Robust portfolios that do not tilt factor exposure

Robust portfolios reduce the uncertainty in portfolio performance. In particular, the worst-case optimization approach is based on the Markowitz model and form portfolios that are more robust compared to mean–variance portfolios. However, since the robust formulation finds a different portfolio from the optimal mean–variance portfolio, the two portfolios may have dissimilar levels of factor exposure. In most cases, investors need a portfolio that is not only robust but also has a desired level of dependency on factor movement for managing the total portfolio risk. Therefore, we introduce new robust formulations that allow investors to control the factor exposure of portfolios. Empirical analysis shows that the robust portfolios from the proposed formulations are more robust than the classical mean–variance approach with comparable levels of exposure on fundamental factors.     

On the equivalence of quadratic optimization problems commonly used in portfolio theory

In the paper, we consider three quadratic optimization problems which are frequently applied in portfolio theory, i.e., the Markowitz mean–variance problem as well as the problems based on the mean–variance utility function and the quadratic utility. Conditions are derived under which the solutions of these three optimization procedures coincide and are lying on the efficient frontier, the set of mean–variance optimal portfolios. It is shown that the solutions of the Markowitz optimization problem and the quadratic utility problem are not always mean–variance efficient.     

Non-linear equity portfolio variance reduction under a mean–variance framework—A delta–gamma approach

To examine the variance reduction from portfolios with both primary and derivative assets we develop a mean–variance Markovitz portfolio management problem. By invoking the delta–gamma approximation we reduce the problem to a well-posed quadratic programming problem. From a practitioner’s perspective, the primary goal is to understand the benefits of adding derivative securities to portfolios of primary assets. Our numerical experiments quantify this variance reduction from sample equity portfolios to mixed portfolios (containing both equities and equity derivatives).     

The opportunity cost of mean–variance choice under estimation risk

Mean–variance portfolio choice is often criticized as sub-optimal in the more general expected utility framework. It is argued that the expected utility framework takes into consideration higher moments ignored by mean variance analysis. A body of research suggests that mean–variance choice, though arguably sub-optimal, provides very close-to-expected utility maximizing portfolios and their expected utilities, basing its evaluation on in-sample analysis where mean–variance choice is sub-optimal by definition. In order to clarify this existing research, this study provides a framework that allows comparing in-sample and out-of-sample performance of the mean variance portfolios against expected utility maximizing portfolios. Our in-sample results confirm the results of earlier studies. On the other hand, our out-of-sample results show that the expected utility model performs worse. The out-of-sample inferiority of the expected utility model is more pronounced for preferences and constraints under which in-sample mean variance approximations are weakest. We argue that, in addition to its elegance and simplicity, the mean–variance model extracts more information from sample data because it uses the covariance matrix of returns. The expected utility model may reach its optimal solution without using information from the covariance matrix.     

Portfolio value-at-risk optimization for asymmetrically distributed asset returns

We propose a new approach to portfolio optimization by separating asset return distributions into positive and negative half-spaces. The approach minimizes a newly-defined Partitioned Value-at-Risk (PVaR) risk measure by using half-space statistical information. Using simulated data, the PVaR approach always generates better risk-return tradeoffs in the optimal portfolios when compared to traditional Markowitz mean–variance approach. When using real financial data, our approach also outperforms the Markowitz approach in the risk-return tradeoff. Given that the PVaR measure is also a robust risk measure, our new approach can be very useful for optimal portfolio allocations when asset return distributions are asymmetrical.     

Mean–variance optimal portfolios in the presence of a benchmark with applications to fraud detection

We first study mean–variance efficient portfolios when there are no trading constraints and show that optimal strategies perform poorly in bear markets. We then assume that investors use a stochastic benchmark (linked to the market) as a reference portfolio. We derive mean–variance efficient portfolios when investors aim to achieve a given correlation (or a given dependence structure) with this benchmark. We also provide upper bounds on Sharpe ratios and show how these bounds can be useful for fraud detection. For example, it is shown that under some conditions it is not possible for investment funds to display a negative correlation with the financial market and to have a positive Sharpe ratio. All the results are illustrated in a Black–Scholes market.     

Geometric representation of the mean–variance–skewness portfolio frontier based upon the shortage function

The literature suggests that investors prefer portfolios based on mean, variance and skewness rather than portfolios based on mean–variance (MV) criteria solely. Furthermore, a small variety of methods have been proposed to determine mean–variance–skewness (MVS) optimal portfolios. Recently, the shortage function has been introduced as a measure of efficiency, allowing to characterize MVS optimal portfolios using non-parametric mathematical programming tools. While tracing the MV portfolio frontier has become trivial, the geometric representation of the MVS frontier is an open challenge. A hitherto unnoticed advantage of the shortage function is that it allows to geometrically represent the MVS portfolio frontier. The purpose of this contribution is to systematically develop geometric representations of the MVS portfolio frontier using the shortage function and related approaches.     

Worst-case robust decisions for multi-period mean–variance portfolio optimization

In this paper, we extend the multi-period mean–variance optimization framework to worst-case design with multiple rival return and risk scenarios. Our approach involves a min–max algorithm and a multi-period mean–variance optimization framework for the stochastic aspects of the scenario tree. Multi-period portfolio optimization entails the construction of a scenario tree representing a discretised estimate of uncertainties and associated probabilities in future stages. The expected value of the portfolio return is maximized simultaneously with the minimization of its variance. There are two sources of further uncertainty that might require a strengthening of the robustness of the decision. The first is that some rival uncertainty scenarios may be too critical to consider in terms of probabilities. The second is that the return variance estimate is usually inaccurate and there are different rival estimates, or scenarios. In either case, the best decision has the additional property that, in terms of risk and return, performance is guaranteed in view of all the rival scenarios. The ex-ante performance of min–max models is tested using historical data and backtesting results are presented.     

Portfolio optimization with an envelope-based multi-objective evolutionary algorithm

The problem of portfolio selection is a standard problem in financial engineering and has received a lot of attention in recent decades. Classical mean–variance portfolio selection aims at simultaneously maximizing the expected return of the portfolio and minimizing portfolio variance. In the case of linear constraints, the problem can be solved efficiently by parametric quadratic programming (i.e., variants of Markowitz’ critical line algorithm). However, there are many real-world constraints that lead to a non-convex search space, e.g., cardinality constraints which limit the number of different assets in a portfolio, or minimum buy-in thresholds. As a consequence, the efficient approaches for the convex problem can no longer be applied, and new solutions are needed.In this paper, we propose to integrate an active set algorithm optimized for portfolio selection into a multi-objective evolutionary algorithm (MOEA). The idea is to let the MOEA come up with some convex subsets of the set of all feasible portfolios, solve a critical line algorithm for each subset, and then merge the partial solutions to form the solution of the original non-convex problem. We show that the resulting envelope-based MOEA significantly outperforms existing MOEAs.     

Mean–variance–skewness efficient surfaces,Stein’s lemma and the multivariate extended skew-Student distribution

Recent advances in Stein’s lemma imply that under elliptically symmetric distributions all rational investors will select a portfolio which lies on Markowitz’ mean–variance efficient frontier. This paper describes extensions to Stein’s lemma for the case when a random vector has the multivariate extended skew-Student distribution. Under this distribution, rational investors will select a portfolio which lies on a single mean–variance–skewness efficient hyper-surface. The same hyper-surface arises under a broad class of models in which returns are defined by the convolution of a multivariate elliptically symmetric distribution and a multivariate distribution of non-negative random variables. Efficient portfolios on the efficient surface may be computed using quadratic programming.     

Fuzzy turnover rate chance constraints portfolio model

One concern of many investors is to own the assets which can be liquidated easily. Thus, in this paper, we incorporate portfolio liquidity in our proposed model. Liquidity is measured by an index called turnover rate. Since the return of an asset is uncertain, we present it as a trapezoidal fuzzy number and its turnover rate is measured by fuzzy credibility theory. The desired portfolio turnover rate is controlled through a fuzzy chance constraint. Furthermore, to manage the portfolios with asymmetric investment return, other than mean and variance, we also utilize the third central moment, the skewness of portfolio return. In fact, we propose a fuzzy portfolio mean–variance–skewness model with cardinality constraint which combines assets limitations with liquidity requirement. To solve the model, we also develop a hybrid algorithm which is the combination of cardinality constraint, genetic algorithm, and fuzzy simulation, called FCTPM.     

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.     

An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution

This paper proposes a unified framework to solve distributionally robust mean-risk optimization problem that simultaneously uses variance, value-at-risk (VaR) and conditional value-at-risk (CVaR) as a triple-risk measure. It provides investors with more flexibility to find portfolios in the sense that it allows investors to optimize a return-risk profile in the presence of estimation error. We derive a closed-form expression for the optimal portfolio strategy to the robust mean-multiple risk portfolio selection model under distribution and mean return ambiguity (RMP). Specially, the robust mean-variance, robust maximum return, robust minimum VaR and robust minimum CVaR efficient portfolios are all special instances of RMP portfolios. We analytically and numerically show that the resulting portfolio weight converges to the minimum variance portfolio when the level of ambiguity aversion is in a high value. Using numerical experiment with simulated data, we demonstrate that our robust portfolios under ambiguity are more stable over time than the non-robust portfolios.     

Risk-adjusted probability measures in portfolio optimization with coherent measures of risk

We consider the problem of optimizing a portfolio of n assets, whose returns are described by a joint discrete distribution. We formulate the mean–risk model, using as risk functionals the semideviation, deviation from quantile, and spectral risk measures. Using the modern theory of measures of risk, we derive an equivalent representation of the portfolio problem as a zero-sum matrix game, and we provide ways to solve it by convex optimization techniques. In this way, we reconstruct new probability measures which constitute part of the saddle point of the game. These risk-adjusted measures always exist, irrespective of the completeness of the market. We provide an illustrative example, in which we derive these measures in a universe of 200 assets and we use them to evaluate the market portfolio and optimal risk-averse portfolios.     

Mean–variance approximations to expected utility

It is often asserted that the application of mean–variance analysis assumes normal (Gaussian) return distributions or quadratic utility functions. This common mistake confuses sufficient versus necessary conditions for the applicability of modern portfolio theory. If one believes (as does the author) that choice should be guided by the expected utility maxim, then the necessary and sufficient condition for the practical use of mean–variance analysis is that a careful choice from a mean–variance efficient frontier will approximately maximize expected utility for a wide variety of concave (risk-averse) utility functions. This paper reviews a half-century of research on mean–variance approximations to expected utility. The many studies in this field have been generally supportive of mean–variance analysis, subject to certain (initially unanticipated) caveats.     

t-分布与不同风险度量下有效投资组合及其边缘的相互关系研究

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivari-ate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

Active allocation of systematic risk and control of risk sensitivity in portfolio optimization

Portfolio risk can be decomposed into two parts, the systematic risk and the nonsystematic risk. It is well known that the nonsystematic risk can be eliminated by diversification, while the systematic risk cannot. Thus, the portfolio risk, except for that of undiversified small portfolios, is always dominated by the systematic risk. In this paper, under the mean–variance framework, we propose a model for actively allocating the systematic risk in portfolio optimization, which can also be interpreted as a model of controlling risk sensitivity in portfolio selection. Although the resulting problem is, in general, a notorious non-convex quadratically constrained quadratic program, the problem formulation is of some special structures due to the features of the defined marginal systematic risk contribution and the way to model the systematic risk via a factor model. By exploiting such special problem characteristics, we design an efficient and globally convergent branch-and-bound solution algorithm, based on a second-order cone relaxation. While empirical study demonstrates that the proposed model is a preferred tool for active portfolio risk management, numerical experiments also show that the proposed solution method is more efficient when compared to the commercial software BARON.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASUREgS

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

Equilibrium in an ambiguity-averse mean–variance investors market

In a financial market composed of n risky assets and a riskless asset, where short sales are allowed and mean–variance investors can be ambiguity averse, i.e., diffident about mean return estimates where confidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a worst case max–min criterion. Then, in a market where all investors are ambiguity-averse mean–variance investors with access to given mean return and variance–covariance estimates, we investigate conditions regarding the existence of an equilibrium price system and give an explicit formula for the equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) mean–variance investors’ market has a deflationary effect on equilibrium prices with respect to a pure mean–variance investors’ market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambiguity-averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral.