The conditional CAPM does not explain asset- pricing anomalies - - PowerPoint PPT Presentation

The conditional CAPM does not explain asset- pricing anomalies Jonathan Lewellen & Stefan Nagel HEC School of Management, March 17, 2005

2

Background Size, B/M, and momentum portfolios, 1964 – 2001 Monthly returns (%)

• Avg. returns

CAPM alphas Portfolio Size B/M R-1,-6 Size B/M R-1,-6 Low 0.71 0.41 0.17 0.07

• 0.20
• 0.41

2 0.74 0.58 0.51 0.16 0.03 0.04 3 0.70 0.66 0.43 0.19 0.17

• 0.01

4 0.69 0.80 0.52 0.21 0.35 0.08 High 0.50 0.88 0.79 0.11 0.39 0.29 Long–short 0.21 0.47 0.61

• 0.03

0.59 0.70 t-stat 0.91 2.98 2.76

• 0.16

4.01 3.14

3

Background Explained by the conditional CAPM w/ time-varying betas? Theory Jensen (1968) Dybvig and Ross (1985) Hansen and Richard (1987) Application to size, B/M, and momentum Zhang (2002) Jagannathan and Wang (1996) Lettau and Ludvigson (2001) Petkova and Zhang (2004) Lustig and Van Nieuwerburgh (2004) Santos and Veronesi (2004) Franzoni (2004), Adrian and Franzoni (2004) Wang (2003)

4

Rolling betas of value stocks, 1930 – 2000

Franzoni (2004)

5

Background Explained by the conditional CAPM w/ time-varying betas? Theory Jensen (1968) Dybvig and Ross (1985) Hansen and Richard (1987) Application to size, B/M, and momentum Zhang (2002) Jagannathan and Wang (1996) Lettau and Ludvigson (2001) Petkova and Zhang (2004) Lustig and Van Nieuwerburgh (2004) Santos and Veronesi (2004) Franzoni (2004), Adrian and Franzoni (2004) Wang (2003)

6

Background Conditional CAPM Rit = αt + βt RMt + εt αt = 0 Empirical tests with constant β Rit = α + β RMt + εt α ≠ 0

7

Intuition 1 Alternate between efficient portfolios A and B

B A Dynamic strategy .5 A + .5 B

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

8

Intuition 2 Rt = βt RMt + εt, βt = β + ηt, γt = Et-1[RMt], ρβ,γ > 0

E[Ri | RM]

• 0.10
• 0.08
• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08 0.10 0.12

• 0.08
• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08

RM

True

• Uncond. regression

9

Overview Perspective on conditional asset-pricing tests A simple empirical test

10

Overview Perspective on conditional asset-pricing tests A simple empirical test Time-variation in betas / expected returns is too small to explain anomalies

11

Theory Excess returns: Rit, RMt No restriction on joint distribution of returns Notation γt = Et-1[RMt],

2 t

σ = vart-1(RMt), βt = covt-1(Rit, RMt) /

2 t

σ γ = E[RMt],

2 M

σ = var(RMt), βu = cov(Rit, RMt) /

2 M

σ β = E[βt]

12

Theory If conditional CAPM holds, what is αu ≡ E[Rit] – βu γ? Et-1[Rit] = βt γt

13

Theory If conditional CAPM holds, what is αu ≡ E[Rit] – βu γ? Et-1[Rit] = βt γt E[Rit] = β γ + cov(βt, γt)

14

Theory If conditional CAPM holds, what is αu ≡ E[Rit] – βu γ? Et-1[Rit] = βt γt E[Rit] = β γ + cov(βt, γt) ⇒ αu = γ(β – βu) + cov(βt, γt)

15

Theory If conditional CAPM holds, what is αu ≡ E[Rit] – βu γ? Et-1[Rit] = βt γt E[Rit] = β γ + cov(βt, γt) ⇒ αu = γ(β – βu) + cov(βt, γt) Conditional beta βu = β + ) , cov( 1 ] ) ( , cov[ 1 ) , cov(

2 t t 2 M 2 t t 2 M t t 2 M

σ β σ + γ − γ β σ + γ β σ γ

16

Theory If conditional CAPM holds, what is αu ≡ E[Rit] – βu γ? Et-1[Rit] = βt γt E[Rit] = β γ + cov(βt, γt) ⇒ αu = γ(β – βu) + cov(βt, γt) Conditional beta βu = β + ) , cov( 1 ] ) ( , cov[ 1 ) , cov(

2 t t 2 M 2 t t 2 M t t 2 M

σ β σ + γ − γ β σ + γ β σ γ Convexity Cubic Volatility

17

Theory If conditional CAPM holds, what is αu ≡ E[Rit] – βu γ? Et-1[Rit] = βt γt E[Rit] = β γ + cov(βt, γt) ⇒ αu = γ(β – βu) + cov(βt, γt) Conditional beta βu = β + ) , cov( 1 ] ) ( , cov[ 1 ) , cov(

2 t t 2 M 2 t t 2 M t t 2 M

σ β σ + γ − γ β σ + γ β σ γ Conditional alpha αu = ) , cov( ] ) ( , cov[ ) , cov( 1

2 t t 2 M 2 t t 2 M t t 2 M 2

σ β σ γ − γ − γ β σ γ − γ β       σ γ −

18

Magnitude αu = ) , cov( ] ) ( , cov[ ) , cov( 1

2 t t 2 M 2 t t 2 M t t 2 M 2

σ β σ γ − γ − γ β σ γ − γ β       σ γ −

19

Magnitude αu = ) , cov( ] ) ( , cov[ ) , cov( 1

2 t t 2 M 2 t t 2 M t t 2 M 2

σ β σ γ − γ − γ β σ γ − γ β       σ γ −

• γ2 /

2 M

σ ? 1964 – 2001: γ = 0.47%, σM = 4.5% ⇒ γ2 /

2 M

σ = 0.011

20

Magnitude αu = ) , cov( ] ) ( , cov[ ) , cov( 1

2 t t 2 M 2 t t 2 M t t 2 M 2

σ β σ γ − γ − γ β σ γ − γ β       σ γ −

• γ2 /

2 M

σ ? 1964 – 2001: γ = 0.47%, σM = 4.5% ⇒ γ2 /

2 M

σ = 0.011

• (γt – γ)2 ?

Suppose γ ≈ 0.5% and 0.0% < γt < 1.0%. Then (γt – γ)2 is at most 0.0052 = 0.000025.

21

Magnitude αu = ) , cov( ] ) ( , cov[ ) , cov( 1

2 t t 2 M 2 t t 2 M t t 2 M 2

σ β σ γ − γ − γ β σ γ − γ β       σ γ −

• γ2 /

2 M

σ ? 1964 – 2001: γ = 0.47%, σM = 4.5% ⇒ γ2 /

2 M

σ = 0.011

• (γt – γ)2 ?

Suppose γ ≈ 0.5% and 0.0% < γt < 1.0%. Then (γt – γ)2 is at most 0.0052 = 0.000025. αu ≈ ) , cov( ) , cov(

t t M t t 2 2

σ β σ γ − γ β

22

1: Constant volatility αu ≈ cov(βt, γt) = ρ σβ σγ

ρ = 0.6 σβ ρ = 1.0 σβ 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σγ = 0.1 σγ = 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5

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1: Constant volatility αu ≈ cov(βt, γt) = ρ σβ σγ

ρ = 0.6 σβ ρ = 1.0 σβ 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σγ = 0.1 σγ = 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5

Economically large Evidence later Fama and French (1992, 1997)

24

1: Constant volatility αu ≈ cov(βt, γt) = ρ σβ σγ

ρ = 0.6 σβ ρ = 1.0 σβ 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σγ = 0.1 σγ = 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5

Economically large Evidence from predictive regressions Campbell and Cochrane (1999)

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1: Constant volatility αu ≈ cov(βt, γt) = ρ σβ σγ

ρ = 0.6 σβ ρ = 1.0 σβ 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σγ = 0.1 σγ = 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5

Arbitrary

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1: Constant volatility αu ≈ cov(βt, γt) = ρ σβ σγ

ρ = 0.6 σβ ρ = 1.0 σβ 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σγ = 0.1 0.02 0.03 0.04 σγ = 0.1 0.03 0.05 0.07 0.2 0.04 0.06 0.08 0.2 0.06 0.10 0.14 0.3 0.05 0.09 0.12 0.3 0.09 0.15 0.21 0.4 0.07 0.12 0.17 0.4 0.12 0.20 0.28 0.5 0.09 0.15 0.21 0.5 0.15 0.25 0.35

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1: Constant volatility αu ≈ cov(βt, γt) = ρ σβ σγ

ρ = 0.6 σβ ρ = 1.0 σβ 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σγ = 0.1 0.02 0.03 0.04 σγ = 0.1 0.03 0.05 0.07 0.2 0.04 0.06 0.08 0.2 0.06 0.10 0.14 0.3 0.05 0.09 0.12 0.3 0.09 0.15 0.21 0.4 0.07 0.12 0.17 0.4 0.12 0.20 0.28 0.5 0.09 0.15 0.21 0.5 0.15 0.25 0.35

B/M portfolio: 0.59% Momentum portfolio: 1.01%

28

1: Constant volatility βt ~ N[1.0, 0.7], γt ~ N[0.5%, 0.5%], ρ = 1.0

E[Ri | RM]

• 0.10
• 0.08
• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08 0.10

• 0.08
• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08

RM

True

• Uncond. regression

29

2: Time-varying volatility αu ≈ ) , cov( ) , cov(

2 t t 2 M t t

σ β σ γ − γ β Effects of time-varying γt and

2 t

σ offset (if they move together)

30

2: Time-varying volatility αu ≈ ) , cov( ) , cov(

2 t t 2 M t t

σ β σ γ − γ β Effects of time-varying γt and

2 t

σ offset (if they move together) Merton (1980): γt = λ

2 t

σ ) , cov(

t t 2 M 2 u

γ β       σ σ ≈ α

γ

< cov(βt, γt)

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2: Time-varying volatility αu ≈ ) , cov(

2 t t 2 M

σ β σ γ − = – γ ρ σβ σv (where vt =

2 t

σ /

2 M

σ )

ρ = 0.2 σβ ρ = 0.5 σβ 0.3 0.5 0.7 0.3 0.5 0.7 Alpha (%) Alpha (%) σv = 1.0

• 0.03
• 0.05
• 0.07

σv = 1.0

• 0.06
• 0.10
• 0.14

1.3

• 0.04
• 0.07
• 0.09

1.3

• 0.08
• 0.13
• 0.18

1.6

• 0.05
• 0.08
• 0.11

1.6

• 0.10
• 0.16
• 0.22

1.9

• 0.06
• 0.10
• 0.13

1.9

• 0.11
• 0.19
• 0.27

2.2

• 0.07
• 0.11
• 0.15

2.2

• 0.13
• 0.22
• 0.31

γ = 0.50

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Testing the conditional CAPM Traditional tests Rit = αit + βit RMt + εit βit = bi0 + bi1 Z1,t-1 + bi2 Z2,t-1 + …

33

Testing the conditional CAPM Traditional tests Rit = αit + βit RMt + εit βit = bi0 + bi1 Z1,t-1 + bi2 Z2,t-1 + … Cochrane (2001) “Models such as the CAPM imply a conditional linear factor model with respect to investors’ information sets. The best we can hope to do is test implications conditioned on variables that we observe. Thus, a conditional factor model is not testable!”

34

Our tests Rit = αit + βit RMt + εit Short-window regressions

• Estimate αit, βit every month, quarter, half-year, or year
• Are conditional alphas zero?

35

Our tests Rit = αit + βit RMt + εit Short-window regressions

• Estimate αit, βit every month, quarter, half-year, or year
• Are conditional alphas zero?
• How volatile are betas?
• Do betas covary with the equity premium and variance?

36

Our tests Rit = αit + βit RMt + εit Short-window regressions

• Estimate αit, βit every month, quarter, half-year, or year
• Are conditional alphas zero?
• How volatile are betas?
• Do betas covary with the equity premium and variance?

37

Our tests Short-window regressions – betas

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 61 121 181 241 301 361 421 481 541

Days

38

Our tests Rit = αit + βit RMt + εit Short-window regressions

• Estimate αit, βit every month, quarter, half-year, or year
• Are conditional alphas zero?

39

Our tests Rit = αit + βit RMt + εit Short-window regressions

• Estimate αit, βit every month, quarter, half-year, or year
• Are conditional alphas zero?
• Assumes only that beta is relatively slow moving

40

Our tests Rit = αit + βit RMt + εit Short-window regressions

• Estimate αit, βit every month, quarter, half-year, or year
• Are conditional alphas zero?
• Assumes only that beta is relatively slow moving
• Don’t need precise estimates of individual αit, βit

41

Our tests Rit = αit + βit RMt + εit Short-window regressions

• Estimate αit, βit every month, quarter, half-year, or year
• Are conditional alphas zero?
• Assumes only that beta is relatively slow moving
• Don’t need precise estimates of individual αit, βit
• Microstructure issues

42

Microstructure issue 1 Horizon effects (compounding) Daily alphas, betas ≠ monthly alphas, betas

43

Microstructure issue 1 Horizon effects (compounding) Daily alphas, betas ≠ monthly alphas, betas

N 2 M N 2 M N M N i N M i i

] R 1 [ E ] ) R 1 [( E ] R 1 [ E ] R 1 [ E )] R 1 )( R 1 [( E ) N ( + − + + + − + + = β

1.49 1.50 1.51 1.52

1 6 11 16 21 26 31 36 41 46 51 56 61

Days (N)

44

Microstructure issue 2 Thin trading / nonsynchronous prices Daily / weekly estimates of beta miss full covariance

45

Microstructure issue 2 Beta estimates, horizons from 1 to 45 days, 1964 – 2001

Small stocks Value stocks

0.6 0.8 1.0 1.2 1.4

1 5 9 13 17 21 25 29 33 37 41 45

Horizon (days)

46

Microstructure issue 2 Partial solution Use value-weighted portfolios and NYSE / Amex stocks Dimson (1979) betas: Ri,t = αi + βi0 RM,t + βi1 RM,t-1 + … + βik RM,t-k + εi,t βi = βi0 + βi1 + … + βik

47

Microstructure issue 2 Beta estimates

• Daily betas

Ri,t = αi + βi0 RM,t + βi1 RM,t-1 + βi2 [(RM,t-2 + RM,t-3 + RM,t-4)/3] + εi,t

• Weekly betas

Ri,t = αi + βi0 RM,t + βi1 RM,t-1 + βi2 RM,t-2 + εi,t

• Monthly betas

Ri,t = αi + βi0 RM,t + βi1 RM,t-1 + εi,t

48

Data NYSE / Amex stocks, 1964 – 2001 VW portfolios 25 size-B/M portfolios (S, B, V, G) 10 momentum portfolios, 6-month returns (W, L)

49

Summary statistics, 1964 – 2001 Monthly, %

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Excess returns Avg. Day 0.57 0.49 0.08 0.32 0.81 0.49

• 0.10

0.87 0.97 Wk 0.63 0.50 0.13 0.37 0.84 0.47

• 0.04

0.91 0.95 Mon 0.71 0.50 0.21 0.41 0.88 0.47 0.01 0.91 0.90 Std err. Day 0.28 0.20 0.19 0.27 0.23 0.13 0.33 0.28 0.26 Wk 0.26 0.18 0.18 0.26 0.22 0.12 0.30 0.26 0.25 Mon 0.34 0.19 0.23 0.30 0.26 0.16 0.35 0.28 0.27

50

Summary statistics, 1964 – 2001

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Unconditional alphas Est. Day 0.09 0.10

• 0.01
• 0.21

0.39 0.60

• 0.64

0.35 0.99 Wk 0.05 0.10

• 0.05
• 0.22

0.37 0.59

• 0.66

0.37 1.03 Mon 0.07 0.11

• 0.03
• 0.20

0.39 0.59

• 0.63

0.38 1.01 Std err. Day 0.15 0.06 0.17 0.10 0.12 0.12 0.18 0.13 0.26 Wk 0.14 0.06 0.16 0.09 0.11 0.11 0.17 0.12 0.25 Mon 0.18 0.07 0.20 0.11 0.13 0.14 0.19 0.13 0.28 Unconditional betas Est. Day 1.07 0.87 0.20 1.18 0.94 -0.25 1.22 1.17

• 0.06

Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16

• 0.17

Mon 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14

• 0.22

Std err. Day 0.03 0.01 0.03 0.02 0.03 0.02 0.03 0.02 0.05 Wk 0.03 0.01 0.04 0.02 0.03 0.03 0.04 0.03 0.06 Mon 0.05 0.02 0.06 0.03 0.04 0.04 0.06 0.04 0.08

51

Test Are conditional alphas zero?

52

Test Are conditional alphas zero? Tests based on the time series of short-window αit Fama-MacBeth approach

53

Test Are conditional alphas zero? Tests based on the time series of short-window αit Fama-MacBeth approach Four versions of the short-window regressions Quarterly (daily returns) Semiannually (daily and weekly returns) Annually (monthly returns)

54

Conditional CAPM, 1964 – 2001

Conditional vs. unconditional alphas (%)

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Unconditional alphas Day 0.09 0.10

• 0.01
• 0.21

0.39 0.60

• 0.64

0.35 0.99 Wk 0.05 0.10

• 0.05
• 0.22

0.37 0.59

• 0.66

0.37 1.03 Month 0.07 0.11

• 0.03
• 0.20

0.39 0.59

• 0.63

0.38 1.01 Average conditional alpha Quarterly 0.42 0.00 0.42

• 0.01

0.49 0.50

• 0.79

0.55 1.33 Semi 1 0.26 0.00 0.26

• 0.08

0.40 0.47

• 0.61

0.39 0.99 Semi 2 0.16 0.01 0.15

• 0.12

0.36 0.48

• 0.83

0.53 1.37 Annual

• 0.06

0.08

• 0.14
• 0.20

0.32 0.53

• 0.56

0.21 0.77

55

Conditional CAPM, 1964 – 2001

Conditional vs. unconditional alphas (%)

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Unconditional alphas Day 0.09 0.10

• 0.01
• 0.21

0.39 0.60

• 0.64

0.35 0.99 Wk 0.05 0.10

• 0.05
• 0.22

0.37 0.59

• 0.66

0.37 1.03 Month 0.07 0.11

• 0.03
• 0.20

0.39 0.59

• 0.63

0.38 1.01 Average conditional alpha Quarterly 0.42 0.00 0.42

• 0.01

0.49 0.50

• 0.79

0.55 1.33 Semi 1 0.26 0.00 0.26

• 0.08

0.40 0.47

• 0.61

0.39 0.99 Semi 2 0.16 0.01 0.15

• 0.12

0.36 0.48

• 0.83

0.53 1.37 Annual

• 0.06

0.08

• 0.14
• 0.20

0.32 0.53

• 0.56

0.21 0.77

56

Conditional CAPM, 1964 – 2001

Conditional alphas and standard errors

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Estimate Quarterly 0.42 0.00 0.42

• 0.01

0.49 0.50

• 0.79

0.55 1.33 Semi 1 0.26 0.00 0.26

• 0.08

0.40 0.47

• 0.61

0.39 0.99 Semi 2 0.16 0.01 0.15

• 0.12

0.36 0.48

• 0.83

0.53 1.37 Annual

• 0.06

0.08

• 0.14
• 0.20

0.32 0.53

• 0.56

0.21 0.77 Standard error Quarterly 0.20 0.06 0.22 0.12 0.14 0.14 0.20 0.13 0.26 Semi 1 0.21 0.06 0.23 0.12 0.14 0.15 0.19 0.14 0.25 Semi 2 0.21 0.06 0.23 0.14 0.15 0.16 0.20 0.15 0.27 Annual 0.26 0.07 0.29 0.16 0.17 0.14 0.21 0.17 0.29

57

Exploring the results Time-varying betas have a small impact on alphas Why? How volatile are betas? Do betas covary with business conditions? Do betas covary with γt and

2 t

σ ?

58

Conditional betas (semiannual, daily returns), 1964 – 2001 Small minus Big

• 0.9
• 0.6
• 0.3

0.0 0.3 0.6 0.9 1.2

1964.2 1971.2 1978.2 1985.2 1992.2 1999.2

59

Conditional betas (semiannual, daily returns), 1964 – 2001 Value minus Growth

• 1.1
• 0.9
• 0.7
• 0.4
• 0.2

0.0 0.2 0.4 0.7

1964.2 1971.2 1978.2 1985.2 1992.2 1999.2

60

Conditional betas (semiannual, daily returns), 1964 – 2001 Winner minus Losers

• 1.6
• 1.1
• 0.5

0.0 0.5 1.1 1.6 2.2

1964.2 1971.2 1978.2 1985.2 1992.2 1999.2

61

Conditional betas, 1964 – 2001

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Unconditional betas Day 1.07 0.87 0.20 1.18 0.94 -0.25 1.22 1.17

• 0.06

Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16

• 0.17

Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14

• 0.22

Average conditional betas Quarterly 1.03 0.93 0.10 1.17 0.98 -0.19 1.19 1.24 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99 -0.20 1.20 1.24 0.05 Semi 2 1.23 0.91 0.32 1.25 1.06 -0.19 1.33 1.19

• 0.14

Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24

• 0.14

Implied std deviation of true betas Quarterly 0.32 0.13 0.33 0.19 0.28 0.25 0.36 0.33 0.63 Semi 1 0.29 0.12 0.30 0.18 0.28 0.24 0.30 0.30 0.55 Semi 2 0.31 0.10 0.32 0.16 0.31 0.29 0.36 0.32 0.62 Annual 0.35

• 0.25

0.04 0.37 0.19 0.19 0.29 0.52

62

Conditional betas, 1964 – 2001

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Unconditional betas Day 1.07 0.87 0.20 1.18 0.94 -0.25 1.22 1.17

• 0.06

Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16

• 0.17

Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14

• 0.22

Average conditional betas Quarterly 1.03 0.93 0.10 1.17 0.98 -0.19 1.19 1.24 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99 -0.20 1.20 1.24 0.05 Semi 2 1.23 0.91 0.32 1.25 1.06 -0.19 1.33 1.19

• 0.14

Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24

• 0.14

Implied std deviation of true betas Quarterly 0.32 0.13 0.33 0.19 0.28 0.25 0.36 0.33 0.63 Semi 1 0.29 0.12 0.30 0.18 0.28 0.24 0.30 0.30 0.55 Semi 2 0.31 0.10 0.32 0.16 0.31 0.29 0.36 0.32 0.62 Annual 0.35

• 0.25

0.04 0.37 0.19 0.19 0.29 0.52

63

Conditional betas, 1964 – 2001

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Unconditional betas Day 1.07 0.87 0.20 1.18 0.94 -0.25 1.22 1.17

• 0.06

Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16

• 0.17

Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14

• 0.22

Average conditional betas Quarterly 1.03 0.93 0.10 1.17 0.98 -0.19 1.19 1.24 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99 -0.20 1.20 1.24 0.05 Semi 2 1.23 0.91 0.32 1.25 1.06 -0.19 1.33 1.19

• 0.14

Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24

• 0.14

Implied std deviation of true betas Quarterly 0.32 0.13 0.33 0.19 0.28 0.25 0.36 0.33 0.63 Semi 1 0.29 0.12 0.30 0.18 0.28 0.24 0.30 0.30 0.55 Semi 2 0.31 0.10 0.32 0.16 0.31 0.29 0.36 0.32 0.62 Annual 0.35

• 0.25

0.04 0.37 0.19 0.19 0.29 0.52

64

Conditional betas, 1964 – 2001

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Unconditional betas Day 1.07 0.87 0.20 1.18 0.94 -0.25 1.22 1.17

• 0.06

Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16

• 0.17

Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14

• 0.22

Average conditional betas Quarterly 1.03 0.93 0.10 1.17 0.98 -0.19 1.19 1.24 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99 -0.20 1.20 1.24 0.05 Semi 2 1.23 0.91 0.32 1.25 1.06 -0.19 1.33 1.19

• 0.14

Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24

• 0.14

Implied std deviation of true betas Quarterly 0.32 0.13 0.33 0.19 0.28 0.25 0.36 0.33 0.63 Semi 1 0.29 0.12 0.30 0.18 0.28 0.24 0.30 0.30 0.55 Semi 2 0.31 0.10 0.32 0.16 0.31 0.29 0.36 0.32 0.62 Annual 0.35

• 0.25

0.04 0.37 0.19 0.19 0.29 0.52

bt = βt + et → var(βt) = var(bt) – var(et)

65

Test Do betas covary with business conditions? Do betas covary with γt and

2 t

σ ?

66

Test Do betas covary with business conditions? Market returns (6 months) Tbill rate Dividend yield Term premium CAY Lagged beta

67

Conditional betas, 1964 – 2001

Correlation between betas and state variables

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L βt-1 0.55 0.68 0.43 0.58 0.67 0.51 0.30 0.45 0.37 RM,-6

• 0.05
• 0.01
• 0.05
• 0.18

0.00 0.14

• 0.53

0.47 0.56 TBILL

• 0.04

0.11

• 0.08

0.15

• 0.12
• 0.25

0.14

• 0.25
• 0.21

DY 0.22 0.64

• 0.04

0.37 0.40 0.18 0.13

• 0.12
• 0.14

TERM

• 0.20

0.19

• 0.27
• 0.12

0.01 0.10

• 0.01
• 0.08
• 0.04

CAY

• 0.12

0.50

• 0.31
• 0.01

0.17 0.20 0.09

• 0.09
• 0.10
• Std. error ≈ 0.116 if no autocorrelation

68

Predicting conditional betas, 1964 – 2001

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Slope estimate βt-1 0.12 0.05 0.11 0.10 0.12 0.08 0.10 0.15 0.22 RM,-6 0.05

• 0.01

0.04 0.02 0.04 0.04

• 0.19

0.20 0.39 TBILL

• 0.13
• 0.02
• 0.11
• 0.03
• 0.14
• 0.13

0.09

• 0.14
• 0.24

DY 0.14 0.05 0.09 0.06 0.16 0.10

• 0.07

0.11 0.19 TERM

• 0.10

0.00

• 0.10
• 0.02
• 0.08
• 0.07

0.07

• 0.11
• 0.19

CAY

• 0.05

0.02

• 0.08
• 0.03
• 0.01

0.03 0.00

• 0.01
• 0.01

t-statistic βt-1 3.53 3.99 2.83 4.24 3.88 2.62 3.03 5.31 4.49 RM,-6 1.52

• 0.45

1.17 0.73 1.58 1.41

• 5.63

7.25 7.63 TBILL

• 2.56
• 1.39
• 2.09
• 1.06
• 3.19
• 2.98

1.79

• 3.41
• 3.22

DY 2.82 3.05 1.74 2.10 3.64 2.65

• 1.50

2.87 2.65 TERM

• 2.40
• 0.25
• 2.21
• 0.81
• 2.40
• 1.99

1.60

• 3.07
• 2.81

CAY

• 1.32

1.86

• 1.81
• 1.34
• 0.17

0.98 0.07

• 0.22
• 0.13

Adj R2 0.37 0.60 0.26 0.34 0.52 0.32 0.35 0.56 0.53

69

Predicting conditional betas, 1964 – 2001

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Slope estimate βt-1 0.12 0.05 0.11 0.10 0.12 0.08 0.10 0.15 0.22 RM,-6 0.05

• 0.01

0.04 0.02 0.04 0.04

• 0.19

0.20 0.39 TBILL

• 0.13
• 0.02
• 0.11
• 0.03
• 0.14
• 0.13

0.09

• 0.14
• 0.24

DY 0.14 0.05 0.09 0.06 0.16 0.10

• 0.07

0.11 0.19 TERM

• 0.10

0.00

• 0.10
• 0.02
• 0.08
• 0.07

0.07

• 0.11
• 0.19

CAY

• 0.05

0.02

• 0.08
• 0.03
• 0.01

0.03 0.00

• 0.01
• 0.01

t-statistic βt-1 3.53 3.99 2.83 4.24 3.88 2.62 3.03 5.31 4.49 RM,-6 1.52

• 0.45

1.17 0.73 1.58 1.41

• 5.63

7.25 7.63 TBILL

• 2.56
• 1.39
• 2.09
• 1.06
• 3.19
• 2.98

1.79

• 3.41
• 3.22

DY 2.82 3.05 1.74 2.10 3.64 2.65

• 1.50

2.87 2.65 TERM

• 2.40
• 0.25
• 2.21
• 0.81
• 2.40
• 1.99

1.60

• 3.07
• 2.81

CAY

• 1.32

1.86

• 1.81
• 1.34
• 0.17

0.98 0.07

• 0.22
• 0.13

Adj R2 0.37 0.60 0.26 0.34 0.52 0.32 0.35 0.56 0.53

70

Test Does beta covary with γt? What is the implied alpha αu ≈ cov(βt, γt)? Two estimates (i) cov(bt, RMt) = cov(βt + et, γt + st) = cov(βt, γt) (ii) cov(

* t

b , RMt) = cov(

* t

b , γt)

71

Beta and the market risk premium, 1964 – 2001

Covariance between estimated betas and market returns Implied αu (%)

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Estimate Quarterly

• 0.32

0.07

• 0.39
• 0.20
• 0.12

0.09 0.16

• 0.23
• 0.38

Semi 1

• 0.17

0.07

• 0.24
• 0.14
• 0.03

0.11

• 0.03
• 0.07
• 0.04

Semi 2

• 0.12

0.07

• 0.19
• 0.10
• 0.03

0.07 0.15

• 0.18
• 0.33

Annual 0.06 0.03 0.03

• 0.03

0.01 0.04

• 0.08

0.11 0.20 Standard error Quarterly 0.08 0.03 0.08 0.05 0.07 0.06 0.09 0.08 0.16 Semi 1 0.07 0.03 0.07 0.04 0.07 0.06 0.08 0.07 0.13 Semi 2 0.08 0.03 0.08 0.04 0.08 0.07 0.10 0.08 0.15 Annual 0.12 0.03 0.13 0.06 0.10 0.09 0.12 0.10 0.19

72

Beta and the market risk premium, 1964 – 2001

Covariance between predicted betas and market returns Implied αu (%)

Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Estimate Quarterly

• 0.06

0.04

• 0.09
• 0.01
• 0.02

0.02 0.06

• 0.05
• 0.12

Semi 1

• 0.07

0.03

• 0.10
• 0.02
• 0.02

0.01 0.05

• 0.07
• 0.12

Semi 2

• 0.04

0.02

• 0.05

0.00

• 0.01

0.00 0.07

• 0.08
• 0.14

Annual 0.03 0.01 0.02 0.00 0.01 0.03 0.05

• 0.03
• 0.08

Standard error Quarterly 0.04 0.02 0.04 0.03 0.05 0.04 0.05 0.06 0.10 Semi 1 0.05 0.02 0.04 0.03 0.05 0.04 0.05 0.06 0.10 Semi 2 0.04 0.02 0.03 0.02 0.05 0.04 0.06 0.05 0.10 Annual 0.05 0.02 0.05 0.03 0.06 0.04 0.06 0.05 0.09

73

Final comments Consumption CAPM Other studies Jagannathan and Wang (1996) Lettau and Ludvigson (2001) Santos and Veronesi (2004) Lustig and Van Nieuwerburgh (2004)

74

Other studies Approach Et-1[Rt] = βt γt ⇒ E[R] = β γ + cov(βt, γt)

75

Other studies Approach Et-1[Rt] = βt γt ⇒ E[R] = β γ + cov(βt, γt) Fama-MacBeth regressions: E[R] = θ0 + θ1 β + θ2 cov(βt, γt)

76

Other studies Approach Et-1[Rt] = βt γt ⇒ E[R] = β γ + cov(βt, γt) Fama-MacBeth regressions: E[R] = θ0 + θ1 β + θ2 cov(βt, γt) Restrictions on θ0, θ1, and θ2 are ignored Estimates of θ2 seem to be much larger than 1

77

Other studies Approach Et-1[Rt] = βt γt ⇒ E[R] = β γ + cov(βt, γt) Fama-MacBeth regressions: E[R] = θ0 + θ1 β + θ2 cov(βt, γt) Restrictions on θ0, θ1, and θ2 are ignored Estimates of θ2 seem to be much larger than 1 Cross-sectional R2s, with restrictions, aren’t meaningful Easy to find high R2s using size-B/M portfolios Simulations 90% confidence interval = [0.12, 0.72]

78

Summary Conditioning relatively unimportant for asset-pricing tests, both in principle and in practice Betas vary significantly over time Conditional alphas are close to unconditional alphas