Robust ranking and portfolio optimization

The portfolio optimization problem has attracted researchers from many disciplines to resolve the issue of poor out-of-sample performance due to estimation errors in the expected returns. A practical method for portfolio construction is to use assets’ ordering information, expressed in the form of preferences over the stocks, instead of the exact expected returns. Due to the fact that the ranking itself is often described with uncertainty, we introduce a generic robust ranking model and apply it to portfolio optimization. In this problem, there are n objects whose ranking is in a discrete uncertainty set. We want to find a weight vector that maximizes some generic objective function for the worst realization of the ranking. This robust ranking problem is a mixed integer minimax problem and is very difficult to solve in general. To solve this robust ranking problem, we apply the constraint generation method, where constraints are efficiently generated by solving a network flow problem. For empirical tests, we use post-earnings-announcement drifts to obtain ranking uncertainty sets for the stocks in the DJIA index. We demonstrate that our robust portfolios produce smaller risk compared to their non-robust counterparts.     

Incorporating the information from direct and indirect neighbors into fitness evaluation enhances the cooperation in the social dilemmas

We propose an improved fitness evaluation method to investigate the evolution of cooperation in the spatial social dilemmas. In our model, a focal player’s fitness is calculated as the linear combination of his own payoff, the average payoffs of direct and indirect neighbors in which two independent selection parameters (α and β) are used to control the proportion of various payoff contribution to the current fitness. Then, the fitness-based strategy update rule is still Fermi-like, and asynchronous update is adopted here. A large plethora of numerical simulations are performed to validate the behaviors of the current model, and the results unambiguously demonstrate that the cooperation level is greatly enhanced by introducing the payoffs from the surrounding players. In particular, the influence of direct neighbors become more evident when compared with indirect neighbors since the correlation between focal players and their direct neighbors is much closer. Current outcomes are significant for us to further illustrate the origin and emergence of cooperation within a wide variety of natural and man-made systems.     

On the multi-dimensional portfolio optimization with stochastic volatility

In a recent paper by Mnif [18], a solution to the portfolio optimization with stochastic volatility and constraints problem has been proposed, in which most of the model parameters are time-homogeneous. However, there are cases where time-dependent parameters are needed, such as in the calibration of financial models. Therefore, the purpose of this paper is to generalize the work of Mnif [18] to the time-inhomogeneous case. We consider a time-dependent exponential utility function of which the objective is to maximize the expected utility from the investor’s terminal wealth. The derived Hamilton-Jacobi-Bellman(HJB) equation, is highly nonlinear and is reduced to a semilinear partial differential equation (PDE) by a suitable transformation. The existence of a smooth solution is proved and a verification theorem presented. A multi-asset stochastic volatility model with jumps and endowed with time-dependent parameters is illustrated.     

CVaR鲁棒均值-CVaR投资组合模型与求解

传统的均值-风险(包括方差、VaR、CVaR等)组合选择模型在计算最优投资组合时,常假定均值是已知的常值,但在实际资产配置中,收益的均值估计会有偏差,即存在着估计风险.在利用CVaR测度估计风险的基础上,研究了CVaR鲁棒均值-CVaR投资组合选择模型,给出了另外两种不同的求解方法,即对偶法和光滑优化方法,并探讨了它们的相关性质及特征,数值实验表明在求解大样本或者大规模投资组合选择问题上,对偶法和光滑优化方法在计算上是可行且有效的.     

Distribution assumptions and risk constraints in portfolio optimization

Empirical distributions are often claimed to be superior to parametric distributions, yet to also increase the computational complexity and are therefore hard to apply in portfolio optimization. In this paper, we approach the portfolio optimization problem under constraints on the portfolios Value at Risk and Expected Tail Loss, respectively, under empirical distributions for the Standard and Poors 100 stocks. We apply a heuristic optimization method which has been found to overcome the restrictions of traditional optimization techniques. Our results indicate that empirical distributions might turn into a Pandoras Box: Though highly reliable for predicting the assets risks, employing these distributions in the optimization process might result in severe mis-estimations of the resulting portfolios actual risk. It is found that even a simple mean-variance approach can be superior despite its known specification errors.AMS Classification: G11, C61Dietmar G. Maringer: Im grateful to two anonymous referees, Peter Winker, Manfred Gilli, Berç Rustem, Erricos Kontoghiorghes, Alfred Lehar, Josef Zechner, Suresh Sundaresan, and conference participants at Aix-en-Provence, Limassol, and Sydney for valuable discussions and comments on earlier versions of this paper.     

Non-separation in the mean–lower-partial-moment portfolio optimization problem

With a number of advantages, lower partial moments (LPM) serve as alternatives to variance as measures of portfolio risk. For two specific targets, a separation property holds in the context of mean–LPM portfolio optimization that allows investors to separate the decision about investment proportions among risky assets from the decision about how much to invest in risky versus risk-free assets. For other targets, however, separation is not guaranteed, and this case has not received much attention in the literature. We show in the case of non-separation that investment curves are not common to all optimizing investors, but that they are convex in (mean, LPM) space and their lower envelope is the efficient frontier. We consider the interesting behavior of investment curves and optimal risky portfolios. We also show empirically that an investor who mistakenly assumes separation holds will not experience significant excess portfolio risk in all practical cases.     

Regularized robust optimization: the optimal portfolio execution case

An uncertainty set is a crucial component in robust optimization. Unfortunately, it is often unclear how to specify it precisely. Thus it is important to study sensitivity of the robust solution to variations in the uncertainty set, and to develop a method which improves stability of the robust solution. In this paper, to address these issues, we focus on uncertainty in the price impact parameters in an optimal portfolio execution problem. We first illustrate that a small variation in the uncertainty set may result in a large change in the robust solution. We then propose a regularized robust optimization formulation which yields a solution with a better stability property than the classical robust solution. In this approach, the uncertainty set is regularized through a regularization constraint, defined by a linear matrix inequality using the Hessian of the objective function and a regularization parameter. The regularized robust solution is then more stable with respect to variation in the uncertainty set specification, in addition to being more robust to estimation errors in the price impact parameters. The regularized robust optimal execution strategy can be computed by an efficient method based on convex optimization. Improvement in the stability of the robust solution is analyzed. We also study implications of the regularization on the optimal execution strategy and its corresponding execution cost. Through the regularization parameter, one can adjust the level of conservatism of the robust solution.     

Ruin problems and myopic portfolio optimization in continuous trading

In continuous trading, ruin problems are important for several reasons. ln the first part of the paper a test criterion for bankruptcy is developed. In the present framework one implicitly assumes the investor's wealth to be different from zero, otherwise the model is not well-defined. It is of practical interest to be able to investigate if a certain stationary Markovian financial strategy may lead to ruin. If ruin can occur, its probability is found to satisfy a partial differential equation.In the second part of the paper, a portfolio optimization problem is investigated and solved using Doléans-Dade's exponential formula. The optimality criterion used is to maximize the expected rate of growth. Because of the special structure of the problem, we avoid the Bellman equation. This fact is fortunate, since the Bellman equation is often very complicated to solve analytically.     

Monte Carlo methods for mean-risk optimization and portfolio selection

Stochastic programming is a well-known instrument to model many risk management problems in finance. In this paper we consider a stochastic programming model where the objective function is the variance of a random function and the constraint function is the expected value of the random function. Instead of using popular scenario tree methods, we apply the well-known sample average approximation (SAA) method to solve it. An advantage of SAA is that it can be implemented without knowing the distribution of the random data. We investigate the asymptotic properties of statistical estimators obtained from the SAA problem including examining the rate of convergence of optimal solutions of the SAA problem as sample size increases. By using the classical penalty function technique and recent results on uniform exponential convergence of sample average random functions, we show that under some mild conditions the statistical estimator of the optimal solution converges to its true counterpart at an exponential rate. We apply the proposed model and the numerical method to a portfolio management problem and present some numerical results.     

Intertemporal portfolio optimization with small transaction costs and stochastic variance

The solution to the intertemporal optimal portfolio selection and consumption rule with small transaction costs is derived via the use of perturbation analysis for the two assets portfolio, one risky and one riskfree. This methodology allows us to apply a broader specification for the function of utility. The additional feature of stochastic variance is also included.     

Convex risk measures for portfolio optimization and concepts of flexibility

Due to their axiomatic foundation and their favorable computational properties convex risk measures are becoming a powerful tool in financial risk management. In this paper we will review the fundamental structural concepts of convex risk measures within the framework of convex analysis. Then we will exploit it for deriving strong duality relations in a generic portfolio optimization context. In particular, the duality relationship can be used for designing new, efficient approximation algorithms based on Nesterov's smoothing techniques for non-smooth convex optimization. Furthermore, the presented concepts enable us to formalize the notion of flexibility as the (marginal) risk absorption capacity of a technology or (available) resources. This paper is dedicated to R.T. Rockafellar for his stimulating and impressive work in convex optimization for decades. We thank you for the insights and inspirations we gained from your fundamental research.     

Robust optimization and portfolio selection: The cost of robustness

Robust optimization is a tractable alternative to stochastic programming particularly suited for problems in which parameter values are unknown, variable and their distributions are uncertain. We evaluate the cost of robustness for the robust counterpart to the maximum return portfolio optimization problem. The uncertainty of asset returns is modelled by polyhedral uncertainty sets as opposed to the earlier proposed ellipsoidal sets. We derive the robust model from a min-regret perspective and examine the properties of robust models with respect to portfolio composition. We investigate the effect of different definitions of the bounds on the uncertainty sets and show that robust models yield well diversified portfolios, in terms of the number of assets and asset weights.     

On the effectiveness of scenario generation techniques in single-period portfolio optimization

In single-period portfolio selection problems the expected value of both the risk measure and the portfolio return have to be estimated. Historical data realizations, used as equally probable scenarios, are frequently used to this aim. Several other parametric and non-parametric methods can be applied. When dealing with scenario generation techniques practitioners are mainly concerned on how reliable and effective such methods are when embedded into portfolio selection models. In this paper we survey different techniques to generate scenarios for the rates of return. We also compare the techniques by providing in-sample and out-of-sample analysis of the portfolios obtained by using these techniques to generate the rates of return. Evidence on the computational burden required by the different techniques is also provided. As reference model we use the Worst Conditional Expectation model with transaction costs. Extensive computational results based on different historical data sets from London Stock Exchange Market (FTSE) are presented and some interesting financial conclusions are drawn.     

A local relaxation method for the cardinality constrained portfolio optimization problem

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a number of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints present in practice. The proposed local relaxation algorithm explores the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using a suitable heuristic method such as the centroids of initial clusters or be warm started based on the last output. Results, using a basket of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm can lead to significant performance gain over popular branch-and-cut methods. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency.     

Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.     

Asset portfolio optimization using support vector machines and real-coded genetic algorithm

This paper presents an integrated approach for portfolio selection in a multicriteria decision making framework. Firstly, we use Support Vector Machines for classifying financial assets in three pre-defined classes, based on their performance on some key financial criteria. Next, we employ Real-Coded Genetic Algorithm to solve a mathematical model of the multicriteria portfolio selection problem in the respective classes incorporating investor-preferences.     

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.     

Asymmetric risk measures and tracking models for portfolio optimization under uncertainty

Traditional asset allocation of the Markowitz type defines risk to be the variance of the return, contradicting the common-sense intuition that higher returns should be preferred to lower. An argument of Levy and Markowitz justifies the mean/variance selection criteria by deriving it from a local quadratic approximation to utility functions. We extend the Levy-Markowitz argument to account for asymmetric risk by basing the local approximation onpiecewise linear-quadratic risk measures, which can be tuned to express a wide range of preferences and adjusted to reject outliers in the data. The implications of this argument lead us to reject the commonly proposed asymmetric alternatives, the mean/lower partial moment efficient frontiers, in favor of the risk tolerance frontier. An alternative model that allows for asymmetry is the tracking model, where a portfolio is sought to reproduce a (possibly) asymmetric distribution at lowest cost.     

Measuring financial risk and portfolio optimization with a non-Gaussian multivariate model

In this paper, we propose a multivariate market model with returns assumed to follow a multivariate normal tempered stable distribution. This distribution, defined by a mixture of the multivariate normal distribution and the tempered stable subordinator, is consistent with two stylized facts that have been observed for asset distributions: fat-tails and an asymmetric dependence structure. Assuming infinitely divisible distributions, we derive closed-form solutions for two important measures used by portfolio managers in portfolio construction: the marginal VaR and the marginal AVaR. We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average, first statistically validating the model based on goodness-of-fit tests and then demonstrating how the marginal VaR and marginal AVaR can be used for portfolio optimization using the model. Based on the empirical evidence presented in this paper, our framework offers more realistic portfolio risk measures and a more tractable method for portfolio optimization.     

60 Years of portfolio optimization: Practical challenges and current trends

The concepts of portfolio optimization and diversification have been instrumental in the development and understanding of financial markets and financial decision making. In light of the 60 year anniversary of Harry Markowitz’s paper “Portfolio Selection,” we review some of the approaches developed to address the challenges encountered when using portfolio optimization in practice, including the inclusion of transaction costs, portfolio management constraints, and the sensitivity to the estimates of expected returns and covariances. In addition, we selectively highlight some of the new trends and developments in the area such as diversification methods, risk-parity portfolios, the mixing of different sources of alpha, and practical multi-period portfolio optimization.