Markus Hirschberger

Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection

In standard portfolio theory, an investor is typically taken as having one stochastic objective, to maximize the random variable of portfolio return. But in this paper, we focus on investors whose purpose is to build, more broadly, a “suitable portfolio” taking additional concerns into account. Such investors would have additional stochastic and deterministic objectives that might include liquidity, dividends, number of securities in a portfolio, social responsibility, and so forth. To accommodate such investors, we develop a multiple criteria portfolio selection formulation, corroborate its appropriateness by examining the sensitivity of the nondominated frontier to various factors, and observe the conversion of the nondominated frontier to a nondominated surface. Furthermore, multiple criteria enable us to provide an explanation as to why the “market portfolio,” so often found deep below the nondominated frontier, is roughly where one would expect it to be with multiple criteria. After commenting on solvability issues, the paper concludes with the idea that what is the “modern portfolio theory” of today might well be interpreted as a projection onto two-space of a real multiple criteria portfolio selection problem from higher dimensional space.

Randomly generating portfolio-selection covariance matrices with specified distributional characteristics

Abstract In portfolio selection, there is often the need for procedures to generate “realistic” covariance matrices for security returns, for example to test and benchmark optimization algorithms. For application in portfolio optimization, such a procedure should allow the entries in the matrices to have distributional characteristics which we would consider “realistic” for security returns. Deriving motivation from the fact that a covariance matrix can be viewed as stemming from a matrix of factor loadings, a procedure is developed for the random generation of covariance matrices (a) whose off-diagonal (covariance) entries possess a pre-specified expected value and standard deviation and (b) whose main diagonal (variance) entries possess a likely different pre-specified expected value and standard deviation. The paper concludes with a discussion about the futility one would likely encounter if one simply tried to invent a valid covariance matrix in the absence of a procedure such as in this paper.

Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds

We present a framework for inverse optimization in a Markowitz portfolio model that is extended to include a third criterion. The third criterion causes the traditional nondominated frontier to become a surface. Until recently, it had not been possible to compute such a surface. But by using a new method that is able to generate the nondominated surfaces of tri-criterion portfolio selection problems, we are able to compute via inverse optimization the implied risk tolerances of given funds that pursue an additional objective beyond risk and return. In applying this capability to a broad sample of conventional and socially responsible (SR) mutual funds, we find that there appears to be no significant evidence that social responsibility issues, after the screening stage, are further taken into account in the asset allocation process, which is a result that is likely to be different from what many SR investors would expect.

Computing the Nondominated Surface in Tri-Criterion Portfolio Selection

Computing the nondominated set of a multiple objective mathematical program has long been a topic in multiple criteria decision making. In this paper, motivated by the desire to extend Markowitz portfolio selection to an additional linear criterion dividends, liquidity, sustainability, etc., we demonstrate an exact method for computing the nondominated set of a tri-criterion program that is all linear except for the fact that one of its objectives is to minimize a convex quadratic function. With the nondominated set of the resulting quad-lin-lin program being a surface composed of curved platelets, a multiparametric algorithm is devised for computing the platelets so that they can be graphed precisely. In this way, graphs of the tri-criterion nondominated surface can be displayed so that, as in traditional portfolio selection, a most preferred portfolio can be selected while in full view of all other contenders for optimality. Finally, by giving an example for socially responsible investors, we demonstrate that our algorithm can outperform standard portfolio strategies for multicriterial decision makers.

Large-scale MV efficient frontier computation via a procedure of parametric quadratic programming

Despite the volume of research conducted on efficient frontiers, in many cases it is still not the easiest thing to compute a mean-variance (MV) efficient frontier even when all constraints are linear. This is particularly true of large-scale problems having dense covariance matrices and hence they are the focus in this paper. Because standard approaches for constructing an efficient frontier one point at a time tend to bog down on dense covariance matrix problems with many more than about 500 securities, we propose as an alternative a procedure of parametric quadratic programming for more effective usage on large-scale applications. With the proposed procedure we demonstrate through computational results on problems in the 1000-3000 security range that the efficient frontiers of dense covariance matrix problems in this range are now not only solvable, but can actually be computed in quite reasonable time.

Comparative issues in large-scale mean-variance efficient frontier computation

One of the functions of a portfolio management system is to return quickly an efficient frontier. However, in the large-scale problems (1000 to 3000 securities) that are beginning to appear with greater frequency, the task of computing the mean-variance efficient frontier, even when all constraints are linear, can range from the significant to the prohibitive. For ease of reference, we call mean-variance problems with all linear constraints Markowitz problems. With little on the time to compute a Markowitz-problem efficient frontier in the literature, we conduct experiments that involve varying problem sizes, methods employed, and optimizers used to present an overall picture of the situation and establish benchmarks in the large-scale arena. One of the conclusions of the experiments is the superiority of the class of techniques that would fall under the title of parametric quadratic programming.

Dotted Representations of Mean-Variance Efficient Frontiers and their Computation

Abstract This paper is about dotted representations of efficient frontiers. Dotted representations, as in portfolio selection, can often be the most practical way of communicating an efficient frontier. The most popular method is to minimize variance subject to fixed levels of expected return. However, even when the fixed levels are evenly dispersed, one can not count on the resulting dots being evenly dispersed. Another method uses fixed values of a risk tolerance parameter, but with this method the resulting dots are even less controllable. In this paper we develop a third approach applicable to what we call Markowitz problems (mean-variance problems with all linear constraints). The approach utilizes the results of algorithms that can compute all hyperbolic segments of a Markowitz efficient frontier. Then the approach can place dots on the hyperbolic segments of the efficient frontier in a variety ways including equally spaced. The advantage of the approach is the speed at which dotted representations can be produced and modified, particularly on large applications.

Extracting from the relaxed for large-scale semi-continuous variable nondominated frontiers

Because of size and covariance matrix problems, computing much of anything along the nondominated frontier of a large-scale (1000–3000 securities) portfolio selection problem with semi-continuous variables is a task that has not previously been achieved. But given (a) the speed at which the nondominated frontier of a classical portfolio problem can now be computed and (b) the possibility that there might be overlaps between the nondominated frontier of the classical problem and that of the same problem but with semi-continuous variables, the paper shows how considerable amounts of the nondominated frontier of a large-scale mean-variance portfolio selection problem with semi-continuous variables can be computed in very little time.

Computation of efficient compromise arcs in Convex Quadratic Multicriteria Optimization

Abstract.Given a Convex Quadratic Multicriteria Optimization Problem, we show the stability of the Domination Problem. By modifying Benson’s single parametric method, which is based on the Domination Problem, we are able to show the existence of an efficient compromise arc connecting any two efficient points. Moreover, we deduce an algorithm which realizes the modification in polynomial time.

Is socially responsible investing just screening

This paper presents the results of an empirical study concerning conventional and socially responsible mutual funds.We apply a sophisticated operations research algorithm embedded in inverse portfolio optimization on financial market data, ESG-scores and CRSP fund data. Due to our results we cannot find strong evidence of differences between conventional and socially responsible mutual funds. In particular, the calculated risk tolerance parameters describing the real portfolio composition best show that socially responsible mutual funds may be even less concerned about the ESG-scores in the preference functional than conventional funds.