Empirical Financial Economics

New developments in asset pricing

Where does m come from?

Stein’s lemmaIf the vector ft+1 and rt+1 are jointly

Normal

11 1

( )( )

( )t

t tt

u cm g f

u c

1 1 1 1 1 1 1

1 1

[( ) ( )] [ ( )] [( ) ]

. . ( [ ( )] )t t t t t t t

t f ft t t

E r g f E g f E r f

r i e the APT assumes E g f exists

Modeling m directly

Typically assume power utility

Equity Premium Puzzle:

Habit persistence:

These models imply

Lettau and Ludvigson (2001)

1 11 1 1

( ), ln

( )t t

t t tt t

u c cm choose f c

u c c

2( ln( ))MV f m

MVMV m

rc

1 ( )

( ) 11 ,

t

t

s

k s t t tt t

t t

c c xm e s

c c

1 1( , ln )t f t t tr Cov r c

1 1 1 1

1 0 1 1 0 1

ln( ),

,t t t t

t t t t

m a b c

a cay b cay

Multivariate Asset Pricing

Consider

Unconditional means are given by

Model for observations is

Shanken result:

m m m mr b f e

r Bf e

fr B

fr r B Bf e 2 2 1(1 ) e

Shanken, J., 1987, Multivariate proxies and asset pricing relations: Living with the Rollcritique Journal of Financial Economics 18, 91-110.

McElroy and Burmeister

Consider

Unconditional means are given by

Model for observations is

Can estimate this model using NLSUR, GMM

m m m mr b f e

r Bf e

1 1f k kr B b

1 1 1f k k k mr r B b Bf b e e

McElroy, M., and E. Burmeister, 1988, Arbitrage pricing theory as a restricted nonlinearregression model Journal of Business and Economic Statistics 6(1), 29-42.

Black, Jensen and Scholes

Jensen, Michael C. and Black, Fischer and Scholes, Myron S., The Capital Asset Pricing Model: Some Empirical Tests. Michael C. Jensen, STUDIES IN THE THEORY OF CAPITAL MARKETS, Praeger

Publishers Inc., 1972. Available at SSRN: http://ssrn.com/abstract=908569

Fama and MacBeth procedure

( )it ft t t i ir r f estimate

0 5 10 15 20 25 30 t

Fama and MacBeth procedure

0 5 10 15 20 25 30 t

( )it ft t t i

t t

r r f

estimate f

Fama and MacBeth procedure

( )it ft t t i ir r f estimate

0 5 10 15 20 25 30 t

( )it ft t t i

t t

r r f

estimate f

Attributes of two pass procedure

Use portfolio returns Lintner (1968) used individual securitiesBlack, Jensen and Scholes (1972) used portfolios Fama and MacBeth (1973) used portfolios out of

sample

Motivated by concern about errors in variables

Inference uses time series of cross section estimates

Use of Ordinary Least Squares in second pass

The Likelihood Function

i

t tf

The market model regression

i

t tf

i

The Fama MacBeth cross section regression

i

t tf

i

ˆt tf

Updating market model

i

t tf

i

ˆt tf

Full Information Maximum Likelihood

i

t tf

i

ˆt tf

Modeling m directly

Typically assume power utility

Equity Premium Puzzle:

Habit persistence:

These models imply

Lettau and Ludvigson (2001)

1 11 1 1

( ), ln

( )t t

t t tt t

u c cm choose f c

u c c

2( ln( ))MV f m

MVMV m

rc

1 ( )

( ) 11 ,

t

t

s

k s t t tt t

t t

c c xm e s

c c

1 1( , ln )t f t t tr Cov r c

1 1 1 1

1 0 1 1 0 1

ln( ),

,t t t t

t t t t

m a b c

a cay b cay

The geometry of mean variance

a

b

a

b

E

2 1a

22

2

2a bE cE

ac b

1

1

1

a

b

c

OLS or GLS?

Out of sample cross section regression Regress average excess returns against

factor loadingsEstimate expected excess returns

soThe covariance matrix of is proportional

to

OLS: Estimate GLS: Estimate

Can use GLS R2 for non-nested model comparison

ty

t B t t ty

t

2 ˆ ˆ, t tt t t

t ty u R y y

12 2

1

1

ˆ ˆ ˆ, t tt t t

t t

y u Ry y

where I and

Lewellan, Nagel and Shanken (2010) Results

Empirical Asset Pricing Model FF 25 Size - B/M portfolios

FF 25 plus 30 industry portfolios

Data from 1963-2004 k OLS R2 GLS R2 OLS R2 GLS R2

CAPM 2 3% 1% 2% 0%

Consumption CAPM 2 5% 1% 2% 0%

Yogo (2006) 4 18% 4% 2% 5%

Santos and Veronesi (2006) 3 27% 2% 8% 2%

Lustig and Van Nieuwerburgh (2004) 4 57% 2% 9% 0%

Lettau and Ludvigson (2001) 4 58% 5% 0% 1%

Fama-French 4 78% 19% 31% 6%

Li, Vassalou, and Xing (2006) 4 80% 26% 42% 4%

Lewellen, Jonathan, Sefan Nagel and Jay Shanken 2010 A skeptical appraisal of assetpricing tests Journal of Financial Economics 96, 175-194.

Choice among alternative benchmarks

Disenchantment with empirical asset pricing

models Fallen out of favor in corporate finance and other

applications

Growing popularity of firm characteristics and

industry controls Limited theoretical or empirical support

These controls can be interpreted in a risk-

class framework Approach has a sound asset pricing justification

New results in asset pricing literature provide basis for a

horserace

Strong asset pricing justification for

industry controls

Brown, Stephen J. and Handley, John C. and Lajbcygier, Paul, Choice Among Alternative Benchmarks: An Asset Pricing Approach (April 17, 2014). Available at SSRN: http://ssrn.com/abstract=2426277

Modigliani and Miller Risk Classes

An asset pricing rationale for MM risk classes:"This process of understanding how the economy allows investors to duplicate the risky return of any individual company should be understood as an expansion of the original MM notion of a risk class. The "risk class" played an important role in the original arbitrage analysis, as Miller explains, but it has subsequently passed from favor. However, I think that it might be time for a revival of a modern perspective on the older views. This is particularly so given the sorry empirical state of our asset pricing theories".

Ross, Stephen A., 1988. “Comment on the Modigliani–Miller propositions” Journal of Economic Perspectives, 2 pp.127–133.

Risk classes

Risk classes imply model for the observations

Consistent with a broad class of asset pricing models

Justifies use of risk class benchmarks

How should we determine affiliation ?Factor sensitivity? (Fama and French

1992)Financial characteristics? (Daniel &

Titman 1997)Industrial affiliation? (Modigliani and

Miller 1958)Basis assets? (Conrad Ahn and Dittmar

2009)

,it ft It It t it It itR r i I B f

i I

Basis Asset Approach

Consider the following model for the observations

Membership classes are ‘basis assets’ (Conrad et al 2007)

Corresponds to k-means model (Hartigan 1975)

Modified Hartigan procedure

Use daily data for a calendar year Start with an initial allocation to risk classes Iteratively reassign securities to minimize sum of

squares (SS) Allow for clustering by date and security (Brown and

Goetzmann 1997)

,it ft It It t it It itR r i I B f

2it

i t

SS 2it

i t i t

SSk k

The horse race

Factor loadingsCharacteristics

IndustriesBasis Assets

The horse race

For every year 1980 – 2010Determine the category membership in

prior yearRegress excess returns against

category membership

Compare models on basis of resulting R2

A valid non-nested model comparison

1,,

0,t

t

t

iI t

it iI It itiI t

i Iy

i I

Attributes of our procedure

Use individual security returns, not portfolios

No concern about errors in variablesRegress on category membership,

not factor loadings

Inference uses time series of cross section estimates

Use of Generalized Least Squares in second pass

Generalized Least Squares?

Sample covariance matrix is singular for

Is GLS infeasible for individual security regressions?

k-factor covariance matrix is nonsingular for

is a better estimator of than is (Fan et al. 2008)

is simple to compute: for

n t

t k

k

1k 1 1ˆ

1k

k It f It B B

1 1 1 1 1 1 1[ ]k It f It It It B B B B

Individual security characteristics do not beat risk factor loadings -- OLS

Out of sample OLS regressing annual returns on factor loadings and characteristics

125 FF Loadings 125 Characteristics Difference

Year N k Rsq Rbar k Rsq Rbar Rsq Rbar

1980 2708 125 9.71% 5.37% 115 12.86% 9.03% 3.15% 3.66%

1981 2907 125 11.43% 7.48% 115 11.84% 8.24% 0.41% 0.75%

1982 3019 125 9.50% 5.62% 124 7.69% 3.77% -1.81% -1.85%

… … … … … … … … … …

2010 4396 125 8.53% 5.87% 125 6.41% 3.70% -2.11% -2.17%

2011 4499 125 9.05% 6.06% 125 4.52% 1.81% -4.54% -4.25%

2012 4501 124 3.87% 0.80% 125 4.12% 1.40% 0.25% 0.60%

  Mean  6.36% 3.30%   6.23% 3.28% -0.13% -0.02%

  t-value  (13.27) (7.03) (10.16) (5.39) (-0.27) (-0.04)

Individual security characteristics DO beat risk factor loadings -- GLS

Out of sample GLS regressing annual returns on factor loadings and characteristics

125 FF Loadings 125 Characteristics Difference

Year N k Rsq Rbar k Rsq Rbar Rsq Rbar

1980 2708 125 6.20% 1.65% 115 10.58% 6.61% 4.38% 4.96%

1981 2907 125 14.65% 10.81% 115 15.16% 11.66% 0.51% 0.85%

1982 3019 125 11.24% 7.40% 124 12.58% 8.83% 1.34% 1.43%

… … … … … … … … … …

2010 4396 125 7.06% 4.34% 125 8.89% 6.23% 1.83% 1.89%

2011 4499 125 19.05% 16.36% 125 10.10% 7.53% -8.95% -8.83%

2012 4501 124 10.97% 8.11% 125 8.22% 5.60% -2.75% -2.51%

  Mean  12.24% 9.35% 13.58% 10.84% 1.34% 1.49%

  t-value  (9.42) (7.03)   (9.71) (7.57) (2.59) (2.82)

Out of sample cross section regression results

Ordinary Least Squares Generalized Least Squares

Risk class methodology R2 Adjusted R2 R2 Adjusted R2

125 Basis Assets13.00% 10.23% 16.64% 13.96%

(14.67) (11.29) (11.43) (9.20)

48 Fama French industry groups7.27% 6.15% 14.20% 13.14%

(10.48) (8.84) (10.49) (9.63)

125 risk classes based on characteristics6.23% 3.28% 13.58% 10.84%

(10.16) (5.39) (9.71) (7.57)

125 risk classes based on loadings6.36% 3.30% 12.24% 9.35%

(13.27) (7.03) (9.42) (7.03)

Out of sample cross section regression results

Ordinary Least Squares Generalized Least Squares

Difference between methods R2 Adjusted R2 R2 Adjusted R2

Basis Assets - 48 Industry groups5.74% 4.09% 2.44% 0.81%

(8.04) (5.53) (2.50) (0.79)

Basis Assets - Characteristics groups6.77% 6.96% 3.06% 3.11%

(8.58) (8.48) (3.28) (3.22)

Basis Assets - Loadings groups6.64% 6.94% 4.40% 4.61%

(11.02) (11.01) (6.79) (6.85)

48 Industry - Characteristics groups1.04% 2.87% 0.62% 2.30%

(1.62) (4.43) (1.25) (4.52)

48 Industry - Loadings groups0.91% 2.85% 1.96% 3.79%

(1.99) (6.23) (3.30) (6.30)

Characteristics - Loadings groups-0.13% -0.02% 1.34% 1.49%

(-0.27) (-0.04) (2.59) (2.82)

Basis Assets – Hoberg-Phillips 100 industries3.33% 2.82% 1.68% 1.18%

(2.30) (1.91) (1.20) (0.83)

Characteristics – Hoberg-Phillips 100 industries-3.37% -4.08% -2.36% -3.00%

(-2.20) (-2.62) (-2.35) (-2.96)

Out of sample cross section regression results

Ordinary Least Squares Generalized Least Squares

Difference between methods R2 Adjusted R2 R2 Adjusted R2

Basis Assets - 48 Industry groups5.74% 4.09% 2.44% 0.81%

(8.04) (5.53) (2.50) (0.79)

Basis Assets - Characteristics groups6.77% 6.96% 3.06% 3.11%

(8.58) (8.48) (3.28) (3.22)

Basis Assets - Loadings groups6.64% 6.94% 4.40% 4.61%

(11.02) (11.01) (6.79) (6.85)

48 Industry - Characteristics groups1.04% 2.87% 0.62% 2.30%

(1.62) (4.43) (1.25) (4.52)

48 Industry - Loadings groups0.91% 2.85% 1.96% 3.79%

(1.99) (6.23) (3.30) (6.30)

Characteristics - Loadings groups-0.13% -0.02% 1.34% 1.49%

(-0.27) (-0.04) (2.59) (2.82)

Basis Assets – Hoberg-Phillips 100 industries3.33% 2.82% 1.68% 1.18%

(2.30) (1.91) (1.20) (0.83)

Characteristics – Hoberg-Phillips 100 industries-3.37% -4.08% -2.36% -3.00%

(-2.20) (-2.62) (-2.35) (-2.96)

Out of sample cross section regression results

Ordinary Least Squares Generalized Least Squares

Difference between methods R2 Adjusted R2 R2 Adjusted R2

Basis Assets - 48 Industry groups5.74% 4.09% 2.44% 0.81%

(8.04) (5.53) (2.50) (0.79)

Basis Assets - Characteristics groups6.77% 6.96% 3.06% 3.11%

(8.58) (8.48) (3.28) (3.22)

Basis Assets - Loadings groups6.64% 6.94% 4.40% 4.61%

(11.02) (11.01) (6.79) (6.85)

48 Industry - Characteristics groups1.04% 2.87% 0.62% 2.30%

(1.62) (4.43) (1.25) (4.52)

48 Industry - Loadings groups0.91% 2.85% 1.96% 3.79%

(1.99) (6.23) (3.30) (6.30)

Characteristics - Loadings groups-0.13% -0.02% 1.34% 1.49%

(-0.27) (-0.04) (2.59) (2.82)

Basis Assets – Hoberg-Phillips 100 industries3.33% 2.82% 1.68% 1.18%

(2.30) (1.91) (1.20) (0.83)

Characteristics – Hoberg-Phillips 100 industries-3.37% -4.08% -2.36% -3.00%

(-2.20) (-2.62) (-2.35) (-2.96)

Out of sample cross section regression results

Ordinary Least Squares Generalized Least Squares

Difference between methods R2 Adjusted R2 R2 Adjusted R2

Basis Assets - 48 Industry groups5.74% 4.09% 2.44% 0.81%

(8.04) (5.53) (2.50) (0.79)

Basis Assets - Characteristics groups6.77% 6.96% 3.06% 3.11%

(8.58) (8.48) (3.28) (3.22)

Basis Assets - Loadings groups6.64% 6.94% 4.40% 4.61%

(11.02) (11.01) (6.79) (6.85)

48 Industry - Characteristics groups1.04% 2.87% 0.62% 2.30%

(1.62) (4.43) (1.25) (4.52)

48 Industry - Loadings groups0.91% 2.85% 1.96% 3.79%

(1.99) (6.23) (3.30) (6.30)

Characteristics - Loadings groups-0.13% -0.02% 1.34% 1.49%

(-0.27) (-0.04) (2.59) (2.82)

Basis Assets – Hoberg-Phillips 100 industries3.33% 2.82% 1.68% 1.18%

(2.30) (1.91) (1.20) (0.83)

Characteristics – Hoberg-Phillips 100 industries-3.37% -4.08% -2.36% -3.00%

(-2.20) (-2.62) (-2.35) (-2.96)

Out of sample cross section regression results

Ordinary Least Squares Generalized Least Squares

Difference between methods R2 Adjusted R2 R2 Adjusted R2

Basis Assets - 48 Industry groups5.74% 4.09% 2.44% 0.81%

(8.04) (5.53) (2.50) (0.79)

Basis Assets - Characteristics groups6.77% 6.96% 3.06% 3.11%

(8.58) (8.48) (3.28) (3.22)

Basis Assets - Loadings groups6.64% 6.94% 4.40% 4.61%

(11.02) (11.01) (6.79) (6.85)

48 Industry - Characteristics groups1.04% 2.87% 0.62% 2.30%

(1.62) (4.43) (1.25) (4.52)

48 Industry - Loadings groups0.91% 2.85% 1.96% 3.79%

(1.99) (6.23) (3.30) (6.30)

Characteristics - Loadings groups-0.13% -0.02% 1.34% 1.49%

(-0.27) (-0.04) (2.59) (2.82)

Basis Assets – Hoberg-Phillips 100 industries3.33% 2.82% 1.68% 1.18%

(2.30) (1.91) (1.20) (0.83)

Characteristics – Hoberg-Phillips 100 industries-3.37% -4.08% -2.36% -3.00%

(-2.20) (-2.62) (-2.35) (-2.96)

Out of sample cross section regression results

Ordinary Least Squares Generalized Least Squares

Difference between methods R2 Adjusted R2 R2 Adjusted R2

Basis Assets - 48 Industry groups5.74% 4.09% 2.44% 0.81%

(8.04) (5.53) (2.50) (0.79)

Basis Assets - Characteristics groups6.77% 6.96% 3.06% 3.11%

(8.58) (8.48) (3.28) (3.22)

Basis Assets - Loadings groups6.64% 6.94% 4.40% 4.61%

(11.02) (11.01) (6.79) (6.85)

48 Industry - Characteristics groups1.04% 2.87% 0.62% 2.30%

(1.62) (4.43) (1.25) (4.52)

48 Industry - Loadings groups0.91% 2.85% 1.96% 3.79%

(1.99) (6.23) (3.30) (6.30)

Characteristics - Loadings groups-0.13% -0.02% 1.34% 1.49%

(-0.27) (-0.04) (2.59) (2.82)

Basis Assets – Hoberg-Phillips 100 industries3.33% 2.82% 1.68% 1.18%

(2.30) (1.91) (1.20) (0.83)

Characteristics – Hoberg-Phillips 100 industries-3.37% -4.08% -2.36% -3.00%

(-2.20) (-2.62) (-2.35) (-2.96)

Kruskal Tau Average Value

Basis assets 48 FF industry 125 characteristics 125 loadings100 HP

industries

Basis assets 1 0.155 0.058 0.045 0.25

48 FF industries 0.155 1 0.023 0.024 0.107

125 characteristics 0.058 0.023 1 0.058 0.394

125 loadings 0.045 0.024 0.058 1 0.427

100 HP industries 0.25 0.107 0.394 0.427 1

Serial dependence 0.175 0.955 0.13 0.067 0.16

Theil U Average Value

Basis assets 48 FF industry 125 characteristics 125 loadings100 HP

industries

Basis assets 1 0.307 0.31 0.289 0.533

48 FF industries 0.307 1 0.197 0.196 0.326

125 characteristics 0.31 0.197 1 0.365 0.695

125 loadings 0.289 0.196 0.365 1 0.724

100 HP industries 0.533 0.326 0.695 0.724 1

Serial dependence 0.426 0.967 0.509 0.406 0.478

Conclusion

Firm specific characteristics commonly used in matched samples

Can be interpreted as basis assets Approach consistent with many asset pricing models Can be applied on an individual security basis

Out of sample, industry classifications explain returns

Superior to risk factor or firm characteristics-based methods Simpler to apply than empirically estimating basis assets Easy to interpret More stable than other classification schemes

Strong endorsement of MM (1958) risk class conjecture