A hybrid heuristic approach to discrete multi-objective optimization of credit portfolios

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Abstract

A hybrid heuristic approach combining multi-objective evolutionary and problem-specific local search methods is proposed to support the risk-return analysis of credit portfolios. Its goal is to compute approximations of discrete sets of Pareto-efficient portfolio structures concerning both the respective portfolio return and the respective portfolio risk using the non-linear, non-convex Credit-Value-at-Risk downside risk measure which is relevant to real world credit portfolio optimization. In addition, constraints like capital budget restrictions are considered in the hybrid heuristic framework. The computational complexity of selected parts of the algorithm is analyzed. Moreover, empirical results indicate that the hybrid method is superior in convergence speed to a non-hybrid evolutionary approach and finds approximations of risk-return efficient portfolios within reasonable time.

Introduction

Since the late 1990s, a number of innovative quantitative approaches to portfolio credit risk modelling and measurement have been developed, cf. e.g. Gupton et al. (1997), CreditSuisse Financial Products (1997), Wilson 1997a, Wilson 1997b and Kealhofer (1998). Moreover, the trade in financial instruments for transferring credit risk like credit default swaps, asset backed transactions, etc. has increased significantly during the last decade, cf. Ferry (2002). In addition, the banking supervision authorities have recently announced important changes in their Basel Committee for Banking Supervision (2003) consultative document concerning capital regulations for financial institutions which are particularly supposed to change the regulation of exposures to credit risk which is an important constraint in real world applications. Due to these developments and other, profit-related considerations there is an increasing demand for constrained optimization of credit portfolios in many financial institutions.

The majority of studies that have been carried out in the portfolio selection context so far focused on stock portfolio optimization (see e.g. Elton and Gruber (1995) for methods based on the work of Markowitz (1952) or Dueck and Winker (1992), Chang et al. (2000), Gilli and Këllezi (2002) for different heuristic approaches) which is significantly different from credit portfolio optimization, mainly because of the asymmetric loss distributions that occur in portfolio credit risk management which require specific methods for the calculation of aggregated portfolio loss distributions (cf. Gupton et al. (1997), p. 7). Only a few studies have focused on the latter problems so far. For a single objective function problem, Andersson et al. (2001) proposed the use of simplex algorithms in a portfolio credit risk simulation model framework while Lehrbass (1999) proposed the use of Kuhn-Tucker optimality constraints in an analytical portfolio credit risk model. We proposed the use of Evolutionary Algorithms for solving credit portfolio optimization problems first in Schlottmann and Seese (2001). In that work, we introduced a hybrid Evolutionary Algorithm to solve a constrained maximization problem that was built upon a single objective function combining both the aggregated return and the aggregated risk of a credit portfolio.

We extend the above studies by focusing on a hybrid heuristic framework for the calculation of a whole set of different risk-return efficient structures for credit portfolios with respect to additional constraints, e.g. capital budget restrictions. For the succeeding considerations the concept of Pareto-optimality is essential, i.e. efficient structures are Pareto-optimal concerning the two distinct (and in most cases contrary) objective functions specifying the aggregated risk and the aggregated return of each potential credit portfolio structure for a given discrete set of investment alternatives.

Since the first reported implementation and test of a multi-objective evolutionary approach, the Vector Evaluated Genetic Algorithm by Schaffer (1984), this special branch of Evolutionary Algorithms (EAs) has attracted many researchers dealing with non-linear and non-convex as well as integer-variable multi-objective optimization problems. Meanwhile, many different EAs have been proposed for such problems, see e.g. Zitzler et al. (2000), Deb (2001), Coello et al. (2002) or Osyczka (2002) for overviews and comparisons of these approaches. We have chosen a Multi-Objective Evolutionary Algorithm (MOEA) as the basis for our hybrid approach since it offers great flexibility concerning hybridization with other methods. Furthermore, unlike traditional methods (cf. e.g. Ehrgott, 2000) a MOEA does neither require the objective functions nor the constraints to be linear or convex, so our hybrid approach allows the use of the Credit-Value-at-Risk, which is a commonly used downside credit risk measure in real world applications. Like other percentile-based risk measures that were e.g. analysed in Pflug (2000) it is neither linear nor convex. A formal definition of the Credit-Value-at-Risk as well as references underlining its importance are provided in the next section.

The paper is organized as follows: In the next section, we describe our portfolio credit risk optimization problem and point out its complexity. Then, we give a short introduction to MOEAs before deriving a proper genetic modelling for portfolio credit risk problems. The succeeding section gives an overview and discusses some elements of our hybrid evolutionary algorithm framework for the approximation of constrained risk-return efficient credit portfolio structures. We provide details of our implementation before describing the parameters and the results of an empirical study where we mainly compare our hybrid implementation to a non-hybrid approach.

Section snippets

Formal definition of the credit portfolio optimization problem

For the formal definition of the constrained discrete credit portfolio optimization problem which is to be solved by our hybrid method, we will now consider an investor, e.g. a bank, who wants to optimize the risk-return structure of her credit portfolio containing m different obligors (borrowers).

Definition 1

Given is a bank that decides in t=0 about investing in a subset of m>1,m∈N obligors to be held in a credit portfolio. The bank has chosen a horizon T∈R+ for its risk calculations.

Each obligor i∈{1,…,m

Overview of our Hybrid Multi-Objective Evolutionary Algorithm (HMOEA)

As pointed out in the introduction, our hybrid approach is based on a MOEA since this concept offers flexibility particularly concerning the objective functions and constraints as well as hybridization support. A MOEA is a randomized heuristic search algorithm reflecting the Darwinian ‘survival of the fittest principle’ that can be observed in many natural evolution processes, cf. Holland (1975). At each discrete time step t∈N, a MOEA works on a set of solutions P(t) called population or

Specification of test cases, parameters and performance criteria

Beside our HMOEA implementation, we have also created a simple enumeration algorithm that investigates all possible portfolio structures to determine the set of feasible Pareto-efficient solutions PE having maximum cardinality for small instances of our Problem 10, i.e. the latter algorithm serves as a proof for the globally optimal portfolio structures that should be discovered by the other search algorithms. For all instances considered in this article, we compared the results of the HMOEA

Conclusion and outlook

We considered a constrained multi-objective portfolio optimization problem based on binary decision variables which represent investment alternatives incorporating credit risk. The aggregated net return from a portfolio and the aggregated downside risk measured by the Credit-Value-at-Risk were considered as objective functions of a bank also having an additional supervisory capital budget restriction.

For the computation of a large set of feasible solutions which approximate Pareto-efficient

Acknowledgements

Support of this work by GILLARDON AG financial software, Germany and by an anonymous German bank is hereby gratefully acknowledged. The authors would like to thank two anonymous referees and the editors of this CSDA Special Issue for their helpful suggestions and comments on an earlier version of this article.

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