Modigliani risk-adjusted performance 1

Modigliani risk-adjusted performanceModigliani risk-adjusted performance or M2 or M2 or Modigliani–Modigliani measure or RAP is a measure ofthe risk-adjusted returns of some investment portfolio. It measures the returns of the portfolio, adjusted for thedeviation of the portfolio (typically referred to as the risk), relative to that of some benchmark (e.g., the market). It isderived from the widely used Sharpe Ratio, but it has the significant advantage of being in units of percent return (asopposed to the Sharpe Ratio – an abstract, dimensionless ratio of limited utility to most investors), which makes itdramatically more intuitive to interpret.

HistoryIn 1966, William Forsyth 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 Franco Modigliani and his granddaughter,Leah Modigliani, developed the Modigliani Risk-Adjusted Performance measure.[3] They originally called it "RAP"(Risk Adjusted Performance). They also defined a related statistic, "RAPA" (presumedly, Risk AdjustedPerformance Alpha), which was defined as RAP minus the risk-free rate (i.e., it only involved the risk-adjustedreturn 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"[4] (because it was developed by the twoModiglianis), but also as the "Modigliani-Modigliani measure" and "M2", for the same reason.

DefinitionModigliani risk-adjusted return for a portfolio P is the difference between (i) the return of a composite portfolio Cwhose risk is the same as that of the market (or benchmark) and (ii) the market (or benchmark) return, where thecomposite portfolio C consists of the portfolio P and a risk free asset with the weights of the two determined by thecondition (i).

Let and be the time series of the returns of the portfolio, the risk free assets and the market, withmeans and and standard deviations and . If the weight of the portfolio in the compositeportfolio is , clearly the mean return of the composite portfolio is

with standard deviation

The weight is determined by the aforementioned condition leading to . (Note that may

be larger than one, in which case one borrows some risk free asset to buy the portfolio in order to increase the risk ofthe composite portfolio up to the level of the market risk.)Hence

This finally yields the excess risk adjusted return - the Modigilani as

.

The connection to other performance measures is made as follows:Let be the excess return of the portfolio (i.e., above the risk-free rate) for some time period :

Modigliani risk-adjusted performance 2

Where is the portfolio return for time period and is the risk-free rate for time period .Then the Sharpe ratio is:

Where is the average of all excess returns over some period and is the standard deviation of those excessreturns.And finally:

Where is the Sharpe Ratio, is the standard deviation of the excess returns for some benchmark portfolioagainst which you are comparing the portfolio in question (often, the benchmark portfolio is the market), and isthe average Market Return for the period in question.For clarity, it may be useful to substitute in for and to rearrange:

The original paper also defined a statistic called "RAPA" (presumedly, Risk Adjusted Performance Alpha).Consistent with the more common terminology of , this would be:

or equivalently,

Thus, the portfolio's excess return is adjusted based on the portfolio's relative riskiness with respect to that of thebenchmark portfolio (i.e., ). 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 Modigliani Risk-Adjusted Performance measure is used to characterize how well a portfolio's return rewards aninvestor for the amount of risk taken, relative to that of some benchmark portfolio and to the risk-free rate. Thus, aninvestment that took a great deal more risk than some benchmark portfolio, but only had a small performanceadvantage, might have lesser risk-adjusted performance than another portfolio that took dramatically less riskrelative to the benchmark, but had similar returns.Because it is directly derived from the Sharpe Ratio, any orderings of investments/portfolios using the ModiglianiRisk-Adjusted Performance measure are exactly the same as orderings using the Sharpe Ratio.

Advantages over the Sharpe Ratio and Other Dimensionless RatiosThe Sharpe Ratio is awkward to interpret when it is negative. Further, it is difficult to directly compare the SharpeRatios of several investments. For example, what does it mean if one investment has a Sharpe Ratio of 0.50 andanother has a Sharpe Ratio of −0.50? How much worse was the second portfolio than the first? These downsidesapply to all risk-adjusted return measures that are ratios (e.g., Sortino ratio, Treynor ratio, Upside-potential ratio,etc.).M2 has the enormous advantage that it is in units of percent return, which is instantly interpretable by virtually allinvestors. Thus, for example, it is easy to recognize the magnitude of the difference between two investmentportfolios which have M2 values of 5.2% and of 5.8%. The difference is 0.6 percentage points of risk-adjustedreturns per year, with the riskiness adjusted to that of the benchmark portfolio (whatever that might be, but usuallythe market).

Modigliani risk-adjusted performance 3

ExtensionsIt is not necessary to utilize standard deviation of excess returns as the measure of risk. This approach is extensible touse of other measures of risk (e.g., Beta), just by substituting the other risk measures for and :

The main idea is that the riskiness of one portfolio's returns is being adjusted for comparison to another portfolio'sreturns.Virtually any benchmark return (e.g., some index or some 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 mightmake sense to compare all such endowments to a benchmark portfolio of 60% stocks and 40% bonds.

References[1] Sharpe, W. F. (1966). "Mutual Fund Performance". Journal of Business 39 (S1): 119–138. doi:10.1086/294846.[2] Sharpe, William F. (1994). "The Sharpe Ratio". Journal of Portfolio Management 1994 (Fall): 49–58.[3] Modigliani, Franco (1997). "Risk-Adjusted Performance". Journal of Portfolio Management 1997 (Winter): 45–54.[4] Modigliani, Leah (1997). "Yes, You Can Eat Risk-Adjusted Returns". Morgan Stanley U.S. Investment Research 1997 (March 17, 1997):

1–4.

External links• The Sharpe ratio (http:/ / www. stanford. edu/ ~wfsharpe/ art/ sr/ sr. htm)

Article Sources and Contributors 4

Article Sources and ContributorsModigliani risk-adjusted performance  Source: http://en.wikipedia.org/w/index.php?oldid=517255224  Contributors: Altruistguy, Ary29, EricObermuhlner, Magioladitis, Ralph Purtcher,Regent of the Seatopians, Saumi a, Tony1, Zfeinst, 6 anonymous edits

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