88 Frequently Asked Questions In Quantitative Finance
If returns are normally distributed then the Sharpe ratio
is related to the probability of making a return in excess
of the risk-free rate. In the expected return versus risk
diagram ofModern Portfolio Theorythe Sharpe ratio is
the slope of the line joining each investment to the risk-
free investment. Choosing the portfolio that maximizes
the Sharpe ratio will give you theMarket Portfolio.We
also know from theCentral Limit Theoremthat 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 stan-
dard deviation incorporates both in its calculation. This
may be important if returns are very skewed.
Modigliani-Modigliani Measure The Modigliani-Modigliani or
M2 measure is a simple linear transformation of the
Sharpe ratio:
M2=r+v×Sharpe
wherevis the standard deviation of returns of the relevant
benchmark. This is easily interpreted as the return you
would expect from your portfolio if it were (de)leveraged
to have the same volatility as the benchmark.
Sortino Ratio 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. How-
ever, if returns are expected to be normally distributed
the semi variance will be statistically noisier than the
