Multifactor models & Linear Empirical Testing Flashcards
The Intertemporal Capital Asset Pricing Model
The Intertemporal Capital Asset Pricing Model (ICAPM) by Merton (1973) generalizes the CAPM
It shares with the CAPM the assumption that investors do not have outside income
But it allows for additional periods and time-varying investment opportunities
in some periods
investment opportunities are good: expected future returns are high in other periods,
investment opportunities are poor: expected future returns are low
A long-term investor’s future marginal utility depends on
1 future wealth from asset payoffs
→ pricing factor RW as in CAPM
2 future quality of investment opportunities
→ additional pricing factors summarizing quality of investment opportunities
Why does marginal utility depend on investment opportunities
A two-period investor simply consumes all wealth next period, ct+1 = Wt+1
But a long-term investor decides how to split wealth next period, Wt+1, into consumption ct+1
→ this affects marginal utility u′(ct+1)
savings Wt+1 − ct+1 → this is affected by investment opportunities
When investment opportunities improve, an investor may want to:
cut back consumption in order to invest more and exploit the opportunities (substitution effect)
increase consumption because the investor is now effectively richer (income effect)
Which effect dominates depends on utility function
→ but unless both cancel, ct+1 is affected by investment opportunities
State variables capturing investment opportunities emerge as factors
Suppose the (random) vector zt summarizes the quality of investment opportunities at t
the statistical distribution of RW conditional on time-t information is a function of z t+1
zt is called a state variable for investment opportunities
Investor’s optimal consumption choice depends on both wealth Wt and zt
The assumption of no outside income in CAPM
CAPM and ICAPM assume that investors have no (non-capital) outside income
We can interpret this assumption in three ways:
1 all sources of income are tradeable
no truly private equity, tradeable human capital unlikely to be a good approximation to reality
2 financial markets are dominated by investors with no or little outside income
e.g. retirees or households close to retirement
perhaps these are the “marginal agents” relevant for pricing?
3 markets are (approximately) complete
payoffs from all sources can be replicated with tradeable asssets this is as if all income sources themselves were tradeable directly
What happens if none of these interpretations applies? - example with quadratic utility and outside income - setup
Quadratic utility example
Quadratic utility example
General lessons from this example
Non-traded outside income can be relevant for asset pricing, e.g. labor income
“service flow” from owner-occupied housing
dividends from ownership of private (non-traded) businesses
State variables that capture future outside income are natural candidates for factors in addition to CAPM factor RW
But keep in mind: only matters if income of most investors is affected simultaneously
otherwise investors could trade and insure each other
in previous example model: implicit representative investor assumption, as only then
investor’s portfolio return must be RW t+1
Conditional vs Unconditional Factor models
Theoretical derivations usually result in conditional factor models
E[Ri ]=γ +β λ +···+β λ
t t+1 t i,1,t 1,t i,J,t J,t
holds for expected returns from t to t + 1 conditional on time-t information
expectations, betas, risk premia can all be time-dependent
In empirical work, we often assume that we have an unconditional factor model
E[Ri ]=γ+β λ +···+β λ t+1 i,1 1 i,J J
holds for expected returns from t to t + 1 without using any information expectations, betas, risk premia are time-invariant
So far, we have ignored the difference
When is the distinction between conditional and Unconditional relevant
Suppose asset returns and factors are i.i.d. over time
(i.i.d. refers to independent and identically distributed)
Then time-t information is no more informative for predicting Rti+1 than no information
→ Et[Rti+1] = E[Rti+1]
… and the joint distribution of returns Rti+1 and factors f1,t+1, …, fJ,t+1 is time-invariant
→ all betas are time-invariant
Hence: if returns and factors are i.i.d.,
conditional and unconditional models are equivalent
Is the i.i.d assumption plausible?
The i.i.d. assumption for asset returns is not literally satisfied,
e.g.
return predictability:
expected returns are (mildly) predictable by certain variables
volatility clustering: high volatility in recent past predictor of high volatility in the future
But assuming i.i.d. returns is also not crazy
asset returns are (close to) uncorrelated over time
for a long time, asset prices were thought to follow random walks
strong predictability unlikely as someone would try to make money out of it
We will therefore keep on ignoring the distinction conditional vs unconditional
Conditional one-factor model can imply unconditional multi factor model
Conditional factor models can sometimes be expressed as unconditional factor models
This usually requires adding “information factors” that capture time variation in investor information
Summary
Three theoretical foundations for factors in addition to CAPM factor RW :
1 ICAPM: state variables for future investment opportunities
2 Outside income: state variables for future outside income
3 Conditional models: state variables for investors’ information sets
Different test portfolios lead to rejection of CAPM
For size-sorted portfolios, the CAPM fit is reasonably good
in fact
However, we see already in previous graph a deterioration of CAPM fit for small firms
(this is the high-beta observation in the upper right part of the figure)
Fama, French (1993) form test portfolios differently:
sort stocks into 5 portfolios by market capitalization (“size”)
sort stocks into 5 portfolios by ratio of book value to market value (“book-to-market”)
form intersection portfolios → leads to 25 test portfolios
With these alternative test portfolios, the CAPM is easily rejected
The Fama-French Three-Factor Model
To explain these portfolios, Fama, French (1993) add two factors to the CAPM
1 size factor SMB (“small minus big”)
excess return on portfolio of small firms minus portfolio of big firms long position in small firms, short position in big firms
2 value factor HML (“high minus low”)
excess return on portfolio of high book-to market firms minus portfolio of low book-to-market firms
long position in “value stocks”, short position in “growth stocks”
Economic interpretation of size and factors
Do size and value factors have economic meaning?
Fama, French (1993) motivate them as state variables in the ICAPM sense but that motivation remains vague and imprecise
they ultimately justify them on empirical grounds: they “work” in sample
There has been a lot of ex-post theorizing to justify these factors by economic models
by now, we have many theories that predict a value-like factor
no broad consensus has formed which ones we should believe
but these theories provide a strong prior that there should be a pricing factor that is highly correlated with the value factor
the theoretical base for the size factor is thinner
theories that predict a size premium exist
but none of them has become widely accepted in finance
empirically, the “small-firm effect” driving the size premium has also largely disappeared
Additional empirical evidence
Subsequent empirical research has tested the Fama-French three-factor model extensively The model works well with various other ways of forming test portfolios
The model can also account for a large number of “pricing anomalies” w.r.t. the CAPM found in the data
However: it does not resolve all such “pricing anomalies” unexplained phenomenon that appears to be very robust: momentum
Momentum factor
Individual stock returns are positively autocorrelated over horizons of 3 to 12 months
these autocorrelations are tiny in any one stock
nevertheless, they imply meaningful outperformance of portfolios grouped by past returns
Positive autocorrelation means:
stocks that did good in recent past are likely to do good in near-term future
can exploit this by buying past winners and selling past loosers
such a strategy is called a momentum strategy
Momentum was discovered by Jegadeesh, Titman (1993) and has since appeared to be robust
Momentum factor MOM (introduced by Carhart, 1997):
excess return on portfolio of stocks with high realized 12-month return minus portfolio of
stocks with low realized 12-month return
Economic interpretation
Momentum is more difficult to justify as a risk factor based on economic theory
There are some behavioral finance models that can generate momentum
(e.g. Barberis, Shleifer, Vishny, 1998; Hong, Stein, 1999)
In addition:
exploiting momentum requires a lot of trading
after taking trading costs into account, there may be no profit opportunity
(e.g. Carhart 1999; Moskowitz, Grinblatt 1999)
hence, presence of rational investors does not necessarily undo momentum However, there is no generally accepted explanation
Summary
Three theoretical foundations for additional factors
ICAPM: state variables for investment opportunities should be factors
outside income: state variables for (aggregate) outside income should be factors conditional models: state variables for investor information should be factors
Common empirical factor models
Fama-French three-factor model: add size and value factors to CAPM
Carhart four-factor model: adds momentum factor to Fama-French model
these models are designed to fix common pricing anomalies of the CAPM\
Remark: the theoretical foundations and empirical practice are only loosely connected not due to the presentation style in this lecture
… but rather reflective of the current state of asset pricing
Basic testing strategy
test the null hypothesis
H0 :α1 =···=αN =0
If a statistical test rejects H0, we have rejected the factor model with factors f1,…,fJ (as always: if the test does not reject, this does not mean that the factor model holds)
How do we construct statistical tests for H0?
we need (sample) estimates αˆ1, …, αˆN of the pricing errors
we need a distribution theory for the estimates αˆ1, …, αˆN
assuming H0 holds
then we can quatify how likely are observable deviations of αˆi from zero (under H0)
Assumptions for time-series regression tests
Suppose the following assumptions hold:
there is a risk-free asset with time-invariant risk-free rate Rf
all factors are excess returns (are of the form fj ,t = R ij − R f for some asset ij ) t
all returns (and consequently factors) are i.i.d. over time
Allows us to recover pricing errors αi directly from the time-series regressions required to estimate betas
Recovering pricing errors from time-series regressions
Time-series regressions with excess returns
This yields a very simply strategy for estimating the pricing error αi :
regress the excess return of asset i on the factors (in a time-series regression)
the estimate of the intercept αˆi in that regression is a direct estimate of i’s pricing error note: if we use instead the raw return Rti , the intercept is ai = αi + Rf
How to test for Ho?
Because the αi s are regression coefficients, there is a simple distribution theory for them:
If N=1, we could test αi =0 with a simple t-test
Regularly N > 1 and so we run more than one regression
GRS test-statistics for One-Factor Models
Reinterpretation of the test statistic: Testing if Factor is one the Frontier
A single-beta representation with excess return factor f means that Rf + f is on the mean-variance frontier (proof is the same as for the CAPM, compare Lecture 4b)
this is the tangency portfolio based on the observed sample distribution of returns
different from the ex-ante tangency portfolio based on the true data-generating distribution
We can therefore interpret the GRS test as a test whether Rf + f is on the frontier
GRS test statistics for Multi Factor models
Motivation for Cross-sectional regression tests
The GRS test based on time-series regressions is very convenient
(have to run those regressions anyway to obtain betas)
But it is only feasible if the factors are excess returns
Otherwise: need the cross-sectional regression to
determine factor risk premia (λs)
We then have to test whether cross-sectional regression residuals are zero this is an unusual test in
econometrics
normally we do not test for the size of residuals
Modified assumptions for cross-sectional regressions
Suppose the following assumptions hold
there is a risk-free asset with time-invariant risk-free rate Rf
factors are arbitrary random variables (not necessarily excess returns)
all returns and factors are i.i.d. over time
Under these assumptions, it is still true that γ = Rf
Hence, the cross-sectional regression equation can be written as
ET[Ri −Rf]=λ1βˆi,1 +···+λJβˆi,J +αi, i =1,…,N
We do not assume that the pricing errors αi are independent across assets
asset returns tend to be highly correlated with one another,
e.g. within industries assuming independence across i would be clearly counterfactual
Cross sectional regression using OLS
We could estimate λˆ1, …, λˆJ using OLS
Know from econometrics: OLS in the presence of correlated errors is:
consistent (λˆ1, …, λˆJ converge to the true λ1, …, λJ as we add more and more data)
but inefficient (it does not use the information in the data in the most efficient way)
We can design a test for pricing errors by combining
OLS formulas for residual covariance matrix in cross-sectional regression
information from time series regression residuals to estimate the error covariance matrix
… instead, we discuss GLS, the standard textbook advice for correlated errors
Generalised least squares
An example of GLS
Pricing erros for GLS regression test
Comparison with GLS test if Factors are excess returns
If all factors are excess returns: test statistic is identical to GRS test
distribution is χ2N−J instead of χ2N
but we can get the J degrees of freedom back by adding the J factors as test assets
The GLS estimates for factor risk premia are simply
λˆ1 =ET[f1], ··· λˆJ =ET[fJ]
Effectively, GLS puts all its weight on the J factor assets
the factor assets are most informative for factor risk premia
all other asset returns do not provide additional information, but are noisier
Regression equations for testing CAPM
can we do without the i.i.d. assumption?
yes
do we need to work with the linear beta representation instead of the linear SDF when estimating and testing models?
no
can one also deal with nonlinear SDF models
yes
Generalised method of moments
All we need to do is replace linear regressions with the Generalized Method of Moments
(GMM)
this is the state-of-the-art econometric method in empirical asset pricing
it generalizes linear regressions
can deal with both nonstandard assumptions on errors and nonlinear estimation
Summarising empirical testing
Regression-based tests of linear factor models under i.i.d. assumptions:
- basic strategy: test null hypothesis that pricing errors are all zero
Test based on time-series regression (GRS test):
- applicable if all factors are excess returns effectively a test for regression intercepts
Test based on cross-sectional regression
- always applicable
- test that residuals vanish OLS and GLS variants
