Factor Models for Asset Returns, Lecture notes of Investment Management and Portfolio Theory

Download Factor Models for Asset Returns and more Lecture notes Investment Management and Portfolio Theory in PDF only on Docsity!

Factor Models for Asset Returns

Eric Zivot

University of Washington

BlackRock Alternative Advisors

March 14, 2011

Outline

1. Introduction
2. Factor Model Specification
3. Macroeconomic factor models
4. Fundamental factor models
5. Statistical factor models

Introduction

Factor models for asset returns are used to

• Decompose risk and return into explanable and unexplainable components
• Generate estimates of abnormal return
• Describe the covariance structure of returns
• Predict returns in specified stress scenarios
• Provide a framework for portfolio risk analysis

Assumptions

1. The factor realizations, ft , are stationary with unconditional moments E[ft] = μf cov(ft) = E[(ft − μf )(ft − μf )^0 ] = Ωf
2. Asset specific error terms, εit, are uncorrelated with each of the common factors, fkt, cov(fkt , εit) = 0, for all k, i and t.
3. Error terms εit are serially uncorrelated and contemporaneously uncorre- lated across assets cov(εit , εjs) = σ^2 i for all i = j and t = s = 0 , otherwise

Notation

Vectors with a subscript t represent the cross-section of all assets

Rt (N×1)

⎛ ⎜⎝^ R ...^1 t RNt

⎞ ⎟⎠ , t = 1,... , T

Vectors with a subscript i represent the time series of a given asset

Ri (T ×1)

⎛ ⎜⎝

Ri 1 ... RiT

⎞ ⎟⎠ , i = 1,... , N

Matrix of all assets over all time periods (columns = assets, rows = time period)

R

(T ×N)

⎛ ⎜⎝

R 11 · · · RN 1

R 1 T · · · RNT

⎞ ⎟⎠

Cross Section Regression

The multifactor model (1) may be rewritten as a cross-sectional regression model at time t by stacking the equations for each asset to give

Rt (N×1)

= α (N×1)

+ B

(N×K)

ft (K×1)

• εt (N×1)

, t = 1,... , T (2)

B

(N×K)

⎡ ⎢⎣

β^01 ... β^0 N

⎤ ⎥⎦ =

⎡ ⎢⎣

β 11 · · · β 1 K ...... ... β (^) N 1 · · · β (^) NK

⎤ ⎥⎦

E[εtε^0 t|ft] = D = diag(σ^21 ,... , σ^2 N )

Note: Cross-sectional heteroskedasticity

Time Series Regression

The multifactor model (1) may also be rewritten as a time-series regression model for asset i by stacking observations for a given asset i to give

Ri (T ×1)

= 1 T

(T ×1)

αi (1×1)

+ F

(T ×K)

βi (K×1)

• εi (T ×1)

, i = 1,... , N (3)

F

(T ×K)

⎡ ⎢⎣

f 10 ... f T^0

⎤ ⎥⎦ =

⎡ ⎢⎣

f 11 · · · fKt ...... ... f 1 T · · · fKT

⎤ ⎥⎦

E[εiε^0 i] = σ^2 i IT

Note: Time series homoskedasticity

Expected Return (α − β) Decomposition

E[Rit] = αi + β^0 i E[ft]

• β^0 i E[ft] = explained expected return due to systematic risk factors
• αi = E[Rit] − β^0 i E[ft] = unexplained expected return (abnormal return)

Note: Equilibrium asset pricing models impose the restriction αi = 0 (no abnormal return) for all assets i = 1,... , N

Covariance Structure

Using the cross-section regression

Rt (N×1)

= α (N×1)

+ B

(N×K)

ft (K×1)

• εt (N×1)

, t = 1,... , T

and the assumptions of the multifactor model, the (N × N) covariance matrix of asset returns has the form

cov(Rt) = ΩF M = BΩf B^0 + D (4)

Note, (4) implies that

var(Rit) = β^0 iΩf βi + σ^2 i cov(Rit , Rjt) = β^0 iΩf βj

Portfolio Analysis

Let w = (w 1 ,... , w (^) n) be a vector of portfolio weights (wi = fraction of wealth in asset i). If Rt is the (N × 1) vector of simple returns then

Rp,t = w^0 Rt =

X^ N i=

wi Rit

Portfolio Factor Model

Rt = α + Bft + εt ⇒ Rp,t = w^0 α + w^0 Bft + w^0 εt = αp + β^0 pft + εp,t αp = w^0 α, β^0 p = w^0 B, εp,t = w^0 εt var(Rp,t) = β^0 pΩf βp + var(εp,t) = w^0 BΩf B^0 w + w^0 Dw

Active and Static Portfolios

• Active portfolios have weights that change over time due to active asset allocation 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 static portfolios

Covariance matrix of assets

ΩF M = σ^2 M ββ^0 + D (6)

where

σ^2 M = var(RMt) β = (β 1 ,... , β (^) N )^0 D = diag(σ^21 ,... , σ^2 N ), σ^2 i = var(εit)

Estimation

Because RMt is observable, the parameters β (^) i and σ^2 i of the single factor model (5) for each asset can be estimated using time series regression (i.e., ordinary least squares) giving

Ri = αbi (^1) T + RM βb (^) i + bεi , i = 1,... , N β^ b (^) i = covd(Rit , RMt)/ vard(RMt) = ˆσ (^) iM /σˆ^2 M α^ bi = R¯i − ˆβ (^) i R¯M σ^ b^2 i = 1 T − 2

εb^0 iεbi

The estimated single factor model covariance matrix is

Ω^ bF M = σb^2 M βb βb^0 + cD

Remarks

1. Computational efficiency may be obtained by using multivariate regression. The coefficients αi and β (^) i and the residual variances σ^2 i may be computed in one step in the multivariate regression model R = XΓ^0 + E The multivariate OLS estimator of Γ^0 is

Γ^ b^0 = (X^0 X)−^1 X^0 R^0. The estimate of the residual covariance matrix is

Σ^ b = 1 T − 2

E^ b^0 Eb

where Eˆ = R − XˆΓ^0 is the multivariate least squares residual matrix. The diagonal elements of Σb are the diagonal elements of Dc.

2. The R^2 from the time series regression is a measure of the proportion of “market” risk, and 1 − R^2 is a measure of asset specific risk. Additionally, σ^ b (^) i is a measure of the typical size of asset specific risk. Given the variance decomposition

var(Rit) = β^2 i var(RMt) + var(εit) = β^2 i σ^2 M + σ^2 i R^2 can be estimated using

R^2 =

ˆβ^2 i ˆσ^2 M var^ d(Rit)

3. Robust regression techniques can be used to estimate β (^) i and σ^2 i. Also, a robust estimate of σ^2 M could be computed.

Estimation

Because the factor realizations are observable, the parameter matrices B and D of the model may be estimated using time series regression:

Ri = αbi (^1) T + F βbi + εbi = Xˆγ + bεi , i = 1,... , N X = [ (^1) T ... F], ˆγ = (ˆαi , βˆ

0 i)

(^0) = (X (^0) X)− (^1) X (^0) Ri

σ^ b^2 i = 1 T − K − 1

bε^0 ibεi

The covariance matrix of the factor realizations may be estimated using the time series sample covariance matrix

Ω^ bf = 1 T − 1

X^ T t=

(ft − f)(ft − f)^0 , f =

T

XT t=

ft

The estimated multifactor model covariance matrix is then

Ω^ bF M = Bb Ωbf Bb^0 + cD (7)

Remarks

1. As with the single factor model, robust regression may be used to compute βi and σ^2 i. A robust covariance matrix estimator may also be used to compute and estimate of Ωf.
2. ΩF M can be made time varying by allowing βi , Ωf and σ^2 i (i = 1,... , N ) to be time varying

Example: Estimation of Single Index Model in R using investment data from Berndt (1991).

Fundamental Factor Models

Fundamental factor models use observable asset specific characteristics (fun- damentals) like industry classification, market capitalization, style classification (value, growth) etc. to determine the common risk factors.

• Factor betas are constructed from observable asset characteristics (i.e., B is known)
• Factor realizations, ft , are estimated/constructed for each t given B
• In practice, fundamental factor models are estimated in two ways.

BARRA-type Single Factor Model

Consider a single factor model in the form of a cross-sectional regression at time t

Rt (N×1)

= β (N×1)

ft (1×1)

• εt (N×1)

, t = 1,... , T

• β is an N × 1 vector of observed values of an asset specific attribute (e.g., market capitalization, industry classification, style classification)
• ft is an unobserved factor realization.
• var(ft) = σ^2 f ; cov(ft , εit) = 0, for all i, t; var(εit) = σ^2 i , i = 1,... , N.

Estimation

For each time period t = 1,... T, the vector of factor betas, β, is treated as data and the factor realization ft , is the parameter to be estimated. Since the error term εt is heteroskedastic, efficient estimation of ft is done by weighted least squares (WLS) (assuming the asset specific variances σ^2 i are known)

f^ ˆt,wls = (β^0 D−^1 β)−^1 β^0 D−^1 Rt , t = 1,... , T (8) D = diag(σ^21 ,... , σ^2 N )

Note 1: σ^2 i can be consistently estimated and a feasible WLS estimate can be computed

f^ ˆt,f wls = (β^0 Dˆ−^1 β)−^1 β^0 Dˆ−^1 Rt , t = 1,... , T D^ ˆ = diag(ˆσ^21 ,... , ˆσ^2 N )

Note 2: Other weights besides ˆσ^2 i could be used

Factor Mimicking Portfolio

The WLS estimate of ft in (8) has an interesting interpretation as the return on a portfolio h = (h 1 ,... , h (^) N )^0 that solves

min h

h^0 Dh subject to h^0 β = 1

The portfolio h minimizes asset return residual variance subject to having unit exposure to the attribute β and is given by

h^0 = (β^0 D−^1 β)−^1 β^0 D−^1

The estimated factor realization is then the portfolio return

f^ ˆt,wls = h^0 Rt

When the portfolio h is normalized such that

PN i hi^ = 1, it is referred to as a factor mimicking portfolio.

BARRA-type Industry Factor Model

Consider a stylized BARRA-type industry factor model with K mutually ex- clusive industries. The factor sensitivities β (^) ik in (1) for each asset are time invariant and of the form

β (^) ik = 1 if asset i is in industry k = 0 , otherwise

and fkt represents the factor realization for the k th^ industry in time period t.

• The factor betas are dummy variables indicating whether a given asset is in a particular industry.
• The estimated value of fkt will be equal to the weighted average excess return in time period t of the firms operating in industry k.

Estimation of Factor Realization Covariance Matrix

Given (bf 1 ,OLS,... , bfT,OLS ), the covariance matrix of the industry factors may be computed as the time series sample covariance

Ω^ bF OLS = 1 T − 1

X^ T t=

(bft,OLS − fOLS )(bft,OLS − fOLS )^0 ,

fOLS =

T

XT t=

bft,OLS

Estimation of Residual Variances

The residual variances, var(εit) = σ^2 i , can be estimated from the time series of residuals from the T cross-section regressions as follows. Let bεt,OLS, t = 1 ,... , T , denote the (N × 1) vector of OLS residuals, and let bεit,OLS denote the ith^ row of εbt,OLS. Then σ^2 i may be estimated using

σ b^2 i,OLS = 1 T − 1

X^ T t=

(bεit,OLS − εi,OLS )^2 , i = 1,... , N

εi,OLS =

T

XT t=

bεit,OLS

Estimation of Industry Factor Model Asset Return Covariance Matrix

The covariance matrix of the N assets is estimated using

Ω^ bOLS = B ΩbF OLSB^0 + DcOLS

where cDOLS is a diagonal matrix with σb^2 i,OLS along the diagonal.

Weighted Least Squares Estimation

• The OLS estimation of the factor realizations ft is inefficient due to the cross-sectional heteroskedasticity in the asset returns.
• The estimates of the residual variances may be used as weights for weighted least squares (feasible GLS) estimation: bft,GLS = (B^0 cD−^1 OLSB)

− (^1) B (^0) cD− 1 OLSRt^ , t^ = 1,... , T

Ω^ bF GLS = 1 T − 1

X^ T t=

(bft,GLS − fGLS )(bft,GLS − fGLS )^0

σ b^2 i,GLS = 1 T − 1

X^ T t=

(bεit,GLS − εi,GLS )^2 , i = 1,... , N

Ω^ bGLS = B ΩbF GLSB^0 + cDGLS