Benchmark Datasets in Portfolio Optimization

Problem Description

The basic model of portfolio optimization problems is based on the seminar work of Markowitz’s mean-variance. Given a set of n assets A = {a₁, …, a_n}. Each asset a_i is associated with an expected return (per period) r_i, and each pair of assets <a_i, a_j> has a covariance &delta_ij. The covariance matrix &delta_iXj is symmetric and each diagonal element &delta_ii represents the variance of asset a_i. A positive value R represents the expected return.

A portfolio can be represented by a set W = {w₁, …, w_n}, where w_i represents the percentage wealth invested on asset a_i. The formulation of the basic problem can thus be defined as the following.

A benchmark set of portfolio optimization problem instances has been provided at the OR Library.

A set of extended portfolio optimization problem with additional constraints and transaction costs has been produced in our research.

Assume the current holds on each asset are denoted as w⁰ = (w⁰₁, …, w⁰_n)^T, which is the percentage wealth investigated on each asset n. The amount transacted in each asset is specified by x = (x₁, ..., x_n)^T. x_i < 0 means selling and x_i > 0 means buying. After the transaction, the adjusted portfolio is w = w⁰ + x and it is held for a fixed period of time. At the end of the time, the return rate of asset i is denoted as r_i and variance is denoted as &delta_ii. &Phi(x) represents the sum of all transaction costs associated with each x . So the portfolio selection problem with transaction cost can be modeled as follows:

F is the set of feasible portfolio subject to the minimum position constraint, minimum trade constraint, and cardinality constraint.