Q^2-\beta^2_Q \sigma^2_B}$$ (see p. 50) where $$\beta_Q=\frac{Cov[r_{Q},r_{B}]}{\sigma^2_B}$$ is the beta of portfolio $$Q$$ and benchmark $$B$$.
\begin{align*} \left(\frac{f_Q}{\sigma_Q}\right)^2 &= \left(\frac{f_B}{\sigma_B}\right)^2 + IR^2 \\ &= \left(\frac{f_B}{\sigma_B}\right)^2 + \left(\frac{f_Q}{\sigma_Q} \right)^2 \left(\frac{\omega_Q}{\sigma_Q} \right)^2 \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} \cdot \frac{\sigma_Q^2-\beta^2_Q \sigma^2_B}{\sigma^2_Q} \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} - \beta^2_B \frac{f^2_Q\sigma^2_B}{\sigma^4_Q} \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} - \left(\frac{f_B\sigma^2_Q}{f_Q\sigma^2_B} \right)^2 \frac{f^2_Q\sigma^2_B}{\sigma^4_Q} \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} - \frac{f^2_B}{\sigma^2_B} \\ &= \frac{f^2_Q}{\sigma^2_Q} \end{align*}
Note that the third last step can be solved using the statement about portfolio $$Q$$'s holdings (which is a mix of benchmark $$B$$ and managed portfolio $$A$$) on page 136.