When do ex-poste (Jensen's alpha) and Treynor disagree with Sharpe/m^2?

Jensen's alpha and Treynor ratio measure performance in different contexts compared to the Sharpe ratio or M² measure, particularly when it comes to risk. Jensen's alpha evaluates how much excess return a portfolio generates relative to its expected return, given its systematic risk, while the Treynor ratio assesses risk-adjusted performance based on systematic risk alone (using beta). The Sharpe ratio, on the other hand, accounts for total risk, combining both systematic and non-systematic risk.

The key aspect in which Jensen's alpha and Treynor may provide differing insights compared to the Sharpe ratio or M² arises when a manager relies heavily on non-systematic risk. If a manager is using a large amount of non-systematic risk, this means that their performance may be influenced more by security selection rather than market direction, which can result in a misleading interpretation of risk-adjusted returns. The Sharpe ratio will capture this total variability, including non-systematic risk, while Jensen's alpha and the Treynor ratio will primarily reflect performance concerning systematic risk alone.

This divergence becomes particularly pronounced because Sharpe and M² evaluate the total risk of the portfolio, making them sensitive to how much diversification is in place. In situations of limited diversification,