Formula, in Risk & Portfolio Construction
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.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
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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Questions people ask next
What is the M-squared (M2) measure formula?
The M-squared formula is M2 = R_f + Sharpe_p x sigma_m, where R_f is the risk-free rate, Sharpe_p is the portfolio's Sharpe ratio, and sigma_m is the market's volatility. It restates the Sharpe ratio as a return rather than a ratio: the return the portfolio would have earned if levered or de-levered to match the market's volatility. The portfolio Sharpe is itself (R_p - R_f) / sigma_p.
How do you calculate the M2 measure with an example?
First compute the portfolio Sharpe ratio, then scale it by market volatility and add the risk-free rate. In the worked example a portfolio with 8% excess return and 16% volatility has a Sharpe of 0.50; with market volatility of 12% and a 3% risk-free rate, M2 = 3% + 0.50 x 12% = 9%. That 9% is what the fund would have returned de-levered to the market's risk level.
What is M2 alpha and how does its formula differ?
M2 alpha (M-squared for beta) is the variation that uses systematic risk, or beta, instead of total volatility. It is the Treynor-ratio analog expressed in return units, whereas the standard M-squared uses total volatility and is the Sharpe-ratio analog in return units. The standard measure matches the portfolio to the market on total risk; the beta version matches on systematic risk.
What is the M-squared excess return?
M-squared excess is M2 minus the market return, which equals (Sharpe_p - Sharpe_m) x sigma_m. It is the volatility-matched outperformance in percentage points. In the worked example M2 is 9% and the market returned 8%, so the M2 excess is +1%, meaning the fund beat the market by only one point once its higher 16% volatility was de-levered to the market's 12%.
Why use M-squared instead of the Sharpe ratio?
M-squared carries the same information as the Sharpe ratio and produces identical rankings, but it is expressed in percentage points over the benchmark rather than as a unitless ratio, which makes it far easier to interpret. Instead of comparing abstract Sharpe numbers, you can say a portfolio beat the market by a given percentage on equal-risk terms.
Sources
- Risk-Adjusted Performance , Franco Modigliani and Leah Modigliani, Journal of Portfolio Management (1997)
- The Sharpe Ratio , William F. Sharpe, Journal of Portfolio Management (1994)
Keep the topic connected
- Formula Sharpe Ratio Formula
The Sharpe ratio formula: excess return over the risk-free rate divided by return volatility, then annualized. Every variable defined, with a worked example.
- Formula Treynor Ratio Formula
The Treynor ratio formula: excess return over the risk-free rate divided by beta. Reward per unit of systematic market risk, with a worked example.
- Formula Annualized Volatility Formula
The annualized volatility formula scales per-period return standard deviation by the square root of periods per year. Root-time scaling explained.
- Learn Sharpe Ratio
Sharpe ratio defined, when it lies (skew, fat tails, autocorrelation), and how to read a Sharpe number you didn't compute yourself.