EricZivot

Report
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
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
R it   i   i1 F1 t 
  ik Fkt   it ,
  i  β i Ft   it
i  1,
, n ; t  ti ,
,T
Ft ~ ( μ F , Σ F )
 it ~ (0,   , i )
2
co v( F jt ,  is )  0 fo r all j , i , s an d t
co v(  it , 
js
)  0 fo r i  j , s an d t
© Eric Zivot 2011
Performance Attribution
E [ R it ]   i   i1 E [ F1t ]     ik E [ F kt ]
Expected return due to systematic “beta” exposure
 i1 E [ F1t ]     ik E [ Fkt ]
Expected return due to firm specific “alpha”
 i  E [ R it ]  (  i1 E [ F1t ]     ik E [ Fkt ])
© Eric Zivot 2011
Factor Model Covariance
R t  α  B Ft  ε t
n 1
n k
n 1
k 1
n 1
var( R t )  Σ F M  B Σ F B   D 
D   diag (  ,1 ,
,   ,n )
2
Note:
2
cov( R it , R jt )  β i var( Ft ) β j  β i Σ F β j
var( R it )  β i Σ F β i    , i
2
© Eric Zivot 2011
Portfolio Linear Factor Model
, w n )   portfolio w eights
w  ( w1 ,
n
w
i
 1, w i  0 for i  1,
,n
i 1
R p , t  w  R t  w  α  w  B Ft  w  ε t
n

wR
i
i 1
n
it
  w i i 
i 1
n
n
 w β F   w 
i
i 1
  p  β p Ft   p , t
© Eric Zivot 2011
i
t
i
i 1
it
Risk Measures
Return Standard Deviation (SD, aka active risk)
  SD ( Rt )   β Σ F β  
2


1/ 2
Value-at-Risk (VaR)
V aR  q   F
1
( ), 0.01    0.10
F  C D F of return R t
Expected Tail Loss (ETL)
ETL  E [ Rt | Rt  VaR ]
© Eric Zivot 2011
5% ETL
5% VaR
10
Density
15
20
25
Risk Measures
0
5
± SD
-0.15
-0.10
-0.05
Returns
© Eric Zivot 2011
0.00
0.05
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, -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
© Eric Zivot 2011
difficult
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 nonparametrically from simulated return data
© Eric Zivot 2011
Simulation Algorithm
• Simulate B values of the risk factors by re-sampling
from full sample empirical distribution:
F
*
1
,
, FB 
*
• Simulate B values of the factor model residuals from
fitted non-normal distribution:
*
ˆ

 i1 ,
, ˆiB  , i  1,
*
,n
• Create factor model returns from factor models fit over
truncated samples, simulated factor variables drawn
from full sample and simulated residuals:
*
*
*
R it  ˆ i  βˆ i Ft  ˆit , t  1,
© Eric Zivot 2011
, B ; i  1,
,n
What to do with  R  ,  F  , ˆ 
*
it
B
t 1
*
it
B
t 1
*
it
B
t 1
• 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
© Eric Zivot 2009
,
?
Factor Risk Budgeting
Given linear factor model for asset or portfolio returns,
R t    β  F t   t    β  F t     z t    β F t
β   ( β  ,   ), Ft  ( Ft, z t ) , z t ~ (0,1)
SD, VaR and ETL are linearly homogenous functions
of factor sensitivities β . Euler’s theorem gives
additive decomposition
k 1
R M (β ) 

j 1
 R M (β )
j

, R M  SD , V aR , E T L
j
© Eric Zivot 2011
Factor Contributions to Risk
Marginal Contribution to
Risk of factor j:
Contribution to Risk
of factor j:
 R M (β )


j
 R M (β )
j
Percent Contribution to Risk 
j
of factor j:
© Eric Zivot 2011

j
 R M (β )

j
R M (β )
Factor Tail Risk Contributions
For RM = VaR, ETL it can be shown that
 VaR ( β )
 j
 ETL ( β )
 j
 E [ F jt | R t  VaR ], j  1,
,k 1
 E [ F jt | R t  VaR ], j  1,
,k 1
Notes:
1. Intuitive interpretations as stress loss scenarios
2. Analytic results are available under normality
© Eric Zivot 2011
Semi-Parametric Estimation
Factor Model Monte Carlo semi-parametric
estimates
Eˆ [ F jt | R t  VaR ] 
B
1

m
F jt  1 VaR     R t  VaR    
*
*
t 1
Eˆ [ F jt | R t  VaR ] 
1
B
F

[ B ]
*
jt
 1  R  VaR 
t 1
© Eric Zivot 2011
*
t
0.00
-0.10
Returns
Hedge fund returns and 5% VaR Violations
1998
2000
5% VaR
2002
2004
2006
Index
0.00
-0.06
Returns
0.06
Risk factor returns when fund return <= 5% VaR
1998
2000
Factor marginal contribution
to 5% ETL
2002
Index
© Eric Zivot 2011
2004
2006
Portfolio Risk Budgeting
Given portfolio returns,
n
R p , t  w R t 
wR
i
it
i 1
SD, VaR and ETL are linearly homogenous functions
of portfolio weights w. Euler’s theorem gives
additive decomposition
R M (w )
n
RM (w ) 
w
i 1
i
 wi
, R M  S D , V a R  , E T L
© Eric Zivot 2011
Fund Contributions to Portfolio Risk
Marginal Contribution to
Risk of asset i:
R M (w )
 wi
wi
Contribution to Risk
of asset i:
Percent Contribution to Risk
of asset i:
R M (w )
wi
© Eric Zivot 2011
 wi
R M (w )
 wi
RM (w )
Portfolio Tail Risk Contributions
For RM = VaR, ETL it can be shown that
 VaR ( w )
 wi
 ETL ( w )
 wi
 E [ R it | R p , t  VaR ( w )], i  1,
,n
 E [ R it | R p , t  VaR ( w )], i  1,
,n
Note: Analytic results are available under normality
© Eric Zivot 2011
Semi-Parametric Estimation
Factor Model Monte Carlo semi-parametric
estimates
Eˆ [ R it | R p ,t  VaR ( w )] 
1
B

m
R it  1 VaR ( w )    R p ,t  VaR  ( w )   
*
*
t 1
Eˆ [ R it | R t  VaR ( w )] 
1
B

[ B ]
R it  1  R p ,t  VaR ( w )
*
*
t 1
© Eric Zivot 2011
0.00 0.04
-0.06
Returns
FoHF Portfolio Returns and 5% VaR Violations
2002
2003
5% VaR
2004
2005
2006
2007
Index
-0.05 0.00 0.05
Returns
Constituant fund returns when FoHF returns <= 5% VaR
2002
2003
2004
Index
Fund marginal contribution to
© Eric Zivot 2011
portfolio 5% ETL
2005
2006
2007
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
1.67% VaR
1.67% ETL
30
FMMC FoHF Returns
FM = 1.42%
FM,EWMA = 1.52%
20
10
0
Density
VaR0.0167 = -3.25%
ETL0.0167 = -4.62%
-0.05
50,000 simulations
0.00
Returns
© Eric Zivot 2011
0.05
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
© Eric Zivot 2011

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