Equilibrium Asset Pricing

1. Equilibrium Asset Pricing Michael J. Brennan June 2008

2. Three standard assumptions • Market prices are efficient • Prices are set by rational expected utility maximizing individuals • Returns are serially independent

3. Three papers • Asset Pricing and Mispricing With Ashley Wang • Agency and Asset Pricing With Feifei Li • Work in progess

4. A. Market prices are efficient

5. Unconditional Rational Prices inconsistent with Unconditional Rational Expected Returns Unconditional rational prices with random mispricing: Proof:

6. 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:

7. A Simple Example • Perpetual bond with coupon $4 and market interest rate 4%: P* = 100 • Bond trades at 90, 100, 110

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

9. A More General Model Ignoring dividends:

10. Components of Return Bias, B = B1 + B2 • B2 > 0 Over-reaction • B2 < 0 Under-reaction • Assuming z is stationary

11. Empirical Analysis • Mispricing model • AR1: Kalman filter estimates • Data • NYSE/AMEX/Nasdaq stocks January 1962-Dec 2004

12. 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 + εt

19. Are returns related to our empirical estimate of ‘theoretical return bias’ B1 ?

20. 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

21. z ranges from 1.08% to 16.70% Annualized FF3 alphas and Bias EstimatesJanuary 1967 to December 2004 Difference (Hi-Lo) = 8.64% p.a. t-stat(Hi-Lo) = 3.25

22. 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

23. B. Prices are set by rational expected utility maximizing individuals

24. 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

25. 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’

26. 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. S&P500 (residual) • Hold for 1 year without rebalancing • Calculate alphas of linked returns • F-M analysis to track rewards to market and S&P500 (residual) betas

28. 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)

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

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

35. 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)

36. First order autocorrelations of 25 FF Size and B/M portfoliosJuly 1926- February 2006

37. Effect of autocorrelation on n-period expected returns • Annualized n month returns: Independent of n if returns iid

38. Standardized annualized returns on FF 25 portfolios as a function of the holding period, n

39. Expected returns vary with holding period Do betas also vary with holding period?

40. Betas of FF 25 portfolios as function of holding period

41. Standardized betas as a function of the holding period (months) for FF25 portfolios 1926-2006

42. 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!

43. Cross-section regressions for n month returns

44. Annualized lam_0 for different holding periods for FF 25 portfolios 1926-2006

45. Scaled Empirical Market Price of Risk as a function of holding period

46. RSQ from Cross Section Regression as function of holding period (months)

47. The 1 month CAPM

48. The 12 month CAPM

49. 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

50. 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