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Question

  • What is the ratio S(p) shown below? Do we have a name for it like 'Sharpe ratio'?image[1]

  • 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.

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  • $\begingroup$ The ratio does not have an official name, but risk-reward ratio is probably generic enough. $\endgroup$ – John Feb 6 at 13:30
  • $\begingroup$ @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? $\endgroup$ – Eiffelbear Feb 6 at 13:52
  • $\begingroup$ 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. $\endgroup$ – John Feb 6 at 16:55

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