A benchmark has a Sharpe ratio of 0.4. An active overlay on top of it delivers 2% alpha with 4% tracking error, uncorrelated with the benchmark. What is the highest Sharpe ratio the combination can reach?

A benchmark has a Sharpe ratio of 0.4. An active overlay on top of it delivers 2% alpha with 4% tracking error, uncorrelated with the benchmark. What is the highest Sharpe ratio the combination can reach?

Approach: Form the information ratio of the overlay, then use the fact that squared Sharpe ratios of uncorrelated return streams add.

0.64. The overlay's information ratio is alpha over tracking error = 2/4 = 0.5. When the active return is uncorrelated with the benchmark the squared Sharpe ratios add, so the best attainable is sqrt(0.4^2 + 0.5^2) = sqrt(0.41) = 0.640. The optimum runs the active risk at IR/S_b = 1.25 times the benchmark volatility, so an overlay sized to 4% tracking error against a 10% benchmark is running at a third of the ideal size and reaches only 0.56. The information ratio is invariant to that scaling while the Sharpe ratio is not, which is why the information ratio measures the skill in the overlay and the Sharpe ratio measures what the client ends up holding.

Follow-up: What is the answer when the active return has a correlation of 0.3 with the benchmark?

Key concepts: information ratio, tracking error, sharpe ratio, active risk.