# Mogdiliani measure

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We also know from the Central Limit Theorem that if you have many different investments all that matters is the mean and the standard deviation. So as long as the CLT is valid the Sharpe ratio makes sense. The Sharpe ratio has been criticized for attaching equal weight to upside 'risk' as downside risk since the standard deviation incorporates both in its calculation. This may be important if returns are very skewed. This is easily interpreted as the return you would expect from your portfolio is it were de leveraged to have the same volatility as the benchmark.

The Sortino ratio is calculated in the same way as the Sharpe ratio except that it uses the square root of the semi-variance as the denominator measuring risk. The semi-variance is measured in the same way as the variance except that all data points with positive return are replaced with zero, or with some target value.

This measure ignores upside 'risk' completely. However, if returns are expected to be normally distributed the semi-variance will be statistically noisier than the variance because fewer data points are used in its calculation.

The Treynor or Reward-to-variability ratio is another Sharpe-like measure, but now the denominator is the systematic risk, measured by the portfolio's beta, see Capital Asset Pricing Model , instead of the total risk:. In a well-diversified portfolio Sharpe and Treynor are similar, but Treynor is more relevant for less diversified portfolios or individual stocks.

The Information ratio is a different type of performance measure in that it uses the idea of tracking error. The numerator is the return in excess of a benchmark again, but the denominator is the standard deviation of the differences between the portfolio returns and the benchmark returns, the tracking error. Journal of Portfolio Management 23 2 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.

Thus, RAPA was effectively the risk-adjusted excess return. The RAP measure has since become more commonly known as "M 2 " [4] because it was developed by the two Modiglianis , but also as the "Modigliani—Modigliani measure" and "M2", for the same reason. The original paper also defined a statistic called "RAPA" presumably, an abbreviation of "risk-adjusted performance alpha".

Thus, the portfolio's excess return is adjusted based on the portfolio's relative riskiness with respect to that of the benchmark portfolio i. 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 M 2 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 risk-free rate.

Thus, an investment that took a great deal more risk than some benchmark portfolio, but only had a small performance advantage, might have lesser risk-adjusted performance than another portfolio that took dramatically less risk relative to the benchmark, but had similar returns. The Sharpe ratio is awkward to interpret when it is negative. Further, it is difficult to directly compare the Sharpe ratios of several investments. For example, what does it mean if one investment has a Sharpe ratio of 0.

How much worse was the second portfolio than the first? These downsides apply to all risk-adjusted return measures that are ratios e. M 2 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 M 2 values of 5. The difference is 0.

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. 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. From Wikipedia, the free encyclopedia. This article has multiple issues. Please help improve it or discuss these issues on the talk page. Learn how and when to remove these template messages.

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