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

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

2 Background Size, B/M, and momenum porfolios, Monhly reurns (%) Avg. reurns CAPM alphas Porfolio Size B/M R -1,-6 Size B/M R -1,-6 Low High Long shor sa

3 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

4 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

5 Inuiion 1 Alernae beween efficien porfolios A and B B A Dynamic sraegy.5 A +.5 B

6 Inuiion R = β R M + ε, β = β + η, γ = E -1 [R M ], ρ β,γ > E[R i R M ] R M True Uncond. regression

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

8 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

9 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

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

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 ) 11

12 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

13 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 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 15 Magniude α u = ), cov( ] ), ( cov[ ), cov( 1 M M M β γ γ γ β γ γ β γ

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

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

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

19 1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ =

20 1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ = Economically large Evidence laer Fama and French (199, 1997) 0

21 1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ = Economically large Evidence from predicive regressions Campbell and Cochrane (1999) 1

22 1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β Monhly alpha (%) Monhly alpha (%) γ = 0.1 γ = Arbirary

23 1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β Monhly alpha (%) Monhly alpha (%) γ = γ =

24 1: Consan volailiy α u cov(β, γ ) = ρ β γ ρ = 0.6 β ρ = 1.0 β Monhly alpha (%) Monhly alpha (%) γ = γ = B/M porfolio: 0.59% Momenum porfolio: 1.01% 4

25 1: Consan volailiy β ~ N[1.0, 0.7], γ ~ N[0.5%, 0.5%], ρ = E[R i R M ] R M True Uncond. regression

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

27 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 )

28 : Time-varying volailiy α u γ cov( β, ) = γ ρ β v (where v = M / M) ρ = 0. β ρ = 0.5 β Alpha (%) Alpha (%) v = v = γ =

29 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

30 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

31 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

32 Our ess Shor-window regressions beas Days 3

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? 33

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 34

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 35

36 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

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

38 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 Days (N)

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

40 Microsrucure issue Bea esimaes, horizons from 1 o 45 days, Small socks 1.0 Value socks Horizon (days) 40

41 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, β ik R M,-k + ε i, β i = β i0 + β i1 + + β ik 41

42 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

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

44 Summary saisics, Monhly, % Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Excess reurns Avg. Day Wk Mon Sd err. Day Wk Mon

45 Summary saisics, Uncondiional alphas Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Es. Day Wk Mon Sd err. Day Wk Mon Uncondiional beas Es. Day Wk Mon Sd err. Day Wk Mon

46 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

47 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

48 Condiional CAPM, Condiional vs. uncondiional alphas (%) Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Uncondiional alphas Day Wk Monh Average condiional alpha Quarerly Semi Semi Annual

49 Condiional CAPM, Condiional vs. uncondiional alphas (%) Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Uncondiional alphas Day Wk Monh Average condiional alpha Quarerly Semi Semi Annual

50 Condiional CAPM, Saisical ess Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Condiional alphas Quarerly Semi Semi Annual Sandard error Quarerly Semi Semi Annual

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

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

53 Condiional beas (semiannual, daily reurns), Small minus Big

54 Condiional beas (semiannual, daily reurns), Value minus Growh

55 Condiional beas (semiannual, daily reurns), Winner minus Losers

56 Condiional beas, Uncondiional beas Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L Day Wk Monh Average condiional beas Quarerly Semi Semi Annual Implied sd deviaion of rue beas Quarerly Semi Semi Annual

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

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

62 Condiional beas, Correlaion beween beas and sae variables Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L β R M, TBILL DY TERM CAY Sd. error if no auocorrelaion 6

63 Predicing condiional beas, Slope esimae Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L β R M, TBILL DY TERM CAY saisic β R M, TBILL DY TERM CAY Adj R

64 Predicing condiional beas, Slope esimae Size B/M Momenum Small Big S-B Grwh Value V-G Losrs Winrs W-L β R M, TBILL DY TERM CAY saisic β R M, TBILL DY TERM CAY Adj R

65 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

66 Bea and he marke risk premium, 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 Semi Semi Annual Sandard error Quarerly Semi Semi Annual

67 Bea and he marke risk premium, 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 Semi Semi Annual Sandard error Quarerly Semi Semi Annual

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

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

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 ) 70

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 71

72 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

73 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