Modigliani risk-adjusted performance

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Modigliani risk-adjusted performance (also known as M², M2, Modigliani–Modigliani measure or RAP) is a measure of the risk-adjusted returns of some

In 1966, William F. Sharpe developed what is now known as the Sharpe ratio.^[1] Sharpe originally called it the "reward-to-variability" ratio before it began being called the Sharpe ratio by later academics and financial operators. Sharpe slightly refined the idea in 1994.^[2]

In 1997, Nobel-prize winner

The RAP measure has since become more commonly known as "M²"^[4] (because it was developed by the two Modiglianis), but also as the "Modigliani–Modigliani measure" and "M2", for the same reason.

Definition

Modigliani risk-adjusted return is defined as follows:

Let ${\displaystyle D_{t}}$ be the excess return of the portfolio (i.e., above the

${\displaystyle t}$

${\displaystyle D_{t}\equiv R_{P_{t}}-R_{F_{t}}}$

where ${\displaystyle R_{P_{t}}}$ is the portfolio return for time period ${\displaystyle t}$ and ${\displaystyle R_{F_{t}}}$ is the

${\displaystyle t}$

Then the Sharpe ratio ${\displaystyle S}$ is

${\displaystyle S\equiv {\frac {\overline {D}}{\sigma _{D}}}}$

where ${\displaystyle {\overline {D}}}$ is the average of all excess returns over some period and ${\displaystyle \sigma _{D}}$ is the standard deviation of those excess returns.

And finally:

${\displaystyle M^{2}\equiv S\times \sigma _{B}+{\overline {R_{F}}}}$

where ${\displaystyle S}$ is the Sharpe ratio, ${\displaystyle \sigma _{B}}$ is the standard deviation of the excess returns for some benchmark portfolio against which you are comparing the portfolio in question (often, the benchmark portfolio is the market), and ${\displaystyle {\overline {R_{F}}}}$ is the

For clarity, one can substitute in for ${\displaystyle S}$ and rearrange:

${\displaystyle M^{2}\equiv {\overline {D}}\times {\frac {\sigma _{B}}{\sigma _{D}}}+{\overline {R_{F}}}.}$

The original paper also defined a statistic called "RAPA" (presumably, an abbreviation of "risk-adjusted performance alpha"). Consistent with the more common terminology of ${\displaystyle M^{2}}$, this would be

${\displaystyle M^{2}\alpha \equiv S\times \sigma _{B}}$

or equivalently,

${\displaystyle M^{2}\alpha \equiv {\overline {D}}\times {\frac {\sigma _{B}}{\sigma _{D}}}.}$

Thus, the portfolio's excess return is adjusted based on the portfolio's relative riskiness with respect to that of the benchmark portfolio (i.e., ${\displaystyle {\frac {\sigma _{B}}{\sigma _{D}}}}$). So if the portfolio's excess return had twice as much risk as that of the benchmark, it would need to have twice as much excess return in order to have the same level of risk-adjusted return.

The M² measure is used to characterize how well a portfolio's return rewards an investor for the amount of risk taken, relative to that of some benchmark portfolio and to the

Because it is directly derived from the Sharpe ratio, any orderings of investments/portfolios using the M² measure are exactly the same as orderings using the Sharpe ratio.

Advantages over the Sharpe ratio and other dimensionless ratios

The

M² has the enormous advantage that it is in units of percentage return, which is instantly interpretable by virtually all investors. Thus, for example, it is easy to recognize the magnitude of the difference between two investment portfolios which have M² values of 5.2% and of 5.8%. The difference is 0.6 percentage points of risk-adjusted return per year, with the riskiness adjusted to that of the benchmark portfolio (whatever that might be, but usually the market).

Extensions

It is not necessary to use standard deviation of excess returns as the measure of risk. This approach is extensible to use of other measures of risk (e.g., beta), just by substituting the other risk measures for ${\displaystyle \sigma _{D}}$ and ${\displaystyle \sigma _{B}}$:

${\displaystyle M_{\beta }^{2}\equiv {\overline {D}}\times {\frac {\beta _{B}}{\beta _{D}}}+{\overline {R_{F}}}}$

The main idea is that the riskiness of one portfolio's returns is being adjusted for comparison to another portfolio's returns.

Virtually any benchmark return (e.g., an index or a particular portfolio) could be used for risk adjustment, though usually it is the market return. For example, if you were comparing performance of endowments, it might make sense to compare all such endowments to a benchmark portfolio of 60% stocks and 40% bonds.

See also

• Capital asset pricing model
• Information ratio
• Jensen's alpha
• Modern portfolio theory
• Roy's safety-first criterion
• Sharpe ratio
• Sortino ratio
• Treynor ratio
• Upside-potential ratio

References