Rachev ratio

According to its original version introduced by the authors in 2004, the Rachev ratio is defined as:

${\displaystyle \rho \left({x'r}\right)={\frac {CVa{R_{(1-\alpha )}}\left({{r_{f}}-x'r}\right)}{CVa{R_{(1-\beta )}}\left({x'r-{r_{f}}}\right)}}}$

or, alternatively,

${\displaystyle \rho \left({x'r}\right)={\frac {ET{L_{\alpha }}\left({{r_{f}}-x'r}\right)}{ET{L_{\beta }}\left({x'r-{r_{f}}}\right)}},}$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ belong to ${\displaystyle \left({0,1}\right)}$, and in the symmetric case: ${\displaystyle \alpha =\beta }$. ${\displaystyle r_{f}}$ is the risk-free rate of return and ${\displaystyle x'r}$ presents the portfolio return. The ETL is the expected tail loss, also known as conditional value at risk (CVaR), is defined as:

${\displaystyle ET{L_{\alpha }}={\frac {1}{\alpha }}\int _{0}^{\alpha }{Va{R_{q}}\left(X\right)dq},}$

and

${\displaystyle Va{R_{\alpha }}=-F_{X}^{-1}\left(\alpha \right)=-\inf \left\{{x\left|P{\left({X\leq x}\right)>\alpha }\right.}\right\}}$

is the value at risk (VaR) of the random return ${\displaystyle X}$.

Thus, the ETL can be interpreted as the average loss beyond VaR:

${\displaystyle ET{L_{\alpha }}\left(X\right)=E\left[{L|L>Va{R_{_{\alpha }}}}\right]}$.

The generalized Rachev ratio is the ratio between the power CVaR of the opposite of the excess return at a given confidence level and the power CVaR of the excess return at another confidence level. That is,

${\displaystyle \rho \left({x'r}\right)={\frac {ET{L_{\left({\gamma ,\alpha }\right)}}\left({{r_{f}}-x'r}\right)}{ET{L_{\left({\delta ,\beta }\right)}}\left({x'r-{r_{f}}}\right)}},}$

where ${\displaystyle ET{L_{\left({\gamma ,\alpha }\right)}}\left(X\right)=E\left[{{\rm {max}}{{\left({L,0}\right)}^{\gamma }}|L>Va{R_{_{\alpha }}}}\right]}$ is the power CVaR of ${\displaystyle X}$, and ${\displaystyle \gamma }$ is a positive constant. The main advantage of the generalized Rachev ratio over the traditional Rachev ratio is conferred by the power indexes ${\displaystyle \gamma }$ and ${\displaystyle \delta }$ that characterize an investor's aversion to risk.

The Rachev ratio can be used in both ex-ante and ex-post analyses.

The 5% ETL and 5% ETR of a non-Gaussian return distribution. Although the most probable return is positive, the Rachev ratio is 0.7 < 1, which means that the excess loss is not balanced by the excess profit in the investment.

In the ex-post analysis, the Rachev ratio is computed by dividing the corresponding two sample AVaR's. Since the performance levels in the Rachev ratio are quantiles of the active return distribution, they are relative levels as they adjust according to the distribution. For example, if the scale is small, then the two performance levels will be closer to each other. As a consequence, the Rachev ratio is always well-defined.

In the ex-ante analysis, optimal portfolio problems based on the Rachev ratio are, generally, numerically hard to solve because the Rachev ratio is a fraction of two CVaRs which are convex functions of portfolio weights. In effect, the Rachev ratio, if viewed as a function of portfolio weights, may have many local extrema.[6]

Several empirical tests of the Rachev ratio and the generalized Rachev ratio have been proposed.[4][7]

An algorithm for solving Rachev ratio optimization problem was provided in Konno, Tanaka, and Yamamoto (2011).[8]

In quantitative finance, non-Gaussian return distributions are common. The Rachev ratio, as a risk-adjusted performance measurement, characterizes the skewness and kurtosis of the return distribution (see picture on the right).

• Post-modern portfolio theory
• Upside potential ratio
• Sharpe ratio
• Sortino ratio
• Omega ratio
• Modern Portfolio Theory