A discussion of stochastic dominance and mean-risk optimal portfolio problems based on mean-variance-mixture models

In risk management and portfolio optimization problems it is vital to model the underlying asset returns by proper distributions. The classical Markowitz portfolio optimization approach is based on the assumption that asset returns are normally distributed. However normality assumption of the asset returns is disputed extensively in the past literature. Numerous past studies on asset returns indicate that models based on normality assumption fail to fit into real-world data. As the seminal paper [7] demonstrates empirical densities of returns have more mass around the origin and in the tails and less in the flanks compared to the normal distributions. One of the noticeable feature of empirical asset returns is its asymmetry, a property of asset returns that is taken to mean skewness. Skewness of asset returns was recognized early on in modern finance theory and since then a great deal of effort was put on developing better models for asset returns with the hope that new models may lead to better financial decisions. It is readily apparent that for reliable financial decisions it is important to specify, as a starting point, a proper multivariate probability distribution for asset returns. Generalized hyperbolic distributions are flexible family of distributions that can capture most of empirical features of asset returns. As it was demonstrated in the paper [7] hyperbolic distributions give almost perfect fit for the empirical return densities. Hyperbolic distributions are subclass of the class of Normal mean-variance mixture (NMVM) models.

A $d-$dimensional random vector $X$ has normal mean-variance mixture distribution with a mixing distribution $Z$ if for any given $Z=z\in\mathbb{R}_{+}=:(0,+\infty)$ the random vector $X$ satisfies $X|_{Z=z}\sim W_{d}(\mu+z\gamma,z\Sigma)$, where $\mu,\gamma\in\mathbb{R}^{d}$ are constant vectors, $\Sigma$ is a constant and positive-definite $d\times d$ matrix, and $W_{d}$ is an $n-$dimensional normal random vector with mean $\mu+z\gamma$ and co-variance matrix $z\Sigma$. It is well known that such random vector $X$ satisfies

where $A=\Sigma^{\frac{1}{2}}$ and $N_{d}$ is an $d-$dimensional normal random vector with mean equal to zero vector in $\mathbb{R}^{d}$ and with co-variance matrix equal to the identity matrix $I_{d}$ in $\mathbb{R}^{d}\times\mathbb{R}^{d}$. The mixing distribution $Z$ in (1) is independent from $N_{d}$.

There are quite popular models within (1) in financial modeling. If $Z$ follows a Generalized Inverse Gaussian (GIG) distribution, the distribution of $X$ follows a multi-dimensional generalized hyperbolic (mGH) distribution. This class of distributions were first introduced in the paper [2] and their numerous financial applications were discussed in the subsequent papers [1], [6], [7], [21], [22], [9], and [8]. For example, the paper [21] demonstrates that the variance-gamma model and its elliptical multivariate generalization can describe asset returns in unite time period exceptionally well. The paper [7] shows that hyperbolic distributions can give almost perfect fit to the empirical density functions of return data. The paper [1] finds that the Generalized Hyperbolic Skew Student’s $t$ distribution matches empirical financial data exceptionally well. In the multivariate financial data setting, the paper [24] calibrates the multivariate generalized hyperbolic (mGH) model to both multi-variate stock and multi-variate exchange rate returns and in their likelihood-ratio test the Gaussian model is always rejected against the general mGH models. The paper [1] applies the multivariate NIG (Normal Inverse Gaussian) distributions in risk management and demonstrates that they provide a much better fit to the empirical distribution of hedge fund returns than the Normal distribution. Generalized hyperbolic distributions were also applied in pricing interest rate derivatives and it was demonstrated that they allow for very accurate pricing of caplets and other interest rate derivatives, see for example [9] and [8].

In this paper, we consider a market with $d$ risky assets and we assume that the return vector $X$ of these $d$ risky assets satisfy (1). The portfolio set is given by $\mathbb{R}^{d}$ and for each portfolio $\omega\in\mathbb{R}^{d}$ the corresponding portfolio return is given by $\omega^{T}X$, where $T$ denotes a transpose. Our main goal of this paper is to study the following problem

for any given expected return level $r\in\mathbb{R}$. In the optimization problem (2), $\rho$ is a risk measure and $e\in\mathbb{R}^{d}$ is the $d-$dimensional vector of ones. In our note we will provide closed form solution to the problem (2) for risk measures $\rho$ that satisfy certain conditions that will be discussed in Section 3 below. As one of the conditions on $\rho$ we requite that $\rho$ is law-invariant, i.e., $\rho(\eta_{1})=\rho(\eta_{2})$ as long as $\eta_{1}\overset{d}{=}\eta_{2}$. Since

where $N(0,1)$ is a standard normal random variable independent from $Z$, we have

We also require that $\rho$ is consistent with second order stochastic dominance relation. Therefore, as we shall see, the discussion of our problem (2) in fact reduces to studying stochastic dominance relations among one dimensional NMVM models.

We assume that all the random variables in this paper are defined in an atomless probability space $(\Omega,\mathcal{F},P)$. We denote by $L^{0}=L^{0}(\Omega,\mathcal{F},P)$ the space of all finite valued random variables and by $L^{k}=L^{k}(\Omega,\mathcal{F},P),1\leq k<+\infty,$ the space of random variables with finite $k$ moments. $L^{\infty}(\Omega,\mathcal{F},P)$ denotes the space of bounded random variables. For vectors $x=(x_{1},x_{2},\cdots,x_{d})^{T}$ and $y=(y_{1},y_{2},\cdots,y_{d})^{T}$ we denote by $x\cdot y=x^{T}y=\sum_{i=1}^{d}x_{i}y_{i}$ the scalar product of them and by $||x||=(x_{1}^{2}+\cdots+x_{d}^{2})^{\frac{1}{2}}$ the Euclidean norm of $x$. As in [10], for each positive integer $d$ we denote by $L_{d}^{k}=(L^{k})^{d}$ the space of all $d-$dimensional random vectors with all the components belong to $L^{k}$. In this paper, unless otherwise stated we assume that the mixing distribution satisfies $Z\in L^{1}$. We also assume that $Z$ has probability density function $f_{Z}(z)$. This latter assumption on $Z$ is in fact is not necessary for our results of this paper. However, we impose this assumption to keep our calculations simple. We use $\phi(\cdot)$ to denote the density function of standard normal random variable and by $\Phi(\cdot)$ its cumulative distribution function throughout the paper.

Our initial motivation in studying the problem (2) was to discuss solutions for (2) under the widely used and popular risk measures $VaR$ and $CVaR$. The value-at-risk (VaR) is defined for any random variable in $L^{0}$. For any $\eta\in L^{0}$, denote by $q_{\alpha}^{-1}(\eta)=\inf\{a\in\mathbb{R}:F_{\eta}(a)\geq\alpha\}$ the left-continuous inverse of the cumulative distribution function $F_{\eta}(\cdot)$ of $\eta$. For each $\alpha\in(0,1)$ the $VaR_{\alpha}$ is defined as $VaR_{\alpha}(\eta)=q_{\alpha}^{-1}(\eta)$. The conditional value-at-risk (CVaR) is defined on $L^{1}$ as $CVaR_{\alpha}(\eta)=(1/(1-\alpha))\int_{\alpha}^{1}q_{\beta}(\eta)d\beta$ for any $\eta\in L^{1}$. In our note we are able to provide closed form solution for (2) when $\rho=CVaR_{\alpha}$ while an analytical solution for (2) when $\rho=VaR_{\alpha}$ seems difficult. The main reason for this is the risk measure $CVaR_{\alpha}$ is consistent with the second order stochastic dominance property. Another important property of CVaR that is essential for our discussions in this paper is its continuity in the $L^{k}$ space for any $k\geq 1$, see Theorem 4.1 of [17] and page 11 of [10] and the references there for this. In fact, while we don’t use this property, it is also known that CVaR is Lipschitz continuous with respect to $L^{k}$ convergence for $1\leq k\leq\infty$, see Proposition 4.4 in [11] for this. As mentioned earlier, the risk measure VaR is well defined on $L^{0}$ and, unlike CVaR, it is not a convex risk measure. It is consistent with the first order stochastic dominance property but it is not consistent with the second order stochastic dominance property. Due to these facts, optimization problems associated with it is challenging.

The paper is organized as follows. In Section 2 below we formulate a sufficient condition for the stochastic dominance relation for NMVM models. Then we use it to derive closed form solutions for frontier portfolios. In Section 3 we study high-order stochastic dominance relations among NMVM models.

In this section we discuss solutions of the problem (2). For the simplicity of our discussions we first fix a positive integer $k\geq 1$ and we consider risk measures $\rho:L^{k}\rightarrow\mathbb{R}\cup\{\infty\}$ for the problem (2). Denoting by $dom(\rho)=\{\eta\in L^{k}:\rho(\eta)<\infty\}$ the domain of $\rho$, we call $\rho$ proper if $\rho(\eta)>-\infty$ for all $\eta\in L^{k}$ and $dom(\rho)\neq\emptyset$. A risk measure on $L^{k}$ is called convex risk measure if it is a proper monetary risk measure (i.e., it is monotone and cash-invariant) which is convex, i.e., $\rho(\lambda\eta_{1}+(1-\lambda)\eta_{2})\leq\lambda\rho(\eta_{1})+(1-\lambda)\rho(\eta_{2}),\forall\lambda\in[0,1],\forall\eta_{1},\eta_{2}\in L^{k}$. A convex risk measure is called coherent if it is positive homogeneous, i.e., $\rho(\lambda\eta)=\lambda\rho(\eta),\forall\lambda>0$. From Corollary 2.3 of [17] any finite convex risk measure on $L^{k}$ is continuous in $L^{k}$. The risk measure $CVaR$ is a coherent risk measure on $L^{k}$ and hence it is continuous on $L^{k}$. This property of $CVaR$ is useful in the proof of the Proposition 2.14 below.

Another property of $\rho$ that is essential for our discussions in this section is its consistency with the second order stochastic dominance property. We say that $\rho$ is consistent with the second order stochastic dominance (SSD-consistent) property if for any two $\eta_{1},\eta_{2}\in L^{k}$ the relation $\eta_{1}\succeq_{(2)}\eta_{2}$ implies that $\rho(\eta_{1})\geq\rho(\eta_{2})$. This property of $\rho$ is needed in the proof of our Theorem 2.4 below and an equivalent version of it is stated in the Lemma 2.13 below. We call $\rho$ is law-invariant if $\rho(\eta_{1})=\rho(\eta_{2})$ whenever $\eta_{1}\overset{d}{=}\eta_{2}$. Any finite valued, law-invariant, convex risk measure is SSD-consistent, see the paper [23] and the references there for this.

Before we state our main result in this section, we first give a short review for the Markovitz mean-variance portfolio optimization problem. In the Markowitz mean-variance portfolio optimization framework the variance is used as a risk measure. In this case, the corresponding optimization problem is

The closed form solution of (3) is standard and can be found in any standard textbooks that discusses the capital asset pricing model, see page 64 of [16] for example. Here we write down the solution of (3) as it is relevant with our main results in this paper. To this end, let $\mu=EX$ denote the mean vector and let $V=Cov(X)$ denote the co-variance matrix of $X$. For each $r\in\mathbb{R}$, the solution of (3) is given by

where

In this section, we will show that the optimal portfolio in our mean-risk portfolio optimization problem (2) also takes a similar form as in (4). Therefore, we first introduce the following notations. For any random vector $\theta$ with mean vector $\mu_{\theta}=E\theta$ and co-variance matrix $\Sigma_{\theta}=Cov(\theta)$ we introduce the following expression

where

Our Theorem 2.4 below shows that, frontier portfolios to the problem (2), under any real-valued, law invariant, and convex risk measure, always exist and they are in the form as in (5) when the return vectors are given by (1).

The risk measure $CVaR_{\alpha}$ for each fixed $\alpha\in(0,1)$ is a finite valued, law-invariant, convex risk measure. Therefore the result of the above Theorem 2.4 holds for this risk measure also. We state this in the following Corollary.

The proof of the Theorem 2.4 above needs some preparation. We first prove the following Lemma. First, observe that from (24) in Section 3 for any $H=a+bZ+c\sqrt{Z}N$ we have

Define

It is easy to check that

Therefore $\Psi_{H}(u,c)$ is an increasing function of $u$ and also of $c$. We use these facts in the proof of the following Lemma.

Before we state our next result we first recall few definitions. Recall that a random variable $\eta_{1}$ second order stochastically dominates another random variable $\eta_{2}$, i.e., $\eta_{1}\succeq_{(2)}\eta_{2}$, iff $\int_{-\infty}^{t}[F_{\eta_{1}}(a)-F_{\eta_{2}}(a)]da\leq 0$ for all $t\in\mathbb{R}$. The dual of SSD is the risk-seeking stochastic dominance (RSSD) relation $\eta_{1}\succeq_{(2^{\prime})}\eta_{2}$ which is defined as $\int_{t}^{\infty}[F_{\eta_{1}}(a)-F_{\eta_{2}}(a)]da\leq 0$ for all $t\in\mathbb{R}$. Since we have $\int_{-\infty}^{t}(F_{\eta_{1}}(a)-F_{\eta_{2}}(a))da=\int_{-t}^{\infty}(F_{-\eta_{2}}(a)-F_{-\eta_{1}}(a))da$ it is clear that the relation $\eta_{1}\succeq_{(2)}\eta_{2}$ is equivalent to the relation $-\eta_{2}\succeq_{(2^{\prime})}-\eta_{1}$. The risk measure $CVaR$ is consistent with RSSD for any $\alpha\in(0,1)$ in the sense that (See page 174 of [5] for this)