Linear Factor Models: Overview Flashcards
Motivation
Empirical asset pricing is dominated by linear factor pricing models
The most widely applied asset pricing models are of this form, e.g.
Capital Asset Pricing Model (CAPM) Fama-French three-factor model Carhart four-factor model
Central equation of linear factor pricing models
Interpretation of this
Interpretation of the Beta representation
Factors f1, …, fJ represent different dimensions of priced risk
Betas measure how an asset’s return covaries with each of the risk factors
simplest example: if J = 1 (one factor), then i = cov(f , Ri )/ var(f )
i,j is a measure of the quantity of type-j risk of the asset
The model tells us that assets earn a risk premium when they comove with risk factors for each unit of type-j risk of an asset, the asset earns an extra expected return of j
Any risk uncorrelated with all risk factors is not priced (i.e. idiosyncratic)
Recalling the risk adjustment equation for excess returns
LFMs and SDFs
In fact, one can show that the linear factor pricing model is equivalent to a SDF model
m = a + b1f1 + · · · + bJ fJ
for some real numbers a, b1, …, bJ
Special Case of Single Factor: Linear SDF Implies Linear Factor Model
Factors Should Proxy Marginal Utility Growth
Factors should measure news not outcomes
Consumption and marginal utility react to news about the future
example: suppose investor gets signal that recession in two years is more likely
in fear of being laid o↵ in the future, the investor may cut back consumption already now
Hence, factors should measure news about bad times rather than their occurence
in above example: variable that forecasts future unemployment rate may be better factor than current unemployment rate itself
Candidate factors forecast asset returns or macroeconomic variables
this logic justifies certain financial variables as factors (e.g. stock returns, term premium) may not be direct measure of bad times, but their forecast them
Factors Should Be Nearly Unpredictable
General Equilibrium Models Can Provide more Definite Guidance
Previous considerations: broad, qualitative guidance how to select factors
More definite guidance results from explicit general equilibrium models
these models relate consumption in equilibrium to a set of measurable variables
they tell us exactly what these variables are and why they should be priced factors
also, they often make additional predictions with regard to the size of factor risk premia
This additional guidance comes at a cost:
- we have to make additional assumptions, which may not hold in reality
- we have to solve a general equilibrium model, which may be di cult
How factors should be selected
- Should proxy marginal utility growth
- Show measure news not outcomes
- Should be nearly unpredictable
How are factors selected in empirical practice
The broad qualitative arguments are regularly invoked to justify a certain set of factors
But the tighter predictions from equilibrium models are often ignored in empirical work taking predictions seriously often harms empirical fit (additional restrictions)
traditional models like CAPM rest on too unrealistic assumptions to be taken literally
it is easy to dismiss their assumptions on theoretical grounds
However, theoretical models do have an important advantage:
empirical finance research has extensively analyzed the same historical sample of prices
many factors that have been found to “work” may be the result of “fishing”: looking long enough in the same sample for something that works in that sample
waiting a few decades to check whether they also work out of sample is not feasible
economically sound mechanisms give a model plausibility and guard against such “fishing”
Ideally, we desire well-founded general equilibrium models that make tight predictions based on plausible assumptions and also work well empirically
! still an area of active research in finance
Measuring Betas
Betass are the coffee cients in a projection of returns on factors
We can interpret this as a population regression
with error terms “episilon” that are uncorrelated with the factors
We can implement this empirically using a time-series (sample) regression
if we observe the following sample:
a time series R1i , …, RTi if asset returns for asset I
for each j, a time series fj,1,…,fj,T of realizations of factor j
Population Regression Versus Sample Regression
Population regression picks coe cients ai, Beta i,1, …, i,Beta J to minimize the true mean-squared error
The sample (time-series) regression
picks coe cient estimates aˆ , Betaˆ , …, BetaJ ˆ to minimize the sum of squared residuals
Measuring Factor Risk Premia ( Lamdas) I: Equation System
Special Case: Factors are Excess Returns
Measuring lamda’s in sample
Special Case of Factors that Are Excess Returns cont.
Measuring Factor Risk Premia ( Lamdas) II: Cross-Sectional Regression
Time-Series vs Cross-Sectional Regression
