Comput Econ (2009) 33:237–262 DOI 10.1007/s10614-008-9158-y Models and Simulations for Portfolio Rebalancing Gianfranco Guastaroba · Renata Mansini · M. Grazia Speranza Accepted: 6 September 2008 / Published online: 5 October 2008 © Springer Science+Business Media, LLC. 2008 Abstract In 1950 Markowitz first formalized the portfolio optimization problem in terms of mean return and variance. Since then, the mean-variance model has played a crucial role in single-period portfolio optimization theory and practice. In this paper we study the optimal portfolio selection problem in a multi-period framework, by considering fixed and proportional transaction costs and evaluating how much they affect a re-investment strategy. Specifically, we modify the single-period portfolio optimization model, based on the Conditional Value at Risk (CVaR) as measure of risk, to introduce portfolio rebalancing. The aim is to provide investors and financial institutions with an effective tool to better exploit new information made available by the market. We then suggest a procedure to use the proposed optimization model in a multi-period framework. Extensive computational results based on different historical data sets from German Stock Exchange Market (XETRA) are presented. Keywords Risk management · Conditional value at risk · Portfolio rebalancing · Multi-period portfolio analysis · Mixed integer linear programming 1 Introduction The original mean-risk portfolio formulation introduced by Markowitz (1952) has provided the fundamental basis for the development of a large part of the modern G. Guastaroba (B) · M. G. Speranza Department of Quantitative Methods, University of Brescia, C. da S.Chiara 50, 25122 Brescia, Italy e-mail: guastaro@eco.unibs.it M. G. Speranza e-mail: speranza@eco.unibs.it R. Mansini Department of Electronics for Automation, University of Brescia, via Branze 38, 25123 Brescia, Italy e-mail: rmansini@ing.unibs.it 123 238 G. Guastaroba et al. financial theory applied to the single-period portfolio optimization problem (also known as buy-and-hold portfolio strategy). In Markowitz (1952) the single-period portfolio optimization problem is modelled as a mean-risk bicriteria optimization problem where the expected return is maximized and the variance as a scalar risk measure is minimized. Since then several alternative bicriteria models have been proposed. For the case of returns distributed as discrete random variables, these models have the relevant advantage to be LP solvable. Whereas some of these LP computable measures, as the mean absolute deviation proposed by Konno and Yamazaki (1991), may be viewed as approximations to the variance, more recently the shortfall or quantile measures are gaining more popularity in various financial applications (see Mansini et al. 2003, 2005 and references therein). Examples of such measures are the Conditional Value at Risk introduced by Rockafellar and Uryasev (2000) (see also Pflug 2000) and the Worst Realization analyzed by Young (1998). The latter are safety measures to be maximized and have found application in various financial optimization problems as demonstrated by several empirical studies that recently appeared in the literature (Andersson et al. 2001; Rockafellar and Uryasev 2000; Mansini et al. 2003). Mansini et al. (2005) introduced some new portfolio optimization models based on the use of multiple CVaR risk measures to allow a better modeling of the risk aversion. In Mansini et al. (2005) both the theoretical properties of the models and their performance on real data are provided. As shown in Mansini et al. (2003), for any risk measure a corresponding safety measure can be defined and vice versa. In this paper we will concentrate on the Worst Conditional Expectation (WCE) measure as defined in Mansini et al. (2003). This measure has several interesting theoretical properties. It satisfies the requirements of the coherent risk measures (see Artzner et al. 1999 for the definition of coherent risk measure) and is consistent with the Second-degree Stochastic Dominance (SSD) (Ogryczak and Ruszczyński 2002). The WCE is closely related to the Conditional Value-at-Risk (CVaR) (Rockafellar and Uryasev 2000). More precisely, they are equivalent in the case of continuous distributions of returns, whereas they can take different values for discrete distributions (see Ogryczak and Ruszczyński 2002). For this reason and for sake of simplicity we will refer to the model based on the WCE as CVaR model. Finally, notice that, according to the aims of this paper, any other portfolio optimization model based on a different risk or safety measure could have been alternatively used. In single-period portfolio optimization theory one basic implication of Markowitz model is that investors hold well diversified portfolios of securities. Nevertheless, in practice, investors typically select portfolios consisting in a small number of securities (see also Blume et al. 1974). There are many reasons for this fact, the most relevant of which is the presence of transaction costs. In financial markets, any movement of money among assets incurs in a transaction cost. In many papers the emphasis has been put on properly modelling transaction costs. Patel and Subrahmanyam (1982) first studied the fixed transaction costs impact on portfolio diversification. Pogue (1970) first dealt with the proportional brokerage fees involved in revising an existing portfolio. Yoshimoto (1996) analyzed the problem of portfolio optimization with variable transaction costs adopting a mean-variance approach. Gennotte and Jung (1994) examined the effect of proportional transaction costs on dynamic portfolio strategies. Finally, recent contributions on optimization models with transaction costs have emphasized 123 Models and Simulations for Portfolio Rebalancing 239 the critical aspect of models computational testing and the need for efficient algorithms to solve them. In Konno and Wijayanayake (2001) and Mansini and Speranza (2005) the authors have proposed exact solution algorithms for portfolio optimization problems with transaction costs, whereas different heuristic solution algorithms can be found in Kellerer et al. (2000) and Chiodi et al. (2003). Transaction costs also affect portfolio optimization problems in a multi-period framework. Due to sudden changes in the market trend an investor might prefer to rebalance her/his portfolio composition to possibly reduce losses or better exploit returns growth. Such adjustments might be desirable even if implying additional costs. Markowitz (1959) recognized the importance of intermediate adjustment opportunities. He suggested that finding the optimal multi-period management strategy is, in principle, a dynamic programming problem. Smith (1967) introduced a model of portfolio revision with transaction costs. His approach modified Markowitz one-period problem and then aimed at applying the solution period by period. In Li et al. (2000) and Li et al. (2001) the authors extended the portfolio optimization setting of the mean-variance approach proposed by Markowitz to the case with transaction costs formulated as a V-shaped function of differences between old and new portfolios avoiding short sales. They also proposed solution algorithms to the studied problems. Elton and Gruber (1974) showed that, under some specific assumptions, even if no new information is expected to arise over time, changing the portfolio will increase the expected utility of the terminal wealth. Li and Ng (2000) and Steinbach (2001) studied extensions of the Markowitz mean-variance approach to multi-period portfolio optimization problems. This paper aims at studying different investment policies as practical tools for the portfolio management in a multi-period framework. In particular, we will analyze a rebalancing portfolio optimization CVaR model and show how optimization could represent a critical tool for financial managers. The remainder of the paper is organized as follows. In Sect. 2 we describe the single-period CVaR optimization model with transaction costs and introduce its variant to deal with portfolio rebalancing. Section 3 is devoted to the experimental analysis and to the comparison of different investment strategies. Alternative data sets are used to analyze all possible market trends. The main objectives are to provide evidence on the effectiveness of the rebalancing optimization model in leading financial decisions under different market conditions and to help decision makers in choosing the proper portfolio rebalancing frequency. Finally, in Sect. 4 some concluding remarks are drawn. 2 The Models In order to make the paper self-contained, we provide a brief introduction to the so called CVaR optimization model. Both the single-period CVaR model with additional side constraints on transaction costs and its variant to consider portfolio rebalancing are discussed later in this section. Let N = {1, 2, . . . , n} denote a set of securities (risky assets) available for an investment. For each security j ∈ N , the rate of return is represented by a random variable R j with a given mean r j = E{R j }. We consider T scenarios with probabilities pt (where t = 1, . . . , T ). We assume that for each 123 240 G. Guastaroba et al. random variable R j its realization r jt under the scenario t is known. The realizations are derived from historical data and the T historical periods are treated as equally probable scenarios (
pt = 1/T ). Thus the
Tmean rate of return for security j is computed as T pt r jt = T1 t=1 r jt . Let us define as w j , j = 1, 2, . . . , n, the r j = E{R j } = t=1 continuous decision variables expressing the portfolio weights and representing the fractions of capital invested in the securities. We assume that the sum of the weights is equal to one and that no short sales are allowed,
i.e. w j ≥ 0 for j = 1, . . . , n. Each portfolio w defines a random variable Rw = nj=1 R j w j representing the portfolio
return with expected value given by µ(w) = E{Rw }. Let µt = nj=1 r jt w j be the
T realization of the portfolio return Rw under scenario t. Then E{Rw } = t=1 pt µt =  
n
T 1
n t=1 T j=1 r jt w j = j=1 r j w j . Recently Young (1998) proposed to select a portfolio based on the maximization of the worst portfolio realization, i.e. max M(w) = {mint=1,...,T µt }. Instead of the worst case scenario, one may maximize the mean of a specified size (quantile) of worst realizations, i.e. the Conditional Value at Risk for a given tolerance level 0 < β ≤ 1 defined as follows: 1 Mβ (w) = β  β 0 Fw(−1) (α)dα (1) (−1) where Fw ( p) = inf {η: Fw (η) ≥ p} is the left-continuous inverse of the cumulative distribution function of the rate of return Fw (η) = P{Rw ≤ η}. Note that M1 (w) = µ(w) and Mβ (w) tends to Young’s M(w) when β tends to 0. If the portfolio return is a discrete random variable represented by its realizations µt under specified scenarios, then the Conditional Value at Risk is defined as the mean portfolio return under the k worst scenarios, where k = βT. We will refer to the portfolio optimization problem based on this safety measure simply as CVaR(β) model. It can be formulated as the following LP problem (see Ogryczak and Ruszczyński (2002)): Mβ (w) = max q − T 1 pt dt β (2) t=1 q − µt ≤ dt t = 1, . . . , T  r j w j ≥ µ0 (3) (4) j∈N  j∈N 123 wj = 1 (5) Models and Simulations for Portfolio Rebalancing 241 wj ≥ 0 j ∈ N (6) dt ≥ 0 t = 1, . . . , T, (7) where q is an auxiliary (unbounded) variable representing the β-quantile at the optimum, whereas µ0 is a parameter representing the minimum required portfolio rate of return. The nonnegative variable dt , t = 1, . . . , T, measures the maximum positive deviation of the portfolio return realization µt from the β-quantile, i.e. dt = max{0, q − µt }. 2.1 The Single-Period CVaR(β) Model with Transaction Costs We use the CVaR measure of risk to model a real case situation where an investor optimally selects a portfolio and holds it until the end of the investment period. For each selected security the investor incurs a fixed and a proportional cost. We assume that securities can be bought in fractions of stock units. Let x = (x j ) j=1,2,...,n denote the vector of decision variables where x j , j ∈ N , represents the fractional value of stock units invested in security j. We define as q j the quotation of security j, j ∈ N , at the date of portfolio selection (time 0) and as q j x j the amount invested in security j at the same date. f j is the fixed transaction cost incurred by the investor when selecting security j and c j is the corresponding proportional transaction cost. Moreover, C is the capital available for the investment whereas u j , j ∈ N , represents the upper limit on the fractional number of units of security j that the investor can purchase. For any 0 < β ≤ 1, the single-period CVaR(β) model with transaction costs can be defined as follows: Basic model max q − T 1 pt dt β (8) t=1 n n   (r jt − c j )q j x j + f j z j ≤ dt t = 1, . . . , T q− j=1 (9) j=1 n n n    (r j − c j )q j x j − f j z j ≥ µ0 qjxj j=1 j=1 n  qjxj = C (10) j=1 (11) j=1 123 242 G. Guastaroba et al. xj ≤ ujzj j = 1, . . . , n (12) dt ≥ 0 t = 1, . . . , T (13) x j ≥ 0 j = 1, . . . , n (14) z j ∈ {0, 1} j = 1, . . . , n. (15) In the following we will refer to the above single-period CVaR(β) model with transaction costs simply as the basic model. The objective function (8) maximizes the Conditional Value at Risk. Constraints (9) along with constraints (13) define the
nonnegative variables dt as max{0, q −
n n (r − c )q x − yt }, where yt = jt j j j j=1 j=1 f j z j is the net portfolio return at time t. Thus, each variable dt measures the deviation of the portfolio net return yt from the β−quantile q when yt < q, whereas it is equal to zero in all the other cases. Constraint (10) establishes that the portfolio net mean return, expressed as difference between the portfolio mean return and the total transaction costs (proportional and fixed), must be at least equal to the portfolio required return µ0 C. Constraint (11) imposes that the total investment in the portfolio must be equal to C. Constraints (12) define the upper bound on the investment in each security. Since the fixed cost f j , j ∈ N , is paid only if security j is selected, we introduce a binary variable z j , j ∈ N , which is forced by constraint (12) to take value 1 if x j > 0. Notice that if x j = 0 then z j is free to take any value. However, at optimum z j will take value zero as the most convenient of the two. Finally, constraints (14) avoid short sales on securities whereas constraints (15) define binary conditions. Some comments are worthwhile on how the transaction costs can be modelled. The transaction costs have to be introduced in the minimum required return constraint (10) to guarantee a minimum net return rate of the portfolio. On the contrary, the transaction costs may be included in the budget constraint (11) or not, depending on whether the capital C is used both to buy the securities and to pay the transaction costs or to buy the securities only. Both modelling alternatives may be interesting to a decision maker. We have chosen the latter that has the advantage of clearly dividing the capital invested in the portfolio and the money spent in transaction costs. In the former alternative, constraint (11) should be modified into n  {(1 + c j )q j x j + f j z j } = C. j=1 We carried out some computational experiments to compare the impact of the two modelling alternatives on the portfolio composition. The results have shown that the portfolios obtained in the two cases are practically identical. 2.2 The Portfolio Rebalancing Model The single-period CVaR(β) model with transaction costs described in the previous section assumes an investor who follows a “buy-and-hold” investment strategy. The 123 Models and Simulations for Portfolio Rebalancing 243 investor allocates the capital among n available securities by constructing a portfolio to be kept over the whole investment horizon. On the contrary, we now assume an investor who may decide to rebalance her/his portfolio to take into account new market information even if this implies additional transaction costs. Let us consider two different investment periods. At the end of the first period (time 0) the investor selects the portfolio and keeps it until the end of the second investment period (time 1) when she/he may decide to rebalance it. The decision variables x 1j , j ∈ N , represent the fractional number of stocks selected in the portfolio after rebalancing. The parameters x 0j , j ∈ N , represent the optimal portfolio selected at time 0 (portfolio composition before rebalancing). To correctly represent a portfolio rebalancing problem we need to consider both a fixed cost f j for each security j sold or purchased and a proportional cost c j . The latter is applied to the absolute difference between the amount invested in security j in the rebalanced portfolio and in the initial one: c j q j |x 1j − x 0j |, where q j represents the quotation of security j at the rebalancing time. To linearize such expression, we introduce a nonnegative variable, δ j , for each security j ∈ N , and the two following constraints: δ j ≥ (x 1j − x 0j ), δ j ≥ −(x 1j − x 0j ). At the optimum δ j , j ∈ N , will take the value |x 1j − x 0j |. Finally, to correctly model the fixed cost f j , j ∈ N , we use the binary variable z j , j ∈ N , that has here a slightly different role with respect to the basic model. z j takes value 1 if the investor buys or sells a fraction of security j and 0 otherwise:  1 if (x 1j − x 0j ) > 0 or (x 1j − x 0j ) < 0, zj = 0 otherwise. Notice that, with respect to the basic model, new upper bounds u 1j on the fractional number of stock units that can be purchased for each security j, j ∈ N , are defined. For sake of simplicity, only the decision variables x 1j and the upper bounds u 1j have been indexed at time 1, since they have a slightly different meaning than in the basic model. Indeed, one may notice that all the parameters, e.g. the set of probabilities pt and the set of quotation q j , and all the variables, e.g. non-negative variables dt and δ j , of the following optimization model are defined with respect to time 1. The rebalancing model can be formulated as follows: Rebalancing model max q − T 1 pt dt β (16) t=1 123 244 G. Guastaroba et al. q− n  r jt q j x 1j + j=1 n  cjqjδj + j=1 − r j q j x 1j j=1 n  n  n  cjqjδj − j=1 q j x 1j = j=1 zj ≥ n  n  f j z j ≤ dt t = 1, ..., T (17) j=1 n  f j z j ≥ µ0 j=1 q j x 0j + α n  q j x 1j (18) j=1 (19) j=1 (x 1j − x 0j ) zj ≥ − u 1j (x 1j − x 0j ) x 1j ≤ u 1j z j u 1j j = 1, . . . , n j = 1, . . . , n j = 1, . . . , n (20) (21) (22) δ j ≥ (x 1j − x 0j ) j = 1, . . . , n (23) δ j ≥ −(x 1j − x 0j ) j = 1, . . . , n (24) dt ≥ 0 t = 1, . . . , T (25) x 1j ≥ 0 j = 1, . . . , n (26) δj ≥ 0 j = 1, . . . , n (27) z j ∈ {0, 1} j = 1, . . . , n. (28) As for the basic model, the objective function (16), along with constraints (17) and (25), determines the maximization of the safety measure and the definition of the nonnegative variables dt , t = 1, . . . , T. The variables dt measure the deviation from the β−quantile. Constraint (18) defines the mean portfolio return based on “buy-and-sell” proportional and fixed transaction costs and imposes
that the net portfolio
return has to be greater than or equal to the required return µ0 nj=1 q j x 1j , where nj=1 q j x 1j is the amount invested in the portfolio at time 1. The budget constraint (19)
requires that the capital has to be equal to the value of the initial portfolio at time 1 ( nj=1 q j x 0j ) plus a nonnegative additional fund α, if available. As already explained, constraints (23) and (24) allow each variable δ j to take the value of the absolute difference |x 1j − x 0j |. Finally, constraints (20) and (21) force variable z j to take value 1 when |x 1j − x 0j | = 0, i.e. when the position of the security j in the portfolio has been changed. Note that if x 1j − x 0j > 0 (the investor increases her/his position in security j - buying case) then (x 1j −x 0j ) u 1j is a positive value in the range (0, 1] since u 1j is an upper bound on x 1j and x 0j . Thus, constraints (20) force z j to 1, whereas constraints (21) do not impose any real restriction requiring z j to be larger than a negative quantity. On the contrary, if x 1j − x 0j < 0 (the investor reduces her/his position in security j-selling case) then 123 Models and Simulations for Portfolio Rebalancing 245 constraints (21) force z j to 1 whereas constraints (20) are not binding. In both cases where z j = 1 the fixed cost f j is included in the net mean return expression (constraint (18)) and in the objective function through constraints (17). When x 1j = x 0j then constraints (20) and (21) let z j free to take any value. In such case, given the presence of the fixed cost in the objective function, z j will take value 0. 3 Experimental Analysis In this section we describe the computational results and compare the basic and the rebalancing models. Computational experiments have been conducted on a PC with a 2,400 MHz Intel Pentium processor and 512 Mb of RAM. The models have been implemented in C++ by means of the Concert Technology 20 and solved with CPLEX 9.0. The results discussion is organized as follows. We first describe the data sets, then we introduce a static investment strategy and analyze its behavior versus different dynamic strategies in both in-sample and out-of-sample periods. 3.1 Data Sets Historical data are represented by daily rates of return computed by using stock prices taken from the German Stock Exchange Market (XETRA), the largest stock market in the EURO area. The rate of return of security j under scenario t has been computed as q −q r jt = jtq jt−1jt−1 , where q jt is the closing price of security j in period t. No dividends have been considered. In order to consider all possible market trends we have constructed four data sets corresponding to different in-sample and out-of-sample time periods selected as follows. The first data set is characterized by a market trend going up in the in-sample period as well as in the out-of-sample one (up-up data set), the second data set by a market increasing in the in-sample period and decreasing in the out-of-sample one (up-down data set), the third data set by a market going down in the in-sample period and going up in the out-of-sample period (down-up data set) and, finally, the last set by a market going down in both the in-sample and the out-of-sample periods (down-down data set). Each of these data sets consists of 6 months of in-sample daily observations and 6 months of out-of-sample ones. The time periods covered by the four data sets are summarized in Table 1, whereas their temporal positioning is shown in Fig. 1. We have considered the 100 securities composing the XETRA DAX100 index at the date of April 1st, 2005. A security is included in the set of available alternatives only if it has been quoted with continuity on the market during the analyzed period. This means that the security has not been suspended for a period longer than 5 successive working days, and that the total number of days of suspension has not been altogether greater than 30 over the in-sample and out-of-sample periods. To reach the number of 100 securities we have substituted the excluded securities with other securities quoted on the German Stock Exchange Market, chosen following the alphabetical order. The reader should be aware that restricting attention to companies that belong to the market 123 246 G. Guastaroba et al. Table 1 The four data sets Data set Up–up Up–down Down–up Down–down In-sample (130 daily observations) Out-of-sample (130 daily observations) Begin End Begin End 01/08/1996 18/09/2001 11/09/2002 09/10/2000 30/01/1997 19/03/2002 12/03/2003 09/04/2001 31/01/1997 20/03/2002 13/03/2003 10/04/2001 31/07/1997 17/09/2002 10/09/2003 08/10/2001 6500 DAX 30 'KURS' - PRICE INDEX FROM 1/1/94 TO 30/11/04 MONTHLY 6000 up - down 5500 5000 4500 4000 up - up 3500 down - up 3000 down - down 2500 2000 1500 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Fig. 1 The four different market data sets index during the entire period of the analysis may give rise to the so called survivorship bias. This means that the selected portfolios performance may be overestimated due to the fact that only companies that were successful enough to survive until the end of the period are included. 3.2 Dynamic Versus Static Strategies: A Computational Comparison In this section we introduce the investment strategy of a static investor and compare it to the investment strategies of more dynamic investors who decide to periodically rebalance their portfolios. We define as BS, i.e. Basic Strategy, the buy-and-hold strategy implemented by the investor who is not interested in dynamically modify its portfolio when market trend changes but quietly waits for the end of the investment period. The main advantage of this static strategy is the lower amount of incurred 123 Models and Simulations for Portfolio Rebalancing 247 Fig. 2 An in-sample and out-of-sample graphical comparison between the BS(β) and the RS5(β) strategies in the down-up data set transaction costs. The Basic Strategy implies only one optimization corresponding to solve the basic model at the date of portfolio selection. On the contrary, a dynamic investor is more sensitive to market changes and accepts to pay additional transaction costs in exchange for a possibly higher portfolio return or a lower portfolio loss. We analyze different levels of dynamism according to the number of times the investor decides to rebalance her/his portfolio. We define as RSk the Rebalancing Strategy where the initial optimal portfolio composition is reoptimized k times. More precisely, in a RSk strategy the initial optimal portfolio is selected running the basic model, while each of the successive k re-optimizations is obtained by solving the rebalancing model at a different future date. We have considered four different RSk strategies corresponding to k = 1, 2, 3 and 5. In the first strategy the investor creates the initial portfolio at time 0 running the basic model on the 6 months in-sample observations, and re-optimizes the portfolio with the rebalancing model only once (k = 1) three months after the selection of the initial portfolio. In the second strategy a more dynamic investor decides to rebalance her/his portfolio twice (k = 2) exactly every two months after the initial portfolio optimization. In the third strategy the investor rebalances three times (k = 3) after 1 month and a half, after 3 months and after 4 months and a half from the initial portfolio optimization. Finally, in the last strategy, the investor rebalances the portfolio every month, that is 5 times (k = 5). The rebalancing model is always solved taking into account only three months of in-sample realizations instead of the six months realizations considered in the basic model. The choice is consistent with the behavior of a dynamic investor who pays more attention to the most recent information provided by the market. As an example, in Fig. 2 we show the in-sample and the out-of-sample periods for the BS strategy and for the RS5 strategy with respect to the down-up data set. In the following, we will denote as BS(β) the BS strategy used by the investor running the basic model with a quantile parameter equal to β. Similarly, we will denote 123 248 G. Guastaroba et al. as RSk(β) the RSk strategy where the initial optimal portfolio and the successive rebalancing optimizations are obtained running the models with the quantile parameter equal to β. For each data set we have implemented and run the BS(β) strategy and the RSk(β) strategies by considering an initial capital equal to 100,000 Euros, three different levels of minimum required return µ0 (0, 0.05 and 0.10 on yearly basis) and four different values of the quantile parameter β (0.01, 0.05, 0.10 and 0.25). We have solved each model considering a fixed cost equal to 12 Euros and a proportional cost for buying (in the basic model) and for buying/selling (in the rebalancing model) equal to the 0.195% of the amount invested. These are real conditions that Italian brokers apply for operations carried out on the German market. Moreover, in our experiments the upper bounds u j for the basic model are calculated as uj = C qj j = 1, . . . , n. (29) The same quantities for the rebalancing model are calculated considering the capital available at the time of rebalancing
n u 1j = 0 s=1 qs x s +α qj j = 1, . . . , n. (30) In our experiments we have assumed α=0. In-Sample Analysis In the following we present the characteristics of the portfolios selected by the investors using the BS(β) and the RSk(β) strategies. Since similar results have been obtained for different values of µ0 and of β, we have decided to only report the results obtained by requiring a minimum rate of return equal to 5% on a yearly basis and a level of the quantile parameter equal to 0.01 and to 0.05. All the remaining results can be found in Guastaroba et al. (2005). Each of the following tables consists of nine columns showing the objective function value (obj.), the portfolio per cent average return on yearly basis including the transaction costs paid by the investor (net return) and without considering the transaction costs (gross return), the number of securities selected by the model (div.), the minimum and the maximum shares within the portfolio, the total transaction costs paid by the investor divided into proportional and fixed transaction costs. The tables for the RSk(β) strategies contain three additional columns (indicated as “number of purchase”, “number of sale”, “cumulative costs”) representing the number of securities purchased and the number of securities sold in the current rebalancing, and the cumulative transaction costs obtained as sum of the transaction costs paid in the current rebalancing and in all the previous ones. In Table 2 we show the complete computational results obtained for the BS(β) strategy implemented in all the four different data sets. The large difference between net 123 Models and Simulations for Portfolio Rebalancing 249 Table 2 BS(β) strategy with µ0 = 5%: optimal portfolio characteristics Instances Up–up BS(0.01) BS(0.05) Up–down BS(0.01) BS(0.05) Down–down BS(0.01) BS(0.05) Down–up BS(0.01) BS(0.05) Obj. 102 Net return Gross return Div. Shares Prop. costs Min Max Fixed costs Total costs −14.297 −12.106 5.00 5.00 162.61 186.24 5 7 0.056 0.072 0.440 0.256 195 195 60 84 255 279 −14.033 −13.678 5.00 5.00 255.01 255.01 12 12 0.012 0.037 0.249 0.152 195 195 144 144 339 339 −29.782 −27.166 5.00 5.00 140.93 151.54 3 4 0.172 0.125 0.443 0.332 195 195 36 48 231 243 −16.736 −15.484 5.00 5.00 198.84 198.84 8 8 0.014 0.018 0.405 0.408 195 195 96 96 291 291 and gross portfolio rate of return is due to the presence of transaction costs and to the formula adopted to convert the rates of return on yearly basis. The number of securities in the portfolios (div.) is lower than in other experiments carried out on the same model (see for instance Mansini et al. (2005)). This can be explained by the presence of fixed transaction costs that increase proportionally to the number of securities selected. One can also observe the higher incidence of proportional transaction costs with respect to fixed ones on the total transaction costs. In terms of computational times, the basic model requires only few seconds, rarely more than one minute to find the optimal solution. In Tables 3–6 we show the results obtained for the RSk(β) strategies. We compare the portfolios obtained by the different strategies at approximately the same date. At this aim we have chosen the unique revision of the dynamic investor who rebalances the portfolio once (after 3 months), the second revision of that who rebalances twice (this means after 4 months), the third revision of the investor who rebalances three times (this means after 4 months and a half), and the fourth of the investor who rebalances five times (this means after 4 months). One may notice that, whereas in the basic model the portfolio net return is always equal to the required one, in the rebalancing model the portfolio net rate of return may yield a larger value. This is especially true for the up-up and down-up data sets and for the investor who rebalances five times. This can be explained by the fact that the in-sample observations belong to a data set where the market was increasing. For the down-up data set and the RS5(β) strategy see Fig. 2. The number of securities selected seems to increase with the number of rebalances. In particular, when the market is increasing (i.e. up-up and down-up data sets) and when new information is provided by the market, the investor purchases more securities than she/he sells. For thoroughness, we have also shown the cumulative cost paid by the investor until the current rebalance. Obviously, this cost increases with the number of rebalances. 123 123 Up–up RS1(0.01) RS1(0.05) Up–down RS1(0.01) RS1(0.05) Down–down RS1(0.01) RS1(0.05) Down–up RS1(0.01) RS1(0.05) Instances 5.00 5.00 5.00 5.00 5.00 5.00 91.88 55.00 −21.642 −22.375 −10.535 −9.390 −7.948 −7.001 Net return −12.515 −10.368 Obj. 102 569.14 424.82 226.85 197.09 99.91 169.43 170.56 195.35 Gross return 14 15 9 10 9 8 9 10 Div. 0.001 0.004 0.026 0.043 0.010 0.036 0.027 0.026 Min Shares Table 3 RS1(β) strategy with µ0 = 5%: optimal portfolio characteristics 0.406 0.409 0.333 0.292 0.268 0.268 0.464 0.270 Max 9 8 6 6 2 2 5 5 Number of purchase 6 6 3 3 4 6 3 4 Number of sale 343.78 249.46 213.93 192.48 94.79 137.03 230.80 233.35 Prop. costs 180 168 108 108 72 96 96 108 Fixed costs 523.78 417.46 321.93 300.48 166.79 233.03 326.80 341.35 Total costs 814.78 708.46 552.93 543.48 505.79 572.03 581.80 620.35 Cumulative costs 250 G. Guastaroba et al. Up–up RS2(0.01) RS2(0.05) Up–down RS2(0.01) RS2(0.05) Down–down RS2(0.01) RS2(0.05) Down–up RS2(0.01) RS2(0.05) Instances 17.71 8.58 5.00 5.00 5.77 6.31 5.00 5.00 −25.421 −24.980 −8.191 −11.251 −8.007 −7.598 Net return −9.194 −8.797 Obj. 102 299.59 336.33 126.71 106.15 270.47 117.28 123.74 102.86 Gross return 15 17 8 8 13 10 14 13 Div. 0.003 0.009 0.009 0.032 0.003 0.017 0.005 0.002 Min Shares 0.351 0.297 0.237 0.384 0.186 0.141 0.246 0.299 Max 6 7 3 4 3 2 6 4 Number of purchase Table 4 RS2(β) strategy with µ0 = 5%: optimal portfolio characteristics of the second revision 7 7 2 1 7 4 3 4 Number of sale 290.36 283.70 152.29 125.17 165.58 89.88 116.73 113.70 Prop. costs 156 168 60 60 120 72 108 96 Fixed costs 446.36 451.70 212.29 185.17 285.58 161.88 224.73 209.70 Total costs 1148.96 1056.36 696.53 647.45 757.41 650.07 914.36 804.46 Cumulative costs Models and Simulations for Portfolio Rebalancing 251 123 123 Up–up RS3(0.01) RS3(0.05) Up–down RS3(0.01) RS3(0.05) Down–down RS3(0.01) RS3(0.05) Down–up RS3(0.01) RS3(0.05) Instances 14.23 5.00 5.00 5.00 5.00 8.52 10.96 5.00 −21.411 −20.751 −10.750 −9.565 −7.653 −7.853 Net return −7.396 −6.908 Obj. 102 132.14 107.19 79.13 57.45 358.26 273.81 147.09 119.37 Gross return 21 19 9 14 11 13 18 14 Div. 0.003 0.006 0.009 0.003 0.005 0.011 0.003 0.002 Min Shares 0.197 0.382 0.360 0.308 0.145 0.169 0.220 0.214 Max 7 5 2 3 4 4 5 3 Number of purchase Table 5 RS3(β) strategy with µ0 = 5%: optimal portfolio characteristics of the third revision 3 4 4 2 6 5 6 6 Number of sale 150.21 121.87 75.30 39.80 204.17 171.19 154.23 144.72 Prop. costs 120 108 72 60 120 108 132 108 Fixed costs 270.21 229.87 147.30 99.80 324.17 279.19 286.23 252.72 Total costs 1320.20 1237.89 887.38 780.30 1022.18 957.98 1334.47 1155.13 Cumulative costs 252 G. Guastaroba et al. Up–up RS5(0.01) RS5(0.05) Up–down RS5(0.01) RS5(0.05) Down–down RS5(0.01) RS5(0.05) Down–up RS5(0.01) RS5(0.05) Instances 103.03 37.85 5.00 5.00 6.91 9.95 39.14 38.47 −18.594 −20.899 −7.774 −8.484 −7.140 −5.472 Net return −6.935 −6.800 Obj. 102 360.91 250.43 121.85 113.60 91.78 42.60 146.81 132.06 Gross return 20 19 13 11 16 20 13 18 Div. 0.010 0.005 0.014 0.009 0.005 0.003 0.001 0.006 Min Shares 0.256 0.254 0.238 0.367 0.157 0.148 0.204 0.174 Max 7 8 3 3 2 1 3 4 Number of purchase Table 6 RS5(β) strategy with µ0 = 5%: optimal portfolio characteristics of the fourth revision 5 3 6 5 5 2 1 4 Number of sale 269.30 162.84 96.38 89.72 53.15 33.64 14.97 74.42 Prop. costs 144 132 108 96 84 36 48 96 Fixed costs 413.30 294.84 204.38 185.72 137.15 69.64 62.97 170.42 Total costs 1560.26 1406.32 1090.93 1044.59 1103.99 817.44 1190.99 1347.27 Cumulative costs Models and Simulations for Portfolio Rebalancing 253 123 254 G. Guastaroba et al. By comparing Tables 3–6, one can see that the objective function value usually increases with the number of rebalances. This is an expected result since to operate more rebalances means to benefit from newer information. Out-of-Sample In the ex-post analysis, we have examined the behavior of all the portfolios selected by using the BS(β) strategy and all the RSk(β) strategies for the six months following the date of the initial portfolio optimization. For each model and data set we have computed a table including nine ex-post parameters defined as follows: the number of times out of 130 the portfolio rate of return outperforms the corresponding required rate of return (N ), the average portfolio rate of return (rav ), the median (rmed ) of the returns, the standard deviation (std) and the semi-standard deviation (s-std), the mean absolute deviation (MAD) and the mean downside deviation (s-MAD), the maximum downside deviation (D-DEV) and the Sortino index. The latter provides a measure of the over-performance of the portfolio mean rate of return with respect to the required one per unit of downside risk (measured by the semi-standard deviation). Such performance criteria are used to compare the out-of-sample behavior of the portfolios selected by using the BS(β) strategy and the RSk(β) strategies, respectively. The average return and the median are expressed on yearly basis. All the dispersion measures (std, s-std, MAD, s-MAD and D-DEV) and the Sortino index have been computed with respect to the minimum required rate of return to make them directly comparable in the different models. In Figs. 3–6 the portfolio performances have been analyzed and compared in terms of cumulative returns. Let us define as r1 , r2 ,…, rm the daily ex-post portfolio rates of return over m periods. Then, the cumulative portfolio rate of return in period t, t ≤ m, is equal to: (31) ((1 + r1 )(1 + r2 ) . . . (1 + rt )) − 1. This allows us to consider the evolution of the portfolio value over time, and allows a complete comparison between the portfolios selected by using the BS(β) strategy with respect to those selected by using the RSk(β) strategies. We first comment Tables 7 and 8, and then Figs. 3–6. In Tables 7 and 8 we show all the performance measures and the dispersion parameters we have computed. In particular, we compare the BS(β) strategy with the RSk(β) strategies, and show their performances with respect to the market behavior of the DAX30 index. It is worth observing that when the market index is decreasing in the ex-post period (see the up-down and down-down data sets), in terms of average returns all the models dominate the market index. This is in particular true for the up-down data set where all the models show an average negative return significantly larger than the market index performance with the gap almost equal to 20% (see also Fig. 4). One may also notice that in the same data sets the Sortino index computed on DAX30 is greater than that obtained using the BS(β) and the RSk(β) strategies. Nevertheless, only following the Sortino index indications may be misleading. In fact, a strategy characterized by higher downside risk and worse negative average return may 123 Models and Simulations for Portfolio Rebalancing 255 Fig. 3 Cumulative returns (up-up data set): A comparison between portfolio optimization models and the market index Fig. 4 Cumulative returns (up-down data set): A comparison between portfolio optimization models and the market index show a better Sortino index than another strategy that shows a better risk measure and a better average return. For this reason we mediate the Sortino index with the cumulative returns. Tables 7 and 8 show that the downside risk incurred by the market index, in particular when the out-of-sample period is down (i.e. in the down-down and the up-down data sets), is greater than that incurred by using both the BS(β) and the RSk(β) strategies. This is also true when the out-of-sample period is up (i.e. in the up-up and the down-up data sets) and can be seen by comparing the risk incurred by the DAX30 123 256 G. Guastaroba et al. Fig. 5 Cumulative returns (down-down data set): A comparison between portfolio optimization models and the market index Fig. 6 Cumulative returns (down-up data set): A comparison between portfolio optimization models and the market index and that incurred by the RSk(β) strategies. This confirms that portfolio optimization models allow a reduction of the risk incurred by the investor. One may also notice that the risk, and in particular the downside risk represented by the s-std and the s-MAD columns, tends to decrease with the number of revisions. This result can be explained considering the attempt of the rebalancing model to reduce the downside risk incurred by the portfolio at each revision. We have also observed that, in the up-down data set, the RS5(β) strategy cannot find a feasible integer solution in the fifth revision. In Tables 7 and 8 we have used symbol “*” to indicate such occurrence. 123 Models and Simulations for Portfolio Rebalancing 257 Table 7 Out-of-sample statistics for µ0 = 0.05 and β = 0.01 Model Up–up BS(0.01) RS1(0.01) RS2(0.01) RS3(0.01) RS5(0.01) dax30 Up–down BS(0.01) RS1(0.01) RS2(0.01) RS3(0.01) RS5(0.01) dax30 Down–down BS(0.01) RS1(0.01) RS2(0.01) RS3(0.01) RS5(0.01) dax30 Down–up BS(0.01) RS1(0.01) RS2(0.01) RS3(0.01) RS5(0.01) dax30 N rav rmed std s-std MAD s-MAD D-DEV Sortino index 66 66 71 68 67 84 196.29 149.18 162.87 187.58 77.58 186.89 0 10.83 58.55 33.04 10.56 321.93 0.0191 0.0166 0.0157 0.0154 0.0109 0.0117 0.0098 0.0080 0.0075 0.0069 0.0062 0.0072 0.0128 0.0105 0.0096 0.0091 0.0078 0.0091 0.0049 0.0041 0.0035 0.0032 0.0032 0.0031 0.0516 0.0396 0.0396 0.0396 0.0326 0.0390 0.0596 0.0872 0.1151 0.1650 0.0865 0.1463 55 55 56 55 * 49 −50.63 −46.99 −55.96 −51.30 * −73.83 −79.91 −58.17 −57.91 −51.43 * −85.18 0.0131 0.0143 0.0136 0.0137 * 0.0248 0.0102 0.0108 0.0106 0.0104 * 0.0185 0.0101 0.0113 0.0108 0.0104 * 0.0189 0.0061 0.0066 0.0066 0.0063 * 0.0114 0.0341 0.0330 0.0339 0.0444 * 0.0584 −0.0147 −0.0112 −0.0140 −0.0139 * −0.0011 58 57 62 60 62 60 −34.18 −31.78 −27.12 −27.18 −36.92 −49.23 3192.81 −37.70 −7.55 −19.74 −1.70 −29.52 0.0146 0.0134 0.0109 0.0120 0.0108 0.0187 0.0118 0.0104 0.0083 0.0094 0.0088 0.0149 0.0101 0.0098 0.0080 0.0086 0.0079 0.0133 0.0057 0.0055 0.0045 0.0048 0.0046 0.0077 0.0911 0.0694 0.0363 0.0571 0.0411 0.0850 −0.0058 −0.0092 −0.0175 −0.0116 −0.0189 −0.0026 75 82 77 82 79 70 425.45 486.93 167.91 198.04 146.60 273.33 −99.46 156.14 120.30 159.04 142.39 98.95 0.0232 0.0149 0.0111 0.0105 0.0104 0.0202 0.0135 0.0050 0.0058 0.0052 0.0057 0.0119 0.0161 0.0095 0.0077 0.0072 0.0075 0.0154 0.0058 0.0023 0.0026 0.0022 0.0025 0.0059 0.0608 0.0224 0.0340 0.0339 0.0296 0.0615 0.0334 0.9666 0.2334 0.3542 0.2073 0.0390 In Figs. 3–6 we present the cumulative returns in the four different market periods with a minimum required rate of return equal to 5% on yearly basis and a quantile parameter β = 0.05. In all the figures the DAX30 shows a cumulative return extremely unstable. See, for instance, Figure 3 where the market index reaches a cumulative return that is greater than that of all the portfolio optimization models in the first 27 ex-post realizations. Later it decreases reaching the worst cumulative return from period 27 to period 57. Then, from period 57 to 120 it yields a cumulative return only better than that yielded using the RS5(β) strategy and finally it has a strong increase but it does not even reach the final cumulative return obtained by the BS(β) strategy. The high volatility of index ex-post returns is also confirmed by the dispersion measures showed in Tables 7 and 8. In Fig. 6 we show the ex-post cumulative returns for the down-up data set. In this data set the market index reaches a cumulative return always better than that yielded by all the optimal portfolios. By comparing the figures in Table 8 it is evident that it incurs a higher level of downside risk. As a consequence, the DAX30 has the worst value for Sortino index, clearly lower than those associated to the optimal portfolios. In particular, still referring to the down-up data set, we have observed that through an appropriate choice of the quantile parameter value (i.e. β = 0.01) the cumulative return reached by the RS1(β) strategy is better than that reached by the market index, 123 258 G. Guastaroba et al. Table 8 Out-of-sample statistics for µ0 = 0.05 and β = 0.05 Model Up–up BS(0.05) RS1(0.05) RS2(0.05) RS3(0.05) RS5(0.05) dax30 Up–down BS(0.05) RS1(0.05) RS2(0.05) RS3(0.05) RS5(0.05) dax30 Down–down BS(0.05) RS1(0.05) RS2(0.05) RS3(0.05) RS5(0.05) dax30 Down–up BS(0.05) RS1(0.05) RS2(0.05) RS3(0.05) RS5(0.05) dax30 N rav rmed std s-std MAD s-MAD D-DEV Sortino index 72 72 76 77 76 84 177.48 144.17 107.18 125.69 127.45 186.89 0 103.59 91.98 130.43 106.42 321.93 0.0134 0.0112 0.0106 0.0102 0.0105 0.0117 0.0078 0.0066 0.0066 0.0062 0.0061 0.0072 0.0099 0.0081 0.0079 0.0079 0.0073 0.0091 0.0036 0.0029 0.0030 0.0029 0.0026 0.0031 0.0402 0.0402 0.0402 0.0348 0.0385 0.0390 0.1126 0.1443 0.1045 0.1470 0.1517 0.1463 48 53 54 51 * 49 −54.01 −54.70 −52.75 −43.83 * −73.83 −70.41 −50.97 −49.02 −50.84 * −85.18 0.0126 0.0142 0.0136 0.0140 * 0.0248 0.0099 0.0110 0.0104 0.0102 * 0.0185 0.0098 0.0111 0.0105 0.0104 * 0.0189 0.0060 0.0067 0.0064 0.0061 * 0.0114 0.0352 0.0326 0.0435 0.0424 * 0.0584 −0.0177 −0.0118 −0.0141 −0.0130 * −0.0011 60 58 64 60 58 60 −31.75 −39.33 −28.01 −33.07 −44.09 −49.23 69731.25 −21.31 0 −20.39 −37.47 −29.52 0.0129 0.0128 0.0117 0.0110 0.0127 0.0187 0.0100 0.0098 0.0088 0.0083 0.0104 0.0149 0.0087 0.0090 0.0083 0.0082 0.0090 0.0133 0.0049 0.0052 0.0047 0.0047 0.0054 0.0077 0.0631 0.0473 0.0406 0.0403 0.0625 0.0850 −0.0105 −0.0137 −0.0145 −0.0203 −0.0120 −0.0026 77 78 80 82 78 70 143.32 165.68 113.14 139.27 87.36 273.33 −90.16 115.90 94.11 119.23 82.29 98.95 0.0135 0.0096 0.0097 0.0091 0.0084 0.0202 0.0084 0.0045 0.0056 0.0051 0.0052 0.0119 0.0098 0.0069 0.0072 0.0064 0.0060 0.0154 0.0038 0.0022 0.0026 0.0021 0.0022 0.0059 0.0406 0.0175 0.0316 0.0371 0.0362 0.0615 0.0723 0.4025 0.1653 0.2548 0.1500 0.0390 with a lower level of downside risk (see the average rate of return and the downside risk measures shown in Table 7). All computational results show that the risk management obtained by means of a portfolio optimization model provides a good strategy when the market is increasing and becomes of crucial importance when the market is decreasing. Figures 4 and 5 show the cumulative returns associated to the market index and to the optimal portfolios in the up-down and in the down-down data sets, respectively. In particular, in Fig. 4 the DAX30 has, on average, a cumulative return clearly worse than that obtained by the optimal portfolios. After the first 7 periods, where the portfolios selected by the optimization models and the DAX30 follow the same trend, the market index approximately yields the worst cumulative returns in all the remaining ex-post periods. This is also true for all the other experiments carried out in the up-down and in the down-down data sets (Fig. 5), for different values of µ0 and of β. The DAX30 shows not only worse cumulative returns than those associated to the portfolios selected by using both the BS(β) and the RSk(β) strategies, but also a higher volatility in the ex-post periodic returns. This can be noticed by analyzing Figs. 4 and 5 and also by comparing Tables 7 and 8. The first aim of this analysis is to understand if the portfolio optimization management represents a valid tool for financial decisions, and then to understand if any 123 Models and Simulations for Portfolio Rebalancing 259 dynamic strategy (i.e. RSk(β) strategy) is better than the static strategy (i.e. BS(β) strategy). The second aim is to provide some general guidelines about the best strategy to follow on the basis of the information available in the in-sample period. As a conclusion, we can say that it is evident that portfolio optimization models represent valuable tools for a rational risk averse investor. Moreover, in the up-down and in the down-down data sets, the downside risk incurred by the market index is clearly higher than that of all the optimal portfolios, with the worst average return and the worst cumulative returns. Similar conclusions can be drawn for the up-up and the down-up data sets. The use of a portfolio optimization model, and in particular the use of a RSk(β) strategy, determines a strong reduction of the instability of the portfolio return. Only in some experiments the cumulative return associated to the market index is better than that reached by an optimal portfolio. If the quantile parameter β is set equal to 0.01 the cumulative return associated to the BS(β) and the RS1(β) strategies are, in almost all the ex-post periods, better than that yielded by the market index. With respect to a comparison between static and dynamic strategies we can draw some conclusions by observing the market trend and the portfolio performances. If the aim of the investor is to minimize the portfolio return instability, rebalancing is the best strategy to follow. By increasing the number of rebalances the investor tends to reduce the incurred downside risk. This is the correct strategy for a very risk averse investor. If the investor is less risk averse and willing to take more risk to possibly yield a higher rate of return we can draw some other conclusions by observing the market trend. If the market is going up the best choice is to implement a BS(β) or RS1(β) strategy (i.e. see Figs. 3 and 6). On the contrary, when the market is going down the investor best choice is to implement a RSk(β) strategy, and in particular the RS2(β) or the RS3(β) strategy (i.e. see Figs. 4 and 5). Obviously, nobody knows what will happen in the future at the time the portfolio is chosen. For this reason we try to draw some guidelines about the strategy to follow for the investor on the basis of the information available in the in-sample period. If the market is increasing in the in-sample period, the very risk averse investor would like to hedge her/his investments from a possibly dramatic fall in market quotations during the out-of-sample period. Thus, as already pointed out, this investor should choose to implement the RS2(β) or the RS3(β) strategy in the up period and also in the down period being the two strategies which can better hedge the investment in case of a possible market fall. On the contrary, a less risk averse investor is willing to take more risk in order to possibly gain from an increasing market trend, thus she/he should implement a BS(β) or RS1(β) strategy because these are the two choices that yield the higher average and cumulative returns. 3.3 Impact of the Amount of Transaction Costs In order to evaluate the impact of the amount of the transaction costs on the portfolio, we have compared the portfolios obtained with different amounts of the transaction costs. We have considered a foreign investor and a local one, who incur in different transaction costs. The former pays the transaction costs considered in Sect. 3.2, 123 260 G. Guastaroba et al. Table 9 Impact of the amount of transaction costs (down–down data set) BS(0.05) 1st rev. RS5(0.05) 2nd rev. RS5(0.05) 3rd rev. RS5(0.05) 4th rev. RS5(0.05) 5th rev. RS5(0.05) Cumulative costs Transaction costs (Transaction costs)/2 Div. Total costs Div. Total costs 4 7 8 12 11 13 243.00 216.70 134.53 264.64 185.72 120.54 1165.13 5 6 7 9 12 12 127.50 110.12 78.79 195.16 153.97 160.47 826.01 Number of common securities 2 4 4 3 4 4 Fig. 7 Cumulative returns (down-down data set): Impact of the amount of transaction costs whereas the latter pays half of that amount (i.e. fixed costs equal to 6 Euros and proportional costs equal to 0.0975%). Both investors implement a rebalancing strategy where the initial portfolio is re-optimized 5 times, with β = 0.05 and µ0 = 5%. The test has been carried out on the down-down data set. The first row of Table 9 corresponds to the initial portfolio, while the successive rows to the re-optimized portfolios. The two central parts of Table 9 report information on the portfolios obtained by the foreign investor and by the local one. For each of the two cases, the number of securities selected (div.) and the total amount of fixed plus proportional transaction costs (total costs) are shown. In the far right column of Table 9 the number of securities selected by both investors (number of common securities) is shown. Comparison of the portfolios obtained by the two investors and the small number of common securities show that the portfolio is very sensitive to the amount of the transaction costs. In Fig. 7 we have compared the out-of-sample net cumulative returns of the portfolios obtained by the two investors and the market index. As expected, the portfolio obtained with lower transaction costs has a better out-of-sample behavior. 123 Models and Simulations for Portfolio Rebalancing 261 4 Conclusions In this paper we have shown how to modify a buy-and-hold optimization model to provide dynamic portfolio optimization strategies. Extensive computational results on real-life data from German Stock Exchange Market have allowed us to draw some interesting conclusions, the most important of which is that portfolio optimization models represent effective tools to help the investor in making financial decisions. More precisely, different models can be used according to the level of risk aversion of the investor. For a very risk averse investor the best choice in order to hedge the portfolio is to implement a RSk(β) strategy and to rebalance two or three times in 6 months. On the other side, a less risk averse investor should prefer to invest through the BS(β) or the RS1(β) strategy. 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