Mean-variance portfolio rebalancing with transaction costs and funding changes

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1 Journal of the Operational Research Society (2011) 62, Operational Research Society Ltd. All rights reserved /11 Mean-variance portfolio rebalancing with transaction costs and funding changes JJ Glen University of Edinburgh Business School, Edinburgh, UK Investment portfolios should be rebalanced to take account of changing market conditions and changes in funding. Standard mean-variance (MV) portfolio selection methods are not appropriate for portfolio rebalancing, as the initial portfolio, change in funding and transaction costs are not considered. A quadratic mixed integer programming portfolio rebalancing model, which takes account of these factors is developed in this paper. The transaction costs in this portfolio rebalancing model are composed of fixed charges and variable costs, including the market impact costs associated with large market trades of individual securities, where these variable transaction costs are assumed to be non-linear functions of traded value. The use of this model is demonstrated and it is shown that when initial portfolio, funding changes and transaction costs are taken into account in portfolio construction and rebalancing, MV efficient portfolios that include risk-free lending do not have the structure expected from portfolio theory. Journal of the Operational Research Society (2011) 62, doi: /jors Published online 15 December 2010 Keywords: finance; portfolio rebalancing; quadratic programming Introduction Both reward and risk should be considered in planning investment portfolios. The most widely used portfolio planning methods are based on the mean-variance (MV) approach to portfolio selection (Markowitz, 1952) in which reward and risk are measured by expected return and variance of return, respectively. MV analysis is used to select efficient portfolios, that is portfolios with minimum variance of return for a specified expected return or maximum expected return for a given variance, where portfolios are normally defined in terms of the proportion by value of securities, for example stocks and bonds, in the portfolio. MV efficient portfolios can be determined by solving a quadratic programming (QP) model with minimisation of variance as its objective and expected return as a constraint. The efficient frontier can be determined either by solving this QP model for different values of expected return or by using a parametric QP model with risk adjusted expected return, that is a function of variance of return, expected return and a risk-return trade-off parameter, as its objective function (Markowitz, 1959). In using standard MV analysis for portfolio construction, the transaction costs associated with purchasing a security are either ignored or are assumed to be proportional to the security s price and this proportional transaction cost is subtracted from the expected return used in the MV model (eg Mulvey, 1993). In practice, however, transaction costs may not be proportional to purchase value, and since these costs must Correspondence: JJ Glen, University of Edinburgh Business School, 50 George Square, Edinburgh, EH8 9J4, UK. generally be paid at or about the time of purchase, transaction costs reduce portfolio return by reducing the amount available for investment rather than by decreasing the return on individual securities. Therefore, even if transaction costs are proportional to security prices, subtracting these costs from expected returns is only an approximate method for taking account of transaction costs in MV analysis. Once a portfolio has been established it should be rebalanced regularly to take account of changes in the market and the availability of new funds, including dividends from the portfolio, or a need to release funds. In rebalancing a portfolio, the initial portfolio, the change in funding and the transaction costs associated with buying and selling securities should be considered. These costs may not be proportional to traded value due to fixed buying and selling charges and market impact costs associated with the adverse price changes that occur in large market trades of individual securities. Market impact costs, which may be regarded as the costs of identifying and providing incentives to sellers or purchasers when the value of a security to be bought or sold is larger than normal traded value, will generally be non-linear functions of traded value. Standard MV methods are not appropriate for portfolio rebalancing, as the initial portfolio, funding changes and transaction costs are not considered. A number of dynamic portfolio models have been proposed, but as with early multiperiod (eg Mossin, 1968) and continuous time (eg Merton, 1969) models, later dynamic models either ignore transaction costs (eg Li and Ng, 2000) or assume these costs are proportional to traded value (eg Muthuraman and Zha, 2008). In

2 668 Journal of the Operational Research Society Vol. 62, No. 4 addition, these dynamic models do not consider exogenous changes in funding and the nature of these models is such that they can only be applied to problems with a small number of securities. As it is simpler to incorporate transaction costs in the single period MV model framework, this model has been used as the foundation for portfolio rebalancing models. For example, Pogue (1970) proposed an extension of the MV model to take account of the initial portfolio, transaction costs, capital gains taxes, dividend taxation and financing costs, but fixed transaction costs were not considered and simultaneous purchase and sale of the same security were not prohibited. The importance of developing methods for solving large single period MV portfolio rebalancing models that incorporate transaction costs has also been recognised (eg Perold, 1984; Best and Hlouskova, 2005). However, Perold (1984) defined portfolios in terms of each security s proportion by value and so this model is not suitable for rebalancing problems in which transaction costs are nonlinear functions of traded value, whereas Best and Hlouskova (2005) did not include an income and expenditure balance constraint in their model. The influence of transaction costs, including market impact costs, in portfolio rebalancing was incorporated in the mixed integer programming approach of Bertsimas et al (1999), which used indirect penalties to take account of transaction costs and costs associated with the number of securities in a portfolio. The number of securities in a portfolio can be incorporated directly into MV analysis by using a quadratic mixed integer programming (QMIP) formulation in which a binary variable is associated with each security. Binary variables of this type were included in the QMIP portfolio selection model proposed by Chang et al (2000), but transaction costs were not included in this model. Jobst et al (2001) incorporated minimum purchase levels, restrictions on the number of securities, and roundlot purchase constraints, that is purchases must be multiples of a specified minimum value, in QMIP models for portfolio construction and rebalancing, but transaction costs were not included. In order to address the limitations of MV methods, particularly in portfolio rebalancing, a QMIP model for MV portfolio rebalancing is developed in this paper. Standard MV methods are first outlined and the QMIP portfolio rebalancing model is then described. This QMIP model takes account of the initial portfolio, funding changes and transaction costs, where transaction costs are composed of the fixed charges and variable costs, including market impact costs, associated with buying and selling securities. The use of this QMIP portfolio rebalancing model is demonstrated and the portfolios generated by the model are compared with the portfolios selected by standard MV analysis. Standard MV analysis Assume there are n securities under consideration for investment in the portfolio, with security i, i = 1, 2,...,n, having Expected Return Standard Deviation of Return Figure 1 The efficient frontier. single period expected return μ i, and let σ ij denote the covariance between the returns on securities i, i = 1, 2,...,n, and j, j = 1, 2,...,n, where, in particular, σ ii denotes the variance in the return on security i. Thenifw i, i = 1, 2,...,n, is the proportion by value of security i in the portfolio, the expected return, E, and variance of return, V, are given by: E = n μ i w i i=1 V = n σ ij w i w j i=1 j=1 Assuming that short selling is not permitted, that is w i 0, i = 1, 2,..., n, the MV efficient portfolio with expected return E can be obtained by solution of a QP model: Minimise σ ij w i w j (1a) subject to i=1 j=1 μ i w i = E i=1 w i = 1 i=1 w i 0, i = 1, 2,...,n (1b) (1c) Note that short selling can be permitted by defining w i, i = 1, 2,...,n, as free variables, although in practice restrictions on short sales would be required (eg Jacobs et al, 2005). The efficient frontier, which can be determined by solving the standard MV model (1) for different values of expected return, E, is generally represented by plotting expected return against standard deviation of return, as in Figure 1. Investment in a risk-free asset, for example short-term lending to a bank, is not considered in MV model (1). The impact of riskfree lending on portfolio construction is usually considered by plotting the efficient frontier without risk-free lending (Figure 1), and drawing the tangent from the risk-free return on the expected return axis to the efficient frontier without risk-free lending (eg Elton et al, 2007). The efficient frontier with risk-free lending is then the line from the risk-free

3 JJ Glen Mean-variance portfolio rebalancing 669 return on the expected return axis to the point of tangency on the efficient frontier without risk-free lending, with the remainder of the efficient frontier with risk-free lending being the same as the efficient frontier without risk-free lending beyond the point of tangency. Portfolios on the linear section of the efficient frontier with risk-free lending are composed of linear combinations of the tangency portfolio and risk-free lending (eg Elton et al, 2007). Alternatively, MV model (1) can be extended to include risk-free lending by, for example, defining risk-free lending as security 0 and letting w 0 denote the proportion by value of risk-free lending in the portfolio, where risk-free lending has single period return μ 0, μ 0 > 0, and zero variance/covariance of return, that is σ 0 j = 0, j = 0, 1, 2,...,n. Transaction costs are not included explicitly in standard MV models, but it is often assumed in MV analysis that transaction costs are proportional to traded value and each security s expected return is reduced by the proportional transaction cost (eg Mulvey, 1993). This approximate approach is, however, not appropriate when transaction costs include a fixed charge or are non-linear functions of traded value. In addition, as standard MV models do not take account of any existing portfolio and funding changes, these models are not suitable for portfolio rebalancing. A portfolio rebalancing model which incorporates initial portfolio, funding changes and transaction costs is now developed. The QMIP portfolio rebalancing model Suppose that a portfolio of known composition is to be rebalanced for MV efficiency with change F in funding, that is new funds of value F available if F > 0 and funds of value F to be withdrawn if F < 0. In this portfolio rebalancing problem, it is assumed that short selling is not permitted and that the transaction costs associated with buying and selling each security may be non-linear functions of the security s traded value. Methods for collecting transaction cost data have been suggested (eg Collins and Fabozzi, 1991), but market impact costs, which are nonlinear functions of traded value, are more difficult to determine than other components of transaction costs (eg Grinold and Kahn, 1999). Several models, some of which are propriety products, have been proposed for assessing market impact costs (eg Almgren, 2003). For example, Grinold and Kahn (1999) argue that market impact cost per share varies with the square root of traded value, whereas in the Deutsche Bank market impact model (Ferraris, 2008) market impact cost per share is a concave function of traded value expressed as a percentage of average daily traded value. For both these market impact models it can be shown that the total market impact costs of buying and selling a security are convex increasing functions of traded value. As other elements of total variable transaction costs such as taxes are also convex, it is assumed in developing and testing the portfolio rebalancing model that the total variable transaction costs of buying and selling a security are convex functions of traded value. Development of the QMIP model For portfolio rebalancing problems of this form, portfolios must be defined in terms of the market value of the investment in each security, with reward and risk measured by the expected total return from the portfolio and the variance of total portfolio return, respectively. In addition, risk-free lending must be included in the set of investment opportunities to ensure productive use of resources. Assume that at the time of rebalancing the market value of the investment in security i, i =0, 1,...,n, in the initial portfolio is H i, H i 0, where, in particular, H 0 is the value of the risk-free component. For security i, i = 0, 1,...,n, let x i, x i 0, denote the market value of the investment in this security in the rebalanced portfolio, with y i, y i 0, denoting the market value of purchases and z i, z i 0, denoting the market value of sales. The market value of each security in the rebalanced portfolio must equal the market value of the security in the initial portfolio plus the market value of purchases less the market value of sales, so that x i = H i + y i z i i = 0, 1,...,n (2) It is assumed that only a fixed charge, which may be zero, is incurred in buying or selling the risk-free asset, while for each risky asset, that is for security i, i = 1, 2,...,n, it is assumed that when the security is traded, the transaction costs are composed of a fixed charge plus variable transaction costs which include market impact costs. It is also assumed that for security i, i = 1, 2,...,n, the variable transaction costs for purchases and sales are convex functions that are approximated by piecewise-linear functions. For simplicity, variable transaction costs as a function of purchase value are represented by the same piecewise-linear function for each security and similarly the variable transaction cost functions for sales are represented by the same piecewise-linear function for each security. Assume that there are q segments in the piecewise-linear approximation of variable transaction costs for purchases of security i, i = 1, 2,...,n, with breakpoints at purchase values B Pk, k = 0, 1,...,q, whereb P0 = 0and B Pk > B P,k 1, k = 1, 2,...,q, and let C Pk, k = 0, 1,...,q, denote the total variable transaction cost for purchase value B Pk (see Figure 2). In the piecewise linearisation of variable transaction costs for purchases of security i, i =1, 2,...,n, let u ik denote the weight attached to breakpoint k,, 1,...,q, where these weights are such that the market value, y i,of purchases of security i is a linear combination of the purchase value at two adjacent breakpoints, that is y i = B Pk u ik (3a) where u ik = 1 (3b)

4 670 Journal of the Operational Research Society Vol. 62, No. 4 Variable Transaction Cost C P 3 C P 2 Figure 2 costs. C P 1 B P 1 B P 2 B P 3 Purchase Value Piecewise-linear approximation of variable transaction with at most two adjacent members of the set of variables {u i0, u i1, u i2,...,u iq } non-zero valued. The total variable transaction cost, t Pi, for purchases of security i, i=1, 2,...,n, is then given by the piecewise-linear approximation, t Pi = C Pk u ik (4) The total variable transaction cost for sales of security i, i = 1, 2,...,n, is approximated in a similar way by a piecewise-linear function in r segments with breakpoints at sales values B Sk, k = 0, 1,...,r, where B S0 = 0 and B Sk > B S,k 1, k = 1, 2,...,r, with C Sk, k = 0, 1,...,r, denoting the total variable transaction cost for sales value B Sk and v ik, k = 0, 1,...,r, denoting the weight of breakpoint kin this piecewise linearisation. Although it has been assumed that variable transaction costs are convex, nonconvex transaction costs can be approximated in the same way by piecewise-linear functions. Let A Pi and A Si, respectively, denote the fixed charges associated with buying and selling security i, i = 0, 1,...,n. To incorporate fixed charges and constraints that prevent simultaneous purchase and sale of the same security in the portfolio rebalancing model, define binary variables δ i and γ i for security i, i = 0, 1,...,n, such that δ i = 1 if purchases of this security are made and δ i = 0 otherwise, while γ i = 1 if sales of this security occur and γ i = 0 otherwise. The logical conditions associated with binary variables δ i and γ i, i =0, 1,...,n, can be modelled by two pairs of constraints (eg Williams, 1993): Assume that the expected total net return required from the rebalanced portfolio is R, where this expected net return is obtained by considering the return required on the nominal initial value, that is the market value of the initial portfolio plus the change in funding. The expected total gross return from the rebalanced portfolio must therefore equal the required expected total net return, R, plus fixed and variable transaction costs, where the variable buying and selling transaction costs for each security i, i = 1, 2,...,n, are approximated by piecewise-linear functions such as (4), so that μ i x i = R + + (A Pi δ i + A Si γ i ) C Pk u ik + r C Sk v ik (6) with constraints such as (3) required for the piecewise linearisation of variable transaction costs. In addition, the total expenditure, that is market value of purchases plus fixed and variable transaction costs, must equal the income from the sale of securities plus the change in funding, F. Therefore, using piecewise-linear functions such as (4) to approximate the variable buying and selling transaction costs for each security i, i = 1, 2,...,n, it is required that (y i + A Pi δ i + A Si γ i ) + + r C Sk v ik = C Pk u ik z i + F (7) The QMIP model for rebalancing a portfolio with funding change F to create an MV efficient portfolio with expected total net return R must include constraint (6) to generate the required return, constraint (7) to balance income and expenditure, and constraint set (2) to balance transactions in each security. The model must also include constraints for piecewise linearisation of transaction costs, for example (3), and the conditions (5) associated with buying and selling securities. Modifying the form of constraints (6), (7) and (2), the QMIP portfolio rebalancing model is then: y i Uδ i 0 and y i εδ i 0 (5a) z i Uγ i 0 and z i εγ i 0 (5b) where U, U > 0, is large, for example upper bound on investment in any security, and ε, ε > 0 is small. Simultaneous purchase and sale of the same security can then be prevented by the constraint set δ i + γ i 1 i = 0, 1,...,n (5c) Minimise subject to σ ij x i x j (8a) i=1 j=1 (μ i x i A Pi δ i A Si γ i ) C Pk u ik r C Sk v ik = R (8b)

5 JJ Glen Mean-variance portfolio rebalancing 671 (y i z i + A Pi δ i + A Si γ i ) + + C Pk u ik r C Sk v ik = F (8c) x i y i + z i = H i i = 0, 1,...,n (8d) y i B Pk u ik = 0 i = 1, 2,...,n (8e) z i r B Sk v ik = 0 i = 1, 2,...,n (8f) u ik = 1 i = 1, 2,...,n (8g) r v ik = 1 i = 1, 2,...,n (8h) y i Uδ i 0 i = 0, 1,...,n (8i) y i εδ i 0 i = 0, 1,...,n (8j) z i Uγ i 0 i = 0, 1,...,n (8k) z i εγ i 0 i = 0, 1,...,n (8l) δ i + γ i 1 i = 0, 1,...,n (8m) x i 0, y i 0, z i 0, δ i = 0, 1, γ i = 0, 1, i = 0, 1,...,n u ik 0, i = 1, 2,...,n, k = 0, 1,...,q; v ik 0 i = 1, 2...,n, k = 0, 1,...,r where for each set of variables {v i0,v i1,v i2,...,v iq } and {u i0, u i1, u i2,...,u ir }, i = 1, 2,...,n, at most two adjacent variables are non-zero. In practice, it is clearly only necessary to apply the portfolio rebalancing model (8) if the required expected total net return, R, from the rebalanced portfolio exceeds the total return, R L, that can be obtained by selling all the risky assets in the initial portfolio and investing entirely in risk-free lending. In solving the portfolio rebalancing model (8) for R > R L,itis not necessary explicitly to represent the requirement that at most two adjacent variables in each set {u i0, u i1, u i2,...,u iq } and {v i0,v i1,v i2,...,v ir }, i = 1, 2,...,n, are non-zero, as the convex form of the variable transaction costs will ensure that these adjacency requirements are satisfied. Note also that the QMIP model (8) can be used to take account of funding level and transaction costs in constructing new portfolios from scratch by setting the initial portfolio to zero, that is H i = 0, i = 0, 1, 2,...,n, and setting F to the value of the new funds available. If required, model (8) can be extended to take account of policy considerations, for example limits on the total expenditure on transaction costs, and factors such as taxation. Use of the portfolio rebalancing model The portfolio rebalancing model (8) has been applied to a small test data set with eight asset classes and six larger data sets in which the number of stocks ranged from 30 to 225. The small test data set, which was used by Elton et al (2007) to illustrate derivation of the efficient frontier, consists of annual return, standard deviation of annual return and the correlation coefficients for eight asset classes. The 30-stock data set comprises monthly returns and covariances of returns for 30 FTSE stocks from Jobst et al (2001). The other five data sets are composed of weekly returns and covariances of returns from five different markets (31 Hang Seng stocks, 85 DAX stocks, 89 FTSE stocks, 98 S&P stocks and 225 Nikkei stocks) as used by Chang et al (2000). For each of these data sets, rebalanced portfolios yielding a specified set of expected returns were generated for a number of different initial portfolios and changes in funding, with transaction costs for each stock consisting of a fixed charge plus variable transaction costs, including market impact costs, where the variable transaction costs were represented by piecewise-linear functions in six segments. The portfolio rebalancing models were solved using Xpress-MP (Dash Optimization, 2007) on a 3.2 GHz Pentium 4 personal computer. In the development environment, the solution times for individual rebalancing problems ranged from less than 1 to 15 s. The results obtained on these seven data sets were broadly similar, but for ease of presentation only results for the small test data set are considered. This small data set (Elton et al, 2007) for eight asset classes numbered 1 8 (US large stocks, US bonds, Canadian stocks, Japanese stocks, emerging market stocks, Pacific stocks, European stocks and US small stocks, respectively) is listed in Table 1. The initial portfolios consisted of equal investment in each asset class, with risk-free lending at 5.0% per annum available for portfolio rebalancing. The eight asset classes were treated as individual stocks in this illustrative application. The portfolio rebalancing model (8) was used to determine the efficient frontier for initial portfolios of value $80K (ie value of investment in each asset class is $10K), $800K and $8000K, with changes in funding of 25% (ie 25% of value of initial portfolio withdrawn), 0% (ie no funds withdrawn or added) and +25% (ie new funding equal to 25% of value of initial portfolio) for each initial portfolio. The portfolio rebalancing model was also used to construct new portfolios from scratch taking account of transaction costs with new funding of $80K, $800K and $8000K. These cases were considered purely to illustrate the use of the model and the effect of transaction costs on investment policy. Although the efficient rebalanced frontier can be constructed for each of these cases (eg Figure 3), greater insight can be gained by considering the composition of rebalanced portfolios. For example, the

6 672 Journal of the Operational Research Society Vol. 62, No. 4 Table 1 Data for test problem Asset class Expected return (%) SD (%) Correlation coefficients Asset class Asset class Asset class Asset class Asset class Asset class Asset class Asset class Source: Elton et al (2007). Expected Annual Return (%) Standard Deviation of Return (% Nominal Value) Figure 3 Efficient rebalanced frontier for initial portfolio of value $8000K and funding change of $2000K. composition of rebalanced portfolios with expected annual returns of 5.0%, 7.5%, 10.0%, 12.5% and 15.0% of nominal initial value, that is the market value of the initial portfolio plus the change in funding, are listed in Tables 2 6, respectively. In Tables 2 6, the portfolio composition is presented in terms of the investment in each asset class expressed as a percentage of the portfolio s market value, together with the portfolio value and the standard deviation of return expressed as a percentage of nominal initial value. To allow comparison with the results from standard MV analysis, the portfolios generated by the standard MV model with risk-free lending for expected annual returns of 5.0%, 7.5%, 10.0%, 12.5% and 15.0% are also shown in Tables 2 6. It should, however, be noted that for a specified initial portfolio and change in funding, the portfolios selected by the standard MV model with risk-free lending would not yield the required expected return from the rebalanced portfolio as transaction costs, which are not considered in the standard MV model, would reduce the funds available for investment. The results in Tables 2 6 demonstrate that when purchase transaction costs are taken into account, portfolios constructed from scratch differ from the corresponding portfolio selected by the standard MV model, unless the required expected return is the risk-free lending rate, that is 5.0% (Table 2). The results in these tables also show that when the initial portfolio, funding changes and transaction costs are taken into account in portfolio rebalancing, the composition of the rebalanced portfolio differs from the corresponding portfolio generated by the standard MV model. From Tables 3 6 it can be seen that many of the rebalanced portfolios, that is excluding portfolios generated from scratch, contain the same investment in more than one asset class. Rebalanced portfolios with equal investment in a number of asset classes occur because the amount invested in each of these asset classes remains unchanged after rebalancing, partly in order to save transaction costs. For example, in the rebalanced portfolio required to generate 7.5% expected return from an initial portfolio of value $80K with $20K increase in funding (Table 3), the 10.07% holding in Asset classes 2, 3, 6 and 8 represents no change in the initial $10K investment in each of these asset classes. Some aspects of the results in Tables 2 6 are clearly a direct consequence of taking transaction costs into account in constructing and rebalancing portfolios. For example, from Table 6 it can be seen that although the portfolio with initial value $800K and 25%, that is $200K, reduction in funding can be rebalanced to yield 15.0% expected return, portfolios with initial values $80K and $8000K and 25% reduction in funding, that is $20K and $2000K, respectively, cannot be rebalanced to yield this expected return. The portfolio with initial value $80K and 25% reduction in funding cannot be rebalanced to yield 15.0% expected return because of the relative importance of fixed transaction costs, whereas the portfolio with initial value $8000K and 25% reduction in funding cannot be rebalanced to yield this return because of the relative importance of variable transaction costs. The results in Tables 2 6 also show that portfolios produced by the portfolio rebalancing model have higher standard deviation of return than the corresponding portfolios generated by the standard MV model with risk-free lending because of the impact of transaction costs. It can also be seen that in portfolios constructed from scratch with specified funding,

7 JJ Glen Mean-variance portfolio rebalancing 673 Table 2 Composition of portfolios for expected annual return of 5.0% Standard MV model No initial portfolio Initial portfolio $80K Initial portfolio $800K Initial portfolio $8000K new funding ($K) with funding change ($K) with funding change ($K) with funding change ($K) Asset class 1 (%) Asset class 2 (%) Asset class 3 (%) Asset class 4 (%) Asset class 5 (%) Asset class 6 (%) Asset class 7 (%) Asset class 8 (%) Risk-free (%) Value (% Nominal Value) SD (% Nominal Value) Table 3 Composition of portfolios for expected annual return of 7.5% Standard MV model No initial portfolio Initial portfolio $80K Initial portfolio $800K Initial portfolio $8000K new funding ($K) with funding change ($K) with funding change ($K) with funding change ($K) Asset class 1 (%) Asset class 2 (%) Asset class 3 (%) Asset class 4 (%) Asset class 5 (%) Asset class 6 (%) Asset class 7 (%) Asset class 8 (%) Risk-free (%) Value (% Nominal Value) SD (% Nominal Value)

8 674 Journal of the Operational Research Society Vol. 62, No. 4 Table 4 Composition of portfolios for expected annual return of 10.0% Standard MV model No initial portfolio Initial portfolio $80K Initial portfolio $800K Initial portfolio $8000K new funding ($K) with funding change ($K) with funding change ($K) with funding change ($K) Asset class 1 (%) Asset class 2 (%) Asset class 3 (%) Asset class 4 (%) Asset class 5 (%) Asset class 6 (%) Asset class 7 (%) Asset class 8 (%) Risk-free (%) Value (% Nominal Value) SD (% Nominal Value) Table 5 Composition of portfolios for expected annual return of 12.5% Standard MV model No initial portfolio Initial portfolio $80K Initial portfolio $800K Initial portfolio $8000K new funding ($K) with funding change ($K) with funding change ($K) with funding change ($K) Asset class 1 (%) Asset class 2 (%) Asset class 3 (%) Asset class 4 (%) Asset class 5 (%) Asset class 6 (%) Asset class 7 (%) Asset class 8 (%) Risk-free (%) Value (% Nominal Value) SD (% Nominal Value)

9 JJ Glen Mean-variance portfolio rebalancing 675 Table 6 Composition of portfolios for expected annual return of 15.0% Standard MV model No initial portfolio Initial portfolio $80K Initial portfolio $800K Initial portfolio $8000K new funding ($K) with funding change ($K) with funding change ($K) with funding change ($K) Asset class 1 (%) * * Asset class 2 (%) * * 0 0 Asset class 3 (%) * * Asset class 4 (%) * * Asset class 5 (%) * * Asset class 6 (%) * * Asset class 7 (%) * * Asset class 8 (%) * * Risk-free (%) * * 0 0 Value (% Nominal Value) * * SD (% Nominal Value) * * Note: denotes no feasible solution. the nominal value of the portfolio decreases as the expected return increases, that is the proportion of the funds invested decreases, due to increased transaction costs in buying risky assets. Since both buying and selling costs may be incurred in rebalancing an existing portfolio, the nominal value of rebalanced portfolios generated from a portfolio with specified initial value and change in funding may not change in a consistent manner as the expected return increases. However, in the results for portfolios with specified initial value and change in funding (Tables 2 6), the nominal value of rebalanced portfolios first increases and then tends to decrease with increasing expected return. In general, broad conclusions about the composition of rebalanced portfolios cannot be obtained and rebalancing decisions should, as noted by Mulvey (1993), be based on specific conditions. For portfolios generated by the standard MV model with risk-free lending, it can be shown that risk-free lending is an element of portfolios with an expected annual return of approximately 11.6% or less. In standard MV portfolios that include a risk-free component, the risky component increases and the risk-free component decreases as the expected annual return is increased from the risk-free rate to approximately 11.6% (see the standard MV portfolios for expected returns of 7.5% and 10.0% in Tables 3 and 4, respectively). By analysing standard MV portfolios that include risk-free lending, it can also be shown that the composition of the risky component of the portfolio, that is the proportion by value of individual asset classes in the risky component, is constant. Thus, as expected from portfolio theory (eg Elton et al, 2007), standard MV portfolios that include risk-free lending are linear combinations of risk-free lending and the tangency portfolio, that is the portfolio with expected annual return of approximately 11.6%. For rebalanced portfolios generated from an initial portfolio with specified initial value and change in funding, it can be seen from Tables 2 6 that the risk-free component of the portfolio decreases, and eventually disappears, as the expected return is increased. It can also be shown that in rebalancing a portfolio with specified initial value and change in funding, the composition of the risky component of rebalanced portfolios that include risk-free lending varies with expected return. Consequently, the efficient frontier for rebalanced portfolios does not have a linear section (eg Figure 3) as found in the efficient frontier of standard MV analysis with risk-free lending (eg Elton et al, 2007). For portfolios constructed from scratch by the portfolio rebalancing model, it can be seen from Tables 2 6 that for a specified level of funding, the risk-free component of the portfolio decreases as the expected return increases. Of the portfolios constructed from scratch in Tables 2 6, only the portfolio with expected annual return of 15% and new funding of $8000K has no risk-free component (Table 6), but it can be shown that the risk-free component disappears from portfolios with new funding of $80K and $800K when expected return is greater than 15%. For portfolios constructed from scratch with specified funding, it can also be shown that in portfolios that include risk-free

10 676 Journal of the Operational Research Society Vol. 62, No. 4 lending, the composition of the risky component of these portfolios varies with expected return. These results show that when transaction costs and funding levels are taken into account in rebalancing existing portfolios or constructing new portfolios, MV efficient portfolios that include risk-free lending do not have the structure expected from portfolio theory (eg Elton et al, 2007). Conclusions Standard MV analysis is not appropriate for portfolio rebalancing since the initial portfolio, funding changes and transaction costs are not considered. A QMIP portfolio rebalancing model which takes account of these factors has been developed in this paper. The transaction costs in this QMIP model are composed of fixed charges and variable transaction costs, where the variable transaction costs include market impact costs. Fixed transaction charges are incorporated in the model by using a pair of binary variables to represent the buy and sell decisions for each security, whereas variable transaction costs are assumed to be convex functions that are approximated by piecewise-linear functions. The QMIP model, which includes risk-free lending in the set of investment opportunities, is used to determine the rebalanced portfolio with minimum variance of total return for a specified expected total net return. This QMIP model can also be used to take account of funding level and transaction costs in constructing new portfolios. The QMIP model has been applied to a number of portfolio construction and rebalancing problems, with computational experience indicating that it is a practical tool for portfolio management. The portfolios generated by this model for problems with equal investment in the same set of eight asset classes, but different initial portfolio values and changes in funding, were examined in more detail and compared with the portfolios generated by standard MV analysis. Although only a limited number of cases were considered, it is clear that when transaction costs and funding changes are taken into account, the composition of a portfolio after rebalancing depends on the value and composition of the initial portfolio, the change in funding, the expected return and the structure of transaction costs. The results from this analysis also show that when initial portfolio, funding changes and transaction costs are taken into account in rebalancing existing portfolios, and when funding levels and transaction costs are taken into account in constructing new portfolios, MV efficient portfolios that include risk-free lending do not have the structure expected from portfolio theory. 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