## 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, …, an}. Each asset ai is associated with an expected return (per period) ri, and each pair of assets <ai, aj> has a covariance &deltaij. The covariance matrix &deltaiXj is symmetric and each diagonal element &deltaii represents the variance of asset ai. A positive value R represents the expected return.

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

Min &sum&deltaijwiwj

s.t. &sumriwi = R
&sumwi = 1
0 <= wi<= 1, i = 0, ..., n

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, …, w0n)T, which is the percentage wealth investigated on each asset n. The amount transacted in each asset is specified by x = (x1, ..., xn)T. xi < 0 means selling and xi > 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 ri and variance is denoted as &deltaii. &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:

Min &sum&deltaijwiwj + &Phi(x)

s.t. &sumriwi = R
&sumwi = 1
w = w0 + x &isin F

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

Last updated date 16/12/2010