Enhanced index tracking based on the stochastic programming model

A portfolio is constructed to reproduce the behavior of the S & P 500 index under prescribed risk constraint. Several popular tracking error measures are compared and finally the downside tracking error measure is chose to penalize the distance when the portfolio return is below the benchmark return. Instead of the full replication of the S & P 500, a cardinality constraint is added on the model to avoid the illiquidity and high transaction fee. The portfolio selection model also contains other market restriction such as buy-in threshold, which limit the number of security bought must exceed some given value. In order to capture the randomness of the security price, the stochastic factor is employed in the model with respect to the distribution of the securities. Under the normal distribution assumption, the stochastic programming problem is rewritten as a deterministic mix-integer problem, which could be solved by the standard commercial solver such as Cplex. Empirical result is given based on the monthly data of S & P 500, which could be obtained from CRSP database.