A Data-Driven Optimization Heuristic for Downside Risk Minimization

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1 A Data-Driven Optimization Heuristic for Downside Risk Minimization Manfred Gilli a,,1, Evis Këllezi b, Hilda Hysi a,2, a Department of Econometrics, University of Geneva b Mirabaud & Cie, Geneva Abstract In practical portfolio choice models risk is often defined as VaR, expected shortfall, maximum loss, Omega function, etc. and is computed from simulated future scenarios of the portfolio value. It is well known that the minimization of these functions can not, in general, be performed with standard methods. We present a multi-purpose data-driven optimization heuristic capable to deal efficiently with a variety of risk functions and practical constraints on the positions in the portfolio. The efficiency and robustness of the heuristic is illustrated by solving a collection of real world portfolio optimization problems using different risk functions such as VaR, expected shortfall, maximum loss and Omega function with the same algorithm. Key words: Portfolio optimization, Heuristic optimization, Threshold accepting, Downside risk 1 Introduction Modern portfolio optimization originated with the mean-variance framework introduced by Markowitz (1952). One of the main reasons of its popularity Corresponding author: Department of Econometrics, University of Geneva, Bd du Pont d Arve 40, 1211 Geneva 4, Switzerland. Tel.: ; fax: addresses: Manfred.Gilli@metri.unige.ch (Manfred Gilli), Evis.Kellezi@Mirabaud.com (Evis Këllezi), Hilda.Hysi@metri.unige.ch (Hilda Hysi). 1 We are grateful to Algorithmics, Inc. ( for providing data and we thank Stan Uryasev and Arun Verma for personal communications. We also thank the editor, an earlier referee and Patrick Burns for their comments. 2 Supported by the Swiss National Science Foundation (project ). Article published in Journal of Risk 8(3), 2006, 1 19.

2 is the fact that it can be performed efficiently using standard quadratic programming techniques. Since then many features of the Markowitz model have been subject to a lot of criticism. Alternative approaches attempt to conform the model s assumptions to reality by, for example, dismissing the normality hypothesis in order to account for the fat-tailedness and the asymmetry of asset returns. As a consequence, other measures of risk, such as value-at-risk, expected shortfall, mean semi-absolute deviation, semi-variance and so on are used, leading to problems that cannot always be reduced to standard linear or quadratic programs. The resulting optimization problem often becomes fairly complex as it exhibits multiple local extrema and discontinuities, in particular when we introduce constraints restricting the trading variables to integers, constraints on the holding size of assets, constraints on the maximum number of different assets in the portfolio, etc. In such situations, classical optimization methods do not work efficiently and heuristic optimization techniques may be the only way out. The use of heuristic optimization techniques to portfolio selection has already been suggested in the literature. Dueck and Winker (1992) are the first to use a local search technique, called threshold accepting, to portfolio choice problems. Gilli and Këllezi (2002a,b) use threshold accepting for minimizing value-at-risk and expected shortfall, Maringer (2005) solves a broad class of portfolio management problems using heuristic optimization. Comparisons of the performance for different heuristic techniques applied to solve portfolio choice problems are given by Chang et al. (2000) and Beasley et al. (2003). As opposed to a standard optimization algorithm, the use of heuristic techniques necessitates the setting of several problem specific parameters. Despite the simplicity of the algorithmic aspect of most heuristics, in particular for local search methods, their efficiency crucially depends on the choice of appropriate values for these parameters. Without some experience, the selection of these parameters can be delicate. By the way, this might be one of the reasons of the so far limited use of heuristic optimization techniques. In the following we provide an algorithm that is suitable to solve a broad class of portfolio optimization problems, where the setting of the problem specific parameters is driven by the data. In Section 2 we give an overview of the kind of portfolio choice problems that can be solved using the heuristic. Details of the heuristic optimization method are given in Section 3. Applications of the algorithm to a variety of risk minimization problems as well as comparisons of the results with other approaches are presented in Section 4. Section 5 concludes. 2

3 2 Models for portfolio choice In the following we introduce the notation used to formalize different portfolio choice models. At time t = 0 we consider an initial wealth v 0 to be invested in a universe A of n A assets with prices p 0j, j = 1,...,n A. The quantities of assets x j, such that j J x j p 0j = v 0, constitute a portfolio with J = {j x j 0} denoting the set of indices of assets in the portfolio. At the planning horizon, chosen to be t = 1, the assets generate returns r j, still unknown at time t = 0. Portfolio choice then consists in finding an optimal allocation x j 0, or equivalently expressed in terms of weights w j = (x j p 0j )/v 0, that minimizes a particular risk function for a given return target under some additional constraints on the holding size of the assets. For a discussion of the proper use of risk measures in portfolio optimization see Ortobelli et al. (2005). In the mean-variance framework, for example, the returns r j are treated as normal random variables with mean E(r) = µ and variance and covariance matrix Σ. Risk is measured as the variance of portfolio return and, for a given return target r d, the mean-variance portfolio is obtained by solving the following quadratic program w inf j min w w Σw (1) w j µ j r d (1 ) j A j A w j = 1 (1 ) w j w sup j j A. (1 ) The vectors wj inf, w sup j, j A represent additional constraints on the minimum and maximum holding size of the individual assets in the portfolio. Generation of price scenarios In the following models the normality assumption of the returns is relaxed and the uncertainty about future returns, i.e. about the future portfolio value v, is modelled through a set of possible realizations, called scenarios. These scenarios can be generated relying on a statistical model, past returns or experts opinions. For the purpose of our analysis, the n S scenarios for the returns r sj, s = 1,...,n S, j = 1,...,n A are bootstrapped from the observed historical returns R = [r tj ]. The corresponding price scenarios for the planning period t = 1 3

4 are computed as p 1sj = p 0j (1 + r sj ), s = 1,...,n S, j = 1,...,n A, giving rise to a different portfolio value v s for each scenario: v s = j J x j p 0j (1 + r sj ) s = 1,...,n S. Downside risk framework This framework takes into consideration the fact that investors are often more concerned about losses, or the risk that their portfolio value falls below a certain target. We therefore define the losses l as l = v 0 v and write the portfolio choice problem as min x Ê n A Φ(l) (2) x inf j E(l) v 0 r d (2 ) x j p 0j = v 0 (2 ) j J x j x sup j j J (2 ) #{J} K (2 ) where Φ(l) is a function defining the risk, (2 ) is the constraint on the portfolio return with r d the desired return target, (2 ) is the budget constraint, (2 ) restricts the holding size of the assets in the portfolio and (2 ) is a cardinality constraint limiting the number of assets in the portfolio to a maximum of K. In the following we enumerate a number of possible definitions of the risk function Φ. Value-at-risk (VaR) is defined as the (1 β)-quantile of the distribution function F of losses of the portfolio, i.e. where VaR (1 β) = F 1 (1 β) β = P(l > VaR) is called the shortfall probability. In other words there is only a probability of β that losses can exceed VaR (1 β). The conditional mean value of the losses given that the losses have exceeded VaR is called expected shortfall (ES) and is defined as ES = E(l l > VaR). 4

5 Another characterization of risk is given by the ratio of the weighted conditional expectation of losses over the weighted conditional expectation of gains Ω = P(l > 0)E(l l > 0) P(l < 0)E(l l < 0). This measure is called Omega and has been introduced by Keating and Shadwick (2002) as a performance measure. Figure 1 illustrates these different risk measures for a continuous distribution f of losses and the corresponding cumulative distribution function F. The expected shortfall is given by ES = VaR + 1 β I 3 (3) where I 3 = VaR (1 F(z))dz. The integral I 1 represents P(l < 0)E(l l < 0), the weighted conditional expectation of losses, and the integral I 2 the weighted conditional expectation of gains P(l > 0)E(l l > 0). These expectations can be computed from the cumulative distribution function as I 1 = 0 The Omega is then computed as F(z) dz and I 2 = 0 (1 F(z)) dz. Ω = I 2 /I 1 (4) and from I 1 and I 2 we also recover the expected loss as 3 E(l) = I 2 I beta F I 2 I 3 = (1 F(z))dz VaR 0 I 1 f 0 VaR β = VaR f(z) dz Fig. 1. Example of a distribution f of losses and the corresponding cumulative distribution function F. As mentioned before, in order to avoid any parametric assumptions about the distribution of losses, we can use scenarios based on observed data. We 3 See (Parzen, 1960, p. 211) for details. 5

6 consider the n S simulated losses ordered such that l (1) l (2) l (ns ). The VaR 1 β can then be defined as the order statistic VaR = l ( (1 β) ns ). (5) An alternative measure of risk can be the maximum loss over all scenarios defined as Max = l (ns ) (6) or Max = max(l) for the unordered loss vector l. The expressions (3) and (4) can be evaluated in two different ways, either by computing I 1, I 2 and I 3 by numerical integration of the empirical cumulative distribution of losses or by estimating I 1, I 2 and ES as an arithmetic mean over discrete scenario losses. The second approach consists in estimating I 1 as Î 1 = P(l < 0) Ê(l < 0) 1{ls<0} 1 = ls 1 {ls<0} n S 1{ls<0} = 1 n S ls 1 {ls<0}. Correspondingly, the expression for I 2 is Î 2 = 1 n S ls 1 {ls>0}. Using expression (4), the Omega value can also be estimated as Ω = ls 1 {ls>0}. (7) l s 1 {ls<0} The expected shortfall can be computed as ÊS = 1 1{ls>VaR} ls 1 {ls>var} (8) and the expected loss as E(l) = 1 n S The functions Φ(l) defined in (3 8) quantify the risk for a given portfolio x in terms of n S scenario losses l s and are all highly non-convex. In order to illustrate the non-convex feature of the problem, we plot in Figure 2 the objective function for the VaR and Omega minimization for a portfolio n S s=1 l s. 6

7 of three equities quoted in the swiss market 4. The objective functions are evaluated based upon 800 price scenarios on a grid of points for quantities x 1 and x 2. The quantity x 3 of the third asset is determined by the budget constraint allowing short selling. The presence of multiple local minima and the non-smoothness of the surfaces indicate that classical gradient based methods cannot be used to solve problem (2) x x x x x2 x 10 4 x x x Fig. 2. Objective functions for VaR (left panel) and Omega (right panel) minimization for a portfolio of three assets. 3 The local search algorithm The non-convexities of the functions for the examples displayed in Figure 2 show the necessity of a global optimization approach. Strategies for global optimization generally resort to local search methods, e.g. simulated annealing (Kirkpatrick et al. (1983)), multiple starting points or sequential local minimization. The latter, also called smoothing/continuation method has been used by Gaivoronski and Pflug (2005) and Verma (2005) for VaR minimization. We use a modified local search method called threshold accepting (TA). The classical local search for minimization of a function f(x) is formalized in algorithm 1. Given f(x), x Ω with Ω R n the search space (possibly discrete) and a current solution x c, a new solution x n is computed in the neighborhood N(x c ) and is accepted if f(x n ) < f(x c ). The stopping criteria is generally defined as a given number of iterations. Different rules for the choice and the acceptance of the neighbor x n (statements 3 4) define a particular heuristic. If we want to escape local minima the algorithm must accept uphill moves 4 The equities are Crealogix, Swiss Steel and Swissquote. 7

8 Algorithm 1 Local search for minimization. 1: Generate current solution x c 2: while stopping criteria not met do 3: Select x n N(x c ) (neighbor to current solution) 4: if f(x n ) < f(x c ) then x c = x n 5: end while which can be achieved by modifying statement 4 as if ( ) f(x n ) f(x c ) < τ then x c = x n where τ is a threshold that is gradually reduced to zero. Modifying the local search in this way led to a heuristic called threshold accepting that was introduced by Dueck and Scheuer (1990). It can be considered as a deterministic analog for simulated annealing where uphill moves are accepted according to a probabilistic criterion. The implementation of the TA algorithm involves the definition of the objective function f, the neighborhood N(x) and the threshold sequence τ which decreases toward zero in a given number of rounds n Rounds. The objective function corresponds to the risk functions Φ(l) defined in (3 8). In each iteration we generate a new element x n in the neighborhood of the current solution x c. In the case of portfolio selection the solutions are vectors representing the positions in each asset and thus the search space Ω can be considered as a subset of a real valued vector space R n A and the concept of neighborhood can be defined using ε-spheres N(x c ) = {x n x n Ω, x n x c < ε} where is a distance measure. The neighbor solutions are generated with Algorithm 2. Algorithm 2 Generation of a neighbor solution. 1: Randomly select asset i J 2: Sell quantity q i of asset i 3: Randomly select asset j A, j i 4: Buy quantity q j of asset j The number of assets q i and q j bought and sold in instructions 2 and 4 in the algorithm 2 are integers and have to be adjusted in order to satisfy the constraints on the holding size (2 ). If the neighbor solution computed with Algorithm 2 violates constraint (2 ) we add the penalty term c max( v 0 r d + E(l), 0) to the objective function, where c is a positive scaling parameter. Althöfer and Koschnick (1991) proved the convergence of the TA algorithm given an appropriate threshold sequence. Their proof however does not al- 8

9 low the construction of the appropriate sequence. In practice the threshold sequence is retrieved from the empirical distribution of n Steps distances between objective function values for successive neighbors. This procedure is detailed in Algorithm 3. Algorithm 3 Generation of threshold sequence. 1: Randomly choose x c Ω 2: for i = 1 to n Steps do 3: Compute x n N(x c ) and i = f(x c ) f(x n ) 4: end for 5: Compute empirical distribution F of ( i, i = 1,...,n Steps 6: Compute threshold sequence τ r = F n Rounds r n Rounds ), r = 1,...,n Rounds The TA algorithm is a so called trajectory method as the current solution is slightly modified by searching within its neighborhood. In order to explore the solution space more efficiently one may restart the algorithm n Restarts times from randomly chosen points in the search space and then take the best solution out of all restarts. Algorithm 4 resumes the complete optimization procedure. Algorithm 4 TA algorithm with restarts. 1: Initialize n Restarts, n Rounds and n Steps 2: Compute threshold sequence τ r (Algorithm 3) 3: for k = 1 : n Restarts do 4: Randomly generate current solution x c X 5: for r = 1 to n Rounds do 6: for i = 1 to n Steps do 7: Generate x n N(x c ) and compute = f(x n ) f(x c ) 8: if < τ r then x c = x n 9: end for 10: end for 11: ξ k = f(x c ), x sol (k) = xc 12: end for 13: x sol = x sol (k), k ξ k = min{ξ 1,...,ξ nrestarts } 4 Applications The TA algorithm has been implemented 5 in Matlab 7.xx and the restarts are executed in a distributed computing environment with 32 PCs. To illustrate its performance we present two applications of portfolio choice minimizing the risk functions discussed in section 2. The first application uses daily observations from 30 June 2002 to 30 June 2005 of the 213 stock prices forming the Swiss Performance Index (SPI). The second application considers a large problem 5 The code can be obtained upon request from the corresponding author. 9

10 of bond portfolio optimization where the future portfolio values, due to credit migration, are given by a set of scenarios. 4.1 Equity portfolio optimization We use daily observations from 30 June 2002 to 30 June 2005 of the 213 stock prices forming the Swiss Performance Index (SPI) and consider a planning period of one month. We construct a set of return scenarios by bootstrapping n S = 800 overlapping blocks of length 20 from the set of 736 daily returns. The sum of the log-returns of each block defines a monthly log-return scenario. The same set of return scenarios is used to compute mean-variance and meandownside risk optimal portfolios. Benchmarking the TA in the mean-variance framework In order to provide a first evidence about the reliability of TA solutions we compute the mean-variance efficient frontier by solving the quadratic program (1) and compare it with the solutions obtained with the TA algorithm. Constraint (1 ) has been set to 0 ω 1, in order to make it tractable by the QP algorithm. The covariance matrix Q and the mean return vector r are estimated from the 800 return scenarios. In Figure 3 we reproduce the efficient frontier for mean-variance portfolios computed with QP and TA for monthly return targets ranging from 0.77% to 3.86%. The TA solution has been computed with 60 restarts with 10 rounds of steps. Figure 4 compares the portfolio weights for a particular portfolio on the frontier. 6 The similarity of the two frontiers and the portfolio compositions confirm the quality of the TA solutions. 6 For a better readability positions inferior to 0.5% have been removed. 10

11 3.5 3 Monthly return in % QP TA Monthly volatility in % Fig. 3. Mean-variance efficient frontier computed with QP and TA QP TA Fig. 4. Mean-variance portfolio weights computed with QP and TA for a target expected return of 0.77% per month. The numbers refer to individual stocks in the SPI index. Downside risk minimization For the same set of scenarios, we compute the efficient frontiers of portfolios minimizing thevar, thees and the Ω for a given return target. We also give the ES corresponding to the VaR-optimized portfolios and the VaR corresponding to the portfolios optimized with respect to ES. We specify an initial wealth of v 0 = 10 7 and the holding sizes as 0.01v 0 x j 0.30v 0, j J. This constraint implies that the portfolio weight should be at least 1% and at most 30% for all the assets that are held. Moreover we constrain the cardinality of the set of assets in the portfolio to be at most 30. The TA algorithm is executed with 60 restarts of 10 rounds and 5000 steps. Figure 5 reproduces the efficient frontiers obtained for VaR, ES and Ω minimization. The frontier obtained for portfolios minimizing VaR and their corresponding ES are marked with a circle, the portfolios minimizing ES and their corresponding VaR with a star and the triangles mark the VaR and ES corresponding to the portfolios optimizing Ω. The cardinality constraint is active only for the first portfolio in the mean-var frontier. The number of assets 11

12 varies from 16 to 30 for the mean-var frontier, from 9 to 22 for the mean-es and from 10 to 27 for the mean-omega frontier. In general, we remark that mean-var portfolios are more diversified than those satisfying the mean-es criteria. These results show that minimizing VaR or ES is not equivalent in terms of tail risk efficiency. For example, for a return target of 2.26% per month, the minimum ES that can be achieved is 1.79%. Minimizing VaR instead of ES would yield a portfolio offering an expected shortfall of 4.68%, more than twice as large as the minimum achievable. 3.5 Monthly expected return in % VaR opt ES VaRopt ES opt VaR ESopt VaR Ω 1.5 ES Ω Monthly loss risk in % Fig. 5. Efficient frontiers for mean-var.95, mean-es.95 and mean-ω portfolios. In Figure 6 we reproduce the empirical distribution for the different efficient portfolios with a monthly return target of 2.26%. The VaR-optimal portfolio is composed of 29 assets, thees-optimal of 15 and the Ω-optimal of 17. Excepting one asset in the ES-optimal portfolio, all asset weights are less than 15%. The right panel shows the distribution for losses and we observe that, for this data set, the ES minimization strategy dominates the other strategies with respect to extreme losses. This at the cost of higher probabilities for smaller losses. 12

13 VaR ES Ω Monthly losses x VaR 0.85 ES Ω Monthly losses x 10 5 Fig. 6. Empirical cumulated distribution of monthly losses for mean-var.95, mean-es.95 and mean-ω portfolios with 2.26% monthly return target. It is interesting to observe that the portfolio minimizing Ω dominates the VaR strategy for almost the entire distribution of losses. The left panel shows the distribution of gains for the different portfolios. 4.2 Bond portfolio optimization In order to further assess the efficiency of the TA algorithm we apply it to a bond portfolio optimization problem introduced by Bucay and Rosen (1999) and Mausser and Rosen (1999). The portfolio is constructed from a set of bonds issued by n A = 80 obligors with a mark-to-market value of 8.8 billion USD. We denote x = [x 1,...,x na ] the vector of positions of the obligors in the portfolio expressed as multiples of the initial holdings, b = [b 1,...,b na ] the mark-to-future value of the instruments if no credit migration occurs and y s = [y s1,...,y s,na ], s = 1,...,n S the scenario values of joint credit states due to events such as default and credit migration. The simulated losses are then l s = n A i=1 x i (b i y si ) s = 1,...,n S. (9) Figure 7 shows the right tail of the empirical distribution of n S = scenarios of the one-year credit losses (9) corresponding to a portfolio with initial holdings x = [1,...,1]. The VaR and expected shortfall (ES) at the 95% and 99% percentiles are VaR.95 = 518 and ES.95 = 824, respectively VaR.99 = and ES.99 = The maximum loss is The credit risk optimization problem for this portfolio has been approached in different ways in the literature. Bucay and Rosen (1999) applied the Cre- 13

14 Fig. 7. Empirical distribution of one-year credit losses (million USD) for the initial positions x = [1,...,1]. ditmetrics 7 methodology, Mausser and Rosen (1999) the regret optimization framework, Andersson et al. (2001) minimize the expected shortfall (called also CVaR) and provide the corresponding VaR and Larsen et al. (2002) suggest a heuristic approach where VaR is minimized by constraining a sequence of CVaR solutions 8.We apply the optimization model (10) for different specifications of the risk function Φ(l) and solve it with the threshold accepting heuristic. min x Ê n A Φ(l) (10) l s = n A i=1 j J x i (b i y si ) s = 1,...,n S (10 ) x j b j n A i=1 b i (10 ) 0 x j 2 j J (10 ) Constraint (10 ) maintains the future portfolio value and constraint (10 ) avoids unrealistic positions in any obligor. Verma (2005) considers the same data set and computes the portfolio minimizing the VaR (without an upper bound for constraints (10 )) using smoothing methods. The settings for the TA algorithm are 64 restarts, 10 rounds and 2000 steps. The computing times, for the distributed execution on 32 Pentium 4 PCs, range from 3 to 7 minutes. For the particular portfolio minimizing VaR.99, Figure 8 shows the distribution of the 64 TA solutions and the obligor weights in the best solution. Constraint (10 ) is stringent for 24 out of the 73 positions in the portfolio. We observe that the 64 solutions lie in a relatively narrow range, confirming again the good functioning of TA. Table 1 and Figures 9 and 10 summarize the results obtained by solving (10 10 ) for different functions Φ. 7 c.f. RiskMetrics Group (1996). 8 Without the cardinality constraint, the CVAR minimization can be formulated as a linear program, see Rockafellar and Uryasev (2000). 14

15 Fig. 8. VaR.99 minimization with upper size constraints 0 x j 2. Upper panel: Empirical distribution of the 64 TA solutions. Lower panel: Ordered obligor weights in the best solution portfolio. Table 1 Optimization results for different risk functions Φ and constraint 0 x j 2. β Φ VaR ES Max Ω #J VaR minimization 0.05 (5) (5) ES minimization 0.05 (8) (8) (3) Max minimization 0.01 (6) Omega minimization 0.01 (4) VaR ES Max 0.95 Omega Fig. 9. Right tails of loss distribution for portfolios minimizing VaR.95, ES.95, Max and Omega. Looking at Figure 9 we observe that the portfolio minimizing the expected shortfall almost completely dominates the other portfolios in the right tail, i.e. has smaller losses with higher probability. If we look at the center of the 15

16 distribution, illustrated in Figure 10, we see that the advantage of the Omega portfolio is its significantly higher probability to produce gains VaR ES Omega Fig. 10. Center of loss distribution for portfolios minimizing VaR.95, ES.95, Max and Omega. The results for portfolios without an upper constraint on the positions, i.e. solving (10 10 ), are given in Table 2. The minimum VaR.99 portfolio weights obtained without upper size constraint on positions are shown in Figure 11. In fact we observe that the portfolio is concentrated in only 34 positions. Table 2 Optimization results for different risk functions Φ and constraint x j 0. Column Φ refers to the equation number. β Φ VaR ES Max Ω #J VaR minimization 0.05 (5) (5) ES minimization 0.05 (3) (3) Max 0.01 (6) Omega minimization 0.01 (4) Fig. 11. Ordered weights for the minimum VaR.99 portfolio without upper size constraint on positions. For the expected shortfall minimization our results coincide with the ones obtained by Andersson et al. (2001), whereas for the VaR.95 minimization we achieve a solution of VaR.95 = 20 instead a value of 94 reported by Verma (2005). Max 16

17 In these applications, we used both ways of computing expected shortfall, Omega and expected loss. The first one, consisting in computing numerically the integrals I 1, I 2 and I 3, can be useful when the loss distribution is defined as a parametric function. In our case, the results based on the numerical integration of the empirical distribution function of scenarios are not significantly different from the the results obtained using arithmetic means. Also, the computational efficiency is not affected by the choice of a particular method. The comments about the relative attractiveness of VaR, expected shortfall or Omega function in this sections should not be interpreted as general conclusions about their usefulness in portfolio optimization. They concern the particular data sets used and general statements would require further investigation. 5 Conclusions Threshold accepting is an optimization heuristic that can be used for a wide class of portfolio choice problems where the objective function is non-convex and has many local minima. This is in particular the case when the risk is expressed as VaR, expected shortfall, Omega, maximum loss etc., and when the future returns of the individual assets are modelled as scenarios. The method has been successfully tested on a set of large real world problems and has been benchmarked with results provided in the literature. We compare portfolios optimized for different risk measures highlighting the features of the different risk functions. A major advantage of the proposed method is its flexibility. The same tool can be used for all sorts of risk functions, side constraints or assumptions on returns. Furthermore, the algorithm parameters are driven by the problem data themselves, which implies that there is no need for particular expertise and makes its use almost as simple as a classical method. References Althöfer, I. and Koschnick, K.-U. (1991). On the Convergence of Threshold Accepting. Applied Mathematics and Optimization, 24: Andersson, F., Mausser, H., Rosen, D., and Uryasev, S. (2001). Credit risk optimization with Conditional Value-at-Risk criterion. Mathematical Programming, 89(2): Beasley, J., Meade, N., and Chang, T.-J. (2003). An evolutionary heuristic for the index tracking problem. European Journal of Operational Research, 148:

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19 Verma, A. (2005). VaR optimal portfolios. A Global Optimization Approach. Workshop on Optimization in Finance, Coimbra. 19