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. Asset Pricing and Mispricing
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% 90
100
110
0.05
0.9
0.3
0.4
0.90
0.20
0.60

9. A More General Model
Ignoring dividends:
Neglecting 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 Data AR1: Kalman filter estimates
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
Portfolio NCV: median size; lower than median beta_HML; close to median beta_SMB and beta_MKT; close to median raw returns and FF3 alphas

21. Annualized FF3 alphas and Bias Estimates January 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

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 complete evaluation must include an assessment of risk….SEC Study of 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 portfolios July 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