The conditional CAPM does not explain assetpricing 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
Low
2
3
4
High
Long–short
t-stat
0.71
0.74
0.70
0.69
0.50
0.21
0.91
0.41
0.58
0.66
0.80
0.88
0.47
2.98
0.17
0.51
0.43
0.52
0.79
0.61
2.76
0.07
0.16
0.19
0.21
0.11
-0.03
-0.16
-0.20
0.03
0.17
0.35
0.39
0.59
4.01
-0.41
0.04
-0.01
0.08
0.29
0.70
3.14
2
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)
3
Rolling betas of value stocks, 1930 – 2000
Franzoni (2004)
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)
5
Background
Conditional CAPM
Rit = t + t RMt + t
t = 0
Empirical tests with constant 
Rit =  +  RMt + t
0
6
Intuition 1
Alternate between efficient portfolios A and B
1.40
1.20
B
1.00
Dynamic
strategy
.5 A + .5 B
0.80
A
0.60
0.40
0.20
0.00
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
7
Intuition 2
Rt = t RMt + t,
t =  + t,
t = Et-1[RMt],
0.12
, > 0
E[Ri | RM]
0.10
0.08
0.06
0.04
0.02
-0.08
-0.06
-0.04
-0.02
0.00
0.00
-0.02
RM
0.02
0.04
0.06
0.08
-0.04
-0.06
-0.08
True
Uncond. regression
-0.10
8
Overview
Perspective on conditional asset-pricing tests
A simple empirical test
9
Overview
Perspective on conditional asset-pricing tests
A simple empirical test
Time-variation in betas / expected returns is too small to
explain anomalies
10
Theory
Excess returns: Rit, RMt
No restriction on joint distribution of returns
Notation
t = Et-1[RMt],
 2t = vart-1(RMt),
t = covt-1(Rit, RMt) /  2t
 = E[RMt],
M2 = var(RMt),
u = cov(Rit, RMt) / M2
 = E[t]
11
Theory
If conditional CAPM holds, what is u  E[Rit] – u ?
Et-1[Rit] = t t
12
Theory
If conditional CAPM holds, what is u  E[Rit] – u ?
Et-1[Rit] = t t
E[Rit] =   + cov(t, t)
13
Theory
If conditional CAPM holds, what is u  E[Rit] – u ?
Et-1[Rit] = t t
E[Rit] =   + cov(t, t)

u = ( – u) + cov(t, t)
14
Theory
If conditional CAPM holds, what is u  E[Rit] – u ?
Et-1[Rit] = t t
E[Rit] =   + cov(t, t)

u = ( – u) + cov(t, t)
Conditional beta
u =  +

1
1
2
2
cov(

,

)

cov[

,
(



)
]

cov(

,

t
t
t
t
t
t )
2
2
2
M
M
M
15
Theory
If conditional CAPM holds, what is u  E[Rit] – u ?
Et-1[Rit] = t t
E[Rit] =   + cov(t, t)

u = ( – u) + cov(t, t)
Conditional beta
u =  +

1
1
2
2
cov(

,

)

cov[

,
(



)
]

cov(

,

t
t
t
t
t
t )
2
2
2
M
M
M
Convexity
Cubic
Volatility
16
Theory
If conditional CAPM holds, what is u  E[Rit] – u ?
Et-1[Rit] = t t
E[Rit] =   + cov(t, t)

u = ( – u) + cov(t, t)
Conditional beta
u =  +

1
1
2
2
cov(

,

)

cov[

,
(



)
]

cov(

,

t
t
t
t
t
t )
2
2
2
M
M
M
Conditional alpha
2





2
u
 = 1  2  cov( t ,  t )  2 cov[ t , (  t   ) ]  2 cov( t ,  2t )
M
M
 M 
17
Magnitude
2





u
 = 1  2  cov( t ,  t )  2 cov[ t , (  t   )2 ]  2 cov( t , 2t )
M
M
 M 
18
Magnitude
2





u
 = 1  2  cov( t ,  t )  2 cov[ t , (  t   )2 ]  2 cov( t , 2t )
M
M
 M 

2
2 /  M
?
1964 – 2001:  = 0.47%, M = 4.5%

2 / M2 = 0.011
19
Magnitude
2





u
 = 1  2  cov( t ,  t )  2 cov[ t , (  t   )2 ]  2 cov( t , 2t )
M
M
 M 

2
2 /  M
?
1964 – 2001:  = 0.47%, M = 4.5%


2 / M2 = 0.011
(t – )2 ?
Suppose   0.5% and 0.0% < t < 1.0%. Then (t – )2 is at most
0.0052 = 0.000025.
20
Magnitude
2





u
 = 1  2  cov( t ,  t )  2 cov[ t , (  t   )2 ]  2 cov( t , 2t )
M
M
 M 

2
2 /  M
?
1964 – 2001:  = 0.47%, M = 4.5%


2 / M2 = 0.011
(t – )2 ?
Suppose   0.5% and 0.0% < t < 1.0%. Then (t – )2 is at most
0.0052 = 0.000025.

  cov( t ,  t )  2 cov( t ,  2t )
M
u
21
1: Constant volatility
u  cov(t, t) =   
 = 0.6
0.3

0.5
 = 1.0
0.7
0.3
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5

0.5
0.7
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5
22
1: Constant volatility
u  cov(t, t) =   

 = 0.6
0.3
0.5

 = 1.0
0.7
0.3
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5
0.5
0.7
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5
Economically large
Evidence later
Fama and French (1992, 1997)
23
1: Constant volatility
u  cov(t, t) =   
 = 0.6
0.3

0.5
 = 1.0
0.7
0.3
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5

0.5
0.7
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5
Economically large
Evidence from predictive regressions
Campbell and Cochrane (1999)
24
1: Constant volatility
u  cov(t, t) =   
 = 0.6
0.3

0.5
 = 1.0
0.7
0.3
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5

0.5
0.7
Monthly alpha (%)
 = 0.1
0.2
0.3
0.4
0.5
Arbitrary
25
1: Constant volatility
u  cov(t, t) =   
 = 0.6
0.3
 = 0.1
0.2
0.3
0.4
0.5

0.5
 = 1.0
0.7
Monthly alpha (%)
0.02
0.03
0.04
0.04
0.06
0.08
0.05
0.09
0.12
0.07
0.12
0.17
0.09
0.15
0.21
0.3
 = 0.1
0.2
0.3
0.4
0.5

0.5
0.7
Monthly alpha (%)
0.03
0.05
0.07
0.06
0.10
0.14
0.09
0.15
0.21
0.12
0.20
0.28
0.15
0.25
0.35
26
1: Constant volatility
u  cov(t, t) =   
 = 0.6
0.3
 = 0.1
0.2
0.3
0.4
0.5

0.5
 = 1.0
0.7
Monthly alpha (%)
0.02
0.03
0.04
0.04
0.06
0.08
0.05
0.09
0.12
0.07
0.12
0.17
0.09
0.15
0.21
0.3
 = 0.1
0.2
0.3
0.4
0.5

0.5
0.7
Monthly alpha (%)
0.03
0.05
0.07
0.06
0.10
0.14
0.09
0.15
0.21
0.12
0.20
0.28
0.15
0.25
0.35
B/M portfolio: 0.59%
Momentum portfolio: 1.01%
27
1: Constant volatility
t ~ N[1.0, 0.7], t ~ N[0.5%, 0.5%],
0.10
 = 1.0
E[Ri | RM]
0.08
0.06
0.04
0.02
-0.08
-0.06
-0.04
-0.02
0.00
0.00
-0.02
RM
0.02
0.04
0.06
0.08
-0.04
-0.06
True
-0.08
Uncond. regression
-0.10
28
2: Time-varying volatility
u  cov( t ,  t ) 

2
cov(

,

t
t )
2
M
Effects of time-varying t and  2t offset (if they move together)
29
2: Time-varying volatility
u  cov( t ,  t ) 

2
cov(

,

t
t )
2
M
Effects of time-varying t and  2t offset (if they move together)
Merton (1980): t =   2t
  2 
   2  cov(  t ,  t ) < cov(t, t)
 M 
u
30
2: Time-varying volatility
u  

2
cov(

,

t
t ) = –    v
2
M
 = 0.2
0.3
v = 1.0
1.3
1.6
1.9
2.2
-0.03
-0.04
-0.05
-0.06
-0.07

0.5
(where vt =  2t / M2 )
 = 0.5
0.7
Alpha (%)
-0.05
-0.07
-0.07
-0.09
-0.08
-0.11
-0.10
-0.13
-0.11
-0.15
0.3
v = 1.0
1.3
1.6
1.9
2.2
-0.06
-0.08
-0.10
-0.11
-0.13

0.5
Alpha (%)
-0.10
-0.13
-0.16
-0.19
-0.22
0.7
-0.14
-0.18
-0.22
-0.27
-0.31
 = 0.50
31
Testing the conditional CAPM
Traditional tests
Rit = it + it RMt + it
it = bi0 + bi1 Z1,t-1 + bi2 Z2,t-1 + …
32
Testing the conditional CAPM
Traditional tests
Rit = it + it RMt + it
it = 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!”
33
Our tests
Rit = it + it RMt + it
Short-window regressions

Estimate it, it every month, quarter, half-year, or year

Are conditional alphas zero?
34
Our tests
Rit = it + it RMt + it
Short-window regressions

Estimate it, it every month, quarter, half-year, or year

Are conditional alphas zero?

How volatile are betas?

Do betas covary with the equity premium and variance?
35
Our tests
Rit = it + it RMt + it
Short-window regressions

Estimate it, it 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
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
37
Our tests
Rit = it + it RMt + it
Short-window regressions

Estimate it, it every month, quarter, half-year, or year

Are conditional alphas zero?
38
Our tests
Rit = it + it RMt + it
Short-window regressions

Estimate it, it every month, quarter, half-year, or year

Are conditional alphas zero?

Assumes only that beta is relatively slow moving
39
Our tests
Rit = it + it RMt + it
Short-window regressions

Estimate it, it 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 it, it
40
Our tests
Rit = it + it RMt + it
Short-window regressions

Estimate it, it 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 it, it

Microstructure issues
41
Microstructure issue 1
Horizon effects (compounding)
Daily alphas, betas  monthly alphas, betas
42
Microstructure issue 1
Horizon effects (compounding)
Daily alphas, betas  monthly alphas, betas
E[(1  Ri )(1  RM )]N  E[1  Ri ]N E[1  RM ]N
i (N) 
E[(1  RM )2 ]N  E[1  RM ]2N
1.52
1.51
1.50
Days (N)
1.49
1
6
11
16
21
26
31
36
41
46
51
56
61
43
Microstructure issue 2
Thin trading / nonsynchronous prices
Daily / weekly estimates of beta miss full covariance
44
Microstructure issue 2
Beta estimates, horizons from 1 to 45 days, 1964 – 2001
1.4
Small stocks
1.2
Value stocks
1.0
0.8
0.6
1
5
9
13
17
21
25
29
33
37
41
45
Horizon (days)
45
Microstructure issue 2
Partial solution
Use value-weighted portfolios and NYSE / Amex stocks
Dimson (1979) betas:
Ri,t = i + i0 RM,t + i1 RM,t-1 + … + ik RM,t-k + i,t
i = i0 + i1 + … + ik
46
Microstructure issue 2
Beta estimates

Daily betas
Ri,t = i + i0 RM,t + i1 RM,t-1 + i2 [(RM,t-2 + RM,t-3 + RM,t-4)/3] + i,t

Weekly betas
Ri,t = i + i0 RM,t + i1 RM,t-1 + i2 RM,t-2 + i,t

Monthly betas
Ri,t = i + i0 RM,t + i1 RM,t-1 + i,t
47
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)
48
Summary statistics, 1964 – 2001
Monthly, %
Size
B/M
Small
Big
S-B
Grwth Value
Excess returns
Avg.
Day
Wk
Mon
0.57
0.63
0.71
0.49
0.50
0.50
0.08
0.13
0.21
0.32
0.37
0.41
Std err. Day
Wk
Mon
0.28
0.26
0.34
0.20
0.18
0.19
0.19
0.18
0.23
0.27
0.26
0.30
Momentum
V-G
Losrs Winrs
W-L
0.81
0.84
0.88
0.49
0.47
0.47
-0.10
-0.04
0.01
0.87
0.91
0.91
0.97
0.95
0.90
0.23
0.22
0.26
0.13
0.12
0.16
0.33
0.30
0.35
0.28
0.26
0.28
0.26
0.25
0.27
49
Summary statistics, 1964 – 2001
Size
Small
Big
B/M
S-B
Unconditional alphas
Est.
Day
0.09
Wk
0.05
Mon
0.07
0.10 -0.01
0.10 -0.05
0.11 -0.03
Std err. Day
Wk
Mon
0.15
0.14
0.18
0.06
0.06
0.07
Unconditional betas
Est.
Day
1.07
Wk
1.25
Mon
1.34
Std err. Day
Wk
Mon
0.03
0.03
0.05
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
-0.21
-0.22
-0.20
0.39
0.37
0.39
0.60
0.59
0.59
-0.64
-0.66
-0.63
0.35
0.37
0.38
0.99
1.03
1.01
0.17
0.16
0.20
0.10
0.09
0.11
0.12
0.11
0.13
0.12
0.11
0.14
0.18
0.17
0.19
0.13
0.12
0.13
0.26
0.25
0.28
0.87
0.86
0.83
0.20
0.39
0.51
1.18
1.27
1.30
0.94 -0.25
1.03 -0.24
1.05 -0.25
1.22
1.33
1.36
1.17 -0.06
1.16 -0.17
1.14 -0.22
0.01
0.01
0.02
0.03
0.04
0.06
0.02
0.02
0.03
0.03
0.03
0.04
0.03
0.04
0.06
0.02
0.03
0.04
0.02
0.03
0.04
0.05
0.06
0.08
50
Test
Are conditional alphas zero?
51
Test
Are conditional alphas zero?
Tests based on the time series of short-window it
Fama-MacBeth approach
52
Test
Are conditional alphas zero?
Tests based on the time series of short-window it
Fama-MacBeth approach
Four versions of the short-window regressions
Quarterly (daily returns)
Semiannually (daily and weekly returns)
Annually (monthly returns)
53
Conditional CAPM, 1964 – 2001
Conditional vs. unconditional alphas (%)
Size
Small
Big
B/M
S-B
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
Unconditional alphas
Day
0.09 0.10 -0.01
Wk
0.05 0.10 -0.05
Month
0.07 0.11 -0.03
-0.21
-0.22
-0.20
0.39
0.37
0.39
0.60
0.59
0.59
-0.64
-0.66
-0.63
0.35
0.37
0.38
0.99
1.03
1.01
Average conditional alpha
Quarterly
0.42 0.00
Semi 1
0.26 0.00
Semi 2
0.16 0.01
Annual
-0.06 0.08
-0.01
-0.08
-0.12
-0.20
0.49
0.40
0.36
0.32
0.50
0.47
0.48
0.53
-0.79
-0.61
-0.83
-0.56
0.55
0.39
0.53
0.21
1.33
0.99
1.37
0.77
0.42
0.26
0.15
-0.14
54
Conditional CAPM, 1964 – 2001
Conditional vs. unconditional alphas (%)
Size
Small
Big
B/M
S-B
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
Unconditional alphas
Day
0.09 0.10 -0.01
Wk
0.05 0.10 -0.05
Month
0.07 0.11 -0.03
-0.21
-0.22
-0.20
0.39
0.37
0.39
0.60
0.59
0.59
-0.64
-0.66
-0.63
0.35
0.37
0.38
0.99
1.03
1.01
Average conditional alpha
Quarterly
0.42 0.00
Semi 1
0.26 0.00
Semi 2
0.16 0.01
Annual
-0.06 0.08
-0.01
-0.08
-0.12
-0.20
0.49
0.40
0.36
0.32
0.50
0.47
0.48
0.53
-0.79
-0.61
-0.83
-0.56
0.55
0.39
0.53
0.21
1.33
0.99
1.37
0.77
0.42
0.26
0.15
-0.14
55
Conditional CAPM, 1964 – 2001
Conditional alphas and standard errors
Size
Small
Estimate
Quarterly
Semi 1
Semi 2
Annual
0.42
0.26
0.16
-0.06
Standard error
Quarterly
0.20
Semi 1
0.21
Semi 2
0.21
Annual
0.26
Big
B/M
S-B
0.00 0.42
0.00 0.26
0.01 0.15
0.08 -0.14
0.06
0.06
0.06
0.07
0.22
0.23
0.23
0.29
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
-0.01
-0.08
-0.12
-0.20
0.49
0.40
0.36
0.32
0.50
0.47
0.48
0.53
-0.79
-0.61
-0.83
-0.56
0.55
0.39
0.53
0.21
1.33
0.99
1.37
0.77
0.12
0.12
0.14
0.16
0.14
0.14
0.15
0.17
0.14
0.15
0.16
0.14
0.20
0.19
0.20
0.21
0.13
0.14
0.15
0.17
0.26
0.25
0.27
0.29
56
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 t and  2t ?
57
Conditional betas (semiannual, daily returns), 1964 – 2001
Small minus Big
1.2
0.9
0.6
0.3
0.0
1964.2
1971.2
1978.2
1985.2
1992.2
1999.2
-0.3
-0.6
-0.9
58
Conditional betas (semiannual, daily returns), 1964 – 2001
Value minus Growth
0.7
0.4
0.2
0.0
1964.2
1971.2
1978.2
1985.2
1992.2
1999.2
-0.2
-0.4
-0.7
-0.9
-1.1
59
Conditional betas (semiannual, daily returns), 1964 – 2001
Winner minus Losers
2.2
1.6
1.1
0.5
0.0
1964.2
1971.2
1978.2
1985.2
1992.2
1999.2
-0.5
-1.1
-1.6
60
Conditional betas, 1964 – 2001
Size
Small
B/M
Big
S-B
Unconditional betas
Day
1.07 0.87
Wk
1.25 0.86
Month
1.34 0.83
0.20
0.39
0.51
1.18
1.27
1.30
0.94 -0.25
1.03 -0.24
1.05 -0.25
1.22
1.33
1.36
1.17 -0.06
1.16 -0.17
1.14 -0.22
Average conditional betas
Quarterly
1.03 0.93
Semi 1
1.07 0.93
Semi 2
1.23 0.91
Annual
1.49 0.83
0.10
0.14
0.32
0.66
1.17
1.19
1.25
1.36
0.98
0.99
1.06
1.17
-0.19
-0.20
-0.19
-0.19
1.19
1.20
1.33
1.38
1.24 0.05
1.24 0.05
1.19 -0.14
1.24 -0.14
0.19
0.18
0.16
0.04
0.28
0.28
0.31
0.37
0.25
0.24
0.29
0.19
0.36
0.30
0.36
0.19
0.33
0.30
0.32
0.29
Implied std deviation of true betas
Quarterly
0.32 0.13 0.33
Semi 1
0.29 0.12 0.30
Semi 2
0.31 0.10 0.32
Annual
0.35
-- 0.25
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
0.63
0.55
0.62
0.52
61
Conditional betas, 1964 – 2001
Size
Small
B/M
Big
S-B
Unconditional betas
Day
1.07 0.87
Wk
1.25 0.86
Month
1.34 0.83
0.20
0.39
0.51
1.18
1.27
1.30
0.94 -0.25
1.03 -0.24
1.05 -0.25
1.22
1.33
1.36
1.17 -0.06
1.16 -0.17
1.14 -0.22
Average conditional betas
Quarterly
1.03 0.93
Semi 1
1.07 0.93
Semi 2
1.23 0.91
Annual
1.49 0.83
0.10
0.14
0.32
0.66
1.17
1.19
1.25
1.36
0.98
0.99
1.06
1.17
-0.19
-0.20
-0.19
-0.19
1.19
1.20
1.33
1.38
1.24 0.05
1.24 0.05
1.19 -0.14
1.24 -0.14
0.19
0.18
0.16
0.04
0.28
0.28
0.31
0.37
0.25
0.24
0.29
0.19
0.36
0.30
0.36
0.19
0.33
0.30
0.32
0.29
Implied std deviation of true betas
Quarterly
0.32 0.13 0.33
Semi 1
0.29 0.12 0.30
Semi 2
0.31 0.10 0.32
Annual
0.35
-- 0.25
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
0.63
0.55
0.62
0.52
62
Conditional betas, 1964 – 2001
Size
Small
B/M
Big
S-B
Unconditional betas
Day
1.07 0.87
Wk
1.25 0.86
Month
1.34 0.83
0.20
0.39
0.51
1.18
1.27
1.30
0.94 -0.25
1.03 -0.24
1.05 -0.25
1.22
1.33
1.36
1.17 -0.06
1.16 -0.17
1.14 -0.22
Average conditional betas
Quarterly
1.03 0.93
Semi 1
1.07 0.93
Semi 2
1.23 0.91
Annual
1.49 0.83
0.10
0.14
0.32
0.66
1.17
1.19
1.25
1.36
0.98
0.99
1.06
1.17
-0.19
-0.20
-0.19
-0.19
1.19
1.20
1.33
1.38
1.24 0.05
1.24 0.05
1.19 -0.14
1.24 -0.14
0.19
0.18
0.16
0.04
0.28
0.28
0.31
0.37
0.25
0.24
0.29
0.19
0.36
0.30
0.36
0.19
0.33
0.30
0.32
0.29
Implied std deviation of true betas
Quarterly
0.32 0.13 0.33
Semi 1
0.29 0.12 0.30
Semi 2
0.31 0.10 0.32
Annual
0.35
-- 0.25
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
0.63
0.55
0.62
0.52
63
Conditional betas, 1964 – 2001
Size
Small
B/M
Big
S-B
Unconditional betas
Day
1.07 0.87
Wk
1.25 0.86
Month
1.34 0.83
0.20
0.39
0.51
1.18
1.27
1.30
0.94 -0.25
1.03 -0.24
1.05 -0.25
1.22
1.33
1.36
1.17 -0.06
1.16 -0.17
1.14 -0.22
Average conditional betas
Quarterly
1.03 0.93
Semi 1
1.07 0.93
Semi 2
1.23 0.91
Annual
1.49 0.83
0.10
0.14
0.32
0.66
1.17
1.19
1.25
1.36
0.98
0.99
1.06
1.17
-0.19
-0.20
-0.19
-0.19
1.19
1.20
1.33
1.38
1.24 0.05
1.24 0.05
1.19 -0.14
1.24 -0.14
0.19
0.18
0.16
0.04
0.28
0.28
0.31
0.37
0.25
0.24
0.29
0.19
0.36
0.30
0.36
0.19
0.33
0.30
0.32
0.29
Implied std deviation of true betas
Quarterly
0.32 0.13 0.33
Semi 1
0.29 0.12 0.30
Semi 2
0.31 0.10 0.32
Annual
0.35
-- 0.25
bt = t + et

Grwth Value
Momentum
V-G
Losrs Winrs
W-L
0.63
0.55
0.62
0.52
var(t) = var(bt) – var(et)
64
Test
Do betas covary with business conditions?
Do betas covary with t and  2t ?
65
Test
Do betas covary with business conditions?
Market returns (6 months)
Tbill rate
Dividend yield
Term premium
CAY
Lagged beta
66
Conditional betas, 1964 – 2001
Correlation between betas and state variables
t-1
RM,-6
TBILL
DY
TERM
CAY
Small
Size
Big
S-B
0.55
-0.05
-0.04
0.22
-0.20
-0.12
0.68
-0.01
0.11
0.64
0.19
0.50
0.43
-0.05
-0.08
-0.04
-0.27
-0.31
B/M
Grwth Value
0.58
-0.18
0.15
0.37
-0.12
-0.01
0.67
0.00
-0.12
0.40
0.01
0.17
V-G
0.51
0.14
-0.25
0.18
0.10
0.20
Momentum
Losrs Winrs W-L
0.30
-0.53
0.14
0.13
-0.01
0.09
0.45
0.47
-0.25
-0.12
-0.08
-0.09
0.37
0.56
-0.21
-0.14
-0.04
-0.10
Std. error  0.116 if no autocorrelation
67
Predicting conditional betas, 1964 – 2001
Small
Size
Big
S-B
Slope estimate
t-1
0.12
RM,-6
0.05
TBILL
-0.13
DY
0.14
TERM
-0.10
CAY
-0.05
0.05
-0.01
-0.02
0.05
0.00
0.02
0.11
0.04
-0.11
0.09
-0.10
-0.08
0.10
0.02
-0.03
0.06
-0.02
-0.03
t-statistic
t-1
3.53
RM,-6
1.52
TBILL
-2.56
DY
2.82
TERM
-2.40
CAY
-1.32
3.99
-0.45
-1.39
3.05
-0.25
1.86
2.83
1.17
-2.09
1.74
-2.21
-1.81
0.60
0.26
Adj R2
0.37
B/M
Grwth Value
V-G
Momentum
Losrs Winrs W-L
0.12
0.04
-0.14
0.16
-0.08
-0.01
0.08
0.04
-0.13
0.10
-0.07
0.03
0.10 0.15 0.22
-0.19 0.20 0.39
0.09 -0.14 -0.24
-0.07 0.11 0.19
0.07 -0.11 -0.19
0.00 -0.01 -0.01
4.24
0.73
-1.06
2.10
-0.81
-1.34
3.88
1.58
-3.19
3.64
-2.40
-0.17
2.62
1.41
-2.98
2.65
-1.99
0.98
3.03 5.31 4.49
-5.63 7.25 7.63
1.79 -3.41 -3.22
-1.50 2.87 2.65
1.60 -3.07 -2.81
0.07 -0.22 -0.13
0.34
0.52
0.32
0.35
0.56
0.53
68
Predicting conditional betas, 1964 – 2001
Small
Size
Big
S-B
Slope estimate
t-1
0.12
RM,-6
0.05
TBILL
-0.13
DY
0.14
TERM
-0.10
CAY
-0.05
0.05
-0.01
-0.02
0.05
0.00
0.02
0.11
0.04
-0.11
0.09
-0.10
-0.08
0.10
0.02
-0.03
0.06
-0.02
-0.03
t-statistic
t-1
3.53
RM,-6
1.52
TBILL
-2.56
DY
2.82
TERM
-2.40
CAY
-1.32
3.99
-0.45
-1.39
3.05
-0.25
1.86
2.83
1.17
-2.09
1.74
-2.21
-1.81
0.60
0.26
Adj R2
0.37
B/M
Grwth Value
V-G
Momentum
Losrs Winrs W-L
0.12
0.04
-0.14
0.16
-0.08
-0.01
0.08
0.04
-0.13
0.10
-0.07
0.03
0.10 0.15 0.22
-0.19 0.20 0.39
0.09 -0.14 -0.24
-0.07 0.11 0.19
0.07 -0.11 -0.19
0.00 -0.01 -0.01
4.24
0.73
-1.06
2.10
-0.81
-1.34
3.88
1.58
-3.19
3.64
-2.40
-0.17
2.62
1.41
-2.98
2.65
-1.99
0.98
3.03 5.31 4.49
-5.63 7.25 7.63
1.79 -3.41 -3.22
-1.50 2.87 2.65
1.60 -3.07 -2.81
0.07 -0.22 -0.13
0.34
0.52
0.32
0.35
0.56
0.53
69
Test
Does beta covary with t?
What is the implied alpha u  cov(t, t)?
Two estimates
(i) cov(bt, RMt) = cov(t + et, t + st) = cov(t, t)
(ii) cov( b*t , RMt) = cov( b *t , t)
70
Beta and the market risk premium, 1964 – 2001
Covariance between estimated betas and market returns
Implied u (%)
Size
Small
Estimate
Quarterly
Semi 1
Semi 2
Annual
-0.32
-0.17
-0.12
0.06
Standard error
Quarterly
0.08
Semi 1
0.07
Semi 2
0.08
Annual
0.12
Big
B/M
Momentum
S-B
Grwth Value
V-G
Losrs Winrs
0.07 -0.39
0.07 -0.24
0.07 -0.19
0.03 0.03
-0.20 -0.12
-0.14 -0.03
-0.10 -0.03
-0.03 0.01
0.09
0.11
0.07
0.04
0.16 -0.23 -0.38
-0.03 -0.07 -0.04
0.15 -0.18 -0.33
-0.08 0.11 0.20
0.03
0.03
0.03
0.03
0.08
0.07
0.08
0.13
0.05
0.04
0.04
0.06
0.07
0.07
0.08
0.10
0.06
0.06
0.07
0.09
0.09
0.08
0.10
0.12
0.08
0.07
0.08
0.10
W-L
0.16
0.13
0.15
0.19
71
Beta and the market risk premium, 1964 – 2001
Covariance between predicted betas and market returns
Implied u (%)
Size
Small
Estimate
Quarterly
Semi 1
Semi 2
Annual
Standard error
Quarterly
Semi 1
Semi 2
Annual
-0.06
-0.07
-0.04
0.03
0.04
0.05
0.04
0.05
Big
B/M
Momentum
S-B
Grwth Value
V-G
0.04 -0.09
0.03 -0.10
0.02 -0.05
0.01 0.02
-0.01 -0.02
-0.02 -0.02
0.00 -0.01
0.00 0.01
0.02
0.01
0.00
0.03
0.06
0.05
0.07
0.05
-0.05
-0.07
-0.08
-0.03
-0.12
-0.12
-0.14
-0.08
0.04
0.04
0.04
0.04
0.05
0.05
0.06
0.06
0.06
0.06
0.05
0.05
0.10
0.10
0.10
0.09
0.02
0.02
0.02
0.02
0.04
0.04
0.03
0.05
0.03
0.03
0.02
0.03
0.05
0.05
0.05
0.06
Losrs Winrs
W-L
72
Final comments
Consumption CAPM
Other studies
Jagannathan and Wang (1996)
Lettau and Ludvigson (2001)
Santos and Veronesi (2004)
Lustig and Van Nieuwerburgh (2004)
73
Other studies
Approach
Et-1[Rt] = t t

E[R] =   + cov(t, t)
74
Other studies
Approach
Et-1[Rt] = t t

E[R] =   + cov(t, t)
Fama-MacBeth regressions:
E[R] = 0 + 1  + 2 cov(t, t)
75
Other studies
Approach
Et-1[Rt] = t t

E[R] =   + cov(t, t)
Fama-MacBeth regressions:
E[R] = 0 + 1  + 2 cov(t, t)
Restrictions on 0, 1, and 2 are ignored
Estimates of 2 seem to be much larger than 1
76
Other studies
Approach
Et-1[Rt] = t t

E[R] =   + cov(t, t)
Fama-MacBeth regressions:
E[R] = 0 + 1  + 2 cov(t, t)
Restrictions on 0, 1, and 2 are ignored
Estimates of 2 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]
77
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
78
Extras …
79
Overview
Conditioning doesn’t explain anomalies
Analysis
Time-varying betas can explain only small pricing errors
Empirical tests
Conditional CAPM performs nearly as poorly as the unconditional
CAPM
80
Time-variation in betas, 1964 – 2001
Size
Small
Big
B/M
S-B
Grwth Value
Momentum
V-G
Losrs Winrs
W-L
Std deviation of estimated betas
Quarterly
0.35 0.15 0.38
Semi 1
0.31 0.13 0.32
Semi 2
0.35 0.13 0.38
Annual
0.54 0.14 0.56
0.22
0.19
0.20
0.27
0.30
0.29
0.33
0.46
0.28
0.25
0.33
0.41
0.41
0.33
0.44
0.52
0.37
0.32
0.36
0.44
0.68
0.58
0.71
0.83
Implied std deviation of true betas
Quarterly
0.32 0.13 0.33
Semi 1
0.29 0.12 0.30
Semi 2
0.31 0.10 0.32
Annual
0.35
-- 0.25
0.19
0.18
0.16
0.04
0.28
0.28
0.31
0.37
0.25
0.24
0.29
0.19
0.36
0.30
0.36
0.19
0.33
0.30
0.32
0.29
0.63
0.55
0.62
0.52
81
Conditional betas
Do betas covary with business conditions?
Lagged variables
Market returns (6 months)
Tbill rate
Dividend yield
Term premium
CAY
Lagged beta
82
Other studies
Cross-sectional R2
Striking improvements in R2 for conditional models
JW: 0.01 to 0.29
LL: 0.13 to 0.66
83
Other studies
Cross-sectional R2
Striking improvements in R2 for conditional models
JW: 0.01 to 0.29
LL: 0.13 to 0.66
Cross-sectional R2s, without restrictions on the slopes, aren’t
very meaningful
Easy to find high R2s using size-B/M portfolios
Simulations 90% confidence interval = [0.12, 0.72]
84
Conditional CAPM can’t explain anomalies
Time-variation in betas and expected returns isn’t large enough
Conditioning doesn’t explain anomalies
Analysis
Empirical tests
85
Theory
How large is a stock’s unconditional alpha if the conditional
CAPM holds?
Rit = t RMt + t
Rit, RMt excess returns
No restriction on joint distribution of returns
t = covt-1(Rit, RMt) / vart-1[RMt]
Market premium and volatility: t = Et-1[RMt],  2t = vart-1[RMt]
86
Theory
If the conditional CAPM holds, what determines a stock’s
unconditional alpha?
Notation
Rit, RMt excess returns
Conditional beta = t = covt-1(Rit, RMt) / vart-1[RMt]
Equity premium and volatility: t = Et-1[RMt],  2t = vart-1[RMt]
No restriction on joint distribution of returns
87
Theory
How large is a stock’s unconditional alpha if the conditional
CAPM holds?
Conditional CAPM: Rit = t RMt + t
Excess returns
t = covt-1(Rit, RMt) / vart-1[RMt]
Market premium and volatility: t = Et-1[RMt],  2t = vart-1[RMt]
No restriction on joint distribution of returns
88
Theory
Unconditional alpha: u  E[Rit] – u 
Rit = t RMt + t
 E[Rit] =   + cov(t, t)
u =  +

u = ( – u) + cov(t, t)

1
1
2
2
cov(

,

)

cov[

,
(



)
]

cov(

,

t
t
t
t
t
t )
2
2
2
M
M
M
2





2
u
 = 1  2  cov( t ,  t )  2 cov[ t , (  t   ) ]  2 cov( t ,  2t )
M
M
 M 
89
Theory
If the conditional CAPM holds, what determines a stock’s
unconditional alpha?
Notation
Rit, RMt excess returns
No restriction on joint distribution of returns
Moments
t = Et-1[RMt],
 2t = vart-1(RMt),
t = covt-1(Rit, RMt) /  2t
 = E[RMt],
M2 = var(RMt),
u = cov(Rit, RMt) / M2
90
Theory
If the conditional CAPM holds, what determines a stock’s
unconditional alpha?
Rit = t RMt + t
No restriction on joint distribution of (excess) returns
Notation
t = covt-1(Rit, RMt) / vart-1[RMt]
Market premium and volatility: t = Et-1[RMt],  2t = vart-1[RMt]
91
Theory
If the conditional CAPM holds, what determines a stock’s
unconditional alpha?
Conditional CAPM: Rit = t RMt + t
Rit, RMt excess returns
No restriction on joint distribution of returns
Notation
t = Et-1[RMt],
 2t = vart-1(RMt),
t = covt-1(Rit, RMt) /  2t
 = E[RMt],
M2 = var(RMt),
u = cov(Rit, RMt) / M2
92
Theory
Rit = t RMt + t
Unconditional beta
u =  +

1
1
2
2
cov(

,

)

cov[

,
(



)
]

cov(

,

t
t
t
t
t
t )
2
2
2
M
M
M
Convexity
Cubic
Volatility
93
Theory
Rit = t RMt + t
Unconditional beta
u =  +

1
1
2
2
cov(

,

)

cov[

,
(



)
]

cov(

,

t
t
t
t
t
t )
2
2
2
M
M
M
Unconditional alpha: u  E[Rit] – u 
E[Rit] =   + cov(t, t)

u = ( – u) + cov(t, t)

2 


 = 1  2  cov( t ,  t )  2 cov[ t , (  t   )2 ]  2 cov( t ,  2t )
M
M
 M 
u
94
Theory
If the conditional CAPM holds, what determines a stock’s
unconditional alpha?
95
Theory
If the conditional CAPM holds, what determines a stock’s
unconditional alpha?
Rit = t RMt + t
No restriction on joint distribution of (excess) returns
Notation
t = Et-1[RMt],
 2t = vart-1(RMt),
t = covt-1(Rit, RMt) /  2t
 = E[RMt],
M2 = var(RMt),
u = cov(Rit, RMt) / M2
96
Microstructure issue 2

Daily betas
Ri,t = i + i0 RM,t + i1 RM,t-1 + i2 [(RM,t-2 + RM,t-3 + RM,t-4)/3] + i,t

Weekly betas
Ri,t = i + i0 RM,t + i1 RM,t-1 + i2 RM,t-2 + i,t

Monthly betas
Ri,t = i + i0 RM,t + i1 RM,t-1 + i,t
97
Our tests
Rit = it + it RMt + it
Short-window regressions
Estimate it, it every month, quarter, half-year, or year
Given the estimates:
Q1: Are conditional alphas zero?
Q2: How volatile are betas?
Q3: Do betas covary with the equity premium and variance?
98
Short-window regressions
Given time series of conditional t, t estimates
Q1: Are conditional alphas zero?
Q2: How volatile are betas?
Q3: Do betas covary with the market risk premium and variance?
99
Short-window regressions
Q1: Are conditional alphas zero?
Q2: How volatile are betas?
Q3: Do betas covary with the market risk premium and variance?
100