SLIDE 1
The conditional CAPM does not explain asset- pricing anomalies Jonathan Lewellen & Stefan Nagel HEC School of Management, March 17, 2005
SLIDE 2 2
Background Size, B/M, and momentum portfolios, 1964 – 2001 Monthly 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
2 0.74 0.58 0.51 0.16 0.03 0.04 3 0.70 0.66 0.43 0.19 0.17
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
0.59 0.70 t-stat 0.91 2.98 2.76
4.01 3.14
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3
Background Explained by the conditional CAPM w/ time-varying betas? Conditional CAPM Et-1[Rit] = βt γt Rit = αt + βt RMt + εt ⇒ αt = 0 Empirical tests w/ constant β E[Rit] ≠ β γ Rit = α + β RMt + εt ⇒ α ≠ 0
SLIDE 4
4
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)
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5
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
SLIDE 6 6
Intuition 2 Rt = βt RMt + εt, βt = β + ηt, γt = Et-1[RMt], ρβ,γ > 0
E[Ri | RM]
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00 0.02 0.04 0.06 0.08
RM
True
SLIDE 7
7
Rolling betas of value stocks, 1930 – 2000
Franzoni (2004)
SLIDE 8
8
Overview Conditional CAPM does not explain anomalies Analysis Perspective on conditional asset-pricing tests Simple empirical test Conditional CAPM performs nearly as poorly as unconditional CAPM
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9
Notation Excess returns: Rit, RMt Moments 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] No restriction on joint distribution of returns
SLIDE 10
10
Theory If conditional CAPM holds, what is αu ≡ E[Rit] – βu γ?
SLIDE 11
11
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)
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12
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
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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) ⇒ 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
SLIDE 14
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) 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 −
SLIDE 15
15
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 −
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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 −
2 M
σ ? 1964 – 2001: γ = 0.47%, σM = 4.5% ⇒ γ2 /
2 M
σ = 0.011
SLIDE 17 17
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 −
2 M
σ ? 1964 – 2001: γ = 0.47%, σM = 4.5% ⇒ γ2 /
2 M
σ = 0.011
Suppose γ ≈ 0.5% and 0.0% < γt < 1.0%. Then (γt – γ)2 is at most 0.0052 = 0.000025.
SLIDE 18 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 −
2 M
σ ? 1964 – 2001: γ = 0.47%, σM = 4.5% ⇒ γ2 /
2 M
σ = 0.011
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
SLIDE 19
19
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
SLIDE 20
20
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)
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21
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)
SLIDE 22
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
Arbitrary
SLIDE 23
23
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
SLIDE 24
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 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%
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1: Constant volatility βt ~ N[1.0, 0.7], γt ~ N[0.5%, 0.5%], ρ = 1.0
E[Ri | RM]
0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.02 0.04 0.06 0.08
RM
True
SLIDE 26
26
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)
SLIDE 27
27
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)
SLIDE 28 28
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
σv = 1.0
1.3
1.3
1.6
1.6
1.9
1.9
2.2
2.2
γ = 0.50
SLIDE 29
29
Testing the conditional CAPM Traditional tests Rit = αit + βit RMt + εit βit = bi0 + bi1 Z1,t-1 + bi2 Z2,t-1 + …
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30
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!”
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Our tests Rit = αit + βit RMt + εit Short-window regressions
- Estimate αit, βit every month, quarter, half-year, or year
- Are conditional alphas zero?
SLIDE 32
32
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
SLIDE 33 33
Our tests Rit = αit + βit RMt + εit Short-window regressions
- Estimate αit, βit every month, quarter, half-year, or year
- Are conditional alphas zero?
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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
SLIDE 35 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?
- Assumes only that beta is relatively slow moving
- Don’t need precise estimates of individual αit, βit
SLIDE 36 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?
- Assumes only that beta is relatively slow moving
- Don’t need precise estimates of individual αit, βit
- Microstructure issues
SLIDE 37
37
Microstructure issue 1 Horizon effects (compounding) Daily alphas, betas ≠ monthly alphas, betas
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38
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)
SLIDE 39
39
Microstructure issue 2 Nonsynchronous prices Daily / weekly estimates of beta miss full covariance
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40
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)
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41
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
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Microstructure issue 2 Beta estimates
Ri,t = αi + βi0 RM,t + βi1 RM,t-1 + βi2 [(RM,t-2 + RM,t-3 + RM,t-4)/3] + εi,t
Ri,t = αi + βi0 RM,t + βi1 RM,t-1 + βi2 RM,t-2 + εi,t
Ri,t = αi + βi0 RM,t + βi1 RM,t-1 + εi,t
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43
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)
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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
0.87 0.97 Wk 0.63 0.50 0.13 0.37 0.84 0.47
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
SLIDE 45 45
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
0.39 0.60
0.35 0.99 Wk 0.05 0.10
0.37 0.59
0.37 1.03 Mon 0.07 0.11
0.39 0.59
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
Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16
Mon 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14
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
SLIDE 46
46
Short-window regressions Tests Q1: Are conditional alphas zero? Q2: How volatile are betas? Q3: Do betas covary with the market risk premium and variance?
SLIDE 47
47
Test 1 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)
SLIDE 48 48
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
0.39 0.60
0.35 0.99 Wk 0.05 0.10
0.37 0.59
0.37 1.03 Month 0.07 0.11
0.39 0.59
0.38 1.01 Average conditional alpha Quarterly 0.42 0.00 0.42
0.49 0.50
0.55 1.33 Semi 1 0.26 0.00 0.26
0.40 0.47
0.39 0.99 Semi 2 0.16 0.01 0.15
0.36 0.48
0.53 1.37 Annual
0.08
0.32 0.53
0.21 0.77
SLIDE 49 49
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
0.39 0.60
0.35 0.99 Wk 0.05 0.10
0.37 0.59
0.37 1.03 Month 0.07 0.11
0.39 0.59
0.38 1.01 Average conditional alpha Quarterly 0.42 0.00 0.42
0.49 0.50
0.55 1.33 Semi 1 0.26 0.00 0.26
0.40 0.47
0.39 0.99 Semi 2 0.16 0.01 0.15
0.36 0.48
0.53 1.37 Annual
0.08
0.32 0.53
0.21 0.77
SLIDE 50 50
Conditional CAPM, 1964 – 2001
Statistical tests
Size B/M Momentum Small Big S-B Grwth Value V-G Losrs Winrs W-L Conditional alphas Quarterly 0.42 0.00 0.42
0.49 0.50
0.55 1.33 Semi 1 0.26 0.00 0.26
0.40 0.47
0.39 0.99 Semi 2 0.16 0.01 0.15
0.36 0.48
0.53 1.37 Annual
0.08
0.32 0.53
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
SLIDE 51
51
Conditional CAPM Why are conditional and unconditional alphas similar? αu ≈ ) , cov( ) , cov(
2 t t 2 M t t
σ β σ γ − γ β
SLIDE 52
52
Test 2 How volatile are betas? bt = βt + et ⇒ var(βt) = var(bt) – var(et)
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Conditional betas (semiannual, daily returns), 1964 – 2001 Small minus Big
0.0 0.3 0.6 0.9 1.2
1964.2 1971.2 1978.2 1985.2 1992.2 1999.2
SLIDE 54 54
Conditional betas (semiannual, daily returns), 1964 – 2001 Value minus Growth
0.0 0.2 0.4 0.7
1964.2 1971.2 1978.2 1985.2 1992.2 1999.2
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Conditional betas (semiannual, daily returns), 1964 – 2001 Winner minus Losers
0.0 0.5 1.1 1.6 2.2
1964.2 1971.2 1978.2 1985.2 1992.2 1999.2
SLIDE 56 56
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
Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16
Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14
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
Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24
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
0.04 0.37 0.19 0.19 0.29 0.52
SLIDE 57 57
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
Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16
Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14
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
Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24
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
0.04 0.37 0.19 0.19 0.29 0.52
SLIDE 58 58
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
Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16
Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14
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
Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24
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
0.04 0.37 0.19 0.19 0.29 0.52
SLIDE 59 59
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
Wk 1.25 0.86 0.39 1.27 1.03 -0.24 1.33 1.16
Month 1.34 0.83 0.51 1.30 1.05 -0.25 1.36 1.14
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
Annual 1.49 0.83 0.66 1.36 1.17 -0.19 1.38 1.24
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
0.04 0.37 0.19 0.19 0.29 0.52
SLIDE 60
60
Test 3 Do betas covary with business conditions? Do betas covary with γt and
2 t
σ ?
SLIDE 61
61
Test 3 Do betas covary with business conditions? Market returns (6 months) Tbill rate Dividend yield Term premium CAY Lagged beta
SLIDE 62 62
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
0.00 0.14
0.47 0.56 TBILL
0.11
0.15
0.14
DY 0.22 0.64
0.37 0.40 0.18 0.13
TERM
0.19
0.01 0.10
CAY
0.50
0.17 0.20 0.09
- 0.09
- 0.10
- Std. error ≈ 0.116 if no autocorrelation
SLIDE 63 63
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
0.04 0.02 0.04 0.04
0.20 0.39 TBILL
- 0.13
- 0.02
- 0.11
- 0.03
- 0.14
- 0.13
0.09
DY 0.14 0.05 0.09 0.06 0.16 0.10
0.11 0.19 TERM
0.00
0.07
CAY
0.02
0.03 0.00
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
1.17 0.73 1.58 1.41
7.25 7.63 TBILL
- 2.56
- 1.39
- 2.09
- 1.06
- 3.19
- 2.98
1.79
DY 2.82 3.05 1.74 2.10 3.64 2.65
2.87 2.65 TERM
- 2.40
- 0.25
- 2.21
- 0.81
- 2.40
- 1.99
1.60
CAY
1.86
0.98 0.07
Adj R2 0.37 0.60 0.26 0.34 0.52 0.32 0.35 0.56 0.53
SLIDE 64 64
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
0.04 0.02 0.04 0.04
0.20 0.39 TBILL
- 0.13
- 0.02
- 0.11
- 0.03
- 0.14
- 0.13
0.09
DY 0.14 0.05 0.09 0.06 0.16 0.10
0.11 0.19 TERM
0.00
0.07
CAY
0.02
0.03 0.00
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
1.17 0.73 1.58 1.41
7.25 7.63 TBILL
- 2.56
- 1.39
- 2.09
- 1.06
- 3.19
- 2.98
1.79
DY 2.82 3.05 1.74 2.10 3.64 2.65
2.87 2.65 TERM
- 2.40
- 0.25
- 2.21
- 0.81
- 2.40
- 1.99
1.60
CAY
1.86
0.98 0.07
Adj R2 0.37 0.60 0.26 0.34 0.52 0.32 0.35 0.56 0.53
SLIDE 65
65
Test 3 Do betas covary with γt? What is αu ≈ cov(βt, γt)? Two estimates (1) cov(bt, RMt) = cov(βt + et, γt + st) = cov(βt, γt) (2) cov(
* t
b , RMt) = cov(
* t
b , γt)
SLIDE 66 66
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
0.07
0.09 0.16
Semi 1
0.07
0.11
Semi 2
0.07
0.07 0.15
Annual 0.06 0.03 0.03
0.01 0.04
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
SLIDE 67 67
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
0.04
0.02 0.06
Semi 1
0.03
0.01 0.05
Semi 2
0.02
0.00
0.00 0.07
Annual 0.03 0.01 0.02 0.00 0.01 0.03 0.05
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
SLIDE 68
68
Final comments Consumption CAPM Other studies Jagannathan and Wang (1996) Lettau and Ludvigson (2001) Santos and Veronesi (2004) Lustig and Van Nieuwerburgh (2004)
SLIDE 69
69
Other studies Approach Et-1[Rt] = βt γt ⇒ E[R] = β γ + cov(βt, γt)
SLIDE 70
70
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)
SLIDE 71
71
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
SLIDE 72
72
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]
SLIDE 73
73
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
