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On the construction of the common optimal market index in the Sharpe model

Kuryłek W.

Control and Cybernetics

1999

Vol. 28, no 4

779-787

The purpose of this paper is to determine one factor which represents the whole market behavior on the basis of the rates of return of all equities traded oo this market. In the seminaal Sharpe model the factor is an exogenous varialble which is not determined by the model itself. This paper extends Sharpe's idea, as it assumes that the factor is a linear combination of all the rates of return of all traded equities. To determine this coefiicients of this linear combination we minimize the loss function which expresses the weighted mean square deviation of all rates of return from their predictions, having given the linear combination form of the market index. It is found that the vector of linear coeffcients has to be a nonzero eigenvector associated with the maximal eigenvalue of the appropriately transformed and estimated covariance matrix. The optimal market index for the Warsaw Stock Exchange was compared with the standard index. It occurs that there is only a very small difference between the standard index of this market and the optimal index.