Portfolio DesignPortfolio Design,Optimization and Stability Analysis

Di th l Wü t d M h d M ht

Optimi ation and Stability Analysis

Diethelm Würtz and Mahendra Mehta

Rmetrics AssociationWebinar, January 26, 2011

Sponsored byp y

Revolution Analytics, Sybase, Finance Online, NeuralTechSoft

Who is Rmetrics ?

The roots of Rmetrics go back to 1997. Rmetrics is a non profit taking Association under  Swiss law working in the public interest in the field of measuring and analyzing risks in finance and related fields

• We operate an Open Source Educational and Teaching Platform• We provide an R Development Environment for Finance• We provide an R Development Environment for Finance• We offer an R Code Archive and a Public Tools Platform• We started to Build a Public Stability and Risk Data Base

Rmetrics has been originated from the Swiss Federal Institute ofTechnology in Zurich, ETH.

2

gy ,

Rmetrics Timeline1997 Starting with a Collection of SPlus Functions for Finance

1999 Moving to the R Environment

2001 Creating Rmetrics Software Packages

2002 Uploading the Rmetrics Packages to CRAN Server 

2003 Introducing R‐sig‐Finance  /  Private Repository – Martin Mächler

2004 Providing Debian Packages – Dirk Eddelbüttel

2007 Organizing the1st Rmetrics User and Developer Workshop2007 Organizing the1 Rmetrics User and Developer Workshop

2008 Founding the Rmetrics Association / Offering Student Internships

2008   2nd Rmetrics Developer Workshop

2008 Joining R‐forge / Starting the Rmetrics Repository

2009 3rd Rmetrics User and Developer Workshop 

2009 Fi t R t i B k “P tf li O ti i ti ith R/R t i ”

3

2009 First Rmetrics eBook “Portfolio Optimization with R/Rmetrics”

2010  Meielisalp Summer School on Computational Finance

Part I

What are the Needs of Portfolio Managers ? 

• EDHEC Business School ReportEDHEC Business School Report• Absolute Risk Objectives• Relative Risk ObjectivesC i M t i E ti ti• Covariance Matrix Estimation

• Estimation Risk Problems

4

Absolute Risk Objectives

When implementing portfolio optimization, do you set absolute risk measures?

* *y

*Source: Felix Goltz, Edhec, 2009 

*

5*Supported by fPortfolio

Relative Risk Objectives

When implementing portfolio optimization, do you set relative risk measures with respect to a benchmark?

*p

**

6*Supported by fPortfolio

Covariance Matrix Estimation

When implementing portfolio optimization, how do you estimate the covariance matrix?

*

*

**

7*Supported by fPortfolio [unpublished] 

Estimation Risk/Problems

How do you deal with estimation risk/problemsof estimating the expected returns ?

*

** **

8*Supported by fPortfolio *Supported by BLCOP   *US Patented

Part II

Portfolio Design

• Portfolio ObjectivesPortfolio Objectives• Quantification of Risk Objectives• Rmetrics Solver FactoryR t i P tf li C t i t• Rmetrics Portfolio Constraints

• Rmetrics Performance Analysis• fPortfolio Package

9

Portfolio Objectives

Mi i i k bj tiMinimum risk objectiveMinimize  Any Risk + Transaction Costssubject to: Return  >  a given level

Any other user defined constraintsy

Maximum return objectiveMaximize  Return  – Transaction Costsbj t t A Ri k i l lsubject to:  Any Risk <  a given level.

Any other user defined constraints

Maximum risk‐adjusted returnjMaximize  Utility = Return – λ*Risk –Transaction Costs

where λ is a risk aversionsubject to: Any user defined constraints

10

Quantification of Risk ObjectivesRisk measures of Stone 1973

[ k = 2, A = Infinity, Y0 = mean (R) ]2Markowitz 1952

S l ti QP 1982 SOCP P i 1994

Pederson and Satchell 1998Rockafeller & Uryasev CVaR 1992  

k = 1, A = VaR, Y0 = 0

Solution: QP 1982, SOCP Programming 1994

for some bounded function W ( )

Solution: LP  

Semi‐VarianceMADLPM...

Artzner, Delbaen, Eber, Heath 1999

11… this makes a coherent risk measure

Rmetrics Solver Factory

fPortfolio default solver interfacesQP    quadprogLP     Rglpk  NLP  Rdonlp2 [ k h l l b l ][Packages: Rsocp, Rsymphony, Rsolnp, Rnlminb2, Rcplex, ...]

Rmetrics2AMPL interface: LP, QP, NLP,  MI[LQNL]POpen Source: Coin OR e g ipopt bonminOpen Source:   Coin‐OR, e.g.  ipopt, bonmin, ... Commercial:     cplex,  donlp2,  gurobi, loqo, minos, snopt, ...Requirement:   AMPL Language

Forthcoming R solver interface: ROI packageVienna Group, Stefan Theussl et al. 

12Supported by Rmetrics packages [unpublished] 

Rmetrics Portfolio Constraints

Performance constraints

Bounds on Assets Transaction Cost Limit ConstraintsLinear Constraints Turnover Constraints Quadratic Constraints Holding ConstraintsNonlinear Constraints Factor ConstraintsInteger ConstraintsRound Lots, Buy‐In, Cardinality, …

New risk constraints

Reserve Ratios for Pension Fund PortfoliosStability Indicators of Financial Markets – Stress Testing Pattern

13Supported by Rmetrics packages [unpublished] 

Mean‐Variance Markowitz Portfolio

0.8

1.0

SBISPISIILMIMPI0.

81.

0

0.318 0.249 0.302 0.432 0.6 0.781 0.969 1.17 1.39Target Risk

Wei

ght

Ris

k =

0.24

8532

WeightsWeights along the Variance Locus | Efficient FrontierSample mean and covariance estimates

Efficient FrontierMV Portfolio | mean-Stdev View

SPIALT

0.0

0.2

0.4

0.6

MPIALT

0.0

0.2

0.4

0.6

0.000102 0.0266 0.053 0.0795 0.106 0.132 0.159 0.185 0.212

MV

| sol

veR

quad

prog

| m

inR

EWP   Equal Weights PortfolioTGP Tangency Portfolio

Efficient Frontier  ‐ Feasible SetSwiss Pension Fund Portfolio

0.15

0.20

n[m

ean]

SPI

MPI

Target Return

0.15

0.20 SBI

SPISIILMIMPIALT

0.15

0.20

0.318 0.249 0.302 0.432 0.6 0.781 0.969 1.17 1.39Target Risk

Wei

ghte

d R

etur

n

| min

Ris

k =

0.24

8532

Weighted Returns

TGP    Tangency PortfolioGMV   Global Minim Risk

MinimumVarianceLocus

Efficient Frontier

Weighted Returns

Return

0.05

0.10

Targ

et R

etur

n

.053

6

0.0

714

SII

0.00

0.05

0.10

0.00

0.05

0.10

0.000102 0.0266 0.053 0.0795 0.106 0.132 0.159 0.185 0.212Target Return

W

MV

| sol

veR

quad

prog

EWP

TGP

Sharpe Ratio

Sample Mean 

0.0 0.5 1.0 1.5 2.0

0.00

0

MV

| sol

veR

quad

prog 0

SBILMI

.40.

60.

81.

0

SBISPISIILMIMPIALT

.40.

60.

81.

0

0.318 0.249 0.302 0.432 0.6 0.781 0.969 1.17 1.39Target Risk

Cov

Ris

k Bu

dget

s

adpr

og |

min

Ris

k =

0.24

9

Cov Risk BudgetsSharpe Ratio

GMV

Covariance Risk Budgets

S l C i i kTarget Risk[Cov]

0.0

0.2

0.0.

00.

20.

0.000102 0.0266 0.053 0.0795 0.106 0.132 0.159 0.185 0.212Target Return

MV

| sol

veR

quaSample Covariance Risk

14

Portfolio FrontierCode Example: Long‐Only Markowitz# LPP Portfolio Example:Data <- as.timeSeries(data(LPP2005REC))[, 1:6]Spec <- portfolioSpec()Constraints <- "LongOnly"

# Portfolio Frontier:

Load Data, set Spec and Constraints

Compute the efficient frontier on 5 # Portfolio Frontier:setNFrontierPoints(Spec) <- 5portfolioFrontier(Data, Spec, Constraints)

Title:MV Portfolio Frontier Estimator: covEstimator

pequidistant points (to reduce the output)

Output:Title

Solver: solveRquadprog

Portfolio Weights:SBI SPI SII LMI MPI ALT

1 1.0000 0.0000 0.0000 0.0000 0 0.00002 0.0193 0.0000 0.1481 0.6665 0 0.16613 0.0000 0.0085 0.2535 0.3386 0 0.3994

The portfolio weights

4 0.0000 0.0210 0.3458 0.0000 0 0.63325 0.0000 0.0000 0.0000 0.0000 0 1.0000

Covariance Risk Budgets:SBI SPI SII LMI MPI ALT

1 1.0000 0.0000 0.0000 0.0000 0 0.00002 0.0064 0.0000 0.1593 0.3359 0 0.4984

The covariance risk budgets

3 0.0000 0.0183 0.1208 -0.0097 0 0.87074 0.0000 0.0286 0.0890 0.0000 0 0.88245 0.0000 0.0000 0.0000 0.0000 0 1.0000

Target Return and Risks:mean mu Cov Sigma CVaR VaR

1 0.0001024718 0.0001024718 0.3177554 0.3177554 0.6949543 0.5485503

The target risk and target return:mean – sample mean

mu – e.g. robust/shrinked meanCov sample covariance 2 0.0541105688 0.0541105688 0.3059123 0.3059123 0.5951772 0.4434369

3 0.1081186658 0.1081186658 0.6146980 0.6146980 1.3293170 0.85487894 0.1621267628 0.1621267628 0.9926528 0.9926528 2.2230719 1.48334985 0.2161348598 0.2161348598 1.4324632 1.4324632 3.3625018 2.2626952

Cov – sample covarianceSigma – e.g. robust/shrinked CovCVaR – Conditional Value-at-Risk

VaR – Value-at-Risk

Constrainted MV PortfoliosCode Example: Add Constraints

# Example:0 339 0 447 0 613 0 793 0 983 1 2

Target Risk

nRis

k =

Weights MVp

Data <- 252*as.timeSeries(data(LPP2005REC))[,1:6]Spec <- portfolioSpec()Constraints <- c("minW[1:6]=0.05",

"maxsumW[c('SBI','LMI']=0.6")

# MV Markowitz Frontier:1

0.0

0.4

0.8 SBI

SPISIILMIMPIALT

0.0

0.4

0.8

0.339 0.447 0.613 0.793 0.983 1.2

0.0486 0.0751 0.102 0.128 0.154 0.181T t R t

Wei

ght

MV

| sol

veR

quad

prog

| m

in64

1

Frontier1 <-portfolioFrontier(Data,Spec,Constraints)

# Robust MV Frontier:setEstimator(Spec) <- "covMcdEstimator"Frontier2 <-

portfolioFrontier(Data Spec Contraints)

Target Return

0.4

0.8

SBISPISIILMIMPIALT

0.4

0.8

0.339 0.449 0.622 0.808 1 1.22Target Risk

Wei

ght

lveR

quad

prog

| m

inR

isk

= 0.

3386

Weights Robust MV

portfolioFrontier(Data,Spec,Contraints)

# QLPM Frontier:setType(Spec) <- "QLPM" setEstimator(Spec) <- "lpmEstimator"Spec@model$param$a <- 1.25Spec@model$param$tau <- "colMeans"

0.0

00.

00

0.0486 0.0751 0.102 0.128 0.154 0.181Target Return

MV

| so

0.345 0.459 0.623 0.817 1.02 1.23Target Risk

ht rog

| min

Ris

k =

0.34

5168

Weights Quadratic LPM a=1.25

p pFrontier3 <-

portfolioFrontier(Data,Spec,Constraints)

# Weights Plot:palette <- seqPalette(7,"OrRd")[-1]weightsPlot(Frontier1, col=palette))

i h l ( i 2 l l ))

0.0

0.4

0.8 SBI

SPISIILMIMPIALT

0.0

0.4

0.8

0.0486 0.0751 0.102 0.128 0.154 0.181Target Return

Wei

gh

QLP

M |

solv

eRqu

adp

weightsPlot(Frontier2, col=palette))weightsPlot(Frontier3, col=palette))

16

Factor Models*

Sharpe’s Single Index Model vs Mean Variance Markowitz

Sharpe's single index model General macroeconomic factor model

Sharpe s Single Index Model vs. Mean Variance Markowitz for a monthly Portfolio of selected US Equities

Barra industry factor model Statistical factor model 

Mean Re

turn

PCA statistical factor model Asymptotic PCA statistical factor model

Sample M

*[unpublished] 

Factor Covariance Risk

17

Estimation Error and Robustification

Sample estimatorImproves diversification of investments 

COVRobust estimatorsMCD, MVE, OGK, …

Other methods:

Shrinkage methodsB S i E iBayes‐Stein EstimatorLedoit‐Wolf Estimator

Random matrix theoryMC  Denoisingg

Factor models

18

Packages: MASS, robustbase, corpcor, tawny, ...

Robust PortfoliosCode Example: Alternative Covariance

# Example:Data <- 252*as.timeSeries(data(LPP2005REC))[,1:6]Spec <- portfolioSpec()Constraints <- "LongOnly"

# Standard Sample Estimator:frontierMarkowitz <-

portfolioFrontier(Data, Spec, Constraints)

# Kendall Rank Estimator:setEstimator(Spec) <- "kendallEstimator”( p )kendallEstimator <-

function (x, spec = NULL,...) {mu <- colMeans(x)Sigma <- cov(x, method = "kendall")list(mu = mu, Sigma = Sigma)}

FrontierKendall <-FrontierKendall <portfolioFrontier(Data, Spec, Constraints)

# Weights Plot:palette <- seqPalette(7,"OrRd")[-1] weightsPlot(FrontierMarkowitz col = palette)weightsPlot(FrontierMarkowitz, col = palette)weightsPlot(FrontierKendall, col = palette)

Seite 19

Rockafeller‐Uryasev: Mean‐CVaR

Mean‐CVaR portfolio 1992Mean‐CVaR Portfolio Optimization p

Linear Programming Problemwith Box and Group Constraints Swiss Pension Fund Portfolio

n Re

turn

where

…Sample Mea

where

20

Negative Conditional Value at Risk

Covariance Risk Budget Constraints

Takes a finite risk resource and decides

Compute from the derivativerisk resource, and decides

how best to allocate it. 

Normalized risk budgets

C t i th tf li ti i tiConstrain the portfolio optimization

21

Packages:  fPortfolio, fAssets

Copulae Tail Risk Budget Constraints

Decreases pair wise tail risk dependence 

SBI CH BondsSPI CH StocksSII CH ImmoLMI World BondsMPI World StocksALT World AltInvest

Copula dependence Coefficient:

Tail Dependence Coefficient:Lower

ALT World AltInvest

Portfolio Design:SBI SPI 0 SBI SII 0.055 SBI LMI 0.064 SBI MPI 0 SBI ALT 0 SPI SII 0 SPI LMI 0 SPI MPI 0 352SPI MPI 0.352 SPI ALT 0.273 SII LMI 0.075 SII MPI 0 LMI MPI 0 LMI ALT 0 MPI ALT 0.124

22Packages: fPortfolio, fCopulae

Part III

New Directions

• Portfolio Risk Surfaces & Risk Profile LinesPortfolio Risk Surfaces & Risk Profile Lines• Rastered Motion Risk Surfaces• Portfolio Shape PictogramsSt bilit M• Stability Measures

23

Reward/Risk SurfacesUse concepts to explore and diversify individual risks andfind more attractive investment opportunities and trading strategies 

Risk surfaces are plots of any risk measures across the feasible set expressed by

Covariance Risk Budget Diversification:Minimize the variance of the individual Risk!

Image PlotsContour PlotsPerspective Plots

d

Efficient Frontier  

Edge or ridge frontiers are lines where the individual risks of each asset or individual are bestSu

ccess o

r Rew

ard

Edge/Risk Profile each asset or individual are best diversified

Notebetter investment strategies can be

g /

better investment strategies can be found on the edge or ridge frontier!Raster Plot – Topo PaletteRisk   

24

Investments Along Risk Profiles

A simple efficient frontier strategy

Smoothly rebalance the investments from the tangency portfolio if it exists, otherwise invest in the global minimum risk portfolio.

Alternative risk profile line strategy

Instead investing on the efficient frontier, we now invest in b tt i k di ifi d tf li ith th t b tbetter risk diversified portfolios with the same return but now on the ridge frontier. 

Remark: These portfolios have higher total risks, but are better diversified

Package: fPortfolioBacktesting25

Portfolio Backtesting

Achieve lower draw‐downs and shorter recovery times 

Investment on Efficient Frontier Investment on Drawdown Risk Profile

Draw

downs

rn

Portfolio

Cumulated

 retu

Benchmark

26

Rastered Risk Surfaces

Rastered risk surface plotsMean Variance Markowitz Portfolio SurfaceDiversification of Weights and Kurtosis Values Rastered risk surface plots 

make multivariate risk displayspossible

Diversification of Weights and Kurtosis ValuesSwiss Pension Fund Portfolio

X‐Axis RiskY‐Axis ReturnColor var(Weights)Size KurtosisM

ean Re

turn

Size Kurtosis

Visualize changes in timei h M i Ch

Sample M

with Motion Charts

Use Parallel ImplementationSample Covariance Risk pp

27

Rmetrics and Google Motion Charts

• Add dynamic

A new understanding in portfolio analytics ?                             

• Add dynamic components tomultivariate data charts.

• Track the evolution of the risk surface.

• Observe velocity and acceleration of a portfolio’s characteristic parameters.

Data Spreadsheets are generated by R/Rmetrics

28

Part IVStable Portfolios  ‐ what does it mean?

Stability of Rolling Portfolios

• Stability Measures for Financial Time Series • Explaining and Understanding Stability Measures• Phase Space Embedding and Rolling IndicationsPhase Space Embedding and Rolling Indications• Bayesian Detection of Intraday Structural Breaks

29

Stability MeasuresRmetrics has software for different stability measures creating selective views on structural breaks and changes, jumps, outliers, and extreme dynamical dependencies

Value viewStructural Changes Breakpoint Detection

Variability viewVolatility and Extreme Value ClusteringStress Scenario LibraryStress Scenario Library

Multiresolution viewTime/Frequency AnalysisWavelet Analysisy

Stability viewPhase Space EmbeddingRobust Statistics

30

USD/EUR StabilityFor example generate an indicative view on the stability status oftime series values, volatilities, multiresolution behavior and stability 

31

Rolling Indications

Log Stock Market Index

Explore instabilities with the phase space embedding approach

Signal ?

low Stability

Sep   7: Federal takeover of Fannie Mae and Freddie Mac Sep 14: Merrill Lynch sold to Bank of America and Lehmann Brothers collapse

high

32

Sep 14: Merrill Lynch sold to Bank of America and Lehmann Brothers collapseSep 15: Lehmann Brothers  files for bankruptcy protectionSep 16: Moody’s and S&P downgrade ratings on AIGSep 17: The US FED lends $85 billion to AIG to avoid bankruptcy. Sep 18: Paulson and Bernanke propose a $700 billion emergency bailout

Intraday Price/Index BreaksExplore the Index/Price by a rolling Bayesian  ApproachYou can do this for Indices/Prices, Returns, Volatilities or any other dynamic quantity

Training Phase Forecasting Phase

abilitie

s

Index/Price 

Prob

a

Forecasting PhaseTraining Phase

S llIndex/Price 

Sell

Buy Sell

33

Portfolio Stability ObjectivesNew concepts  to optimize and stabilize dynamic systemslike portfolios , peer groups, or trading strategies

ObjectiveObjectiveMaximize Stability

Subject to:Success/Reward ConstraintsLoss/Risk ConstraintsStress Resistance Constraints

34

Part VHow to valuate and compare correlations and dependency structures in portfolios, in peer groups, and in trading strategies ?

Looking for Peer Group InstabilitiesLooking for Peer Group Instabilities

• Factor Shape Modeling • Portfolio Shape Pictograms• Portfolio Shape Pictograms• Dynamical Evolution of Geometrical Shapes• Shape Orientation Cycles• Portfolio Stability ObjectivesPortfolio Stability Objectives

35

Factor Shape ModelingA new geometric factorization approach in portfolio design 

Classification of feasible sets of portfolios, peer groups, trading strategies by shape pictograms:

area                    mass centerorientation excentricity

Geometric factor modellingGeometric factor modellingGenerates new kind of factor interactions and models or allows for additional factor constraints – use shape pictograms

Black‐Litterman investment like viewsIncorporates forecasts and expectations on the economic development which we can use in the pBlack‐Litterman approach – use shape pictograms

36

Dynamical Evolution of ShapesModel and forecast economic behaviour, explore peer groups, or evaluate trading strategies using dynamic shape factors

R lli d/ i k hRolling reward/risk shapes Geometric shape factors 

Area

enterR

atio        A

 Orie

ntation   Ce

Date

Eccentricity

   

37

Date

Shape Orientation CyclesUse the shape orientation cycle indicator or other derived indicators from geometric factor analysis to find structures in your portfolios, peer groups or trading strategies

Peer group analysis: MSCI developed market indexPeer group analysis: MSCI developed market index

Hongkong Sub Prime9/11Black MondayStock Crisis 19871973/1974

on Angle 

Orie

ntatio

The orange lines present identifiable patterns 

38

Use this approach to look for patterns in automated trading strategies and/or to analyze the behaviour of your models and human traders! 

Making Live Easier

h l

U R t i ’ Ad d P tf li E i t

The Tools …

Use Rmetrics’ Advanced Portfolio EnvironmentDevelop with Revolution’s GUIUse its Powerful Time Consuming DebuggingUse Parallel Computing with R/Revolution

39

t iTh k t

Special offer for webinar participants: www.rmetrics.org/ebooks‐bundle

www.rmetrics.orgThanks to:Yohan ChalabiWilliam ChenChristine Dong

Th k Y

info@rmetrics.orgChristine DongAndrew EllisSebastian Pérez SaaibiDavid Scott Thank YouDavid ScottStefan Theussl

Questions?Questions?

• Slides and Replay from today’s webinar:Slides and Replay from today s webinar:– bit.ly/portfolio‐webinar

• Download R: www.inside‐r.org/download• Visit Rmetrics: www.rmetrics.org

– Download: www.rmetrics.org/software– Ebooks: www.rmetrics.org/ebooks

41