Background

Problem 1. Minimizing Koenker and Basset error

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 2. Minimizing Partial Moment Penalty

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

 

Background

This case study applies percentile regression to the return-based style classification of a mutual fund. The procedure regresses fund return by several indices as explanatory variables. The estimated coefficients represent the fund’s style with respect to each of the indices. This problem was considered by Carhart (1997) and Sharpe (1992). They estimated conditional expectation of a fund return distribution (under the condition that a realization of explanatory variables is observed).

Bassett and Chen (2001) extended this approach and conducted style analyses of quantiles of the return distribution. This extension is based on the quantile regression approach suggested by Koenker and Bassett (1978). The quantile regression model is more flexible compared to the standard least squares regression because it can identify dependence of various parts of the distribution from explanatory variables. A portfolio style depends on how a factor influences the entire return distribution, and this influence cannot be described by a single number. The single number given by the least squares regression may obscure the tail behavior (which could be of a prime interest to a manager). With the quantile regression we can estimate, for instance, the impact of explanatory variables on the 99-th percentile of the loss distribution. Portfolios having exposures to derivatives may have very different regression coefficients of the mean value and tail quantiles. For instance, let us consider a strategy in investing into naked deep out-of-the-money options. This strategy in most cases behaves like a bond paying some interest, however, in rare cases the strategy loses some amount of money (may be quite significant). Therefore, the mean value and 99-th percentile may have very different regression coefficients for the explanatory variables.

 

Problem 1

 

Simplified Problem Statement

 

Minimize kb_err

 

where

 

kb_err = Koenker and Basset error function

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with riskprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS:

 

 

Problem 2

 

Simplified Problem Statement

 

Minimize alpha*Pm_pen + (1-alpha)*Pm_pen_g

 

where

 

Pm_pen = Partial Moment Penalty for Loss

Pm_pen_g = Partial Moment Penalty for Gain

 

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with riskprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS: