Financial Econometrics Econ 40357 Factors Single-Factor ModelsThe - - PowerPoint PPT Presentation

Financial Econometrics Econ 40357 Factors Single-Factor Models–The market model and the CAPM

N.C. Mark

University of Notre Dame and NBER

October 19, 2020

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Textbook

Brooks pp. 586-588.

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The market model and the CAPM

Finance people like to talk about (common) factors Factor is systematic component driving the cross-section over time. Factor may be observed or latent (unobservered) Returns driven by common and idiosyncratic factors Investors are paid to bear systematic risk (part driven by common factors) CAPM is a single-factor model. Factor is the market return. Later, we talk about multi-factor models. Finance people like to embed factor models within the beta-risk framework.

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The Beta-Risk Model

Question is: Which assets pay high returns and which pay low returns over long periods of time, and why? e.g., Big versus small firms. Do small firms pay more or less? If more, what’s the risk in small firms that make people afraid of them? Answer is those assets with greater exposure to the risk factor. Measure exposure with beta. The big question here, is what is (are) risk factor(s)?

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The Beta-Risk Model

Asset pricing model: In finance, all models take the form E

• r e

t,i

= βiλ + αi where r e

t,i = αi + βift + ǫt,i

In the CAPM, factor ft = r e

t,m is the market excess return.

i

• (

)

ei

E R

Slope

i

• 1

( ) E f

• ff

ff ff

• ⇒

Picture shows relation between risk and return. Risk is

• covariance. Excess returns vary proportionally to βi .

αi is the deviation (Jensen’s alpha). β is the asset’s exposure to the risk factor, f. It says, the risk-premium (expected excess return) varies in proportion to the asset’s exposure to risk factor. λ is that factor of proportionality. 5 / 15

The Beta-Risk Model

Let ft = re

t,m. Each asset’s return i = 1, ..., N, is assumed to be generated by

re

t,i = αi + βift + ǫt,i

Take time-series expectation E

• re

t,i

• = E (ft) βi + αi

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Jensen’s alpha, αi, is risk-adjusted performance measure. The average return on security (portfolio) above or below that predicted by theory (e.g., CAPM).

2

αi = 0, portfolio manager has no value. αi > 0, manager has special talent.

3

Key implication from model: Excess return explained entirely by exposure to risk factor λ = E(ft) αi = 0

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All finance models take beta-risk form (short version)

Investor’s Euler equation. xt+1,j is payoff from asset j that costs pt,j. pt,ju′ (ct) = Et

• βu′ (ct+1) xt+1,j
• (1)

If asset is stock, xt+1,j = pt+1,j + dt. If asset is coupon bond, replace dt with

• coupon. If asset is discount bond, xt+1,j = 1.

Express in return form, 1 = Et βu′(ct+1) u′(ct) xt+1,j pt,j

• Change notation: mt+1 = βu′(ct+1)/u′(ct) is the stochastic discount factor.

(1 + rt+1,j) = xt+1,j/pt,j is the gross return. Rewrite the Euler equation one more time 1 = Et(mt+1(1 + rt+1,j)) (2) Holds for all traded assets j = 1, ..., N. Also holds for the risk free asset whose return is 1 + rf

t .

1 = Et(mt+1(1 + rr

t ))

(3)

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All finance models take beta-risk form (short version)

Subtract (3) from (2) to get 0 = Et(mt+1re

t+1,j)

Take unconditional expectations of both sides, 0 = E(mt+1re

t+1,j)

Now the timing t + 1, t doesn’t matter. Assume a one-factor representation for the SDF. ⇐ this is key mt = 1 − b(ft − µf ) (4) What is factor ft? Could be consumption growth, could be asset returns. Substitute (4) into Euler equation to get the beta-risk representation = E(re

t (1 − b(ft − µf )))

= E(re

t ) − bCov(re t , ft)

= E(re

t ) − bVar(ft)Cov(re t , ft)

Var(ft) (5) Hence, E(re

t ) = λf β

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Estimate and Test the CAPM with Time-Series Method

This method works when factor is an excess return. Preliminary analysis

Estimate and test if price of risk E(ft) = λ is statistically significant: Run the regression ft = c + ǫt

• f the factor (excess return) on constant.

Constant is estimate of λ. Do Newey-West on the constant, test if it is greater than 0.

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Estimate and test CAPM with Time-Series Method

Run the time-series regression r e

t,i = αi + βift + ǫt,i

for each asset i = 1, ..., n, using Newey-West. Do individual t-tests

• n the αi.

A cheap and not entirely correct joint test: If all the αi are zero, then the sum of the αi is zero. If the αi estimates are independent, then t2

1 + t2 2 + · · · t2 n ∼ χ2 n

where t2

i is the squared value of the Newey-West t-ratio on αi.

This test is not entirely right because it ignores possible correlation across the αi

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A correct joint test on the α’s

Let ˆ α =      ˆ α1 ˆ α2 . . . ˆ αN      Σǫ =        Var(ǫ1) Cov(ǫ1, ǫ2) · · · Cov(ǫ1, ǫN) Cov(ǫ2, ǫ1) Var(ǫ2) · · · Cov(ǫ2, ǫN) . . . Cov(ǫN, ǫ1) · · · Var(ǫN)        Var(ǫi) = 1 T

T

∑

t=1

ǫ2

t,i,

Cov(ǫi, ǫj) = 1 T

T

∑

t=1

ǫt,iǫt,j ¯ f = 1 T

T

∑

t=1

ft, Σf = 1 T

T

∑

t=1

(ft − µf )(ft − µf )′ Then test statistic is, T

• 1 + ¯

f ′Σ−1

f

¯ f −1 ˆ α′Σ−1

ǫ

ˆ α

• ∼ χ2

N

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Chances are, you didn’t learn this in econometrics class. Why? Because in econometrics, you learned about constructing standard errors (and t-ratios) for a single regression. Here we are looking at the joint distribution of αi and αj across different regressions How to do this in Eviews? Estimate as system, ask for the joint test.

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Object → New Object → System Write down the system model Estimate by Ordinary Least Squares → View → Coefficient Diagnostics

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Estimate/test CAPM with Time-Series Method

In the time-series regression, r e

i,t = αi + βift + ǫi,t

Let us impose the restriction that mean returns are proportional to betas, E(r e

i,t) = βiλf = αi + βiE(ft)

Then the intercept should be αi = βi(λ − E(ft)) The intercept in the regression controls the mean return. If λ = E(ft), the intercept will be zero. In order to test this restriction, you need an estimate of λ, and this only works if the factor is a return. If the model is true, αi = 0. (why is that?)

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