Second-order stochastic dominance constrained portfolio optimization: Theory and computational tests

Due to the definition of second-order stochastic dominance (SSD) in terms of utility theory, portfolio optimization with SSD constraints is of major practical interest. We contribute to the field in two ways: first, we present a self-contained theory with some new results and new proofs of known results; second, we perform a set of tests for computational efficiency. We provide new and simple arguments for the formulation of SSD constraints in a mathematical programming framework. For many individuals, an SSD constraint may seem too severe wherefore various relaxations (ASSD), have been proposed. We introduce yet another relaxation, directional SSD, where a candidate portfolio is admissible if a step from the benchmark in the direction of the candidate yields a dominating portfolio. Optimal step size depends on individual preferences reflected by the objective function. We compare computational efficiency of seven approaches for SD constrained portfolio problems, including SSD and ASSD constrained cases.     

Portfolio construction based on stochastic dominance and target return distributions

Mean-risk models have been widely used in portfolio optimization. However, such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties. The problem is multi-objective and is transformed into a single objective problem by using the reference point method, in which target levels, known as aspiration points, are specified for the objective functions. A model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return. This concept is extended by additionally specifying reservation points, which act pre-emptively in the optimization model. The theoretical properties of the models are studied. The performance of the models on real data drawn from the Hang Seng index is also investigated.     

A note on Bernoulli's principle and probability dominance

The concept of -dominance proposed by Wrather and Yu (Ref. 1) is shown to be not equivalent to that of expected utilities, which depend on the chosen alternative and the not chosen alternative.This work was performed under the auspices of GNAFA-CNR, Rome, Italy.The author thanks Professor P. L. Yu for his helpful comments.on leave at the Department of Mathematics, University of Arizona, Tucson, Arizona.     

Enhanced indexation based on second-order stochastic dominance

Second order Stochastic Dominance (SSD) has a well recognised importance in portfolio selection, since it provides a natural interpretation of the theory of risk-averse investor behaviour. Recently, SSD-based models of portfolio choice have been proposed; these assume that a reference distribution is available and a portfolio is constructed, whose return distribution dominates the reference distribution with respect to SSD. We present an empirical study which analyses the effectiveness of such strategies in the context of enhanced indexation. Several datasets, drawn from FTSE 100, SP 500 and Nikkei 225 are investigated through portfolio rebalancing and backtesting. Three main conclusions are drawn. First, the portfolios chosen by the SSD based models consistently outperformed the indices and the traditional index trackers. Secondly, the SSD based models do not require imposition of cardinality constraints since naturally a small number of stocks are selected. Thus, they do not present the computational difficulty normally associated with index tracking models. Finally, the SSD based models are robust with respect to small changes in the scenario set and little or no rebalancing is necessary.     

Necessary conditions for stochastic dominance: A general approach

In this paper a general method for developing necessary conditions for all degrees of stochastic dominance is derived. The method, a minimization of the expected value of certain functions of the random variable, is used to rederive known necessary conditions for dominance and is then used to derive new necessary conditions. Some of the old and new conditions are then compared empirically using a data set of security returns.     

Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization

Inspired by the successful applications of the stochastic optimization with second order stochastic dominance (SSD) model in portfolio optimization, we study new numerical methods for a general SSD model where the underlying functions are not necessarily linear. Specifically, we penalize the SSD constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation (SA) method and the level function method to solve the penalized problem. Both methods are iterative: the former requires to calculate an approximate subgradient of the objective function of the penalized problem at each iterate while the latter requires to calculate a subgradient. Under some moderate conditions, we show that w.p.1 the sequence of approximated solutions generated by the SA method converges to an optimal solution of the true problem. As for the level function method, the convergence is deterministic and in some cases we are able to estimate the number of iterations for a given precision. Both methods are applied to portfolio optimization problem where the return functions are not necessarily linear and some numerical test results are reported.     

Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

Stochastic dominance relations are well studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance constraints can be solved by linear programming. However, problems involving first order stochastic dominance constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete first order constrained optimization model and present a relaxation based on second order constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional second order constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio optimization problem. This research was supported by the NSF awards DMS-0603728 and DMI-0354678.     

Inverse stochastic dominance constraints and rank dependent expected utility theory

We consider optimization problems with second order stochastic dominance constraints formulated as a relation of Lorenz curves. We characterize the relation in terms of rank dependent utility functions, which generalize Yaari's utility functions. We develop optimality conditions and duality theory for problems with Lorenz dominance constraints. We prove that Lagrange multipliers associated with these constraints can be identified with rank dependent utility functions. The problem is numerically tractable in the case of discrete distributions with equally probable realizations. Research supported by the NSF awards DMS-0303545, DMS-0303728, DMI-0354500 and DMI-0354678.     

Data envelopment analysis of mutual funds based on second-order stochastic dominance

Although data envelopment analysis (DEA) has been extensively used to assess the performance of mutual funds (MF), most of the approaches overestimate the risk associated to the endogenous benchmark portfolio. This is because in the conventional DEA technology the risk of the target portfolio is computed as a linear combination of the risk of the assessed MF. This neglects the important effects of portfolio diversification. Other approaches based on mean–variance or mean–variance–skewness are non-linear. We propose to combine DEA with stochastic dominance criteria. Thus, in this paper, six distinct DEA-like linear programming (LP) models are proposed for computing relative efficiency scores consistent (in the sense of necessity) with second-order stochastic dominance (SSD). The aim is that, being SSD efficient, the obtained target portfolio should be an optimal benchmark for any rational risk-averse investor. The proposed models are compared with several related approaches from the literature.     

Expected utility approximation and portfolio optimisation

Classical portfolio selection problems that optimise expected utility can usually not be solved in closed form. It is natural to approximate the utility function, and we investigate the accuracy of this approximation when using Taylor polynomials. In the important case of a Merton market and power utility we show analytically that increasing the order of the polynomial does not necessarily improve the approximation of the expected utility. The proofs use methods from the theory of parabolic second-order partial differential equations. All results are illustrated by numerical examples.     

A new stochastic factor model: General explicit solutions

We present a new stochastic factor model. In doing so, we provide general, explicit solutions to the portfolio optimization problem.     

Stochastic dominance and parameter estimation: The case of symmetric stable distributions

Stochastic dominance rules can be applied to the selection of statistical estimators. This paper applies the procedure to estimators of location parameters of stable distributions: the mean and the median. It was found that the preference of one estimator over another depends on the characteristic exponent and on the sample size. Furthermore for some combinations of characteristic exponents and sample size we found that the stochastic dominance rule yields no preference implying that depending on one's utility function one may prefer the mean over the median or vise versa. This result differs from the common mean squared error criterion.     

Multiperiod portfolio optimization models in stochastic markets using the mean–variance approach

We consider several multiperiod portfolio optimization models where the market consists of a riskless asset and several risky assets. The returns in any period are random with a mean vector and a covariance matrix that depend on the prevailing economic conditions in the market during that period. An important feature of our model is that the stochastic evolution of the market is described by a Markov chain with perfectly observable states. Various models involving the safety-first approach, coefficient of variation and quadratic utility functions are considered where the objective functions depend only on the mean and the variance of the final wealth. An auxiliary problem that generates the same efficient frontier as our formulations is solved using dynamic programming to identify optimal portfolio management policies for each problem. Illustrative cases are presented to demonstrate the solution procedure with an interpretation of the optimal policies.     

Optimal strategies for asset allocation and consumption under stochastic volatility

Selecting optimal asset allocation and consumption strategies is an important, but difficult, topic in modern finance. The dynamics is governed by a nonlinear partial differential equation. Stochastic volatility adds further complication. Even to obtain a numerical solution is challenging. Here, we develop a closed-form approximate solution. We show that our theoretical predictions for the optimal asset allocation strategy and the optimal consumption strategy are in surprisingly good agreement with the results from full numerical computations.     

Continuous-time portfolio selection with liability: Mean–variance model and stochastic LQ approach

In this paper we formulate a continuous-time mean–variance portfolio selection model with multiple risky assets and one liability in an incomplete market. The risky assets’ prices are governed by geometric Brownian motions while the liability evolves according to a Brownian motion with drift. The correlations between the risky assets and the liability are considered. The objective is to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. We derive explicitly the optimal dynamic strategy and the mean–variance efficient frontier in closed forms by using the general stochastic linear-quadratic (LQ) control technique. Several special cases are discussed and a numerical example is also given.     

Multistage stochastic convex programs: Duality and its implications

In this paper, we study alternative primal and dual formulations of multistage stochastic convex programs (SP). The alternative dual problems which can be traced to the alternative primal representations, lead to stochastic analogs of standard deterministic constructs such as conjugate functions and Lagrangians. One of the by-products of this approach is that the development does not depend on dynamic programming (DP) type recursive arguments, and is therefore applicable to problems in which the objective function is non-separable (in the DP sense). Moreover, the treatment allows us to handle both continuous and discrete random variables with equal ease. We also investigate properties of the expected value of perfect information (EVPI) within the context of SP, and the connection between EVPI and nonanticipativity of optimal multipliers. Our study reveals that there exist optimal multipliers that are nonanticipative if, and only if, the EVPI is zero. Finally, we provide interpretations of the retroactive nature of the dual multipliers. This work was supported by NSF grant DMII-9414680.     

Robustness of optimal portfolios under risk and stochastic dominance constraints

Solutions of portfolio optimization problems are often influenced by a model misspecification or by errors due to approximation, estimation and incomplete information. The obtained results, recommendations for the risk and portfolio manager, should be then carefully analyzed. We shall deal with output analysis and stress testing with respect to uncertainty or perturbations of input data for static risk constrained portfolio optimization problems by means of the contamination technique. Dependence of the set of feasible solutions on the probability distribution rules out the straightforward construction of convexity-based global contamination bounds. Results obtained in our paper [Dupa?ová, J., & Kopa, M. (2012). Robustness in stochastic programs with risk constraints. Annals of Operations Research, 200, 55–74.] were derived for the risk and second order stochastic dominance constraints under suitable smoothness and/or convexity assumptions that are fulfilled, e.g. for the Markowitz mean–variance model. In this paper we relax these assumptions having in mind the first order stochastic dominance and probabilistic risk constraints. Local bounds for problems of a special structure are obtained. Under suitable conditions on the structure of the problem and for discrete distributions we shall exploit the contamination technique to derive a new robust first order stochastic dominance portfolio efficiency test.     

Stochastic dominance and risk measure: A decision-theoretic foundation for VaR and C-VaR

Is it possible to obtain an objective and quantifiable measure of risk backed up by choices made by some specific groups of rational investors? To answer this question, in this paper we establish some behavior foundations for various types of VaR models, including VaR and conditional-VaR, as measures of downside risk. In this paper, we will establish some logical connections among VaRs, conditional-VaR, stochastic dominance, and utility maximization. Though supported to some extents with unanimous choices by some specific groups of expected or non-expected-utility investors, VaRs as profiles of risk measures at various levels of risk tolerance are not quantifiable – they can only provide partial and incomplete risk assessments for risky prospects.     

Decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance

A problem of decision making under uncertainty in which the choice must be made between two sets of alternatives instead of two single ones is considered. A number of choice rules are proposed and their main properties are investigated, focusing particularly on the generalizations of stochastic dominance and statistical preference. The particular cases where imprecision is present in the utilities or in the beliefs associated to two alternatives are considered.     

Forward dynamic utility functions: A new model and new results

A major obstacle in the existing models of forward dynamic utilities and investment performance evaluation is to establish the existence and uniqueness of the optimal solutions. Consequently, we present a new model of forward dynamic utilities. In doing so, we establish the existence and uniqueness of the solutions for a general (smooth) utility function, and we show that the assumptions needed for such solutions are similar to those under the backward formulation. Moreover, we provide unique viscosity solutions. We also provide discontinuous viscosity solutions. In addition, we introduce Hausdorff-continuous viscosity solutions to the portfolio model.