The CAPM Strikes Back? An Investment Model with Disasters

The CAPM Strikes Back? An Investment Model with Disasters Hang Bai 1 Kewei Hou 1 Howard Kung 2 Lu Zhang 3 1 The Ohio State University 2 London Business School 3 The Ohio State University and NBER Federal Reserve Bank of New York September 17, 2015

Introduction Insight An investment model with disasters replicates: The failure of the CAPM in capturing the value premium in nite samples in which disasters are not materialized; The relative success of the CAPM in samples in which disasters are materialized

Introduction Literature Early quantitative theories of cross-sectional asset pricing rely on single-factor models: Gomes, Kogan, and Zhang (2003); Carlson, Fisher, and Giammarino (2004); Zhang (2005); Cooper (2006) Recent quantitative theories introduce two-shock models: Ai and Kiku (2013); Kogan and Papanikolaou (2013); Belo, Lin, and Bazdresch (2014); Koh (2014) Prior disaster models: Rietz (1988); Barro (2006, 2009); Barro and Ursua (2008); Gourio (2012); Gabaix (2012); Wachter (2013)

Outline 1 Stylized Facts 2 The Model 3 Failing the CAPM 4 The Beta Anomaly

Outline 1 Stylized Facts 2 The Model 3 Failing the CAPM 4 The Beta Anomaly

Stylized Facts The CAPM regressions for the b/m deciles, July 1963June 2014 Fama and French (1992, 1993) L 2 3 4 5 6 7 8 9 H H L m 0.42 0.52 0.55 0.55 0.54 0.59 0.68 0.71 0.79 0.93 0.51 t m 2.00 2.72 2.95 2.89 2.97 3.25 3.81 3.88 4.07 3.91 2.75 α 0.12 0.01 0.06 0.06 0.08 0.13 0.24 0.27 0.32 0.39 0.51 t α 1.28 0.15 0.92 0.57 0.83 1.45 2.27 2.28 2.92 2.50 2.26 β 1.06 1.01 0.98 0.99 0.91 0.93 0.88 0.88 0.94 1.07 0.01 t β 40.79 46.28 34.55 29.35 27.50 28.34 22.89 17.53 20.85 15.44 0.07 R 2 0.86 0.92 0.90 0.87 0.83 0.85 0.78 0.76 0.76 0.66 0.00

Stylized Facts The CAPM regressions for the b/m deciles, July 1926June 2014 Ang and Chen (2007) L 2 3 4 5 6 7 8 9 H H L m 0.57 0.68 0.68 0.68 0.73 0.76 0.77 0.92 1.03 1.09 0.52 t m 3.24 4.09 4.05 3.67 4.12 4.04 3.86 4.43 4.36 3.82 2.60 α 0.09 0.06 0.05 0.01 0.08 0.07 0.05 0.18 0.21 0.14 0.23 t α 1.27 1.20 0.89 0.13 1.02 0.93 0.58 1.93 1.83 0.94 1.18 β 1.00 0.95 0.97 1.06 1.00 1.05 1.09 1.14 1.27 1.45 0.45 t β 47.76 28.52 59.63 20.06 27.74 16.04 16.14 15.40 13.46 13.38 3.52 R 2 0.90 0.92 0.92 0.90 0.89 0.87 0.84 0.82 0.79 0.72 0.14

80 60 Stylized Facts The value premium vs. MKT, July 1926June 2014 80 32Aug 39Sep 60 The value premium 40 20 0 20 29Oct 98Aug 32May 87Oct 32Apr 31Sep 30Jun 31May 33Feb 37Sep 80Mar 08Oct 32Oct 40May 31Dec 38Mar 33May 34Jan 75Jan 76Jan 31Jun 38Jun 33Jun 38Apr 33Aug 28Nov 87Jan 74Oct 32Jul 33Apr The value premium 40 20 0 20 40 40 20 0 20 40 The market excess return 40 40 20 0 20 40 The market excess return

Stylized Facts Large swings in the stock market and the value premium MKT H L MKT H L November 1928 11.79 0.41 August 1933 12.03 4.92 October 1929 20.07 7.57 January 1934 12.63 34.10 June 1930 16.25 3.54 September 1937 13.57 10.90 May 1931 13.16 3.09 March 1938 23.80 22.67 June 1931 13.75 14.80 April 1938 14.49 8.76 September 1931 29.07 5.03 June 1938 23.77 15.22 December 1931 13.42 16.73 September 1939 16.94 56.61 April 1932 17.98 2.85 May 1940 21.93 15.49 May 1932 20.44 3.61 October 1974 16.10 13.58 July 1932 33.47 45.73 January 1975 13.66 19.70 August 1932 36.41 69.99 January 1976 12.16 15.04 October 1932 13.09 12.97 March 1980 12.90 9.02 February 1933 15.06 7.45 January 1987 12.47 2.98 April 1933 37.93 22.41 October 1987 23.24 1.21 May 1933 21.36 45.01 August 1998 16.08 6.33 June 1933 13.05 10.29 October 2008 17.23 11.93

Stylized Facts The CAPM's general problem, the beta anomaly, July 1963June 2014, Fama and French (2006) L 2 3 4 5 6 7 8 9 H H L m 0.51 0.52 0.52 0.56 0.66 0.54 0.68 0.53 0.62 0.63 0.12 t m 3.64 3.46 3.11 3.15 3.46 2.67 3.06 2.25 2.33 1.92 0.43 α 0.22 0.17 0.11 0.12 0.17 0.02 0.10 0.08 0.06 0.18 0.40 t α 2.03 1.75 1.32 1.39 1.89 0.22 1.17 0.83 0.47 0.90 1.48 β 0.57 0.68 0.81 0.87 0.98 1.03 1.14 1.22 1.35 1.61 1.04 t β 12.29 16.79 19.13 20.74 27.23 30.22 46.72 41.42 34.60 30.04 11.41 R 2 0.54 0.68 0.77 0.79 0.86 0.86 0.88 0.86 0.84 0.78 0.43

Stylized Facts The CAPM's general problem, the beta anomaly, July 1928June 2014, Fama and French (2006) L 2 3 4 5 6 7 8 9 H H L m 0.57 0.63 0.64 0.73 0.82 0.71 0.80 0.72 0.82 0.81 0.24 t m 4.80 4.51 4.23 4.33 4.36 3.55 3.69 2.99 3.04 2.59 0.94 α 0.21 0.17 0.12 0.14 0.16 0.01 0.04 0.13 0.11 0.26 0.47 t α 2.68 2.18 2.04 2.37 2.29 0.11 0.54 1.53 1.09 1.80 2.40 β 0.57 0.74 0.82 0.93 1.05 1.12 1.22 1.36 1.49 1.70 1.12 t β 22.94 29.62 35.56 40.62 41.07 40.12 46.51 36.08 26.55 40.59 18.46 R 2 0.67 0.81 0.85 0.88 0.90 0.90 0.91 0.90 0.88 0.85 0.58

Outline 1 Stylized Facts 2 The Model 3 Failing the CAPM 4 The Beta Anomaly

The Model Highlights Embedding disasters into a standard investment model: Rare disasters in consumption (productivity) growth Asymmetric adjustment costs: Value rms are more exposed to disaster risk than growth rms Recursive preferences In a sample without disasters, estimated betas only reect risk in normal times, but the value premium is driven by disaster risk

The Model Recursive utility The pricing kernel: M t+1 = ι ( Ct+1 C t ) 1 ψ U1 γ [ t+1 E t U 1 γ t+1 ] 1/ψ γ 1 γ

The Model Consumption dynamics Log consumption growth: g ct = ḡ + g t Normal states follow a discretized autoregressive process: Five states: {g 1, g 2, g 3, g 4, g 5 } Transition matrix: p ij Prob(g t+1 = g i g t = g j ): p 11 p 12... p 15 p 21 p 22... p 25 P =...... p 51 p 52... p 55

The Model Consumption dynamics Insert the disaster state, g 0 = λ D (disaster size < 0), and the recovery state, g 6 = λ R (recovery size > 0) Modify transition matrix: θ 0 0... 0 1 θ η p 11 η p 12... p 15 0 η p 21 p 22 η... p 25 0 P =........ η p 51 p 52... p 55 η 0 0 (1 ν)/5 (1 ν)/5... (1 ν)/5 ν η: disaster probability; θ: disaster persistence; ν: recovery persistence

The Model Firms, technology Operating prots: Π it = (X t Z it ) 1 ξ K ξ it fk it Aggregate productivity growth: g xt = g + φg t Firm-specic productivity: z it+1 = (1 ρ z ) z + ρ z z it + σ z e it+1

The Model Firms, asymmetric adjustment costs Capital accumulation: K it+1 = I it + (1 δ)k it Asymmetric capital adjustment costs: ( ) 2 a + K it + c+ Iit 2 K Kit for it I it > 0 Φ(I it, K it ) = 0 for I it = 0 ( ) 2 a K it + c Iit 2 K Kit for it I it < 0 in which c > c + > 0 and a > a + > 0 capture asymmetry

The Model Firms, value maximization Source of funds constraint: D it = Π it I it Φ(I it, K it ) Value maximization: ( V it = max {χ it } max {I it } D it + E t [M t+1 V (K it+1, X t+1, Z it+1 )], sk it in which s 0 is the liquidation value parameter Entry and exit, delisting return, reorganizational costs ),

Outline 1 Stylized Facts 2 The Model 3 Failing the CAPM 4 The Beta Anomaly

Failing the CAPM Calibration, preferences Parameters Value Description ι 0.99035 Time discount factor γ 5 The relative risk aversion ψ 1.5 The elasticity of intertemporal substitution

Failing the CAPM Calibration, consumption dynamics Parameters Value Description ḡ 0.019/12 The average consumption growth ρ g 0.6 The persistence of consumption growth σ g 0.0025 The conditional volatility of consumption growth η 0.028/12 The disaster probability λ D 0.0275 The disaster size θ 0.914 1/3 The disaster persistence λ R 0.0325 The recovery size ν 0.95 The recovery persistence

Failing the CAPM The impulse response of log consumption to a disaster shock mimics that in Nakamura, Steinsson, Barro, and Ursua (2013) 0 0.05 0.1 0.15 0.2 0.25 0 5 10 15 20 25

Failing the CAPM Calibration, technology Parameters Value Description ξ 0.65 The curvature parameter in the production function δ 0.01 The capital depreciation rate f 0.005 Fixed costs of production φ 1 The leverage of productivity growth z 9.75 The long-run mean of log rm-specic productivity ρ z 0.985 The persistence of log rm-specic productivity σ z 0.5 The conditional volatility of log rm-specic productivity a + 0.035 Upward nonconvex adjustment costs a 0.05 Downward nonconvex adjustment costs c + 75 Upward convex adjustment costs c 150 Downward convex adjustment costs s 0 The liquidation value parameter κ 0.25 The reorganizational cost parameter R 0.425 The delisting return

Failing the CAPM The CAPM regressions for the b/m deciles, no-disaster samples G 2 3 4 5 6 7 8 9 V V G m 0.77 0.76 0.75 0.75 0.75 0.77 0.80 0.85 0.95 1.21 0.45 t m 18.58 18.43 18.09 17.98 18.11 18.57 19.32 20.52 22.55 24.99 7.10 α 0.01 0.02 0.06 0.11 0.10 0.08 0.02 0.08 0.23 0.46 0.47 t α 0.07 0.24 0.73 1.26 1.24 0.95 0.18 0.99 2.83 5.02 3.71 β 0.95 0.96 1.00 1.05 1.05 1.05 1.00 0.95 0.89 0.92 0.03 t β 11.07 10.87 11.32 11.98 11.89 11.88 11.47 10.95 9.85 9.00 0.24 R 2 0.11 0.12 0.12 0.14 0.14 0.13 0.13 0.11 0.10 0.08 0.00

Failing the CAPM The CAPM regressions for the b/m deciles, disaster samples G 2 3 4 5 6 7 8 9 V V G m 0.74 0.74 0.73 0.74 0.75 0.77 0.81 0.86 0.96 1.19 0.45 t m 13.83 13.61 13.43 13.24 13.15 13.07 13.10 13.07 13.17 13.61 5.83 α 0.08 0.06 0.04 0.01 0.00 0.03 0.05 0.09 0.13 0.13 0.21 t α 1.38 1.09 0.76 0.32 0.08 0.53 0.82 1.15 1.40 1.38 1.72 β 0.82 0.85 0.87 0.90 0.94 1.00 1.08 1.19 1.37 1.64 0.82 t β 18.82 23.74 28.64 33.03 33.72 30.60 26.47 20.63 16.00 18.05 6.82 R 2 0.45 0.47 0.49 0.50 0.52 0.55 0.57 0.60 0.64 0.65 0.24

Failing the CAPM Value is more exposed to disaster risk than growth 30 30 20 20 10 10 0 0 20 Capital 10 0 20 10 z 0 20 10 Capital 0 20 10 z 0

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Failing the CAPM Impulse responses of risk and risk premiums for value and growth deciles to a disaster shock 5 4 3 2 1 0 0 0 5 10 15 20 25 1 0 5 10 15 20 25

Failing the CAPM Nonlinearity in the CAPM regressions 60 40 The value premium 20 0 20 40 20 0 20 40 60 The market excess return

Failing the CAPM Nonlinearity in the pricing kernel 25 20 The pricing kernel 15 10 5 0 20 0 20 40 60 The market excess return

Failing the CAPM Comparative statics λ D θ η ν λ R 0.025 0.03 0.955 0.985 0.13% 0.33% 0.935 0.965 2.75% 3.75% Disaster samples m 0.34 0.55 0.29 0.47 0.42 0.46 0.46 0.43 0.45 0.44 t m 4.78 6.75 4.49 5.62 5.72 5.78 5.94 5.61 5.83 5.70 α 0.22 0.20 0.21 0.16 0.21 0.21 0.20 0.22 0.21 0.21 t α 1.98 1.51 2.08 1.33 1.60 1.89 1.66 1.80 1.75 1.78 β 0.77 0.86 0.74 0.77 0.79 0.86 0.85 0.78 0.85 0.81 t β 6.65 7.11 6.56 7.39 6.01 7.80 6.74 6.75 6.74 7.04 No-disaster samples m 0.33 0.54 0.28 0.55 0.42 0.46 0.45 0.43 0.44 0.45 t m 5.63 8.08 5.24 7.89 6.74 7.29 7.09 6.90 6.99 7.06 α 0.24 0.71 0.07 0.86 0.43 0.50 0.49 0.45 0.47 0.47 t α 2.14 4.96 0.67 5.66 3.38 3.97 3.88 3.52 3.72 3.77 β 0.12 0.19 0.32 0.33 0.02 0.05 0.05 0.02 0.04 0.04 t β 0.89 1.35 2.56 2.32 0.13 0.35 0.41 0.19 0.31 0.34

Failing the CAPM Comparative statics a + a c + c f 0.025 0.045 0.035 0.065 50 100 100 200 0 0.015 Disaster samples m 0.48 0.29 0.25 0.47 0.37 0.49 0.39 0.46 0.47 0.40 t m 6.57 3.75 3.73 5.97 4.64 6.59 5.25 5.90 6.30 4.86 α 0.25 0.23 0.21 0.23 0.24 0.20 0.21 0.22 0.21 0.22 t α 1.91 2.05 1.66 1.82 2.04 1.61 1.82 1.80 1.71 1.89 β 0.96 0.64 0.61 0.89 0.73 0.90 0.76 0.85 0.87 0.75 t β 6.42 6.19 4.32 6.77 7.23 6.57 6.57 6.85 6.61 7.25 No-disaster samples m 0.45 0.28 0.22 0.46 0.38 0.49 0.39 0.46 0.45 0.40 t m 8.48 4.24 3.84 7.32 5.54 8.42 6.35 7.29 7.78 5.81 α 0.63 0.14 0.26 0.49 0.27 0.62 0.41 0.49 0.54 0.31 t α 5.39 1.10 2.16 3.83 1.98 5.12 3.27 3.89 4.42 2.31 β 0.23 0.16 0.04 0.04 0.13 0.17 0.02 0.04 0.10 0.11 t β 1.71 1.20 0.32 0.31 0.96 1.25 0.15 0.36 0.77 0.76

Failing the CAPM Comparative statics s κ R γ ψ 0.15 0.3 0 0.5 0.3 0.55 3.5 6.5 1 2 Disaster samples m 0.20 0.03 0.45 0.45 0.47 0.44 0.18 0.57 0.06 0.51 t m 2.95 0.28 5.79 5.87 6.08 5.64 2.61 7.19 2.67 5.74 α 0.27 0.35 0.21 0.21 0.19 0.23 0.23 0.11 0.29 0.18 t α 2.72 3.80 1.72 1.73 1.58 1.89 2.48 0.81 10.00 1.60 β 0.63 0.48 0.82 0.83 0.83 0.83 0.75 0.67 1.74 0.67 t β 7.06 6.57 6.84 6.80 6.75 6.85 6.24 5.29 10.34 8.43 No-disaster samples m 0.27 0.10 0.44 0.45 0.46 0.44 0.15 0.60 0.07 0.50 t m 4.37 1.68 7.07 7.11 7.27 7.03 3.18 8.47 3.15 7.10 α 0.34 0.20 0.47 0.47 0.48 0.47 0.12 0.96 0.31 0.82 t α 2.78 1.74 3.68 3.69 3.76 3.65 1.64 5.87 11.70 5.23 β 0.09 0.13 0.03 0.03 0.03 0.03 0.51 0.34 1.98 0.30 t β 0.66 1.01 0.22 0.23 0.22 0.25 4.35 2.45 17.67 2.27

Outline 1 Stylized Facts 2 The Model 3 Failing the CAPM 4 The Beta Anomaly

The Beta Anomaly Deciles formed on rolling market betas, disaster samples L 2 3 4 5 6 7 8 9 H H L m 0.76 0.78 0.81 0.83 0.85 0.86 0.86 0.85 0.83 0.79 0.04 t m 13.72 14.09 14.04 13.89 13.55 13.41 13.08 12.65 11.79 11.50 0.53 α 0.04 0.06 0.06 0.04 0.01 0.01 0.04 0.08 0.16 0.17 0.21 t α 0.69 1.29 1.17 0.82 0.29 0.05 0.49 0.89 1.45 2.10 1.73 β 0.90 0.90 0.95 0.99 1.04 1.08 1.12 1.16 1.23 1.20 0.30 t β 19.75 25.57 33.62 34.40 31.60 25.92 21.23 18.56 15.11 17.35 2.49 R 2 0.53 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.06

The Beta Anomaly Deciles formed on rolling market betas, no-disaster samples L 2 3 4 5 6 7 8 9 H H L m 0.80 0.82 0.84 0.85 0.85 0.86 0.84 0.82 0.79 0.74 0.06 t m 20.12 20.36 20.48 20.45 19.82 20.00 19.58 18.98 18.24 16.65 0.93 α 0.01 0.12 0.17 0.19 0.15 0.16 0.11 0.03 0.09 0.42 0.44 t α 0.16 1.55 2.15 2.28 1.75 1.88 1.30 0.39 1.06 4.98 3.49 β 0.97 0.86 0.82 0.82 0.86 0.86 0.90 0.97 1.08 1.43 0.47 t β 11.79 10.20 9.50 9.27 9.30 9.15 9.72 10.37 11.67 15.74 3.49 R 2 0.13 0.10 0.09 0.09 0.09 0.09 0.10 0.11 0.14 0.23 0.01

The Beta Anomaly Measurement errors in rolling market betas

Conclusion Summary An investment model with disasters replicates the failure of the CAPM in capturing the value premium in no-disaster samples, and its relative success in disaster samples The beta anomaly largely due to measurement errors in pre-ranking rolling betas A rst step in integrating the disaster literature with investment-based asset pricing