M-Squared (M2) Formula
M-squared (the Modigliani-Modigliani measure) restates the Sharpe ratio as a return rather than a ratio. It is the return a portfolio would have earned if it had been levered or de-levered to match the market's volatility. Because it is expressed in percentage points over the benchmark, it is far easier to interpret than a raw Sharpe number while preserving the same ranking.
Formula
Copy the exact expression or work through it step by step below.
M2 = R_f + Sharpe_p x sigma_m
M2 excess = M2 - R_m = (Sharpe_p - Sharpe_m) x sigma_m
where Sharpe_p = (R_p - R_f)/sigma_p Variables
Sharpe_p
Portfolio Sharpe ratio
The portfolio's excess return over the risk-free rate divided by its own volatility. M-squared scales this by the market's volatility to put it in return units.
sigma_m
Market volatility
Standard deviation of the benchmark's returns. The portfolio is hypothetically levered until its volatility equals sigma_m, making it directly comparable to the market.
R_f
Risk-free rate
Return on the riskless asset, used both in the Sharpe ratio and as the base the scaled risk premium is added to.
R_m
Market return
Benchmark return. Subtracting it from M2 gives the M-squared excess, the volatility-matched outperformance in percentage points.
Step By Step
- 1
Compute the portfolio Sharpe ratio from its excess return and volatility.
Portfolio excess return 8%, volatility 16%: Sharpe = 0.08/0.16 = 0.50.
- 2
Identify the market's volatility.
Market volatility is 12%.
- 3
Scale the portfolio Sharpe by market volatility and add the risk-free rate.
M2 = 3% + 0.50 x 12% = 3% + 6% = 9%.
- 4
Subtract the market return for the M-squared excess, the volatility-matched outperformance.
If the market returned 8%, M2 excess = 9% - 8% = +1%.
Worked Example
Comparing a volatile fund to the market on equal-risk terms
Portfolio excess return / volatility
8% / 16%
Market return / volatility
8% / 12%
Risk-free rate
3%
Portfolio Sharpe = 0.08 / 0.16 = 0.50. M2 = R_f + Sharpe_p x sigma_m = 0.03 + 0.50 x 0.12 = 0.03 + 0.06 = 0.09 = 9%. Market return is 8%, so M2 excess = 9% - 8% = +1%.
M-squared of 9%, a +1% volatility-matched advantage over the market. The fund's raw return (8% + 3% = 11% total) looked better than the market only because it took more risk (16% vs 12% volatility). Once de-levered to the market's 12% volatility, it would have returned 9%, just one point above the market: the honest, risk-equalized comparison the M-squared measure is built to deliver.
Common Variations
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Sources & References
- Risk-Adjusted Performance — Franco Modigliani and Leah Modigliani, Journal of Portfolio Management (1997)
- The Sharpe Ratio — William F. Sharpe, Journal of Portfolio Management (1994)
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