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

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 0.17 Low 0.71 0.41 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 2

Background Explained by the conditional CAPM w/ time-varying betas? Conditional CAPM E t-1 [R it ] = β t γ t R it = α t + β t R Mt + ε t ⇒ α t = 0 Empirical tests w/ constant β E[R it ] ≠ β γ R it = α + β R Mt + ε t ⇒ α ≠ 0 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

Intuition 1 Alternate between efficient portfolios A and B 1.40 1.20 B 1.00 Dynamic 0.80 strategy A .5 A + .5 B 0.60 0.40 0.20 0.00 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 5

Intuition 2 R t = β t R Mt + ε t , β t = β + η t , γ t = E t-1 [R Mt ], ρ β , γ > 0 0.12 E[R i | R M ] 0.10 0.08 0.06 0.04 0.02 R M 0.00 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 -0.02 -0.04 True -0.06 Uncond. regression -0.08 -0.10 6

Rolling betas of value stocks, 1930 – 2000 Franzoni (2004) 7

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 8

Notation Excess returns: R it , R Mt Moments γ t = E t-1 [R Mt ], σ = var t-1 (R Mt ), β t = cov t-1 (R it , R Mt ) / σ 2 2 t t σ = var(R Mt ), β u = cov(R it , R Mt ) / γ = E[R Mt ], σ 2 2 M M β = E[ β t ] No restriction on joint distribution of returns 9

Theory If conditional CAPM holds, what is α u ≡ E[R it ] – β u γ ? 10

Theory If conditional CAPM holds, what is α u ≡ E[R it ] – β u γ ? E t-1 [R it ] = β t γ t E[R it ] = β γ + cov( β t , γ t ) ⇒ α u = γ ( β – β u ) + cov( β t , γ t ) 11

Theory If conditional CAPM holds, what is α u ≡ E[R it ] – β u γ ? E t-1 [R it ] = β t γ t E[R it ] = β γ + cov( β t , γ t ) ⇒ α u = γ ( β – β u ) + cov( β t , γ t ) Conditional beta γ 1 1 β u = β + β γ + β γ − γ + β σ 2 2 cov( , ) cov[ , ( ) ] cov( , ) σ t t σ t t σ t t 2 2 2 M M M 12

Theory If conditional CAPM holds, what is α u ≡ E[R it ] – β u γ ? E t-1 [R it ] = β t γ t E[R it ] = β γ + cov( β t , γ t ) ⇒ α u = γ ( β – β u ) + cov( β t , γ t ) Conditional beta γ 1 1 β u = β + β γ + β γ − γ + β σ 2 2 cov( , ) cov[ , ( ) ] cov( , ) σ t t σ t t σ t t 2 2 2 M M M Convexity Cubic Volatility 13

Theory If conditional CAPM holds, what is α u ≡ E[R it ] – β u γ ? E t-1 [R it ] = β t γ t E[R it ] = β γ + cov( β t , γ t ) ⇒ α u = γ ( β – β u ) + cov( β t , γ t ) Conditional beta γ 1 1 β u = β + β γ + β γ − γ + β σ 2 2 cov( , ) cov[ , ( ) ] cov( , ) σ t t σ t t σ t t 2 2 2 M M M Conditional alpha  γ  γ γ 2 α u = − β γ − β γ − γ − β σ 2 2 1 cov( , ) cov[ , ( ) ] cov( , )   t t t t t t σ σ σ 2 2 2   M M M 14

Magnitude  γ  γ γ 2 α u = − β γ − β γ − γ − β σ 2 2 1 cov( , ) cov[ , ( ) ] cov( , )   t t t t t t σ σ σ 2 2 2   M M M 15

Magnitude  γ  γ γ 2 α u = − β γ − β γ − γ − β σ 2 2 1 cov( , ) cov[ , ( ) ] cov( , )   t t t t t t σ σ σ 2 2 2   M M M • γ 2 / σ ? 2 M γ 2 / 1964 – 2001: γ = 0.47%, σ M = 4.5% ⇒ σ = 0.011 2 M 16

Magnitude  γ  γ γ 2 α u = − β γ − β γ − γ − β σ 2 2 1 cov( , ) cov[ , ( ) ] cov( , )   t t t t t t σ σ σ 2 2 2   M M M • γ 2 / σ ? 2 M γ 2 / 1964 – 2001: γ = 0.47%, σ M = 4.5% ⇒ σ = 0.011 2 M • ( γ t – γ ) 2 ? Suppose γ ≈ 0.5% and 0.0% < γ t < 1.0%. Then ( γ t – γ ) 2 is at most 0.005 2 = 0.000025. 17

Magnitude  γ  γ γ 2 α u = − β γ − β γ − γ − β σ 2 2 1 cov( , ) cov[ , ( ) ] cov( , )   t t t t t t σ σ σ 2 2 2   M M M • γ 2 / σ ? 2 M γ 2 / 1964 – 2001: γ = 0.47%, σ M = 4.5% ⇒ σ = 0.011 2 M • ( γ t – γ ) 2 ? Suppose γ ≈ 0.5% and 0.0% < γ t < 1.0%. Then ( γ t – γ ) 2 is at most 0.005 2 = 0.000025. γ α u ≈ β γ − β σ 2 cov( , ) cov( , ) t t t t σ 2 M 18

1: Constant volatility α u ≈ cov( β t , γ t ) = ρ σ β σ γ ρ = 0.6 σ β ρ = 1.0 σ β 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σ γ = 0.1 σ γ = 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 19

1: Constant volatility α u ≈ cov( β t , γ t ) = ρ σ β σ γ ρ = 0.6 σ β ρ = 1.0 σ β 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σ γ = 0.1 σ γ = 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) 20

1: Constant volatility α u ≈ cov( β t , γ t ) = ρ σ β σ γ ρ = 0.6 σ β ρ = 1.0 σ β 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σ γ = 0.1 σ γ = 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) 21

1: Constant volatility α u ≈ cov( β t , γ t ) = ρ σ β σ γ ρ = 0.6 σ β ρ = 1.0 σ β 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σ γ = 0.1 σ γ = 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 Arbitrary 22

1: Constant volatility α u ≈ cov( β t , γ t ) = ρ σ β σ γ ρ = 0.6 σ β ρ = 1.0 σ β 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σ γ = 0.1 σ γ = 0.1 0.02 0.03 0.04 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 23

1: Constant volatility α u ≈ cov( β t , γ t ) = ρ σ β σ γ ρ = 0.6 σ β ρ = 1.0 σ β 0.3 0.5 0.7 0.3 0.5 0.7 Monthly alpha (%) Monthly alpha (%) σ γ = 0.1 σ γ = 0.1 0.02 0.03 0.04 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% 24

1: Constant volatility β t ~ N[1.0, 0.7], γ t ~ N[0.5%, 0.5%], ρ = 1.0 0.10 E[R i | R M ] 0.08 0.06 0.04 0.02 R M 0.00 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 -0.02 -0.04 True -0.06 Uncond. regression -0.08 -0.10 25

2: Time-varying volatility γ α u ≈ β γ − β σ 2 cov( , ) cov( , ) t t σ t t 2 M Effects of time-varying γ t and σ offset (if they move together) 2 t 26

2: Time-varying volatility γ α u ≈ β γ − β σ 2 cov( , ) cov( , ) t t σ t t 2 M Effects of time-varying γ t and σ offset (if they move together) 2 t Merton (1980): γ t = λ σ 2 t  σ  2 γ α ≈ β γ < cov( β t , γ t ) u cov( , )   σ t t 2   M 27

2: Time-varying volatility γ α u ≈ − β σ 2 = – γ ρ σ β σ v (where v t = σ / σ ) 2 2 cov( , ) t t M t σ 2 M ρ = 0.2 σ β ρ = 0.5 σ β 0.3 0.5 0.7 0.3 0.5 0.7 Alpha (%) Alpha (%) σ v = 1.0 σ v = 1.0 -0.03 -0.05 -0.07 -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 28

Testing the conditional CAPM Traditional tests R it = α it + β it R Mt + ε it β it = b i0 + b i1 Z 1,t-1 + b i2 Z 2,t-1 + … 29

Testing the conditional CAPM Traditional tests R it = α it + β it R Mt + ε it β it = b i0 + b i1 Z 1,t-1 + b i2 Z 2,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 !” 30

Our tests R it = α it + β it R Mt + ε it Short-window regressions • Estimate α it , β it every month, quarter, half-year, or year • Are conditional alphas zero? 31

Our tests Short-window regressions – betas 1.4 1.2 1.0 0.8 0.6 0.4 0.2 1 61 121 181 241 301 361 421 481 541 Days 32