Factor Model Based Risk Measurement and Management R/Finance 2011: Applied Finance with R April 30,...
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•Factor Model Based Risk Measurement and Management
•R/Finance 2011: Applied Finance with R•April 30, 2011
• Eric Zivot• Robert Richards Chaired Professor of Economics• Adjunct Professor, Departments of Applied Mathematics,
Finance and Statistics, University of Washington• BlackRock Alternative Advisors
© Eric Zivot 2011
Risk Measurement and Management• Quantify asset and portfolio exposures to risk
factors– Equity, rates, credit, volatility, currency– Style, geography, industry, etc.
• Quantify asset and portfolio risk– SD, VaR, ETL
• Perform risk decomposition– Contribution of risk factors, contribution of
constituent assets to portfolio risk
• Stress testing and scenario analysis
© Eric Zivot 2011
Asset Level Linear Factor Model
1 1
2,
,
1, , ; , ,
~ ( , )
~ (0, )
cov( , ) 0 for all , , and
cov( , ) 0 for , and
it i i t ik kt it
i i t it
i
t F F
it i
jt is
it js
R F F
i n t t T
F j i s t
i j s t
β F
F μ Σ
© Eric Zivot 2011
Performance Attribution
][][][ 11 ktiktiiit FEFERE
][][ 11 ktikti FEFE
])[][(][ 11 ktiktiiti FEFERE
Expected return due to systematic “beta” exposure
Expected return due to firm specific “alpha”
© Eric Zivot 2011
Factor Model Covariance
2 2,1 ,
var( )
( , , )
t FM
ndiag
FR Σ BΣ B D
D
11 1 1t t
n knn k n R α B F εt
2,
cov( , ) var( )
var( )
it jt i t j i j
it i i i
R R
R
F
F
β F β β Σ β
β Σ β
Note:
© Eric Zivot 2011
Portfolio Linear Factor Model
,
1 1 1 1
,
p t t t t
n n n n
i it i i i i t i iti i i i
p p t p t
R
w R w w w
w R w α w BF w ε
β F
β F
1
1
( , , ) portfolio weights
1, 0 for 1, ,
n
n
i ii
w w
w w i n
w
© Eric Zivot 2011
Risk Measures
1( ), 0.01 0.10
of return t
VaR q F
F CDF R
Value-at-Risk (VaR)
Expected Tail Loss (ETL)
[ | ]t tETL E R R VaR
Return Standard Deviation (SD, aka active risk)
1/22( )t FSD R β Σ β
© Eric Zivot 2011
Risk Measures
Returns
De
nsity
-0.15 -0.10 -0.05 0.00 0.05
05
10
15
20
25
± SD
5% VaR5% ETL
© Eric Zivot 2011
Tail Risk Measures: Non-Normal Distributions
• Asset returns are typically non-normal• Many possible univariate non-normal
distributions– Student’s-t, skewed-t, generalized hyperbolic,
Gram-Charlier, a-stable, generalized Pareto, etc.
• Need multivariate non-normal distributions for portfolio analysis and risk budgeting.
• Large number of assets, small samples and unequal histories make multivariate modeling difficult
© Eric Zivot 2011
Factor Model Monte Carlo (FMMC)
• Use fitted factor model to simulate pseudo asset return data preserving empirical characteristics of risk factors and residuals– Use full data for factors and unequal history for
assets to deal with missing data
• Estimate tail risk and related measures non-parametrically from simulated return data
© Eric Zivot 2011
Simulation Algorithm• Simulate B values of the risk factors by re-sampling
from full sample empirical distribution:
• Simulate B values of the factor model residuals from fitted non-normal distribution:
• Create factor model returns from factor models fit over truncated samples, simulated factor variables drawn from full sample and simulated residuals:
1 , , B* *F F
* *1
ˆ ˆ, , , 1, ,i iB i n
* * *ˆˆ ˆ , 1, , ; 1, ,it i i t itR t B i n β F
© Eric Zivot 2009
What to do with ?
• Backfill missing asset performance• Compute asset and portfolio performance
measures (e.g., Sharpe ratios)• Compute non-parametric estimates of asset
and portfolio tail risk measures• Compute non-parametric estimates of asset
and factor contributions to portfolio tail risk measures
* *
1 1 1ˆ, , ,
B B B
it it itt t tR
*F
© Eric Zivot 2011
Factor Risk Budgeting
Given linear factor model for asset or portfolio returns,
SD, VaR and ETL are linearly homogenous functions of factor sensitivities . Euler’s theorem gives additive decomposition
( , ), ( , ) , ~ (0,1)
t t t t t t
t t t t
R z
z z
β F β F β F
β β F F
1
1
( )( ) , , ,
k
jj j
RMRM RM SD VaR ETL
ββ
β
© Eric Zivot 2011
Factor Contributions to Risk
Marginal Contribution to Risk of factor j:
Contribution to Risk of factor j:
Percent Contribution to Risk of factor j:
( )
j
RM
β
( )j
j
RM
β
( )( )j
j
RMRM
β
β
© Eric Zivot 2011
Factor Tail Risk Contributions
For RM = VaR, ETL it can be shown that
( )[ | ], 1, , 1
( )[ | ], 1, , 1
jt tj
jt tj
VaRE F R VaR j k
ETLE F R VaR j k
β
β
Notes: 1. Intuitive interpretations as stress loss scenarios2. Analytic results are available under normality
© Eric Zivot 2011
Semi-Parametric Estimation
Factor Model Monte Carlo semi-parametric estimates
* *
1
* *
1
1ˆ[ | ] 1
1ˆ[ | ] 1[ ]
B
jt t jt tt
B
jt t jt tt
E F R VaR F VaR R VaRm
E F R VaR F R VaRB
© Eric Zivot 2011
1998 2000 2002 2004 2006
-0.1
00
.00
Index
Re
turn
sHedge fund returns and 5% VaR Violations
1998 2000 2002 2004 2006
-0.0
60
.00
0.0
6
Index
Re
turn
s
Risk factor returns when fund return <= 5% VaR
5% VaR
Factor marginal contribution to 5% ETL
© Eric Zivot 2011
Portfolio Risk Budgeting
Given portfolio returns,
SD, VaR and ETL are linearly homogenous functions of portfolio weights w. Euler’s theorem gives additive decomposition
,1
n
p t t i iti
R w R
w R
1
( )( ) , , ,
n
ii i
RMRM w RM SD VaR ETL
w
ww
© Eric Zivot 2011
Fund Contributions to Portfolio Risk
Marginal Contribution to Risk of asset i:
Contribution to Risk of asset i:
Percent Contribution to Risk of asset i:
( )
i
RM
w
w
( )i
i
RMw
w
w
( )( )i
i
RMw RM
w
w
w
© Eric Zivot 2011
Portfolio Tail Risk Contributions
For RM = VaR, ETL it can be shown that
,
,
( )[ | ( )], 1, ,
( )[ | ( )], 1, ,
it p ti
it p ti
VaRE R R VaR i n
w
ETLE R R VaR i n
w
ww
ww
Note: Analytic results are available under normality
© Eric Zivot 2011
Semi-Parametric Estimation
Factor Model Monte Carlo semi-parametric estimates
* *, ,
1
* *,
1
1ˆ[ | ( )] 1 ( ) ( )
1ˆ[ | ( )] 1 ( )[ ]
B
it p t it p tt
B
it t it p tt
E R R VaR R VaR R VaRm
E R R VaR R R VaRB
w w w
w w
© Eric Zivot 2011
2002 2003 2004 2005 2006 2007
-0.0
60
.00
0.0
4
Index
Re
turn
sFoHF Portfolio Returns and 5% VaR Violations
2002 2003 2004 2005 2006 2007
-0.0
50
.00
0.0
5
Index
Re
turn
s
Constituant fund returns when FoHF returns <= 5% VaR
5% VaR
Fund marginal contribution to portfolio 5% ETL
© Eric Zivot 2011
Example FoHF Portfolio Analysis
• Equally weighted portfolio of 12 large hedge funds
• Strategy disciplines: 3 long-short equity (LS-E), 3 event driven multi-strat (EV-MS), 3 direction trading (DT), 3 relative value (RV)
• Factor universe: 52 potential risk factors• R2 of factor model for portfolio ≈ 75%, average
R2 of factor models for individual hedge funds ≈ 45%
© Eric Zivot 2011
FMMC FoHF Returns
Returns
De
nsity
-0.05 0.00 0.05
010
20
30 1.6
7%
ET
L
1.6
7%
VaR
sFM = 1.42%sFM,EWMA = 1.52%VaR0.0167 = -3.25%ETL0.0167 = -4.62%
50,000 simulations
© Eric Zivot 2011
Factor Risk Contributions
© Eric Zivot 2011
Hedge Fund Risk Contributions
© Eric Zivot 2011
Hedge Fund Risk Contribution
© Eric Zivot 2011
Summary and Conclusions
• Factor models are widely used in academic research and industry practice and are well suited to modeling asset returns
• Tail risk measurement and management of portfolios poses unique challenges that can be overcome using Factor Model Monte Carlo methods