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Factor Model Based RiskMeasurement andManagement

R/Finance 2011: Applied Finance with R

April 30, 2011

Eric Zivot

Robert Richards Chaired Professor of EconomicsAdjunct Professor, Departments of Applied Mathematics,

Finance and Statistics, University of WashingtonBlackRock Alternative Advisors

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Risk Measurement and Management

Quantify asset and portfolio exposures to riskfactors

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

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Asset Level Linear Factor Model

Eric Zivot 2011

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

I

E F F I

E I

I W

I

I I

!

d!

! !

!

! {

F

F

L

K K

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Performance Attribution

Eric Zivot 2011

][][][ 11 ktiktiiit FEFERE FFE ! .

][][ 11 ktikti FF .

])[][(][ 11 ktiktiiti FFE ! .

Expected return due to systematic beta exposure

Expected return due to firm specific alpha

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Portfolio Linear Factor Model

Eric Zivot 2011

,

1 1 1 1

,

p t t t t

n n n n

i it i i i i t i it

i i i i

p p t p t

R

w R w w wE I

E I! ! ! !

d d d d! !

d! !

d!

w R w w BF w

F

F

1

1

( , , ) port olio weights

1, 0 or 1, ,

n

n

i i

i

i n!

d! !

! u !

w K

K

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Risk Measures

Eric Zivot 2011

1( ), 0.01 0.10

of return t

VaR q F

F CDF R

E E E E! ! e e

!

Value-at-Risk (VaR)

Expected Tail Loss (ETL)

[ | ]t t ETL E R R VaRE E! e

Return Standard Deviation (SD, aka active risk)

1/2

2( )t F

SD R IW Wd! !

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Risk Measures

Eric Zivot 2011

Returns

Density

-0 .15 -0.10 -0.05 0.00 0.05

0

5

10

15

20

25

SD

5% VaR5% ETL

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Tail Risk Measures: Non-Normal

Distributions Asset returns are typically non-normal

Many possible univariate non-normal

distributions Students-t, ske ed-t, generalized hyperbolic,

Gram-Charlier, E-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

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Factor Model Monte Carlo (FMMC)

Use fitted factor model to simulate pseudo

asset return data preserving empirical

characteristics of risk factors and residuals

Use fulldata for factors and unequalhistory for

assets to deal ith missing data

Estimate tail risk and related measures non-

parametrically from simulated return data

Eric Zivot 2011

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Simulation Algorithm

Simulate B values of the risk factors by re-samplingfrom fullsample empirical distribution:

SimulateB

values of the factor model residuals fromfitted non-normal distribution:

Create factor model returns from factor models fit over

truncated samples, simulated factor variables dra nfrom fullsample and simulated residuals:

Eric Zivot 2011

_ a1 , , B* *F FK

_ a* *1 , , , 1, ,i iB i nI I !K K

* * * , 1, , ; 1, ,it i i t it t B i nE Id! ! ! F K K

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What to do ith ?

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 assetand factor contributions to portfolio tail risk

measures

Eric Zivot 2009

_ a _ a _ a* *1 1 1

, , , B B B

it it it t t t

R I! ! !

*F

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Factor Risk Budgeting

Eric Zivot 2011

Given linear factor model for asset or portfolio returns,

SD, VaR andETL are linearly homogenous functions

of factor sensitivities . Eulers theorem gives

additive decomposition

( , ), ( , ) , ~ (0,1)

t t t t t t

t t t t

R z

z z

I

I

E I E W E

W

d d d! ! v !

dd d d! !

F F F

F F

%%

% %

1

1

( )( ) , , ,

k

j

j j

RM RM RM S D VaR ETLE EF

F

!

x! !

x

%% %

%

%

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Factor Contributions to Risk

Eric Zivot 2011

Marginal Contribution to

Risk of factorj:

Contribution to Risk

of factorj:

Percent Contribution to Riskof factorj:

( )

j

RM

F

x

x

%

%

( )j

j

RMF

F

x

x

%%%

( ) ( )jj

RM RMFF

xx

%

% %%

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Factor Tail Risk Contributions

Eric Zivot 2011

ForRM=VaR,ETL it can be sho n that

( )[ ], 1, , 1

( )[ ], 1, , 1

jt t j

jt t

j

VaR E F R VaR j k

ETL E F R VaR j k

E

E

EE

F

F

x! ! !

x

x! e !

x

%% K

%

%% K

%

Notes:

1. Intuitive interpretations as stress loss scenarios

2. Analytic results are available under normality

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Semi-Parametric Estimation

Eric Zivot 2011

Factor Model Monte Carlo semi-parametric

estimates

_ a

_ a

* *

1

* *

1

1

[ | ] 1

1[ | ] 1[ ]

B

jt t jt t t

B

jt t jt t

t

a a a

a aB

E E E

E E

I I

E

!

!

! ! e e

e ! e

% %

% %

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Eric Zivot 2011

1 8 2 0 0 0 2 0 0 2 2 0 0 4 2 0 0 6

010

00

0

I d x

tr

Hedge fund returns and 5% VaR Violations

1 8 2 0 0 0 2 0 0 2 2 0 0 4 2 0 0 6

006

000

006

I d x

tr

Risk factor returns when fund return

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Portfolio Risk Budgeting

Eric Zivot 2011

Given portfolio returns,

SD, VaR andETL are linearly homogenous functions

of portfolio eights w. Eulers theorem gives

additive decomposition

,

1

n

p t t i it

i!

d! ! w R

1

( )( ) , , ,

n

i

i i

SD VaR ETLw

E E

!

x! !

x

ww

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Fund Contributions to Portfolio Risk

Eric Zivot 2011

Marginal Contribution to

Risk of asset i:

Contribution to Risk

of asset i:

Percent Contribution to Riskof asset i:

( )

i

RM

w

x

x

w

( )i

i

R Mw

w

x

x

w

( ) ( )ii

RMw RMw

xx

w w

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Portfolio Tail Risk Contributions

Eric Zivot 2011

ForRM=VaR,ETL it can be sho n that

,

,

( )[ | ( )], 1, ,

( )[ | ( )], 1, ,

it p ti

it p t

i

VaR E R R VaR i n

w

ETL E R R VaR i n

w

E

E

EE

x! ! !

x

x! e !

x

ww

ww

K

K

Note: Analytic results are available under normality

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Semi-Parametric Estimation

Eric Zivot 2011

Factor Model Monte Carlo semi-parametric

estimates

_ a

_ a

* *

, ,

1

* *

,

1

1[ | ( )] 1 ( ) ( )

1[ | ( )] 1 ( )[ ]

B

it p t it p t

t

B

it t it p t

t

E R R VaR R VaR R VaRm

E R R VaR R R VaRB

E E E

E E

I I

E

!

!

! ! e e

e ! e

w w w

w w

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Eric Zivot 2011

2 0 0 2 2 0 0 3 2 0 0 2 0 0 2 0 0 2 0 0 7

00

000

00

tr

FoHF Portfolio Returns and 5% VaR Violations

2 0 0 2 2 0 0 3 2 0 0 2 0 0 2 0 0 2 0 0 7

00

00

0

00

t

r

Constituant fund returns when FoHF returns

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Example FoHF Portfolio Analysis

Equally eighted 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

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Eric Zivot 2011

FMMC FoHF Returns

t r

it

0 0 0 0 0 0 0

0

10

2

0

0

1

E

1

WFM= 1.42%WFM,EWMA = 1.52%VaR0.0167 = -3.25%

ETL0.0167 = -4.62%

50,000 simulations

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Factor Risk Contributions

Eric Zivot 2011

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Hedge Fund Risk Contributions

Eric Zivot 2011

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Hedge Fund Risk Contribution

Eric Zivot 2011

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Summary and Conclusions

Factor models are idely used in academic

research and industry practice and are ell

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