Background

Problem 1. Minimizing Koenker and Bassett 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

References

 

 

Background

 

This case study applies percentile regression to the return-based style classification of a mutual fund.   The procedure regresses fund returns on the returns of 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 returns distribution, under the condition that a realization of explanatory variables is observed.

 

Basset and Chen (2001) extended this approach and conducted style analyses of quantiles of the returns distribution. This extension is based on the quantile regression approach suggested by Koenker and Basset (1978).  The quantile regression model is more powerful than 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 risk 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 the strategy  of writing 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 (that may be quite significant). Therefore, the mean value and 99-th percentile may have very different regression coefficients for the explanatory variables.

 

This case study regresses the S&P 500 index and Fidelity Magellan Fund on the Russell Value Index (RUJ), RUSSELL 1000 VALUE INDEX (RLV), Russell 2000 Growth Index (RUO) and Russell 1000 Growth Index (RLG). The confidence level in quantile regression varies from 0.1 to 0.9. The idea is to identify if the tails have different explanatory coefficients compared to the median of the residual distribution. Numerical experiments indeed show (see Figure 3)  that the lower and upper parts of the Magellan Fund returns distribution has somewhat different style explanatory coefficients than the median. In particular, RLG and RLV funds have approximately similar impact on the median; RLV has the largest positive impact on the upper part of the Magellan Fund returns distribution, while the RLG fund has the largest positive impact on the lower part. At the same time the RUJ fund  has the larger negative influence on the upper part of the Magellan Fund returns distribution than on the lower part.

 

Problem 1

 

Simplified Problem Statement

 

Minimize Kb_err

 

where

 

Kb_err = Koenker and Bassett error

 

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 PSG MATLAB function tbpsg_run (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 PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

 

 

Input Files to run CS:

 

 

References

 

[1]  Basset G.W., Chen H-L. (2001): Portfolio Style: Return-based Attribution Using Quantile Regression. Empirical Economics 26, 293-305.

[2]  Carhart M.M. (1997): On Persistence in Mutual Fund Performance. Journal of Finance 52, 57-82.

[3]  Koenker R, Basset G. (1978): Regression Quantiles.  Econometrica 46, 33-50.

[4]  Sharpe W.F. (1992): Asset Allocation: Management Style and Performance Measurement. Journal of Portfolio Management (Winter), 7-19.