•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