Treynor Ratio, Jensen's Alpha and M² Calculator
Three ways to judge a portfolio's return against its risk: Jensen's alpha, the Treynor ratio and the Modigliani M², from the portfolio and market returns, beta and volatility.
Results
The dashed line marks the value you entered.
What this tool does
Jensen's alpha is what is left of the return after subtracting what the CAPM says the portfolio's market exposure deserved, so a positive alpha is return not explained by beta. The Treynor ratio divides the excess return by beta, so it treats only market risk as risk — fitting for one fund among many — whereas the Sharpe ratio and M² use total volatility. M² rescales the portfolio to the market's volatility by mixing it with cash or leverage, giving a return you can compare directly with the market's. Over short periods all three are mostly noise, so use several years.
Formula
Jensen α = Rp − [Rf + β(Rm − Rf)] · Treynor = (Rp − Rf) ÷ β · M² = Rf + (Rp − Rf) × σm ÷ σp
Variables
| Symbol | Meaning | Unit |
|---|---|---|
rp | Portfolio return over the period | % |
rf | Risk-free return over the same period | % |
b | Beta of the portfolio | — |
rm | Market (benchmark) return over the same period | % |
sp | Volatility of the portfolio (standard deviation) | % |
sm | Volatility of the market (standard deviation) | % |
A | Jensen's alpha | % |
TR | Treynor ratio (excess return per unit of beta) | % |
CE | Return the CAPM says it should have earned | % |
M2 | M² (return at the market's level of risk) | % |
MG | M² minus the market return | % |
Worked example
- Portfolio return over the period11 %
- Risk-free return over the same period3 %
- Beta of the portfolio1.2
- Market (benchmark) return over the same period9 %
- Volatility of the portfolio (standard deviation)16 %
- Volatility of the market (standard deviation)14 %
- Jensen's alpha0.80 %
- Treynor ratio (excess return per unit of beta)6.67 %
- Return the CAPM says it should have earned10.20 %
- M² (return at the market's level of risk)10.00 %
- M² minus the market return1.00 %
Limitations
- This is an informational calculator, not personalised financial advice. Rates, fees, taxes and contract conditions vary between institutions and countries.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.