Mean-risk Models Research Articles

The column generation approach to the mean-risk model for the portfolio selection problem with spillover risk aversion

In this article, we propose extensions of the general mean-risk portfolio selection models to limit the spillover effect of risk. We use the conditional value at risk (CoVaR) as a measure to identify the spillover risk contribution of securities. The goal is to find a portfolio that minimizes general risk measures while the spillover risk contribution is limited. Based on the frameworks of the general mean-risk models, we additionally guarantee that a portfolio has limited spillover risk contributions. To solve these problems efficiently, we propose a Dantzig-Wolfe decomposition reformulation that yields a strong continuous relaxation bound by introducing a set of feasible investment patterns. We develop an algorithm that incorporates a column generation procedure to reduce the spillover effect of tail risk in subproblems. Finally, we present the performance of the portfolios using real historical data. The results show that our model can provide better portfolios than the general mean-risk models.

Risk minimisation using options and risky assets

We consider mean-risk portfolio optimisation models, with risk measured by symmetric measures (variance) as well as downside or tail measures (lower partial moments, conditional value at risk). A framework for including index options in the universe of assets, in addition to stocks, is provided. The exercise of index options is settled in cash, making this implementable with a variety of strike prices and maturities. We use a dataset with stocks from FTSE 100 and index options on FTSE100. Numerical results show that, for low risk-low return and to medium risk-medium return portfolios, the addition of an index put further reduces the risk to a considerable extent, particularly in the case of mean-CVaR efficient portfolios, where the left tail of the portfolio return distribution is dramatically improved. For high risk-high return portfolios, the inclusion of an index call improves the right tail of the return distribution, creating thus the opportunity for considerably higher returns.

On Risk-Averse Stochastic Semidefinite Programs with Continuous Recourse

The vast majority of the literature on stochastic semidefinite programs (stochastic SDPs) with recourse is concerned with risk-neutral models. In this paper, we introduce mean-risk models for stochastic SDPs and study structural properties as convexity and (Lipschitz) continuity. Special emphasis is placed on stability with respect to changes of the underlying probability distribution. Perturbations of the true distribution may arise from incomplete information or working with (finite discrete) approximations for the sake of computational efficiency. We discuss extended formulations for stochastic SDPs under finite discrete distributions, which turn out to be deterministic (mixed-integer) SDPs that are (almost) block-structured for many popular risk measures.

A practical assessment of risk-averse approaches in production lot-sizing problems

ABSTRACTThis paper presents an empirical assessment of four state-of-the-art risk-averse approaches to deal with the capacitated lot-sizing problem under stochastic demand. We analyse two mean-risk models based on the semideviation and on the conditional value-at-risk risk measures, and alternate first and second-order stochastic dominance approaches. The extensive computational experiments based on different instances characteristics and on a case-study suggest that CVaR exhibits a good trade-off between risk and performance, followed by the semideviation and first-order stochastic dominance approach. For all approaches, enforcing risk-aversion helps to reduce the cost-standard deviation substantially, which is usually accomplished via increasing production rates. Overall, we can say that very risk-averse decision-makers would be willing to pay an increased price to have a much less risky solution given by CVaR. In less risk-averse settings, though, semideviation and first-order stochastic dominance can be appealing alternatives to provide significantly more stable production planning costs with a marginal increase of the expected costs.

A mean-risk mixed integer nonlinear program for transportation network protection

This paper focuses on transportation network protection to hedge against extreme events such as earthquakes. Traditional two-stage stochastic programming has been widely adopted to obtain solutions under a risk-neutral preference through the use of expectations in the recourse function. In reality, decision makers hold different risk preferences. We develop a mean-risk two-stage stochastic programming model that allows for greater flexibility in handling risk preferences when allocating limited resources. In particular, the first stage minimizes the retrofitting cost by making strategic retrofit decisions whereas the second stage minimizes the travel cost. The conditional value-at-risk (CVaR) is included as the risk measure for the total system cost. The two-stage model is equivalent to a nonconvex mixed integer nonlinear program (MINLP). To solve this model using the Generalized Benders Decomposition (GBD) method, we derive a convex reformulation of the second-stage problem to overcome algorithmic challenges embedded in the non-convexity, nonlinearity, and non-separability of first- and second-stage variables. The model is used for developing retrofit strategies for networked highway bridges, which is one of the research areas that can significantly benefit from mean-risk models. We first justify the model using a hypothetical nine-node network. Then we evaluate our decomposition algorithm by applying the model to the Sioux Falls network, which is a large-scale benchmark network in the transportation research community. The effects of the chosen risk measure and critical parameters on optimal solutions are empirically explored.

Weak Continuity of Risk Functionals with Applications to Stochastic Programming

Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean risk models are essentially defined by a parametric recourse problem and a quantification of risk. From the perspective of qualitative robustness theory, we discuss sufficient conditions for continuity of the resulting objective functions with respect to perturbation of the underlying probability measure. Our approach covers a fairly comprehensive class of both stochastic programming related risk measures and relevant recourse models. Not only does this unify previous approaches but also extends known stability results for two-stage stochastic programs to models with mixed-integer quadratic recourse and mixed-integer convex recourse, respectively.

Chemotherapy appointment scheduling under uncertainty using mean-risk stochastic integer programming.

Oncology clinics are often burdened with scheduling large volumes of cancer patients for chemotherapy treatments under limited resources such as the number of nurses and chairs. These cancer patients require a series of appointments over several weeks or months and the timing of these appointments is critical to the treatment's effectiveness. Additionally, the appointment duration, the acuity levels of each appointment, and the availability of clinic nurses are uncertain. The timing constraints, stochastic parameters, rising treatment costs, and increased demand of outpatient oncology clinic services motivate the need for efficient appointment schedules and clinic operations. In this paper, we develop three mean-risk stochastic integer programming (SIP) models, referred to as SIP-CHEMO, for the problem of scheduling individual chemotherapy patient appointments and resources. These mean-risk models are presented and an algorithm is devised to improve computational speed. Computational results were conducted using a simulation model and results indicate that the risk-averse SIP-CHEMO model with the expected excess mean-risk measure can decrease patient waiting times and nurse overtime when compared to deterministic scheduling algorithms by 42 % and 27 %, respectively.

Portfolio optimization with disutility-based risk measure

In this paper we propose a quantile-based risk measure which is defined using the modified loss distribution according to the decision maker’s risk and loss aversion. The properties related to different classes of disutility functions are established. A portfolio selection model in the Mean-Risk framework is proposed and equivalent formulations of the model generating the same efficient frontier are given. The advantages of this approach are investigated using real world data from NYSE. The differences between the efficient frontier of the proposed model and the classical Mean-Variance and Mean-CVaR are quantified and interpreted. Extensive experiments show that the efficient portfolios obtained by using the proposed model exhibit lower risk levels and an increased satisfaction compared to the other two Mean-Risk models.

Multiobjective Mean-Risk Models for Optimization in Finance and Insurance

Abstract In this paper we propose some models for solving optimization problems which arise in finance and insurance. First the general framework for Mean-Risk models is introduced. Then several approaches for multiobjective programming, such as Mean-Value-at-Risk and Mean-Conditional Value-at-Risk are used for building the model Mean-Value-at-Risk-Conditional Value-at-Risk using both Value-at-Risk and Conditional Value-at-Risk simultaneously for risk assessment. A two stage portfolio optimization model is developed, using Value-at-Risk and also Conditional Value-at-Risk measures in two stages separately.

Sufficient conditions under which SSD- and MR-efficient sets are identical

Three approaches are commonly used for analyzing decisions under uncertainty: expected utility (EU), second-degree stochastic dominance (SSD), and mean-risk (MR) models, with the mean–standard deviation (MS) being the best-known MR model. Because MR models generally lead to different efficient sets and thus are a continuing source of controversy, the specific concern of this article is not to suggest another MR model. Instead, we show that the SSD- and MR-efficient sets are identical, as long as (a) the risk measure satisfies both positive homogeneity and consistency with respect to the Rothschild and Stiglitz (1970) definition(s) of increasing risk and (b) the choice set includes the riskless asset and satisfies a generalized location and scale property, which can be interpreted as a market model. Under these conditions, there is no controversy among MR models and they all have a decision-theoretic foundation. They also offer a convenient way to compare the estimation error related to the empirical implementation of different MR models.

Incorporating risk measures in closed-loop supply chain network design

This paper considers a location-allocation problem in a closed-loop supply chain (CLSC) with two extensions: first, demand and prices of new and return products are regarded as non-deterministic parameters and second, the objective function is developed from expected profit to three types of mean-risk ones. Indeed, design and planning an integrated CLSC in real-world volatile markets is an important and necessary issue. Further, risk-neutral approaches, which are considered expected values, are not efficient for such uncertain conditions. Hence, this paper, copes with the design and planning problem of a CLSC in a two-stage stochastic structure. Besides, risk criteria are considered through using three types of popular and well-behaved risk measures: mean absolute deviation, value at risk and conditional value at risk (CVaR). Consequently, three types of mean-risk models are developed as objective functions and decision-making procedures are undertaken based on the expected values and risk adversity criteria. Finally, performances of the developed mean-risk models are evaluated in various aspects. Results reveal that the inefficiencies of risk-neutral approaches can be overcome. In addition, in terms of quality of solutions, the acceptability of CVaR is proved when it is compared to other risk measures.

The Optimization and Evaluation of Investment Portfolio

Investment to assets is getting more and more importance today. Shares of stock options have become one of the region’s economic indicators illustrating current economic situation. The main subject in researches of investment portfolios forecasting became a discussion about what values are best to measure portfolio risk and expected returns. Also there is a wide range of researches on macroeconomic and social events influence on evolution of stock returns. The main problem of today’s professional investment is to achieve the master level of complex investment decision adoption, included macroeconomic and historical data return analysis. The main aim of this work is to overview the portfolio optimization and evaluation models and to apply them to form the different portfolios of securities on the Vilnius Stock Market and compare them in the factor of conservativeness. There are used mean-risk models, such as Markowitz classic model, Mean Absolute Deviation model and MiniMax model, to optimize the investment portfolio in this research. Also there is performed the approbation of the multifactor model to evaluate the formed portfolios and to see the influence of the macroeconomic indexes to stock returns. Studies of this research showed 8 indicators, which have strongest linear dependence on shares return: VILIBOR interbank interest rate, consumer prices index, producer price index, construction expenses price index, trade balance, foreign direct investments, gross domestic product and inflation. Multifactorial analysis results showed that approved model to shares return data is not statistically reliable. Also, after further analysis there were found that Markowitz and Mean Absolute Deviation models create more conservative and optimal portfolio compared to MiniMax model, which is more suitable for aggressive and speculative investor. DOI: http://dx.doi.org/10.5755/j01.ee.24.4.5416

Stochastic dominance, regret dominance and regret-theoretic dominance

It is well known that stochastic dominance alone is insufficient for ensuring preferences when individuals experience regret. In this paper, we study two additional notions of dominance: (a) regret-theoretic dominance, which characterizes preferences in regret theory and (b) regret dominance, which characterizes preferences in mean-risk models with regret-based risk measures. We (a) extend our understanding of preferences in regret theory to problems with multiple choices under an infinite number of scenarios, (b) highlight that some notions of regret in the normative literature, specifically relative regret, can lead to unreasonable preferences within a mean-risk framework and (c) illustrate how regret dominance can help reduce the size of conventional efficient sets. Conditions where stochastic dominance, regret dominance and regret-theoretic dominance are equivalent are also presented.

The Bayesian Roots of Risk Parity in a Mean-Risk World

Risk parity has been considered a heuristic asset allocation method. In this paper, we show that, to the contrary, risk parity is a special case of a mean-risk type of a portfolio optimization problem with log-regularization to constrain weights. We show that log-regularization leads to a fund separation type property where the optimal solution lies between an unconstrained mean-risk portfolio and a risk parity (or risk budgeted in general) portfolio. We also demonstrate in a Bayesian framework that these log-regularizations are actually equivalent to putting prior distributions on the optimal portfolio weights. While leading to non-convex solutions, we further provide applications of these log-regularized mean-risk models in long/short asset allocation so as to generalize risk parity beyond long-only investments. We provide numerical examples on a typical institutional asset universe, and finally, we prove that there are efficient risk parity portfolios not necessarily satisfying equal Sharpe and constant correlation conditions, which have been widely mistaken as the only efficiency conditions for risk parity portfolios in the industry.

Portfolio selection using mean-risk model and mean-risk diversification model

With mean-risk and mean-risk diversification models, return distributions are characterised and compared using two statistics: the expected value and the value of a risk measure. This paper uses mean-risk model and risk curve obtained from the same model for portfolio selection problem. Security returns are assumed to be normally distributed. Further, the same mean-risk model is modified by using entropy to diversify the risk; this model can be called as mean-risk diversification model. In both the models, normal distribution is used to calculate the probability of likely losses of portfolio. The idea of mean-risk model is to regard expected return of a portfolio as the investment return and risk curve thus formed as investment risk, and the idea of mean-risk diversification model is to ensure that the portfolio thus formed is well diversified. The objective is to maximise the investor’s return at a preset confidence level and minimise the risk. Two numerical examples are presented for the sake of illustr...

Postoptimality for mean-risk stochastic mixed-integer programs and its application

The mean-risk stochastic mixed-integer programs can better model complex decision problems under uncertainty than usual stochastic (integer) programming models. In order to derive theoretical results in a numerically tractable way, the contamination technique is adopted in this paper for the postoptimality analysis of the mean-risk models with respect to changes in the scenario set, here the risk is measured by the lower partial moment. We first study the continuity of the objective function and the differentiability, with respect to the parameter contained in the contaminated distribution, of the optimal value function of the mean-risk model when the recourse cost vector, the technology matrix and the right-hand side vector in the second stage problem are all random. The postoptimality conclusions of the model are then established. The obtained results are applied to two-stage stochastic mixed-integer programs with risk objectives where the objective function is nonlinear with respect to the probability distribution. The current postoptimality results for stochastic programs are improved.

Optimizing the Societal Benefits of the Annual Influenza Vaccine: A Stochastic Programming Approach

Seasonal influenza is a major public health concern, and the first line of defense is the flu shot. Antigenic drifts and the high rate of influenza transmission require annual updates to the flu shot composition. The World Health Organization recommends which flu strains to include in the annual vaccine, based on surveillance and epidemiological analysis. There are two critical decisions regarding the flu shot design. One is its composition; currently, three strains constitute the flu shot, and they influence vaccine effectiveness. Another critical decision is the timing of the composition decisions, which affects the flu shot production. Both of these decisions have to be made under uncertainty many months before the flu season starts. We quantify the trade-offs involved through a multistage stochastic mixed-integer program that determines the optimal flu shot composition and its timing in a stochastic and dynamic environment.We incorporate risk sensitivity through mean-risk models. Our results provide valuable insights for pressing policy issues.

Constrained mean-risk portfolio optimisation: an application of multiobjective simulated annealing

We solve different constrained mean-risk portfolio optimisation models using a recently developed simulated annealing-based multiobjective optimisation algorithm. We consider practical and widely used constraints in portfolio modelling, i.e., the cardinality constraint which imposes a limit on the number of assets in the portfolio and the quantity constraints which restrict the proportion of each asset in the portfolio to lie between lower and upper bounds. Various risk measures are employed – the classical variance and three risk measures that form the so called downside-risk family, i.e., expected shortfall, value-at-risk and semivariance. The experimental results demonstrate that the algorithm generates a number of efficient portfolios capturing a great range of the trade-offs between mean and risk independently of the risk function used. Comparison with benchmark (when possible) efficient frontiers provides strong evidence that the algorithm effectively solves the discrete constrained mean-risk problems. Furthermore, a computational comparison with a population-based evolutionary algorithm, namely SPEA2, shows that the simulated annealing-based multiobjective optimisation technique achieves almost as good performance in a timely manner.

Mean-risk models using two risk measures: a multi-objective approach

This paper proposes a model for portfolio optimization, in which distributions are characterized and compared on the basis of three statistics: the expected value, the variance and the CVaR at a specified confidence level. The problem is multi-objective and transformed into a single objective problem in which variance is minimized while constraints are imposed on the expected value and CVaR. In the case of discrete random variables, the problem is a quadratic program. The mean-variance (mean-CVaR) efficient solutions that are not dominated with respect to CVaR (variance) are particular efficient solutions of the proposed model. In addition, the model has efficient solutions that are discarded by both mean-variance and mean-CVaR models, although they may improve the return distribution. The model is tested on real data drawn from the FTSE 100 index. An analysis of the return distribution of the chosen portfolios is presented.

Expected Utility and Mean-Risk Asset Pricing Models

Many people find the mean-variance assumption of the CAPM hard to justify. This paper replaces the mean-variance assumption with a weaker assumption based on the class of consistent risk measures. This class of risk measures is chosen because there is a weak form of equivalence between the mean consistent risk and the expected utility approaches to portfolio optimization. The general form for mean-consistent risk asset pricing models is derived, and the generalized mean-risk beta provided. The CAPM and a number of other well-known mean-risk asset pricing models arise as special cases.