TOC
1. Topics and Methodologies
Empirically, financial economists try to understand a wide range of topics, mainly include
- Asset pricing:
- time series behaviors: the behavior of a particular asset over time
- cross-section behaviors across different assets
- Corporate finance:
- firm decisions: from born to die (entry to exit)
- even before born: VC, PE, Entrepreneurship, aggragation
- link to macroeconomics
Methodologically, we will fall into two broader topics:
- causal inference: most attention by academics but less by investors
- forecasting: most attention by investors but less by academics
2. Forecast
In empirical asset pricing, we try to understand the cross-sectional behavior of asset return.
We can formulate a forecast question:
- : the return of firm at period
- : the predicted return based on the infromation available until
- : the information set available to us up till period , containing:
- macro information: monetary policy, macro variables, international new etc.
- individual information like stock return in the history, earning news, comments, news papers, analysts’ updates etc.
- all other relevant information
We usually denote as .
In general, in portfolio management, we care about
where , the portfolio weights we try to estimate using the information until ; , the return of firm at period .
As for problem (1), the solution (best estimation) is
In literature, abbreviation: , the expectation of the return conditional on information set .
3. Stochastic Discount Factor
We simplify the forecast problem using domain knowledge.
First, no arbitrage condition, then , s.t.
where is the stochastic discount factor (SDF).
Assume the price of an asset at period (after dividend payment) is . The random payoff at period is . Then based on no arbitrage assuption:
Denote , no arbitrage implies
Denote the risk-free rate in the market, then we have
where is the excess return of asset .
Since , which also holds for conditional expectation, then
Using the condition , we have
4. Factor Pricing Model
Without loss of generality, we can always write
(note that ).
Otherwise, we can write it as
with conditional on and .
The orthogonal component will not be priced in.
Theorem: The market is complete iff
In theory, we can estimate
which is exactly the mean-variance efficient portfolio (MVE), but is an infeasible estimator.
is non-estimable empirically. To handle this problem, two approaches are reasonable:
- approach 1: prune
- approach 2: prune SDF
Approach 1: impose restriction on
Suppose we can observe firm (one stock, one firm) characteristics and assume
Thus
where and
Then we can estimate
Empirically, we can estimate
where
The expected return takes the form of
Thus, the cross-sectional return takes a form of
Approach 2: simplify SDF using a factor model
Recall that
Assume takes a factor model
where . Under structure model, there are common factors affecting SDF.
Thus, the expected return can be written as
and we can get a factor pricing model
In matrix notation, we can write it as
5. Anomalies
The cross-section expected return accrued to the risk premium is
We can write it as
However, time-varying is not estimable.
We can always normalize , then we can consider two step estimation:
- estimate the factor exposure
- test the equation
In the real world, literature documents
with , and is just a function of observable up till . If , we refer to it as anomaly.
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