The tail mean–variance optimal portfolio selection under generalized skew-elliptical distribution

Abstract

In the insurance and financial markets, events of extreme losses happen in the tail of return distributions, and investors are sensitive to these losses. The Tail Mean–Variance (TMV) criterion focuses on the rare risk but large losses, and it has recently been used in financial management for portfolio selection. In this paper, the proposed TMV criterion is based on the two measures of risk, i.e., the Tail Conditional Expectation (TCE) and Tail Variance (TV) under Generalized Skew-Elliptical (GSE) distribution. We obtain an explicit solution with simple implementation and use a convex optimization approach for the TMV optimization problem under the GSE distribution. We also provide a practical example of a portfolio optimization problem using the proposed TMV criterion. The empirical results show that the optimal portfolio performance can be improved by controlling the tail variability of returns distribution.

Introduction

The Tail Mean–Variance (TMV) criterion for portfolio selection, introduced by Landsman (2010), is defined as follows:

$$TM{V}_q\left(L\right)=\mathbb{E}\left(L|L>Va{R}_q\left(L\right)\right)+\lambda Var\left(L|L>Va{R}_q\left(L\right)\right)$$

where $L$ is a random loss, with a continuous distribution on a portfolio, and $\lambda >0$. Value at Risk (VaR) is the most well-known measure of risk, introduced by Morgan/Reuters (1996), and at confidence level $q$ is equal to:

$$Va{R}_q\left(L\right)=inf\left(x\in R:F_L\left(x\right)\ge q\right)$$

$q\in (0,1)$ and $F_L\left(x\right)$ is the Cumulative Distribution Function (CDF) of loss $L$. The computation of VaR is based on the normal distribution of financial data. Although the normal distribution is the most popular distribution, which is used for modeling in finance and economics, it is not fit for modeling portfolio losses or financial risks. If the data set is non-symmetric, or it has a heavy tail, the normal distribution may not fit well; see, e.g., Ignatieva and Platen, 2010, McNeil et al., 2005, and Granger (2003). Therefore, we need an appropriate and non-symmetric distribution for a detailed analysis of real data.

The TMV criterion in Eq. (1) is composed of a weighted sum of the two risk measures. The first risk measure is the Tail Conditional Expectation (TCE), introduced by Artzner et al. (1999), which gives information about the mean of the tail of the loss distribution. The TCE measure, compared to the VaR, offers a more conservative measure of risk at the same level of confidence $q$. Panjer (2002) introduced the TCE measure for a multivariate normal family, and Landsman and Valdez (2003) presented it for elliptical distribution. Also, Landsman et al. (2013a) derived the TCE measure for a family of skew-elliptical distributions, named the generalized skew-elliptical (GSE) distributions. The second risk measure is the Tail Variance (TV), proposed by Furman and Landsman (2006). Furman and Landsman observed in many samples that the TCE alone does not present enough information about the risks on the right tail, so they introduced the TV measure. The TV criterion measures the deviation of the loss from the mean along the tail of the distribution. Landsman et al. (2013b) analyzed results related to the tail variance and the tail variance premium (TVP) of risks for the class of log-elliptical distributions. Ignatieva and Landsman (2015) studied TCE and TVP for the class of symmetric generalized hyperbolic distribution. Jamshidi Eini and Khaloozadeh (2020) derived the TV measure for the generalized skew-elliptical distributions.

Optimal portfolio selection theory based on the classicalMean–Variance (MV) criterion is initiated by Markowitz (1952) and expanded by many authors. The MV criterion is defined as follows:

$$MV\left(L\right)=\mathbb{E}\left(L\right)+\frac{1}{2}\tau Var\left(L\right)=-\mathbb{E}\left(R\right)+\frac{1}{2}\tau Var\left(R\right)$$

where $\tau >0$, and portfolio return $R=-\mathrm{L}$. The TMV criterion, unlike the MV criterion, focuses on the behavior of the tail of returns distribution through the $q$-quantile defined in the VaR. This issue is very important for financial managers because they are worried about the performance of the portfolio in the event of extreme losses in the capital markets. Xu and Mao (2013) introduced a novel capital allocation rule based on the TMV principle for multivariate elliptical distributions and applied the results to various business units for an insurance company. Owadally and Landsman (2013) obtained an explicit solution for the optimal portfolio on the TMV criterion and compared their solution with previous work in Landsman (2010). Jiang et al. (2016) obtained an explicit solution of the TV measure for the generalized Laplace distribution and optimization of the TMV portfolio. Landsman et al. (2020) considered the maximization of functional of expected portfolio return and variance portfolio return in its most general form. They presented an explicit closed-form solution of the optimal portfolio selection for concave and non-concave functionals. Much research has been done to measure the risk and portfolio selection for symmetric distributions in recent years. However, despite the asymmetry in the real data, less attention has been paid to asymmetric distributions.

The purpose of this paper is to obtain an optimal portfolio by minimizing the proposed TMV criterion subject to a budget constraint, which is based on the two risk measures, i.e., the TCE and the TV under generalized skew-elliptical distribution. The proposed TMV criterion is sensitive to extreme losses that happen in the tail of return distributions. We follow this goal by using a convex optimization method and characteristics of the classical MV criterion, presented by Owadally and Landsman (2013). We also present an explicit solution for the optimal portfolio by providing both insight and computational convenience, which helps to compare the optimal portfolio with other criteria, such as the MV and Minimum Variance criteria.

This paper is classified as follows: The next section defines the class of generalized skew-elliptical distributions. Section 3 formulates the tail conditional expectation and the tail variance measures of risk for the GSE distributions. Section 4 discusses the characterization of the optimal portfolio under the TMV criterion using MV efficiency and finds an explicit solution for the TMV optimization problem under the GSE distributions. The proposed solution is also examined by presenting a general form of the optimization function and its solution. Section 5 shows a practical example of an application to the TMV portfolio optimization on the Nasdaq stock market. Section 6 offers a conclusion to the paper, and several proofs appear in the Appendix.