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Benchmark Datasets in Portfolio OptimizationProblem 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 = {a_{1}, …, 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 = {w_{1}, …, 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. &sumw_{i} = 1 0 <= w_{i}<= 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 w^{0} = (w^{0}_{1}, …, w^{0}_{n})^{T}, which is the percentage wealth investigated on each asset n. The amount transacted in each asset is specified by x = (x_{1}, ..., x_{n})^{T}. x_{i} < 0 means selling and x_{i} > 0 means buying. After the transaction, the adjusted portfolio is w = w^{0} + 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: &sumw_{i} = 1 w = w^{0} + 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