Case Study: Portfolio Replication with Risk Constraint

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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: problem_CS_Portfolio_Replication_CVaR_0p001.txt

Minimize Meanabs_err (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)
Meanabs_pen = Mean Absolute Error
Cvar_risk = CVaR Risk for Loss
Box constraints = constraints on individual decision variables
# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset 30 600 0.01743 0.06
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines  Matlab Code Data
R Subroutines R Code Data
R rpsg_solver R Code Data


The case study demonstrates optimization setting for a portfolio replication problem with the replication error measured by Mean Absolute Penalty. Underperformance of the portfolio compared to S&P100 index is measures 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).


• Andrews, C., Ford, D., Mallinson, K. (1986): The design of index funds and alternative methods of replication, The Investment Analyst, 82, 16–23.
• Beasley, J.E., Meade, N., Chang, T.-J. (1999): Index tracking, Working Paper, Imperial College, London.
• 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.
• Connor, G., Leland, H. (1995): Cash management for index tracking, Financial Analysts Journal 51 (6), 75–80.
• Dalh, H., Meeraus, A., Zenios, S.A. (1993): Some financial optimization models: I risk management. In: Zeniosh, S.A. (Ed.), Financial Optimization. Cambridge University Press, Cambridge, 3–36.
• 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.
• Rockafellar, R.T. and Uryasev, S. (2002): Conditional Value-at-Risk for General Loss Distributions, Journal of Banking and Finance, 27/7.
• 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.