# Another variation of the 'Sharpe ratio' in CVaR-based portfolio optimization?

## Question

• 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.

• The ratio does not have an official name, but risk-reward ratio is probably generic enough. – John Feb 6 '19 at 13:30
• @John Thanks for your helpful comment! Then do we have any portfolio performance measure as popular as Sharpe ratio, which can be applied to CVaR based portfolios? – Eiffelbear Feb 6 '19 at 13:52
• There's nothing stopping you from calculating the risk-reward ratio using CVaR as your measure of risk. However, it can be tricky to interpret, particularly relative to a Sharpe ratio. You may find better luck using CVaR deviation, which is CVaR minus the expected return. – John Feb 6 '19 at 16:55