A comparison of MAD and CVaR models with real features

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Journal of Banking & Finance 32 (2008) 1188–1197 www.elsevier.com/locate/jbf

A comparison of MAD and CVaR models with real features Enrico Angelelli a,*, Renata Mansini b, M. Grazia Speranza a b

a Dipartimento Metodi Quantitativi, University of Brescia, Italy Dipartimento di Elettronica per l’Automazione, University of Brescia, Italy

Received 31 October 2005; accepted 12 July 2006 Available online 5 October 2007

Abstract In this paper we consider two different mixed integer linear programming models for solving the single period portfolio selection problem when integer stock units, transaction costs and a cardinality constraint are taken into account. The first model has been formulated by using the maximization of the worst conditional expectation as objective function. The second model is based on the maximization of the safety measure corresponding to the mean absolute deviation. Extensive computational results are provided to compare the financial characteristics of the optimal portfolios selected by the two models on real data from European stock exchange markets. Some simple heuristics are also introduced that provide efficient and effective solutions when an optimal integer solution cannot be found in a reasonable amount of time. Ó 2007 Elsevier B.V. All rights reserved. JEL classification: C61; G11 Keywords: Portfolio optimization; Worst conditional expectation; CVaR; MAD; Real features; Mixed integer linear programming

1. Introduction Portfolio selection problems have been largely discussed both in a deterministic and in a stochastic domain, either in a single period or in a multi-period framework. In the single period portfolio theory, the main contribution is due to Markowitz (1952, 1959) who first formalized the portfolio selection problem as a mean-risk bicriteria optimization problem where the expected return is maximized and the variance is minimized. Whereas the original Markowitz model is a quadratic programming problem, since (Sharpe, 1971) many models have been proposed with linear formulations for the case of discrete random variables. Linear programming (LP) computable risk measures are computationally very attractive and thus of crucial importance in solving complex real-life financial applications. In the last decades, many LP solvable portfolio optimization models have been analyzed in the literature making

*

Corresponding author. Tel.: +390302988583; fax: +390302400925. E-mail address: [email protected] (E. Angelelli).

0378-4266/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2006.07.015

their classification and comparison a critical issue. In Mansini et al. (2003), the authors have introduced a systematic overview of all such LP solvable models providing a wide discussion of their theoretical properties and classifying them with respect to the types of risk or safety measures. In this paper, we analyze the problem of an investor who decides, in a single period framework, to allocate his/her capital on securities by assigning a nonnegative weight (share of the capital) to each security, while meeting some necessary or desired conditions on the investment (portfolio real features). In particular, we assume that the securities can be bought only in multiples of a minimum lot as usually imposed by the Market Stock Exchange (this is the case for several European markets as well as for the Japanese one) and that fixed and proportional transaction costs have to be paid for any selected security. Moreover, as it frequently happens in practice, since the investor prefers to manage a portfolio with few securities, we introduce a maximum threshold on the number of securities to be selected in the optimal portfolio (cardinality constraint). The main objective is to provide a computational analysis of portfolio selection models that take into account the

E. Angelelli et al. / Journal of Banking & Finance 32 (2008) 1188–1197

described real features. At this aim, among all LP solvable models, we have decided to focus our attention on two performance measures. The first one is a well known measure of risk frequently used in portfolio optimization problems, i.e. the mean absolute deviation (MAD), proposed by Konno and Yamazaki (1991). The second measure is the worst conditional expectation as defined in Mansini et al. (2003) which represents the portfolio mean return under a specified size (quantile) of worst realizations. The two chosen performance measures are among the most extensively studied measures in portfolio optimization theory and practice. Whereas the MAD is known as a risk measure to be minimized, the worst conditional expectation is a safety measure, to be maximized. It has been shown (see Mansini et al., 2003) that for each risk measure there exists a well-defined safety measure and vice versa. In order to make the two models comparable, we have considered the safety measure corresponding to the MAD known as mean downside underachievement which, for simplicity, we will still refer to as MAD. In order to make the paper self-contained, we briefly recall some basic concepts on the worst conditional expectation and on MAD measures. The corresponding models with additional real features on minimum lots, transaction costs and a cardinality constraint are discussed in Section 2. Let J ¼ f1; 2; . . . ; ng be the set of securities available for the investment. The rate of return for each security j 2 J is represented by a random variable Rj with a given mean rj ¼ EfRj g. Further, let wj ; j ¼ 1; 2; . . . ; n; denote the decision variables expressing the weights defining the portfolio. In the continuous case (i.e. in the case without minimum lots) portfolio weights represent the fractions of capital invested in the different securities and their sum has to be equal to one. Moreover, no short sales are usually allowed, thus wj P 0 for j ¼ 1; . . . ; n. Each portfolio w defines a corresponding random variPn able Rw ¼ j¼1 Rj wj that represents the portfolio rate of return. The mean rate Pnof return for portfolio w is given by lðwÞ ¼ EfRw g ¼ j¼1 rj wj . Hence, the mean rate of return is a linear function of portfolio w. We consider T scenarios with probabilities pt (where t ¼ 1; . . . ; T ) and assume that for each random variable Rj its realization rjt under scenario t is known. The problem of scenarios generation is out of the scope of this paper. Thus, we take the most recent T historical periods as equally probable scenarios and set pt ¼ 1=T ; t ¼ 1; . . . ; T . The realizations of the Pn portfolio return Rw are thus given by lt ¼ j¼1 rjt wj and the portfolio expected value can be computed as: " # T n T X X X EfRw g ¼ rjt wj pt ¼ pt lt : t¼1

j¼1

t¼1

Recently, (Young, 1998) proposed an LP portfolio optimization model based on the maximization of the worst portfolio realization. Given the portfolio realization lt under scenario t; then the worst realization is defined as MðwÞ ¼ mint¼1;...;T flt g. If, instead of the worst case scenario, we generalize the measure to the mean of a specified

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size (quantile) of worst realizations, then we obtain the worst conditional expectation. For the simplest case of equally probable scenarios (pt ¼ 1=T ), one may define the worst conditional expectation M Tk ðwÞ as the portfolio mean return under the k worst scenarios. In general, the worst conditional expectation for any real tolerance level 0 < b 6 1 is defined as: Z 1 b ð1Þ M b ðwÞ ¼ F ðaÞda; ð1Þ b 0 w where F wð1Þ ðpÞ ¼ inffg : F w ðgÞ P pg is the left-continuous inverse of the cumulative distribution function of the rate of return F w ðgÞ ¼ PfRw 6 gg: For any 0 < b 6 1, the conditional worst realization M b ðwÞ is a Second degree Stochastic Dominance (SSD) consistent measure, while M b ðwÞ is coherent in the sense of Artzner et al. (1999) (for details see Mansini et al., 2003). Note that M 1 ðwÞ ¼ lðwÞ and M b ðwÞ tends to MðwÞ when b tends to 0. For a discrete random variable represented by its realizations lt , the worst conditional expectation is LP computable as: ( ) T 1X d t pt M b ðwÞ ¼ max g  b t¼1 s:t: d t P g  lt ;

d t P 0;

for t ¼ 1; . . . ; T ;

ð2Þ

where g is an auxiliary unbounded variable representing the b-quantile, while variable d t ¼ maxf0; g  lt g measures the deviation of portfolio realization lt from g when lt < g: For details on the formulation of the worst conditional expectation as an optimization problem, see Mansini et al. (2003) and references therein. The worst conditional expectation is closely related to the measure called Conditional Concentration (Shalit and Yitzhaki, 1994), Expected Shortfall (Embrechts et al., 1997) or Conditional Value-at-Risk (CVaRb ðwÞ) (Rockafellar and Uryasev, 2000). More precisely, M b ðwÞ is equivalent to CVaRb ðwÞ in the case of continuous distributions of returns, while they can take different values for discrete distributions (see Ogryczak and Ruszczyn´ski, 2002). Since recently (see Rockafellar and Uryasev, 2000) the LP formula of the worst conditional expectation has been used by models for portfolio optimization as a computational approximation to CVaR for continuous distributions, we have decided to refer to the worst conditional expectation as the CVaR model. For a further study on the worst conditional expectation model and its variants see also (Mansini et al., 2007). The second measure of risk we study is the mean absolute deviation (EfjRw  lðwÞjg) introduced by Konno and Yamazaki (1991) (MAD model). More precisely, we consider the downside mean semi-deviation from the mean, i.e. dðwÞ ¼ EfmaxflðwÞ  Rw ; 0gg. The downside mean semi-deviation is always equal to half of the mean absolute deviation from the mean, i.e. EfjRw  lðwÞjg ¼ 2 dðwÞ (see Kenyon et al., 1999). Thus, as shown by Speranza (1993), the corresponding mean-risk model is equivalent to the MAD model with half the number of constraints. For a

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discrete random variable represented by its realizations, the mean semi-deviation  dðwÞ is a convex P piecewise linear T function of the realizationsPlt , given by t¼1 maxflðwÞ lt ; 0gpt . Hence, since lt ¼ j2J rjt wj , the mean semi-deviation is also a convex piecewise linear function of the portfolio w itself and is LP computable as: T X  dðwÞ ¼ min pt dt subject to t¼1

dt P lðwÞ  lt ;

dt P 0;

t ¼ 1; . . . ; T :

ð3Þ

The corresponding safety measure, known as mean downside underachievement, and from now simply referred to as MAD, is given by: lðwÞ   dðwÞ ¼ EflðwÞ  maxflðwÞ  Rw ; 0gg ¼ EfminfRw ; lðwÞgg and is LP computable as follows (see Mansini et al., 2003 for details): ( ) T X X max rj wj  pt dt j2J

t¼1

subject to dt P lðwÞ  lt ;

dt P 0;

t ¼ 1; . . . ; T :

ð4Þ

In real-life financial applications where investors look for portfolios meeting different real features such as minimum transaction lots and transaction costs, the LP solvability of the model becomes a crucial aspect. Even if the solution of the quadratic Mean-Variance model has been tackled in presence of buy-in thresholds, cardinality constraints and transaction lot restrictions (see Chang et al., 2000; Jobst et al., 2001), the computational challenge of solving large real portfolio problems has justified a long tradition in the literature of mixed integer LP models for portfolio selection with real features (see Bertsimas et al., 1999; Chiodi et al., 2003; Kellerer et al., 2000; Konno, 1990; Konno and Wijayanayake, 2001; Mansini and Speranza, 1999; Mansini and Speranza, 2005). In this paper, we aim at providing a financial and computational comparison of MAD and CVaR models with real features analyzing their performance on real size instances. This will be achieved by means of simple and effective heuristics to be used when integer optimal solutions cannot be found in a reasonable amount of time. Moreover, we provide theoretical results on the complexity of the models and comments on the relation between the optimal integer portfolio and the portfolio obtained through the solution of the corresponding continuous model (obtained by removing integrality conditions on the variables). The paper is organized as follows. In Section 2 the portfolio selection models with real features are described. In Section 3 the computational complexity of finding a feasible solution to the models is shown, the heuristic algorithms are introduced and the relation between the optimal integer solution of the models and the optimal solution of the corresponding continuous models is discussed. In Section 4 the computational results obtained

by solving the mixed integer models on real instances are described and compared. Some interesting financial conclusions are also drawn through ex-post analysis. In Section 5 conclusions and future developments are drawn.

2. The models with real features In this section we formulate the MAD and CVaR portfolio optimization models taking into account real features and side constraints. Let qj ; j 2 J ; be the quotation of security j at the date of portfolio selection, C 0 and C 1 be the lower and the upper bound on the capital available for the portfolio investment while parameter l0 represents the minimum required expected portfolio return. Each security can be bought only in terms of an integer number of stocks (minimum lot requirement). Let xj , j 2 J ; denote the decision variable representing the number of stock units selected for security j in the portfolio. We assume that a fixed cost fj and a proportional cost cj have to be paid if any positive amount is invested in security j: Moreover, the total proportional cost to be paid for security j cannot be lower than a given amount K j : This means that when the total amount of the proportional cost that has to be paid for security j is strictly lower than K j , the investor is obliged to pay K j : To correctly model such transaction costs we assume that when investing in a security j the investor will always buy a number of stock units xj such that cj qj xj P K j : Since variables xj are integer, the following lower bound is introduced for each security j 2 J : & ’ Kj lj ¼ : cj qj Moreover, since the total capital invested cannot exceed C 1 the following upper bound applies to the maximum number of units for each security: $ % C1 uj ¼ : qj Given the quantile size b; 0 < b 6 1, the CVaR model with real features can be formulated as follows: T 1X p dt b t¼1 t n n X X u ðrjt  cj Þqj xj þ fj zj 6 d t t ¼ 1;. ..;T

maximize u 

j¼1 n X

ðrj  cj Þqj xj 

j¼1 n X

ð5Þ ð6Þ

j¼1 n X

fj zj P l0

j¼1

zj 6 m

n X

qj x j

ð7Þ

j¼1

ð8Þ

j¼1

lj zj 6 xj 6 uj zj j ¼ 1;.. .;n

ð9Þ

E. Angelelli et al. / Journal of Banking & Finance 32 (2008) 1188–1197 n X

qj x j 6 C 1

ð10Þ

d t P 0 t ¼ 1;.. .;T

ð11Þ

xj P 0 integer j ¼ 1; ...; n

ð12Þ

zj 2 f0;1g j ¼ 1;.. .;n:

ð13Þ

C0 6

j¼1

Variable u is an independent free variable which at optimality represents the value of the b-quantile. Binary variable zj , j 2 J ; is set equal to 1 when the corresponding security is selected in the portfolio and 0 otherwise. The objective function expresses the maximization of the difference between the b-quantile u and the worst downside deviation. The set of constraints (6) along with nonnegativity conditions (11) defines each variable d t as the deviation of the portfolio realization y t from u when y t < u and zero Pn otherwise, i.e. d ¼ maxf0; u  y g; where y ¼ t t t j¼1 ðrjt  Pn cj Þqj xj  j¼1 fj zj is the portfolio realization under scenario t when fixed and proportional costs are taken into account (net portfolio realization). Constraint (7) imposes that the net portfolio mean return has to be greater than or equal to the required return l0 applied to the portfolio investment. Constraint (8) expresses the cardinality condition by imposing that the number of securities selected in the portfolio cannot be greater than m. If a security j is selected in the portfolio, i.e. xj > 0; then constraint (9) forces the number of stock units for security j to be included between lj and uj : Constraint (10) says that the capital invested in the portfolio must be included between the lower bound C 0 and the upper bound C 1 : Finally, constraints (12) and (13) are nonnegative and binary conditions. The MAD model with real features, in the safety version, can be formulated as follows: maximize

n X

ðrj  cj Þqj xj 

j¼1

dt þ

n X

ðrjt  rj Þqj xj P 0

n X

fj zj 

T X

j¼1

t ¼ 1; . . . ; T

pt dt

ð14Þ

t¼1

ð15Þ

j¼1

constraints (7)–(10), (12) and (13) dt P 0

t ¼ 1; . . . ; T ;

ð16Þ

where all parameters and variables have the already explained meaning. Notice that, since the fixed and proportional costs are both applied to the mean return portfolio and to each of its realizations, then variable dt ; t ¼ 1; . . . ; T ; which represents the portfolio absolute semideviation under scenario t, does not depend on the costs. 3. Properties and algorithms It has been shown (see Mansini and Speranza, 1999) that the problem of identifying a feasible solution to a portfolio optimization problem with minimum transaction lots only is NP-complete. Moreover, in Kellerer et al. (2000), the authors have shown that finding a feasible solution for a problem with fixed and proportional transaction costs only

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is NP-complete. The results shown hold for portfolio optimization problems with any risk or safety performance measure. The problem of finding a feasible solution to the CVaR (and MAD) model with real features includes, as special case, each of the feasibility problems above mentioned. Therefore, the following result follows. Proposition 1. The problem of finding a feasible solution to the CVaR and MAD models with real features is NPcomplete. Therefore, it is very unlikely that efficient (i.e. polynomial time) exact procedures to solve such problems exist. This is the reason why we propose heuristic procedures. Our scope is to design heuristics that have some important characteristics that make them of theoretical and practical value. First of all, the heuristics should be efficient, that is they should require a reasonable amount of time to solve problems of realistic size. Secondly, the heuristics should be effective, that is they should find portfolios”almost” as good as the optimal portfolio. Finally, the heuristics should be simple, that is easy to implement and generalize to variants of the models we discuss here and to other portfolio optimization models. We call continuous relaxation of the CVaR (and MAD) model the same model with the only difference that the integrality conditions are relaxed, that is the constraint xj P 0 integer, is substituted by the constraint xj P 0 and the constraint zj 2 f0; 1g is substituted by the constraint 0 6 zj 6 1. The optimal solution of the continuous relaxation of the CVaR and MAD models can be efficiently found by means of a standard commercial software for the solution of linear programming models, such as CPLEX. The time required is very small even on problems of realistic size. This is the reason why we propose heuristics that take advantage of the optimal solutions of the continuous relaxation. The basic idea of such heuristics is that a positive value of a variable xj in the optimal solution of the continuous relaxation identifies a certainly interesting security. Other interesting securities are those with 0 value but small reduced cost. Then, we discard the non-interesting securities and the set of interesting securities is taken as the only set on which we solve the CVaR (and MAD) model with real features. The size of the models becomes in this way much smaller and the optimal solution can be obtained by means of a commercial software in a reasonable time. Heuristic H ð0Þ considers as interesting securities only those that have a positive value (xj > 0) in the optimal solution of the corresponding continuous relaxation model. Heuristic H ðmÞ extends the subset of interesting securities considered in H ð0Þ by adding the m securities that correspond to those variables xj which are null (xj ¼ 0) and have the smallest (in absolute value) reduced costs in the solution of the linear relaxation. The value m is the parameter representing the maximum number of securities allowed in the portfolio, due to the cardinality constraint. Such a

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choice is arbitrary, but follows the idea that the more securities are allowed in the optimal portfolio, the more securities should be taken into account. Any positive integer value might substitute m in H ðmÞ. Let J LP be the set of securities selected by the continuous relaxation of the MAD model and let J IP be the set of securities selected by the optimal solution of the MAD model with real features. It can be shown that there is no guarantee that J IP  J LP . In general, it may happen that a security discarded by the continuous relaxation solution appears in the optimal solution of the model with real features and that an interesting selected security does not. Proposition 2. In general, J IP is not included in J LP . Proof. Consider the following problem instance. Let J consist of two securities only, T ¼ 2 and m ¼ 2. Let C 0 ¼ 100; C 1 ¼ 101 and l0 ¼ q while fixed and proportional transaction costs are assumed to be zero for all securities, i.e. cj ¼ 0 and fj ¼ 0; j ¼ 1; 2: Security 1 has r11 ¼ r12 ¼ r1 ¼ q þ ; with  > 0; and a minimum lot value q1 ¼ 50: Security 2 has r21 ¼ r22 ¼ r2 ¼ q; and a minimum lot value q2 ¼ 1. The optimal solution of the continuous relaxation model selects security 1 with a fractional number of lots equal to x1 ¼ 101=50 ¼ 2:02 and an investment amount equal to 101. The objective function value is equal to 101ðq þ Þ, that is the expected portfolio return. On the contrary, the optimal portfolio of the MAD model with real features includes both securities. The optimal solution is x1 ¼ 2 and x2 ¼ 1 for a total investment equal to 2  50 þ 1 ¼ 101 and an objective function value equal to 100ðq þ Þ þ q. h Similar results can be proved for the CVaR model. In spite of this negative result, we will see in the next section that the heuristics are efficient and effective. 4. Experimental analysis This section is devoted to the computational analysis of the described models in a real-life decision environment. The models have been solved with CPLEX(R) 9.0 on a AMD Athlon(TM) XP 2000+. We provide and comment only the main results of the extensive computational analysis performed. Interested readers can refer to Angelelli et al. (2005) for more details on the computational results. Since even finding a feasible solution for these problems may be computationally cumbersome, large size instances may require huge computational time to be solved to optimality. For this reason we have decided to set to 1 hour the maximum running time allowed to CPLEX routine. When the time limit is reached the best solution found is provided. The CVaR model with real features has been solved for three different values of b, i.e. b ¼ 0:05 (a strong downside risk aversion), b ¼ 0:25 and b ¼ 0:5 (corresponding to the median), respectively referred in the following tables as

CVaR(0.05), CVaR(0.25) and CVaR(0.5) model. All these models have been compared to the MAD model. The historical realizations consist of weekly rates of return over the period (1999–2001). The rates of return have been computed as relative variations of the quotation prices without considering dividends. We have tested the models and their continuous relaxations as well as the performance of the proposed heuristics by using real data from Milan, Paris and Frankfurt Stock Exchanges. The in-sample analysis has been conducted on the first two years (T ¼ 104), while the last year has been used as outof-sample period. Different instances have been created according to the number of securities used (four different sets have been created containing n ¼ 200; 300; 400 and 600 securities, respectively) and the number of securities allowed in the portfolio (parameter m for the cardinality constraint has been set to 10 and 20). In terms of markets composition the first set contains 152 Italian and 48 French securities. The second set consists of 152 Italian and 148 French securities, while the third set with n ¼ 400 contains 152 Italian, 224 French and 24 German securities. Finally, the last set contains 152 Italian, 224 French and 224 German securities. We have assumed to solve the portfolio optimization problem for an Italian investor who will pay lower commissions when buying Italian stocks with respect to foreign ones. Thus, fixed transaction costs have been set to 5 Euro for all the securities, i.e. fj ¼ 5 8j; while proportional costs cj are equal to 0.7% for each Italian security and to 0.9% for each foreign (French or German) security. The fixed amount of transaction commission K j ; used to compute the lower bound on stock units investment for each security, has been set to 12 Euro for the Italian market and to 50 Euro for French and German ones. The capital range has been taken from C 0 ¼ 100; 000 to C 1 ¼ C 0 ð1 þ 1%Þ Euro, while the minimum required return has been fixed to l0 ¼ 0% (Maximum Safety Portfolio – MSP). In the following we present in and out-of-sample analysis of the portfolios obtained by the models and results on the performance of the heuristics. 4.1. In-sample analysis Table 1 provides a detailed comparison between integer and relaxed portfolios composition for the case in which the maximum number of securities allowed in the portfolio is set to 20. For sake of synthesis we have omitted the results obtained when the maximum number of allowed securities is 10. The substantial difference is in the computing times required for the exact solution which is, on average, larger for instances with m ¼ 10. The table is divided into three parts. The first part (columns 1–3) contains the name of the model solved (column 1) and the value of the main parameters of the instance, namely the number n of the securities (column 2) and the threshold m on the number of securities which can be

E. Angelelli et al. / Journal of Banking & Finance 32 (2008) 1188–1197

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Table 1 Optimal solutions: continuous relaxation vs. integer solutions with maximum cardinality equal to 20 Model parameters Model

CVaR(0.05) CVaR(0.25) CVaR(0.50) MAD CVaR(0.05) CVaR(0.25) CVaR(0.50) MAD CVaR(0.05) CVaR(0.25) CVaR(0.50) MAD CVaR(0.05) CVaR(0.25) CVaR(0.50) MAD

n

200 200 200 200 300 300 300 300 400 400 400 400 600 600 600 600

Continuous relaxation solution m

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

# LP

24 28 25 18 31 33 36 23 29 40 44 27 37 42 42 24

Mixed integer solution

# LP below lb

# LP above lb

in IP

out IP

in IP

out IP

3 1 2 3 4 5 3 1 5 8 10 4 7 5 8 6

9 11 9 3 15 13 20 10 17 22 25 13 21 24 25 13

8 13 14 8 11 11 11 11 6 7 5 6 8 10 7 5

4 3 0 4 1 4 2 1 1 3 4 4 1 3 2 0

selected (column 3). The second part (columns 4–8) shows the diversification of the optimal solutions of the continuous relaxations. In particular, column 4 (identified as # LP) indicates the number of securities selected by the continuous relaxation of each model. Columns 5–6 and columns 7–8 show the number of securities with a number of selected stock units lower than (# LP below lb) or greater than/equal to (# LP above lb) the corresponding lower bound (indicated as lb). For each case, we specify the number of securities that are or are not included in the corresponding integer optimal solution (identified as ‘‘in IP” and ‘‘out IP”, respectively). For instance, by reading the first line of Table 1 (model CVaR(0.05) with n ¼ 200 and m ¼ 20) we see that the relaxed optimal solution contains 24 securities of which 12 are below their lower bounds and 12 above. Moreover, 3 out of the 12 securities which do not reach the lower bound level have been also selected in the corresponding integer solution while the remaining 9 have been not. Similarly, 8 out of the 12 securities which have reached or exceeded the lower bound level have been selected in the corresponding integer solution while the remaining 4 have been not. This table helps in evaluating how good the subset of the securities selected by the relaxed model is to identify the subset of securities selected by the mixed integer model. When comparing the composition of the portfolios obtained by the continuous relaxations, it is evident that CVaR models propose the most diversified portfolios (the number of selected securities ranges between 24 and 44 for CVaR models and between 18 and 27 for the MAD model). The third part of Table 1 concerns the integer optimal solutions (columns 9–14). Column 9 provides the per cent absolute gap between the integer and the relaxed optimal solution values. More precisely, by denoting as zLP the value of the relaxed optimal solution and as z the integer  optimal value, the gap is computed as zLPzz : Column 10 

Gap LP/IP (%)

Comp. time (s)

# IP

NOT in LP

in LP

at lb

12.11 12.67 12.58 28.82 19.34 17.52 21.70 63.88 23.34 22.85 30.52 92.70 31.99 29.73 35.83 67.43

108 259 121 9 645 2852 648 7 2033 3600 3600 21 3600 3600 3600 18

13 15 17 11 17 16 16 12 14 16 17 10 18 18 17 11

2 1 1 0 2 0 2 0 3 1 2 0 3 3 2 0

11 14 16 11 15 16 14 12 11 15 15 10 15 15 15 11

1 0 2 0 2 1 2 0 5 5 5 0 3 2 4 3

shows the computational time required to solve the problem; if such time exceeds the fixed time limit (1 h) the computation is stopped and the corresponding 3600 s are reported. In such case the value used as z in the gap computation is given by the best integer solution value found so far. Finally, while column 11 (titled as ‘‘# IP”) shows the number of selected securities, the following one (”NOT in LP”) indicates the number of securities that are not in the corresponding continuous solution (column 12). The last entries show the number of securities selected in the continuous solution that are also included in the integer solution (”in LP”) and the number of securities that are set at their lower bound (”at lb”). Notice that in the case of the MAD model all the securities selected in the optimal integer solution were always contained in the continuous optimal solution. When analyzing computing times one notices that, within one hour of computational time, CPLEX is able to find the optimal solution in only 7 out of 12 CVaR instances (2 out of 12 in the case m ¼ 10). On the contrary, the optimal solution of the MAD model is always found within the specified time limit. In order to understand how much time CPLEX needs to converge to an optimal solution in the cases where one hour is not sufficient, in Table 2 we have shown the computational time required to solve to optimality 6 of the 8 instances of model CVaR(0.25). For 2 out of the 8 instances, the cases with n ¼ 400; 600 and m ¼ 10, 60 h (216,000 s) were not sufficient and we decided to stop CPLEX. We show the objective function value (Safety (IP)) and the objective function value (Safety (3600 s)) of the solution found within one hour. The best solution found within one hour in 5 out of the 8 instances is the optimal one. For the instance n ¼ 600 and m ¼ 20, CPLEX needed about 45 h of computational time to prove it. Finally, it may be interesting to know how many securities selected by the integer MAD

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E. Angelelli et al. / Journal of Banking & Finance 32 (2008) 1188–1197

Table 2 Optimal integer solutions for model CVaR(0.25) Model parameters

Integer optimal solution

Model

n

m

Comp. time (s)

Safety (3600 s)

Safety (IP)

Common Sec. in MAD and CVaR

CVaR(0.25) CVaR(0.25) CVaR(0.25) CVaR(0.25) CVaR(0.25) CVaR(0.25) CVaR(0.25) CVaR(0.25)

200 300 400 600 200 300 400 600

10 10 10 10 20 20 20 20

4976 54,872 216,000 216,000 259 2852 7318 163,089

1380.98 1240.87 1205.01 1203.19 1294.23 1156.60 1075.76 985.94

1380.98 1237.52 1161.49 1110.84 1294.23 1156.60 1075.76 985.94

2 2 3 1 4 5 6 3

model are also included in the integer CVaR(0.25) model. At this aim we have added the last column in Table 2 providing such information. Additional information on in-sample analysis can be found in Table 3 where the main results obtained by solving the different models (in-sample part of the table) and testing their ex-post behaviour (out-of-sample part of the table) are shown. In Table 3, for each model, we show the value of the safety measure, the amount of capital invested and the transaction costs paid as well as the mean portfolio return without costs (Gross Return) and that obtained by subtracting all the proportional and fixed costs (Net Return). Notice that for all models the amount paid for costs tends to increase when n grows although the number of selected securities in the different portfolios is more or less the same. This is due to the presence, in the optimal solution, of a larger number of foreign securities for which transaction costs are typically higher. Finally comparing the capital invested by the different portfolios it can also be noticed that when the value of the safety measure is positive (see, for instance, the MAD model in Table 3 for n ¼ 400 and 600) then the capital invested in the portfolio is as large as possible (very close to the upper bound C 1 ) to increase the objective function value. The opposite is true when the value of the safety is negative.

4.2. Out-of-sample analysis In this subsection we provide the analysis of the ex-post performance of the selected portfolios. To this aim we consider both the portfolio return evaluated in each of the 52 periods following the portfolio selection date (periodic returns) and the cumulative returns obtained when assuming to sell the portfolio after 1; 2 . . . ; 52 periods. By analyzing the second part of Table 3 one can see that all the portfolios show an ex-post average negative return (column Net Return). To clarify this behaviour we have reported the ex-post returns for some instances and compared their performance to two of the main European market indices (German index DAX30 and Italian index MIB30). Fig. 1 shows the cumulative returns for n ¼ 600 with cardinality constraint set to m ¼ 20. For sake of clarity we compare MAD model to only one of the CVaR model, that for b ¼ 0:05: We do not report the results for CVaR(0.25) and CVaR(0.5) models being very similar to those of CVaR(0.05) model. By observing the cumulative returns, one can see that CVaR portfolios are more stable while the MAD portfolios tend to have larger and quicker drops. While no clear dominance can be claimed among the CVaR models, all such models dominate the MAD model

Table 3 Optimal portfolios: in-sample and out-of-sample analysis Model parameters

In-sample analysis

Out-of-sample analysis

Model

n

m

#

Safety

Capital

Cost

Gross Ret.

Net Ret.

Gross Ret.

Net Ret.

Std dev.

CVaR(0.05) CVaR(0.25) CVaR(0.5) MAD CVaR(0.05) CVaR(0.25) CVaR(0.5) MAD CVaR(0.05) CVaR(0.25) CVaR(0.5) MAD CVaR(0.05) CVaR(0.25) CVaR(0.5) MAD

200 200 200 200 300 300 300 300 400 400 400 400 600 600 600 600

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

13 15 17 11 17 16 16 12 14 16 17 10 18 18 17 11

1688.14 1294.23 821.24 225.76 1399.57 1156.60 670.08 105.40 1328.87 1075.76 620.93 85.60 1109.40 985.94 569.25 137.53

100,001 100,000 100,001 100,000 100,001 100,000 100,000 100,001 100,000 100,001 100,000 101,000 100,000 100,000 100,001 101,000

781 775 785 796 842 840 847 827 861 900 885 890 906 894 904 920

781 775 836 1499 842 840 937 1515 865 900 1063 2060 906 894 1190 2088

0 0 51 703 0 0 90 688 4 0 178 1170 0 0 286 1168

342 370 292 613 511 376 497 917 170 304 340 1010 400 350 209 778

1123 1145 1077 1400 1352 1215 1344 1744 1031 1204 1224 1900 1306 1244 1113 1698

3177 3203 3195 4501 3143 2797 3086 4523 2179 2703 3330 5247 2592 2815 3295 5486

E. Angelelli et al. / Journal of Banking & Finance 32 (2008) 1188–1197

1195

Fig. 1. Out-of-sample analysis: cumulative returns for n ¼ 600 and m ¼ 20.

and usually provide a better behaviour with respect to market indices (MIB30 and DAX30), while this is typically not the case for MAD model. In Fig. 2, as an example, we show the periodic ex-post returns for n ¼ 600 and m ¼ 20: The CVaR model tends to select more diversified portfolios with respect to the MAD model, thus guaranteeing a more stable behaviour. To conclude, the MAD model creates in-sample higher mean return portfolios with respect to CVaR models. However, due to a smaller degree of diversification, the MAD portfolios are more risky and, in the case of negative market trends, that is the case analyzed in this paper, the ex-post performance of MAD portfolios is not satisfactory.

4.3. Heuristics performance We have observed that the solution of the MAD model with side constraints never requires more than 21 s (see Table 1) while in many instances the optimal solution of the CVaR model cannot be obtained within one hour of time. This is especially true when the cardinality constraint

is set to 10. When the constraint is relaxed to 20 the problem seems to be much easier. In the following, we provide the results obtained on the efficiency and effectiveness of Heuristics H ð0Þ and H ðmÞ with m ¼ 10 and m ¼ 20. In Fig. 3, as an example, we compare the shares of the securities in the portfolio selected by Heuristic H ð0Þ with respect to the integer (IP) and relaxed (LP) optimal portfolio shares for the instance with n ¼ 300; m ¼ 20 and b ¼ 0:05. The securities included in the continuous portfolio with a medium/large share are more likely to be included in the integer portfolio with respect to the securities included in the continuous solution at a low level. However there are some exceptions: Security 87 has a very little share in the continuous solution but has been included in the integer portfolio. Other securities (e.g. 150), even if not included in the continuous solution, have been selected in the integer portfolio. Finally, it may also happen that some securities (e.g. 266) have a relevant share in the continuous solution, but are excluded either by the integer solution or by the heuristic portfolio found by using H ð0Þ: Table 4 shows the computational results obtained by applying the proposed heuristics to CVaR models. The

Fig. 2. Out-of-sample analysis: periodic returns for n ¼ 600 and m ¼ 20.

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E. Angelelli et al. / Journal of Banking & Finance 32 (2008) 1188–1197 n=300; m=20; beta=0.05 0.18

0.16

0.14 LP IP H(0)

0.12

0.10

0.08

0.06

0.04

0.02

0.00 10 11 21 22 29 42 62 68 76 80 86 87 90 109 121 123 124 150 151 161 162 185 189 200 209 214 223 232 243 266 277 285 294

Fig. 3. Instance n ¼ 300; m ¼ 20: portfolio composition comparison between model CVaR(0.05), its continuous relaxation and heuristic H ð0Þ.

Table 4 Heuristic solutions: A comparison Model

n

m

Hð0Þ #

Safety

Time

GAP (%)

#

Safety

Time

GAP (%)

CVaR(0.05) CVaR(0.25) CVaR(0.5) CVaR(0.05) CVaR(0.25) CVaR(0.5) CVaR(0.05) CVaR(0.25) CVaR(0.5) CVaR(0.05) CVaR(0.25) CVaR(0.5) CVaR(0.05) CVaR(0.25) CVaR(0.5) CVaR(0.05) CVaR(0.25) CVaR(0.5) CVaR(0.05) CVaR(0.25) CVaR(0.5) CVaR(0.05) CVaR(0.25) CVaR(0.5)

200 200 200 300 300 300 400 400 400 600 600 600 200 200 200 300 300 300 400 400 400 600 600 600

10 10 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 20 20

10 10 10 10 10 10 10 10 10 10 10 10 14 16 16 17 16 14 15 16 16 17 17 17

1754.80 1381.81 861.26 1621.45 1240.84 763.71 1593.97 1185.06 692.19 1514.87 1195.73 664.26 1712.06 1300.57 821.54 1419.65 1156.60 681.33 1456.42 1094.80 629.42 1222.32 997.23 600.01

2 20 6 26 62 276 16 356 346 606 3600 1429 1 4 2 3 13 28 3 42 108 13 58 75

0.00 0.06 0.00 0.22 0.00 0.92 7.09 1.68 0.52 3.91 0.62 0.07 1.40 0.49 0.04 1.41 0.00 1.65 8.76 1.74 1.35 9.24 1.13 5.13

10 10 10 10 10 10 10 10 10 10 10 10 15 16 17 19 16 17 17 16 16 20 18 15

1754.76 1381.85 861.26 1614.77 1240.85 763.72 1584.97 1185.10 688.62 1492.10 1195.78 664.24 1696.52 1300.54 821.24 1401.56 1156.60 671.05 1379.23 1094.77 622.09 1180.30 993.57 597.44

6 21 31 105 133 618 66 1350 684 1159 3600 2349 4 18 6 15 48 69 7 136 182 35 153 309

0.00 0.06 0.00 0.63 0.00 0.92 6.56 1.68 0.01 5.49 0.62 0.07 0.50 0.49 0.00 0.14 0.00 0.15 3.79 1.77 0.19 6.39 0.77 4.95

HðmÞ

Average GAP

computational time is strongly reduced below one hour for all but one instance. Column GAP(%) provides the percentage relative errors between the integer optimal solution value (or the best solution value found within 3600 s if optimization was not terminated) and the value of the solution found by the heuristics. Positive values of the gap indicate that the solution found by the heuristic algorithm is better than the one found by the exact algorithm within one hour

1.36

1.59

of computational time. The average value of the gap is 1:36% for H ð0Þ and 1:59% for H ðmÞ. 5. Conclusions and future research In this paper we have analyzed and compared the performance of two single period models, the CVaR model and the MAD model. The MAD model has been studied

E. Angelelli et al. / Journal of Banking & Finance 32 (2008) 1188–1197

in its safety version to make the models comparable. Both models are mixed integer linear programming models, due to the inclusion of side constraints to model fixed costs, transaction lots and cardinality constraints. The computational results have shown a larger computational time required to solve the CVaR model with respect to the MAD model. Even on small instances (200 securities) the CVaR model may require more than one hour of computing time, while the MAD model has been always solved in less than one minute. From a financial point of view, the CVaR model offers more stable portfolios with respect to the MAD model and is thus preferable when the market trend is unstable or negative. Good solutions can be obtained through simple heuristics in a reasonable amount of time. As future development, we will concentrate on improving the computational time required to solve the CVaR model with real features with both exact and heuristic solution algorithms. Acknowledgement The authors wish to thank two anonymous referees for their useful comments and suggestions. References Angelelli, E., Mansini, R., Speranza, M.G., 2005. A comparison of MAD and CVaR with side constraints. Technical Report No. 238, Dipartimento Metodi Quantitativi. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measures of risk. Mathematical Finance 9, 203–228. Bertsimas, D., Darnell, C., Soucy, R., 1999. Portfolio construction through mixed-integer programming at Grantham. Mayo, Van Otterloo and Company, Interfaces 29, 49–66. Chang, T.-J., Meade, N., Beasley, J.E., Sharaiha, Y.M., 2000. Heuristics for cardinality constrained portfolio optimisation. Computers and Operations Research 27, 1271–1302. Chiodi, L., Mansini, R., Speranza, M.G., 2003. Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research 124, 245–265.

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