Lecture 11 Multi-factor Models: Statistical Factor Models (PCA)
. Lochstoer
UCLA Anderson School of Management
Winter 2022
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. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Overview of Lecture 11
1 Statistical Factor Models: Principal Components Analysis
2 Properties
3 Estimation
4 Application: Modeling the Yield Curve
5 Application: Factors in stock returns
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Motivation
use factor models to Öght the curse of dimensionality
Önd a couple of factors that explain most of the variation
factor models can naturally arise from economic theory (next lecture)
up to this slide, factors are observable. I market index
I portfolios sorted on accounting variables
statistical factor models: choice of factors determined by data.
I principal component analysis (PCA)
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Factor Model
model returns
rit=αi+βi0ft+εit, i=1,…,N, t=1,…,T
using matrix notation
rt =α+βft+εt, t=1,…,T
rt isaN1vectorofreturns.
β is a N K matrix of factor loadings. the covariance matrix of returns:
Cov(rt) = Σr = βΣf β0 +Σε assume Σε is diagonal.
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Principal Components Analysis (PCA)
given a vector of returns rt = (r1t,r2t,…,rNt), PCA tries to use a few linear combinations to explain the structure of Σr
a few means less than N
let wi be a N 1 vector of weights, i = 1,2,…,N
we construct linear combinations of the random vectors:
y i t = w i0 r t = ∑ w i j r j t
in Önancial applications, we can conveniently interpret yit as a portfolio
we normalize the weights such that wi0 wi = 1
. Lochstoer UCLA Anderson School of Management () Winter 2022
Properties
moments of yit :
V ( y i t ) = w i0 Σ r w i , i = 1 , 2 , . . . , N Cov yit,yjt = wi0Σrwj, i,j = 1,2,…,N
idea of PCA: Önd linear combinations (the vectors wi ) that are uncorrelated and with variances of yit that are as large as possible
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PCA (contíd)
1 1st principal component: maximizes the variance of y1 subject to w10 w1 = 1
2 2nd principal component: maximizes the variance of y2 subject to w20 w2 = 1
and Cov(y1,y2) = 0
3 3rd principal component: maximizes the variance of y3 subject to w30 w3 = 1
and Cov(y3,yi) = 0, i = 1,2
4 k-th principal component: maximizes the variance of yk subject to
wk0 wk = 1 and Cov(yr,yi) = 0, i = 1,2,…,k 1
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Spectral Decomposition: Covariance Matrix DeÖnition
A real, symmetric m m matrix B has a spectral decomposition given by B = PΛP0
Λ is a diagonal matrix with eigenvalues λ on the diagonal that are all real and positive
P is an m m orthogonal matrix consisting of the m eigenvectors any orthogonal matrix C satisÖes C0C = I and C0 = C 1
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Eigenvalues and -vectors
let ei denote the i-th column of P. This is the eigenvector associated with the i-th eigenvalue λi.
each eigenvector/eigenvalue pair are solutions to the equation
in general, the m eigenvalues λ may be real or complex numbers.
if λi is a complex eigenvalue, then its associated eigenvector ei is also complex, but for a covariance matrix eigenvalues are always real.
each eigenvector ei is not unique up to scale, i.e. γei is also a eigenvector where γ is any real number.
consequently, the eigenvectors are typically normalized to have unit length: ei0ei =1.
. Lochstoer UCLA Anderson School of Management () Winter 2022
Principal Components Result
Let (λi , ei ) be the i-th (eigenvalue, eigenvector) pair for the covariance matrix Σr . The i-th principal component is given by:
y i t = e i0 r t = ∑ e i j r j t
The variance of the i-th principal component is given by: ei0Σrei =λi, i=1,2,…,N
because ei0ei = 1 and ei0ej = 0 choose the vectors wi = ei
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Variance Accounting
the total variance accounted for by the i-th principal component is V(yit) = λi
∑ Ni = 1 V ( y i t ) λ 1 + . . . + λ N where ∑Ni=1 V(yit) = λ1 +…+λN
Typically, we order eigenvalues and -vectors according to the magnitude of the λi
. Lochstoer UCLA Anderson School of Management () Winter 2022
Principal Components (R and MatLab)
princomp() is the R function that corresponds exactly to the Spectral Decomposition we have discussed here.
prcomp() can also be used. It relies on the Singular Value Decomposition, which is numerically more stable. For the typical covariance matrix (symmetric positive deÖnite) this shouldnít matter, but R on a general basis recommends using prcomp().
Implementation in R will be further discussed in the TA session In MatLab, you can use the eig() function
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Statistical Factor Model DeÖnition
Consider a N-dimensional time series of returns rt = (r1t , r2t , . . . , rNt ). and assume the series is covariance stationary. For a covariance stationary time series we can deÖne its mean
E [rt] = μ, Σr = E h(rt μ)(rt μ)0i
where Σr is a N N matrix. The statistical factor model postulates that the
return is linearly dependent on a few unobservable random variables:
rt =μ+βft+εt
where ft = (f1t,f2t,…,fKt) with K < N, and β is a N K matrix of factor
The factors can be found using PCA
Stock iís beta with respect to factor j is the iíth element of eigenvector j
I See PCA_Example.xls
I This depends on normalization of eigenvectors. In general, betas are
proportional to this element.
. Lochstoer UCLA Anderson School of Management () Winter 2022
Statistical Factor Model: IdentiÖcation
the covariance matrix of returns is
Σr =βΣfβ0+Σε
the factors ft are latent and not directly observable. We estimate them from
assume Σε is diagonal.
the data rt = (r1t,r2t,...,rNt).
we cannot identify both Σf and β from the data. Why? note that for any K K orthogonal matrix P
r t μ = β f t + ε t = β ? f t? + ε t
whereβ?=βPandft?=P0ft.Wecaníttellifitisft orft?.
Thus: PCA-derived factors do not by themselves have economic meaning
I However, we may derive some meaning by investigating the eigenvectors I more on this in a couple of slides (yield example)
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Applications of PCA
There are many applications of PCA in economics and Önance
1 yield curves: Litterman and Scheinkman (1991), Cochrane and Piazzesi (2005)
2 macroeconomic indicators: Stock and Watson (1999), Stock and Watson (2002)
3 currency returns: Lustig, Roussanov, and Verdelhan (2011)
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Factor Models and the Yield Curve
There are lots of di§erent yields and interest rates.
It would be helpful if we could summarize the variation bond yields using a couple of factors.
Litterman and Scheinkman (1991) and Knez, Litterman and Scheinkman (1994) Önd, using principal component analysis, that there is a level, slope and curvature factor in bond yields and bond returns.
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Bond Notation
an n-period zero coupon bond pays one dollar n periods from now notation:
I P(n) denotes the price of an n-period zero-coupon bond t
I p(n) denotes the log price of an n-period zero-coupon bond t
p(n) =logP(n) tt
I the log yield of an n-period zero-coupon bond is: y(n) = 1p(n)
I the log holding period return:
hpr(n) =p(n 1) p(n) t+1 t+1 t
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Winter 2022
Principal Components of Yields
12 24 36 48 60
Principal component
0.362 -0.569 0.387 -0.455
0.395 -0.184 0.391 0.104 0.380 0.272 0.371 0.385 0.360 0.455
98.01 1.71
0.620 -0.139
-0.581 -0.304 -0.058
0.193 0.355
The sample period is 6/1952-12/2014. Zero coupon bonds with maturity in months. CRSP Fama- .
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Factor Loadings
The sample period is 6/1952-12/2014. CRSP Fama- .
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Factors in the Yield Curve
The sample period is 6/1952-12/2014. Yields (left) and level factor (right). CRSP Fama- .
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Winter 2022 20 / 30
Factors in the Yield Curve (contíd)
The sample period is 6/1952-12/2014. Yields (left) and level factor (right). CRSP Fama- .
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Winter 2022 21 / 30
Replicating Yields: 1 month
Replicate log yield as linear in Örst 3 PCs:
yˆ(1)=a+β PC +β PC +β PC
t 1 1,1 1,t 1,2 2,t 1,3 3,t
The sample period is 6/1952-12/2014. CRSP Fama- .
. Lochstoer UCLA Anderson School of Management () Winter 2022
Replicating Yields: 1 year
Replicate log yield as linear in Örst 3 PCs:
yˆ(12) =a +β PC +β PC +β
The sample period is 6/1952-12/2014. CRSP Fama- .
. Lochstoer UCLA Anderson School of Management () Winter 2022
t 12 12,1 1,t 12,2 2,t
PC 12,3 3,t
Replicating Yields: 5 years
Replicate log yield as linear in Örst 3 PCs:
yˆ(60) =a +β PC +β PC +β
The sample period is 6/1952-12/2014. CRSP Fama- .
. Lochstoer UCLA Anderson School of Management () Winter 2022
t 60 60,1 1,t 60,2 2,t
PC 60,3 3,t
PCs in the cross-section of stock returns
Monthly data from July 1969 through December 2016 on 138 portfolios from ́s webpage
Industry and characteristic sorted portfolios (nearly all) Next slides show:
1 The Örst PC is by far the most important and it has correlation 0.985 with the market
2 The next PCs decline in importance (variance) quite slowly.
3 The Örst 20 PCs accounts for 91% of the variation in the return of these
portfolios (the Örst 10 accounts for 86%)
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PCs in the cross-section of stock returns
Variance of PCs
0 20 40 60 80 100 120 140 PC nr
Variance of PCs (2 to 138)
0 20 40 60 80 100 120 140 PC nr minus 1
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
PCs in the cross-section of stock returns
0.1 0 -0.1 -0.2 -0.3 -0.4
Weights(betas) in Eigenvectors 1 and 2
0 20 40 60 80 100 120 140 Stock i
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
References Lecture Note 10 and 11
Black, F. and R. Litterman (1992). Global portfolio optimization. Financial Analysts Journal 48(5), pp. 28-43.
Blume, M. E. (1975). Betas and their regression tendencies. The Journal of Finance 30(3), 115-143.
Campbell, J. Y., A. W. Lo, and A. C. MacKinlay (1997). The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press.
Chen, N.-F., R. Roll, and S. A. Ross (1986). Economic forces and the stock market. The Journal of Business 59(3), pp. 383-403.
Cochrane, J. H. and M. Piazzesi (2005). Bond risk premia. American Economic Review 95(1), 138-160.
DeMiguel, V., L. Garlappi, and R. Uppal (2009). Optimal versus naive diversiÖcation: How ine§cient is the 1/N portfolio strategy? The Review of Financial Studies 22(5), 1915-1953.
Fama, E. F. and K. French (1992). The cross-section of expected stock returns. The Journal of Finance 47(2), 427-465.
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References (contíd)
Fama, E. F. and K. French (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33(1), 23-49.
Hastie, T., R. Tibshirani, and J. Friedman (2009). The Elements of Statistical Learning (Second ed.). , NY: Springer Press.
Knez, P. J., R. Litterman, and J. Scheinkman (1994). Explorations into factors explaining money market returns. The Journal of Finance 49(5), pp. 1861-1882.
Lintner, J. (1965). The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. The Review of Economics and Statistics 47(1), 13-37.
Litterman, R. and J. Scheinkman (1991). Common factors a§ecting bond returns. The Journal of Fixed Income 1(1), 54-61.
Lustig, H., N. Roussanov, and A. Verdelhan (2011). Common risk factors in currency markets. The Review of Financial Studies 24(11), 3731-3777.
Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance 7(1), 77-91.
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References (contíd)
Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. The Journal of Finance 19(3), 425-442.
Stock, J. H. and M. W. Watson (1999). Forecasting ináation. Journal of Monetary Economics 44(2), 293-335.
Stock, J. H. and M. W. Watson (2002). Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97, 1167-79.
Vasicek, O. A. (1973). A note on using cross-sectional information in Bayesian estimation of security betas. The Journal of Finance 28(5), 1233-1239.
. Lochstoer UCLA Anderson School of Management () Winter 2022
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