Portfolio Optimization with R/Rmetrics Diethelm Wrtz Yohan - PowerPoint PPT Presentation

Portfolio Optimization with R/Rmetrics Diethelm Würtz Yohan Chalabi, Andrew Ellis, Martin Hanf ETH Zurich, Rmetrics Association, Finance Online GmbH Thanks to William Chen, Alexios Ghalanos, Francisco Gochez, Dominik Locher, Stephan Theussl useR! Conference Rennes, July 2009 Chicago, April 2009 www.rmetrics.org

The Problem … Portfolio Optimization Problem … return, risk, performance ratio Example Swiss Pension Fund Portfolio For a given set of financial assets let us find the composition 1) which minimizes the risk for a given return (reward), 2) which maximizes the return for a given risk, 3) which optimizes a reward/risk performance ratio, 4) which finds the global minimum risk, subject to certain constraints and preferences . Chicago, April 2009 www.rmetrics.org Page 2

How to quantify Risk ? Stone 1973 y are the financial returns, f ( ) their multivariate distribution A, Y 0 , and k parameters Includes: Mean - Covariance Risk Mean - CVaR Measure: k = 1 , A = VaR, Y 0 = 0 Pederson and Satchell 1998 for some bounded function W ( ) Artzner, Delbaen, Eber, Heath 1999 … this makes a coherent risk measure www.rmetrics.org Page 3

fPortfolio Models Topics Managing Data Sets of Assets Exploratory Data Analysis of Assets fPortfolio Zoo: Portfolio Framework Mean-Variance Portfolios Rmetrics Software Mean-CVaR Portfolios Portfolio Backtesting Portfolio Optimization with R/Rmetrics eBook I Efficient Portfolio Maximize Return Minimize Risk Scenario … Optimization Non-Linear Objectives Rglpk Rsymphony RlpSolve Rnlminb Rdonlp2 Mean-Var robMV Q LPM Short selling Black Litterman quadprog Ripop RshortExact Linear Constraints Copula Pooling quadprog Ripop Quadratic Constraints Rsocp Non-linear Constraints Rdonlp2 Mixed Integer Advanced Rsymphony Portfolio Optimization with R/Rmetrics eBook II www.rmetrics.org Page 4

Plotting Frontiers and Weights … Efficient Frontier MV Portfolio | mean-Stdev View 0.20 Target Return[mean] 0.15 0.10 0.05 MV | solveRquadprog 0.00 0.0 0.5 1.0 1.5 2.0 Target Risk[Cov]

Reducing Estimation Errors … www.rmetrics.org Page 6

Copulae Tail Risks … Copulae Lower Tail Risk Dependence Budgets SBI CH Bonds SPI CH Stocks SII CH Immo LMI World Bonds MPI World Stocks ALT World AltInvest Tail Dependence: Lower … Quadratic Constraints use Rsocp (not yet fully implemented) www.rmetrics.org Page 7

GCC Country Rotation 0.5 Series 0.0 -1.0 2005-06-01 2006-12-20 2008-07-09 www.rmetrics.org Page 8