Equilibrium Asset Pricing

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Equilibrium Asset Pricing

• Michael J. Brennan
• June 2008

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Three standard assumptions

• Market prices are efficient
• Prices are set by rational expected utility
  maximizing individuals
• Returns are serially independent

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Three papers

• Asset Pricing and Mispricing
• With Ashley Wang
• Agency and Asset Pricing
• With Feifei Li
• Work in progess

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A. Market prices are efficient
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Unconditional Rational Prices inconsistent with
Unconditional Rational Expected Returns

• Unconditional rational prices with random
  mispricing
• Proof

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A Basic Result

• If
• mispricing is uncorrelated with fundamentals, and
• prices are unconditionally rational, then
• Expected returns exceed rational expected returns
  a mispricing return premium

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• A Simple Example
• Perpetual bond with coupon 4 and market
  interest rate 4 P 100
• Bond trades at 90, 100, 110

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• A Simple Example
• Annual Transition probabilities
• Steady state probabilities
• Expected Price 100 (Unconditional rational
  pricing)
• Expected rate of return 4.42

90 100 110
110 0.05 0.9 0.05
100 0.3 0.4 0.3
90 0.05 0.90 0.05
0.20 0.60 0.20
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A More General Model

• Ignoring dividends

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Components of Return Bias, B B1 B2

• B2 gt 0 Over-reaction
• B2 lt 0 Under-reaction
• Assuming z is stationary

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Empirical Analysis

• Mispricing model
• AR1 Kalman filter estimates
• Data
• NYSE/AMEX/Nasdaq stocks January 1962-Dec 2004

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AR1 Mispricing Estimates

• Each January from 1967 to 2004 KF used to
  estimate mispricing return bias from FF3
  residuals (e) over previous 60 months assuming
  AR1 model

Assumes mispricing uncorrelated with
fundamentals FF3 et
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Are returns related to our empirical estimate of
theoretical return bias B1 ?
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10 Portfolios formed in January of each year

• Based on estimates of B1
• Equally weighted and not rebalanced during year
• Estimated MRP of portfolios runs from 14bp to 6
  p.a.
• High bias portfolios
• Higher
• No difference in
• ?Firms with highest fundamental volatility have
  most mispricing

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Annualized FF3 alphas and Bias EstimatesJanuary
1967 to December 2004

• ?z ranges from 1.08 to 16.70

Difference (Hi-Lo) 8.64 p.a. t-stat(Hi-Lo)
3.25
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Conclusions

• A mean zero stochastic mispricing error can drive
  expected return away from fundamental return
• Lower
• For mispricing independent of fundamentals, more
  transient and volatile mispricing leads to bigger
  return premium
• Slow adjustment to information can potentially
  explain
• very high liquidity premium since illiquid stocks
  are those most subject to mispricing

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B. Prices are set by rational expected
utility maximizing individuals
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Agency and Asset Pricing

• CAPM with
• Individual mean-variance investors
• Agents
• also mean-variance but with respect to return
  relative to (individual) benchmark portfolio
• Equilibrium
• Two beta capm
• market beta positive risk premium
• (aggregate) benchmark beta negative risk
  premium

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• betas w.r.t market
  and benchmark residual
• Note the benchmark portfolio is riskless
  asset for agents
• different agents may have different benchmarks
  aggregate benchmark portfolio

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Empirical Analysis

• Form 25 value weighted portfolios in January each
  year from 1931 to 2006 based on
• CRSP value market weighted beta
• beta w.r.t. SP500 (residual)
• Hold for 1 year without rebalancing
• Calculate alphas of linked returns
• F-M analysis to track rewards to market and
  SP500 (residual) betas

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• The agency induced benchmark effect is
• Confined to large firms and shows up only in
  value weighted portfolios
• Correlation between proportional institutional
  ownership and log firm size is 0.63 (Gompers and
  Metrick, 2001)
• Confined to post 1970 period
• in recent years risk-adjusted measures of
  performance have been receiving considerable
  attention outside the academic journals.. Bank
  Administration Institute study of 1968..complete
  evaluation must include an assessment of
  risk.SEC Study of 1971 ..performance measures
  must be adjusted for volatility.. (Klemkosky,
  1973)

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The results (for value weighted portfolios) are
robust to measurement wrt FF 3-factor model
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Conclusion

• Significant agency/benchmark effect
• Starts from around 1970
• Only apparent for large firms
• Robust to FF 3-factor model

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C. Security Returns are iid

• One period expected return is sufficient
  statistic for n period expected return
• Risk should be measured using one period returns
• How long is period
• Instantaneous Merton (1971)
• One month (CRSP)

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First order autocorrelations of 25 FF Size and
B/M portfoliosJuly 1926- February 2006
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Effect of autocorrelation on n-period expected
returns

• Annualized n month returns
• Independent of n if returns iid

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Standardized annualized returns on FF 25
portfolios as a function of the holding period, n
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Expected returns vary with holding period Do
betas also vary with holding period?
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Betas of FF 25 portfolios as function of holding
period
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Standardized betas as a function of the holding
period (months) for FF25 portfolios 1926-2006
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The issue

• At what frequency (if any) do we expect CAPM to
  hold?
• High frequency if low transaction costs
• Low frequency if high transaction costs
• High and low frequency ??
• An empirical issue!

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Cross-section regressions for n month returns
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Annualized lam_0 for different holding periods
for FF 25 portfolios 1926-2006
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Scaled Empirical Market Price of Risk as a
function of holding period
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RSQ from Cross Section Regression as function of
holding period (months)
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The 1 month CAPM
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The 12 month CAPM
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Conclusion

• Single period of CAPM is arbitrary
• Returns are not iid
• Betas and expected returns both depend on holding
  period
• Fit of CAPM improves with assumed holding period

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Summary

• Random mispricing affects (risk-adjusted) average
  returns
• Average returns affected by agency/benchmark
  effects
• Returns not iid
• Expected returns and betas depend on holding
  period
• Fit of CAPM improves with assumed holding period