Modigliani risk-adjusted performance (also known as M2, M2, Modigliani–Modigliani measure or RAP) is a measure of the risk-adjusted returns of some investment portfolio. It measures the returns of the portfolio, adjusted for the risk of the portfolio relative to that of some benchmark (e.g., the market). It can be interpreted as the difference between the scaled excess return of the portfolio and that of the market, where the scaled portfolio has the same volatility as the market. It is derived from the widely used Sharpe ratio, but it has the significant advantage of being in units of percent return (as opposed to the Sharpe ratio – an abstract, dimensionless ratio of limited utility to most investors), which makes it dramatically more intuitive to interpret.
History In 1966, William F. Sharpe developed what is now known as the Sharpe ratio. 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. In 1997, Nobel-prize winner Franco Modigliani and his granddaughter, Leah Modigliani, developed what is now called the Modigliani risk-adjusted performance measure. They originally called it "RAP" (risk-adjusted performance). They also defined a related statistic, "RAPA" (presumably, an abbreviation of "risk-adjusted performance alpha"), which was defined as RAP minus the risk-free rate (i.e., it only involved the risk-adjusted return above the risk-free rate). Thus, RAPA was effectively the risk-adjusted excess return. The RAP measure has since become more commonly known as "M2" (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 D t {\displaystyle D_{t}} be the excess return of the portfolio (i.e., above the risk-free rate) for some time period t {\displaystyle t} :
D t ≡ R P t − R F t {\displaystyle D_{t}\equiv R_{P_{t}}-R_{F_{t}}}
where R P t {\displaystyle R_{P_{t}}} is the portfolio return for time period t {\displaystyle t} and R F t {\displaystyle R_{F_{t}}} is the risk-free rate for time period t {\displaystyle t} . Then the Sharpe ratio S {\displaystyle S} is
S ≡ D ¯ σ D {\displaystyle S\equiv {\frac {\overline {D}}{\sigma _{D}}}}
where D ¯ {\displaystyle {\overline {D}}} is the average of all excess returns over some period and σ D {\displaystyle \sigma _{D}} is the standard deviation of those excess returns. And finally:
M 2 ≡ S × σ B + R F ¯ {\displaystyle M^{2}\equiv S\times \sigma _{B}+{\overline {R_{F}}}}
where S {\displaystyle S} is the Sharpe ratio, σ B {\displaystyle \sigma _{B}} is the standard deviation of the excess returns for some benchmark portfolio (often, the market) against which the portfolio in question is being compared, and R F ¯ {\displaystyle {\overline {R_{F}}}} is the average risk-free rate for the period in question. For clarity, one can substitute in for S {\displaystyle S} and rearrange:
M 2 ≡ D ¯ × σ B σ D + R F ¯ . {\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 M 2 {\displaystyle M^{2}} , this would be
M 2 α ≡ S × σ B {\displaystyle M^{2}\alpha \equiv S\times \sigma _{B}}
or equivalently,
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