Multifactor models & Linear Empirical Testing

Question 1

The Intertemporal Capital Asset Pricing Model

Answer 1

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

Question 2

A long-term investor’s future marginal utility depends on

Answer 2

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

Question 3

Why does marginal utility depend on investment opportunities

Answer 3

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

Question 4

State variables capturing investment opportunities emerge as factors

Answer 4

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

Question 5

The assumption of no outside income in CAPM

Answer 5

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

Question 6

What happens if none of these interpretations applies? - example with quadratic utility and outside income - setup

Answer 6

Question 7

Quadratic utility example

Question 7

Quadratic utility example

Answer 7

Question 8

General lessons from this example

Answer 8

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

Question 9

Conditional vs Unconditional Factor models

Answer 9

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

Question 10

When is the distinction between conditional and Unconditional relevant

Answer 10

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

Question 11

Is the i.i.d assumption plausible?

Answer 11

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

Question 12

Conditional one-factor model can imply unconditional multi factor model

Answer 12

Conditional factor models can sometimes be expressed as unconditional factor models

This usually requires adding “information factors” that capture time variation in investor information

Question 13

Summary

Answer 13

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

Question 14

Different test portfolios lead to rejection of CAPM

Answer 14

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

Question 15

The Fama-French Three-Factor Model

Answer 15

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”

Question 16

Economic interpretation of size and factors

Answer 16

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

Question 17

Additional empirical evidence

Answer 17

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

Question 18

Momentum factor

Answer 18

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

Question 19

Economic interpretation

Answer 19

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

Question 20

Summary

Answer 20

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

Question 21

Basic testing strategy

Answer 21

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)

Question 22

How do we construct statistical tests for H0?

Answer 22

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)

Question 23

Assumptions for time-series regression tests

Answer 23

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

Question 24

Recovering pricing errors from time-series regressions

Answer 24

Question 25

Time-series regressions with excess returns

Answer 25

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

Question 26

How to test for Ho?

Answer 26

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

Question 27

GRS test-statistics for One-Factor Models

Answer 27

Question 28

Reinterpretation of the test statistic: Testing if Factor is one the Frontier

Answer 28

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

Question 29

GRS test statistics for Multi Factor models

Answer 29

Question 30

Motivation for Cross-sectional regression tests

Answer 30

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

Question 31

Modified assumptions for cross-sectional regressions

Answer 31

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

Question 32

Cross sectional regression using OLS

Answer 32

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

Question 33

Generalised least squares

Answer 33

Question 34

An example of GLS

Answer 34

Question 35

Pricing erros for GLS regression test

Answer 35

Question 36

Comparison with GLS test if Factors are excess returns

Answer 36

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

Question 37

Regression equations for testing CAPM

Answer 37

Question 38

can we do without the i.i.d. assumption?

Answer 38

yes

Question 39

do we need to work with the linear beta representation instead of the linear SDF when estimating and testing models?

Answer 39

no

Question 40

can one also deal with nonlinear SDF models

Answer 40

yes

Question 41

Generalised method of moments

Answer 41

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

Question 42

Summarising empirical testing

Answer 42

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