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

www.rmetrics.org Chicago, April 2009

The Problem …

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.

Portfolio Optimization Problem

… return, risk, performance ratio

Chicago, April 2009 www.rmetrics.org Page 2

How to quantify Risk ?

Stone 1973 Pederson and Satchell 1998 Artzner, Delbaen, Eber, Heath 1999

… this makes a coherent risk measure y are the financial returns, f ( ) their multivariate distribution A, Y0, and k parameters for some bounded function W ( )

www.rmetrics.org Page 3

Includes: Mean - Covariance Risk Mean - CVaR Measure: k = 1, A = VaR, Y0 = 0

fPortfolio Models

fPortfolio Zoo:

Rmetrics Software

Short selling RshortExact Linear Constraints quadprog Ripop Quadratic Constraints Rsocp

Minimize Risk Mean-Var robMV

Non-Linear Objectives Rnlminb Rdonlp2

Efficient Portfolio

Q LPM quadprog Ripop

Maximize Return

…

Black Litterman Copula Pooling Non-linear Constraints Rdonlp2

Scenario Optimization

Rglpk Rsymphony RlpSolve

Portfolio Optimization with R/Rmetrics eBook I Advanced Portfolio Optimization with R/Rmetrics eBook II

www.rmetrics.org

Mixed Integer Rsymphony Topics Managing Data Sets of Assets Exploratory Data Analysis of Assets Portfolio Framework Mean-Variance Portfolios Mean-CVaR Portfolios Portfolio Backtesting

Page 4

Plotting Frontiers and Weights …

0.0 0.5 1.0 1.5 2.0 0.00 0.05 0.10 0.15 0.20

MV | solveRquadprog

Efficient Frontier

Target Risk[Cov] Target Return[mean] MV Portfolio | mean-Stdev View

Reducing Estimation Errors …

www.rmetrics.org Page 6

Copulae Tail Risks …

Copulae Lower Tail Risk Dependence Budgets

Tail Dependence: Lower SBI CH Bonds SPI CH Stocks SII CH Immo LMI World Bonds MPI World Stocks ALT World AltInvest www.rmetrics.org

… Quadratic Constraints use Rsocp (not yet fully implemented)

Page 7

GCC Country Rotation

www.rmetrics.org Page 8

Series 2005-06-01 2006-12-20 2008-07-09

• 1.0

0.0 0.5