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

Computational Statistics & Data Analysis 47 (2004) 373–399www.elsevier.com/locate/csda

A hybrid heuristic approach to discretemulti-objective optimization of credit portfolios

Frank Schlottmann∗ , Detlef SeeseInstitute AIFB, Department of Economics, University Karlsruhe (TH), D-76128 Karlsruhe, Germany

Accepted 8 November 2003

Abstract

A hybrid heuristic approach combining multi-objective evolutionary and problem-speci.c localsearch methods is proposed to support the risk-return analysis of credit portfolios. Its goal is tocompute approximations of discrete sets of Pareto-e2cient portfolio structures concerning boththe respective portfolio return and the respective portfolio risk using the non-linear, non-convexCredit-Value-at-Risk downside risk measure which is relevant to real world credit portfoliooptimization. In addition, constraints like capital budget restrictions are considered in the hy-brid heuristic framework. The computational complexity of selected parts of the algorithm isanalyzed. Moreover, empirical results indicate that the hybrid method is superior in convergencespeed to a non-hybrid evolutionary approach and .nds approximations of risk-return e2cientportfolios within reasonable time.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Hybrid heuristic approach; Multi-Objective evolutionary Algorithm; Local search; Portfolio creditrisk; Credit-Value-At-Risk; Constrained Pareto-e2cient portfolio

1. Introduction

Since the late 1990s, a number of innovative quantitative approaches to portfoliocredit risk modelling and measurement have been developed, cf. e.g. Gupton et al.(1997), CreditSuisse Financial Products (1997), Wilson (1997a, b) and Kealhofer(1998). Moreover, the trade in .nancial instruments for transferring credit risk likecredit default swaps, asset backed transactions, etc. has increased signi.cantly dur-ing the last decade, cf. Ferry (2002). In addition, the banking supervision authorities

∗ Corresponding author. Tel.: +49-172-619-6203; fax: +49-7225-983194.E-mail address: [email protected] (F. Schlottmann).

0167-9473/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2003.11.016

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374 F. Schlottmann, D. Seese / Computational Statistics & Data Analysis 47 (2004) 373–399

have recently announced important changes in their Basel Committee for BankingSupervision (2003) consultative document concerning capital regulations for .nancialinstitutions which are particularly supposed to change the regulation of exposures tocredit risk which is an important constraint in real world applications. Due to thesedevelopments and other, pro.t-related considerations there is an increasing demand forconstrained optimization of credit portfolios in many .nancial institutions.

The majority of studies that have been carried out in the portfolio selection contextso far focused on stock portfolio optimization (see e.g. Elton and Gruber (1995) formethods based on the work of Markowitz (1952) or Dueck and Winker (1992), Changet al. (2000), Gilli and KHellezi (2002) for diIerent heuristic approaches) which is sig-ni.cantly diIerent from credit portfolio optimization, mainly because of the asymmetricloss distributions that occur in portfolio credit risk management which require speci.cmethods 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 asingle objective function problem, Andersson et al. (2001) proposed the use of sim-plex algorithms in a portfolio credit risk simulation model framework while Lehrbass(1999) proposed the use of Kuhn-Tucker optimality constraints in an analytical port-folio credit risk model. We proposed the use of Evolutionary Algorithms for solvingcredit portfolio optimization problems .rst in Schlottmann and Seese (2001). In thatwork, we introduced a hybrid Evolutionary Algorithm to solve a constrained maxi-mization problem that was built upon a single objective function combining both theaggregated return and the aggregated risk of a credit portfolio.

We extend the above studies by focusing on a hybrid heuristic framework for thecalculation of a whole set of diIerent risk-return e2cient structures for credit portfolioswith respect to additional constraints, e.g. capital budget restrictions. For the succeed-ing considerations the concept of Pareto-optimality is essential, i.e. e2cient structuresare Pareto-optimal concerning the two distinct (and in most cases contrary) objectivefunctions specifying the aggregated risk and the aggregated return of each potentialcredit portfolio structure for a given discrete set of investment alternatives.

Since the .rst reported implementation and test of a multi-objective evolutionaryapproach, the Vector Evaluated Genetic Algorithm by SchaIer (1984), this specialbranch of Evolutionary Algorithms (EAs) has attracted many researchers dealing withnon-linear and non-convex as well as integer-variable multi-objective optimization prob-lems. Meanwhile, many diIerent EAs have been proposed for such problems, see e.g.Zitzler et al. (2000), Deb (2001), Coello et al. (2002) or Osyczka (2002) for overviewsand comparisons of these approaches. We have chosen a Multi-Objective Evolution-ary Algorithm (MOEA) as the basis for our hybrid approach since it oIers greatKexibility concerning hybridization with other methods. Furthermore, unlike traditionalmethods (cf. e.g. Ehrgott, 2000) a MOEA does neither require the objective functionsnor the constraints to be linear or convex, so our hybrid approach allows the useof the Credit-Value-at-Risk, which is a commonly used downside credit risk measurein real world applications. Like other percentile-based risk measures that were e.g.analysed in PKug (2000) it is neither linear nor convex. A formal de.nition of theCredit-Value-at-Risk as well as references underlining its importance are provided inthe next section.

F. Schlottmann, D. Seese / Computational Statistics & Data Analysis 47 (2004) 373–399 375

The paper is organized as follows: In the next section, we describe our portfoliocredit risk optimization problem and point out its complexity. Then, we give a shortintroduction to MOEAs before deriving a proper genetic modelling for portfolio creditrisk problems. The succeeding section gives an overview and discusses some elementsof our hybrid evolutionary algorithm framework for the approximation of constrainedrisk-return e2cient credit portfolio structures. We provide details of our implementationbefore describing the parameters and the results of an empirical study where we mainlycompare our hybrid implementation to a non-hybrid approach.

2. Formal de�nition of the credit portfolio optimization problem

For the formal de.nition of the constrained discrete credit portfolio optimizationproblem 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 portfoliocontaining m diIerent obligors (borrowers).

De�nition 1. Given is a bank that decides in t = 0 about investing in a subset ofm¿ 1; m∈N obligors to be held in a credit portfolio. The bank has chosen a horizonT ∈R+ for its risk calculations.

Each obligor i ∈ {1; : : : ; m} is subject to the risk of default and it is characterized bythe following data which is considered to be constant within the time period [0; T ]: lossgiven default of obligor i (amount of money/credit exposure which is lost if obligori defaults) ei ∈R+, expected rate of return ri ∈R based on ei (net of cost), expecteddefault probability pi ∈R+ within [0; T ], supervisory capital requirement percentagewi ∈R based on ei.

The dependence structure between joint defaults of obligors i; j ∈ {1; : : : ; m} is givenby an undirected graph G := (V; E) and a function h : E → R, where V := {1; : : : ; m}is the set of obligors, E := V × V is the complete edge set of potential defaultdependencies between obligors (i; j) and h : E → R is a function expressing the strengthof the default dependency between each pair of two obligors (i; j).

The loss given default ei describes the loss (e.g. being equal to the amount lent toobligor i) for the bank which occurs if obligor i is held in the credit portfolio and failsto pay his obligations within [0; T ]. This default event of obligor i has an expectedprobability of pi. The wi parameters are used to de.ne a constraint in our optimizationproblem which is described in more detail below, and the ri parameters specify theexpected return for the bank after re.nancing cost if obligor i is held in the portfolio.

The above representation of the default dependence structure, which is the mainmodelling issue in portfolio credit risk problems, is discussed and analysed further inSeese and Schlottmann (2003). The function h can be a correlation based function ormore general, a copula based function, see Frey and McNeil (2001) for details aboutmathematical modelling of default dependencies.

When required, we will abbreviate the respective set of scalar variables ei; ri; pi; wi

of all obligors by vectors e := (e1; : : : ; em)T; r := (r1; : : : ; rm)T; p := (p1; : : : ; pm)T;w := (w1; : : : ; wm)T.


We assume a .xed capital budget for the bank that is given by the bank’s maximumsupervisory capital which is required to be provided by the bank due to the supervi-sory regulations (cf. e.g. Basel Committee for Banking Supervision, 2003). This con-straint serves as an illustrative example for additional constraints which are relevant toreal-world applications and therefore have to be considered by optimization algorithms.

De�nition 2. The bank’s supervisory capital budget is given by K ¿ 0.

We now consider an arbitrary element from the search space of portfolio structuresfor the given data from De.nition 1.

De�nition 3. A portfolio structure is given by a vector

x = (x1; x2; : : : ; xm)T; xi ∈ {0; ei}: (1)

Since every xi can only take the values 0 or ei, the bank has to decide whether tohold the whole amount of loss given default ei in its portfolio. In many real worldportfolio optimization problems the decision is e.g. either keeping the obligor i inthe credit portfolio or selling the entire risk of obligor i to a risk buyer (the sameholds vice versa for the risk buyer). This is particularly true for formerly non-tradedinstruments like corporate loans in a bank’s credit portfolio. Even if there are morethan two decision alternatives for each potential investment i, the decision variableswill still consist of a .nite, discrete number of choices.

Given the speci.ed decision alternatives, the bank has to consider two conKictingobjective functions: the aggregated return and the aggregated risk from its portfolio.Usually, there is a trade-oI between both objectives since any rational investor willask for a premium (additional return) to take risk.

De�nition 4. The aggregated expected return from a portfolio structure x is calculatedby

ret(x; p; r) =m∑

i=1

rixi −m∑

i=1

pixi =m∑

i=1

(ri − pi)xi: (2)

This is a common net risk adjusted return calculation for performance measurement,cf. e.g. Ong (1999, p. 218). We will now de.ne a speci.c measure of credit risk whichis commonly used in many models for contemporary real world credit risk management,cf. Gupton et al. (1997), CreditSuisse Financial Products (1997), Wilson (1997a, b).

De�nition 5. For a given portfolio structure x the Credit-Value-at-Risk (CVaR) at thearbitrary, but .xed con.dence level �∈ (0; 1) is obtained by calculating

risk(x; p; h) := q�pf (x; p; h) − �pf (x; p); (3)

where q�pf (x; p; h) is the �-percentile of the cumulative distribution function of ag-

gregated losses calculated from the portfolio for the given parameters x; p and the


dependency structure speci.ed by h. Moreover, �pf (x; p) is the expected loss calculatedby �pf (x; p) =

∑mi=1 xi pi.

For the problem speci.cation within this section, we do not need a speci.cation ofthe calculation procedure for the cumulative distribution function of aggregated lossesor q�

pf . We will return to these details in Section 3.3 which describes the portfoliocredit risk model used in our empirical study.

The risk measure computes the amount of internal risk capital (also called economiccapital) to be reserved by the bank for the potential losses arising from portfolio struc-ture x. In real-world applications there are signi.cant diIerences between bank-internalrisk capital budgeting processes and supervisory capital limits which are external rulesto be obeyed. Therefore, the supervisory capital budget K from De.nition 2 is now usedin conjunction with the wi supervisory capital requirement percentages from De.nition1 to build an additional constraint on portfolio structures.

De�nition 6. The required amount of supervisory capital for a given portfolio structurex is

cap(x; w) :=m∑

i=1

xiwi: (4)

De�nition 7. A given portfolio structure x is feasible if and only if

cap(x; w)6K: (5)

Based upon the above de.nition of feasibility, the following de.nition is essentialfor the concept of Pareto-e2ciency.

De�nition 8. Given are two distinct feasible portfolio structures x and y. x dominatesy if and only if one of the following cases is true:

ret(x; p; r) ¿ ret(y; p; r) ∧ risk(x; p; h)6 risk(y; p; h): (6)

ret(x; p; r)¿ ret(y; p; r) ∧ risk(x; p; h) ¡ risk(y; p; h): (7)

If x dominates y, we will denote this relationship by xd¿y.

According to De.nition 8, a feasible portfolio structure x is better than a feasibleportfolio structure y if and only if x is better in at least one of the two criteria and notworse than y in the other criterion. Obviously, a rational investor will prefer x over y

if xd¿y.

De�nition 9. Given is the set S of all possible portfolio structures for the speci.eddata from De.nition 1 and the subset S ′ ⊆ S of all feasible structures in S. A solution


x ∈ S ′ is a feasible global non-dominated portfolio structure if and only if it satis.esthe following condition:

∀y ∈ S ′ : @(yd¿x): (8)

The bank’s goal is to identify the set of feasible non-dominated portfolio structureshaving maximum cardinality. If this set—or at least a su2cient approximation of it—isknown, she can choose between a large number of feasible, risk-return e2cient invest-ment alternatives using her preferences or utility function. This is similar to the e2cientfrontier considerations of Markowitz (1952), but in a discrete decision space using thenon-linear, non-convex Credit-Value-at-Risk measure.

Problem 10. The problem of 4nding the set of feasible Pareto-e7cient portfolio struc-tures having maximum cardinality for the set of investment alternatives S can beformulated as: Calculate the set

PE∗ := {x ∈ S ′: ∀y ∈ S ′ : @(yd¿x)}: (9)

Since Problem 10 is NP-hard (cf. Seese and Schlottmann, 2003) and contains anon-linear, non-convex objective function, we have opted for a heuristic optimizationalgorithm, see e.g. Winker (2001) for an overview and a discussion of diIerent heuristicoptimization approaches. Our approach is described within the next section.

3. A hybrid heuristic approach to the discrete credit portfolio optimization problem

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

As pointed out in the introduction, our hybrid approach is based on a MOEA sincethis concept oIers Kexibility particularly concerning the objective functions and con-straints as well as hybridization support. A MOEA is a randomized heuristic searchalgorithm reKecting the Darwinian ‘survival of the .ttest principle’ that can be ob-served in many natural evolution processes, cf. Holland (1975). At each discrete timestep t ∈N, a MOEA works on a set of solutions P(t) called population or gener-ation. A single solution x ∈P(t) is called an individual. To apply a MOEA to acertain problem the decision variables have to be transformed into genes, i.e. therepresentation of possible solutions by contents of the decision variables has to betransformed into a string of characters from a given alphabet �. The original rep-resentation of a solution is called phenotype, the genetic counterpart is called geno-type. We will cover the genetic representation issues for our Problem 10 in a ded-icated subsection below, so that we can .rst concentrate on the main algorithmicelements.


Algorithm 1 shows an overview of our HMOEA which combines ideas from dif-ferent MOEA schemes with a problem-speci.c additional local search variationoperator.

Algorithm 1. Pseudo code for HMOEA.1: Initialize e; p; r; h; w and K (de.ned in Section 2) and t = 02: Generate initial population P(t)3: Initialize elite population Q(t) = ∅4: Evaluate P(t)5: repeat6: Select individuals from P(t)7: Recombine selected individuals (variation operator 1)8: Mutate recombined individuals (variation operator 2)9: Apply local search to mutated individuals (variation operator 3)

10: Create oIspring population P′(t)11: Evaluate joint population J (t) = P(t) ∪ P′(t)12: Update elite population Q(t) from J (t)13: Generate P(t + 1) from J (t)14: t = t + 115: until (Q(t) = Q(max{0; t − tdiI})) ∨ (t ¿ tmax)

The initial population P(0) is generated by random initialization of every individualto obtain a diverse population in the search space of potential solutions and beyondthat, to support a fair benchmarking of the diIerent Evolutionary Algorithms in ourempirical study, cf. Fogel and Michalewicz (2000, p. 183).

We propose the use of an elite population Q(t) in our algorithm that contains thefeasible, non-dominated solutions found so far at each population step t. This elitepopulation has many advantages: Rudolph and Agapie (2000) have shown that thesolutions within such an elite population (‘archive population’ in their work) convergealmost surely to the solutions in PE∗ for t → ∞. Moreover, the number of individualsin the population P(t) that has to be chosen a priori before running the algorithm, islimiting the number of best solutions to be kept by the algorithm during time. If wemaintain an elite population, the choice of the size of P(t) is not crucial concerning thenumber of solutions to be kept during the evolution process. Furthermore, since we haveto deal with a discrete non-linear, non-convex problem, which can have many localoptima that are not uniformly distributed in the two-dimensional objective functionspace, it is di2cult to satisfy both the requirement of .nding a well-distributed setof diverse solutions, and the requirement of approximating the largest possible set offeasible Pareto-e2cient portfolio structures if we use only one population. We willpresent an example distribution of the maximum Pareto-e2cient set in our empiricalstudy to visualize these facts.

The evaluation of P(t) and J (t) is based on the non-domination concept proposedby Goldberg (1989) and explicitly formulated for constrained problems e.g. by Deb(2001). In our context, it leads to the following type of domination check (cf. Deb,2001, p. 288) which extends De.nition 8 by cases 12 and 13.


De�nition 11. Given are two distinct portfolio structures x and y: x constraint-dominatesy if and only if one of the following cases is true:

cap(x; w)6K ∧ cap(y; w)6K ∧ret(x; p; r) ¿ ret(y; p; r) ∧ risk(x; p; h)6 risk(y; p; h) (10)

cap(x; w)6K ∧ cap(y; w)6K ∧ret(x; p; r)¿ ret(y; p; r) ∧ risk(x; p; h) ¡ risk(y; p; h) (11)

cap(x; w)6K ∧ cap(y; w) ¿K (12)

cap(x; w) ¿K ∧ cap(y; w) ¿K ∧ cap(x; w) ¡ cap(y; w): (13)

If x constraint-dominates y, we will denote this relationship by xc¿y.

The .rst two cases in De.nition 11 refer to the cases from De.nition 8 where onlyfeasible solutions were considered. Case 12 expresses a preference for feasible overinfeasible solutions and case 13 prefers the solution that has lower constraint violation.

The non-dominated sorting procedure in our HMOEA uses the dominance crite-rion from De.nition 11 to classify the solutions in a given population, e.g. P(t), intodiIerent levels of constraint-domination. The best solutions which are not constraint-dominated by any other solution in the population, obtain non-domination level 1 (bestrank). After that, only the remaining solutions are checked for constraint-domination,and the non-constraint-dominated solutions among these obtain non-domination level 2(second best rank). This process is repeated until each solution has obtained an asso-ciated non-domination level.

The selection operator in Algorithm 1 is performed using a binary tournament basedon De.nition 11. Two individuals x and y are randomly drawn from the current pop-ulation P(t), using uniform probability of psel := 1=|P(t)| for each individual. Theseindividuals are checked for constraint-domination according to De.nition 11 and if,

without loss of generality, xc¿y then x wins the tournament and is considered for re-

production. If none of the two solutions dominates the other, they cannot be comparedusing the constraint-domination criterion, and the winning solution is .nally determinedusing a draw from a uniform distribution over both possibilities. Tournament selectionis favourable because it is invariant of .tness function scaling and transformation and itsupports potential parallel implementations of our hybrid algorithm, cf. Blickle (2000).Osyczka (2002) provides a number of examples showing further advantages of tour-nament selection in diIerent problem settings.

The selected individuals from the current population P(t) are modi.ed using geneticvariation operators (see e.g. Baeck, 2000, p. 235 I. for an overview of such operators).The .rst variation operator is the standard one-point crossover for discrete decisionvariables, i.e. the gene strings of two selected individuals are cut at a randomly chosenposition and the resulting tail parts are exchanged with each other to produce two newoIspring. This operation is performed with crossover probability pcross on individuals


selected for reproduction. The main goal of this variation operator is to move thepopulation through the space of possible solutions.

In analogy to natural mutation, the second variation operator changes the genes ofselected individuals randomly with probability pmut (mutation rate) per gene to allowthe invention of new, previously undiscovered solutions in the population. Its secondtask is the prevention of the HMOEA stalling in local optima as there is always apositive probability to leave a local optimum if the mutation rate is greater than zero.

Our third variation operator in Algorithm 1 represents a problem-speci.c local searchprocedure that is applied with probability plocal to each selected solution x after cross-over and mutation. This local search procedure can exploit the structure of a givensolution x to perform an additional local optimization of x towards PE∗, e.g. by usinga so-called hill climbing algorithm that changes x according to local information aboutour objective functions in the region around x. We consider this to be a signi.cantimprovement compared a standard, non-hybrid MOEA since the randomized searchprocess of the MOEA can be guided a bit more towards feasible Pareto-e2cient solu-tions and therefore, such a local search operator can improve the convergence speed ofthe overall algorithm towards the desired solutions. This is particularly important forreal world applications, where speed matters when large portfolios are to be consid-ered. In addition to these arguments, e.g. the CreditRisk+ portfolio credit risk modelprovides additional local structure information for a current solution x beyond theobjective function values that can be exploited very e2ciently from a computationalcomplexity perspective. This is described in more detail in a dedicated subsectionbelow.

By application of the variation operators to the selected individuals we obtain anoIspring population P′(t). The members of the joint population J (t) containing allparent solutions from P(t) and all oIspring solutions from P′(t) are evaluated using thenon-dominated sorting procedure described above. In the next step, the elite populationQ(t) is updated.

Before .nishing the population step t and setting t → t + 1 the members of thenew parent population P(t + 1) have to be selected from J (t) since in the majority ofiterations |J (t)|¿ |P(t + 1)| by de.nition of J (t) := P(t) ∪ P′(t). Because elitist EAs,which preserve the best solutions from both parents and oIspring, usually show betterconvergence properties, we also use this mechanism in our algorithm. Besides elitism,we also need a diversity preserving concept to achieve a good distribution of thesolutions in the whole objective space. We incorporate the concept of crowding-sortproposed in Deb (2001, p. 236). This diversity-preserving mechanism is favourableover other proposals, e.g. niche counting based on Euclidean -regions in the decisionvariable space or the objective function space since the crowding-sort does not requirean additional parameter ¿ 0 which is di2cult to estimate particularly for our dis-crete, non-linear and non-convex Problem 10, and in our case the crowding sort hasa smaller computational complexity of O(|J (t)|log |J (t)|) compared to the quadraticcomplexity which is required by other mechanisms, cf. Deb (2001, p. 237). Since thenumber of objective functions is constant for m → ∞ we have opted for crowding-sortaccording to diversity in the two-dimensional objective function space and not in them-dimensional decision variable space.


The algorithm is terminated if Q(t) has not been improved for a certain numbertdiI of population steps or if a maximum number of tmax population steps has beenperformed.

3.2. Choosing an appropriate genetic representation

An important design question for the HMOEA is to choose a proper genetic repre-sentation of the decision variables. We assume that the decision variables xi will beconnected to obtain gene strings representing potential solutions. The resulting geno-types consist of real-valued genes which are connected to strings and take either value0 or ei depending on the absence or presence of investment alternative i in the currentsolution. The gene strings have length m and represent some of the 2m combinations ofpossible portfolio structures. Of course, such binary-style variables can also be storede2ciently by using single-bit variables in an implementation, but since we use par-tial derivatives with respect to xi in our hybrid algorithm, we consistently keep thereal-valued variables in our notation.

Since we use the one-point crossover to conduct the search for good solutions inthe simulated evolution process there are some important issues of portfolio creditrisk modelling which suggest a well adapted genetic representation for the phenotypes.The one-point crossover cuts two gene strings at a random position and crosses thetails of the strings to produce two oIspring with crossover probability pcross. Theprobability of two genes i; j (these variables represent the index of the genes associatedto investment alternative i and j, respectively) from one individual being cut by thecrossover increases proportional to the distance |i − j| between the two genes in thegene string as the cut position is determined by a draw from a uniform distributionover m − 1 cut possibilities:

prob(crossover cuts gene i and j) =1

m − 1|i − j|: (14)

For better results of the crossover operator we must ensure that there is a high prob-ability of good partial solutions being recombined with other solutions and not beingdestroyed by the crossover’s cut operation. More formally, we search for a permuta-tion #(i) of the portfolio data represented by our genes ensuring a high probability ofsuccess for crossover. Therefore, we have to remember that the degree of dependencebetween two diIerent investment alternatives i; j plays the central role in aggregatedportfolio credit risk calculations, cf. our remarks in Section 2.

In our Problem 10, the dependence structure has no inKuence on the aggregatedreturn objective function given in De.nition 4, so it is su2cient to concentrate solelyon the risk objective function when considering the possible inKuence between diIerentgene positions.

According to De.nition 1, the dependence structure between investment alternativesis determined by the function h(i; j). We can exploit this information by determiningthe maximum strength of the dependency of an investment alternative from all othersand by building a permutation based on this measure. This greedy algorithm ensuresthat the genes of the more dependent investment alternatives are located closely to


each other, and it has a very low computational complexity compared to combinatorialproblems arising from the question of .nding the best of m! possible permutations.

De�nition 12. The maximum strength of the dependency of investment alternative ifrom all other investment alternatives j �= i is given by

s(i) := maxj �=i

{h(i; j)}: (15)

We can build our requested permutation #(i) by calculating and sorting these strengthvalues.

Lemma 13. For the given graph G = (V; E) and the function h from De4nition 1, wecan calculate a permutation #(i) of the portfolio data based on the strength fromDe4nition 12 in O(m2h) computations where h is the number of necessary steps tocompute h(i; j).

Proof. We construct a greedy algorithm Perm that takes each vertex i ∈V once andcomputes the (m − 1) values of h(i; j) for each j ∈V; j �= i to .nd the maximums(i) for the given i. Since G has m vertices, the computational complexity of thisoperation is O(m2h). Afterwards, algorithm Perm sorts the m number pairs (s(i); i)in ascending order using the s(i) values as primary sorting criterion. This operationrequires O(m log m) computational steps. The sorted array of number pairs (s(i); i)represents the permutation. If k is the index of (s(i); i) after sorting, the permutationof i is #(i) := k. The overall complexity of the algorithm Perm is O(m2h).

If our algorithm is provided with all values of h(i; j) at its start, e.g. this is the case ifpairwise correlations between investment alternatives are speci.ed, then the calculationof the permutation requires only m2 computational steps. We use the algorithm fromLemma 13 as a preprocessing stage before running the HMOEA as described in theprevious section.

3.3. Integration of the CreditRisk+ portfolio credit risk model

To avoid unneccessary details in Section 2, we did not specify the calculation proce-dure for the objective function risk(: : :) in our derivation of Problem 10. Moreover, weused a graph and a function h to describe the dependencies between diIerent obligorsin a given portfolio.

However, when implementing an algorithm for risk-return optimization of credit port-folios, we have to choose a model that provides us a function h according to De.nition1 and beyond that, an algorithm for computing risk(: : :) according to De.nition 5. Inthe literature, there are diIerent alternatives for modelling the dependencies betweenobligors and for calculating portfolio credit risk measures. Among these alternatives,CreditMetrics by Gupton et al. (1997), CreditRisk+ by CreditSuisse Financial Products(1997), the model by Wilson (1997a, b) and the KMV option based approach (seeKealhofer, 1998) are intensively discussed in many academic and application-oriented


publications. Since we set our focus on loan portfolios according to the real world dataused in our empirical study, we concentrate on CreditRisk+ here.

In the following paragraphs, we will give a brief description of the CreditRisk+General Sector Analysis model for a one year horizon (i.e. T = 1 in De.nition 1) thatconcentrates on the main issues concerning our algorithm, see CreditSuisse FinancialProducts (1997, pp. 32–57) for a more detailed derivation of the model. It is anactuarial approach that uses an intensity based modelling of defaults, i.e. the defaultof each obligor in the portfolio is considered to be a stopping time of a hazard rateprocess expressed by a Poisson-like process. In case of a default of obligor i before T ,i.e. if the .rst jump of the hazard rate process occurs before T , the loss given defaultamount ei will be entirely lost.

Given is the data from De.nition 1 of m obligors in the portfolio. Particularly, eachobligor has a loss given default ei, an associated annual mean default rate pi (typically,pi is small: 0 ¡pi ¡ 0:1) and an annual default rate volatility (i¿ 0. Furthermore,there is a total of n independent sectors as common risk factors, where the .rst sector(k = 1) is obligor speci.c, i.e. in this sector there is no implicit default correlationbetween obligors (k = 1; : : : ; n unless otherwise noted). The obligors are allocated tothe sectors according to sector weights *ik ∈ [0; 1]; ∀i :

∑nk=1 *ik = 1.

The probability generating function (abbreviated PGF) for the losses from the entireportfolio is de.ned by

G(z) :=∞∑i=0

prob(aggregated losses = iL)zi; (16)

where L is a constant de.ning loss given default bands of constant width and prob(: : :)represents the probability of losing i times the value of L from the whole portfolio.

Since the sectors are independent, Eq. (16) can be decomposed to

G(z) =n∏

k=1

Gk(z); (17)

where Gk(z) is the PGF for the losses from the portfolio in sector k.To obtain the approximated cumulative loss distribution function for the portfolio, a

recurrence relation, the recursion by Panjer (1981), can be applied to evaluate the co-e2cients of the PGF (for a more detailed background see Panjer and Willmot, 1992).After that, risk measures, e.g. an �-percentile of the portfolio loss distribution for agiven, .xed �∈ (0; 1), and the Credit-Value-at-Risk at this con.dence level � can beevaluated using De.nition 5. The choice of � := 0:99 is common in real-world appli-cations (cf. the references provided prior to De.nition 5 in Section 2), and therefore,we will also use it in our empirical study.

An interesting feature of the model concerning the portfolio optimization task arethe marginal risk contributions of obligor i to the standard deviation of portfolio creditrisk:

RC(i := ei

@(pf

@ei=

eipi

(pf

ei +

n∑k=1

*ik

((k

�k

)2 m∑

j=1

ejpj*jk

; (18)


where (pf is the portfolio standard deviation derived from the PGF of the portfoliolosses, �k ; (k are sector speci.c parameters calculated directly from the input parametersei; pi; (i; *ik using formula (19) below (note that (1 = 0 by de.nition of sector 1).

∀k : �k :=m∑

i=1

*ikpi; (1 := 0; ∀k ¿ 1 : (k :=m∑

i=1

*ik(i: (19)

Alternatively, by setting (k := !k�k for k ¿ 1 using parameters (variation coe2cients)!k; k = 2; : : : ; n only �k has to be calculated according to (19) and in this case, noobligor-speci.c default rate volatilities (i are required to calculate the sector speci.cparameters.

To calculate an approximation of the marginal risk contributions to the chosen�-percentile, a scaling factor is de.ned as /pf := (q�

pf − �pf )=(pf where �pf ; (pf ; q�pf

are the expectation, standard deviation and �-percentile of the portfolio loss distribu-tion, respectively.

The .gures calculated by applying formula (18) can be used as a basis for theapproximate marginal risk contribution of obligor i to the �-percentile by scaling therisk contribution obtained from (18) according to /pf and adding it to the obligorspeci.c expected loss:

RC�i := eipi + /pfRC(

i : (20)

The calculation of (20) for all investment alternatives i ∈ {1; : : : ; m} requires onlyO(mn) additional operations after the calculation of the coe2cients of the PGF fromformula (17) which is mandatory to evaluate the risk(: : :) target function for each in-dividual. Note that the number of sectors n is constant in a given problem instanceand usually small (n¡ 10), so the computation of (20) requires only linear computingtime O(m) measured by the number of investment alternatives m. We exploit this lowcomplexity to ensure a computationally e2cient calculation within our local searchvariation operator which is derived in the next paragraph.

3.4. Implementation of local search variation operator

We use the following local search target function based on the quotient betweenaggregated net return and aggregated risk for a given portfolio structure x:

f(x; p; r; h) :=ret(x; p; r)

risk(x; p; h): (21)

Considering De.nitions 4 and 5 as well as the CreditRisk+ calculation method for the�-percentile q�

pf (x; p; (;*) of the cumulative distribution of aggregated losses from theportfolio structure x under the given data p; (;*; r the function f can be written as

f(x; p; (;*; r) :=∑m

i=1 xi(ri − pi)q�

pf (x; p; (;*) − ∑mi=1 xipi

: (22)

If we maximize this function f we will implicitly maximize ret(x; p; r) and min-imise risk(x; p; h), and this will drive the portfolio structure x towards the set of


Pareto-e2cient portfolio structures (cf. the domination criteria speci.ed in De.nition8). In addition to that, we have to consider our constraints to ensure the local searchvariation operator keeps the portfolio structure x feasible or moves an infeasible portfo-lio structure x back into the feasible region. An overview of our local search operatorscheme based on these considerations is shown in Algorithm 2.

Algorithm 2. Pseudo code for Local Search Variation Operator.1: for all x ∈P(t) do2: Draw independent uniform random variate Z ∼ U (0; 1)3: if Z6plocal then4: if cap(x; w) ¿K then5: D = −16: else7: Choose D between D = 1 or D = −1 with uniform probability 0.58: end if9: for all i ∈ {1; : : : ; m} do

10: xi = xi

11: end for12: Step = 013: repeat14: for all i ∈ {1; : : : ; m} do15: xi = xi

16: end for17: retold =

∑mi=1 xi(ri − pi)

18: riskold = q�pf (x; p; (;*) − ∑m

i=1 xipi

19: for all j ∈ {1; : : : ; m} do20: Calculate the partial derivatives dj = @

@xjf(x; p; (;*; r)

21: end for22: if D = −1 then23: Choose the minimal gradient component i = argminj{dj|xj ¿ 0}24: Remove this obligor: xi = 025: else26: Choose the maximal gradient component i = argmaxj{dj|xj = 0}27: Add this obligor to portfolio: xi = ei

28: end if29: retnew =

∑mi=1 xi(ri − pi)

30: risknew = q�pf (x; p; (;*) − ∑m

i=1 xipi

31: Step = Step + 132: until (Step = Stepmax) ∨ (∀i : xi = 0) ∨ (∀j : xj ¿ 0)∨

(x ∈P(t)) ∨ (x ∈Q(t)) ∨ (D = −1 ∧ cap(x; w)6K)∨(D = 1 ∧ (cap(x; w) ¿K ∨ (retnew6 retold ∧ risknew¿ riskold)))

33: Replace x in P(t) by its optimized version34: end if35: end for


If the current solution x from P(t) to be optimized with probability plocal is infeasiblebecause the capital restriction is violated (cf. line 4 in Algorithm 2), the algorithmwill remove the investment alternative having the minimum gradient component valuefrom the portfolio. This condition drives the hybrid search algorithm towards feasiblesolutions. In case of a feasible solution that is to be optimized, the direction of searchfor a better solution is determined by a draw of a uniformly distributed U (0; 1)-randomvariable (cf. line 7 in Algorithm 2). This stochastic behaviour helps preventing the localsearch variation operator from stalling into the same local optima.

The local search algorithm terminates if a maximum number Stepmax of local searchoptimization steps has been performed, if the current solution cannot be modi.ed fur-ther, if it is already included in the populations P(t) or Q(t) or if no local improvementconcerning the violation of constraints or the target functions can be made. While theother termination conditions are quite natural for a local search algorithm in our setting,the number of iterations of the local search loop is bounded by parameter Stepmax toprevent the local search operator from discovering the same local optima again andagain. Furthermore, a small value of Stepmax yields a better runtime performance es-pecially for large instances of Problem 10 because the local search variation operatordoes not run a potentially large number of iterations in each of its applications.

We prefer the gradient-based search over other local search methods in Algorithm2 because the gradient provides valuable information about potentially good searchdirections while being computed e2ciently: Remembering the fact that the marginal riskcontributions of all obligors can be calculated in O(m) for a given portfolio structurex and .xed n∈N (cf. Section 3.3) we can exploit these .gures to obtain an e2cientcomputation of the gradient’s components:

Lemma 14. The partial derivative dj for obligor j required in Algorithm 2 can beapproximated by evaluation of

dj :=xj(rj − pj)(/pf(pf ) − (∑m

i=1 xi(ri − pi))

(/pf RC(j )

xj(/pf(pf )2 : (23)

Proof. See Appendix A.

By using this result in the analysis of Algorithm 2 we obtain:

Lemma 15. Assuming q�pf (x; p; (;*) can be evaluated in O(m) computational steps for

any arbitrarily given solution x ∈ S under 4xed parameters n; p; (;* and Stepmax6m,the worst case complexity of Algorithm 2 is

O(|P(t)| × m × Stepmax): (24)

Proof. The body of the for all loop starting at line 2 in Algorithm 2 is executed |P(t)|times. Thus, for plocal6 1 the maximum number of iterations of the instructions afterline 3 is O(|P(t)|).

Under the assumption that an evaluation of the objective function risk(: : :) :=q�

pf (x; p; (;*) − ∑mi=1 xipi for any possible solution x of the given problem instance


does not require more than O(m) computational eIort (which was made particularlybecause we do not intend to discuss the CreditRisk+ algorithm here), the largest com-putational eIort within each iteration of the repeat : : : until loop is O(m) which is alsospent on the calculation of the partial derivatives for all obligors, cf. our remarksprior to Algorithm 2. This is particularly due to the fact that checking the existence ofx ∈P(t) within the until condition requires only a number of computational steps whichis logarithmic in the number of elements of P(t) if we use binary search e.g. basedon trees storing the objective function values. The same holds for verifying x ∈Q(t).Since there are at most 2m diIerent solutions to the given problem which have to bestored in Q(t) the inequality max{log |P(t)|; log |Q(t)|}6m is a tautology, such thatthe search for x ∈P(t) and for x ∈Q(t) does not raise the complexity bound of therepeat : : : until loop found so far.

Finally, the repeat : : : until loop is terminated after a maximum of Stepmax itera-tions since for Stepmax6m all other possible situations that violate the until conditionlead to termination of the loop before or exactly until Stepmax iterations have beenperformed.

If we inspected the objective function values of all portfolio structures in the neigh-bourhood of x (i.e. all portfolio structures which could be derived from x by mod-i.cation of a single decision variable) using another local search strategy instead ofexploiting the gradient of x, the computational complexity of the local search varia-tion operator would be higher since we would require m evaluations of the objectivefunctions in each single local search iteration.

Besides this advantage of the chosen gradient-based search strategy, the above re-sults indicate appropriate choices of the parameters plocal and Stepmax. To obtain alow average computational complexity of the local search variation operator which isimportant for real world applications, plocal should be chosen signi.cantly smaller than1 to lower the |P(t)| term from expression (24) because the instructions within theouter for all loop after line 3 are executed plocal × |P(t)| times on average. More-over, to ensure a low average computational eIort to be spent by Algorithm 2, alsoStepmax should be kept small to avoid too many iterations of the repeat : : : untilloop.

From the perspective of the overall HMOEA algorithm, the choice of plocal can bemade by the respective user of the HMOEA depending on his or her preferences: Ifone is interested in .nding better solutions in earlier populations, the probability shallbe set higher, and in this case more computational eIort is spent by the algorithmon the local improvement of the solutions. However, the local search optimizationpressure should not be too high since one is usually also interested in .nding a diverseset of solutions. This is particularly important for relatively small problem instanceshaving many isolated local optima. Therefore, a choice of 0 ¡plocal6 0:2 yieldedgood results in our tests during the development of the hybrid system. Analogously,the parameter Stepmax can be set to large values by the user to enforce a strong localsearch towards better solutions, but this might lead to less diversity. To achieve anappropriate relation between computational eIort and solution quality, we identi.eda range of 16Stepmax6 4 that yielded promising results in our test cases. Beyond


the above considerations, we will provide examples for adequate parameter choicesdepending on the problem size in the next section.

4. Description of empirical study and results

4.1. Speci4cation of test cases, parameters and performance criteria

Beside our HMOEA implementation, we have also created a simple enumerationalgorithm that investigates all possible portfolio structures to determine the set of fea-sible Pareto-e2cient solutions PE∗ having maximum cardinality for small instances ofour Problem 10, i.e. the latter algorithm serves as a proof for the globally optimalportfolio structures that should be discovered by the other search algorithms. For allinstances considered in this article, we compared the results of the HMOEA to therespective results of a non-hybrid MOEA that incorporates all features of the HMOEAexcept for the local search operator which is disabled in the non-hybrid algorithm.Particularly, the MOEA also bene.ts from all problem speci.c algorithmic featuresthat we have proposed for the HMOEA in the previous sections, e.g. the presence ofthe elite population and the chosen genetic representation. All implementations werewritten in the C programming language, and the tests were carried out on a standarddesktop PC (2 GHz single CPU). In the randomized algorithms, we implemented acombined multiple linear congruential pseudorandom generator proposed and tested inL’Ecuyer (1999) having a period length of about 2191 and yielding satisfactory spectraltest results. For all evolutionary algorithms, we performed 50 independent runs of thesame algorithm on the respective test problem using diIerent pseudorandom numbergenerator seeds, which particularly yielded diIerent initial populations P(0). To ensurea fair comparison between the EAs we used the same initial population for both theHMOEA and the non-hybrid MOEA given a speci.c pseudorandom generator seed.

Although more test cases were examined during the development of the system (e.g.for estimating the parameters like |P(t)|; pcross; plocal; Stepmax etc.) we focus on twosample loan portfolios and a given supervisory capital budget for each portfolio inthe following text. The speci.cation of the portfolio m25n2 is shown in the appendixwhereas portfolio m386n2 contains con.dential real-world portfolio data kindly pro-vided by a German bank.

Table 1 shows their parameters, the level of the supervisory capital budget K com-pared to the sum of all supervisory capital requirements

∑mi=1 wiei in the corresponding

test case as well as the standard parameter settings for the EAs depending on the re-spective portfolio which are a basis for our empirical comparison between the diIerentalgorithms.

In both test cases, we chose a quite common parameter setting of pcross := 0:95and pmut := 1=m, which is reported to work well in many other EA studies (cf.Spears (2000) for a discussion of pcross and Baeck (2000) for pmut), and this was alsosupported by test results during our development of the HMOEA and the non-hybridMOEA. Of course, the population size was chosen depending on the size of the searchspace. Furthermore, for the HMOEA, we chose the parameter plocal depending on the


Table 1Overview of test portfolios and parameter settings

Parameter Name of portfolio

m25n2 m386n2

m (# obligors) 25 386n (# sectors) 2 2K=(

∑mi=1 wiei) (constraint level) 50% 71%

|P(t)| (population size) 30 100pcross (crossover probability) 0.95 0.95pmut (mutation probability) 1=25 1=386plocal (local search probability) 0:05 0:10Stepmax (max. local search iterations) 4 4tdiI (termination condition parameter) 100 100tmax (termination condition parameter) 50; 000 50; 000

problem size and held Stepmax := 4 constant to provide a better comparison between theresults of the algorithm depending on the problem instance. The respective evolutionaryprocess is stopped after a total of tmax := 50; 000 population steps or if Q(t) has notbeen improved for tdiI := 100 population steps.

We calculate performance measures for the output of the heuristic optimization al-gorithms based on the set coverage metric from Zitzler (1999) which is de.ned asfollows:

De�nition 16. Given are two sets of portfolio structures PE1; PE2 which are approxi-mations for PE∗ de.ned in Problem 10. The pair of set coverage metric values C1;2 :=(C1; C2) is calculated by

C1 := C(PE2; PE1) =|{x ∈PE1|∃y ∈PE2 : y

c¿x}|

|PE1| (25)

C2 := C(PE1; PE2) =|{y ∈PE2|∃x ∈PE1 : x

c¿y}|

|PE2| : (26)

This metric provides us a criterion for comparing two diIerent sets of solutionsproduced by diIerent algorithms. We have chosen this metric since it allows the com-parison of approximation sets having diIerent cardinalities, and particularly in our largetest case, we do not need PE∗ for the evaluation of the results. An algorithm com-puting PE2 is considered to be better in convergence to PE∗ than another algorithmthat computes PE1 if C1 ¿C2, i.e. if the fraction of solutions in PE2 which are dom-inated by solutions from PE1 is smaller than the fraction of solutions in PE1 that aredominated by solutions from PE2. To be more transparent, we investigate both thenominator and the denominator of (25) and (26) separately. Therefore, two importantgoals of multi-objective approximation algorithms are evaluated: Finding an approxi-mation set whose elements are closer to corresponding members of PE∗ and which


also has a high cardinality. So we can compare both the quantity and the quality oftwo alternative approximations for PE∗.

In addition to the evaluation of these goals, we compute the following performancemetric (cf. Zitzler, 1999):

De�nition 17. Given is an approximation set of portfolio structures PE1 for PE∗

de.ned in Problem 10. The maximum spread 5(PE1) is obtained by evaluation of

5(PE1) :=√

dist2ret + dist2risk ; (27)

where

distret := maxx∈PE1

(ret(x; p; r)) − minx∈PE1

(ret(x; p; r))

distrisk := maxx∈PE1

(risk(x; p; h)) − minx∈PE1

(risk(x; p; h)):

The maximum spread allows a comparison between diIerent approximation setsbased on the largest Euclidean distance between two solutions in the two-dimensionalobjective function space. We have chosen this additional metric because the set cov-erage metric does not cover the largest spread between the found solutions which isalso an important goal in multi-objective optimization. A larger spread is preferable,i.e. an approximation set PE1 is better than another set PE2 concerning this criterionif 5(PE1) ¿5(PE2). We have normalized the 5 values by 10−4 to obtain a range of06 5¡ 10 for the presentation purposes in the next section. This does not inKuencethe results.

Furthermore, we analyse the runtime behaviour of the respective Evolutionary Al-gorithm both by the number of performed target function calls and by the requiredcomputing time on the reference PC. The number of target function calls is a popu-lar criterion in many studies comparing diIerent optimization algorithms, but in ourparticular problem setting, the computational eIort that has actually to be spent toevaluate the .tness function depends heavily on the characteristics of the individual.This is mainly due to the downside risk target function which is evaluated using theCreditRisk+ algorithm. Therefore, we also measured the actual computing time whichis required to solve the given problem instance on the reference machine. On onehand, this number should not be over-interpreted because it is dependent on the refer-ence machine and on implementation details but on the other hand, it is an importantindication for real-world applications.

In the next subsection, we present the means and the standard deviations of theabove performance measures for the randomized algorithms (HMOEA and non-hybridMOEA). These were calculated from the 50 independent runs for a given probleminstance which yielded an evaluation of the members of the .nal elite population Q(t)after termination of the respective algorithm as well as the required computation timeand the number of target function calls in each run.


risk(...) objective function

ret(

...)

obje

ctiv

e fu

nctio

n

50000400003000020000100000

HMOEA enumeration

0

1000

2000

3000

4000

5000

6000

Fig. 1. Comparison of PE∗ and PE2 for portfolio m25n2. ◦, HMOEA; ×, enumeration.

4.2. Empirical results

In all test cases, the approximation set computed by the non-hybrid MOEA is denotedby PE1 and the HMOEA’s output is denoted by PE2.

First of all, for the portfolio m25n2 test data set, we compare the result PE2 of oneHMOEA run that required 33 s to PE∗ which was obtained by a complete enumerationof the search space that required 31; 292 s (≈ 8:7 h). Fig. 1 shows both results. A visualinspection reveals that PE2 is a good approximation set for PE∗ since the points of PE∗

(indicated by ‘x’) are approximated by mostly identical or at least very close points ofPE2 which are marked by a respective circle in Fig. 1. As we use the crowding-sortprocedure based on the Euclidean distance in the objective function space to selectindividuals from |J (t)| to |P(t + 1)| (cf. Section 3.1) it is not surprising that someof the solutions obtained by enumeration which are located very closely to anothersolution in the objective function space might be approximated by a single HMOEAsolution.

By comparing the HMOEA’s runtime to the upper bound given by the enumerationruntime we obtain a ratio of 33=31; 292 ≈ 0:11%. We consider this to be a very reason-able ratio in favour of the HMOEA, particularly if we look at the close approximationof PE∗ by PE2 in Fig. 1. In addition, we have to point out that each independent runof the HMOEA (as well as each run of the non-hybrid MOEA) required less than 1min for the computation of an approximation set PE2.

Table 2 shows the comparison between the HMOEA and the non-hybrid MOEA forthis small portfolio. The numbers in the ‘rel. diI.’ columns always display the relativediIerence between PE2 and PE1 based on PE1 concerning the performance criteria (alldigits of intermediate results were considered for calculating the rel. diI.).

The results in Table 2 indicate that the HMOEA computed an approximation setwhich on average contained more solutions than PE1. Moreover, the mean number ofdominated solutions within the respective approximation set was lower for the HMOEA.


Table 2Results for portfolio m25n2.

Measure Mean Standard deviation

PE1 PE2 rel. diI. (%) PE1 PE2 rel. diI. (%)

# solutions 140.86 144.84 +2:83 5.74 5.08 −11:60# dominated 12.96 10.94 −15:59 7.06 4.93 −30:17C 9.13% 7.59% −16:87 4.86% 3.57% −26:595 3.12 3.31 +6:33 0.017 0.005 −71:76# risk() calls 35063 41337 +17:89 7140 8852 +23:97runtime [s] 26.12 29.66 +13:55 5.11 6.36 +24:38

Table 3Results for portfolio m386n2.

Measure Mean Standard deviation

PE1 PE2 rel. diI. (%) PE1 PE2 rel. diI. (%)

# solutions 1472.64 1715.44 +16:49 75.36 70.98 −5:82# dominated 1374.86 62.10 −95:48 87.94 57.16 −35:00C 93.39% 3.65% −96:09 4.27% 3.44% −19:505 2.00 2.03 +1:13 0.09 0.06 −32:61# risk() calls 1199900 1297465 +8:13 343781 302633 −11:97runtime [s] 3387.96 3732.62 +10:17 941.68 849.92 −9:74

Hence, the average set coverage metric value was also better for the HMOEA. The av-erage maximum spread of the solutions within the respective resulting set was improvedby the hybridization, too.

In addition to these improvements of the average solution quality, also the observedstandard deviations of these criteria were lower for the HMOEA. This indicates thatthe solution quality was less volatile over the 50 independent runs which is of coursea desired feature of a stochastic search algorithm.

The signi.cant improvement of the criteria concerning solution quality was achievedat the cost of a moderately higher mean and standard deviation of the number of targetfunction calls due to the additional local search variation operator, and this is alsoreKected by the observed runtime diIerences.

Compared to the improvement of particularly the domination-related criteria whichare naturally very important in risk-return optimization, the additional computationaleIort required is quite small (especially if we look at the absolute runtime of about30 s on average).

In the larger test case which has a signi.cantly larger search space, we expect animprovement of the ratios between improvements of the solution quality performancecriteria and additional computational eIort to be spent due to the hybridization.Table 3 displays the results in the portfolio m386n2 test case which meet this ex-pectation.


The strong diIerences between the algorithms concerning the .rst three criteria inTable 3 are due to the following facts: The HMOEA found more solutions than theMOEA in 49 of 50 independent runs. Moreover, in each of the 50 independent runsthe number of dominated solutions in PE2 was signi.cantly lower than the number ofdominated solutions in PE1. As a consequence, the set coverage metric values weresigni.cantly better for the HMOEA in all 50 runs. These facts are reKected both bythe mean and the standard deviation of the respective criterion.

The average maximum spread was slightly improved by the hybrid approach whilethe improvement of the standard deviation of this performance measure by the HMOEAwas much better. Due to the supervisory capital constraint and the binary-style decisionvariables, the potential improvement of the maximum spread is always bounded forinstances of our Problem 10.

Looking to the additional computational eIort which was spent due to the hybridiza-tion, both the increase in the average number of objective function calls and the increasein the average runtime were fairly small and particularly smaller than in the portfoliom25n2 test case. Moreover, in contrast to the results for the small portfolio, the ob-served standard deviations of the number of objective function calls and the runtimewere even smaller for the HMOEA in the portfolio m386n2 test runs. So the relationbetween improvements of the criteria concerning the solution quality and the additionalcomputational eIort to be spent is better for the larger test case.

By relating the required average runtime for an HMOEA run to the average numberof feasible approximation solutions found, we obtain a ratio of 3732:62=1715:44 ≈2:18 s per feasible approximation solution which we consider to be a reasonable resultfor the HMOEA.

We conclude that the above results support our claim that the hybridization of theMOEA improves the convergence properties of the algorithm. Especially when dealingwith very large search spaces, the exploitation of local information around a solutionis valuable in the evolutionary process since it drives the evolutionary process fastertowards the most promising solutions. On the other hand, the other variation operatorsare also very important when using such local information since a strong local searchprocess can stall into a small number of local optima which are only a few pointscompared to a large set of feasible, Pareto-e2cient portfolio structures. So a hybridapproach is preferable.

5. Conclusion and outlook

We considered a constrained multi-objective portfolio optimization problem based onbinary decision variables which represent investment alternatives incorporating creditrisk. The aggregated net return from a portfolio and the aggregated downside riskmeasured by the Credit-Value-at-Risk were considered as objective functions of a bankalso having an additional supervisory capital budget restriction.

For the computation of a large set of feasible solutions which approximate Pareto-e2cient solutions to our portfolio optimization problem, we proposed a hybrid ap-proach that combines concepts from diIerent Multi-Objective Evolutionary Algorithm


(MOEA) schemes with a problem speci.c local search operator. This hybrid approachis not restricted to linear or convex objective functions and also Kexible concerning theconstraints. We chose the CreditRisk+ portfolio credit risk model for our implemen-tation of the algorithm and derived a local search operator that exploits model-speci.cfeatures. Furthermore, we investigated the computational complexity of diIerent ele-ments of our hybrid algorithm and provided reasonable bounds supporting the choiceof the speci.c parameters for the local search operator.

The empirical results of our implementation indicated that the convergence speedtowards the feasible, global Pareto-e2cient set could be improved by applying theadditional local search variation operator. Particularly for the large case consisting ofreal-world data from a German bank, the hybridization of the MOEA and the localsearch algorithm yielded a better quantity and a better quality of the solutions as wellas a higher spread of the solutions in the objective function space both on averageand in the vast majority of the single, independent algorithm runs. The additionalcomputational cost of the local search variation operator are low compared to theadvantages, and the user can decide about the amount of additional computational costto be invested in favour of a higher convergence speed by setting two parameters, theprobability for the application of the local search variation operator to each individualin a given population and the maximum number of local search iterations given a localsearch variation operator application.

Although our implementations were executed on a single standard desktop PC, par-ticularly the HMOEA found approximations of many feasible, Pareto-optimal solutionswithin reasonable time. Remembering the fact that Evolutionary Algorithms are wellsuited for distributed computing or parallel implementation (see e.g. Schmeck et al.,2001) there is a perspective for improving the speed of future implementations of ourapproach by using more than one CPU at least for some tasks in our hybrid algorithm.

Further research from the viewpoint of risk modelling can e.g. integrate another vari-ant of the CreditRisk+ model (cf. Buergisser et al. (2001), Gordy (2001) or Giese(2003)) or a completely diIerent portfolio credit risk model like CreditMetrics (cf.Gupton et al., 1997) into our hybrid algorithm scheme. Of course, other choices ofthe risk(: : :) objective function like Expected Shortfall, Tail Conditional Expectationand related measures (see e.g. Artzner et al. (1999) for an overview and a discus-sion) are possible. Due to the Kexibility of our algorithm, many further constraints ofpractical interest could be integrated, for instance the simultaneous use of diIerent cap-ital budgets or Credit-Value-at-Risk based limits on subsets of investment alternatives(e.g. depending on obligor speci.c criteria like country or industry) in the optimizationprocess. Even more sophisticated restrictions can be handled, e.g. a minimum overallquality of the parts of a portfolio to be sold in an Asset-Backed Security transactionwhich is itself calculated using a non-linear pricing model.

Acknowledgements

Support of this work by GILLARDON AG .nancial software, Germany and by ananonymous German bank is hereby gratefully acknowledged. The authors would like


to thank two anonymous referees and the editors of this CSDA Special Issue for theirhelpful suggestions and comments on an earlier version of this article.

Appendix A.

A.1. Proof of formula (23)

Proof. Given is an arbitrary, but .xed portfolio speci.ed by the vectors x; p; (;*; r.The function f is de.ned as follows:

f(x; p; (;*; r) :=∑m

i=1 xi(ri − pi)q�

pf (x; p; (;*) − ∑mi=1 xipi

: (A.1)

If we calculate a constant multiplier for the given portfolio data

/pf :=q�

pf (x; p; (;*) − �pf (x; p)

(pf (x; p; (;*); (A.2)

which can be abbreviated by /pf := (q�pf −�pf )=(pf in analogy to CreditSuisse Financial

Products (1997, p. 63), the �-percentile function can be reformulated by

q�pf = �pf + /pf(pf : (A.3)

By substituting the �-percentile function in formula (A.1) according to expression (A.3)we obtain

f(x; p; (;*; r) =∑m

i=1 xi(ri − pi)�pf + /pf(pf − ∑m

i=1 xipi: (A.4)

Taking into account that �pf :=∑m

i=1 xipi, formula (A.4) can be simpli.ed to

f(x; p; (;*; r) =∑m

i=1 xi(ri − pi)/pf(pf

: (A.5)

The partial derivative of f is calculated by deriving (A.5) using quotient rule:

dj :=@

@xjf(x; p; (;*; r)

=(rj − pj)(/pf(pf ) − (∑m

i=1 xi(ri − pi)) (

@@xj

(/pf(pf ))

(/pf(pf )2 (A.6)

For xj �= 0 formula (A.6) is equivalent to

xj(rj − pj)(/pf(pf ) − xj(∑m

i=1 xi(ri − pi)) (

@@xj

(/pf(pf ))

xj(/pf(pf )2

=xj(rj − pj)(/pf(pf ) − (∑m

i=1 xi(ri − pi)) (

xj@

@xj(/pf(pf )

)xj(/pf(pf )2


=xj(rj − pj)(/pf(pf ) − (∑m

i=1 xi(ri − pi)) (

/pf xj@

@xj(pf

)xj(/pf(pf )2 : (A.7)

Finally, remembering that

RC(j := xj

@(pf

@xj;

the substitution of the partial derivative yields

dj =xj(rj − pj)(/pf(pf ) − (∑m

i=1 xi(ri − pi))

(/pfRC(j )

xj(/pf(pf )2 : (A.8)

A.2. Speci4cation of portfolio m25n2

The ri and wi parameters in Table 4 are calculated in relation to ei. The variationcoe2cient in the .rst sector is !1 := 0 by de.nition (i.e. it does not incorporate

Table 4Speci.cation of portfolio m25n2.

i ei *i1 (%) *i2 (%) pi (%) ri (%) wi (%)

1 3000 9 91 2.0 3.10 8.432 7000 7 93 2.0 4.97 21.073 2000 6 94 2.0 3.63 12.904 2000 5 95 2.0 3.18 9.405 29000 2 98 2.0 4.43 11.866 6000 9 91 3.0 5.03 15.427 16000 6 94 3.0 3.62 10.988 19000 6 94 3.0 5.96 16.549 37000 6 94 3.0 4.70 16.08

10 8000 5 95 3.0 4.08 13.4911 22000 5 95 3.0 3.33 13.0612 23000 5 95 3.0 3.62 11.5213 8000 4 96 3.0 5.13 19.4014 35000 4 96 3.0 5.76 11.5115 9000 3 97 3.0 5.20 13.3116 26000 3 97 3.0 3.26 19.0717 8000 2 98 3.0 4.86 10.6418 12000 10 90 4.0 5.47 11.8019 10000 10 90 4.0 4.95 23.0920 17000 10 90 4.0 5.86 12.1721 16000 9 91 4.0 7.00 15.9022 29000 9 91 4.0 6.00 15.7123 14000 7 93 4.0 6.15 15.1524 21000 7 93 4.0 4.70 18.6225 7000 5 95 4.0 4.69 11.46


correlation between obligors, cf. our remarks in Section 3.3), and !2 := 0:75 for sector2 according to real world variation coe2cients of default rates.

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

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