# Case Study: Style Classification with Quantile Regression

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Case study background and problem formulations

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

PROBLEM 1: problem_kb_err_Style_Classification_Fidelity_Magellan_0p1
Minimize kb_err
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kb_err = Koenker and Basset error function
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Data and solution in Run-File Environment

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec)
Dataset1 Problem Statement Data Solution 5 1,264 0.001221 0.01

Data and solution in MATLAB Environment

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 Matlab code Data Solution 5 1,264 0.00122086 <0.01

Data and solution in R Subroutines

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 R code Data 5 1,264 0.00122086 <0.01

Data and solution in R rpsg_solver

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 R code Data 5 1,264 0.00122086 <0.01
PROBLEM 2: problem_pm_Style_Classification_Fidelity_Magellan_0p1
Minimize alpha*Pm_pen + (1-alpha)*Pm_pen_g
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Pm_pen = Partial Moment Penalty for Loss
Pm_pen_g = Partial Moment Penalty for Gain
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Data and solution in Run-File Environment

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 2.66GHz (sec)
Dataset1 Problem Statement Data Solution 5 1,264 0.001221 0.01

Data and solution in MATLAB Environment

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 Matlab code Data Solution 5 1,264 0.00122086 <0.01

Data and solution in R Subroutines

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 R code Data 5 1,264 0.00122086 <0.01

Data and solution in R rpsg_solver

Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 R code Data 5 1,264 0.00122086 <0.01
Minimize cvar_dev
Value:
Var_risk
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Cvar_dev = CVaR Deviation for Loss
Var_risk = VaR Risk for Loss
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Dataset1 5 1,264 0.00122086 <0.01 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data

CASE STUDY SUMMARY

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.
We regresses quantile of the return distribution of the 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). We want to calculate coefficients for the explanatory variables of the tail of the distribution of residuals (these coefficients may differ from the regression coefficients for the mean and the median of the distribution). The confidence level in quantile regression is 0.1.
References
• Bassett G.W., Chen H-L. (2001): Portfolio Style: Return-based Attribution Using Quantile Regression. Empirical Economics 26, 293-305.
• Carhart M.M. (1997): On Persistence in Mutual Fund Performance. Journal of Finance 52, 57-82.
• Koenker R, Bassett G. (1978): Regression Quantiles. Econometrica 46, 33-50.
• Sharpe W.F. (1992): Asset Allocation: Management Style and Performance Measurement. Journal of Portfolio Management (Winter), 7-19.
• Rockafellar R.T. and S. Uryasev. The Fundamental Risk Quadrangle in Risk Management, Optimization, and Statistical Estimation. Research Report 2011-5, ISE Dept., University of Florida, November 2011.
• Rockafellar, R.T. and S. Uryasev (2013): The Fundamental Risk Quadrangle in Risk Management, Optimization, and Statistical Estimation. Surveys in Operations Research and Management Science, 18 (to appear).
• Rockafellar, R.T., Uryasev, S., and M. Zabarankin (2008): Risk Tuning with Generalized Linear Regression. Mathematics of Operations Research, 33(3), 712–729.