Value-at-Risk portfolio problem

European Journal of Operational Research 176 (2007) 423–434 www.elsevier.com/locate/ejor

Decision Support

A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem Stefano Benati b

a,*

, Romeo Rizzi

b

a Dipartimento di Informatica e Studi Aziendali, Universita` di Trento, Via Inama 5, 38100 Trento, Italy Dipartimento di Informatica e Telecomunicazioni, Universita` di Trento, Via Sommarie 14, 38100 Trento, Italy

Received 29 April 2004; accepted 6 July 2005 Available online 22 November 2005

Abstract In this paper, we consider an extension of the Markovitz model, in which the variance has been replaced with the Value-at-Risk. So a new portfolio optimization problem is formulated. We showed that the model leads to an NP-hard problem, but if the number of past observation T or the number of assets K are low, e.g. fixed to a constant, polynomial time algorithms exist. Furthermore, we showed that the problem can be formulated as an integer programming instance. When K and T are large and aVaR is small—as common in financial practice—the computational results show that the problem can be solved in a reasonable amount of time. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Portfolio optimization; Complexity theory; Linear integer programming

1. Introduction The optimal stock selection is a classic financial problem since the seminal work of Markowitz [24]. It consists of picking the best amount of stocks, with the aim of maximizing future returns. It is a typical multivariate problem: the only way to improve future returns is to increase the risk level * Corresponding author. Tel.: +39 461 88 21 06; fax: +39 461 88 21 24. E-mail addresses: [email protected] (S. Benati), [email protected] (R. Rizzi).

that the decision maker is disposed to accept. In the Markowitz approach, future returns are random variables that can be controlled by two parameters: the portfolio efficiency is measured by the expectation, while risk is calculated by the standard deviation. Therefore, the problem consists in a quadratic programming instance with some side constraints. There are several assumptions and consequences behind the Markovitz mean/variance model, such as returns are normally distributed, so that mean and variance are sufficient to fully describe the portfolio return distribution function. But in some occurrences this

0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.07.020

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assumption is not respected by data. For example, a big amount of research pointed out that real financial data are characterized by fat tails. In other words, the probabilities of incurring on extreme losses or gains are much higher than predicted by the normality assumption. Another assumption that can assure the use the mean–variance approach is the quadratic shape of the decision maker utility function. But in this case one must accept that the utility function is decreasing with respect to wealth when a threshold is overcome. These observations lead to new research directions on portfolio models. In the last years, some scholars developed new models for the optimal portfolio problem, taking into account the return non-normality. The simplest models are straightforward extensions of the Markowitz model. It is commonly accepted that efficiency is measured by the portfolio return expectation, but variance is now replaced by some lower tail return distribution function statistics. The simplest extension considers the expectation of the below-target deviation, that is, E[min(R  u, 0)], where R is the random variable representing future returns and u is a fixed threshold. Such contributions can be found in [13,28] and applications can be found in [31,18]. Another risk measure extension can be found in [32], where the risk measure is the portfolio maximum loss, that is computed by historical data. In [21], direct utility maximization with bigger penalties associated to higher losses is introduced and solved by linear programming. An axiomatic foundation of risk measures is proposed in [3]. The theory of the coherent risk measures implies that the most appropriate way of risk measuring must be the worst return expected value over a range of scenarios, e.g. the Conditional Value-at-Risk (CVaR). The operational consequences are considered in [2,26,29]. The model has also been extended in [1] to the case of multiple CVaR constraints. A similar risk measure is the Worst Conditional Expectation (WCE), whose operational consequences were developed in [4,5]. We refer to [23] for a recent and updated survey on portfolio models. All these streams of research did not consider the risk measure that most prominently imposed itself within the financial community in the last

10 years, that is, the Value-at-Risk (VaR). The formal definition of Value-at-Risk is nothing else but the a-quantile of the return distribution function, a 2 (0, 1), where a is usually chosen to be 0.01, 0.05 or 0.1. The mathematical properties of VaR are not that appealing (at least from a mathematical point of view), since it is not linear nor convex and can have many local minima and maxima, see [15]. Moreover, it can prevent portfolio diversification, as can be seen in [3]. In any case, it was proposed by the Basel committee (a Committee of International Bankers that suggests and dictates new rules for financial management) and J.P. Morgan helped its diffusion with the implementation of specific software, see [27], so it is much more employed in finance than the mathematical more correct CVaR or WCE. This diffusion justifies the efforts of some authors to develop optimization models where, in the stream of the Markovitz model extensions, variance is replaced by VaR, like the model contained in [6]. Unfortunately, VaR is a piece-wise linear function, whose graph can display many local minima and maxima, see [15], therefore an algorithm that quickly solves the problem has not been found yet. The approach that was pursued up to now is to implement heuristic procedures, like the ones contained in [7,17], or optimal exponential algorithms that are practical only for a very small number of assets, e.g. less than or equal to 3, see [15,6]. Another possibility is that the assets satisfy some monotonic assumptions, as in [10], but this is a condition that is not easily met in practice. In this paper, we show that the optimal mean/ Value-at-Risk portfolio problem is NP-hard even when future returns are described by discrete uniform distributions. Furthermore, we propose a mixed integer linear programming formulation, which allows us to solve medium size yet practical instances using Cplex. Other contributions about the use of an integer linear formulation for the VaR constrained portfolio can be found in [20,30]. We also report on our experimental evaluation of this solution approach: for some input data the answer is obtained in a few seconds, while for other data the computational times sharply increase. Luckily, the financial instances are usually tractable. We show that in an application to

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Italian data the model overperformed the index, suggesting that it provides useful information to the decision maker. Finally, we propose a polynomial time algorithm whenever the number of assets K is fixed, considerably extending and improving the results in [6].

425

Suppose that a number of K assets is available in the financial market. Let Rj be the random variable representing the future return of asset j. The portfolio optimization problem with VaR constraint is formulated as the classic mean–variance approach, but with VaR instead of variance as risk measure. The decision maker fixes two parameters, the probability aVaR and the return rVaR, and he will not accept any investment whose Value-atRisk is less than rVaR, i.e., no investment in which Pr[X 6 rVaR] P aVaR is taken into consideration. A general formulation [6,15] is the following (Problem P1):

(5) prevents short-selling, X is the random variable representing the portfolio return, so that it is the convex combination of stocks Rj according to (3). The objective function (1) to be maximized is the portfolio expected return, but two parameters must be fixed: rVaR and aVaR. The decision maker is willing to accept only portfolios for which the probability of going under rVaR is less than or equal to the threshold aVaR—this constraint is described in (2) and is equivalent to say that VaRaVaR ðX Þ P rVaR . Let Z be a random variable known through the set of past observations {z1, . . . , zT} and a 2 (0,1) be given. The methods to estimate VaRa(Z) from the finite set of observations can be roughly distinguished between parametric and non-parametric. The first case applies when Z is assumed to be distributed as a known family of functions, e.g. normal, student-t, and so on. In this case, VaR is calculated after the parameter estimation of the distribution function and strictly depends on it. For example, if we assume that stock returns are normally distributed, then replacing the Markovitz mean–variance model with a mean–VaR model does not produce any noticeable effect, since VaR is determined by the same parameters. If we do not make any assumption about the shape of the cdf, then we can rely on order estimators, that is: let a 2 (0, 1), the a-quantile is estimated by the position of the observation that has the a-percent of the data on the left. More formally, let the observations be ranked in z1:T, . . . , zi:T, . . . , zT:T, where zi:T 6 zj:T if i < j. Let t ¼ mini fij Ti P ag. Then the estimated VaR is:

max E½X ;

ð1Þ

VaRa ðZÞ ¼ zt:T .

ð2Þ

See [12] for the properties of this estimator. In the financial jargon, when no assumption is explicitly made about FZ(Æ), then we say that VaRa(Z) has been estimated through ‘‘historical simulation’’ [7]. Consider the portfolio problem. According to (4), the portfolio X is the convex combination of random variables Rj. Let rij be the observed return of Rj in time i, then P the observed portfolio return in time i is xi ¼ Kj¼1 kj rij . If we do not make any assumptions on the distribution functions of Rj, j = 1, . . . , K, then the distribution function of X

2. The Portfolio optimization model Let X be the random variable representing the investment future return and let FX(Æ) be its cumulative distribution function. The Value-at-Risk with threshold a, denoted by VaRa(X), is the aquantile of the distribution. Definition 1. Let a 2 (0, 1). The Value-at-Risk with threshold a of X is defined as: VaRa ðX Þ ¼ inffxjF X ðxÞ P ag. In particular, if FX(Æ) is continuous and strictly increasing, then VaRa ðX Þ ¼ F 1 X ðaÞ.

k

Pr½X 6 r X ¼

K X

VaR

6a

VaR

;

k j Rj ;

ð3Þ

j¼1 K X

kj ¼ 1;

ð4Þ

j¼1

kj P 0 for every j ¼ 1; . . . ; K.

ð5Þ

Constraint (4) requires that one unit of wealth must be allocated on different assets and constraint

ð6Þ

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is unknown, therefore, to estimate the VaR, we can only rely on the order statistic (6). The VaR calculation can be easily extended to the case that there are probabilities pi associated to observation xi. Assume that Rj, j = 1, . . . , K follows a discrete joint distribution function, defined over V elementary events 1,. . .,T, such that Pr½ Kj¼1 ðRj ¼ rij Þ ¼ pi . The case applies when rij are return past data and pi is the probability that is assigned to the occurrence of past realization i. Then Problem P1 can be formulated by the following mixed integer linear programming instance. Let rMin be the minimum return that can be observed in the market (for example rMin = 100%). Let rVaR, aVaR be the pair of parameters that are fixed by the decision maker to control risk. Then the problem is (Problem P2): max k;x;y

xi ¼

T X

min aVaR

aVaR ;k;x;y T X

p i xi ;

kj rij for every i ¼ 1;...;T ;

p i xi P r 

ð14Þ ð15Þ

i¼1

xi ¼

K X

kj rij for every i ¼ 1;.. .;T

ð16Þ

j¼1

ð7Þ

xi P rMin þ ðrVaR  rMin Þy i for every i ¼ 1; ...; T

i¼1 K X

In the literature, it is common to shift from ‘‘Max Return/Fixed Risk’’ problems, as Problem P2 can be considered, to ‘‘Min Risk/Fixed Return’’ problems. Under risk convexity conditions, the two problems are equivalent. Since function VaRa(X) is not concave nor convex, it is convenient to write the ‘‘Min Risk/Fixed Return’’ problem explicitly. Let r* be the minimum expected return that the decision maker is prepared to accept, then we obtain a mixed integer problem again (Problem P3):

ð8Þ

T X

j¼1

pi ð1  y i Þ 6 aVaR

i¼1

xi P rMin þ ðrVaR  rMin Þy i for every i ¼ 1;...T ; T X

pi ð1  y i Þ 6 aVaR ;

ð9Þ ð10Þ

i¼1 K X

K X

kj ¼ 1

ð17Þ

j¼1

y i 2 f0;1g for every i ¼ 1; ...; T kj ¼ 1;

ð11Þ

j¼1

y i 2 f0;1g for every i ¼ 1;...;T ;

ð12Þ

kj P 0 for every j ¼ 1;...;K.

ð13Þ

Variables kj are the percentage of wealth that is allocated to asset j, variables xi are the portfolio observed return in time i, the objective function (7) is the maximum of the expected value. Constraint (8) sets xi as the linear combination of rij, constraints (11) and (13) allocate 1 unit of wealth and prevent short-selling. Finally, constraints (9) and (10) prevent the choice of portfolios whose VaR is below the fixed threshold. Every time xi is below rVaR, then yi must be equal to 0 and 1  yi = 1 in constraint (10). Therefore, all probabilities of events i whose returns are below the VaR threshold are summed up. If the result is greater than aVaR, then the portfolio is not feasible.

kj P 0 for every j. The only difference with Problem P2 is that constraint (15) imposes the minimum portfolio expected return r*, and that aVaR is the decision variable of the objective function (14). We will see that there are two main advantages of using a mixed integer linear programming formulation of the problem. First, we improve on the computational results that are available in [15], so that problems of medium size can now be solved. Moreover, integer linear models can be easily modelled and solved using an algebraic modelling tool (like MPL), paired with a linear optimization software (like Cplex). The experienced coding time was two days for writing the basic subroutines, while it would have been much higher if one has to write the code from scratch, coding in C or Fortran the global optimization approach of [15].

S. Benati, R. Rizzi / European Journal of Operational Research 176 (2007) 423–434

3. Problem complexity In this section, we show that already the feasibility problem associated with Problem 2 is NPcomplete in the strong sense. From this, it follows that both Problem 2 and Problem 3 are NP-hard in the strong sense, i.e., they are both intractable even if we restrict attention to those instances in which the range of the numbers involved is at most polynomial in the input length. We refer to [16] for a precise definition and a detailed explanation of the notion of NP-completeness in the strong sense, and only point out that this strong negative result rules out not only the existence of polynomial time algorithms but also of pseudopolynomial time algorithms and of FPTAS for both Problem 2 and Problem 3. Moreover, even assuming we were willing to accept an approximation of the optimal value, there is no hope (unless P = NP) to obtain an approximation algorithm for these problems either, since such an algorithm should return at least one feasible solution whenever there exists one, regardless of its objective function value. We further notice that these negative results do hold already under the assumption that all the pis are identical, that is, even in the special (but prominent) case in which a same probability is assigned to each past observation. To achieve what stated above, we reduce MINIMUM NODE COVER to our feasibility problem. It is well known [16,8] that MINIMUM NODE COVER is NP-complete, since this was one of the six problems shown NP-complete in the celebrated paper by Karp [19]. In a generic instance of MINIMUM NODE COVER, we are given as input a graph G = (V, E), where V = {vi : i = 1, . . . , n} is the vertex set and E = {ei : i = 1, . . . , m} is the arc set, and an integer h, and we are asked to decide whether G admits a node cover of size h, that is, whether there exists an XV with jXj = h such that every edge of G has at least one endnode in X. The reduction. Given an instance hG, hi of MINIMUM NODE COVER, build up an associated instance of Problem 2 as follows. Let G = (V, E), n = jVj and m = jEj. First, take the nodes of G as the set of available stocks, that is, set K := n and introduce a stock Rj for every node of G. Intuitively, investing on stock Rj will correspond to take node

427

vj into the node cover X. In other words, assume the nodes in V are re-labelled so that V = {R1, R2, . . . , RK}. Next, set T := n2 + m and introduce the T observations by defining the entries rij as follows. For j = 1, 2, . . . , K, and i = 1, 2, . . . , T, the entry rij, which specifies the return on stock Rj observed at time i, is set as follows: for i = 1, 2, . . . , n2, set rij := h if i  j is a multiple of n and rij := 0 otherwise; for i = n2 + 1, n2 + 2, . . . , T = n2 + m, set rij := h if the node Rj belongs to the edge ein2 and rij := 0 otherwise. 1 Finally, take rMin = 0, rVaR = 1, pi ¼ mþn 2 and nðnhÞ VaR a ¼ mþn2 . The reduction is complete: we have by now fully specified an instance of Problem P2 on the basis of the input instance hG,hi of problem MINIMUM NODE COVER. Clearly, all the actions prescribed in the above reduction can be performed in polynomial time. The following lemma proves the correctness of the reduction, from which all the negative results stated in the beginning of this section follow. Lemma 1. The instance of Problem 2 associated to a graph G = (V, E) and an integer h by means of the reduction described above is feasible if and only if G admits a node cover of size h. Proof. Let X  V be a node cover of G with jXj = h. Then the following choice for the decision variables shows that the associated instance of Problem 2 is feasible: take kj ¼ 1h if Rj 2 X and kj = 0 if Rj 62 X; furthermore, if i > n2, then take all yi = 1, else if i 6 n2 then yi = 1 if and only if i = an + j, for some a 2 N and Rj 2 X, otherwise yi = 0. We will show that all constraints of type (9) are satisfied. If i 6 n2 and kj ¼ 1h, then it follows ri = 1 for every i = an + j, a 2 N, therefore constraints (9) are satisfied with our choice yi = 1. If i 6 n2 and kj = 0, then ri = 0 for every i = an + j, a 2 N, then constraints (9) are satisfied with our choice yi = 0. Moreover, all constraints of type (9) with i > n2 are satisfied with yi = 1, since X is a node cover. Notice that the number of variables yi set to 0 is n(n  h). 1 Since all pis are equal to T1 ¼ mþn 2 , it follows that X

pi ð1  y i Þ ¼

1 nðn  hÞ nðn  hÞ ¼ ¼ aVaR . 2 mþn m þ n2

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This implies that constraint (10) is also satisfied (with strict equality). Now for the converse direction. Assume the associated instance of Problem P2 is feasible and let k1, k2, . . . , kK and y1, y2, . . . , yT be a feasible solution. By constraint (9), we know that the number of yis set toP0 is at most n(n  h). On the other side, since kj 6 1, at most h among the values k1, k2, . . . , kK are not strictly less than 2 1 h. From the first n constraints of type (9), it hence follows that at least n(n  h) among y1, y 2 ; . . . ; y n2 are set to 0. Therefore, precisely n(n  h) among y 1 ; y 2 ; . . . ; y n2 are set to 0 and, moreover, if n < i < n2, then yi = yian for some a 2 N. Define X as the set of those Rj such that yj = 1. By what said, jXj = h. Since yi = 1 for i > n2, it follows that X is a node cover of G. The proof is complete. h

small number of assets. This is reasonable: the investor may choose the optimal portfolio allocation between a bond and a stock fund, or local and international asset funds, and so on. Indeed, the case K = 2 was considered in [6] and it was solved there by means of complete enumeration of all numbers in (0, 1). We will show that if K is fixed to a small constant, then the problem is polynomially solvable. Consider first the case that the choice is between two assets only (as it was the case in [6]), therefore K = 2, and suppose that pi ¼ T1 ; i ¼ 1; . . . ; T . The past observations of R1 and R2 are the arrays r1 ¼ ½r11 ; . . . ; r1T  and r2 ¼ ½r21 ; . . . ; r2T . The portfolio past observations are the convex combination x(k) = kr1 + (1  k)r2. Let t ¼ mini fi j Ti P aVaR g. The Value-at-Risk constraint on x(k) can be rewritten as: minfxðkÞg P rVaR t

4. Two polynomial time solvable special cases The negative result of the previous section is due to the fact that both the number T of past observations and the number K of feasible assets may vary from input to input and grow arbitrarily large. We may wonder whether the problem remains NP-hard even in the case that T or K are bounded by a constant. We remark that Problems 2 and 3 are polynomially time solvable in the following two cases. Case 1: The number of observations, T, is bounded by a constant. This is easy to see. There are 2T ways of fixing the decisions variables yi. If T is a constant, so is 2T. Once the values of these variables are decided, we remain with an LP problem. The interest on case 1 is mainly theoretical. First of all, the resulting polynomial algorithm is practical only for very small values of T, say T 6 10. Moreover, we cannot imagine an application where T can be reasonably fixed to such a small value. Financial time series are usually much larger. Case 2: The number of assets, K, is bounded by a constant. Suppose that K is fixed to a small value. This corresponds to an investor that wants to find the optimal wealth allocation between a

ð18Þ

where mint{x(k)} is the t smallest value of array x(k). Function (18) is piecewise linear with respect to k, as can be seen in Fig. 1. In Fig. 2, Problem P2 is represented. The objective function is the line connecting E[r1] and E[r2]. Function y = mint{x(k)} crosses the line y = rVaR a number of times, determining k values that are feasible or unfeasible. Now the optimal k value is easily calculated. It is a point of the intersection between y = mint{x(k)} and y = rVaR. Moreover, since in the

Fig. 1. r(ij) is the observed return of asset i, time j. Function y = min2(x(k)) is drawn in bold line.

S. Benati, R. Rizzi / European Journal of Operational Research 176 (2007) 423–434

Fig. 2. Description of the portfolio problem: The feasible region is the union of k(4) 6 k 6 k(3) and k(2) 6 k 6 k(1).

example E[r1] > E[r2], the best one is the feasible point that is closest to k = 1. To calculate its value, we can proceed as follows. Calculate the intersection between the line y = rVaR and each line connecting points r2j and r1j . There are at most T intersection points, giving a set of k1 > k2 >    > kT candidate optimal solutions. Calculate the Value-at-Risk of x(ki) for increasing values of i, the first feasible one is optimal. Now, for the case of general (but fixed K), we rely on a more abstract viewpoint. In Problem 2 we are asked to optimize a given linear objective function on a space of feasible k vectors. The linear objective function can be expressed as a linear combination of kj, j = 1, . . . , K, it is sufficient to combine the objective function (7) and the linear forms (8). Let K :¼ fk 2 RK : 1k ¼ 1; k P 0g, where k is the vector k = (k1, k2, . . . , kK). For any iP= 1, . . . , T, consider the linear functional xi ðkÞ ¼ K j¼1 kj rij : when the linear functional crosses the value rVaR, it determines a partition of K into two regions, separated by the hyperplane xi(k) = rVaR and characterized by yi = 0 on one side, and yi = 1 on the other. Repeating this process for every i = 1, . . . , T, a finer partition of K is obtained. In this way, the family of hyperplanes {xi(k) = rVaR : i = 1, . . . , T}, partitions K into polyhedral regions, where each region is characterized by a vector y = [y1, . . . , yT], with yi = 1 if xi(k) P rVaR, yi = 0 otherwise. Moreover, any

429

point k 2 K is feasible if and only if it belongs to a region for which constraint (10) is satisfied. This allows us to speak of feasible (or infeasible) regions: each single region of the above partition is feasible if constraint (10) is satisfied by the corresponding vector y = [y1, . . . , yT], otherwise it is unfeasible. Moreover, every set of this partition is a polytope—of which we possess an explicit description in terms of defining inequalities—we can hence resort on LP to maximize the objective function on each feasible region and take the best of the optima as an optimal solution to our problem. The important observation which enters at this point is a classical result in Computational Geometry stating that the number of regions separated by n hyperplanes in a d-dimensional space is O(nd), this implies that the number of regions of the partition of K end hence the number of the LP problems we need to solve is polynomial once d—in our case K—is fixed. References to this classical results and to algorithms for efficiently enumerating these regions can be found in [11,9,25].

5. Computational results The results of the previous section show that optimal solutions to Problems P2 and P3 can be obtained using exponential algorithms, therefore we cannot expect to be able to solve them for arbitrary large T and K, and for every parameters set. However, the integer linear programming formulation can be a useful tool, since we know that if the problem size is not too high, commercial software can be very powerful. Problems P2 and P3 were coded and solved using Cplex 7.5. Preliminary time analysis suggested to use the depth-first strategy to explore the Branch&Bound tree, with variable selection left to ‘‘automatic’’. Moreover, no degree of tolerance error has been accepted. Data came from the closing daily price of 25 main assets of the Milan stock market, reported for 100 and 200 days after January 1, 2001. Both models show that the parameters values affect the computational times to a great extent. For low values of aVaR, e.g. 0.05, the problem is

S. Benati, R. Rizzi / European Journal of Operational Research 176 (2007) 423–434

quickly solved (some seconds), but the computational time can rise sharply when rVaR is close to the median, that is when aVaR = 0.5. Luckily enough, the financial practice calculates risk using the first alternative. Consider Problem P3 first. Tables 1, 2 and Fig. 3 report the computational experience when 100 daily returns are calculated (that is, the problem is solved using 100 binary variables). The parameter r* that is used for Table 1 is the index expectation, while rVaR ranges from low to high values. As can be seen, computational times range from few seconds to some minutes. Note that when they are higher, the resulting objective Table 1 Computational results for model P3, when 100 past data are considered and the minimum required return is high r*

rVaR

Objective function

B&B nodes

Seconds

0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 0.9 1 1.1 1.2 1.3 1.4

0.06 0.08 0.09 0.09 0.1 0.11 0.12 0.13 0.15 0.16 0.19 0.2 0.22 0.23 0.25 0.27 0.29 0.3 0.32 0.35 0.37 0.39 0.39 0.42 0.45 0.47 0.48 0.48 0.5 0.52 0.54 0.55 0.57

92 209 182 149 147 166 175 205 503 487 562 854 1080 1499 2891 2601 1662 3008 3343 3403 7385 27,355 27,355 34,615 11,573 13,946 14,215 17,551 9453 25,098 36,315 33,436 8805

0.93 1.37 1.21 1.43 1.54 1.37 1.54 2.03 2.53 2.53 3.46 4.45 4.95 5.77 11.15 10.27 6.16 10.49 11.26 11.53 23.18 98.86 100.07 140.01 42.68 54.04 60.58 69.26 46.9 106.45 155.93 152.75 34.44

Table 2 Computational results for model P3, when 100 past data are considered and minimum required return is low r*

rVaR

Objective function

B&B nodes

Seconds

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.01 0.01 0.01 0.01 0.02 0.03 0.03 0.05 0.07 0.08 0.09 0.1 0.11 0.13 0.15 0.17 0.17 0.18 0.2 0.22 0.24

0 0 0 0 39 78 91 1292 3980 1991 6264 11,266 45,850 46,704 40,724 84,962 54,720 11,0748 60,882 235,488 353,834

1.05 0.11 0.05 0.11 0.82 0.93 1.54 6.1 13.85 8.73 23.01 35.64 128.8 130.55 112.81 264.35 174.06 310 172.91 651.25 1128.01

Times Times 200 150

Seconds

430

100 50

-2

-1.5

-1

0 -0.5 0 0.5 Return Value-at-Risk

1

1.5

2

Fig. 3. Computational times for different values of rVaR (data from Table 1). It can be easily seen that there is a range of values where the problem is computationally difficult.

function aVaR is close to 0.5, so that rVaR is close to the median. The parameter r* that is used in Table 2 is less than the expectation of the market index, so that the resulting problem is less constrained. As a consequence, the computational times are much higher. Again, we can see that times range from some seconds to many minutes. Table 3 reports the computational times when 200 past data are considered (so that the model

S. Benati, R. Rizzi / European Journal of Operational Research 176 (2007) 423–434 Table 3 Computational results for model P3, when 200 past data are considered r*

rVaR

Objective function

B&B Nodes

Seconds

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8

0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

0.045 0.05 0.055 0.06 0.07 0.08 0.085 0.09 0.105 0.115 0.135 0.145

582 707 2213 1229 11,434 9584 28,478 16,818 21,424 16,704 92,336 235,284

8.68 9.89 20.65 16.04 106.83 85.08 274.52 144.84 177.9 153.68 719.25 2192.24 P1 hour

is solved with 200 binary variables). The expectation is fixed to some point more than the index expectation. But it can be seen that the model is quickly solvable only when the objective function is low, that is when aVaR is at most the 0.1-quantile of the distribution function. When aVaR is greater than 0.135, we truncated the computation after 1 hours of calculations. Now consider Problem P2. Tables 4–7 and Fig. 4 report our computational experience. We limited our attention to aVaR fixed to 0.05 and the 0.1 quantiles and let rVaR vary. Problems that

Table 4 Computational results for model P2, when 100 past data are considered rVaR

aVaR

Objective function

B&B nodes

Seconds

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.627811743 0.622751146 0.6176783 0.609232804 0.580067556 0.561035516 0.513912334 0.362105839 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible

8 11 23 44 79 234 265 5125 1206 2281 1554 1043 781 308 290

0.11 0.11 0.16 0.22 0.39 0.71 0.87 9.33 3.29 4.17 2.09 1.98 1.48 0.93 0.88

431

Table 5 Computational results for model P2, when 100 past data are considered rVaR

aVaR

Objective function

Nodes

Seconds

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.639120421 0.63780908 0.636586603 0.635156701 0.631437849 0.626073286 0.622710084 0.619347357 0.608870853 0.591232823 0.527471854 0.384727288 Unfeasible Unfeasible Unfeasible

13 15 8 7 9 26 24 77 48 504 6177 49,634 54,708 18,452 22,906

0.16 0.11 0.11 0.17 0.16 0.22 0.22 0.44 0.38 1.27 12.3 87.38 95.35 36.03 44.05

Table 6 Computational results of model P2 when 200 past data are considered rVaR

aVaR

Objective function

Nodes

Seconds

3 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.471323043 0.468226913 0.463566693 0.459440231 0.453357989 0.449244086 0.445650432 0.440939388 0.433608592 0.424155551 0.415803743 0.403701217 0.394219997 0.38321618 0.365031177 0.34444625 0.310949151 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible

33 48 55 108 209 365 223 241 429 306 279 668 1677 1207 4575 29,570 40,084 153,889 162,212 145,957 33,331 19,494 7783 4641 2000

0.44 0.44 0.55 0.72 1.1 1.64 1.26 1.54 1.98 1.76 1.92 3.46 7.85 5.55 18.07 110.18 189.93 728.59 758.8 687.72 126.33 77.12 35.38 21.69 9.5

we discovered trivial to solve because only one asset is selected are discarded from the analysis.

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Table 7 Computational results of model P2 when 200 past data are considered rVaR

aVaR

Objective function

B&B nodes

Seconds

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.465750093 0.459899961 0.452569894 0.444182605 0.438690922 0.432675418 0.425549043 0.418416506 0.399933017 0.374557147

56 113 346 1326 2042 1915 3755 4230 9136 61,284

0.88 0.99 1.81 4.83 7.74 7.69 14.72 17.68 37.02 242.61 >1 hour

Times Times

Seconds

800 700 600 500 400 300 200 100 0

-3.5

-3

-2.5

-2 -1.5 Return Value-at-Risk

-1

-0.5

0

Fig. 4. Computational times for different values of rVaR (data from Table 6). It can be easily seen that there is a range of values where the problem is computationally difficult.

Tables 4, 5 and Fig. 4 show that the problem is solvable in a reasonable computational time, if 100 past data are considered (100 binary variables). But times can vary: there is a threshold rVaR that separates feasible from unfeasible instances of the problem. When the parameter rVaR is close to that value, times are the highest. The difficulty of the problem emerges when 200 past data are considered. If aVaR is 0.05 the same behavior is observed: the highest computational times are observed when it is difficult to decide whether the problem is feasible or unfeasible. But when aVaR is equal to 0.1, times can be more than 1 hours, showing a tendency towards an exponential behavior. We want to analyze the severity of this potential difficulty as the problem dimensions increase further. We extended the data set to 500 daily observations of the 30 main stocks of the Milan market

and we considered a sample of reasonable values of rVaR. We assume that a fund manager fixes the parameter rVaR to obtain a portfolio that is less risky than the market portfolio (that is, buying all the assets). If the optimized portfolio has the greatest expected value, than it is convenient to buy it. Therefore, the 500 data time series of the Dow Jones Italian Index is considered and the value rVaR is computed: for our computational experiDJ ments we suppose that rVaR P rVaR DJ . For every stock i, i = 1, . . . , 30, the value rVaR is computed. i Consider problem version P2, in the range: VaR rVaR 6 maxfrVaR g. DJ 6 r i i

Clearly, a feasible solution exists. The computational experiments are reported in Table 8. As can be seen, if a is 0.05, the problem is solvable, but it becomes intractable for a = 0.1. As a conclusion, we can say that our Integer Linear Programming formulation of the problem is effective under the condition that the risk quantile a is equal to 0.05. We mention that some previous simulation papers [4,14,22,23], considered from 60 to 250 past data to find optimal portfolios. 5.1. Financial analysis Now we want to test whether Value-at-Risk optimization is a model that provides useful information to the decision maker. We make the following experiment. We started on January 1st, 2003 and solved a problem with the following parameters: 100 past daily returns of the 30 main assets of the Milan market, aVaR = 0.1, rVaR ¼ rVaR DJ , pi = 1/100, i = 1, . . . , 100. That is, we want a portfolio that is not riskier than the index, but it has the greatest expected value. We buy the portfolio and hold it for 20 market days. After that we repeat the optimization with Table 8 Time (in seconds) and branch&bound nodes for 500 past data a

rVaR

Time

Nodes

0.05 0.05 0.1 0.1

rVaR DJ maxi frVaR g i rVaR DJ maxi frVaR g i

25.43 448.02 911 >8 hours

2051 28192 62972

S. Benati, R. Rizzi / European Journal of Operational Research 176 (2007) 423–434 Table 9 Comparison of the VaRPortfolio with the Market Index Date

Index

Portfolio

01/01/2003 27/01/2003 24/02/2003 24/03/2003 21/04/2003 19/05/2003 16/06/2003 14/07/2003 11/08/2003 08/09/2003 06/10/2003 03/11/2003 01/12/2003 29/12/2003 26/01/2004 23/02/2004 22/03/2004 19/04/2004 17/05/2004 14/06/2004 12/07/2004 09/08/2004 06/09/2004 04/10/2004 01/11/2004

6.81 +3.74 1.75 +5.74 0.01 +8.84 2.08 +0.64 +1.78 1.70 +4.20 +4.63 1.87 +4.88 1.13 4.23 +7.90 4.64 +3.71 0.66 4.13 +3.90 +3.60 +1.75

0.94 +12.39 7.32 +4.40 +3.48 +8.10 2.22 +3.36 +1.53 0.17 +5.96 +5.36 5.44 +10.96 6.05 5.91 +6.76 3.27 +3.92 2.98 1.61 +4.85 +7.70 0.79

Index wealth

Portfolio wealth

100.00 93.19 96.67 94.98 100.43 100.42 109.30 107.02 107.71 109.63 107.76 112.28 117.48 115.28 120.91 119.54 114.48 123.52 117.79 122.16 121.35 116.34 120.88 125.23 127.42

100.00 99.06 111.33 103.18 107.72 111.47 120.50 117.82 121.78 123.64 123.43 130.79 137.91 130.40 144.70 135.94 127.91 136.55 132.09 137.27 133.18 131.03 137.38 147.95 146.78

updated parameters and buy a new portfolio from scratch, assuming no transaction costs. The results that we obtained are reported in Table 9. In 8 periods the Market Index was clearly superior, in 7 period there was by and large a tie (the difference between the two portfolios were less than 1%), in 10 periods the optimized portfolio was best. It is worth to note that 100 euros, if invested in the index portfolio at the beginning of 2003, became 127.42 euros at the end of 2004, but 146.78 if invested in the VaR portfolio. Clearly, this is not the proof that the optimized portfolio is better, but, as a conclusion, we can say that the optimized portfolio provides useful information to the decision maker.

6. Conclusion In this paper, we consider an extension of the Markovitz model, in which the variance has been replaced with the Value-at-Risk. So a new port-

433

folio optimization problem is formulated. We showed that the model leads to an NP-hard problem, but if the number of past observation T or the number of assets K are low, e.g. fixed to a constant, polynomial time algorithms exist. Furthermore, we showed that the problem can be formulated as an integer programming instance. When K and T are large—as common in financial practice—the computational results show that in some cases the problem can be quickly solved. The condition is that aVaR must be small, that is 0.05. We think that there are two main problem features that deserve more research. First, it is interesting to implement a branch&bound algorithm that exploits the problem structure, in order to improve the computational times provided by the Cplex solver. Moreover, heuristic algorithms must be devised to solve the problem for the case aVaR P 0.1. The second path for a new research is to improve the financial models of optimal portfolio. For example, the model can be easily adjusted to include more than one Value-at-Risk constraint, to simulate better the decision maker risk attitude. For example, when several constraints are added to the model, the result is an approximation of a decision that is driven by the First Order Stochastic Dominance. Another interesting application is using the model when aVaR = 0.5, that is, when the median is considered as a risk or an efficiency criterion. The median has many interesting properties, for example it is a robust statistic, that is a location index that is not affected by outliers, as happens to the mean. Financial data has a lot of outliers, so it is interesting to see what happens if the mean is replaced by the median in a portfolio model.

Acknowledgements The present work benefited from helps and suggestions from Renata Mansini and Maria Grazia Speranza. Prof Stefano Benati has been supported by the MIUR project ‘‘Computational methods for Portfolio Optimization and Financial Analysis’’.

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