Tests of linear beta pricing models (I)

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Tests of linear beta pricing models (I)

• FINA790C
• Spring 2006
• HKUST

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Outline

• Linear beta pricing models
• Conditional and unconditional MV efficiency
• Base case Maximum likelihood methods with
  conditional homoscedasticity
• Non-traded factors
• Traded factors
• Example riskfree rate, traded factors
• Extensions
• GMM
• Cross-sectional regression methodology

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SDF once again

• Suppose there is a stochastic discount factor
  mt1 such that Etmt1Rit1 1
• Consider the portfolio p with maximal correlation
  with m
• mt1 ?0t ?1tRpt1 ?t1
• The asset z whose return is uncorrelated with m
  and p has conditional expected return 1/Etmt1
  (riskfree rate if it exists)

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Linear beta pricing

• This gives pricing relation in terms of z and p
• EtRit1 EtRzt1 ßipt1(EtRpt1-EtRzt1
  )
• From efficient set mathematics this is equivalent
  to saying that portfolio p is on the minimum
  variance frontier
• OR we can just assume that mt1 is a linear
  function of pre-specified factors

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Identifying the factors

• Theoretical arguments or economic intuition
• Sharpe-Lintner-Black CAPM
• Intertemporal CAPM
• Consumption CAPM
• Statistical factors
• Empirical anomalies
• Size, book-to-market, momentum

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The role of conditioning information

• Theory is usually stated in terms of conditional
  expectations but we dont know what is in
  investors information sets
• What is the relationship between unconditional
  minimum variance and conditional minimum
  variance?
• If a portfolio is unconditionally minimum
  variance then it must be conditionally minimum
  variance, but if it is conditionally minimum
  variance that does not imply that it is
  unconditionally minimum variance

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Linear beta pricing models

• Suppose the return generating process for each of
  N assets follows a K-factor linear model
• Rit ai ftßi uit
• Euitft 0 , i1,,N
• where ßi is the Kx1 vector of betas for asset i
• ßi ?f-1 E(Rit E(Rit))(ft E(ft)) and
• ?f E(ft-E(ft))(ft-E(ft))

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Linear beta pricing models

• Then the linear beta pricing model says
• E(Rit) ?0 ?ßi
• where ?0 , ? are constants
• ?j is the risk premium for factor j (the
  expected return on a portfolio of the N assets
  which has beta of 1 on factor j and beta of 0 for
  all other factors)

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Estimating testing linear beta pricing model

• We can estimate and test the model using maximum
  likelihood methods if we make distributional
  assumptions for returns and factors
• Collect the N returns and K factors
• Rt a Bft ut where a µR - BµF
• and suppose that ut is iid multivariate normal
  N(0, ?u) conditional on contemporaneous and past
  values of the factors

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Maximum likelihood estimation

• The log likelihood of the unconstrained model is
• L -½NTln(2p) -½Tln(?u)-½?(Rt-a-Bft)?u-1(Rt-a
  -Bft)
• So the estimators are
• a µR Bµf
• B (1/T)?(RtµR)(ftµf)?f-1
• where µR (1/T) ?Rt µf (1/T) ?ft
• ?f (1/T)?(ftµf)(ftµf)
• ?u (1/T)?(Rt-a-Bft)(Rt-a-Bft)

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Maximum likelihood estimation

• For the constrained model a ?01B(?- µf)
• The constrained ML estimators are let ? ?- µf
• Bc (1/T)?(Rt?01)(ft?)(1/T)?
  (ft?)((ft?)-1
• ?uc (1/T)?(Rt- ?01-Bc(ft?)) (Rt-
  ?01-Bc(ft?))
• ? (?0 ?) Z?uc -1Z-1Z?uc-1(µR
  Bcµf)
• where Z 1 Bc
• The estimators can be computed by successive
  iteration

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Testing

• The asymptotic variance of ? is
• (1/T)1(?µf)?f-1(?µf)Z?uc-1Z-1
• The LRT statistic is
• JLR -T ln(?u) ln(?uc)
• We can recover the factor risk premiums ? from
  ? ? µf which has asymptotic variance
  (1/T)?f var(?)

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Traded factors

• When the factors are traded then
• Factors ft are returns on benchmark portfolios.
  Estimators for the unconstrained model are
  exactly the same (under this new interpretation
  of the factors)
• ? is the vector of expected returns on the
  benchmark portfolios in excess of the zero-beta
  rate
• So the restriction on a is ?01-?0B1
• ?0(1-B1)

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Traded factors - constrained model

• Estimators for the constrained model are
• Bc(1/T)?(Rt?01)(ft?01)(1/T)?(ft?01
  )(ft?01)-1
• ?uc (1/T)?(Rt- ?01-Bc(ft?01))(Rt-?01-Bc
  (ft?01))
• ?0 (1- Bc1)?uc -1(1- Bc1)-1(1-
  Bc1)?uc -1(µRBcµf)
• The asymptotic variance of ?0
• (1/T)1(µf-?01)?f-1(µf-?01)(1-Bc1)?uc-
  1(1-Bc1)-1
• We can recover the factor risk premiums ? from
• ? µf-?01 which has asymptotic variance
  (1/T)?f var(?0)11

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Simple case

• Suppose there is a riskfree asset with rate rf,
  and there is only one factor rM (the
  Sharpe-Lintner CAPM)