INVESTMENTS Faculty:Bernard DUMAS

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Master of Science April 2008
INVESTMENTSFaculty Bernard DUMAS The
CAPM session 2-2
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Overview

• CAPM
• Proof
• Is the CAPM a true statement?
• Deviations
• The equilibrium approach to investment
• Performance measurement
• (This has been the main impact of the CAPM on the
  investment industry)

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Proof of CAPM
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Reminder on optimal diversification at individual
investor level condition of optimality

• How can you tell whether a portfolio p is well
  diversified or efficient ?
• For each security i, E(Ri) - r must be lined up
  with cov(Ri,Rp) or, equivalently, with
• ?i/p cov(Ri,Rp)/var(Rp)

?i/p
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Market level syllogism

• All weighted averages of efficient portfolios
  are efficient
• At equilibrium, the market portfolio, m, is an
  average of individually held portfolios, each of
  which is assumed to be efficient
• Therefore, the market portfolio is efficient

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Market equilibrium

• For each security i, E(Ri) - r must be lined up
  with cov(Ri,Rm) or, equivalently, with ?i/m
  cov(Ri,Rm)/var(Rm)
• CAPM can be extended to the case in which there
  exists no risk-less asset.

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The CAPM logic
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How do you test the CAPM?
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Is the CAPM true?

• Content cross-sectional relationship
• when comparing securities to each other, linear,
  positive-slope relationship of mean excess return
  (risk premium) with beta
• zero intercept
• no variable, other than beta, matters as a
  measure of risk or has an influence on expected
  return differences

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Is the CAPM true?

• How can you tell?
• Two-pass approach
• First, for each security, measurement of mean
  excess return and beta using history of returns
  (time series)
• Second, relate mean excess return to beta (cross
  section)
• First pass has no economic meaning, just a
  measurement (a statistical model). Second pass is
  embodiment of CAPM (a pricing model).
• Note, however, that, if CAPM is right, it follows
  from second pass that ? in the first pass should
  have been equal to zero.

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Example
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First pass Security A
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Second pass CAPM line
Best fit Intercept 3.82Slope
5.21Adjusted R2 0.4 Theoretical
line Intercept 0 Slope 8.12
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Discussion CAPM may not even be testable

• 1. the market portfolio is not observable (Roll
  critique)
• 2. should use time-varying version, based on the
  information set of the investors. The latter is
  not observable (Hansen and Richard critique).

E(Ri) - r
A
date 2
B
B
A
B
date 1
A
?i
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Observed deviations from the CAPM
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There are deviations from CAPM

• Fama and French (1992) investigate 100 NYSE
  portfolios for the period 1963-1990
• The portfolios are grouped into 10 size classes
  and 10 beta classes
• They find that return differential (risk premium)
  on ? is negative (and non significant)
• whereas return differential on size is large and
  significant.

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beta is dead ?
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Book/market also
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Recent thinking

• Question is Fama-French evidence reliable?
• Returns not IID return differentials may come in
  waves.
• Perhaps, CAPM is right at each point in time
• But CAPM line moves about
• When indicator variables are used to track
  these changes over time (such as variables we
  shall list in lecture on predictability), size
  and B/M no longer show up in CAPM
• So, these variables were showing up in the
  Fama-French analysis, not because CAPM was wrong,
  but only because movements in the line had not
  been properly accounted for

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Other criticisms of CAPM

• No account of re-investment risk (multi-period
  aspects)
• inter-temporal hedging
• No account of investors non traded wealth
  (similar to Roll critique)
• when human capital is included, the CAPM so
  revised holds up better

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The equilibrium approach to investment
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In equilibrium

• If all investors were mean-variance investors and
  had identical expectations,
• the CAPM would be right,
• And every investor could just hold the market
  portfolio (index fund), adjusting the level of
  risk by mixing it with riskless asset.
• If there are deviations from the CAPM,
• Then, some investors are not mean-variance
  investors,
• there may be room for tilted index funds,
• a form of active portfolio management.
• You need a specific reason to deviate from the
  market portfolio

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Equilibrium interplay us vs. them

• If everyone is a rational investor but have
  different expectations,
• People who are bullish on a security hold more of
  it
• If you are rational and others are not,
• For instance, others may display aversion to some
  type of securities (e.g., small firms)
• They require higher return on these securities
• In equilibrium, you profit from their aversion by
  holding more of these securities
• And, of course, they hold less
• It is very important to know what you are
  (presumably rational) and what others are (If
  you do not know who the fools are, you are the
  fool  ) so that
• By being yourself rational, you can exploit
  behavioral traits of others
• You are aware of the reason for which you are
  likely to make a gain
• This is a great justification for an investment
  strategy
• If irrational behavior is very prevalent, that is
  no reason to discard concept of optimizing
  behavior
• To the opposite, that is a great reason for you
  to know what it means to be rational
• And for you to be rational
• Most crucial piece of information how do you
  differ from the average investor?

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Performance measurement
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Main impact of CAPM performance measurement

• Jensens alpha of the fund p
• Only sign (not value) of ? matters
• Cannot rank funds by ?
• Use instead
• Appraisal (or information) ratio ?p/?(?p)

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What is the source of ? ?

• Positive ? (if benchmark market) arises from
• Either CAPM is incorrect
• E.g., some investors are not mean-variance
  investors
• In that case, the ? of any security reflects
  ability of mean-variance investors to take
  advantage of these investors by tilting their
  portfolio permanently
• Or CAPM is correct but the portfolio manager has
  more information than the market
• he/she receives signals
• conducts time-varying portfolio strategy/tactic
• then, ? of fund is truly a measure of expertise
  precision of signals ? number of signals received
  and acted upon
• For measure of expertise, benchmark can be
• Either market portfolio (if known to be
  efficient)
• Or efficient portfolio calculated ex post, but
  with constant weight
• In which case, ? measures ability of time-varying
  portfolio policy based on signals to beat
  constant-weight policy

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Performance results

• The vast majority of active managers has
  difficulty beating the SP500