Sharpe ratio, Treynor ratio, M² and Jensen's alpha

How do you judge whether a return was worth the risk?

Raw return is not enough: a fund that earned 12% by taking twice the market’s risk has not necessarily done well. Performance evaluation asks how a portfolio did, and four measures turn the answer into a risk-adjusted return . Two use total risk and two use systematic risk.

Risk-adjusted performance measuresexcess return relative to risk

• Total risk (standard deviation) Memorize use for a portfolio that is the investor’s whole wealth
  • Sharpe ratio Memorize (Rp − Rf) ÷ σp; a ranking
  • M² Memorize Sharpe × σm + Rf; a return, compare with Rm
• Systematic risk (beta) Memorize use for one part of a well-diversified portfolio
  • Treynor ratio Memorize (Rp − Rf) ÷ βp; a ranking
  • Jensen’s alpha Memorize Rp − CAPM return; a return, compare with zero

What is the Sharpe ratio?

The Sharpe ratio is the portfolio’s excess return per unit of total risk. It is the slope of the capital allocation line through the portfolio. A higher Sharpe ratio means more return for each unit of standard deviation.

$$S = \frac{R_p - R_f}{\sigma_p}$$

What is the Treynor ratio?

The Treynor ratio is the same idea with beta as the risk: excess return per unit of systematic risk.

$$T = \frac{R_p - R_f}{\beta_p}$$

What is M²?

A ratio ranks funds but does not say by how much they beat the market. M² turns the Sharpe ratio back into a return. Lever or de-lever the fund with the risk-free asset until its standard deviation equals the market’s, then read its return. The excess return is rescaled by σ_m ÷ σ_p.

$$M^2 = (R_p - R_f)\,\frac{\sigma_m}{\sigma_p} + R_f$$

Compare M² with the market return, not with zero. The difference is the M² alpha .

$$M^2\ \text{alpha} = M^2 - R_m$$

What is Jensen’s alpha?

Jensen’s alpha is the portfolio’s return minus the return the capital asset pricing model (CAPM) requires for its beta. It is the vertical distance between the portfolio and the security market line.

$$\alpha_p = R_p - \left[R_f + \beta_p\,(R_m - R_f)\right]$$

Which measure should you use?

Compare two funds with the market. The risk-free rate is 3%.

	Return	σ	β
Fund A	11%	18%	1.1
Fund B	9.4%	10%	0.8
Market	10%	15%	1.0

	Sharpe	Treynor	M² (M² alpha)	Jensen’s α
Fund A	8 ÷ 18 = 0.444	8 ÷ 1.1 = 7.27	0.444 × 15 + 3 = 9.67% (−0.33%)	11 − (3 + 1.1 × 7) = +0.30%
Fund B	6.4 ÷ 10 = 0.640	6.4 ÷ 0.8 = 8.00	0.64 × 15 + 3 = 12.60% (+2.60%)	9.4 − (3 + 0.8 × 7) = +0.80%
Market	0.467	7.00	10.00% (0)	0

Edit the table of returns, risks and betas and watch the lines move. The left chart ranks by total risk, the right chart by beta. Total-risk measures punish undiversified risk; beta-based measures ignore it.

Fund A beats the CAPM (positive alpha, Treynor above the market’s 7.00) but carries a lot of company-specific risk, so on total risk (Sharpe, M²) it lags the market. Fund B wins on every measure. If Fund A were an investor’s only holding, the Sharpe ratio says no. As one small sleeve of a diversified portfolio, Treynor and alpha say it is fine.

Sharpe and Treynor need a positive excess return to rank sensibly, and the Treynor ratio fails for negative betas.

Go deeper Why M² is a levered return, and where the Sharpe ratio comes from

To match the market’s risk, hold the fraction σm ÷ σp of the fund and put the rest in the risk-free asset (a fraction above 1 means borrowing). The return is (σm ÷ σp) × Rp + (1 − σm ÷ σp) × Rf, which rearranges to the M² formula. Because every fund is brought to the same risk, M² ranks funds exactly as the Sharpe ratio does.

The Sharpe ratio is the slope of the capital allocation line through the fund, and the market’s Sharpe ratio is the slope of the capital market line. Benchmark-relative measures such as the information ratio come later in the curriculum.

Exam tip Sigma goes with Sharpe and M². Beta goes with Treynor and Jensen. Compare M² with the market return and alpha with zero.