What is the ratio S(p) shown below? Do we have a name for it like 'Sharpe ratio'?
The ratio above is introduced in the academic paper Optimal portfolio selection in a Value-at-Risk framework
- The author of the paper said that from the CVaR point of view, the best portfolio is the one which maximizes the ratio S(p).
What notation means
- r(p) : the expected total return on a portfolio p for a certain period.
- rf : the risk-free rate
- W(0) : the amount of the initial wealth
- q(c,p) : the quantile that corresponds to probability (1-c) of occurrence, which can be read off the cdf of the expected return distribution for portfolio p
One example helping to understand the ratio S(p)
- Let's suppose that the expected annual return of your portfolio is 5%, r(p) = 5%.
- The annual risk-rate is 2%, rf = 2%.
- The bottom 10% expected annual return of the probability distribution of your portfolio is -50%. (It means the probability distribution is left-tailed very much). q(0.9,p) = -50%
- My initial wealth is $ 1,000.
Then S(p) is (5% - 2%) / (1000 * 2% - 1000 * (-50%) ) = 3% / 520 = 0.006%
If the probability distribution of my portfolio is less left-tailed, let's say q(0.9,p) = 1%, then S(p) = 0.3% ( 3% / 10 ), which is much bigger than 0.006% previously.
So I can see why the author of the academic paper said to maximize the ratio S(p) to get the best portfolio from the CVaR viewpoint, but I am curious to know if the ratio has any official name, such as Sharpe ratio.