Relating Benchmark Sharpe, Portfolio Sharpe, and Information Ratio
Summary
The document outlines a derivation connecting a benchmark's Sharpe ratio, a portfolio's Sharpe ratio, and the portfolio's information ratio. It begins with the identity that the squared portfolio Sharpe ratio is the sum of the squared benchmark Sharpe ratio and squared information ratio. It then expresses information ratio using residual risk and relates that risk to total portfolio volatility, benchmark volatility, and portfolio beta.
Substitution into the identity is intended to show how the residual-risk and beta terms reconcile with the benchmark contribution, leaving the portfolio Sharpe relationship. The answer attributes the result to a portfolio management text and refers to a construction in which a portfolio combines benchmark and managed holdings. The excerpt gives an algebraic argument rather than empirical evidence, and it does not spell out all assumptions behind the identity or the portfolio construction. Its notation also shifts in the displayed steps, so readers should check definitions and intermediate algebra against the cited source before applying the result.
Key ideas
- The stated identity decomposes squared portfolio Sharpe into benchmark Sharpe and information ratio contributions.
- Information ratio is related to residual risk and the portfolio's Sharpe ratio.
- Residual risk is defined using portfolio variance, benchmark variance, and portfolio beta.
- The derivation is theoretical and relies on specific portfolio definitions and assumptions.
- The displayed notation warrants checking against the cited reference before reuse.
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Full text
# Proof that Sharpe ratio of the benchmark is related to the maximal information ratio and Sharpe ratio
# 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 $
## Answer by oronimbus (score 2)
https://quant.stackexchange.com/a/76880
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 $B$.
Then, it follows that:
$$ \begin{align*} \left(\frac{f_Q}{\sigma_Q}\right)^2 &= \left(\frac{f_B}{\sigma_B}\right)^2 + IR^2 \\ &= \left(\frac{f_B}{\sigma_B}\right)^2 + \left(\frac{f_Q}{\sigma_Q} \right)^2 \left(\frac{\omega_Q}{\sigma_Q} \right)^2 \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} \cdot \frac{\sigma_Q^2-\beta^2_Q \sigma^2_B}{\sigma^2_Q} \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} - \beta^2_B \frac{f^2_Q\sigma^2_B}{\sigma^4_Q} \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} - \left(\frac{f_B\sigma^2_Q}{f_Q\sigma^2_B} \right)^2 \frac{f^2_Q\sigma^2_B}{\sigma^4_Q} \\ &= \frac{f^2_B}{\sigma^2_B} + \frac{f^2_Q}{\sigma^2_Q} - \frac{f^2_B}{\sigma^2_B} \\ &= \frac{f^2_Q}{\sigma^2_Q} \end{align*} $$
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.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.