The Rachev Ratio (or R-Ratio) is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Dr. Svetlozar Rachev and has been extensively studied in quantitative finance. Unlike the reward-to-variability ratios, such as Sharpe ratio and Sortino ratio, the Rachev ratio is a reward-to-risk ratio, which is designed to measure the right tail reward potential relative to the left tail risk in a non-Gaussian setting. Intuitively, it represents the potential for extreme positive returns compared to the risk of extreme losses (negative returns), at a rarity frequency q (quantile level) defined by the user. The ratio is defined as the Expected Tail Return (ETR) in the best q% cases divided by the Expected tail loss (ETL) in the worst q% cases. The ETL is the average loss incurred when losses exceed the Value at Risk at a predefined quantile level. The ETR, defined by symmetry to the ETL, is the average profit gained when profits exceed the Profit at risk at a predefined quantile level. For more tailored applications, the generalized Rachev Ratio has been defined with different powers and/or different confidence levels of the ETR and ETL.
Definition According to its original version introduced by the authors in 2004, the Rachev ratio is defined as:
ρ ( x ′ r ) = C V a R ( 1 − α ) ( r f − x ′ r ) C V a R ( 1 − β ) ( x ′ r − r f ) {\displaystyle \rho \left({x'r}\right)={\frac {CVa{R_{(1-\alpha )}}\left({{r_{f}}-x'r}\right)}{CVa{R_{(1-\beta )}}\left({x'r-{r_{f}}}\right)}}}
or, alternatively,
ρ ( x ′ r ) = E T L α ( r f − x ′ r ) E T L β ( x ′ r − r f ) , {\displaystyle \rho \left({x'r}\right)={\frac {ET{L_{\alpha }}\left({{r_{f}}-x'r}\right)}{ET{L_{\beta }}\left({x'r-{r_{f}}}\right)}},}
where α {\displaystyle \alpha } and β {\displaystyle \beta } belong to ( 0 , 1 ) {\displaystyle \left({0,1}\right)} , and in the symmetric case: α = β {\displaystyle \alpha =\beta } . r f {\displaystyle r_{f}} is the risk-free rate of return and x ′ r {\displaystyle x'r} presents the portfolio return. The ETL is the expected tail loss, also known as conditional value at risk (CVaR), is defined as:
E T L α = 1 α ∫ 0 α V a R q ( X ) d q , {\displaystyle ET{L_{\alpha }}={\frac {1}{\alpha }}\int _{0}^{\alpha }{Va{R_{q}}\left(X\right)dq},}
and
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