M² (Modigliani-Modigliani): risk-adjusting returns to a common volatility for direct comparison

The Sharpe ratio tells you how much return a portfolio earned per unit of risk, but it does not tell you what that risk-adjusted return translates to in actual percentage terms. M²—pronounced M-squared or Modigliani-squared—bridges this gap. Developed by Franco Modigliani and his granddaughter Leah Modigliani in 1997, it rescales a portfolio's return to the same volatility as the benchmark. The result is a return figure that can be compared directly against the benchmark return, making the performance comparison intuitive even for audiences unfamiliar with Sharpe ratios.

What M² measures

M² answers a specific question: if this portfolio had been levered up or deleveraged to match the benchmark's volatility exactly, what return would it have earned? A portfolio with lower volatility than the benchmark is hypothetically levered up (using the risk-free rate to fund the leverage). A portfolio with higher volatility is hypothetically deleveraged (partially invested in the risk-free asset). The result is a risk-normalised return expressed in percentage terms, directly comparable to the benchmark's actual return. If M² exceeds the benchmark return, the portfolio outperformed on a risk-adjusted basis. If M² falls below the benchmark return, the portfolio underperformed despite whatever its raw return was.

The formula

M² = Sharpe × σm + Rf

Where Sharpe is the portfolio's Sharpe ratio, σm is the benchmark's standard deviation, and Rf is the risk-free rate. Equivalently, M² can be written as:
M² = Rp + Sharpe × (σm − σp)
Where σp is the portfolio's standard deviation. This form makes the adjustment intuitive: if σp = σm, M² equals the portfolio return exactly. If σp < σm, the extra term adds return (leverage up the portfolio). If σp > σm, the term subtracts return (deleverage the portfolio).

How to interpret M²

A portfolio with M² of 9.5 per cent compared to a benchmark return of 8.0 per cent has outperformed on a risk-adjusted basis by 1.5 percentage points. This is directly interpretable by investors who understand returns but not Sharpe ratios. The spread between M² and the benchmark return is sometimes called the M²-alpha, and it maps monotonically to the Sharpe ratio differential: a higher M² always corresponds to a higher Sharpe ratio. The advantage of M² over the Sharpe ratio is presentation—the percentage form is more accessible and more actionable for non-specialist audiences.

Rolling M²

Rolling M² computed over 12- or 36-month windows reveals whether a portfolio's risk-adjusted outperformance is stable or concentrated in specific market regimes. A portfolio with a positive long-run M² but negative rolling M² in recent periods is exhibiting deteriorating risk-adjusted performance that would not be visible from cumulative return alone. Rolling M² is particularly useful for detecting strategy decay in systematic portfolios where factor premia may be contracting.

Limitations

M² inherits all of the Sharpe ratio's limitations. It uses total volatility as the risk measure, which penalises upside volatility equally with downside volatility—a property addressed by the Sortino ratio. It assumes that leverage and deleveraging at the risk-free rate is practically achievable, which may not hold in all contexts. Like the Sharpe ratio, M² is sensitive to the measurement period and the choice of risk-free rate. For portfolios with non-normal return distributions—strategies that sell optionality, for example—M² can misrepresent risk in the same way the Sharpe ratio does.

M² in pfolio

M² is not currently displayed in pfolio Insights. Sharpe ratio and benchmark standard deviation are both available in Insights; M² can be calculated from these using the formula M² = Sharpe × σm + Rf.

Related metrics

• Sharpe ratio
• Treynor ratio
• Sortino ratio
• Alpha
• Active return