Portfolio selection under uncertainty: a new methodology for computing relative‐robust solutions

• Date: 01 May 2021
• Author: Joana Matos Dias,Sandra Caçador,Pedro Godinho
• DOI: http://doi.org/10.1111/itor.12674
• Published date: 01 May 2021

Intl. Trans. in Op. Res. 28 (2021) 1296–1329

DOI: 10.1111/itor.12674

INTERNATIONAL

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IN OPERATIONAL

RESEARCH

Portfolio selection under uncertainty: a new methodology for

computing relative-robust solutions

Sandra Cac¸adora,b, c,∗, Joana Matos Diasb,c, d and Pedro Godinhob,c

aHigher Institute for Accountancy and Administration of the Universityof Aveiro, R. Associac¸˜

ao Humanit´

aria dos

Bombeiros Volunt´

arios de Aveiro,Aveiro 3810-500, Portugal

bCentre for Business and Economics Research(CeBER), Faculty of Economics, University of Coimbra, Av. Dias da Silva,

165, Coimbra 3004–512, Portugal

cFaculty of Economics, Universityof Coimbra, Av. Dias da Silva, 165, Coimbra 3004–512, Portugal

dInstitute for Systems Engineering and Computers at Coimbra, Rua Antero de Quental, N°199, Coimbra3000–033,

Portugal

E-mail: sandracacador@ua.pt [Cac¸ ador]; joana@fe.uc.pt [Dias]; pgodinho@fe.uc.pt [Godinho]

Received 19 June2018; received in revised form 24 January 2019; accepted 16 April 2019

Abstract

In this paper, a new methodology for computing relative-robust portfolios based on minimax regret is

proposed. Regret is defined as the utility loss for the investor resulting from choosing a given portfolio

instead of choosing the optimal portfolio of the realized scenario. The absolute-robust strategy was also

considered and, in this case, the minimum investor’s expected utility in the worst-case scenario is maximized.

Several subsamples are gathered from the in-sample data and for each subsample a minimax regret and a

maximin solution are computed, to avoid the risk of overfitting. Robust portfolios are computed using a

genetic algorithm, allowing the transformation of a three-level optimization problem in a two-level problem.

Results show that the proposed relative-robust portfolio generally outperforms (other) relative-robust and

non-robust portfolios, except for the global minimum variance portfolio. Furthermore, the relative-robust

portfolio generally outperforms the absolute-robust portfolio, even considering higher risk aversion levels.

Keywords:robust optimization; portfolio selection; relative robustness;minimax regret

1. Introduction

Deciding how to optimally allocate the investor’s wealth among all possible investment alternatives

requires the consideration of the tradeoff between expected return and risk, and this is a difficult

problem. Actually, the investor has to decide what to do today, without knowing what the future

will bring regarding the different investment alternatives. In this context, the concept of optimal

solution is not well defined. We can think of representing the inherent uncertainty by using a set

∗Corresponding author.

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S. Cac¸ador et al. / Intl. Trans. in Op. Res. 28 (2021) 1296–1329 1297

of parameters that are, most of the times, calculated by observing past data. However, as it is not

possible to foresee all potential realizations of the uncertain values, it is also not possible to assume

that these parameters are a truthful representation of future uncertainty. A given portfolio can be

optimal considering a given set of problem parameters and still present a poor performance as

the uncertainty is resolved. If the uncertainty that arises from the impossibility to know the future

values of the parameters at the decision-making moment is ignored, then the calculated optimal

portfolio will be extremely sensitive to small variations in the problem parameters and may thus

show a poor performance when applied to new data (Best and Grauer, 1991a, 1991b; Chopra and

Ziemba, 1993; Jagannathan and Ma, 2003).

In this regard, some methodologies that explicitly incorporate uncertainty into the optimization

model can be applied in order to mitigate the impact of the estimation errors in the classical mean-

variance optimization problems. An example is the robust optimization methodology, whose roots

can be found in the field of robust control theory.

Robustoptimization has emerged as a computationally attractive alternative to other methodolo-

gies, like stochastic programming or dynamic programming, since it requires relatively general and

simple assumptions about the probability distributions of the uncertain parameters (Fabozzi et al.,

2007). The robust formulation of an optimization problem considers not only the nominal values

of the uncertain parameters but also the deviations from these nominal values. Uncertainty in the

parameters can be described by uncertainty sets that contain all possible values or only the most

likely values of the uncertain parameters, with their sizes defining the level of uncertainty admitted

or, equivalently, the desired level of robustness.

The first to apply this idea was Soyster (1973), who presented a linear optimization model to

compute a solution that was feasible for all possible values of an uncertain parameter belonging

to a convex set (Ben-Tal et al., 2009). While Soyster’s approach achieved the desired outcome

of immunizing the optimal solution against parameter uncertainty, it was widely considered too

conservative for practical implementation (Bertsimas and Thiele, 2006). More than 20 years later,

other authors addressed the overconservatism within Soyster’s approach and robust optimization

began to establish itself as a methodology with applications in many fields of knowledge, including

portfolio theory.

Two main approaches are considered in the robust portfolio optimization literature. The most

prevalent is the absolute-robust optimization approach where the optimal solution is computed

assuming the worst possible realization within the uncertainty set for the uncertain parameters.

Since assuming that the worst scenario will happen might result in too conservative decisions, one

could analyze robustness in a relative manner. In the relative-robust optimization approach the

objective is to guarantee that the maximum difference between the optimal objective function value

for eachs cenario (considering theoptimal solutionfor that scenario) and the objective function value

obtained for the same scenario by the robust solution (i.e., not scenario dependent) is minimized.

In this paper, a new methodology for computing relative-robust portfolios is presented, based on

the optimization of the investor’s expected utility. Since empirical applicationsof the relative-robust

models are lacking within the portfolio optimization theory, an analysis of the real benefits of the

relative-robust methodology is performed. For that purpose, an empirical application is conducted

in order to assess and compare the performance of the proposed relative-robust portfolio to (other)

relative-robust and non-robust portfolios already described in the portfolio theory literature. A

comparison of the performance of relative-robust and absolute-robust portfolios is also presented

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and conclusions are drawn regarding the real benefits of the proposed methodology from the

investors’ perspective.

This paper contributes for this field of research in several ways. First, a new methodology for

computing relative-robust portfolios based on minimax regret is presented. Second, a three-level

optimization problemis transformed into a two-level problem by using an evolutionaryalgorithm to

solve the relative-robust portfolio optimizationproblem and find the relative-robustsolution. Third,

the real benefits of applying the relative-robustness approach in portfolio selection are analyzed by

comparing in-sample and out-of-sample performances of relative-robust and non-robust portfolios,

which highlight the main strengths of the proposed methodology. Fourth, the effect of considering

long-term past data over short-term past data in the definition of the uncertainty set is investigated

by using in-sample sets of different lengths in the empirical analysis. Finally, the relevance of the

proposed robust models is analyzed for different levels of risk aversion and the performances of

relative-robust and absolute-robust approaches are compared.

Forthe empirical analysis, historical daily data from January 1992 to December 2016 (25 years) of

the stocks of the DAX indexwere collected from Thompson Reuters Datastream. Differentmethod-

ological approaches are implemented and the corresponding optimal portfolios are computed, in

particular the proposed relative-robust (RR) portfolio, the absolute-robust(AR) portfolio, the clas-

sical Mean-Variance (MV) portfolio, the Global Minimum Variance (GMV) portfolio, the Equally

Weighted (EW) portfolio, and the relative-robust (WS) portfoliopresented by Xidonas et al. (2017).

The latter is, to the best of the authors’ knowledge, the only known contribution with empirical

applications within the relative-robust portfolio optimization approach. The performances of the

different computed portfolios are compared considering both in-sample and out-of-sample data,

for return, risk, modified Sharpe ratio, and regret.

The results of this study suggest that reducing the in-sample period length increases the exposure

of the computed portfolios to individual stocks while it seems to improve the overall out-of-sample

performance of the AR, the RR, and the GMV portfolios and substantially deteriorates the out-

of-sample performance of the MV portfolio. Regardless of the in-sample period length, it can be

observed that the RR portfolio is highly diversified, assigning nonzero weights to the majority of

the assets.

The overall results support previous findings concerning the sensitivity of the MV portfolio to

the estimation error and the effects of the input uncertainty in the optimization process (Best and

Grauer, 1991b; Chopra and Ziemba, 1993; Jagannathan and Ma, 2003; DeMiguel et al., 2009), as

well as the outperformance of the GMV portfolio (Chan et al., 1999; Jagannathan and Ma, 2003).

Furthermore, the results of this study suggest that the proposed relative-robust model generates

optimal portfolios that consistently presentlow risk, and (non-negative) attractive returns,standing

out as one of the very few optimal portfolios with no poor performances. In fact, the RR portfolio

outperforms the MV portfolio and the EW portfolio in many of the windows under analysis, even

outperforming the GMV portfolio in some windows, which enhance the relevance of the proposed

methodology among non-robust portfolio optimization methodologies. When compared to the

minimax regret model presented by Xidonas et al. (2017), the proposed relative-robust approach

seems to providemore consistent results concerning the out-of-sample performance of the generated

portfolios.

Finally, the comparison of the proposed relative-robust and absolute-robust models leads to

important conclusions concerning the real benefits of the proposed methodology fromthe investors’

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2019 The Authors.

International Transactionsin Operational Research C

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