The Hull, the Feasible Set, and the Risk Surface A Review of the Portfolio Modelling Infrastructure in R/Rmetrics
g
/
Diethelm Würtz
Institute for Theoretical Physics ETH Zurich
Institute
for Theoretical Physics ETH Zurich
Curriculum for Computational Science ETH Zurich
Finance Online & Rmetrics Association Zurich
UseR! Gaithersburg, July 2010
Joint work with
Yohan Chalabi, William Chen, Christine Dong, Andrew Ellis, Sebastian Pérez Saaibi, David Scott, Stefan Theussl
www.itp.phys.ethz.ch | www.rmetrics.org | www.finance.ch
Overview
Part I Rmetrics and Our Visions
Part II
Portfolio Analysis with R/Rmetrics
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Part III
Part III
New Directions
New Directions
2
Part I
Who is Rmetrics and what are our Visions ?
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
• W
We operate an Educational and Teaching Platform
t
Ed ti
l d T hi Pl tf
• We offer an R Code Archive and a Public Tools Platform
• We started to Build a Public Stability and Risk Data Base
We started to Build a Public Stability and Risk Data Base
about 50 packages and 25 developers on r‐forge
3
Vision No 1: An Open Educational and Teaching Platform
An Open Educational and Teaching Platform
Rmetrics Open Source Teaching Platform and Community
•
•
•
•
•
•
Software Packages Serving as Code Archive
Datafeeds for Public Available Data on the Web
Datafeeds for Public Available Data on the Web
High Quality Documentation R/Rmetrics eBooks
Support Student Internships for Training at ETHZ
Support Student Internships for Training at ETHZ
Rmetrics Meielisalp Summer School
Rmetrics Meielisalp User and Developer Workshop
Rmetrics Meielisalp User and Developer Workshop
4
Vision No 2: An Open Code Archive
An Open Code Archive
Why we Maintain an R Code Archive
• We need a platform which provides algorithms and software tools to measure and control the risk and
software tools to measure and control the risk and
unstabilities of financial investments.
• We need more graphical tools which allow for better views on the performance risk attributions and
views on the performance, risk attributions and stability of financial investments.
5
Vision No 3: An Open Stability and Risk Data Platform
An Open Stability and Risk Data Platform
Why we Create a Stability and Risk Data Platform
• Financial stability and risk data are in the public interest.
• We need an independent data base and platform which allows to make investments more transparent
which allows to make investments more transparent and reproducible for everybody.
• We believe it is time to start with such a project, feel
j
free to join us.
6
Part II
Portfolio Design with R/Rmetrics
• What is a Financial Portfolio?
What is a Financial Portfolio?
• Portfolio Objectives and Constraints
• Rmetrics Portfolio Solver Interfaces
Examples
• Absolute and Relative Risk Objectives
Ab l t
d R l ti Ri k Obj ti
• Covariance Matrix Estimation
• Extreme Risk Measures
Extreme Risk Measures
• Estimation Risk/Problems
7
Example
Swiss Pension Funds
Swiss Pension Funds
Performance of Swiss Pension Funds
… based on Global Custody Data of Credit Suisse, as at December 31, 2009
This Index is not an artificially constructed performance index but an index that is based on actual pension fund data.
Example
Swiss Pension Fund Portfolio
DJIA @14000
2Y Rolling Risk‐Return
Nasdaq all time high
2000‐03‐10
9‐11
Lehman failed
2008‐09‐15
5Y Rolling Risk Return
5Y Rolling Risk‐Return
8
Example
Swiss Pension Funds Benchmark
Example
Swiss Pension Fund Portfolio
http://www.pictet.com/en/home/lpp_indices/lpp2005.html
High Risk
Medium
m Risk
Low
w Risk
The Benchmark is Risk not Stability
9
Fund & Portfolio Objectives
j
Minimum Risk Objective
Minimize subject to
subject to:
Any Risk + Transaction Costs
Return > a given level
Return > a given level
Any other user defined constraints
Maximum Return Objective
Maximum Return Objective
Maximize subject to: Return – Transaction Costs
Any Risk < a given level.
Any other user defined constraints
Any other user defined constraints
Maximum Risk‐Adjusted Return
Maximize
Maximize subject to:
Utility = Return –
Utility
= Return – λλ*Risk
Risk –Transaction Costs
–Transaction Costs
where λ is a risk aversion
Any user defined constraints
Rmetrics Solver Interfaces
fPortfolio Default Solver Interfaces fPortfolio
Default Solver Interfaces
QP quadprog
LP Rglpk NLP Rdonlp2
NLP Rdonlp2 [Packages: Rsocp, Rsymphony, Rsolnp, Rnlminb2, Rcplex, ...]
Rmetrics2AMPL Interface: LP, QP, NLP, MI[LQNL]P
R
i 2AMPL I
f
LP QP NLP MI[LQNL]P
Open Source: Coin‐OR, e.g. ipopt, bonmin, ... Commercial: cplex, donlp2, gurobi, loqo, minos, snopt, ...
Disadvantage: Requires to learn AMPL
Forthcoming Solver Interface: ROI Package g
g
Vienna Group, Stefan Theussl et al. 11
Rmetrics Portfolio Constraints
Performance Constraints
Performance Constraints
Bounds on Assets
Linear Constraints
Linear Constraints
Quadratic Constraints
Nonlinear Constraints
Integer Constraints
Integer Constraints
Transaction Cost Limit Constraints
Turnover Constraints
Turnover Constraints Holding Constraints
Factor Constraints
Round Lots, Buy‐In, Cardinality, …
New Risk Constraints
e.g. Reserve Ratios for Pension Fund Portfolios
Stability Indicators of Financial Markets – Stress Testing Pattern
12
Rmetrics fPortfolio Examples
The Needs of Portfolio Managers
The Needs of Portfolio Managers
EDHEC Business School Report
Felix Goltz, Edhec, 2009 13
Absolute Risk Objectives
j
When implementing portfolio optimization, do you set absolute risk measures?
*
*
*
*
*Supported by fPortfolio
14
Relative Risk Objectives
j
When implementing portfolio optimization, do you set relative risk measures with respect to a benchmark?
*
*
Source: Felix Goltz, Edhec, 2009 *Supported by fPortfolio
*
15
Example
Q
Quantification of Risk Objectives
ifi i
f Ri k Obj i
Risk Measures of Stone 1973
Markowitz 1952
[ k = 22, A = IInfinity,
fi it Y0 = mean (R) ]
2
Solution: QP 1982, SOCP Programming 1994
Rockafeller & Uryasev CVaR 1992 Pederson and Satchell 1998
Pederson and
k = 1, A = VaR, Y0 = 0
Solution: LP for some bounded function W ( )
Semi‐Variance
Semi
Variance
MAD
LPM
...
Artzner Delbaen Eber Heath 1999
Artzner, Delbaen, Eber, Heath 1999
… this makes a coherent risk measure
16
Covariance Matrix Estimation
When implementing portfolio optimization, h d
how do you estimate the covariance matrix?
i
h
i
i ?
*
*
*
*Supported by fPortfolio [unpublished] *
17
Example
Mean Variance Markowitz Portfolio
Mean‐Variance Markowitz Portfolio
0.302
0.432
0.6
0.781
0.969
1.17
1.39
SBI
SPI
SII
LMI
MPI
ALT
0.2
0
0.0
EWP Equal Weights Portfolio
TGP Tangency Portfolio
GMV Global Minim Risk
ALT
0.000102
SPI
0.0266
0.053
0.0795
0.106
0.132
0.159
0.185
0.212
Target Return
Weighted Return
0.15
0.20
Minimum
Variance
Locus
Target Risk
0.302
0.432
0.6
0.781
0.969
1.17
1.39
SBI
SPI
SII
LMI
MPI
ALT
Efficient Frontier
0.10
EWP
TGP
0.000102
1.0
1.5
Sample Covariance Risk
Target Risk[Cov]
2.0
0.106
0.132
0.159
0.185
0.212
Target Return
Covariance Risk Budgets
Cov Risk Budgets
0 318
0.318
0 249
0.249
Target Risk
0 302
0.302
0 432
0.432
06
0.6
0 781
0.781
0 969
0.969
1 17
1.17
1 39
1.39
Cov Risk Budgets
0.6
0.8
1.0
SBI
0.0795
SBI
SPI
SII
LMI
MPI
ALT
0.4
LMI
0.053
0.2
GMV
0.0266
0.0
MV | solveRquadprog
Sharpe Ratio
0.0536
0.05
SII
0.5
0.00
0.0714
0.0
05
0.10
0
0.249
0.000102
0.0266
0.053
0.0795
0.106
0.132
0.159
0.185
MV | solveR
Rquadprog | minRisk = 0.249
0.15
MPI
0.00
Sample
e Mean Return
Targe
et Return[mean]
0.318
MV | solvveRquadprog | minRisk = 0.248
8532
Weighted Returns
Weighted Returns
0.0
MV | so
olveRquadprog | minRisk = 0.2
248532
Target Risk
0.249
Weigh
ht
0.8
1.0
1
0.318
0.4
Efficient Frontier ‐ Feasible Set
Efficient Frontier
Swiss Pension Fund Poretfolio
MV Portfolio | mean-Stdev View
0.20
Weights along the Variance Locus | Efficient Frontier
Weights
0.6
Sample Mean and Covariance Estimates
0.212
Target Return
18
Example
Factor Models
Factor Models
Sharpe’s Single Index Model vs. Mean Variance Markowitz for a monthly Portfolio of selected US Equities
Sharpe's single index model General macroeconomic factor model
Saample Mean Retu
urn
Barra industry factor model Statistical factor model PCA statistical factor model Asymptotic PCA statistical factor model y p
Factor Covariance Risk
19
Example
Estimation Error and Robustification
Estimation Error and Robustification
Improves Diversification of Investments SSample Estimator
l
i
COV
Robust Estimators
MCD MVE OGK
MCD, MVE, OGK, …
Other Methods:
Shrinkage Methods
Bayes‐Stein Estimator
Ledoit‐Wolf
Ledoit
Wolf Estimator
Estimator
Random Matrix Theory
MC Denoising
Factor Models
Factor Models
Packages: MASS robustbase corpcor tawny
Packages: MASS, robustbase, corpcor, tawny, ...
20
Extreme Risk Measures
When implementing portfolio optimization, how do you calculate extreme risk?
*
*
*
*
*Supported by fPortfolio
21
Example
R k f ll U
Rockafeller‐Uryasev: Mean‐CVaR
M
CV R
Samp
ple Mean Return
Mean‐CVaR Portfolio Optimization
with Box and Group Constraints Swiss Pension Fund Portfolio
Mean‐CVaR Portfolio 1992
Linear Programmig Problem
…
where
Negative Conditional Value at Risk
g
22
Estimation Risk/Problems
/
How do you deal with estimation risk/problems
of estimating the expected returns ?
f
h
d
?
*
* * *
*Supported by fPortfolio *Supported by BLCOP *US Patented
23
Example
Covariance Risk Budget Constraints
Covariance Risk Budget Constraints
Takes a finite Takes
a finite
risk resource, and decides
how best to allocate it. Compute from the derivative
p
Normalized risk budgets
Constrain the portfolio optimization
Packages: fPortfolio, fAssets
24
Example
Copulae Tail Risk Budget Constraints
Tail Risk Budget Constraints
Decreases pair wise tail risk dependence SBI CH Bonds
SPI CH Stocks
SII CH Immo
LMI World Bonds
MPI World Stocks
ALT World AltInvest
Tail Dependence Coefficient:
Lower
SBI
SBI
SBI
SBI
SBI
SPI
SPI
SPI
SPI
SII
SII
LMI
LMI
MPI
SPI
SII
LMI
MPI
ALT
SII
LMI
MPI
ALT
LMI
MPI
MPI
ALT
ALT
Copula dependence Coefficient:
Portfolio Design:
0
0.055
0.064
0
0
0
0
0.352
0.273
0.075
0
0
0
0.124
Packages: fPortfolio, fCopulae
25
Part III
New Directions
•
•
•
•
Portfolio Risk Surfaces & Risk Profile Lines
Rastered Motion Risk Surfaces
Portfolio Shape Pictograms
Stability Measures
Risk Surfaces & Risk Profile Lines Mean Variance Markowitz Portfolio with Covariance Risk Budget Constraints
Swiss Pension Fund Portfolio
Risk Profiles
Sample Mean Reeturn
An edge or ridge is a line on the An
edge or ridge is a line on the
surface where the risk measure is best diversified.
Covariance Risk Budget Profile
On this risk profile (black thick line) the portfolios with the best
line) the portfolios with the best diversified covariance risk budgets can be found.
Sample Covariance Risk
Example
Investments along Risk Profiles
Investments along Risk Profiles
A simple Efficient Frontier Strategy
Smoothly rebalance the investments from the tangency portfolio Smoothly
rebalance the investments from the tangency portfolio
if it exists, otherwise invest in the global minimum risk portfolio.
Alternative Risk Profile Line Strategy
Alternative Risk Profile Line Strategy
Instead investing on the efficient frontier, we now invest in better risk diversified portfolios with the same return but now on p
the ridge frontier. Remark: These portfolios have higher total risks, but are better diversified
Package: fPortfolioBacktesting
28
Example
Portfolio Backtesting
Portfolio Backtesting
Achieve lower drawdowns and shorter recovery times Draawdowns
Investment on Efficient Frontier
Cumulated return
Portfolio
Benchmark
Investment on Drawdown Risk Profile
Rastered Motion Risk Surfaces
M
Mean Variance Markowitz Portfolio Surface
V i
M k it P tf li S f
Diversification of Weights and Kurtosis Values
Swiss Pension Fund Portfolio
Rastered Risk Surface Plots make multivariate risk displays
possible
Sample Mean Return
X‐Axis
YA i
Y‐Axis
Color
Size
Risk
R
Return
var(Weights)
Kurtosis
Visualize changes in time
with Motion Charts
with Motion Charts
Sample Covariance Risk
Example
Rmetrics and Google Motion Charts
Rmetrics and Google Motion Charts
A new understanding in portfolio analytics ? US Patented ?
• Add dynamic components to
multivariate data charts.
• Track the evolution T k h
l i
of the risk surface.
• Observe velocity Observe velocity
and acceleration of a portfolio’s characteristic characteristic
parameters.
Data Spreadsheets are generated by R/Rmetrics
Portfolio Shape Pictograms
p
g
Rastered Feasible Set and Shape Picogram
Mean Variance Markowitz Portfolio
Swiss Pension Fund Portfolio
Classification of feasible sets by shape pictograms Factor Modelling
Generates new kind of factor models or allows for additional factor constraints
Return
Factors:
Area
Center
Orientation Eccentricity
Orientation Eccentricity
Risk
Investment Views
Enables forecasts of economic developments, which we can use in the Black‐Litterman approach
32
Example:
Peer Group Analysis Evolution of Portfolio Shapes
Peer Group Analysis ‐
Evolution of Portfolio Shapes
Idea
Model and forecast economic behaviour
Model
and forecast economic behaviour
using portfolio shape factors
Shape Factors Eccentricity Orientatio
on Center Ratio
o Area
Rolling Portfolio Shapes
Date
33
Example:
Shape Orientation Cycle MSCI EU Stock Index Universe
Shape Orientation Cycle –
MSCI EU Stock Index Universe
The orientation factor is a good indicator of economic turmoil
Peer Group Analysis: MSCI Developed Market Index
Hongkong
Stock Crisis 1987
Black Monday 9/11
Sub Prime
Orieentation An
ngle 1973/1974
The orange lines present identifiable patterns 34
Example:
Orientation Eccentricity Breakout MSCI EU Indices
Orientation‐Eccentricity Breakout –
MSCI EU Indices
Orientation Factor Eccentricity Factor
35
Stability Measures
y
Value View
Structural Changes Breakpoint Detection
Volatility View
Volatility and Extreme Value Clustering
Stress Scenario Library
Multiresolution View
l
l
Time/Frequency Analysis
Wavelet Analysis
Stability View
Phase Space Embedding
R b S i i
Robust Statistics
36
Example
Stability Measures USD EUR FX Rate
Stability Measures –
USD EUR FX Rate
37
Example
Stability Measures Greece Gvt
Stability Measures –
Greece Gvt Bonds
38
Example
Stability Measures – British Petroleum
39
Our Proposal
Stability as New Peer Group & Portfolio Objectives
f
Objective
Maximize Stability
Subject to:
j
Return Constraints
Risk Constraints
Stress Resistance Constraints
Stress Resistance Constraints
40
Joint work with:
Yohan Chalabi
William Chen
Christine Dong
A d
Andrew Ellis
Elli
Sebastian Pérez Saaibi
David Scott
Stefan Theussl
www.rmetrics.org
www.ethz.ch
[email protected]
h
h h
Thank You
Thank You