[SOLVED] CS代写 Lecture Note 10

Lecture Note 10
The Cross-Section of Stock Returns: Factor Models
. Lochstoer
UCLA Anderson School of Management

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Winter 2022
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Overview of Lecture Note 10: The Cross-Section
single-factor model
I Portfolio choice and the curse of dimensionality
I The simplest covariance matrix: A single factor
I Single-factor model math
I Portfolios or single stocks: estimation error and beta stability
2 Multi-factor models
I Properties I Estimation
3 Applications
I Firm characteristics-based factors (the Fama-French model)
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Portfolio Problem
Portfolio of N securities with N  N variance-covariance matrix Σ
Vector of N  1 portfolio weights: w
Variance of portfolio return is given by:
σ2P = w0Σw N
= ∑wiwjσij i,j=1
= ∑ w i 2 σ 2i + ∑ ∑ w i w j σ i σ j ρ i j
i=1 i=1 j=1,j6=i
Markowitz (1952): choose w to minimize the variance of the portfolio for a given
. Lochstoer UCLA Anderson School of Management () Winter 2022 3 / 42

Inputs to Portfolio Analyssis
Standard deviation of portfolio return:
“N #1/2 ∑ wiwjσij
“N 22 N N #1/2
∑wi σi +∑ ∑ wiwjσiσjρij i=1 i=1 j=1,j6=i
= [w0Σw]1/2 Mean of portfolio return:
E[RP] = ∑wiE[Ri] = ∑wiμi = μP
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Inputs to Portfolio Analysis
1 Expected returns E [Ri ] = μi and risk Σ:
I we could use historical data to estimate expected returns
I we could use equilibrium models to estimate expected returns (see Black and Litterman (1992))
2 Pairwise correlations: I hard to estimate
I unstable through time
. Lochstoer UCLA Anderson School of Management () Winter 2022 5 / 42

Curse of Dimensionality
number of parameters to estimate necessary to carry out portfolio analysis is staggering.
suppose you follow between 150 and 250 stocks
we need all correlation coe¢ cients ρij for stocks i and j
total number of free correlations in an N  N covariance matrix Σ is N (N +1)
(11,325 in the case of N =150 stocks)
in the case of a large number of assets: traditional mean-variance analysis tends to perform very poorly out of sample; see, e.g. DeMiguel, Garlappi, and Uppal (2009)
For more on the impact of the curse of dimensionality in statistics, see Chapter 2 of Hastie, Tibshirani, and Friedman (2009)
. Lochstoer UCLA Anderson School of Management () Winter 2022 6 / 42

Single Factor Model
single factor models: co-movement is due to a single factor.
economic factor models: use economic theory or economic intuition to determine the factor.
statistical factor models: use the data to determine the factor.
some textbooks call this a single index model. The factor is often a stock
market index.
Capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Mossin (1966) is the most famous single-factor model.
I Theímarketreturníisthefactor
. Lochstoer UCLA Anderson School of Management () Winter 2022 7 / 42

Stock Returns
0.5 0.4 0.3 0.2 0.1
0 -0.1 -0.2 -0.3 -0.4
1987 1990 1993
Monthly data from CRSP: The Market Return is the CRSP Value-Weighted Return on NYSE-AMEX-NASDAQ. Sample 1987:1-2007.12
. Lochstoer UCLA Anderson School of Management () Winter 2022

Single Factor Model: The Market Model
stocks move in tandem due to a common factor ft the return on a stock i can be written as:
Rit = αi + βi ft + εit let Rmt denote the market return.
suppose the common factor is the market. Then, we have
Rit = αi + βi Rmt + εit
I εit is the component of the return that is independent of the market return Rmt.
I assume E [εit] = 0. I βi is a constant.
. Lochstoer UCLA Anderson School of Management () Winter 2022

Single Factor Model
Two key (simplifying) assumptions:
disturbances are independent of the market return: Cov[εit,Rmt] = 0
εit is independent of εjt :
Cov[εit , εjt ] = 0
) no e§ects beyond the market that account for covariation (e.g. industry e§ects)
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Moments of the Single Factor Model
This model then implies:
E[Rit]=αi +βiE[Rmt], i =1,…,N
μi = αi + βi μm , i = 1, . . . , N
V [Rit] = β2i V [Rmt]+V [εit], i = 1,…,N
σ2i =β2iσ2m+σ2εi, CovRit,Rjt=βiβjσ2m,
i =1,…,N
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Single Factor Model Portfolio Math
The expected return on a portfolio is:
μp = ∑wiμi
= ∑wiαi +μm ∑wiβi i=1 i=1
Recall that the variance of the portfolio return is: “N2222NN 2#
V [RP] = ∑wi (βi σm +σεi)+ ∑ ∑wiwjβiβjσm i=1 i=1 j=1
Note the dramatic decrease in number of parameters to be estimated 3N +2 estimates: αi,βi,σ2ei ,i = 1,…,N and σ2m,E [Rm]
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Single Factor Model Portfolio Math (contíd)
The beta of a portfolio βp is:
βp = ∑ wi βi
where wi are the portfolio weights The alpha of a portfolio αp is:
αp = ∑wiαi
i=1 where wi are the portfolio weights
then, given the factor model, the expected return on the portfolio is:
μp = αp + βp μm
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Singel Factor Model: Market Model
if the factor ft is the market return Rmt, then the factor is a traded asset.
the model should apply to the market return as well.
then the expected return on the market portfolio is:
μm = αm + βm μm this immediately implies that:
I αm=0 I βm=1
the market portfolio has a beta of one and an alpha of zero.
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Single Factor Model Parameters
to use the single-factor model, we need estimates of αi and βi
what you need, is the betas going forward (i.e. the beta for a security that
applies over the next say 5 years)
we typically use historical betas as estimates of future betas I these are backward looking
I but they are informative about future betas
. Lochstoer UCLA Anderson School of Management () Winter 2022 15 / 42

Single Factor Model Estimation
R m , 1 37 26 R i , 1 37
Rm,2 7 6 Ri,2 7 . 75,yi =64 . 75
Rm,T write the single factor model as:
withβi=[αi βi ]0
Use least squares to estimate βi for each asset i
yi = Xβi + εi
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Sampling Error
consider the case of individual securities the estimator of the slope coe¢ cient is :
bβ = ∑Tt=1(Rmt Rm)(Rit Ri) i ∑ Tt = 1 R m t R m
2
the variance of the estimator of the slope coe¢ cient is : Vbβiβi=1 var(εˆit)
T var (Rmt )
measure of sampling error
sampling error is ináuenced by σ2εi , which can be large for individual stocks
. Lochstoer UCLA Anderson School of Management () Winter 2022 17 / 42

IBM and MSFT
0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1
RM SFT= 1.4*Rm+ 0.011
R = 1.1*R – 0.0005 IBM m
-0.1 -0.2 -0.3 -0.4
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Market Return
-0.1 -0.2 -0.3 -0.4
Monthly data from CRSP. Sample 1987.1-2007.12. IBM is left and MSFT is right.
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Market Return
. Lochstoer UCLA Anderson School of Management () Winter 2022 18 / 42
IBM Return
MSFT Return

Time-Variation in Betas
empirical evidence suggests there is quite a lot of time variation in historical estimates of βís over di§erent samples
I could be because true betas are varying over time I could be sampling error
for individual securities, little relation between estimated betas in subsequent time samples
for portfolios of securities sorted on certain characteristics ) more stability for portfolios of securities ) less sampling error
. Lochstoer UCLA Anderson School of Management () Winter 2022 19 / 42

Instability in IBM Betas
26-32 33-39 40-46 47-53 54-60 61-67 68-74 75-81 82-88 89-95 96-02
Historical IBMBetas
1.4 1.3 1.2 1.1
1 0.9 0.8 0.7 0.6
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Historical Beta IMB period t
Monthly data from CRSP. Sample 1926.1-2002.12. Estimates over 7-year non-overlapping intervals.
. Lochstoer UCLA Anderson School of Management () Winter 2022 20 / 42
Historical beta IBM period t+1

Portfolio sorts
the academic Önance literature sorts stocks into portfolios statistically, this reduces idiosyncratic volatility of the ëassetí
I more noisy measurement of the true beta at the individual security level (larger ësampling errorí)
I less at the portfolio level
economically, risk characteristics of individual securities might change over
I key idea: sort individual securities into portfolios based on characteristics I examples:
1 size (market cap)
2 book-to-market
3 industries
. Lochstoer UCLA Anderson School of Management () Winter 2022

Portfolio Sampling Error
consider the case of a portfolio with N assets. the estimator of the slope coe¢ cient is :
bβ = ∑Tt=1(Rmt Rm)(Rpt Rp) p ∑ Tt = 1 R m t R m
2
the variance of the estimator of the slope coe¢ cient is :
Vbβpβp= 1 var(εˆpt) T var (Rmt )
var(εˆpt)= ∑wi2var(εˆit) i=1
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Sampling Error (contíd)
originally, we have N total Örms.
separate N Örms into a smaller set of j = 1, . . . , J portfolios. Each portfolio has N(j) assets in it. Assume no Örm shows up in more than 1 portfolio at a given time.
consider an equally-weighted portfolio with w = 1 , i = 1,…,N(j) i N(j)
by building portfolios we reduce sampling error, by pushing this term 1 N(j)
σ 2ε p = ( j )
2 ∑ σ 2ε i N i=1
moreover, we might create more stable betas
to zero as N(j) increases.
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Size and Book-to-Market
Fama and French (1992, 1993) argue that size and book/market are key risk characteristics
They construct portfolios by sorting stocks on these characteristics
The portfolios, which are constructed at the end of each June, are the intersections of 2 portfolios formed on size (market equity, ME) and 3 portfolios formed on the ratio of book equity to market equity (BE/ME).
I The size breakpoint for year t is the median NYSE market equity at the end of June of year t.
I BE/ME for June of year t is the book equity for the last Öscal year end in t-1 divided by ME for December of t-1.
I The BE/ME breakpoints are the 30th and 70th NYSE percentiles.
. Lochstoer UCLA Anderson School of Management () Winter 2022 24 / 42

Size and Book-to-Market (contíd)
The six portfolios are
size vs. B/M low medium high small 1 2 3 big 4 5 6
We can give them nice labels size vs. B/M
medium Small Neutral Big Neutral
low Small Growth Big Growth
high Small Value Big Value
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Portfolio Betas
Portfolio 1 22
00 1 2 3 4 5 6 7 8 9 10 11
Portfolio 3 22
00 1 2 3 4 5 6 7 8 9 10 11
Portfolio 5 22
00 1 2 3 4 5 6 7 8 9 10 11
Portfolio 2
6 Fama-French portfolios sorted on size and book-to-market. Monthly data from CRSP. Sample 1926.1-2002.12. Estimates over 11 non-overlapping 7-year intervals.
. Lochstoer UCLA Anderson School of Management ()
Winter 2022 26 / 42
1 2 3 4 5 6 7 Portfolio 4
1 2 3 4 5 6 7 Portfolio 6
1 2 3 4 5 6 7

Stability in Portfolio Betas
Portfolio 1
Portfolio 2
1.1 1 0.9 0.8
1 1.2 1.4 0.8 1 1.2 1.4
betas at t Portfolio 3
betas at t Portfolio 4
betas at t Portfolio 5
1.1 1. 05 1 0. 95 0.9
1.6 1.4 1.2
betas at t Portolio 6
. Lochstoer UCLA Anderson School of Management ()
Winter 2022 27 / 42
0.8 0.9 1 1.1 1.2
betas at t
0.8 1 1.2 1.4 1.6
betas at t
6 Fama-French portfolios sorted on size and book-to-market. Monthly data from CRSP. Sample 1926.1-2002.12. Estimates over 11 non-overlapping 7-year intervals.
betas at t+1 betas at t+1 betas at t+1
betas at t+1 betas at t+1 betas at t+1

Accuracy in Historical Betas
empirical evidence suggests historical portfolio betas are better predictors of future portfolio betas
substantial gain in stability
sorting stocks into portfolios is one way to produce stability.
there are other ways to produce stability. In statistics, these are called ëshrinkage estimatorsíor Bayesian estimators
I the basic idea is to push our parameter estimates to a prior value that is economically plausible.
I Vasicek (1976) and Blume (1975) are early applications of this idea
. Lochstoer UCLA Anderson School of Management () Winter 2022

Multi-factor Models
so far, we have assumed that all of the covariation among securities is due to a single common factor ft
the common factor ft is typically taken to be the market return Rmt however, there might be other reasons security prices co-move ) multiple
what should the extra factors be?
some examples:
1 industry factors
2 macro-economic factors
3 factors built from accounting variables
4 statistical (& latent) factors
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Multi-factor Models (contíd)
stocks move in tandem
the return on a stock i can be written as:
Rit =αi +βi1f1t +βi2f2t +…+βiKfKt +εit, fit are correlated with each other.
for example:
I f1t is a market index
I f2t is an interest rate index
i =1,…,N
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Multi-factor Models in Matrix Notation
let ft denote a K  1 vector of factors
let βi denote a K  1 vector of factor loadings we can write the model in matrix notation as
R i t = α i + β i0 f t + ε i t , i = 1 , . . . , N we can then stack all N assets together
Rt = α + βft + εt
Rt is a N 1 vector of returns, β is a N K matrix of factor loadings
let Σf denote the K  K covariance matrix of the factors.
let Σε denote the N  N covariance matrix of the idiosyncratic errors
. Lochstoer UCLA Anderson School of Management () Winter 2022

Multi-factor Model: Assumptions
1 disturbances are independent of the factor: Cov[εit , fjt ] = 0
2 εit is independent of εjt :
Cov[εit , εjt ] = 0
no e§ects beyond the factors that we have included that account for
covariation. This means that Σε is diagonal.
. Lochstoer UCLA Anderson School of Management () Winter 2022 32 / 42

Multi-factor Model: Moments
Covariance matrix:
E[Rt] = α+βE[ft] = α+βμf
Var[Rt] = βVar[ft]β0+Var[εt] = βΣfβ0+Σε
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Number of Parameters to be Estimated
suppose the factors ft are observable
total number of parameters to be estimated:
K + 2N + NK + K (K + 1) 2
I we need estimates of the K  1 vector μf
I we need estimates of α and the diagonal elements of Σε
I we need estimates of the N  K matrix of factor loadings β
I we need estimates of the covariance matrix Σf . It has K (K +1) free parameters. 2
. Lochstoer UCLA Anderson School of Management () Winter 2022 34 / 42

Industry Factor Model
common co-movement between stocks not driven by the market some of this was related to industry e§ects
I 30 to 50 % of variation in stock returns is due to the market I 10 % of variation in stock returns is due to industry e§ects
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Industry Factor Model: General
Seems most appropriate for Örms that operate in multiple industries The return on a stock i can be written as:
Rit =αi +βimRmt +βi1R1t +…+βiKRKt +εit, i =1,…,N the Rjt,j = 1,…,K are returns on industries
Rmt is the market return
. Lochstoer UCLA Anderson School of Management () Winter 2022 36 / 42

Industry Factor Model: Simple
Seems more appropriate for Örms that operate in a single industry The return on a stock i can be written as:
Rit =αi +βimRmt +βijRjt +εit, i =1,…,N
the Rjt is the return on an industry based index in which Örm i operates
Rmt is the market index
. Lochstoer UCLA Anderson School of Management () Winter 2022

Fama and French (1993) Model
The Fama/French factors are constructed using the 6 value-weighted portfolios formed on size and book-to-market.
SMB (Small Minus Big)
I SMB = 1/3 (Small Value + Small Neutral + Small Growth) – 1/3 (Big Value + Big Neutral + Big Growth).
HML (High Minus Low)
I HML = 1/2 (Small Value + Big Value) – 1/2 (Small Growth + Big Growth).
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).
. Lochstoer UCLA Anderson School of Management () Winter 2022 38 / 42

Fama and French (1993) Model (contíd)
multi-factor model for monthly returns:
R = α + β Rm + β RSMB + β RHML + ε it i i,M t i,SMB t i,HML t it
we will use N = 10 test assets formed on industries
. Lochstoer UCLA Anderson School of Management () Winter 2022 39 / 42

Fama and French (1993): Industry Loadings
Sample: 1926.7 ñ2008.12
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HML Loadings
bHML for each industry:
Sample: 1926.7 ñ2008.12
. Lochstoer UCLA Anderson School of Management () Winter 2022 41 / 42

SMB Loadings
bSMB for each industry:
Sample: 1926.7 ñ2008.12
. Lochstoer UCLA Anderson School of Management () Winter 2022 42 / 42

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