Omega-CVaR portfolio optimization and its worst case analysis

Abstract

This paper presents a novel framework for optimizing portfolios using distribution dependent thresholds in Omega ratio to control the downside risk. Portfolios resulting from the maximization of the classical Omega ratio simultaneously maximize the probability of superior performance compared to a threshold point set by an investor and minimize the probability of a worse performance compared to the same threshold. However, there is no mandatory rule or mechanism to choose this threshold point in the Omega ratio optimization model yet. In this paper, we redefine the Omega ratio for a loss averse investor by taking the distribution dependent threshold point as the conditional value-at-risk at an $\alpha$ confidence level (${\mathrm{CVaR}_{\alpha }}$) of the benchmark market. The $\alpha$-value reflects the attitude of an investor towards losses. We then embed this new Omega-${\mathrm{CVaR}_{\alpha }}$ model in a robust portfolio optimization framework and present its worst case analysis under three uncertainty sets. The robustness is introduced both in the Omega measure and the ${\mathrm{CVaR}_{\alpha }}$ measure. We show that the worst case Omega-${\mathrm{CVaR}_{\alpha }}$ robust optimization models are linear programs for mixed and box uncertainty sets and a second order cone program under ellipsoidal sets, and hence tractable in all three cases. We conduct a comprehensive empirical investigation of the classical ${\mathrm{CVaR}_{\alpha }}$ model, the STARR$_{\alpha }$ model, the Omega-${\mathrm{CVaR}_{\alpha }}$ model, and robust Omega-${\mathrm{CVaR}_{\alpha }}$ model under a mixed uncertainty set for listed stocks of the S&P 500. The optimal portfolios resulting from the Omega-${\mathrm{CVaR}_{\alpha }}$ model exhibit a superior performance compared to the classical ${\mathrm{CVaR}_{\alpha }}$ model in the sense of higher expected returns, Sharpe ratios, modified Sharpe ratios, and lesser losses in terms of ${\mathrm{VaR}_{\alpha }}$ and ${\mathrm{CVaR}_{\alpha }}$ values. The robust Omega-${\mathrm{CVaR}_{\alpha }}$ model under mixed uncertainty set is shown to dominate the Omega-${\mathrm{CVaR}_{\alpha }}$ model in terms of all performance measures. Furthermore, both the Omega-${\mathrm{CVaR}_{\alpha }}$ and robust Omega-${\mathrm{CVaR}_{\alpha }}$ model under a mixed uncertainty set yield significantly lower risk compared to STARR$_{\alpha }$ model in terms of $\mathrm{CVaR}_{\alpha }$ and variance values.

Appendices

Appendix A: Worst Omega ratio in continuous case under mixed uncertainty set

Define

$$\begin{aligned} H= & {} \frac{E_p(L - \ell _x)^+}{E_p(\ell _x - L)^+} \,=\, \frac{\displaystyle \int _{\ell }(L - \ell _x)^+p(\ell )\,\mathrm{d}\ell }{\displaystyle \int _{\ell }(\ell _x - L)^+p(\ell )\,\mathrm{d}\ell } = \frac{\displaystyle \int _{\ell }\displaystyle \sum \nolimits _{k=1}^{s}w_k(L - \ell _x)^+p^k(\ell )\,\mathrm{d}\ell }{\displaystyle \int _{\ell }\displaystyle \sum \nolimits _{k=1}^{s}w_k(\ell _x - L)^+p^k(\ell )\,\mathrm{d}\ell }. \end{aligned}$$

Applying the Charnes and Cooper (1962) transformation, with $\gamma \, > \, 0$ as a homogenization variable, we have,

$$\begin{aligned} H= & {} \displaystyle \int _{\ell }\displaystyle \sum _{k=1}^{s}w_k(L\gamma - \ell _{\widetilde{x}})^+p^k(\ell )\,\mathrm{d}\ell \;\text {with}\; \displaystyle \int _{\ell }\displaystyle \sum _{k=1}^{s}w_k(\ell _{\tilde{x}} - L\gamma )^+p^k(\ell )\,d\ell = 1, \;\; \widetilde{x} \!= \!x\gamma ,\\= & {} \displaystyle \sum _{k=1}^{s}w_kG^k_1 \quad \;\;\text {with}\; \quad \displaystyle \sum _{k=1}^{s}w_kG^k_2 = 1, \end{aligned}$$

where $G^k_1 = \displaystyle \int _{\ell }(L\gamma - \ell _{\widetilde{x}})^+p^k(\ell )\,\mathrm{d}\ell , \; G^k_2 = \displaystyle \int _{\ell }(\ell _{\widetilde{x}} - L\gamma )^+p^k(\ell )\,\mathrm{d}\ell$.

Appendix B: Mixed uncertainty set

The mixed uncertainty set for the distribution of benchmark market loss $\ell _z$ is as follows:

$$\begin{aligned} \mathcal {Q}_M = \left\{ q = \displaystyle \sum _{k=1}^{\hat{s}}\hat{w}_kq^k;\; \displaystyle \sum _{k=1}^{\hat{s}}\hat{w}_k=1,\;\hat{w}_k \ge 0,\;k=1,\ldots ,\hat{s}\right\} , \end{aligned}$$

where $q^k$ is the kth likelihood density function of portfolio loss $\hat{\ell }_z$. The worst case analysis of ${P_{\mathrm{CVaR}_{\alpha }(\hat{\ell }_z)}}$ under the mixed uncertainty set for continuous case is already discussed in Zhu and Fukushima (2009). Here we re-state its discrete version for reader comprehension. Let $\hat{T}^k$ be the finite number of scenarios of $\hat{\ell }_z$ (using sampling techniques) with the kth, $k=1,\ldots ,\hat{s}$ likelihood probability vector $q^k = ((q^k_1,\ldots ,q^k_{\hat{T}^k})^t; \; (q^k)^te =1,\; q^k_i \ge 0,\, \forall \,i=1,\ldots ,\hat{T}^k)$. Then following Zhu and Fukushima (2009), the ${P_{\mathrm{CVaR}_{\alpha }(\hat{\ell }_z)}}$ model under $\mathcal {Q}_M$ as follows:

where $\hat{B}^k = [\hat{\ell }^k_{ij}]_{\hat{T}^k \times m}$ is the loss matrix of portfolio z corresponding to the likelihood probability function $q^k$.

Therefore, the dual of ${P_{\mathrm{MCVaR}_{\alpha }(\hat{\ell }_z)}}$ is derived as follows:

where $v_k,\, k=1,\ldots ,\hat{s}$, is the kth component of vector v.

Appendix C: Box uncertainty set

Let $\mathcal {Q}_B = \{q = q^0 + \hat{\pi };\; \hat{\pi }^te = 0,\; \underline{\hat{\pi }} \le \hat{\pi } \le \bar{\hat{\pi }} \}$ be a box uncertainty set for the distribution of benchmark market loss $\hat{\ell _z}$, then ${P_{\mathrm{CVaR}_{\alpha }(\hat{\ell }_z)}}$ under $\mathcal {Q}_B$ is given according to Zhu and Fukushima (2009):

$$\begin{aligned} {P_{\mathrm{BCVaR}_{\alpha }(\hat{\ell }_z)}}\qquad \min \quad \hat{\theta } \end{aligned}$$

$$\begin{aligned} \text {subject to:} \;\; \tau + \frac{1}{1-\alpha }(q^0)^t\hat{u} + \frac{1}{1-\alpha }(\bar{\hat{\pi }}^t\xi + \underline{\hat{\pi }}^t\varrho )\le & {} \hat{\theta } \\ \epsilon e + \xi - \varrho= & {} \hat{u} \\ \hat{u} + \tau e - \hat{B}z\ge & {} 0 \\ e^tz= & {} 1,\;\;\tau \in \mathbb {R},\;\; z \in \mathbb {R}^{\hat{m}}_+ \qquad \\ \epsilon \in \mathbb {R},\;\hat{u}, \, \xi ,\,\varrho\in & {} \mathbb {R}^{\hat{T}}_{+}, \end{aligned}$$

where $\hat{B} = [\hat{\ell }_{ij}]_{\hat{T} \times m}$ is the loss matrix of portfolio z. The dual of ${P_{\mathrm{BCVaR}_{\alpha }(\hat{\ell }_z)}}$ is as follows:

Appendix D: Ellipsoidal uncertainty set

For fixed values of $\theta , \widetilde{u}, \widetilde{d}$, here, we first derive the dual of $P_{10}$ in the following steps: The Lagrange of $P_{10}$ with Lagrange multipliers $\lambda _1,\,\lambda _2,\,\lambda _3,\,\lambda _4,\,\lambda _5$, is given as follows:

$$\begin{aligned} L(.) \!= & {} \! \widetilde{u}^tp^0\! +\! u^tB\pi - \lambda _1 (\widetilde{d}^tp^0 + \widetilde{d}^tA\pi -1) - \lambda ^t_2(p^0 + A\pi ) -\lambda _4(e^tA\pi ) + \lambda _5\pi - \lambda _3.\nonumber \\= & {} (A^t\widetilde{u} - \lambda _1A^t\widetilde{d} - \lambda _4A^te - A^t\lambda _2 + \lambda _5)\pi + \widetilde{u}^tp^0 - \lambda _1\widetilde{d}^tp^0 + \lambda _1 - \lambda ^t_2p^0 - \lambda _3.\nonumber \\ \end{aligned}$$

(24)

Using the minimax representation of the primal problem $P_{11}$ as

$$\begin{aligned} \displaystyle \min _{\pi }\displaystyle \max _{\{||\lambda _5||_2 \le \lambda _3,\lambda _1,\lambda _2,\lambda _4\}}L(.) = \displaystyle \min _{\{||\lambda _5||_2 \le \lambda _3,\lambda _1,\lambda _2,\lambda _4\}}\displaystyle \max _{\pi }L(.), \end{aligned}$$

(25)

the inner problem of the latter one is affine in $\pi$ and can be solved by taking its derivative with respect to $\pi$ which leads to the following dual constraint derived from Eq. (24):

$$\begin{aligned} A^t\widetilde{u} - \lambda _1A^t\widetilde{d} - \lambda _4A^te - A^t\lambda _2 + \lambda _5 = 0 \end{aligned}$$

Therefore, the dual of $P_{11}$ is given as follows:

$$\begin{aligned} P_{12}\qquad \max \quad \widetilde{u}^tp^0 - \lambda _1\widetilde{d}^tp^0 + \lambda _1 - \lambda ^t_2p^0 - \lambda _3 \end{aligned}$$

$$\begin{aligned} \text {subject to:} \;\;A^t\widetilde{u} - \lambda _1A^t\widetilde{d} - \lambda _4A^te - A^t\lambda _2 + \lambda _5 = 0\end{aligned}$$

(26)

$$\begin{aligned} ||\lambda _5||_2 \le \lambda _3 \end{aligned}$$

(27)

$$\begin{aligned} \lambda _1, \lambda _3, \lambda _4 \in \mathbb {R};\,\,\,\,\, \lambda _2 \in \mathbb {R}_{+}^T,\,\,\,\, \lambda _5 \in \mathbb {R}^T. \end{aligned}$$

(28)

$P_{12}$ is an SOCP problem.

Dual of worst case of ${\mathrm{CVaR}_{\alpha }}$ model under ellipsoidal case

For the ellipsoidal set $\mathcal {Q}_E = \{ q = q^0 + \hat{A}\hat{\pi };\; e^t\hat{A}\hat{\pi } = 0,\; q^0 + \hat{A}\hat{\pi } \ge 0,\; ||\hat{\pi }||_2 \le 1 \}$ of portfolio z, the worst case of ${P_{\mathrm{CVaR}_{\alpha }(\ell _z)}}$ is as follows (Zhu and Fukushima 2009):

${P_{\mathrm{ECVaR}_{\alpha }(\hat{\ell }_z)}}$ is an SOCP problem. Analogously to the dual derivation $P_{12}$ from $P_{11}$, we can also obtain the dual of ${P_{\mathrm{ECVaR}_{\alpha }(\hat{\ell }_z)}}$ as follows: