Capital Asset Pricing Model Part 2: The Empirics

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Capital Asset Pricing ModelPart 2 The Empirics
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RECAP Preference

• Ingredients
• 5 Axioms for Expected Utility Theorem
• Prefer more to less (Greedy)
• Risk aversion
• Assets jointly normally distributed

Expected Return E(Rp)
Increasing Utility
Standard Deviation s(Rp)
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RECAP Min-Variance opp. set
E(Rp)
- Portfolios along the efficient set/frontier are
referred to as mean-variance efficient
Efficient frontier
Individual risky assets
Min-variance opp. set
s(Rp)
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RECAP Capital Market Line (a.k.a Linear
efficient set)
E(Rp)
CML
M
E(RM)

• Ingredients
• Homogenous Belief
• Unlimited Lending/borrowing

Rf
s(Rp)
sM
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RECAP 2-fund separation
Everyones U-maximizing portfolio consists of a
combination of 2 assets only Risk-free asset and
the market portfolio. This is true irrespective
of the difference of their risk-preferences
CML
E(Rp)
B
E(RM)
(M) Market Portfolio
CML Equation E(Rp) Rf (E(RM)- Rf)/sMs(Rp)
A
Rf
sM
s(Rp)
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RECAP CAPM SML
E(return) Risk-free rate of return Risk
premium specific to asset i Rf (Market
price of risk)x(quantity of risk of asset i)
E(Ri) Rf E(RM)-Rf x COV(Ri,
RM)/Var(RM) E(Ri) Rf E(RM)-Rf x ßi
E(Ri)
SML
E(RM)
slope E(RM) - Rf Eqm. Price of risk
Rf
ßi COV(Ri, RM)/Var(RM)
ßM 1
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Empirical Studies of CAPM

• Is CAPM useful?
• Given many unrealistic assumptions, how good does
  the model fit into the reality?
• Think about the following questions
• 1 What exactly are the predictions of the
  CAPM?
• 2 Are they testable?
• 3 What is a regression?
• 4 How to test hypothesis? What is t-test?

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1 What are the predictions ?

• a CAPM says more risk, more rewards
• b HOWEVER, reward-able risk ? asset total
  risk, but systematic risk (beta)
• c We ONLY need Beta to predict returns
• d return LINEARLY depends on Beta

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2 Testable ?
E(Ri) Rf E(RM)-Rf x COV(Ri,
RM)/Var(RM) E(Ri) Rf E(RM)-Rf x ßi

• Ideally, we need the following inputs
• a Risk-free borrowing/lending rate Rf
• b Expected return on the market E(RM)
• c The exposure to market risk
• ßi cov(Ri,RM)/var(RM)

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2 Testable ?
E(Ri) Rf E(RM)-Rf x COV(Ri,
RM)/Var(RM) E(Ri) Rf E(RM)-Rf x ßi

• In reality, we make compromises
• a short-term T-bill (not entirely risk-free)
  Rf
• b Proxy of market-portfolio (not the true
  market) E(RM)
• c Historical beta
• ßi cov(Ri,RM)/var(RM)

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2 Testable ?

• Problem 1
• What is the market portfolio? We never truly
  observe the entire market.
• We use stock market index to proxy market, but
• i only 1/3 non-governmental tangible assets are
  owned by corporate sector. Among them, only 1/3
  is financed by equity.
• ii what about intangible assets, like human
  capital?

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2 Testable ?

• Problem 2
• Without a valid market proxy, do we really
  observe the true beta?
• i suggesting beta is destined to be estimated
  with measurement errors.
• ii how would such measurement errors bias our
  estimation?

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2 Testable ?

• Problem 3
• Borrowing restriction.
• Problem 4
• Expected return measurement.
• i are historical returns good proxies for
  future expected returns? Ex Ante VS Ex Post

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3 Regression
E(Ri) Rf E(RM)-Rf x COV(Ri,
RM)/Var(RM) E(Ri) Rf E(RM)-Rf x
ßi E(Ri) Rf E(RM)-Rf x ßi
With our compromises, we test
Ri Rf RM-Rf x ßi
Using the following regression equation
Rit Rft ?0 ?1ßi eit
In words, Excess return of asset i at time t over
risk-free rate is a linear function of beta plus
an error (e). Cross-sectional Regressions to be
performed!!!
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3 Regression
Rit Rft ?0 ?1ßi eit

• CAPM predicts
• a ?0 should NOT be significantly different from
  zero.
• b ?1 (RMt - Rft)
• c Over long-period of time ?1 gt 0
• d ß should be the only factor that explains the
  return
• e Linearity

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4 Generally agreed results
Rit Rft ?0 ?1ßi eit

• a ?0 gt 0
• b ?1 lt (RMt - Rft)
• c Over long-period of time, we have ?1 gt 0
• d ß may not be the ONLY factor that explains
  the return
• (firm size, p/e ratio, dividend yield,
  seasonality)
• e Linearity holds, ß2 unsystematic risk
  become insignificant under the presence of ß.

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4 Generally agreed results
Rit Rft
CAPM Predicts
Actual
?1 (RMt - Rft)
?0 0
ßi
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Rolls Critique

• Message We arent really testing CAPM.
• Argument Quote from Fama French (2004)
• Market portfolio at the heart of the model is
  theoretically and empirically elusive. It is not
  theoretically clear which assets (e.g., human
  capital) can legitimately be excluded from the
  market portfolio, and data availability
  substantially limits the assets that are
  included. As a result, tests of CAPM are forced
  to use compromised proxies for market portfolio,
  in effect testing whether the proxies are on the
  min-variance frontier.
• Viewpoint essentially, implications from CAPM
  arent independently testable. We do not have the
  benchmark market to base on. Every implications
  are tested jointly with whether the proxy is
  efficient or not.