Modigliani Risk-Adjusted Performance (M²)

A measure that adjusts portfolio returns to match market volatility, allowing direct comparison with benchmark returns

Overview

Modigliani Risk-Adjusted Performance (also known as M², M-squared, or RAP) is an innovative risk-adjusted performance measure developed by Nobel Prize-winning economist Franco Modigliani and his granddaughter Leah Modigliani in 1997. Unlike abstract ratios like the Sharpe ratio, M² expresses risk-adjusted performance in percentage terms, making it exceptionally intuitive for investors to interpret and compare.

The core concept behind M² is elegant: it adjusts a portfolio's returns to match the volatility level of some benchmark (typically the market), allowing for direct, apples-to-apples comparison between portfolio performance and benchmark performance. This is achieved by creating a theoretical leveraged or deleveraged version of the original portfolio that has the same risk level as the benchmark.

M² solves a fundamental problem in performance measurement by translating the Sharpe ratio's dimensionless value into percentage returns that investors can immediately understand. For instance, an M² of 2% indicates that the portfolio, when adjusted to have the same risk as the benchmark, outperformed that benchmark by 2 percentage points—a clear and actionable insight.

Intuitive Explanation

Imagine you're comparing two different marathon runners with different strategies:

Runner A sprints at full speed but takes frequent rest breaks when exhausted.

Runner B maintains a consistent, moderate pace throughout the race.

Both runners might complete the marathon in the same total time, but their approach to managing energy (risk) is completely different. To fairly compare their performance, you might ask: "If Runner A were forced to run with the same consistency (risk level) as Runner B, how would their finishing time compare?"

That's essentially what M² does. It asks: "If we adjusted this portfolio to have exactly the same volatility as the market benchmark, what would its returns be?" This allows for direct comparison between different investment strategies, regardless of their original risk levels.

Financial analogy: M² is like normalizing cars' fuel efficiency at a standard speed of 55 mph. Even if one car was driven aggressively and another conservatively, adjusting both to the same "risk level" (speed) allows for fair comparison of their inherent efficiency. Similarly, M² adjusts portfolios to the same risk level as the benchmark, revealing their true performance edge or shortfall in percentage terms that any investor can understand.

Detailed Mathematical Explanation

M² builds upon the Sharpe ratio's foundation but transforms it into a more intuitive percentage return measure. Let's explore its mathematical formulation:

Core Formula

The Modigliani Risk-Adjusted Performance (M²) is defined as:

M² Formula
M2=Sp×σm+Rf‾M^2 = S_p \times \sigma_m + \overline{R_f}

where S_p is the portfolio's Sharpe ratio, σ_m is the standard deviation of the market (benchmark), and R̄_f is the average risk-free rate.

Derivation from the Sharpe Ratio

The Sharpe ratio for a portfolio (S_p) is calculated as:

Sp=Rp‾−Rf‾σpS_p = \frac{\overline{R_p} - \overline{R_f}}{\sigma_p}

Where:

  • Rp‾\overline{R_p} is the average return of the portfolio

  • Rf‾\overline{R_f} is the average risk-free rate

  • σp\sigma_p is the standard deviation of portfolio returns (volatility)

By substituting and rearranging, M² can also be expressed as:

M2=Rp∗‾=(Rp‾−Rf‾)×σmσp+Rf‾M^2 = \overline{R_p^*} = (\overline{R_p} - \overline{R_f}) \times \frac{\sigma_m}{\sigma_p} + \overline{R_f}

This formulation directly shows how M² represents the return of a risk-adjusted portfolio, where:

  • Rp∗‾\overline{R_p^*} is the return of the portfolio adjusted to the market's volatility level

  • (Rp‾−Rf‾)×σmσp(\overline{R_p} - \overline{R_f}) \times \frac{\sigma_m}{\sigma_p} is the excess return adjusted for relative volatility

Alternative Expression: M² Alpha

A related measure, sometimes called M² Alpha or RAPA (Risk-Adjusted Performance Alpha), represents just the risk-adjusted excess return:

M² Alpha Formula
Mα2=Sp×σm=(Rp‾−Rf‾)×σmσpM^2_{\alpha} = S_p \times \sigma_m = (\overline{R_p} - \overline{R_f}) \times \frac{\sigma_m}{\sigma_p}

This represents the risk-adjusted excess return above the risk-free rate.

Practical Interpretation

The key insight is that M² measures the return that would be achieved if the portfolio were adjusted (through leverage or de-leveraging) to have the same volatility as the benchmark. This allows for direct comparison with the benchmark return:

If M² >R_m: The portfolio outperformed the benchmark on a risk-adjusted basis

If M² <R_m: The portfolio underperformed the benchmark on a risk-adjusted basis

If M² = R_m: The portfolio performed exactly as well as the benchmark on a risk-adjusted basis

The difference (M² - R_m) represents the portfolio's risk-adjusted excess return compared to the benchmark.

Implementation in Portfolio Analysis

Our implementation of the Modigliani Risk-Adjusted Performance involves the following key steps:

  1. Data Preparation: Collect historical return data for the portfolio, benchmark, and risk-free rate over the same time period.

  2. Calculate Return Statistics: Compute the average returns and standard deviations for the portfolio and benchmark.

  3. Calculate the Sharpe Ratio: Determine the portfolio's Sharpe ratio by dividing excess return by portfolio volatility.

  4. Apply the M² Formula: Multiply the Sharpe ratio by the benchmark's standard deviation and add the risk-free rate.

  5. Interpretation: Compare the resulting M² value directly with the benchmark's average return to determine risk-adjusted outperformance or underperformance.

In portfolio optimization applications, M² can be used in several ways:

  • Performance Evaluation: Comparing the risk-adjusted performance of different portfolios or strategies against a common benchmark.

  • Manager Selection: Evaluating different portfolio managers' risk-adjusted performance in percentage terms that are intuitive to clients.

  • Portfolio Optimization: While not typically used as a direct optimization objective, M² can be used to evaluate outcomes from other optimization methods.

  • Risk Management: Understanding how changes in portfolio composition affect risk-adjusted returns relative to a benchmark.

Worked Example

Let's calculate the M² measure for two different investment portfolios and compare their risk-adjusted performance against a market benchmark.

Step 1: Historical Data

Assume we have the following annual performance data:

Risk-free rate: 3% per annum

Market benchmark: Average return: 10%, Standard deviation (volatility): 15%

Portfolio A: Average return: 12%, Standard deviation: 20%

Portfolio B: Average return: 8%, Standard deviation: 10%

Step 2: Calculate Sharpe Ratios

First, we calculate the Sharpe ratio for each portfolio:

Portfolio A

S_A = (12% - 3%) / 20% = 9% / 20% = 0.45

Portfolio B

S_B = (8% - 3%) / 10% = 5% / 10% = 0.50

Step 3: Calculate M² Values

Using the formula M² = S × σ_m + R_f:

Portfolio A

M²_A = 0.45 × 15% + 3% = 6.75% + 3% = 9.75%

Portfolio B

M²_B = 0.50 × 15% + 3% = 7.5% + 3% = 10.5%

Step 4: Compare with Benchmark

Now we can directly compare the M² values with the market benchmark return of 10%:

Portfolio A: M²_A = 9.75% (underperforms benchmark by 0.25%)

Portfolio B: M²_B = 10.5% (outperforms benchmark by 0.5%)

Step 5: Interpretation

This example reveals some important insights:

Portfolio A has a higher raw return (12%) than both Portfolio B (8%) and the market (10%). However, when adjusted for risk, its performance (M² = 9.75%) actually falls slightly below the market's return of 10%. This suggests that Portfolio A's higher returns don't fully compensate for its higher volatility.

Portfolio B has a lower raw return (8%) than both Portfolio A (12%) and the market (10%). Yet, when adjusted for risk, it demonstrates the best performance (M² = 10.5%), outperforming the market by 0.5%. This indicates that Portfolio B achieves its returns with remarkably low risk, making it the most efficient portfolio on a risk-adjusted basis.

This is precisely the kind of insight that makes M² valuable—it transforms abstract risk-adjusted comparisons into concrete percentage returns that can be directly compared with benchmark performance, revealing which investment approach truly adds value when accounting for risk.

Practical Applications

Performance Reporting

M² excels in client reporting contexts where portfolio managers need to explain risk-adjusted performance to non-technical audiences. The percentage-based format makes it immediately comprehensible to investors without requiring technical knowledge of statistics or finance theory.

Fund Selection

When comparing multiple funds, M² allows investors to see which would provide the best return at a standardized risk level. This facilitates better-informed investment decisions by separating manager skill from risk-taking.

Risk Budgeting

In institutional settings, M² helps allocate risk budgets by identifying which strategies deliver the most return per unit of risk. This is particularly valuable when working with a diverse range of investment approaches across multiple asset classes.

Strategy Evaluation

For systematic trading strategies, M² provides a framework for comparing performance across various market environments. It helps answer whether a strategy's returns justify its risk profile compared to a passive benchmark approach.

Asset Allocation

At the portfolio construction level, M² can inform allocation decisions by highlighting which asset classes or factors deliver superior risk-adjusted returns relative to broad market exposure. This helps build more efficient investment portfolios.

Advantages and Limitations

Advantages

  • Intuitive Interpretation: Expresses risk-adjusted performance in percentage returns rather than an abstract ratio, making it immediately meaningful to all investors.

  • Direct Benchmark Comparison: Allows for straightforward comparison against benchmark returns, as both are expressed in the same units and adjusted to the same risk level.

  • Client Communication: Simplifies the explanation of risk-adjusted performance to clients without requiring technical knowledge of finance or statistics.

  • Negative Returns: Unlike the Sharpe ratio, M² remains meaningful and easy to interpret even when dealing with negative returns or Sharpe ratios.

  • Customizable Benchmark: Can use any relevant benchmark, not just the market, allowing for appropriate comparisons within specific investment mandates or strategies.

Limitations

  • Total Volatility Focus: Like the Sharpe ratio it's derived from, M² treats all volatility equally, without distinguishing between upside and downside risk.

  • Normal Distribution Assumption: Implicitly assumes returns are normally distributed, which often doesn't hold for many investment strategies, particularly those with options or alternative assets.

  • Benchmark Dependency: Results are highly dependent on the chosen benchmark, potentially leading to different conclusions when different benchmarks are used.

  • Historical Data Limitations: As with all backward-looking metrics, past performance data may not reliably predict future results.

  • Leverage Practicality: The theoretical leveraging or de-leveraging implied in the calculation may not be practically implementable due to constraints, costs, or leverage limitations.

  • Time Period Sensitivity: Results can vary significantly depending on the time period chosen, potentially leading to period-selection bias.

References

  • Sharpe, W. F. (1966). "Mutual Fund Performance". Journal of Business. 39(S1): 119–138. doi:10.1086/294846.

  • Sharpe, William F. (1994). "The Sharpe Ratio". Journal of Portfolio Management. 1994(Fall): 49–58. doi:10.3905/jpm.1994.409501. S2CID 55394403.

  • Modigliani, Franco (1997). "Risk-Adjusted Performance". Journal of Portfolio Management. 1997(Winter): 45–54. doi:10.3905/jpm.23.2.45. S2CID 154490980.

  • Modigliani, Leah (1997). "Yes, You Can Eat Risk-Adjusted Returns". Morgan Stanley U.S. Investment Research. 1997(March 17, 1997): 1–4.

  • Bacon, Carl (2013). "Practical Risk-Adjusted Performance Measurement". Wiley Finance.

  • Le Sourd, V. (2007). "Performance Measurement for Traditional Investment". EDHEC Risk and Asset Management Research Centre.

Related Topics

Sharpe Ratio

The classic risk‐adjusted return metric that divides excess portfolio return by total volatility.

Information Ratio

A performance metric that evaluates active return per unit of risk relative to a benchmark index.

Treynor Ratio

A portfolio performance metric that measures returns earned in excess of the risk-free rate per unit of market risk (beta).