Notations

 

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

 

J = 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; scenarios are equally probable (in the current case study are the monthly historical returns of the Fidelity Magellan Fund);

= monthly rate of return of i-th style index (i = 1,2,…,I) under scenario, scenarios are equally probable;

= random value having J equally probable scenarios, , i = 1,2,…,I;

= random scenario vector;

= vector of regression coefficients (loading factors);

= vector of auxiliary variables;

= loss function, which is the residual of the regression;

= loss function with auxiliary variables;

where = rescaled Koenker and Basset error function, where  is the realization of random value on scenario j, ;

= Rockafellar error function. The statistic of this error function equals to the mixed percentile with  coefficients k=1,…,K, (see Rockafellar and Uryasev (2011), Example 10: A Mixed-Quantile-Based  Quadrangle);

k=1,…,K = weighting coefficients for the mixed percentile,, ;

= PSG Partial Moment Penalty for Loss function with parameter ;

= PSG Partial Moment Penalty for Gain function with parameter ;

= rescaled Koenker and Basset error applied to the residual of the regression;

= Koenker and Basset  error applied to the residual of the regression;

= Rockafellar error function applied to the residual of the regression.

 

 

Optimization Problem 1

 

minimizing Rockafellar error function

 

                       (CS.1)

 

 

Optimization Problem 2

 

minimizing Rockafellar error function implemented with Partial Moments

                       (CS.2)

 

subject to

 

linear constraint on additional variables

                       (CS.3)