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
   i

F i 1 1 t
  i
 β F i t

 
 it


1, , ;
 t i
, , T
F ik kt
F t
~ (
μ Σ
F F
)
 it
~ (0,


2
, i
) cov( F ,
 jt is
)
 j i s t cov(
  it
, js
)

0 for i
 j s t
  it
,
© Eric Zivot 2011

Performance Attribution
E [ R it
]
  i
  i 1
E [ F
1 t
]
    ik
E [ F kt
]
Expected return due to systematic “beta” exposure
 i 1
E [ F
1 t
]
    ik
E [ F kt
]
Expected return due to firm specific “alpha”
 i

E [ R it
]

(
 i 1
E [ F
1 t
]
    ik
E [ F kt
])
© Eric Zivot 2011

Factor Model Covariance
R
  B F ε t n

1 n

1 k
 t
1
 n
 t
1 var( R t
)
 Σ
FM

F
 

D

 diag (


2
,1
, ,


2
, n
)
Note: cov( R R it jt
)
 β  i
F β t j
  i F var( R it
)
  i F i
 

2
, i j
© Eric Zivot 2011

R
Portfolio Linear Factor Model w
 w
1
, w n
)
 
portfolio weights i n 

1 w i

1, w i

0 for i

1, , n
  t
    t
  t
 i n 

1 w R i it
 i n 

1 w
 i i
 i n 

1
  p
 β F p t
  w
 i i t
 i n 

1 w
 i it
© Eric Zivot 2011

Risk Measures
Return Standard Deviation ( SD, aka active risk )
 
SD R t



F
 

2

1/2
Value-at-Risk ( VaR )
VaR

 q


F

1
F

CDF of return R t
Expected Tail Loss ( ETL )
ETL


[ t
| t

VaR

]
© Eric Zivot 2011
0.10

5% ETL
Risk Measures
5% VaR
± 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’st , skewedt , 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 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 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
*
, , F
*
B

• Simulate B values of the factor model residuals from fitted non-normal distribution:

 ˆ * i 1
, ,
 ˆ * iB

, 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
 i t
*   ˆ , 1, , ; it

1, , n
© Eric Zivot 2011

     

1
F
* it
B

1
 ˆ * it
B B
What to do with ?
t t 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 t
 z t
  t
β  
( ,


), F t
 F  z t t
 z t
SD , VaR and ETL are linearly homogenous functions additive decomposition
RM
 k j

1 

1
 j

RM
  j
, RM

,

, ETL

© Eric Zivot 2011

Factor Contributions to Risk
Marginal Contribution to
Risk of factor j :

RM
  j
Contribution to Risk of factor j :
 j

RM
 
Percent Contribution to Risk  j of factor j :

RM
  j j
RM
© Eric Zivot 2011

Factor Tail Risk Contributions
For RM = VaR , ETL it can be shown that

VaR

  j

ETL

  j

[ jt
| R t

VaR

], j

1, , k

1

[ 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
ˆ[ |
R jt t

VaR

]

1 m t
B 

1
F jt
* 
1

VaR


R t
* 
VaR

ˆ[ |
R jt t

VaR

]

1
[ B

] t
B 

1
F jt
*

1

R t
*

VaR


 

© Eric Zivot 2011

Hedge fund returns and 5% VaR Violations
5% VaR
1998 2000 2002
Index
2004 2006
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,
R
 t
 i n 

1 w R i it
SD , VaR and ETL are linearly homogenous functions of portfolio weights w
. Euler’s theorem gives additive decomposition
RM
 i n 

1 w i

RM
 w i
, RM

,

, ETL

© Eric Zivot 2011

Fund Contributions to Portfolio Risk
Marginal Contribution to
Risk of asset i :

RM ( w )
 w i w i

RM ( w )
 w i
Contribution to Risk of asset i :
Percent Contribution to Risk of asset i : w i

RM
 w i
( w )
RM ( w )
© Eric Zivot 2011

Portfolio Tail Risk Contributions
For RM = VaR , ETL it can be shown that

VaR

 w i

ETL

 w i


[ it
| R

VaR

[ it
| R

VaR
 w i

1, , n w i

1, , n
Note: Analytic results are available under normality
© Eric Zivot 2011

Semi-Parametric Estimation
Factor Model Monte Carlo semi-parametric estimates
ˆ[ | it ,

VaR

ˆ[ | it t

VaR


1 m t
B 

1
R it
* 
1

VaR
 w

1
[ B

] t
B 

1
R
* it

1

R
* 
VaR


R
* 
VaR


 

© Eric Zivot 2011

FoHF Portfolio Returns and 5% VaR Violations
5% VaR
2002 2003 2004
Index
2005 2006 2007
Constituant fund returns when FoHF returns <= 5% VaR
2002 2003 2004
Fund marginal contribution to portfolio 5% ETL
Index
© Eric Zivot 2011
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
• R 2 of factor model for portfolio ≈ 75%, average
R 2 of factor models for individual hedge funds ≈
45%
© Eric Zivot 2011

FMMC FoHF Returns

FM
= 1.42%

FM,EWMA
= 1.52%
VaR
0.0167
ETL
0.0167
= -3.25%
= -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