An analysis of a mean-variance enhanced index tracking problem with weights constraints ”

ARTICLE INFO Wanderlei Lima de Paulo, Marta Ines Velazco Fontova and Renato Canil de Souza (2018). An analysis of a mean-variance enhanced index tracking problem with weights constraints. Investment Management and Financial Innovations, 15(4), 183-192. doi:10.21511/imfi.15(4).2018.15 DOI http://dx.doi.org/10.21511/imfi.15(4).2018.15 RELEASED ON Monday, 19 November 2018 RECEIVED ON Wednesday, 22 August 2018 ACCEPTED ON Friday, 02 November 2018


INTRODUCTION
In general, an index tracking problem aims at establishing an optimal allocation so that the return of the portfolio replicates the return of a market index (passive management strategy), without purchasing all of the assets that compose the market index. On the other hand, the so-called enhanced index tracking (EIT) problem consists of constructing a portfolio that replicates and outperforms the market index by generating excess return. The literature presents different approaches and methods to construct an indexed portfolio, considering transaction cost, cardinality and weights constraints (see, for example, Canakgoz  Usually, the studies on tracking problem use the historical look-back approach, in which the tracking portfolio is constructed considering a sample of past observations. The classical uni-period mean-variance approach is considered in Roll (1992), Alexander and Baptista (2010) and Paulo et al. (2016), for example. Specifically, Paulo et al. (2016) studied an unconstrained EIT problem (e.g., allow short-selling), for which the authors derive an analytical solution. Differently from the cardinality constraint approach, they consider an approach in which the EIT portfolio is composed of a previously selected subset of assets belonging to the market index portfolio. Following the same approach, in this paper, we deal with a mean-variance EIT problem with weights constraints. In this case, it is not possible to obtain an analytic solution, but the problem can be easily solved using a quadratic programming algorithm.
Particularly, the weights constraints play an important rule in the asset allocation problem, since it allows to control the level of short and long positions, as well as to avoid the concentration risk (when a portfolio has a large exposure to an asset or few assets), for example. In the work of Jagannathan and Ma (2003), the authors studied the effects of weights constraints on the global mean-variance portfolio using a shrinkage approach. Considering the same framework, we have shown that constructing a constrained mean-variance EIT portfolio is equivalent to constructing an unconstrained EIT portfolio (as studied in Paulo et al., 2016). This equivalence allows us to study the effect of weights constraints on the covariance matrix and on the EIT portfolio.
The remainder of the paper is organized as follows. Section 1 includes the formulation of the EIT problem with weights constraints and its optimality conditions. In section 2, it is shown that constructing the constrained EIT portfolio is equivalent to constructing the unconstrained EIT portfolio using a shrinkage covariance matrix. A numerical example for an EIT portfolio using the S&P 500 index as market index is presented in section 3. Final section presents some remarks.

PROBLEM FORMULATION AND OPTIMALITY CONDITIONS
We recall that an EIT problem aims at constructing a portfolio that replicates a market index and at the same time obtaining positive excess return (on average). In this case, this problem can be seen as a dual-objective optimization problem, a tradeoff between maximizing the mean excess return and minimizing a tracking error measure (measure of how closely a portfolio outperforms the index).
Let us consider a portfolio composed of n assets (usually belonging to the market index). We denote by    : i) achieve an ave rage re0turn rate higher than a market index (active man-agement strategy), ii) replicate the return rate of a market index (index tracking strategy) or iii) track a market index with a positive excess return rate (EIT strategy).
In the following, we present the optimality conditions for the constrained problem (1)-(4) and show that its solution solves the unconstrained problem (e.g., n ω ∈ℜ ) using a shrinkage covariance matrix proposed by Jagannathan and Ma (2003). Studies related to the shrinkage covariance matrix estimators can be seen in Ledoit As the problems (1)-(4) are strictly convex quadratic optimization problem, the system (5) where 0 λ is the Lagrange multiplier for the constraint 1, e ω ′ = so that the optimal solution can be written as

SOME THEORETICAL RESULTS
In the work of Jagannathan and Ma (2003), the authors have shown that constructing a constrained global minimum variance portfolio is equivalent to constructing an unconstrained minimum variance portfolio after modifying the covariance matrix in a particular way (shrinkage approach). In this section, following the same analysis and procedure as in Roncalli (2011) and Jagannathan and Ma (2003), we show the same equivalence for the mean-variance EIT problem (subsection 2.1) and we analyze the effects of weights constraints on the EIT portfolio (subsection 2.2).

Shrinkage covariance matrix approach
In the following proposition, we show that the solution of the constrained problem (1)-(4) solves the unconstrained problem (e.g., n ω ∈ℜ ) using a shrinkage covariance matrix as proposed by Jagannathan and Ma (2003). e ω ′ = so that by the transformation (12), From conditions (7) and (8), we have Now, from condition (5) and by (14), it follows that Finally, by (15) and condition (9), taking Σ = Σ  and , ω ω showing that solution ω  solves the unconstrained problem with , Σ = Σ  completing the proof.
From Proposition 1, we have that constructing the constrained problem from Σ is equivalent to con-structing the unconstrained problem from (12).
The elements of the matrix Σ  can be written as Table 1, considering the conditions (7) and (8). Notice that , In the following, we illustrate the impact of portfolio weights constraints through a numerical example, considering an EIT portfolio composed of five assets (the parameters are presented in Table   2). The column ω * shows the optimal solution for the unconstrained problem ( ) 5 , ω ∈ℜ calculated by (11)  ), we obtain the results presented in Table 3. The column ω  shows the weights for the constrained EIT   (11) with notice that the first and fifth assets reached the upper bounds, whereas the fourth asset reached its lower bound (the second and third assets did not reach the lower and upper bounds). Finally, notice that the implied volatility of the first and fifth assets increased ( )

The effect of the weights constraints
As presented in Jagannathan and Ma (2003), for the case of an unconstrained global minimum variance portfolio, an asset tends to receive low portfolio weight at the optimum if it has higher covariances with other assets. Likewise, an asset with low covariances with other assets tends to receive high weights. On the other hand, when we impose 0, ω > the variance of an asset i and its covariances with other assets are reduced, so that the asset i tends to receive high weight even if it has higher variance and higher covariances with other assets. Similarly, imposing , i i w ω + ≤ the variance of the asset i and its covariances with other assets are increased, so that the asset i tends to receive low portfolio weight even if it has lower variance and lower covariances with other assets.
In the following, we show that the previous findings do apply to a special case of the EIT problem (1)-(4), in which ϕσ β λ + + = Let us consider that asset i tends to have higher covariances with other assets, so that the -th i row of Σ tends to have larger elements than other rows. Therefore, from condition (17), to achieve optimality, it is necessary to reduce asset i 's portfolio weight ( ), i ω so that the asset i may even have negative weight if its covariances with other assets are sufficiently high. Thus, an asset tends to receive low weight at the optimum if it has higher covariances with other assets. Likewise, if an asset has low covariances with other assets, it tends to receive high portfolio weights. Now, let us consider that we impose the lower bonds ( ) .  r ϕσ β + and the covariances between the assets (notice now that the problem aims at minimizing the tracking error and at the same time maximizing the excess return).

ASSET ALLOCATION EXAMPLE
Following a similar analysis and procedure as in Paulo et al. (2016), in this section, we study the performance of the constrained and unconstrained EIT portfolios. We consider the S&P 500 index as the benchmark target to be tracked using a portfolio composed of the following ten stocks ( ) V t be the position value of the position associated with the S&P 500 index, given by  ). This result occurs due to the significant changes in market parameter values during out-of-sample period (compared to the insample period). In this situation, the performance of the portfolios can be improved if we rebalance them periodically. Figure 3 shows the cumulative monthly portfolio values in out-of-sample period, in which we rebalance the weights ω  and ω * monthly. For this propose, we reestimate the parameters r and M µ using a moving average cri-  , ω * considering an out-of-sample analysis a good performance level by setting the parameter , ϕ from the practical point of view, the constrained EIT problem is more appropriate to construct a EIT portfolio than the unconstrained EIT problem, since the first one allows to control the level of short and long positions, as well as to avoid the concentration risk (when a portfolio has a large exposure to an asset or few assets).

FINAL REMARKS
In this paper, we consider an uni-period mean-variance enhanced index tracking (EIT) problem with weights constraints. As theoretical result, we show that constructing the constrained EIT portfolio is equivalent to constructing the unconstrained EIT portfolio (as studied in Paulo et al., 2016), when we modify the covariance matrix in a particular way (shrinkage covariance matrix). From this equivalence, we analyze the impact of weights constraints on the covariance matrix and on the EIT portfolio. In general, the effects of weights constraints on the EIT portfolio are different from those observed in the case of global minimum variance portfolio (as studied in Jagannathan & Ma, 2003). We present a numerical example using the S&P 500 index as the market index to be tracked using a portfolio composed of ten stocks. We observe that the constrained EIT portfolio performs better than the unconstrained EIT portfolio. This particular result cannot be generalized, since the performance of the unconstrained EIT portfolio can be improved by choosing an appropriate parameter .
ϕ However, we highlight that a constrained problem is more appropriate to construct a portfolio than an unconstrained problem, since the first one allows to control the level of short and long positions and avoids the concentration risk.