Empirical Financial Economics New developments in asset pricing

Where does m come from?  Stein’s lemma  If the vector f t+1 and r t+1 are jointly Normal

Modeling m directly  Typically assume power utility  Equity Premium Puzzle:  Habit persistence:  These models imply  Lettau and Ludvigson (2001)

Multivariate Asset Pricing  Consider  Unconditional means are given by  Model for observations is  Shanken result: Shanken, J., 1987, Multivariate proxies and asset pricing relations: Living with the Roll critique Journal of Financial Economics 18,

McElroy and Burmeister  Consider  Unconditional means are given by  Model for observations is  Can estimate this model using NLSUR, GMM McElroy, M., and E. Burmeister, 1988, Arbitrage pricing theory as a restricted nonlinear regression model Journal of Business and Economic Statistics 6(1),

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., Available at SSRN:

Fama and MacBeth procedure t

Fama and MacBeth procedure t

Fama and MacBeth procedure t

Attributes of two pass procedure  Use portfolio returns  Lintner (1968) used individual securities  Black, 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

The market model regression

The Fama MacBeth cross section regression

Updating market model

Full Information Maximum Likelihood

Modeling m directly  Typically assume power utility  Equity Premium Puzzle:  Habit persistence:  These models imply  Lettau and Ludvigson (2001)

The geometry of mean variance

OLS or GLS?  Out of sample cross section regression  Regress average excess returns against factor loadings  Estimate expected excess returns so  The covariance matrix of is proportional to  OLS: Estimate  GLS: Estimate  Can use GLS R 2 for non-nested model comparison

Lewellan, Nagel and Shanken (2010) Results Empirical Asset Pricing Model FF 25 Size - B/M portfolios FF 25 plus 30 industry portfolios Data from kOLS R 2 GLS R 2 OLS R 2 GLS R 2 CAPM23%1%2%0% Consumption CAPM25%1%2%0% Yogo (2006)418%4%2%5% Santos and Veronesi (2006)327%2%8%2% Lustig and Van Nieuwerburgh (2004)457%2%9%0% Lettau and Ludvigson (2001)458%5%0%1% Fama-French478%19%31%6% Li, Vassalou, and Xing (2006)480%26%42%4% Lewellen, Jonathan, Sefan Nagel and Jay Shanken 2010 A skeptical appraisal of asset pricing tests Journal of Financial Economics 96,

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:

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., “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)

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)

The horse race Factor loadings Characteristics Industries Basis Assets

The horse race  For every year 1980 – 2010  Determine the category membership in prior year  Regress excess returns against category membership  Compare models on basis of resulting R 2  A valid non-nested model comparison

Attributes of our procedure  Use individual security returns, not portfolios  No concern about errors in variables  Regress 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

Individual security characteristics do not beat risk factor loadings -- OLS Out of sample OLS regressing annual returns on factor loadings and characteristics 125 FF Loadings125 CharacteristicsDifference YearNkRsqRbarkRsqRbarRsqRbar %5.37% %9.03%3.15%3.66% %7.48% %8.24%0.41%0.75% %5.62% %3.77%-1.81%-1.85% ………………………… %5.87% %3.70%-2.11%-2.17% %6.06% %1.81%-4.54%-4.25% %0.80% %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 Loadings125 CharacteristicsDifference YearNkRsqRbarkRsqRbarRsqRbar %1.65% %6.61%4.38%4.96% %10.81% %11.66%0.51%0.85% %7.40% %8.83%1.34%1.43% ………………………… %4.34% %6.23%1.83%1.89% %16.36% %7.53%-8.95%-8.83% %8.11% %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 SquaresGeneralized Least Squares Risk class methodologyR2R2 Adjusted R 2 R2R2 125 Basis Assets 13.00%10.23%16.64%13.96% (14.67)(11.29)(11.43)(9.20) 48 Fama French industry groups 7.27%6.15%14.20%13.14% (10.48)(8.84)(10.49)(9.63) 125 risk classes based on characteristics 6.23%3.28%13.58%10.84% (10.16)(5.39)(9.71)(7.57) 125 risk classes based on loadings 6.36%3.30%12.24%9.35% (13.27)(7.03)(9.42)(7.03)

Out of sample cross section regression results Ordinary Least SquaresGeneralized Least Squares Difference between methodsR2R2 Adjusted R 2 R2R2 Basis Assets - 48 Industry groups 5.74%4.09%2.44%0.81% (8.04)(5.53)(2.50)(0.79) Basis Assets - Characteristics groups 6.77%6.96%3.06%3.11% (8.58)(8.48)(3.28)(3.22) Basis Assets - Loadings groups 6.64%6.94%4.40%4.61% (11.02)(11.01)(6.79)(6.85) 48 Industry - Characteristics groups 1.04%2.87%0.62%2.30% (1.62)(4.43)(1.25)(4.52) 48 Industry - Loadings groups 0.91%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 industries 3.33%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 SquaresGeneralized Least Squares Difference between methodsR2R2 Adjusted R 2 R2R2 Basis Assets - 48 Industry groups 5.74%4.09%2.44%0.81% (8.04)(5.53)(2.50)(0.79) Basis Assets - Characteristics groups 6.77%6.96%3.06%3.11% (8.58)(8.48)(3.28)(3.22) Basis Assets - Loadings groups 6.64%6.94%4.40%4.61% (11.02)(11.01)(6.79)(6.85) 48 Industry - Characteristics groups 1.04%2.87%0.62%2.30% (1.62)(4.43)(1.25)(4.52) 48 Industry - Loadings groups 0.91%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 industries 3.33%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 SquaresGeneralized Least Squares Difference between methodsR2R2 Adjusted R 2 R2R2 Basis Assets - 48 Industry groups 5.74%4.09%2.44%0.81% (8.04)(5.53)(2.50)(0.79) Basis Assets - Characteristics groups 6.77%6.96%3.06%3.11% (8.58)(8.48)(3.28)(3.22) Basis Assets - Loadings groups 6.64%6.94%4.40%4.61% (11.02)(11.01)(6.79)(6.85) 48 Industry - Characteristics groups 1.04%2.87%0.62%2.30% (1.62)(4.43)(1.25)(4.52) 48 Industry - Loadings groups 0.91%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 industries 3.33%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 SquaresGeneralized Least Squares Difference between methodsR2R2 Adjusted R 2 R2R2 Basis Assets - 48 Industry groups 5.74%4.09%2.44%0.81% (8.04)(5.53)(2.50)(0.79) Basis Assets - Characteristics groups 6.77%6.96%3.06%3.11% (8.58)(8.48)(3.28)(3.22) Basis Assets - Loadings groups 6.64%6.94%4.40%4.61% (11.02)(11.01)(6.79)(6.85) 48 Industry - Characteristics groups 1.04%2.87%0.62%2.30% (1.62)(4.43)(1.25)(4.52) 48 Industry - Loadings groups 0.91%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 industries 3.33%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 SquaresGeneralized Least Squares Difference between methodsR2R2 Adjusted R 2 R2R2 Basis Assets - 48 Industry groups 5.74%4.09%2.44%0.81% (8.04)(5.53)(2.50)(0.79) Basis Assets - Characteristics groups 6.77%6.96%3.06%3.11% (8.58)(8.48)(3.28)(3.22) Basis Assets - Loadings groups 6.64%6.94%4.40%4.61% (11.02)(11.01)(6.79)(6.85) 48 Industry - Characteristics groups 1.04%2.87%0.62%2.30% (1.62)(4.43)(1.25)(4.52) 48 Industry - Loadings groups 0.91%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 industries 3.33%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 SquaresGeneralized Least Squares Difference between methodsR2R2 Adjusted R 2 R2R2 Basis Assets - 48 Industry groups 5.74%4.09%2.44%0.81% (8.04)(5.53)(2.50)(0.79) Basis Assets - Characteristics groups 6.77%6.96%3.06%3.11% (8.58)(8.48)(3.28)(3.22) Basis Assets - Loadings groups 6.64%6.94%4.40%4.61% (11.02)(11.01)(6.79)(6.85) 48 Industry - Characteristics groups 1.04%2.87%0.62%2.30% (1.62)(4.43)(1.25)(4.52) 48 Industry - Loadings groups 0.91%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 industries 3.33%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 assets48 FF industry125 characteristics125 loadings 100 HP industries Basis assets FF industries characteristics loadings HP industries Serial dependence

Theil U Average Value Basis assets48 FF industry125 characteristics125 loadings 100 HP industries Basis assets FF industries characteristics loadings HP industries Serial dependence

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