A Return-to-CVaR Ratio for Portfolio Selection
Summary
The document presents a portfolio score from an academic paper on selection in a value-at-risk framework and asks whether it has a recognized name comparable to the Sharpe ratio. The score uses expected portfolio return relative to the risk-free rate, scaled by initial wealth times the difference between a specified return quantile and the risk-free rate. The author of the paper is described as recommending that portfolios be ranked by maximizing this measure from a CVaR perspective.
An illustrative calculation compares portfolios with the same expected and risk-free returns but different lower-tail quantiles: the portfolio with the more adverse tail receives a much smaller score. This example conveys how downside-tail outcomes affect the denominator, but it is not empirical validation. The document does not provide the formula image or a full derivation, and the notation and sign convention for the quantile-based risk term would need checking against the cited paper before implementation. It reports no established name for the ratio.
Key ideas
- The described portfolio score scales excess expected return by a quantile-based tail-risk term.
- The source paper is said to favor maximizing the score for CVaR-oriented portfolio selection.
- The example shows that a more adverse lower-tail outcome reduces the calculated score.
- The document asks whether the measure has a standard name but does not answer that question.
Tags
Full text
# Another variation of the 'Sharpe ratio' in CVaR-based portfolio optimization? # 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.
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