Background

Problem. optimization setting for a portfolio replication problem with the replication error measured by Mean Absolute Penalty

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

References

 

 

Background

 

The case study demonstrates optimization setting for a portfolio replication problem with the replication error measured by Mean Absolute Penalty. Under performance of the portfolio compared to S&P100 index is measured by CVaR. Distribution of residuals is shaped with a CVaR constraint (several constraints can be specified, if of interest). We replicated S&P100 index using 30 stocks belonging to this index (tickers: GD, UIS, NSM, ORCL, CSCO, HET, BS, TXN, HM, INTC, RAL, NT, MER, KM, BHI, GEN, HAL, BDK, HWP, LTD, BAC, AVP, AXP, AA, BA, AGC, BAX, AIG, AN, AEP). Historical data on stock prices are used for building scenario matrices.

This case study was considered in Rockafellar and Uryasev (2002). For other references on portfolio

replication, see, for instance, Andrews et al. (1986), Beasley and Meade (1999), Buckley and Korn

(1998), Connor and Leland (1995), Dalh et al. (1993), Konno and Wijayanayake (2000), Rudd (1980),

and Toy and Zurack (1989).

 

Problem

 

Simplified Problem Statement

 

Minimize Meanabs_pen (minimizing replication error)

 subject to

Cvar_risk ≤ Const1 (CVaR constraint on the underperformance of the portfolio compared to the index)

Lenear = Const2 (budget constraint)

Box constraints (no-short constraints on exposures)

 

where

 

Meanabs_pen = Mean Absolute Penalty

Cvar_risk = CVaR Risk for Loss

Box constraints = constraints on individual decision variables

 

 

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 subroutine riskconstrparam (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS:

 

 

References

 

[1]  Andrews, C., Ford, D., Mallinson, K. (1986): The design of index funds and alternative methods of replication, The Investment Analyst, 82, 16–23.

[2]  Beasley, J.E., Meade, N., Chang, T.-J. (1999): Index tracking, Working Paper, Imperial College, London.

[3]  Buckley, I.R.C., Korn, R. (1998): Optimal index tracking under transaction costs and impulse control, International Journal of Theoretical and Applied Finance, 315–330.

[4]  Connor, G., Leland, H. (1995): Cash management for index tracking, Financial Analysts Journal 51 (6), 75–80.

[5]  Dahl, H., Meeraus, A., Zenios, S.A. (1993): Some financial optimization models: I Risk management. In: Zenios, S.A. (Ed.), Financial Optimization. Cambridge University Press, Cambridge, 3-36.

[6]  Konno, H., Wijayanayake, A. (2000): Minimal Cost Index Tracking under Nonlinear Transaction Costs and Minimal Transaction Unit Constraints, Tokyo Institute of Technology, CRAFT Working paper 00-07.

[7]  Rockafellar, R.T. and Uryasev, S. (2002): Conditional Value-at-Risk for General Loss Distributions, Journal of Banking and Finance, 27/7.

[8]  Rudd, A. (1980): Optimal selection of passive portfolios. Financial Management, 57–66.

Toy, W.M., Zurack, M.A. (1989): Tracking the Euro-Pac index, The Journal of Portfolio Management ,15, (2), 55–58.