Lecture 12
Equilibrium Factor Models: Estimation and Tests
. Lochstoer
UCLA Anderson School of Management
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Winter 2022
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Overview of Lecture 12
Equilibrium factor models
Time-series regression tests of factor models
1 Alpha and mispricing
2 The 25 Fama-French portfolios as test assets
1 Testing the CAPM
2 Testing the Fama-French 3-factor model
3 Alpha and mean-variance e¢ ciency (max Sharpe ratios)
4 Appendix: The Arbitrage Pricing Theory (APT)
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Risk comes in many áavors
we used to think that risk comes in a single áavor a stockís βi with the market factor
βi measures the ëquantity of riskíor amount of exposure of asset i to the market.
early empirical evidence originally supported the CAPM…now it is resoundingly rejected
now, we know that risk comes in many áavors
that complicates portfolio advice
it makes performance analysis more challenging!
. Lochstoer UCLA Anderson School of Management () Winter 2022
Why should we expect multiple factors?
the average investor has a job
compare two stocks with the same market beta stock A does well in a recession
stock B does poorly in a recession
CAPM says we are indi§erent between the two stocks are we?
some evidence for a missing recession risk factor!
. Lochstoer UCLA Anderson School of Management () Winter 2022
Why should we expect multiple factors?
suppose some less sophisticated investors have portfolios that are biased towards large, growth Örms (because these are more glamorous)
then, the sophisticated investors have to overweight the small, value Örms in their portfolio of risky assets
these sophisticated investors do not and cannot hold the market portfolio in fact, the market portfolio is no longer e¢ cient
this gives rise to new factors (like value and size)
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Why estimate factor models?
test your model
can it account for interesting cross-sectional variation in average returns on assets in a particular asset class?
once you have a ëgood modelí
estimate cost of capital for company
estimate risk-adjusted returns on trading strategy
do performance attribution, style analysis
look for skill in risk-adjusted returns of pension funds and hedge funds
. Lochstoer UCLA Anderson School of Management () Winter 2022
An expected return-beta pricing model
The CAPM is the example you already know:
E[Ri,t Rf,t]=βM,iE[RM,t Rf,t] What is the property of the market portfolio?
It is mean-variance e¢ cient ñit has the maximal Sharpe ratio In fact, mathematically it is true that:
E[Ri,t Rf,t]=βMVE,iE[RMVE,t Rf,t]. where RMVE is the return to the mean-variance e¢ cient portfolio!
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
The unicorn: The Mean-Variance E¢ cient Portfolio
íAllíwe need is to Önd the mean-variance e¢ cient portfolio.
This is what multifactor expected return models are all about
Find, say, three excess return factors that span the excess return mean-variance e¢ cient portfolio (the íeísuperscript is for excess returns):
ERe = i,t
How do you estimate the betas? Linear regression, as usual
RMe VE,t = a1RFe1,t +a2RFe2,t +a3RFe3,t.
β ERe MVE,i MVE,t
βMVE,ia1E RFe1,t+βMVE,ia2E RFe2,t+βMVE,ia3E RFe3,t
βMVE ,i E a1RFe 1,t + a2RFe 2,t + a3RFe 3,t β1E RFe1,t+β2E RFe2,t+β3E RFe3,t
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Di§erent Estimation Methods for Linear Factor Model
1 Time series regressions (this lecture) I OLS
2 Cross-sectional regressions (next lecture) I OLS
I Fama and MacBeth (1973)
. Lochstoer UCLA Anderson School of Management () Winter 2022
Single Factor Equilibrium Model: APT version
Assume there is only one factor that explains covariation in stock returns:
Re =α +βf +ε , it i it it
where εit are uncorrelated across stocks and time, and ft is a traded factor (e.g., excess market returns)
Since ε-risk can be diversiÖed away in large portfolios, it cannot will not earn a risk premium if markets are perfectly competitive
Two sources of variation in returns: βi ft and εit
I Risk premium on βift is βiE [ft]
I Risk premium on εit is zero
I Thus,riskpremiumonstockshouldbeE[Re]=βE[ft] andso… αi =0! it i
Letís test this model using the 25 Fama-French portfolios! Let the factor be the market factor
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The 25 Fama-French portfolios
Fama and French (1993) sort the data into 25 portfolios.
they construct portfolios by sorting them along two dimensions:
1 5 bins sorted on size
2 5 bins sorted on book-to-market
25 Portfolios: large spread in average excess returns
the goal is to maximize the variation in the expected excess returns E Re vs.
E Re , i.e. the left hand side variables in the regression. jt
Then, we want to see if the factors ft can capture this variation.
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Fama-French 25 b/m and size sorted portfolios
25 Fama-French portfolios. Monthly data. 1960-2015.
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Testing the CAPM
the CAPM predicts that αi = 0 for all assets and that variation in the cross-section of expected returns can be explained by variation in the market times an assets βi
one-step procedure: run a time series regression of excess returns on the factor to estimate the βi ís
Re=α+βRe +ε it i imt it
letís plot the predicted excess returns Rˆ ie = βˆ i T1 ∑Tt =1 Rme t realized excess returns R ̄ e = 1 ∑T Re
against the
i T t=1 i,t
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Failure of the CAPM
25 Fama-French portfolios. Monthly data. 1960-2015. Plot of the predicted excess returns βˆ 1 ΣT Re against the realized average excess returns 1 ΣT Re
i T t=1 mt T t=1 it
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CAPM alphas
CAPM αís for the 25 Fama-French portfolios from a time series regression of returns on the factors. Monthly data. 1960-2015.
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CAPM betas
CAPM βís for the 25 Fama-French portfolios from a time series regression of returns on the factors. Monthly data. 1960-2015.
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CAPM betas (bar chart)
CAPM βís for the 25 Fama-French portfolios from a time series regression of returns on the factors. Monthly data. 1960-2015.
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Multifactor model
the CAPM fails to capture the cross-section of expected returns built from the FF 25 portfolios sorted on size and book-to-market.
we need more factors ft to explain variation in returns. Fama and French: construct factors ft from returns
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Fama and French (1993) 3-factor model
The Fama/French factors are constructed using the 6 value-weight portfolios formed on size and book-to-market.
SMB (Small Minus Big) :
SMB = 1/3(SmallValue + SmallNeutral + SmallGrowth) 1/3(BigValue + BigNeutral + BigGrowth).
HML (High Minus Low) :
HML = 1/2(SmallValue + BigValue) 1/2(SmallGrowth + BigGrowth).
Rtm Rmt Rft , the excess return on the market, is the value-weight return on all NYSE, AMEX, and NASDAQ stocks (from CRSP) minus the one-month Treasury bill rate (from Ibbotson Associates).
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Time Series Regression
one-step procedure: run a time series regression of excess returns on the factor to estimate the βi ís
Re =α +βmRm+βsmbRsmb+βhmlRhml+ε it i i t i t i t it
DeÖne the factor “risk premium” or “risk price” as λˆ = λm , λsmb , λhml where λˆj =T1∑Tt=1Rtjforj=m,smb,hml
letís plot the predicted excess returns βˆ i0 λˆ against the realized excess returns R ̄e=1∑T Re
i T t=1 it
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FF 3-factor model
25 Fama-French portfolios. Monthly data. 1960-2015. Plot of the predicted excess returns βˆ 0 λˆ against the realized average excess returns 1 ΣT R e . The risk
i T t=1 it prices, λˆ , are the means of the factors.
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FF 3-factor alphas
Fama-French 3-factors αˆís for the 25 Fama-French portfolios from a time series regression on the factors. Monthly data. 1960-2015.
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FF 3-factor betas
Fama-French 3-factors βˆís for the 25 Fama-French portfolios from a time series regression on the factors. Monthly data. 1960-2015.
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Testing the model: Single factor
We can test the model
by running time series regressions:
Re =α +βf +ε , t=1,…,T it i it it
With i.i.d errors, the asymptotic test statistic for the pricing errors is: ” f ̄ 2 # 1
T 1+ 2 αˆ0Σˆ 1αˆχ2N. σˆ f
where Σˆ denotes the covariance matrix of εt.
The CAPM is resoundingly rejected by Fama-French (1993)
This is the íGRS testí(from Gibbons, Ross, and Shanken (1987))
. Lochstoer UCLA Anderson School of Management () Winter 2022
E [Re ] = β E [ft ] it i
Testing the model: Multiple factors
We can test the model
by running time series regressions:
Re =α +β0f +ε , t=1,…,T it i it it
With i.i.d Normally distributed errors, the exact small-sample test statistic for the pricing errors is:
h ̄0 ˆ 1 ̄i 1 0 ˆ 1 (T N K)/N 1+fΣf f αˆΣε αˆFN,T N K
where Σˆ ε denotes the covariance matrix of εt, Σˆ f denotes the covariance matrix of the factors ft , f ̄ is the average factor, and αˆ are the OLS estimates of α.
The Fama-French 3-factor model is also rejected…! (though not nearly at the same signiÖcance level as the CAPM)
E[Re] = β0E[ft] it i
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Factor Models and Mean-Variance E¢ ciency: The Data Mining Concern
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Properties of the in-sample MVE
We can, given a set of assets, easily compute the in-sample MVE portfolio: minw0Ωˆw suchthat w0R ̄e =m
where w = [w1 w2 … wN ]0, Ωˆ is the sample variance-covariance matrix of excess returns, and R ̄ te = R ̄ 1et R ̄ 2et … R ̄ Ne t 0 is the vector of sample average excess returns for each asset.
The optimal portfolio weights are (up to a constant of proportionality): w M V E ∝ Ωˆ 1 R ̄ e
The in-sample mean variance e¢ cient portfolio therefore has squared Sharpe ratio: SRM2VE =R ̄e0Ωˆ 1R ̄e
Next slide has the derivations
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MVE derivations
Objective function in Lagrangian form: min1w0Ωˆw k w0R ̄e m
First order condition wrt w (an N 1 vector):
Ωˆ w k R ̄ e = 0 w M V E = k Ωˆ 1 R ̄ e .
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
MVE derivations (contíd)
Recall,wMVE =kΩˆ 1R ̄e.
Average excess return on MVE portfolio
R ̄ Me V E = w M V E 0 R ̄ e = k R ̄ e 0 Ωˆ 1 R ̄ e Variance of excess return on MVE portfolio:
var(RMe VE) = wMVE0ΩˆwMVE =k2R ̄e0Ωˆ 1ΩˆΩˆ 1R ̄e = k 2 R ̄ e 0 Ωˆ 1 R ̄ e
Squared Sharpe ratio of MVE:
k R ̄ e 0 Ωˆ 1 R ̄ e 2
SR2(RMeVE)= k2R ̄e0Ωˆ 1R ̄e =R ̄e0Ωˆ 1R ̄e . Lochstoer UCLA Anderson School of Management ()
Winter 2022
The GRS statistic revisited
Consider an asymptotic version (with i.i.d. residuals) of the Gibbons-Ross-Shanken (GRS) statistic for testing whether a factor model is rejected or not:
αˆ 0 Σˆ 1 αˆ
T ε χ2 (N)
1 + R ̄ e 0 Σˆ 1 R ̄ e FFF
NotethatR ̄e0Σˆ 1R ̄e isthein-samplemaximumSharperatiosquaredobtained FFF
using the factor portfolios only.
ΣF is the K K variance-covariance matrix of the factors
Note that αˆ0Σˆ 1αˆ is the maximum Sharpe ratio squared of hedged stock returns. ε
E[αˆ+εˆ] = EhRe ^β0Rei=αˆ foralliand iit itiFti
var(αˆ+εˆt) = Σˆε (anNNmatrix) . Lochstoer UCLA Anderson School of Management ()
Winter 2022
A Mean-Variance Decomposition
Since hedged stock returns are uncorrelated with the factor returns (by construction), we have that
R ̄ e 0 Ωˆ 1 R ̄ e = αˆ 0 Σˆ 1 αˆ + R ̄ e 0 Σˆ 1 R ̄ e ε FFF
We show this is true in a couple of slides Then, we have that
αˆ0Σˆ 1αˆ 1+R ̄e0Σˆ 1R ̄e +αˆ0Σˆ 1αˆ ε=FFFε 1
1 + R ̄ e 0 Σˆ 1 R ̄ e 1 + R ̄ e 0 Σˆ 1 R ̄ e FFF FFF
1 + R ̄ e 0 Ωˆ 1 R ̄ e
= 1+R ̄e0Σˆ 1R ̄e 1
Thus, all alphas are zero if the maximum in-sample Sharpe ratio obtained using the factors equals the maximum in-sample Sharpe ratio of the test assets!
In other words, if a linear combination of the factors is the in-sample mean-variance e¢ cient portfolio, the factor model prices all assets perfectly in the sense that all alphas equal zero
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The failure of the CAPM revisited
The fact that the CAPM does not work means the Sharpe ratio of the market portfolio is far from the maximum Sharpe ratio obtainable using the test assets
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A corollary on the implication of alpha
If a strategy has íalphaídi§erent from zero, it means it can be combined with the factor portfolios to obtain a higher Sharpe ratio than what one could get using the factor portfolios alone
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Implications of alpha: Implementation of R Portfolio
The investable set of assets are N stocks, labelled i = 1, …, N and K factor portfolios, labelled j = 1, …, K .
DeÖne the factor-neutral assets as:
RαRe ^β0Re =αˆ+εˆ, foralli
it it Ft i it
Put all of the investable asset returns in an (N + K ) 1 vector
Rt = hRFe t RFe t … RFe t R1αt R2αt … RNαti0. Note that we only use the 12K
factor-neutral assets and the factors. Otherwise, there would be collinearity between Rie , Riα, and RFe .
Let Ωˆ denote the sample variance covariance matrix of Rt . Note that it is block
Ωˆ=ΣˆF ˆ0 0 Σε
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Winter 2022 34 / 54
Implementation of R Portfolio (contíd)
Now, letís Önd the portfolio weights that makes the maximum Sharpe ratio: w M V E ∝ Ωˆ 1 R t
Note that Rt = R ̄F0 αˆ00, where R ̄F is the K 1 vector of sample factor means and αˆ is the N 1 vector of estimated alpha for the individual assets. Thus:
MVE Σˆ 1 0 Σˆ 1R ̄F w ∝ F 1 Rt= F 1
0 Σˆε Σˆε αˆ From our earlier results, the max Sharpe ratio squared is:
SRM2 VE = Rt0Ωˆ 1Rt
00Σˆ 1 0R ̄F 0 1 0 1
=R ̄Fαˆ0FΣˆ 1 αˆ=R ̄FΣˆFR ̄F+αˆΣˆεαˆ ε
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Weíre always only one factor away from the MVE portfolio
Consider the misspeciÖed factor model
Re =α +β0f +ε . it i it it
whereαi 6=0foralli. Which factor is missing?
The factor with portfolio weights
Adding this factor mechanically ensures, in-sample, that the factors span the in-sample mean-variance e¢ cient portfolio
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The Missing Factor and Data Mining
Letís think more about the portfolio weights of the íÖnalífactor w ∝Σ 1α
Letís assume, for discussion purposes, that Σε is diagonal.
The íÖnalífactor goes long positive alpha assets and short negative alpha
Seems a lot like how we do our characteristics-based factor. E.g., value and momentum
I Go long value stocks, short growth stocks I Go long winners, short losers, etc.
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Data Mining and Economic Rationale
The fact that we know how to make our model work in-sample, for a given set of test assets, makes introducing this factor vacuous in itself
An economic story of what risk or behavioral phenomenon the factor represents
Out-of-sample, cross-country and/or cross-asset corroborating evidence
Thus, given how easy it is to data-mine, we are looking for economics that explain why we would think the phenomenon persists out-of-sample
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Additional reading
I have written a note on factor model testingó time-series and cross-sectional regressions, as well as the mean-variance math laid out in the previous slides
Itís posted on BruinLearn under íWeek 9íand may serve as a useful background reading in addition to the slides
Iíve also posted the Fama-French (1993) paper, as well as the Gibbons, Ross, Shanken (1987) paper.
. Lochstoer UCLA Anderson School of Management () Winter 2022 39 / 54
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Arbitrage Pricing Theory (APT) of Ross (1976)
assume the data are generated by a multi-factor model
Re =α+βf+βf+…+βf+ε, i=1,…,N
it i i1 1t i2 2t iK Kt it = αi+βi0ft+εit
the APT does not identify what the factors are
the factors ft could be traded assets, macro variables, or latent factors
assume the errors are uncorrelated with each factor: Eεit(fjt fjt)=0 8i,j
some textbooks/authors (implicitly) assume the factors are uncorrelated with one another…an orthogonal factor model.
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Arbitrage Pricing Theory of Ross (1976)
assumptions:
1 disturbances are independent of the factors:
2 εit is independent of εjt : This implies Σε is diagonal.
Cov[εit , fjt ] = 0 Cov[εit , εjt ] = 0
this is ëlikeíthe multiple factor model
contribution of Ross (1976) and APT: derive equilibrium implications see also Chamberlain and Rothschild (1983)
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Arbitrage Pricing Theory of Ross (1976): Equilibrium
Suppose that excess returns are generated by a linear factor model: Re=a+bf +bf +…+b f +ε, i=1,…,N
it i i1 1t i2 2t iK Kt it and assume no risk-free arbitrage opportunities exist.
Then, there exist risk prices λj for each factor such that the expected return on any security j can be stated as:
E[Re]=λ +b λ +b λ +…+b λ , i=1,…,N it0i11i22 iKK
The theory puts no restrictions on these risk prices, except when the factors are traded assets.
(for details; see Chapter 9 of Cochrane (2005) and Chapter 6 of Campbell, Lo, and MacKinley (1997))
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Example with two factors
assume now the data are generated by a two-factor model: Re=α+βf +βf +ε, i=1,…,N
it i i1 1t i2 2t it
suppose we build an equally weighted wi = 1 portfolio using NP assets
Rpt = NP∑αi+NP∑βi1f1t+NP∑βi2f2t+NP∑εit
e 1 NP 1 NP i=1 i=1
1 NP 1 NP i=1 i=1
= αp +βp1f1t +βp2f2t + NP ∑εit
in a ëwell-diversiÖedíportfolio, the residual risk disappears because of the Law
of Large Numbers
only systematic risk is left: Rpet αp + βp1f1t + βp2f2t well-diversiÖed ) weights wi cannot be too extreme
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Example with two factors
in a well-diversiÖed portfolio, the residual risk disappears only systematic risk is left:
Reα+βf +βf, i=1,…,P it i i1 1t i2 2t
1 zero aggregate risk portfolio
1 construct a portfolio with β1 = 0 and β2 = 0
2 the expected excess return on this portfolio is 0
2 factor-1-mimicking portfolio
1 construct a zero-investment portfolio with β1 = 1 and β2 = 0
2 the expected (excess) return on this portfolio is: E [R1et ]
3 factor-2-mimicking portf
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