Factor Model Risk Analysis: A Comprehensive Guide for Asset Returns, Lecture notes of Credit and Risk Management

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• Amath 546/Econ 589Factor Model Risk AnalysisEric ZivotUniversity of WashingtonJune 3,

Outline^ •^ Factor Model Specification^ •^ Factor Risk Budgeting^ •^ Portfolio Risk Budgeting^ •^ Factor Model Monte Carlo

Three Types of Asset Return Factor Models^ 1. Macroeconomic factor model(a) Factors are observable economic and

financial time series

2. Fundamental factor model(a) Factors are created from observerable asset characteristics3. Statistical factor model(a) Factors are unobservable and extracted from asset returns

Factor Model Specification The three types of multifactor models for asset returns have the general form^ =^ +^ +^ ^1 ^1 ^

+^ · · ·^ +^ ^ +^ ^2 ^ ^ ^2 ^ ^

0 = + βf+^ ^    • is the simple return (real or in excess of the risk-free rate) on asset

(^ = 1     )^ in time period^ ^ (

= 1     ^ ),

^ • is the  common factor^ (

-^ ^ is the^ factor loading^ or^ factor beta^

^ for asset  on the  factor,

-^ is the asset^ specific factor.^

lated across assets^ ( )^ =^ ^ 

2 for all^ ^ =^ ^ and^ ^ =^  ^ = 0 ^ otherwise

Remarks:^ •^ Statistical modeling of returns involves statistical modeling of factors andresiduals^ •^ Typical factor models have a small number of factors (e.g.,

 ^ 10)

-^ Multivariate modeling of factors is a relatively low dimensional problem^ —^ Copula models are feasible for factors^ —^ Multivariate GARCH (e.g. DCC) is feasible for factor covariances

Notation Vectors with a subscript^ ^ represent the cross-section of all assets^ ⎛ ⎜^ R=⎝^ (×1)

⎞  1 ⎟ ... ^ = 1     ⎠^ 

Vectors with a subscript^ ^ represent the time series of a given asset^ ⎛^ ⎜^ R=⎝^ (^ ×1)

⎞ 1 ⎟ ... ^ = 1     ⎠^ 

Matrix of all assets over all time periods (columns = assets, rows = time period)^ ⎛ ⎜^ R=⎝^ (^ ×)^

Cross Section Regression The multifactor model (1) may be rewritten as a

cross-sectional^ regression

model at time^ ^ by stacking the equations for each asset to give^ R=^ α+^ ^ (×1)^ (×1)

Bf+^ ε^  ^ = 1     ^  (×)^ (×1)^ (×1)

⎡^ ⎤^ ⎡^0 β^ · · ·^11 1 ⎢⎥⎢^ .. B=.=⎣⎦^ ⎣ (×) 0 β^ 

02 [εε|f]^ =^ D^ =^ (      ^1

2 ) ^

Note: Cross-sectional heteroskedasticityThis representation is useful for risk analysis across assets.

Multivariate Regression Collecting data from^ ^ = 1     

allows the model (3) to be expressed as the

multivariate regression^ [R    ^ R] =^1 [     ^11 ^ ^

] +^ F[β    ^ β] + [ε    ^ ε^1  1 ^

]

or^ R=^1 ^ (^ ×)^ (^ ×1)

00 α+^ FB+^ E (1×)^ (^ ×)^ (×)^ (^ ×) 0 = XΓ+ E^ "^ #^0 α.^0 .X= [ 1. F] Γ=^  0 B ( ×(+1)) ((+1)×)^

Alternatively, collecting data from

^ = 1     ^ allows the model (2) to be

expressed as the multivariate regression^ [R    ^ R] = [α    ^1 ^

α] +^ B[f    ^ f] + [ε    ^ ε]^1 ^1 ^

or^0 R=^ α^ (×^ )^ (×1)

0001 +^ BF+^ E  (×)^ (×^ )^ (×^ ) (1×^ ) 0 0 = ΓX+ E " # 010. . X=  Γ= [α^.^ B] 0 F ((+1)× ) (×(+1))^

Covariance Structure Using the cross-section regression^ R=^ α+^ B^ (×1)^ (×^ (×1)

f+^ ε^  ^ = 1      ) (×1)^ (×1)

and the assumptions of the multifactor model, the

(^ ×^ )^ covariance matrix

of asset returns has the form^ (R) =^ Ω

0 = BΩB+^ D^   

Note, (4) implies that^0 ()^ =^ βΩ^ 

2 β+    0 ( ) = βΩβ   ^0 βΩβ^ ^ ( ) =   h³^ ´i^1 ^20022 βΩβ+^ ^ (βΩβ+^ ) ^  ^ ^ ^ ^ ^

Conditional Covariance Structure Let^ denote the information available at time^

^ We can allow the factor

covariances and residual variances to be time varying^ f=^ μ+^ ε|−^1 ×^11 ^2 ε=^ Ωz⇒^ ^ ^ ^

(ε|) =^ Ω−^1 ×^2 =  ⇒ (|) =^  ^ = 1        −^1 ^

Then the factor model conditional covariance matrix is^ (R|) =^ −^1

0 Ω= BΩB+^ D  

Note: We can also allow the factor betas to be time varying (i.e.,

B^ =^ B)

Active and Static Portfolios^ •^ Active portfolios have weights that change over time due to active assetallocation decisions^ •^ Static portfolios have weights that are

fixed over time (e.g. equally weighted portfolio) • Factor models can be used to analyze the risk of both active and staticportfolios

Unconditional Asset Risk Measures: Factor Model and Normal Distrib-ution^ =^ ^

0 + βf+^ ^    2 f∼  (μ Ω) () =  ( ) = 0^ for all^    ^  

Then^ []^ =^ 

0 = + βμ     202 () = = βΩβ+^   ^    r 02  = βΩβ+        = +  ×^      1   = −  ()     

Note: In practice,^ = 0^ is typically imposed so that^

0 = βμ.  ^ 