Treynor Ratio

The numerator is the portfolio's realised return over the measurement period less the average risk-free rate over the same period, so both terms must be stated on a consistent frequency and, if annualised, annualised the same way. The denominator is the portfolio's beta, normally estimated by regressing portfolio excess returns on benchmark excess returns; the slope coefficient of that regression is the beta used.

Geometrically, the ratio is the slope of the line joining the risk-free asset to the portfolio in mean-return versus beta space. The security market line is the equivalent slope for the market portfolio, whose beta is one, so the market's Treynor ratio simply equals the market risk premium. A portfolio plotting above the security market line has a steeper slope from the risk-free point and therefore a higher Treynor ratio than the market.

The measure rests on the CAPM proposition that only non-diversifiable risk earns compensation. Residual, firm-specific volatility is assumed to vanish once the portfolio is combined with other holdings, so it does not belong in the risk denominator. This is why the measure is appropriate for evaluating a sleeve, mandate or fund that sits inside a broader diversified portfolio, and inappropriate for an investor whose entire wealth is in that single portfolio.

Jensen's alpha and the Treynor ratio share the same risk model but answer different questions. Alpha is the vertical distance above the security market line, expressed in return units, and is therefore scale-dependent on beta. Dividing alpha by beta and adding the market risk premium reproduces the Treynor ratio, which is why alpha alone can rank portfolios differently once leverage or beta exposure differs across candidates.

Practitioners select the risk measure that matches the role the portfolio plays. Sharpe divides excess return by total standard deviation and suits a stand-alone allocation or an entire fund. Treynor divides by beta and suits a component holding. M-squared restates Sharpe performance as a risk-adjusted return in percentage terms against the benchmark's volatility, which many committees find easier to interpret than a bare ratio. The appraisal ratio, alpha divided by residual standard error, isolates skill relative to the unsystematic risk taken to obtain it and is common in evaluating active satellite managers.

All of these are ex post statistics computed from a return history. Their reliability depends on sample length, the stability of beta, and whether the benchmark genuinely reflects the manager's opportunity set. A misspecified benchmark distorts beta and therefore the ranking, an issue Roll's critique raises for every beta-based measure.

Candidates should be able to compute the ratio from a table of returns, a risk-free rate and a beta, and to recover any missing input by rearranging the relationship. Equally common are comparison questions: given Sharpe and Treynor figures for several funds, identify which manager is preferred under a stated mandate, and explain why the two measures may disagree when portfolios differ in diversification.

Interpretation matters more than arithmetic. A higher value means more excess return per unit of market exposure, but the number carries no meaning in isolation; it must be compared with the market's own ratio or with peer portfolios measured over the same window with the same benchmark. Candidates should also be able to state the conditions under which the measure fails: betas near zero, negative betas, negative excess returns, poorly diversified portfolios and short or unstable estimation samples.