Formal Problem Statement

Notations

=  number of style indices used for classification. We consider four indices: Russell 1000 value index (variable, i =1), Russell 1000 growth index (variable, i = 2), Russell 2000 value index (variable, i = 3), and Russell 2000 growth index (variable, i = 4);

=  number of scenarios (time periods), j={1,…,J} index of scenarios;

= monthly rate of return of the fund, for which the classification is conducted, under scenario j; scenarios are equally probable (in the current case study scenarios are monthly historical returns of the S&P 500 index or Fidelity Magellan Fund);

= monthly rate of return of i-th style index under scenario j, scenarios are equally probable;

= random value having J equally probable scenarios,;

=  random scenario vector;

= vector of regression coefficients (loading factors);

= loss function;

= gain function.

Optimization Problem

minimizing error function

(CS.2)