Lecture 13
Factor Construction and Fama-Mac
. Lochstoer
UCLA Anderson School of Management
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Winter 2022
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Overview of Lecture 13
Constructing your own factors / trading strategies
1 Fama-MacBeth regressions and factor-mimicking portfolios
I Trading on a stock characteristic
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Applications of Fama-Mac
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Cross-sectional, expected return relation
Let Xi,t denote a 1 K vector of stock characteristics known at time t.
An example is a stockís market beta estimated using historical data from
time t j to time t.
Another example is a stockís log book-to-market ratio, bmi,t
Xi,t can be a vector and can include a 1 if an intercept term is desired
The Fama-MacBeth procedure we will consider seeks to estimate the K 1 vector
λ in the below relation:
Et Re = Xi,tλ i,t+1
Note the conditional nature of this statement. We are asking how, if at all, the conditional expected excess returns on stock i are related to the characteristics,
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The Fama-MacBeth procedure we will consider seeks to estimate λ in the below
i = 0. We donít care about the sign of ηi,t+1; itís just an error term, so we write the
Recall that Re i,t+1
= Et hRe i + η where E hEt hRe i η i,t+1 i,t+1 i,t+1
Re =Xλ η . i,t+1 i,t i,t+1
regression
Re =Xλ+ε . i,t+1 i,t i,t+1
Et Re = Xi,tλ i,t+1
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Winter 2022
For each time t, run the regression (with N observations):
Collect λˆt+1 The Önal estimate is
λ + ε for i = 1, …, N i,t t+1 i,t+1
across time, is:
var λˆ = var λˆt+1 T
ˆ1Tˆ λ=T ∑λt+1
The squared standard error of the estimate, assuming λt+1 are uncorrelated
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Consider the estimate λˆt+1:
λˆ t + 1 = X t 0 X t 1 X t 0 R t e + 1
where Xt = [X10,t X20,t … XN0 ,t]0 is an N K matrix. Rte+1 = hR1e,t+1 R2e,t+1 … RNe ,t+1i is an N 1 vector.
Note that we could easily make the number of stocks time-dependent, Nt .
Now, note that we can write this in terms of portfolio weights (on excess returns): λˆt+1 =wt0Rte+1
where wt0 = (Xt0Xt ) 1 Xt0 is a K N matrix of portfolio weights.
The kíth row of wt0 are the portfolio loadings for the kíth row of λˆt+1
As is usual, the portfolio loadings do not need to sum to one since we are operating with excess returns
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FM Interpretation: Example
Consider the following Fama-MacBeth regression:
Re =λintercept+λbm bm +ε i,t+1 t+1 t+1 i,t
In this case, OLS implies that:
fori=1,…,N
bm N 1 bmi,t Eti [bmi,t] e λt+1=∑N vari(bm ) Ri,t+1
i=1| t{zi,t} =wi,t
where Eti [] and varti () denote the cross-sectional mean and variance of bmi,t at time t.
Note that λbm is a long-short portfolio return t+1
The (excess) return to a portfolio that is long high book-to-market stocks and short low book-to-market stocks.
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FM Interpretation: Example (contíd)
The Önal estimate of the Fama-MacBeth regression is ˆ1Tˆ
λ=T ∑λt+1 t=1
Thus, the estimated coe¢ cient λˆbm is the mean return to the portfolio that goes long high bm stocks and short low bm stocks.
The expected excess return to a factor-mimicking portfolio
Note that the intercept is capturing the average return to all stocks as well as a
term related to the mean of the explanatory variable, bmi,t:
bm2 1N bm 1N
λintercept= i,t
t+1 bm2 bm 2Ni=1 i,t+1 bm2 bm 2Ni=1 i,t i,t+1
i,t ∑bm Re i,t i,t i,t i,t
The math behind these results are on the next two slides.
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Regression Math
Note: A bar denotes a cross-sectional average
X0X =N” 1 bmi,t #
t t bmi,t bm2 i,t
0 1 1 1 ” bm2 XtXt= 2i,t
bmi,t # 1
X t0 X t 1 X t0 =
1 1 ” bm2 bmi,t # 1 1 … 1
Nbm2 bmi,t bmi,t i,t
Nbm2 bm 2 bmi,t 1 bm1,t bm2,t … bmN,t
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Winter 2022
Regression Math (contíd)
X t0 X t 1 X t0 =
” bm2 bmi,tbm1,t bm2 bmi,tbm2,t … bm2 bmi,tbm
i,t i,t i,t
bm1,t bmi,t bm2,t bmi,t … bmN,t bmi,
i,t So, Önally:
1 bmi,t ∑ Ri,t+1 bmi,t ∑ bmi,tRi,t+1
λintercept
i =1 i =1 N bm2 bm 2
= i,t ∑Re i,t ∑bmRe
bm2 1N bm 1N
bm2 bm 2Ni=1 i,t+1 bm2 bm 2Ni=1 i,t i,t+1
If bmi ,t = 0, λintercept is the equal-weighted excess stock return at time t + 1.
i,t i,t i,t
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The long-short portfolio sorts we typically see in Önance are motivated by
cross-sectional forecasting regressions
Find a characteristic that you think forecasts next period stock return in the cross-section
It is cross-sectional as the intercept captures the average return
I Thus, we are looking for di§erences in return across assets, while not attempting to forecast the average movement
Letís look at an example of such regressions
Novy-Marx (2013) and the ProÖtability Anomaly
Anomaly, by the way, means alpha relative to existing model(s)
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The ProÖtability Factor
Novy-Marx argues:
The typical measure of future proÖtability, current earnings, is inferior to gross proÖtability
For instance, investments that are treated as expenses (advertising, R&D, human capital development) are subtracted of proÖts to get the current earnings, but these investments actually signal higher future proÖtability
Earnings are often manipulated, which obscures their relation to future earnings
Of these reasons, Novy-Marx suggests that a better proxy for future proÖtability is Gross ProÖtability divided by Book Value of Assets
Gross ProÖtability = Total Revenue (RevT) – Cost of Goods Sold (COGS). Book Assets (AT)
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The ProÖtability Factor (contíd)
Fama-MacBeth regressions showing the the GP characteristics is indeed related to average returns
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Why do these factors íworkí?
First, simplest model possible (constant growth and discount rates): P=1
Thus, high valuations mean low discount rates (expected returns) and/or high
Thus, with a good estimate of future growth, value investing will íworkíin terms of giving a spread in returns
Of course, b/m may not continue to work; perhaps we will need better estimates of g
CAPM alpha may not continue, perhaps in the future variation in r is all explained by the CAPM
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Why do these factors íworkí? (contíd)
Using “clean surplus” accounting (see Ohlson (1995)), we can write the market to
book ratio as:
Mt = 1 ∑∞ Yt+τ ∆Bt+τ. τ=1 (1+r)τ
where Y is earnings, B is book value of equity and r is a long-run discount rate The above equation makes a couple of statements about future expected returns:
1 Holding earnings Y and investment ∆B constant, a higher market value (size or m/b ratio) means lower expected returns.
2 Holding market value and investment constant, higher earnings Y (proÖtability) means higher expected returns.
3 Holding proÖtability and market value constant, higher investment means lower expected returns.
In sum: whether of behavioral or risk-based reasons, there are reasons to think portfolio sorts based on value, proÖtability, and investment may continue to give spreads in average returns
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The Fama-French 5-factor Model
FF sorts into factors in several di§erent ways in order to show robustness of the results. The new factors RMV and CMA (Robust minus Weak proÖtability and Conservative minus Aggressive investment, respectively) are formed in a similar fashion as the HML sort.
The model is:
Ri,t Rf,t =αi+bi(RM,t RF,t)+siSMBt+hiHMLt+riRMWt+ciCMAt+εi,t.
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The Fama-French 5-factor Model (contíd)
All factors have signiÖcant positive risk premiums (Panel A)
All three last factors (HML, RMW, CMA) have high returns coming mainly from small Örms (Panel B)
The paper has many tables and shows the model can account for expected returns on large set of íanomalyísorted portfolios, though the low return on small stocks that invest a lot despite low proÖtability remains a puzzle. Also, the authors show that the HML factor in fact becomes redundant, so really this may be re-couched as a 4-factor model.
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One more thing…
Note that a regression beta can be a characteristic.
For instance, consider regressing Örm iís returns on ináation.
Get the íináation betaíof stock i βιná . i,t
Ask: if we go long high ináation beta stocks, short low ináation beta stocks do we earn a premium?
For instance, if high ináation is bad, the premium on such a strategy should be negative (an ináation hedge portfolio)
Fama-MacBeth regressions estimate this premium as the average return to the ináation beta long-short portfolio
λináation
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References
Eugene F. Fama, . French, A Öve-factor asset pricing model, Journal of Financial Economics, Volume 116, Issue 1, April 2015, Pages 1-22
-Marx, The other side of value: The gross proÖtability premium, Journal of Financial Economics, Volume 108, Issue 1, April 2013, Pages 1-28
Ohlson, Contemporary accounting research, 1995, Earnings, Book Values, and Dividends in Equity Valuation.
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