The Appraisal Ratio

A manager who simply holds more of the market and gets lucky in an up year can look skillful on a raw-return basis, even with no genuine stock-picking ability. The appraisal ratio was designed, originally by Treynor and Black, to separate skill from market exposure by measuring risk-adjusted alpha, the return that isn't explained by the manager's market beta at all.

The formula is $\text{Appraisal Ratio} = \dfrac{\alpha}{\sigma_\epsilon}$, where $\alpha$ (Jensen's alpha) is the manager's average return in excess of what their market beta alone would predict, and $\sigma_\epsilon$ is the standard deviation of the residual, the leftover, unexplained return after stripping out the market component. Dividing by residual risk rather than total risk means a manager isn't penalized for taking on ordinary market risk; only unrewarded, idiosyncratic noise around their alpha counts against them.

Suppose a manager's regression against the market produces an alpha of 2% per year, with a residual standard deviation of 4% (the year-to-year noise in that alpha once market moves are stripped out). Appraisal ratio = 2 / 4 = 0.5, a decent, though not exceptional, figure; ratios above roughly 0.5 are generally considered attractive for an active manager, since sustaining even a modest, low-noise alpha consistently is hard.

The appraisal ratio is closely related to the information ratio (which uses tracking error against a benchmark rather than beta-adjusted residual risk) and is the metric underlying the Treynor-Black model for how much a portfolio should tilt toward an active manager's views versus a passive index.