Identifying and Compensating for Model Mis-Specification in Factor Risk Models

Abstract

Techniques for using factor risk models to more accurately estimate the risk or active risk of an investment portfolio are disclosed. Inherent “modeling error” in factor risk models is identified and compensated for. One or more factors are added to compensate for factors that are unspecified or unattributed in the original factor risk model and which lead to modeling error. The approach can be used with a variety of different factor risk models, and for a variety of securities. Knowledge of the risk associated with modeling error can be utilized when estimating risk or active risk using factor risk models or when constructing optimal portfolios by mean-variance optimization or other portfolio construction strategies using factor risk models.

1 . A computer-based method comprising:
 electronically inputting a set of N possible investment opportunities;   defining a vector space of N-dimensional vectors of all possible portfolios in the N investment opportunities, with vector elements corresponding to investment holdings in any investment portfolio;   electronically receiving by a programmed computer a factor risk model defined and calibrated for all possible portfolios in the N investment opportunities, said factor risk model comprising a matrix of factor exposures, a matrix of factor covariances, and a matrix of specific covariances;   electronically receiving by the programmed computer an estimate of expected investment returns defined by an N-dimensional vector;   defining an original portfolio optimization strategy for determining an optimized portfolio of investment holdings comprising an objective function that includes a utility measurement of the portfolio defined by the sum of a first constant times the vector inner product of the investment holdings of the optimized portfolio and the expected investment returns added to a second constant times the variance predicted by the factor risk model for the optimized portfolio;   electronically receiving by the programmed computer an N-dimensional target vector;   determining utilizing the programmed computer an N-dimensional new factor vector substantially representing the projection of the target vector into the null space of the transpose of said matrix of factor exposures so that the vector inner product of said new factor and any column of said matrix of factor exposure substantially vanishes;   determining a modified portfolio optimization strategy in which the objective function is altered by adding a constant times the square of the vector inner product of the new factor vector and the investment holdings of the optimized portfolio; and   electronically outputting the optimized portfolio.   
     
     
         2 . The method of  claim 1  where said target vector is the vector of expected investment returns. 
     
     
         3 . The method of  claim 1  further comprising the step of utilizing the modified portfolio optimization strategy to determine a set of optimized portfolios at a series of different dates. 
     
     
         4 . The method of  claim 1  where said target vector is the implied alpha of the optimized portfolio defined by the original portfolio optimization strategy. 
     
     
         5 . The method of  claim 4  further comprising the step of utilizing the modified portfolio optimization strategy to determine a set of optimized portfolios at a series of different dates. 
     
     
         6 . The method of  claim 1  wherein the first constant is zero and the second constant is non-zero. 
     
     
         7 . The method of  claim 1  wherein the first constant is non-zero and the second constant is zero. 
     
     
         8 . A programmed computer-based apparatus comprising:
 means for electronically inputting a set of N possible investment opportunities;   means for defining a vector space of N-dimensional vectors of all possible portfolios in the N investment opportunities, with vector elements corresponding to investment holdings in any investment portfolio;   means for electronically receiving by a programmed computer a factor risk model defined and calibrated for all possible portfolios in the N investment opportunities, said factor risk model comprising a matrix of factor exposures, a matrix of factor covariances, and a matrix of specific covariances;   means for electronically receiving by the programmed computer an estimate of expected investment returns defined by an N-dimensional vector;   means for defining an original portfolio optimization strategy for determining an optimized portfolio of investment holdings comprising an objective function that includes a utility measurement of the portfolio defined by the sum of a first constant times the vector inner product of the investment holdings of the optimized portfolio and the expected investment returns added to a second constant times the variance predicted by the factor risk model for the optimized portfolio;   means for electronically receiving by the programmed computer an N-dimensional target vector;   means for determining utilizing the programmed computer an N-dimensional new factor vector substantially representing the projection of the target vector into the null space of the transpose of said matrix of factor exposures so that the vector inner product of said new factor and any column of said matrix of factor exposure substantially vanishes;   means for determining a modified portfolio optimization strategy in which the objective function is altered by adding a constant times the square of the vector inner product of the new factor vector and the investment holdings of the optimized portfolio; and   means for electronically outputting the optimized portfolio.   
     
     
         9 . The apparatus of  claim 8  where said target vector is the vector of expected investment returns. 
     
     
         10 . The apparatus of  claim 9  further comprising:
 means for the step of utilizing the modified portfolio optimization strategy to determine a set of optimized portfolios at a series of different dates. 
 
     
     
         11 . The apparatus of  claim 8  where said target vector is the implied alpha of the optimized portfolio defined by the original portfolio optimization strategy. 
     
     
         12 . The apparatus of  claim 11  further comprising:
 means for utilizing the modified portfolio optimization strategy to determine a set of optimized portfolios at a series of different dates. 
 
     
     
         13 . The apparatus of  claim 8  wherein the first constant is zero and the second constant is non-zero. 
     
     
         14 . The apparatus of  claim 8  wherein the first constant is non-zero and the second constant is zero.