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What Is M-squared (Modigliani-Modigliani Measure)?

The M-squared (Modigliani-Modigliani Measure), often abbreviated as M², is a measure of risk-adjusted return for investment portfolios. It falls under the broader category of portfolio performance measurement metrics. The M-squared measure provides a percentage return that can be directly compared to a benchmark portfolio, typically the market portfolio, after adjusting for differences in total risk. This makes it intuitive for investors to understand whether a portfolio has truly outperformed its benchmark given the level of risk taken. Unlike other ratio-based measures, the M-squared is expressed in units of percentage return, facilitating a more straightforward interpretation.

History and Origin

The M-squared measure was developed by Nobel laureate Franco Modigliani and his granddaughter Leah Modigliani in 1997. ⁹Their seminal paper, "Risk-Adjusted Performance: How to Measure It and Why," introduced this metric as a way to improve the interpretability of risk-adjusted return comparisons. ⁸Prior to M-squared, measures like the Sharpe Ratio provided valuable insights but were expressed as a ratio, which could be less intuitive for direct comparison of performance across different risk levels. The Modiglianis sought to create a measure that effectively "re-scaled" a portfolio to match the risk of a benchmark, thereby allowing for a direct comparison of their returns in percentage terms. This innovation aimed to provide a clearer answer to the question of whether a portfolio manager genuinely added value, taking into account the total risk assumed.

Key Takeaways

M-squared (M²) is a risk-adjusted return measure that presents performance in percentage terms, making it highly intuitive.
It scales a portfolio's risk to match a benchmark, allowing for a direct comparison of returns.
A positive M-squared indicates outperformance relative to the benchmark, while a negative value suggests underperformance.
M-squared is derived from the Sharpe Ratio but offers a more readily interpretable result.
It considers a portfolio's total risk, encompassing both systematic risk and unsystematic risk.

Formula and Calculation

The M-squared (Modigliani-Modigliani Measure) is derived from the Sharpe Ratio and involves scaling the portfolio to match the standard deviation of the market portfolio.

The formula for M-squared is:

$M^2 = (R_P - R_f) \left( \frac{\sigma_M}{\sigma_P} \right) + R_f - R_M$

Alternatively, it can be expressed in terms of the Sharpe Ratio:

$M^2 = SR_P \cdot \sigma_M + R_f - R_M$

Where:

(R_P) = Portfolio return (the actual return of the portfolio being evaluated)
(R_M) = Market portfolio return (the return of the chosen benchmark market portfolio)
(R_f) = Risk-free rate (e.g., the return on a U.S. Treasury bill)
(\sigma_P) = Standard deviation of the portfolio (a measure of its total risk)
(\sigma_M) = Standard deviation of the market portfolio (a measure of the benchmark's total risk)
(SR_P) = Sharpe Ratio of the portfolio, calculated as (\frac{R_P - R_f}{\sigma_P})

The calculation effectively creates a hypothetical "adjusted" portfolio that combines the actual portfolio with the risk-free rate to achieve the same total risk as the benchmark. The M-squared value then represents the difference in returns between this adjusted portfolio and the benchmark.

Interpreting the M-squared (Modigliani-Modigliani Measure)

Interpreting the M-squared measure is straightforward due to its expression in percentage terms. A positive M-squared value indicates that the portfolio generated a higher return on investment than the market portfolio, after adjusting for the same level of total risk. Conversely, a negative M-squared implies that the portfolio underperformed the market on a risk-adjusted return basis.

For instance, an M-squared of 2% means that the evaluated portfolio, when risk-adjusted to the market's total risk, would have returned 2 percentage points more than the market. This direct percentage comparison makes it easy to rank various investment strategy options or portfolio managers, as it provides a clear indication of their ability to generate excess returns for a given level of total risk. It is particularly useful for investors who are concerned with total risk, including both systematic risk and unsystematic risk, as they may not be fully diversified across numerous assets.

⁷## Hypothetical Example

Consider an investor evaluating two portfolios, Portfolio A and Portfolio B, against a market benchmark.

Market Benchmark (M):

Return ((R_M)) = 10%
Standard Deviation ((\sigma_M)) = 15%

Risk-Free Rate ((R_f)) = 3%

Portfolio A (PA):

Return ((R_{PA})) = 15%
Standard Deviation ((\sigma_{PA})) = 20%

Portfolio B (PB):

Return ((R_{PB})) = 12%
Standard Deviation ((\sigma_{PB})) = 10%

Calculate M-squared for Portfolio A:

First, calculate Portfolio A's Sharpe Ratio:
(SR_{PA} = (15% - 3%) / 20% = 0.6)

Now, calculate M-squared for Portfolio A:
$M^2_A = SR_{PA} \cdot \sigma_M + R_f - R_M$
$M^2_A = 0.6 \cdot 15\% + 3\% - 10\%$
$M^2_A = 9\% + 3\% - 10\% = 2\%$

Calculate M-squared for Portfolio B:

First, calculate Portfolio B's Sharpe Ratio:
(SR_{PB} = (12% - 3%) / 10% = 0.9)

Now, calculate M-squared for Portfolio B:
$M^2_B = SR_{PB} \cdot \sigma_M + R_f - R_M$
$M^2_B = 0.9 \cdot 15\% + 3\% - 10\%$
$M^2_B = 13.5\% + 3\% - 10\% = 6.5\%$

Interpretation:

Portfolio A has an M-squared of 2%. This means that if Portfolio A were adjusted to have the same total risk as the market, it would have outperformed the market by 2 percentage points.
Portfolio B has an M-squared of 6.5%. If Portfolio B were adjusted to have the same total risk as the market, it would have outperformed the market by 6.5 percentage points.

Despite Portfolio A having a higher raw return (15% vs. 12%), Portfolio B demonstrates superior portfolio performance on a risk-adjusted return basis according to the M-squared measure. This highlights the importance of considering risk when evaluating returns, particularly for investors focused on total risk.

Practical Applications

The M-squared (Modigliani-Modigliani Measure) finds several practical applications in the financial world, particularly within portfolio performance evaluation and investment strategy selection.

Fund Comparison: Fund managers and analysts use M-squared to compare the risk-adjusted return of different mutual funds, exchange-traded funds (ETFs), or other investment vehicles. Because it expresses performance in percentage terms adjusted for risk, it allows for a more intuitive ranking than ratios alone. For example, Morningstar uses risk-adjusted ratings in their fund analysis.
*⁶ Manager Evaluation: Institutional investors and wealth managers utilize M-squared to assess the effectiveness of active management. It helps determine if a manager's skill (alpha) truly translates into superior returns after accounting for the risk they undertake.
Asset Allocation Decisions: Investors can use M-squared to evaluate the efficiency of their asset allocation strategies. By comparing the M-squared of their overall portfolio against various benchmarks, they can gauge whether their current allocation effectively balances risk and return.
Performance Reporting: Financial advisors often incorporate M-squared into client reports to provide a clear and understandable metric of how a client's portfolio has performed relative to a relevant benchmark, adjusted for risk. This enhances transparency and helps clients grasp the value added by their portfolio's composition.

Limitations and Criticisms

While the M-squared (Modigliani-Modigliani Measure) offers a clear and intuitive way to assess risk-adjusted return, it shares some of the limitations inherent in its underlying metric, the Sharpe Ratio, and other traditional portfolio performance measures.

Reliance on Historical Data: M-squared is calculated using historical returns and standard deviation. Past performance is not indicative of future results, and relying solely on historical data may not accurately predict a portfolio's future risk-adjusted performance. M⁵arket conditions can change, altering the relationship between risk and return.
Sensitivity to Data Period: The value of M-squared can be sensitive to the specific time period over which it is calculated. Different starting and ending dates can yield varying results, potentially misleading investors about consistent outperformance or underperformance.
Assumption of Normal Distribution: Like the Sharpe Ratio, M-squared assumes that asset returns are normally distributed. In reality, financial returns often exhibit "fat tails" (more extreme positive and negative events than a normal distribution would predict) and skewness, meaning that standard deviation may not fully capture the true risk of extreme losses or gains.
*⁴ Manipulation Potential: Portfolio managers could potentially manipulate M-squared by taking on excessive, unrewarded risk if they believe their returns will temporarily increase, thereby artificially inflating the measure.
*³ Focus on Total Risk: While M-squared considers total risk (both systematic risk and unsystematic risk), it might be less relevant for highly diversification portfolios. For an investor with a fully diversified portfolio, unsystematic risk is largely eliminated, and beta (a measure of systematic risk) might be a more appropriate risk metric.

These limitations suggest that M-squared should be used as one of several tools in a comprehensive evaluation framework, rather than as a standalone determinant of investment strategy effectiveness.

M-squared (Modigliani-Modigliani Measure) vs. Sharpe Ratio

The M-squared (Modigliani-Modigliani Measure) and the Sharpe Ratio are both widely used metrics for evaluating risk-adjusted return in portfolio performance. They share a fundamental connection, as M-squared is directly derived from the Sharpe Ratio. T²he key difference lies in their interpretability.

The Sharpe Ratio measures the excess return (return above the risk-free rate) per unit of total risk (as measured by standard deviation). It is a ratio, indicating the steepness of the capital market line for a given portfolio. A higher Sharpe Ratio indicates better risk-adjusted performance. However, comparing a Sharpe Ratio of 0.8 to 1.2, while indicating superior performance, doesn't immediately tell an investor how many percentage points of return difference that implies after risk adjustment.

The M-squared (Modigliani-Modigliani Measure) takes the Sharpe Ratio a step further by transforming it into a percentage return that can be directly compared to a benchmark's return. It scales the portfolio to have the same standard deviation as the market portfolio, effectively "normalizing" the risk. The resulting M-squared value then shows the additional return the portfolio would have generated above or below the market benchmark, if both had assumed the same total risk. This makes M-squared particularly intuitive for investors, as it provides a clear percentage figure of outperformance or underperformance.

In essence, M-squared addresses a common criticism of the Sharpe Ratio by converting its output into a more easily understandable and comparable metric—a percentage return. While the Sharpe Ratio provides the foundation, M-squared provides the user-friendly comparison.

##¹ FAQs

What does a high M-squared value mean?

A high M-squared value indicates that a portfolio has achieved superior risk-adjusted return compared to its benchmark. It means the portfolio would have generated more percentage points of return than the benchmark, even if both had taken on the same level of total risk.

Is M-squared better than other performance measures?

M-squared offers the advantage of being expressed in percentage terms, making it more intuitive for direct comparison against a benchmark than ratio-based measures like the Sharpe Ratio. However, no single portfolio performance measure is perfect. It is often beneficial to use M-squared in conjunction with other metrics, such as Jensen's Alpha or the Treynor Ratio, to get a comprehensive view of a portfolio's performance across different risk perspectives (total risk vs. systematic risk).

Can M-squared be used for all types of portfolios?

M-squared is suitable for evaluating diversified portfolios where total risk (as measured by standard deviation) is a relevant concern. It is particularly useful when comparing portfolios that have different risk levels and for investors who are not fully diversified. For highly diversified portfolios, where unsystematic risk is largely eliminated, other measures focusing solely on systematic risk might also be considered.

What is the ideal benchmark for M-squared?

The ideal benchmark for M-squared is a broadly diversified market portfolio or an index that represents the investable universe relevant to the portfolio being evaluated. For example, a broad market index like the S&P 500 is often used for U.S. equity portfolios, while a global index might be more appropriate for an international portfolio. The benchmark should reflect the investment objectives and constraints of the portfolio.