CS计算机代考程序代写 finance Observed Factor Models
Observed Factor Models
Chris Hansman
Empirical Finance: Methods and Applications Imperial College Business School
Topic 5
February 4th
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Today
1. General Framing of Linear Factor Models
2. Single Index Model and the CAPM
3. Multi-Factor Models Fama-French
Macroeconomic Factors
4. Barra approach
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Part 1: Linear Factor Models
1. Clarifying the Assumptions Behind the Linear Factor Model 2. Time-Series and Cross-Sectional Notation
3. Conditional and Unconditional Covariances of Factor Returns
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Linear Factor Models
Suppose we observe the returns on m assets (i = 1,2,··· ,m) Often excess returns (rit − rf )
And often in logs: rit = log Pt Pt −1
Over T time periods (t = 1,2,··· ,T)
Can think of this as a panel of returns
Denote each return by xi,t, so every t we see a vector of length m: x1,t
x2,t xt= .
. xm,t
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Linear Factor Models
Assume that returns xit are driven by K common factors:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit The set of common factors is:
f1,t f2,t
ft = . .
fK ,t
These are the same for all assets (constant over i) But change over time (different for t, t+1)
Each ft has dimension (K × 1)
T different versions of this vector in sample
One for each time period
In some applications we will assume we know ft —in others we will estimate it
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Linear Factor Models
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit The set of factor loadings is
β1,i β2,i
βi= . .
βK ,i
An asset has a fixed relationship with each factor
Do not change over time
This means K different parameters for each asset i
Each βi has dimension (K × 1)
m different versions of this vector in sample:
One for each asset
In some applications we will assume we know βi , in most we will estimate it
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Linear Factor Models
Assume that returns xit are driven by K common factors:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit αi is the intercept for each asset
εit is the error or asset specific factor
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Linear Factor Models: Cross Sectional
Suppose we focus on a single cross section of the data: time t For each individual asset i we have:
xi,t =αi +βi,1f1,t +βi,2f2,t +···+βi,KfK,t +εit Or written for all m assets at once in matrix notation:
xt =α+Bft +εt Looks similar to OLS—but not quite
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Linear Factor Models: Cross Sectional
xt =α+Bft +εt
m×1 m×1 K×1 m×1 Looks similar to OLS—but not quite
Menti…
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Linear Factor Models: Cross Sectional
Assume that returns xt are driven by K unobserved common factors:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
For all m assets can be written more concisely for each period t as:
xt =α+Bft+εt
x1,t α1 β1,1 ··· x2,t α2 β2,1 ···
β1,K f1,t ε1,t
β2,K f2,t ε2,t . . + . ... . ...
. = . + . ..
xm,t αm
βm,1
··· βm,K
m×K
fK,t εm,t
K×1 m×1
m×1 m×1
α and B are constant for all t!
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Linear Factor Model
xt =α+Bft+εt
Assumptions:
{ft} is k-variate covariance stationary: for all t:
E[ft]= μf
K×1
Cov[ft] = Ωf = E[(ft −μf )(ft −μf )′]
Menti…
E[εit|fkt] = 0 for all i,k,t
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Linear Factor Model
xt =α+Bft+εt
Assumptions:
εit has the following properties:
E[εt]= 0
m×1
Cov[εt] = E[εtεt′] = Ψ
m×m Cov[εt,εt′]=E[εtεt′′]= 0 fort̸=t′
m×m
σ12 0 ··· 0
0 σ2 ··· 0
Cov[εt]=Ψ= . . . . .
. .. .
0 0 ··· σm2
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Linear Factor Model
xt =α+Bft+εt Summary of Parameters
α: (m×1) intercepts for m assets
B:(m×K)loadings(βik)onK factorsformassets μf : (K × 1) vector of means for K factors
Ωf : (K × K ) variance covariance matrix of factors
Ψ: (m×m) diagonal matrix of asset specific variances
Given our assumptions xt is m-variate covariance stationary with: E[xt|ft] =?
Cov[xt|ft]=? E[xt]=μx =? Cov[xt] = Σx =?
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Linear Factor Model
xt =α+Bft+εt Summary of Parameters
α: (m×1) intercepts for m assets
B:(m×K)loadings(βik)onK factorsformassets μf : (K × 1) vector of means for K factors
Ωf : (K × K ) variance covariance matrix of factors
Ψ: (m×m) diagonal matrix of asset specific variances
Given our assumptions xt is m-variate covariance stationary with: E[xt|ft] = α +Bft
Cov[xt|ft] = Cov(εt) = Ψ E[xt]=μx =α+Bμf Cov[xt]=Σx =BΩfB′+Ψ
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Linear Factor Model: Time Series
Can also write the linear factor model for each asset i’s time series: xi = αi 1T + Fβi + εi
xi,1 1 f1,1 xi,2 1 f1,2
. =αi.+ .
f2,1 f2,2 .
··· ···
..
fK,1 β1,i εi,1
fK,2 β2,i εi,2 . . + . . . .
. xi,T
T×1
. .
.
.
1
T×1
f1,T
··· T×K
fK,T
f2,T
βK,i εi,T
K×1 T×1
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Linear Factor Model: Time Series
Can also write the linear factor model for each asset i’s time series: xi = αi 1T + Fβi + εi
Much closer to the OLS specifications we have seen in the past: We are used to Y=Xβ+v
xi is analogous to Y
Our factor realizations F are analogous to X
αi 1T is just an explicit way of specifying the constant term
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Linear Factor Model: Time Series
Can also write the linear factor model for each asset i’s time series: xi = αi 1T + Fβi + εi
What about the covariance of εi ?
σi2 0 ··· 0
0 σ2 ··· 0 i
Cov(εi) = . . … .
0 0 ··· σi2
T×T
This follows from our assumption: Cov[εt,εt′]=E[εtεt′′]= 0 fort̸=t′
m×m
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Part 2: The Index Model/CAPM
1. The Index Model as a Special Case of the General Framework
2. Review of Two-Pass Approach and Testing CAPM
3. Estimating Covariances of Factor Returns Why are factor models useful
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Applications: The Index Model/Testing CAPM
Much empirical work testing the CAPM/multifactor models applies this general framework
Flashback:
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Applications: The Index Model/Testing CAPM
xi,t = αi +β1,if1,t +εit
Test of the CAPM consider models of this form with a single factor What do we use for f1,t?
Menti
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Applications: The Index Model/Testing CAPM
xi,t = αi +β1,iRm,t +εit
Test of the CAPM consider models of this form with a single factor
We often perform “two pass” strategies:
First pass estimates β1,i , αi for each asset
Second pass uses these estimated βs to test the CAPM These two passes use different aspects of the data
(cross-section/time series)
Does first pass use cross-sectional or time series approach? Menti
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The Index Model/Testing CAPM: Cross-Section
xt =α+BRmt+εt
m×1 m×1 m×1 1×1 m×1
Suppose we only had one cross-section of data (one period)
Need to estimate m different αs and m different βs But only have m observations (of xit )!
And only one Rmt
Can’t estimate more than m parameters with m data points!
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The Index Model/Testing CAPM: First Pass
xi =αi1T +Rmβi +εi
Estimate OLS regression on time-series version of our factor specification
One regression for each asset i
Recover two parameters αˆi and βˆi for each asset i OLS estimates are like always, if we define
Then our estimates are just:
αˆi ′ −1 ′
1 1
Z=. .
1
Rm1
Rm2 . .
Rmt
βˆ =(ZZ) (Zxi) i
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Estimating The Index Model
xi =αi1T +Rmβi +εi
We will estimate this model on log montly returns for a set of 5 assets
1. SPY: S&P 500 ETF
2. EFA: A non US equities ETF
3. IJS: A small-cap value ETF
4. EEM: an emerging markets ETF 5. AGG: A bond fund
Monthly from January 2013-December 2017
Thanks to Jonathan Regenstein for the example (check his great R for finance tutorials)
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Estimating The Index Model
xi =αi1T +Rmβi +εi
As a proxy for the market portfolio, we use the S&P
What does this imply about β1?
Data available on insendi: returns.csv
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The Index Model/Testing CAPM: First Pass
xi =αi1T +Rmβi +εi With αˆi and βˆi, then for each i we can:
Estimate residuals
Use these to estimate asset specific variances (for each i):
(T − 2)
Write Ψˆ as a diagonal matrix of all these variances:
σˆ12 0 ··· 0 0 σˆ2 ··· 0
ˆˆ
εˆ = x − αˆ 1 − R βˆ iiiTmi
εˆ ′ εˆ σˆi2= i i
2 Cov[εt]=Ψ= . . … .
0 0 ··· σˆm2
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Estimating Covariance Matrix
xi =αi1T +Rmβi +εi A major benefit of having βˆi, and Ψˆ?
We can now estimate the covariance of asset returns! Cov[Xt]=Σx =BΩfB′+Ψ
In this case B is just the vector of βi s for all m assets: Bˆ = [βˆ1,βˆ2,··· ,βˆm]′
The only missing piece is Ωf
Because we have only one factor Rmt , it’s easy to estimate:
ˆ 2 ∑Tt=1(RMt −R ̄m)2
Ωf =Var(Rmt)=σˆR =
T −1
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Estimating Covariance Matrix
So we may estimate: Or, written out fully:
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
xi =αi1T +Rmβi +εi Cov[Xt]=Σx =BΩfB′+Ψ
βˆ σˆ2 0 ··· 0 11
ˆ2 β22ˆˆ ˆ 0σˆ2···0
Σˆx = . ·σˆR ·(β1,β2,··· ,βm)+ . .
. ….
. . . . . . .
βˆm 0 0 ··· σˆm2
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Estimating Covariance Matrix
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
Why do we want to estimate the covariance matrix?
Natural to want to understand relationship between returns of
different assets
In general, hopeless without some structure:
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Estimating Covariance Matrix–Flashback
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
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Estimating Covariance Matrix
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
Σx is an m×m symmetric matrix
Without any structure on Σx how many different parameters are
there to estimate?
How many parameters are included in our Σˆx? Menti
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Estimating Covariance Matrix
Significantly fewer parameters in Σˆx
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
m1m
Number of parameters grows exponentially faster with m in general
Assets Parameters in Sample Σx Parameters in Model Σˆx 235
5 15 11
10 55 21 100 5050 201 1000 500500 2001
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Estimating Covariance Matrix
More technically, Σx is m×m
We have m×T data points (one for each asset in each period)
If we estimate the sample analogue of Σx directly (e.g. by computing each sample variance and covariance directly):
Our estimated matrix can’t be more than rank T
Notinvertibleifm>T!
See if you can work this out…
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The Index Model/Testing CAPM: Second Pass
xit = αi +βiRmt +εit
Take expectations of both sides:
E[xi] = E[αi]+βiE[Rm]
CAPM predicts E [αi ] = 0 so should be the case that: E[xi]=βiE[Rm] (1)
Now we have αˆi, βˆi, σˆi in hand for each i
Second pass uses these parameters to test (1)
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The Index Model/Testing CAPM: Second Pass
x ̄ i = γ 0 + γ 1 βˆ i + γ 2 σˆ i + η i
CAPM tests: expected excess return should be determined only by systemic risk (β)
1. γ0=0oraverageαis0
2. γ2 = 0 (idiosyncratic risk shouldn’t be priced) 3. γ1=R ̄m
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Part 3: Multi-Factor Models
1. The Fama-French Three Factor Model Details of Construction
Calculating the Covariance of Asset Returns 2. Extensions of the Three Factor Model
3. Macroeconomic Factors
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Fama-French Three Factor Model
Recall our general linear factor model:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
Fama-French is just another version of this with three particular factors:
xi,t = αi +β1,if1,t +β2,if2,t +β3,if3,t +εit
The factors are:
1. f1,i = Rmt : proxy for excess market return—same as before 2. f2,i = SMBt : size factor
3. f2,i = HMLt : value factor
How do we get these last two?
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How to get the Fama-French Factors
First, rank assets according to two variables:
1. Market equity (ME)
2. Book-to-market (book equity over market equity)
Cut them into buckets:
Source: Ken French’s website (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_portfolios.html)
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Fama-French Three Factor Model
Source: Ken French’s website (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/bench_m_buy.html)
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Fama-French Three Factor Model
To construct the size factor SMB (small minus big):
Average return on the three small portfloios minus average return on
the three big portfolios
1 3
3
To construct the value factor HML (high minus low):
Average return on the two value portfolios minus average return on
the two growth portfolios
1 2
− 1 (Small Growth+Big Growth) 2
(Small Value+Small Neutral+Small Growth)
SMB =
− 1 (Big Value+Big Neutral+Big Growth)
HML =
(Small Value+Big Value)
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Fama-French Three Factor Model
So the general form:
xi,t = αi +βR,iRm,t +βSMB,iSMBt +βHML,iHMLt +εit
Just like before, (first pass) time series regression to estimate αi, βs:
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt For each i, can collect αˆi, βˆR,i βˆSMB,i and βˆHML,i
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Estimating The Fama French Model
xi,t = αi +βR,iRm,t +βSMB,iSMBt +βHML,iHMLt +εit Data available on the hub: ff returns.csv
Use R excess this time… What is the SMB β for EFA? Menti
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Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt
With αˆi, βˆR,i, βˆSMB,i, and βˆHML,i (for each i) can do two things
Second pass regression to assess model
Construct the covariance matrix of asset returns: Σˆx
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
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Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
What are the dimensions of these objects now? Menti
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Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
Very similar to single factor model, but Ωˆf takes a bit more work:
σˆR2 Ωˆf =σˆR,SMB
σˆR ,HML
Each entry is just a sample variance or covariance e.g.:
T ̄ ̄ σˆSMB,HML = ∑t=1(SMBt −SMB)(HMLt −HML)
T −1
σˆSMB,R σˆ2
σˆHML,R σˆHML,SMB
SMB σˆSMB ,HML
σˆ 2 HML
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Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
βˆ βˆ ··· βˆ R,1 R,2 R,m
Written out fully:
βˆ βˆ βˆ
R,1 SMB,1 HML,1 βˆR,2 βˆSMB,2 βˆHML,2
R
σˆ12
0
0 ···
σˆ 2 2 · · ·
0
0
σˆ2
σˆSMB,R σˆHML,R
βˆSMB,1 βˆSMB,2 · · · βˆSMB,m
Σˆx=. . .σˆ σˆ2 σˆ . . . R,SMB SMB HML,SMB … 2… . . .σˆR,HMLσˆSMB,HML σˆHML. . .
βˆR ,m
βˆSMB ,m βˆHML,m
βˆHML,1 βˆHML,2
· · ·
βˆHML,m
+…
. …. ….
0 0 ··· σˆm2
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Fama- French—Covariance Matrix
More parameters in Σˆx in 3 factor vs. 1 factor model: 3m βs
mσis
6 factor variance/covariance parameters
Still way fewer than the general form of Σx
Assets Parameters in Sample Σx Parameters in Model Σˆx 2 3 14
5 15 26
10 55 46 100 5050 406 1000 500500 4006
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Fama French—Extensions
Numerous other factors have been proposed
Fama and French have suggested a two additional factors:
1. Profitability (robust minus weak operating profitablity) 2. Investment (conservative minus aggressive asset growth)
Another, momentum, has been one of the most popular
Tendency of good or bad performance to persist over several months
Usually defined as the average returns of winners minus average returns of losers in the last x months
Actual implementation is the same with more factors—just a few more parameters
Can also let xit represent excess returns on portfolios of assets, rather than assets themselves
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Macroeconomic Factors
An alternative approach uses key macro variables as factors For example, Chen, Roll, and Ross use:
IP: Growth rate in industrial production
EI: Changes in expected inflation
UI: Unexpected inflation
CG: Unexpected changes in risk premiums GB: Unexpected changes in term premia
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Macroeconomic Factors
In this case, our general model becomes:
xi,t =αi +βR,iRm,t +βIp,iIPt +βEI,iEIt +βUI,iUIt +βCG,iCGt +βGB,iGCt +εit
Can use two-pass procedure to estimate βˆs, evaluate the model
Like before, can use estimated βˆs, asset specific variances, and factor covariances to estimate asset covariance matrix
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Part 4: BARRA approach
1. Details of the BARRA model Flipped roles for β, f
2. Estimation details A review of GLS
3. An application in R
BARRA industry model
4. Factor mimicking portfolios
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BARRA approach
Developed by Bar Rosenburg for BARRA (now owned by MSCI) Flipped approach to our linear factor model:
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit Instead of knowing fk,t, suppose we know all the β’s
We know asset i′s exposure to underlying factor fk,t Do not know the value of fk,t in period t
The goal is then to estimate fk,t in each period Rather than to estimate some β
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BARRA approach
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit Silly example
Suppose the only thing that matters for returns is a “tech factor” f1,t Define βi = 1 if asset i is a tech stock, 0 otherwise
Note: I’ve written x ̃it instead of xit
This is just the demeaned excess return for each xit
x ̃it = xit − ∑Tt=1 xit T
Lets us drop αi, interpret fi,t as mean 0
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BARRA approach
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
For each cross section, we can write this in matrix notation: ̃x t = B f t + ε t
The difference is that here B is our “data” —
β1,1 ··· βK,1
β1,2 ··· βK,2 B=. .. .
… β1,m ··· βK,m
m×K
A matrix of (fixed) asset specific attributes
Market-cap, industry classification, style classification, etc
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BARRA approach
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
For each cross section, we can write this in matrix notation: ̃x t = B f t + ε t
ft = [f1,t f2,t · · · fK ,t ]′ parameters to be estimated in each t
K×1
̃xt = [x1,t x2,t · · · xm,t ]′ is a vector of de-meaned returns
M×1
Var (εit ) = σi2 is different for each asset i Different assets have different variances
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BARRA approach
̃x t = B f t + ε t
This looks just like the standard OLS matrix notation And we can estimate our ft like always:
ˆfOLS = (B′B)−1B′ ̃x tt
A bit weird conceptually because the role of the βs flips But no technical difference
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BARRA approach—Issue with OLS
One small issue here:
̃x t = B f t + ε t ˆfOLS = (B′B)−1B′ ̃x
tt
σ12 0 ··· 0 0 σ2 ··· 0
2 Cov(εt)=Ψ= . . … .
0 0 ··· σm2
Heteroskedasticity!
The classic assumptions for OLS to be efficient require
σ 12 = σ 2 2 · · · = σ m2
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BARRA approach—Solution: GLS
Flashback (he uses u instead of εt )
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BARRA approach—Solution: GLS
Flashback (he uses Ω instead of Ψ)
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BARRA approach—Implementing GLS
̃x t = B f t + ε t Our GLS estimator is just:
ˆfGLS = (B′Ψ−1B)−1B′Ψ−1 ̃x tt
Issue: we don’t know Ψ
Solution: three step procedure (special case of feasible GLS)
(1) Estimate ˆfOLS using regular OLS for each t t
Compute residuals ˆεit for each t and m
(2) Estimate Ψˆ using the time series of residuals
(3) EstimateˆfFGLS usingΨˆ t
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BARRA approach—Implementing GLS: Step 1
̃x t = B f t + ε t
Estimate ˆfOLS for each of the t cross-sections:
ˆfOLS = (B′B)−1B′ ̃x tt
t
This gives T different versions of the vector ˆfOLS t
For each i and t, compute:
εˆ =x ̃ −β fˆOLS−β fˆOLS−···−β fˆOLS
it i,t 1,i 1,t 2,i 2,t K,i K,t
This gives m×T different versions of the scalar εˆit Oneforeachi andt
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BARRA approach—Implementing GLS: Step 2
̃x t = B f t + ε t
Use the times series of εˆit for each i to compute σˆi2
Create the Ψˆ matrix:
∑T εˆ2 σˆi2= t=1 it
T −1
σˆ12 0 ··· 0 0 σˆ2 ··· 0
2 Cov(εt)=Ψ= . . … .
ˆ
0 0 ··· σˆm2
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BARRA approach—Implementing GLS: Step 3
̃x t = B f t + ε t
Use Ψˆ to compute ˆfFGLS separately for each time t
t
ˆfFGLS =(B′Ψˆ−1B)−1B′Ψˆ−1 ̃x
tt This gives a vector of length K for each period t
ˆfFGLS =[fˆGLS fˆGLS ··· fˆGLS]′ t 1,t 2,t K,t
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BARRA approach—Covariance of Asset Returns
̃x t = B f t + ε t
Σˆ x = B Ωˆ f B ′ + Ψˆ
Almost the same as before–but this time we know B
Have to estimate Ωf = Cov (ft ) using our GLS estimates
Where
σˆ2 σˆf1,f2 ··· σˆf1,fK f1
σˆf 2,f 1 σˆ2 ··· σˆf 2,fK ˆf2
Ωf = . . … .
σˆfK,f 1 σˆfK,f 2 ··· σˆ2 fK
1 T ˆFGLS ̄FGLS σˆfk,fl = T −1 ∑(fk,t −fk
ˆFGLS ̄FGLS )(fl,t −fl )
t=1
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Application: BARRA Industry Model
̃x t = B f t + ε t
Let’s take the silly example seriously: suppose we believe there are industry specific factors driving asset returns
But we don’t know what the factors are in any given month…
Lets suppose we have 10 stocks (m=10) in three industries (K=3)
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Application: BARRA Industry Model
Three industries: Financial Services, Tech, and Other
Source: Tsay, R.S. (2010) Analysis of Financial Time Series. 3rd Edition, John Wiley & Sons, Hoboken.
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Application: BARRA Industry Model
x ̃i,t = βfin,iffin,t +βtech,iftech,t +βo,ifo,t +εit
For each asset i, define the factor loading (βi,k) for industry k as:
1 if asset i is in industry k βi,k = 0 otherwise
For example βfin,i = 1 for Citigroup while other elements of βi are 0: βi =(1,0,0)
Note that these are known for each i and fixed over time Dell Inc. is always a tech company
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Application: BARRA Industry Model
x ̃i,t = βfin,iffin,t +βtech,iftech,t +βo,ifo,t +εit Lets estimate the industry model
First, load data on 10 stocks
10 stocks in three categories
Monthly data from 1990-2003 (168 months)
Here m=10, k =3, T =168
Source: Tsay, R.S. (2010) Analysis of Financial Time Series. 3rd Edition, John Wiley & Sons, Hoboken.
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Application: BARRA Industry Model
x ̃i,t = βfin,iffin,t +βtech,iftech,t +βo,ifo,t +εit
Next, demean the data (so we have x ̃i,t instead of xit)
And lets calculate the sample covariance/correlation of returns
cov return <- var(X) corr return <- cor(X)
Because T >> m here, calculating these directly is no issue
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Application: BARRA Industry Model
̃x t = B f t + ε t
Finally, generate B the matrix of loadings: 4 Financials, 3 Tech, 3, other
This is just a matrix of dummy variables
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Application: BARRA Industry Model
̃x t = B f t + ε t Now, step 1 of the GLS procedure: OLS
ˆfOLS = (B′B)−1B′ ̃x tt
F hat<-solve(t(B)%*%B)%*%t(B)%*%t(X) NotethatXisT×m
Each cross section ̃xt is a row
So we transpose: X′
And calculate ˆfOLS for all T periods in one line t
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Application: BARRA Industry Model
̃x t = B f t + ε t
Step 2 of the GLS procedure : Calculate residuals:
εˆOLS =x ̃ −BˆfOLS ttt
Use these residuals to calculate
σˆ 12 = ∑ εˆ 12 t
t T−1
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Application: BARRA Industry Model
̃x t = B f t + ε t
Step 2 of the GLS procedure Continued: Calculate Ψˆ:
σˆ12 0 ··· 0 0 σˆ2 ··· 0
2 Cov(εt)=Ψ= . . ... .
ˆ
0 0 ··· σˆm2 Psi hat <- diag(apply(e hat gls,1,var))
Note that we take the transpose of e hat ols because we want a time series variance within asset
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Application: BARRA Industry Model
̃x t = B f t + ε t
Step 3 of the GLS procedure: estimate ˆfFGLS :
t
ˆfFGLS =(B′Ψˆ−1B)−1B′Ψˆ−1 ̃x
tt
F hat gls<-solve(t(B)%*%Psi inv%*%B)%*%t(B)%*%Psi inv%*%t(X)
Once again each cross section ̃xt is a row
So we transpose: X′
And calculate ˆfFGLS for all T periods in one line t
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Application: BARRA Industry Model
̃x t = B f t + ε t
Σˆ x = B Ωˆ f B ′ + Ψˆ
Finally, lets compute the covariance of asset returns
First: Ωˆf =Cov(ft)
Then,usingBandourestimatedΨˆ,wecancalculateσˆ x
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BARRA Models and Factor Mimicking Portfolios
Consider a single factor model with m assets ̃x t = β 1 f t + ε t
Imagine we want to choose weights ωi for each asset to find the minimum (residual) variance portfolio such that:
∑ωiβ1 =1 i
Interpretation: the minimum (residual) variance portfolio whose return moves exactly with the factor
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BARRA Models and Factor Mimicking Portfolios
If ω =(ω1 ω2 ··· ωm), we can restate this as argmin 1ω′Ψω
ω′β1 =1 This problem has solution:
ω′ = (β1′ Ψ−1β1)−1β1′ Ψ−1
These are the weights on the minimum variance portfolio
ω2 Subject to
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BARRA Models and Factor Mimicking Portfolios
So our portfolio weights are:
ω′ =(β1′Ψ−1β1)β1′Ψ−1
And the return on this portfolio is just:
ω′x ̃ = (B′Ψ−1B)B′Ψ−1 ̃x = ˆfGLS
it tt
This return is just the estimated factor realization
When scaled so that ∑t ωi = 1, it is called the Factor Mimicking Portfolio
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Factor Mimicking Portfolios–Application
Our portfolio weights are:
ω′ =(β1′Ψ−1β1)β1′Ψ−1
omega=(inv(B’*inv(Psi hat)*B)*B’*inv(Psi hat))’;
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Today
1. General Framing of Linear Factor Models
2. Single Index Model and the CAPM
3. Multi-Factor Models Fama-French
Macroeconomic Factors
4. Barra approach
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