The Capital Asset Pricing Model Session 5

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The Capital Asset Pricing ModelSession 5

• Andrei Simonov

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Assumptions

• Single holding period
• Investors are risk-averse
• Investors are small
• The information about asset payoffs is common
  knowledge
• Assets are in unlimited supply
• Assets are perfectly divisible
• No transaction cost
• Wealth W is invested in assets

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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 cov(Ri,Rp)/var(Rp)

?i
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Market level syllogism
port 1
E(Ri) - r

• All weighted averages of
• efficient portfolios
• are efficient
• Assume each person holds an efficient portfolio
• At equilibrium, the market portfolio, m, is an
  average of individually held portfolios
• Therefore, the market portfolio must be efficient

avg 12
port 2
cov(Ri,Rp)
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Market level Math (1)

• Conditions of optimality (Remember
  Microeconomics?)

Risk Aversion
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Market level Math (2)

• But Market Portfolio is a legidimate asset,
• Market price of risk

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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
  cov(Ri,Rm)/var(Rm)
• CAPM can be extended to the case in which there
  exists no risk-less asset.

?m ?m var(Rm) E(Rm) - r
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Extensions

• No risk-free assets
• No short sales
• Different lending and borrowing rate
• Restricted opportunity sets (Hietala, JF89)
• Personal taxes
• Multi-period extensions
• The simple form of CAPM is rather robust.
  Modification of basic assumptions leads to
  changes in existing terms and appearance of new
  terms (induced factors)
• However, sumultaneous modification of multiple
  assumptions leads to SERIOUS departure from
  standard CAPM.

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Is CAPM right?

• 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

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Is CAPM right?

• How can you tell?
• Two-pass approach
• for each security, measurement of mean excess
  return and beta using history of returns (time
  series)
• relate mean excess return to beta (cross section)
• First pass has no economic meaning, just a
  measurement. Second pass is embodiment of CAPM.

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Example
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First pass Security A
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Second pass CAPM line
Intercept 3.82Slope ?m 5.21Adjusted R2
0.44
?G
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Discussion CAPM may not be testable

• 1. the market 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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There are deviations from CAPM (or ?s)

• 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?
  Return differential 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 criticism 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 included, revised CAPM holds
  up better

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Conclusion

• If CAPM were right every mean-variance investor
  could just hold the market portfolio (index
  fund), adjusting the level of risk by mixing it
  with riskless asset.
• If CAPM is not right, there is room for tilted
  index funds. See Dimensional Fund Advisors case.

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Multidimensional Investments and APT Session 6

• Andrei Simonov

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Outline

• Pitfalls in portfolio construction
• Ideas that have survived
• Statistical models
• Streamlining the investment process
• Estimating statistical factor models
• Pricing models
• Multi-factor pricing models
• Arbitrage Pricing Theory

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Pitfalls in portfolio construction

• Minor deviations in inputs (especially expected
  returns) lead to large changes in decisions.
  Generally, large, crazy positions.
• When estimating from the past, estimation risk.
• Problem especially severe when there are many
  line items. In fact, optimization becomes
  meaningless when more items than observations.

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Jorion simulation
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Ideas that have survived

• A statistical concept exposure
• beta is an exposure to market risk
• need to keep track of exposures, when
  constructing a portfolio
• use of exposures to classify securities and
  systematize portfolio construction process
• beta measures each assets contribution to total
  portfolio risk
• idea useful for risk management need to develop
  accounting of risk (or breakdown of risks)
• but beta needs a generalization find more common
  factors
• A pricing concept
• In pricing, only common risk factors matter.
  Other risks can be diversified away.

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Statistical models streamlining the investment
process

• Covariance matrix is huge (many entries)
• Leads to imprecisely computed portfolios
• Let us impose structure on the matrix
• We may have 10000 securities to choose from
• In fact, there may be only 20 basic sources of
  risks
• A particular security can be seen as a portfolio
  of these basic risks and must be identified as
  such

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Example one factor model

• One-factor model ( bs called loadings or
  exposures)
• where residuals ?i,t are independent
• Then

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Example construct common factor
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Example compare correlations
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Estimating beta of a security

• Use model
• (example market 50 large stocks 50
  small stocks)

negligible
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Extension multi-factor statistical models

• Multi-factor model
• When I hold security i, I am truly holding b1,i
  of risk 1, b2,i of risk 2 etc..
• Then

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Estimating Statistical Factor Models three
approaches

• Capture the way in which securities returns move
  together
• Factor analysis
• Factor analysis constructs a set of abstract
  factors that best explain the estimated
  covariances
• Throws no light on underlying economic
  determinants of the covariances
• Use of macroeconomic variables
• Use of firm specific variables

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Estimating Statistical Factor Modelsmacroeconomic
variables

• Business cycle risk
• unanticipated growth in industrial production
• Confidence risk
• default spread (Baa - Aaa) which is a proxy for
  unanticipated changes in risk premia
• Term premium risk
• return on long bonds minus short bonds, which is
  a proxy for unanticipated shifts in slope of
  yield curve
• Other Oil prices
• Get exposures (loadings) of each stock by
  multiple regression

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Estimating Statistical Factor Models use of
firm-specific information

• Form factor-mimicking portfolios which capture
  factors
• Market RM - r
• B/M High-minus-low (HML) RHML RH - RL
• Size Small-minus-big (SMB) RSMB RS - RB
• Estimate exposures by regression

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Example calculation of multi-factor statistical
model
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BARRA and other quant shops

• Similar approach
• measure exposures onto any number of risk
  categories (country, industry, small vs. big
  etc..).
• get value of factor return for each category,
  each time period, by comparing across firms
• interpret this months factor return as a way of
  pinpointing the category of firms that are
  currently most profitable
• Up to 80 factors!

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Statistical factor models in the standard CAPM
vs. Multi-factor pricing models

• Multi-factor statistical models can be used to
  estimate parameters of the standard CAPM. E.g.,
  securities betas from exposures times factor
  betas
• One may still use standard CAPM still one risk
  premium (single-factor pricing model)
• Opposite several risk premia. Multi-factor
  pricing models

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Multi-factor pricing models state-dependent
preferences

• Investor cares about portfolio variance, and
  also about performance in a recession
• investors try to buy stocks that do well in a
  recession
• this drives down expected return of those stocks
  beyond the market beta effect (?recession lt 0)

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Numerical example

• Exposures obtained by multiple regression of
  individual security on the factors, as in
  statistical models
• Prices of risk obtained by second-pass
  cross-sectional regression like in CAPM.

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Example calculation multi-factor pricing model
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Example E(R) on security B
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Other justification for this sort of pricing
Arbitrage Pricing Theory

• Large number of securities, finite number of
  factors
• Residuals become irrelevant. Only exposures b to
  common factors matter for pricing
• Dichotomy of risk variables
• some (factors, in finite number) affect all
  securities
• others (residuals, in large number) affect only
  one security each
• Pricing equation there must exist premia ?
• Otherwise, could work out approximate arbitrage

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Math Derivation of APT(1)

• Main requirement No possibility to create
  something out of nothing
• Certainty case n1 assets, K factors, ngtK, 0
  asset is risk-free one
• Consider portfolio of yj0(1-Skbjk) units of
  riskless asset and yjkbjk of risky asset
• Looks similar to returns of asset j! Let us
  exploit it by buying 1 worth of this portfolio
  and shorting 1 of asset j. No-arbitrage
  requirement tells that the return on such
  portfolio should be 0

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Math Derivation of APT(2)

• Uncertainty case
• For any small e, there exists at most N assets,
  Nltn, for which arbitrage condition is violated
• The no-arbitrage condition requires that N/n?0 as
  n??. Moreover, as Dybvig(1983) shows, the error
  also decreases as the number of players in
  economy increases

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CAPM vs. APT
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CAPM vs. APT estimates
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Conclusions

• Securities are analyzed as portfolios of
  underlying risks
• Clearly draw the distinction between
• statistical model that uses risk factors which
  are common to returns as a description,
• and a multi-factor pricing model that prices each
  risk factor separately
• Less restrictive approach to asset pricing
  several dimensions of risk are priced
• general multi-factor model based on incompletely
  specified investor behavior
• APT relies on absence of approximate arbitrage