The conditional CAPM does not explain assetpricing. Jonathan Lewellen & Stefan Nagel. HEC School of Management, March 17, 2005

The condiional CAPM does no explain assepricing anomalies Jonahan Lewellen & Sefan Nagel HEC School of Managemen, March 17, 005

Background Size, B/M, and momenum porfolios, 1964 001 Monhly reurns (%) Avg. reurns CAPM alphas Porfolio Size B/M R -1,-6 Size B/M R -1,-6 Low 0.71 0.41 0.17 0.07-0.0-0.41 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 0.5 0.1 0.35 0.08 High 0.50 0.88 0.79 0.11 0.39 0.9 Long shor 0.1 0.47 0.61-0.03 0.59 0.70 -sa 0.91.98.76-0.16 4.01 3.14

Background Explained by he condiional CAPM w/ ime-varying beas? Condiional CAPM E -1 [R i ] = β γ R i = α + β R M + ε α = 0 Empirical ess w/ consan β E[R i ] β γ R i = α + β R M + ε α 0 3

Background Explained by he condiional CAPM w/ ime-varying beas? Theory Jensen (1968) Dybvig and Ross (1985) Hansen and Richard (1987) Applicaion o size, B/M, and momenum Zhang (00) Jagannahan and Wang (1996) Leau and Ludvigson (001) Pekova and Zhang (004) Lusig and Van Nieuwerburgh (004) Sanos and Veronesi (004) Franzoni (004), Adrian and Franzoni (004) Wang (003) 4

Inuiion 1 Alernae beween efficien porfolios A and B 1.40 1.0 B 1.00 0.80 0.60 A Dynamic sraegy.5 A +.5 B 0.40 0.0 0.00.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 5

Inuiion R = β R M + ε, β = β + η, γ = E -1 [R M ], ρ β,γ > 0 0.1 0.10 E[R i R M ] 0.08 0.06 0.04 0.0 R M 0.00-0.08-0.06-0.04-0.0 0.00-0.0 0.0 0.04 0.06 0.08-0.04-0.06-0.08 True Uncond. regression -0.10 6

Rolling beas of value socks, 1930 000 Franzoni (004) 7

Overview Condiional CAPM does no explain anomalies Analysis Perspecive on condiional asse-pricing ess Simple empirical es Condiional CAPM performs nearly as poorly as uncondiional CAPM 8

Noaion Excess reurns: R i, R M Momens γ = E -1 [R M ], = var -1 (R M ), β = cov -1 (R i, R M ) / γ = E[R M ], M = var(r M ), β u = cov(r i, R M ) / β = E[β ] No resricion on join disribuion of reurns M 9

Theory If condiional CAPM holds, wha is α u E[R i ] β u γ? 10

Theory If condiional CAPM holds, wha is α u E[R i ] β u γ? E -1 [R i ] = β γ E[R i ] = β γ + cov(β, γ ) α u = γ(β β u ) + cov(β, γ ) 11

Theory If condiional CAPM holds, wha is α u E[R i ] β u γ? E -1 [R i ] = β γ E[R i ] = β γ + cov(β, γ ) α u = γ(β β u ) + cov(β, γ ) Condiional bea β u γ 1 1 = β + cov( β, γ ) + cov[ β, ( γ γ) ] + cov( β, ) M M M 1

Theory If condiional CAPM holds, wha is α u E[R i ] β u γ? E -1 [R i ] = β γ E[R i ] = β γ + cov(β, γ ) α u = γ(β β u ) + cov(β, γ ) Condiional bea β u γ 1 1 = β + cov( β, γ ) + cov[ β, ( γ γ) ] + cov( β, ) M M Convexiy Cubic Volailiy M 13

14 Theory If condiional CAPM holds, wha is α u E[R i ] β u γ? E -1 [R i ] = β γ E[R i ] = β γ + cov(β, γ ) α u = γ(β β u ) + cov(β, γ ) Condiional bea β u = β + ), cov( 1 ] ) (, cov[ 1 ), cov( M M M β + γ γ β + γ β γ Condiional alpha α u = ), cov( ] ) (, cov[ ), cov( 1 M M M β γ γ γ β γ γ β γ

15 Magniude α u = ), cov( ] ), ( cov[ ), cov( 1 M M M β γ γ γ β γ γ β γ

Magniude α u γ γ γ = 1 cov(, ) cov[ β, ( γ γ) ] cov( β, ) β γ M M M γ / M? 1964 001: γ = 0.47%, M = 4.5% γ / M = 0.011 16

Magniude α u γ γ γ = 1 cov(, ) cov[ β, ( γ γ) ] cov( β, ) β γ M M M γ / M? 1964 001: γ = 0.47%, M = 4.5% γ / M = 0.011 (γ γ)? Suppose γ 0.5% and 0.0% < γ < 1.0%. Then (γ γ) is a mos 0.005 = 0.00005. 17

Magniude α u γ γ γ = 1 cov(, ) cov[ β, ( γ γ) ] cov( β, ) β γ M M M γ / M? 1964 001: γ = 0.47%, M = 4.5% γ / M = 0.011 (γ γ)? Suppose γ 0.5% and 0.0% < γ < 1.0%. Then (γ γ) is a mos 0.005 = 0.00005. α u γ cov( β, γ ) cov( β, ) M 18

1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β 0.3 0.5 0.7 0.3 0.5 0.7 Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ = 0.1 0. 0. 0.3 0.3 0.4 0.4 0.5 0.5 19

1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β 0.3 0.5 0.7 0.3 0.5 0.7 Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ = 0.1 0. 0. 0.3 0.3 0.4 0.4 0.5 0.5 Economically large Evidence laer Fama and French (199, 1997) 0

1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β 0.3 0.5 0.7 0.3 0.5 0.7 Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ = 0.1 0. 0. 0.3 0.3 0.4 0.4 0.5 0.5 Economically large Evidence from predicive regressions Campbell and Cochrane (1999) 1

1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β 0.3 0.5 0.7 0.3 0.5 0.7 Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ = 0.1 0. 0. 0.3 0.3 0.4 0.4 0.5 0.5 Arbirary

1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β 0.3 0.5 0.7 0.3 0.5 0.7 Monhly alpha (%) Monhly alpha (%) γ = 0.1 0.0 0.03 0.04 γ = 0.1 0.03 0.05 0.07 0. 0.04 0.06 0.08 0. 0.06 0.10 0.14 0.3 0.05 0.09 0.1 0.3 0.09 0.15 0.1 0.4 0.07 0.1 0.17 0.4 0.1 0.0 0.8 0.5 0.09 0.15 0.1 0.5 0.15 0.5 0.35 3

1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β 0.3 0.5 0.7 0.3 0.5 0.7 Monhly alpha (%) Monhly alpha (%) γ = 0.1 0.0 0.03 0.04 γ = 0.1 0.03 0.05 0.07 0. 0.04 0.06 0.08 0. 0.06 0.10 0.14 0.3 0.05 0.09 0.1 0.3 0.09 0.15 0.1 0.4 0.07 0.1 0.17 0.4 0.1 0.0 0.8 0.5 0.09 0.15 0.1 0.5 0.15 0.5 0.35 B/M porfolio: 0.59% Momenum porfolio: 1.01% 4

1: Consan volailiy β ~ N[1.0, 0.7], γ ~ N[0.5%, 0.5%], ρ = 1.0 0.10 0.08 E[R i R M ] 0.06 0.04 0.0 0.00 R M -0.08-0.06-0.04-0.0 0.00-0.0 0.0 0.04 0.06 0.08-0.04-0.06-0.08 True Uncond. regression -0.10 5

: Time-varying volailiy α u γ cov( β, γ ) cov( β, ) M Effecs of ime-varying γ and offse (if hey move ogeher) 6

7 : Time-varying volailiy α u ), cov( ), cov( M β γ γ β Effecs of ime-varying γ and offse (if hey move ogeher) Meron (1980): γ = λ ), cov( M u γ β α γ < cov(β, γ )

: Time-varying volailiy α u γ cov( β, ) = γ ρ β v (where v = M / M) ρ = 0. β ρ = 0.5 β 0.3 0.5 0.7 0.3 0.5 0.7 Alpha (%) Alpha (%) v = 1.0-0.03-0.05-0.07 v = 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. 1.9-0.06-0.10-0.13 1.9-0.11-0.19-0.7. -0.07-0.11-0.15. -0.13-0. -0.31 γ = 0.50 8

Tesing he condiional CAPM Tradiional ess R i = α i + β i R M + ε i β i = b i0 + b i1 Z 1,-1 + b i Z,-1 + 9

Tesing he condiional CAPM Tradiional ess R i = α i + β i R M + ε i β i = b i0 + b i1 Z 1,-1 + b i Z,-1 + Cochrane (001) Models such as he CAPM imply a condiional linear facor model wih respec o invesors informaion ses. The bes we can hope o do is es implicaions condiioned on variables ha we observe. Thus, a condiional facor model is no esable! 30

Our ess R i = α i + β i R M + ε i Shor-window regressions Esimae α i, β i every monh, quarer, half-year, or year Are condiional alphas zero? 31

Our ess Shor-window regressions beas 1.4 1. 1.0 0.8 0.6 0.4 0. 1 61 11 181 41 301 361 41 481 541 Days 3

Our ess R i = α i + β i R M + ε i Shor-window regressions Esimae α i, β i every monh, quarer, half-year, or year Are condiional alphas zero? 33

Our ess R i = α i + β i R M + ε i Shor-window regressions Esimae α i, β i every monh, quarer, half-year, or year Are condiional alphas zero? Assumes only ha bea is relaively slow moving 34

Our ess R i = α i + β i R M + ε i Shor-window regressions Esimae α i, β i every monh, quarer, half-year, or year Are condiional alphas zero? Assumes only ha bea is relaively slow moving Don need precise esimaes of individual α i, β i 35

Our ess R i = α i + β i R M + ε i Shor-window regressions Esimae α i, β i every monh, quarer, half-year, or year Are condiional alphas zero? Assumes only ha bea is relaively slow moving Don need precise esimaes of individual α i, β i Microsrucure issues 36

Microsrucure issue 1 Horizon effecs (compounding) Daily alphas, beas monhly alphas, beas 37

Microsrucure issue 1 Horizon effecs (compounding) Daily alphas, beas monhly alphas, beas β (N) i 1.5 = E[(1+ R i )(1 + R E[(1+ R M M )] ) N E[1 + R i] N ] E[1 + R N M E[1 + R N ] M ] N 1.51 1.50 1.49 Days (N) 1 6 11 16 1 6 31 36 41 46 51 56 61 38

Microsrucure issue Nonsynchronous prices Daily / weekly esimaes of bea miss full covariance 39

Microsrucure issue Bea esimaes, horizons from 1 o 45 days, 1964 001 1.4 1. Small socks 1.0 Value socks 0.8 0.6 1 5 9 13 17 1 5 9 33 37 41 45 Horizon (days) 40

Microsrucure issue Parial soluion Use value-weighed porfolios and NYSE / Amex socks Dimson (1979) beas: R i, = α i + β i0 R M, + β i1 R M,-1 + + β ik R M,-k + ε i, β i = β i0 + β i1 + + β ik 41

Microsrucure issue Bea esimaes Daily beas R i, = α i + β i0 R M, + β i1 R M,-1 + β i [(R M,- + R M,-3 + R M,-4 )/3] + ε i, Weekly beas R i, = α i + β i0 R M, + β i1 R M,-1 + β i R M,- + ε i, Monhly beas R i, = α i + β i0 R M, + β i1 R M,-1 + ε i, 4

Daa NYSE / Amex socks, 1964 001 VW porfolios 5 size-b/m porfolios (S, B, V, G) 10 momenum porfolios, 6-monh reurns (W, L) 43

Summary saisics, 1964 001 Monhly, % Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Excess reurns Avg. Day 0.57 0.49 0.08 0.3 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 0.1 0.41 0.88 0.47 0.01 0.91 0.90 Sd err. Day 0.8 0.0 0.19 0.7 0.3 0.13 0.33 0.8 0.6 Wk 0.6 0.18 0.18 0.6 0. 0.1 0.30 0.6 0.5 Mon 0.34 0.19 0.3 0.30 0.6 0.16 0.35 0.8 0.7 44

Summary saisics, 1964 001 Uncondiional alphas Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Es. Day 0.09 0.10-0.01-0.1 0.39 0.60-0.64 0.35 0.99 Wk 0.05 0.10-0.05-0. 0.37 0.59-0.66 0.37 1.03 Mon 0.07 0.11-0.03-0.0 0.39 0.59-0.63 0.38 1.01 Sd err. Day 0.15 0.06 0.17 0.10 0.1 0.1 0.18 0.13 0.6 Wk 0.14 0.06 0.16 0.09 0.11 0.11 0.17 0.1 0.5 Mon 0.18 0.07 0.0 0.11 0.13 0.14 0.19 0.13 0.8 Uncondiional beas Es. Day 1.07 0.87 0.0 1.18 0.94-0.5 1. 1.17-0.06 Wk 1.5 0.86 0.39 1.7 1.03-0.4 1.33 1.16-0.17 Mon 1.34 0.83 0.51 1.30 1.05-0.5 1.36 1.14-0. Sd err. Day 0.03 0.01 0.03 0.0 0.03 0.0 0.03 0.0 0.05 Wk 0.03 0.01 0.04 0.0 0.03 0.03 0.04 0.03 0.06 Mon 0.05 0.0 0.06 0.03 0.04 0.04 0.06 0.04 0.08 45

Shor-window regressions Tess Q1: Are condiional alphas zero? Q: How volaile are beas? Q3: Do beas covary wih he marke risk premium and variance? 46

Tes 1 Are condiional alphas zero? Tess based on he ime series of shor-window α i Fama-MacBeh approach Four versions of he shor-window regressions Quarerly (daily reurns) Semiannually (daily and weekly reurns) Annually (monhly reurns) 47

Condiional CAPM, 1964 001 Condiional vs. uncondiional alphas (%) Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Uncondiional alphas Day 0.09 0.10-0.01-0.1 0.39 0.60-0.64 0.35 0.99 Wk 0.05 0.10-0.05-0. 0.37 0.59-0.66 0.37 1.03 Monh 0.07 0.11-0.03-0.0 0.39 0.59-0.63 0.38 1.01 Average condiional alpha Quarerly 0.4 0.00 0.4-0.01 0.49 0.50-0.79 0.55 1.33 Semi 1 0.6 0.00 0.6-0.08 0.40 0.47-0.61 0.39 0.99 Semi 0.16 0.01 0.15-0.1 0.36 0.48-0.83 0.53 1.37 Annual -0.06 0.08-0.14-0.0 0.3 0.53-0.56 0.1 0.77 48

Condiional CAPM, 1964 001 Condiional vs. uncondiional alphas (%) Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Uncondiional alphas Day 0.09 0.10-0.01-0.1 0.39 0.60-0.64 0.35 0.99 Wk 0.05 0.10-0.05-0. 0.37 0.59-0.66 0.37 1.03 Monh 0.07 0.11-0.03-0.0 0.39 0.59-0.63 0.38 1.01 Average condiional alpha Quarerly 0.4 0.00 0.4-0.01 0.49 0.50-0.79 0.55 1.33 Semi 1 0.6 0.00 0.6-0.08 0.40 0.47-0.61 0.39 0.99 Semi 0.16 0.01 0.15-0.1 0.36 0.48-0.83 0.53 1.37 Annual -0.06 0.08-0.14-0.0 0.3 0.53-0.56 0.1 0.77 49

Condiional CAPM, 1964 001 Saisical ess Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Condiional alphas Quarerly 0.4 0.00 0.4-0.01 0.49 0.50-0.79 0.55 1.33 Semi 1 0.6 0.00 0.6-0.08 0.40 0.47-0.61 0.39 0.99 Semi 0.16 0.01 0.15-0.1 0.36 0.48-0.83 0.53 1.37 Annual -0.06 0.08-0.14-0.0 0.3 0.53-0.56 0.1 0.77 Sandard error Quarerly 0.0 0.06 0. 0.1 0.14 0.14 0.0 0.13 0.6 Semi 1 0.1 0.06 0.3 0.1 0.14 0.15 0.19 0.14 0.5 Semi 0.1 0.06 0.3 0.14 0.15 0.16 0.0 0.15 0.7 Annual 0.6 0.07 0.9 0.16 0.17 0.14 0.1 0.17 0.9 50

Condiional CAPM Why are condiional and uncondiional alphas similar? α u γ cov( β, γ ) cov( β, ) M 51

Tes How volaile are beas? b = β + e var(β ) = var(b ) var(e ) 5

Condiional beas (semiannual, daily reurns), 1964 001 Small minus Big 1. 0.9 0.6 0.3 0.0-0.3-0.6-0.9 1964. 1971. 1978. 1985. 199. 1999. 53

Condiional beas (semiannual, daily reurns), 1964 001 Value minus Growh 0.7 0.4 0. 0.0-0. -0.4-0.7-0.9-1.1 1964. 1971. 1978. 1985. 199. 1999. 54

Condiional beas (semiannual, daily reurns), 1964 001 Winner minus Losers. 1.6 1.1 0.5 0.0-0.5-1.1-1.6 1964. 1971. 1978. 1985. 199. 1999. 55

Condiional beas, 1964 001 Uncondiional beas Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Day 1.07 0.87 0.0 1.18 0.94-0.5 1. 1.17-0.06 Wk 1.5 0.86 0.39 1.7 1.03-0.4 1.33 1.16-0.17 Monh 1.34 0.83 0.51 1.30 1.05-0.5 1.36 1.14-0. Average condiional beas Quarerly 1.03 0.93 0.10 1.17 0.98-0.19 1.19 1.4 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99-0.0 1.0 1.4 0.05 Semi 1.3 0.91 0.3 1.5 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 1.4-0.14 Implied sd deviaion of rue beas Quarerly 0.3 0.13 0.33 0.19 0.8 0.5 0.36 0.33 0.63 Semi 1 0.9 0.1 0.30 0.18 0.8 0.4 0.30 0.30 0.55 Semi 0.31 0.10 0.3 0.16 0.31 0.9 0.36 0.3 0.6 Annual 0.35 -- 0.5 0.04 0.37 0.19 0.19 0.9 0.5 56

Condiional beas, 1964 001 Uncondiional beas Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Day 1.07 0.87 0.0 1.18 0.94-0.5 1. 1.17-0.06 Wk 1.5 0.86 0.39 1.7 1.03-0.4 1.33 1.16-0.17 Monh 1.34 0.83 0.51 1.30 1.05-0.5 1.36 1.14-0. Average condiional beas Quarerly 1.03 0.93 0.10 1.17 0.98-0.19 1.19 1.4 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99-0.0 1.0 1.4 0.05 Semi 1.3 0.91 0.3 1.5 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 1.4-0.14 Implied sd deviaion of rue beas Quarerly 0.3 0.13 0.33 0.19 0.8 0.5 0.36 0.33 0.63 Semi 1 0.9 0.1 0.30 0.18 0.8 0.4 0.30 0.30 0.55 Semi 0.31 0.10 0.3 0.16 0.31 0.9 0.36 0.3 0.6 Annual 0.35 -- 0.5 0.04 0.37 0.19 0.19 0.9 0.5 57

Condiional beas, 1964 001 Uncondiional beas Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Day 1.07 0.87 0.0 1.18 0.94-0.5 1. 1.17-0.06 Wk 1.5 0.86 0.39 1.7 1.03-0.4 1.33 1.16-0.17 Monh 1.34 0.83 0.51 1.30 1.05-0.5 1.36 1.14-0. Average condiional beas Quarerly 1.03 0.93 0.10 1.17 0.98-0.19 1.19 1.4 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99-0.0 1.0 1.4 0.05 Semi 1.3 0.91 0.3 1.5 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 1.4-0.14 Implied sd deviaion of rue beas Quarerly 0.3 0.13 0.33 0.19 0.8 0.5 0.36 0.33 0.63 Semi 1 0.9 0.1 0.30 0.18 0.8 0.4 0.30 0.30 0.55 Semi 0.31 0.10 0.3 0.16 0.31 0.9 0.36 0.3 0.6 Annual 0.35 -- 0.5 0.04 0.37 0.19 0.19 0.9 0.5 58

Condiional beas, 1964 001 Uncondiional beas Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Day 1.07 0.87 0.0 1.18 0.94-0.5 1. 1.17-0.06 Wk 1.5 0.86 0.39 1.7 1.03-0.4 1.33 1.16-0.17 Monh 1.34 0.83 0.51 1.30 1.05-0.5 1.36 1.14-0. Average condiional beas Quarerly 1.03 0.93 0.10 1.17 0.98-0.19 1.19 1.4 0.05 Semi 1 1.07 0.93 0.14 1.19 0.99-0.0 1.0 1.4 0.05 Semi 1.3 0.91 0.3 1.5 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 1.4-0.14 Implied sd deviaion of rue beas Quarerly 0.3 0.13 0.33 0.19 0.8 0.5 0.36 0.33 0.63 Semi 1 0.9 0.1 0.30 0.18 0.8 0.4 0.30 0.30 0.55 Semi 0.31 0.10 0.3 0.16 0.31 0.9 0.36 0.3 0.6 Annual 0.35 -- 0.5 0.04 0.37 0.19 0.19 0.9 0.5 59

Tes 3 Do beas covary wih business condiions? Do beas covary wih γ and? 60

Tes 3 Do beas covary wih business condiions? Marke reurns (6 monhs) Tbill rae Dividend yield Term premium CAY Lagged bea 61

Condiional beas, 1964 001 Correlaion beween beas and sae variables Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L β -1 0.55 0.68 0.43 0.58 0.67 0.51 0.30 0.45 0.37 R M,-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 0.15-0.1-0.5 0.14-0.5-0.1 DY 0. 0.64-0.04 0.37 0.40 0.18 0.13-0.1-0.14 TERM -0.0 0.19-0.7-0.1 0.01 0.10-0.01-0.08-0.04 CAY -0.1 0.50-0.31-0.01 0.17 0.0 0.09-0.09-0.10 Sd. error 0.116 if no auocorrelaion 6

Predicing condiional beas, 1964 001 Slope esimae Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L β -1 0.1 0.05 0.11 0.10 0.1 0.08 0.10 0.15 0. R M,-6 0.05-0.01 0.04 0.0 0.04 0.04-0.19 0.0 0.39 TBILL -0.13-0.0-0.11-0.03-0.14-0.13 0.09-0.14-0.4 DY 0.14 0.05 0.09 0.06 0.16 0.10-0.07 0.11 0.19 TERM -0.10 0.00-0.10-0.0-0.08-0.07 0.07-0.11-0.19 CAY -0.05 0.0-0.08-0.03-0.01 0.03 0.00-0.01-0.01 -saisic β -1 3.53 3.99.83 4.4 3.88.6 3.03 5.31 4.49 R M,-6 1.5-0.45 1.17 0.73 1.58 1.41-5.63 7.5 7.63 TBILL -.56-1.39 -.09-1.06-3.19 -.98 1.79-3.41-3. DY.8 3.05 1.74.10 3.64.65-1.50.87.65 TERM -.40-0.5 -.1-0.81 -.40-1.99 1.60-3.07 -.81 CAY -1.3 1.86-1.81-1.34-0.17 0.98 0.07-0. -0.13 Adj R 0.37 0.60 0.6 0.34 0.5 0.3 0.35 0.56 0.53 63

Predicing condiional beas, 1964 001 Slope esimae Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L β -1 0.1 0.05 0.11 0.10 0.1 0.08 0.10 0.15 0. R M,-6 0.05-0.01 0.04 0.0 0.04 0.04-0.19 0.0 0.39 TBILL -0.13-0.0-0.11-0.03-0.14-0.13 0.09-0.14-0.4 DY 0.14 0.05 0.09 0.06 0.16 0.10-0.07 0.11 0.19 TERM -0.10 0.00-0.10-0.0-0.08-0.07 0.07-0.11-0.19 CAY -0.05 0.0-0.08-0.03-0.01 0.03 0.00-0.01-0.01 -saisic β -1 3.53 3.99.83 4.4 3.88.6 3.03 5.31 4.49 R M,-6 1.5-0.45 1.17 0.73 1.58 1.41-5.63 7.5 7.63 TBILL -.56-1.39 -.09-1.06-3.19 -.98 1.79-3.41-3. DY.8 3.05 1.74.10 3.64.65-1.50.87.65 TERM -.40-0.5 -.1-0.81 -.40-1.99 1.60-3.07 -.81 CAY -1.3 1.86-1.81-1.34-0.17 0.98 0.07-0. -0.13 Adj R 0.37 0.60 0.6 0.34 0.5 0.3 0.35 0.56 0.53 64

Tes 3 Do beas covary wih γ? Wha is α u cov(β, γ )? Two esimaes (1) cov(b, R M ) = cov(β + e, γ + s ) = cov(β, γ ) () cov( b *, R M ) = cov( b *, γ ) 65

Bea and he marke risk premium, 1964 001 Covariance beween esimaed beas and marke reurns Implied α u (%) Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Esimae Quarerly -0.3 0.07-0.39-0.0-0.1 0.09 0.16-0.3-0.38 Semi 1-0.17 0.07-0.4-0.14-0.03 0.11-0.03-0.07-0.04 Semi -0.1 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 0.0 Sandard error Quarerly 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 0.08 0.03 0.08 0.04 0.08 0.07 0.10 0.08 0.15 Annual 0.1 0.03 0.13 0.06 0.10 0.09 0.1 0.10 0.19 66

Bea and he marke risk premium, 1964 001 Covariance beween prediced beas and marke reurns Implied α u (%) Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Esimae Quarerly -0.06 0.04-0.09-0.01-0.0 0.0 0.06-0.05-0.1 Semi 1-0.07 0.03-0.10-0.0-0.0 0.01 0.05-0.07-0.1 Semi -0.04 0.0-0.05 0.00-0.01 0.00 0.07-0.08-0.14 Annual 0.03 0.01 0.0 0.00 0.01 0.03 0.05-0.03-0.08 Sandard error Quarerly 0.04 0.0 0.04 0.03 0.05 0.04 0.05 0.06 0.10 Semi 1 0.05 0.0 0.04 0.03 0.05 0.04 0.05 0.06 0.10 Semi 0.04 0.0 0.03 0.0 0.05 0.04 0.06 0.05 0.10 Annual 0.05 0.0 0.05 0.03 0.06 0.04 0.06 0.05 0.09 67

Final commens Consumpion CAPM Oher sudies Jagannahan and Wang (1996) Leau and Ludvigson (001) Sanos and Veronesi (004) Lusig and Van Nieuwerburgh (004) 68

Oher sudies Approach E -1 [R ] = β γ E[R] = β γ + cov(β, γ ) 69

Oher sudies Approach E -1 [R ] = β γ E[R] = β γ + cov(β, γ ) Fama-MacBeh regressions: E[R] = θ 0 + θ 1 β + θ cov(β, γ ) 70

Oher sudies Approach E -1 [R ] = β γ E[R] = β γ + cov(β, γ ) Fama-MacBeh regressions: E[R] = θ 0 + θ 1 β + θ cov(β, γ ) Resricions on θ 0, θ 1, and θ are ignored Esimaes of θ seem o be much larger han 1 71

Oher sudies Approach E -1 [R ] = β γ E[R] = β γ + cov(β, γ ) Fama-MacBeh regressions: E[R] = θ 0 + θ 1 β + θ cov(β, γ ) Resricions on θ 0, θ 1, and θ are ignored Esimaes of θ seem o be much larger han 1 Cross-secional R s, wih resricions, aren meaningful Easy o find high R s using size-b/m porfolios Simulaions 90% confidence inerval = [0.1, 0.7] 7

Summary Condiioning relaively unimporan for asse-pricing ess, boh in principle and in pracice Beas vary significanly over ime Condiional alphas are close o uncondiional alphas 73