Aggregation, Capital Heterogeneity, and the Investment CAPM

Transcription

1 Aggregation, Capital Heterogeneity, and the Investment CAPM Andrei S. Gonçalves 1 Chen Xue 2 Lu Zhang 3 1 UNC 2 University of Cincinnati 3 Ohio State and NBER PBCSF November 21, 218

2 Introduction Theme A detailed treatment of aggregation and capital heterogeneity substantially improves the performance of the investment CAPM

3 Introduction Traditional asset pricing based on the demand of risky assets Markowitz (1952) Treynor (1962), Sharpe (1964), Lintner (1965), Mossin (1966) Merton (1973), Ross (1976) Rubinstein (1976), Lucas (1978), Breeden (1979) Hansen and Singleton (1982, 1983), Breeden, Gibbons, and Litzenberger (1989) Cochrane (25), Back (21), Campbell (217) Berk and DeMarzo (213), Bodie, Kane, and Marcus (214)

4 Introduction An emerging framework based on the supply of risky assets Böhm-Bawert (1891) Fisher (193), Hirshleifer (1958, 1965, 197) Modigliani and Miller (1958) Cochrane (1991) Zhang (25, 217) Liu, Whited, and Zhang (29), Liu and Zhang (214) Hou, Xue, and Zhang (215), Hou, Mo, Xue, and Zhang (218)

5 Introduction Marshall's scissors: Marshall 189 [1961, 9th edition, p. 348] Ricardo and Mill: Costs of production determine value Jevons, Menger, and Walras: Marginal utility determines value The water versus diamond debate We might as reasonably dispute whether it is the upper or under blade of a pair of scissors that cuts a piece of paper, as whether value is governed by utility or costs of production. It is true that when one blade is held still, and the cutting is aected by moving the other, we may say with careless brevity that the cutting is done by the second; but the statement is not strictly accurate, and is to be excused only so long as it claims to be merely a popular and not a strictly scientic account of what happens (our emphasis).

6 Introduction Campbell (217): An entire chapter on the investment model An empirical challenge facing the structural investment model: This problem, that dierent parameters are needed to t each anomaly, is a pervasive one in the q-theoretic asset pricing literature (p. 275).

7 Introduction The empirical challenge facing the structural investment CAPM Liu, Whited, and Zhang (29): Liu and Zhang (214): journal of political economy TABLE 2 Parameter Estimates and Tests of Overidentification SUE B/M CI A. Matching Expected Returns a [1.7] [25.5] [.3] a [.] [.3] [.] x d.f p m.a.e B. Matching Expected Returns and Variances Average predicted returns L 2L 1M 3L 2M 3M 1W 2W 3W Average realized returns

8 Introduction Average predicted versus realized stock returns, vw value and momentum deciles H 1 L

9 Outline 1 The Model of the Firms 2 Econometric Methods 3 Data 4 GMM Estimation and Tests 5 Diagnostics: Dynamics of Factor Premiums 6 Out-of-sample Tests

10 Outline 1 The Model of the Firms 2 Econometric Methods 3 Data 4 GMM Estimation and Tests 5 Diagnostics: Dynamics of Factor Premiums 6 Out-of-sample Tests

11 The Model Setup Operating prots: Π(K it, W it, X it ) K it : Physical capital; W it : Working capital X it : A vector of exogenous shocks Constant returns to scale, Cobb-Douglas Capital accumulation: K it+1 = I it + (1 δ it )K it W it+1 = W it + W it Adjustment costs on physical (not working) capital: ( Iit ) 2 K it Φ(I it, K it ) = a 2 K it

12 The Model Optimal investments Optimal physical capital investment: E t [M t+1 r K ] = 1, in which it+1 the physical capital investment return: [ r K it+1 = (1 τ t+1 ) +(1 δ it+1 ) Y γ it+1 K K it+1 + a 2 ( Iit+1 K it+1 ) 2 ] + τ t+1 δ it+1 [ 1 + (1 τ t+1 )a ( Iit+1 K it+1 )] 1 + (1 τ t )a ( Iit K it ) Optimal working capital investment: E t [M t+1 r W ] = 1, in which it+1 the working capital investment return: r J it (1 τ t+1 )γ W Y it+1 W it+1

13 The Model A supply theory of asset pricing The weighted average of the investment returns equals the weighted average of the cost of equity and after-tax cost of debt: w K it r K it+1 + (1 w K it )r W it+1 = w B it r Ba it+1 + (1 w B it ) r S it+1 w K it = q it K it+1 /(q it K it+1 + W it+1 ) and w B it = B it+1/(p it + B it+1 ) Modigliani and Miller (1958, Proposition III) The investment CAPM: r S it+1 = w it K r K + (1 w K it+1 it )r W w B it+1 it r Ba it+1 1 wit B }{{} The fundamental return, rit+1 F

14 Outline 1 The Model of the Firms 2 Econometric Methods 3 Data 4 GMM Estimation and Tests 5 Diagnostics: Dynamics of Factor Premiums 6 Out-of-sample Tests

15 Econometric Methods Generalized Method of Moments, GMM Test the expected return implications of the investment CAPM: E[r S pt+1 r F pt+1] =, r S : Portfolio p's stock return, r F : The fundamental return pt+1 pt+1 The investment CAPM alpha: α p = E T [r S r F ], with pt+1 pt+1 E T [ ] the sample mean The supply counterpart of Hansen and Singleton (1982)

16 Econometric Methods A technical point γ K and γ W enter the moment condition only in the form of γ = γ K + γ W : w K it r K it+1 + (1 w K it )r W it+1 = (1 τ t+1 )(γ K + γ W )Y it+1 /(K it+1 + W it+1 ) q it K it+1 /(K it+1 + W it+1 ) + W it+1 /(K it+1 + W it+1 ) + w K it (1 τ t+1 )(a/2) (I it+1 /K it+1 ) 2 + τ t+1 δ it+1 + (1 δ it+1 )q it+1 q it +(1 w K it ) The 2-capital model as parsimonious as the physical capital model

17 Econometric Methods GMM methodology, based on Hansen (1982) Let c (γ, a), g T the sample moments, D = g T / c The GMM objective function: g T Wg T, in which W = I Var(^c) = (D WD) 1 D WSWD(D WD) 1 /T Var(g T ) = [ I D(D WD) 1 D W ] S [ I D(D WD) 1 D W ] /T The overidentication test: g T [var(g T )] + g T χ 2 (# moments # parameters)

18 Econometric Methods Aggregation in prior studies Portfolio-level fundamental returns are constructed from portfolio-level accounting variables aggregated from the rm level: [ Npt i=1 E w iptr S ] ( ipt+1 ) r F γ pt+1 K, a; Y pt+1, K pt+1, I pt+1, δ pt+1, I pt, K pt, r Ba, w B = pt+1 pt N pt : The number of rms in portfolio p at the start of t, w ipt : Stock i's weight in portfolio p, r S : The return of stock i in ipt+1 p over time t, r F : The fundamental return of p pt+1 Aggregating rm-level characteristics to the portfolio level: I pt+1 = N pt I i=1 ipt+1, wpt B = N pt B i=1 ipt+1/ N pt (P i=1 ipt + B ipt+1 ), etc

19 Econometric Methods Exact aggregation Construct rm-level fundamental returns from rm-level accounting variables, then aggregate to portfolio-level fundamental returns: [ Npt i=1 E w iptr S N pt ( ipt+1 r F ipt+1 i=1 w ipt γ, a; Y ipt+1, K ipt+1, I ipt+1, δ ipt+1, I ipt, K ipt, r Ba ipt+1, w B ipt ) ] = Why? r F ipt+1 : Firm i's fundamental return, r F pt+1 varies with w ipt Economics: Firms can make dierent investment choices Econometrics: The substantial rm-level heterogeneity helps identify structural parameters

20 Outline 1 The Model of the Firms 2 Econometric Methods 3 Data 4 GMM Estimation and Tests 5 Diagnostics: Dynamics of Factor Premiums 6 Out-of-sample Tests

21 Data Testing deciles 4 testing deciles formed on: Book-to-market: Bm Momentum (prior 11-month returns, 1-month horizon): R 11 Asset growth: I/A Return on equity: Roe NYSE breakpoints and value-weighted returns

22 Data Average returns of the 4 testing deciles, January 1967June 217 L H H L The Bm deciles R t R The R 11 deciles R t R The I/A deciles R t R The Roe deciles R t R

23 Data Variable measurement Y it : Sales K it : Net property, plant, and equipment W it : Current assets B it+1 : Long-term debt plus short-term debt (zero if missing) P it : Market equity, from CRSP τ t : The statutory corporate income tax rate from the Commerce Cleaning House δ it : The amount of depreciation and amortization minus amortization, scaled by net PPE I it : K it+1 (1 δ it )K it rit B : Total interest and related expenses, scaled by total debt

24 Data Timing alignment Construct monthly fundamental returns from annual accounting variables to match with monthly stock returns For each month t, take rm-level accounting variables from the scal year end closest to month t to measure time-t variables in the model, and to take accounting variables from the subsequent scal year end to measure time-t + 1 variables Compound the portfolio stock returns within a 12-month rolling window with month t in the middle of the window to match with the portfolio fundamental return for month t

25 Data Descriptive statistics, rm-level variables in the fundamental returns, June 1967December 216 Mean σ 5% 25% 5% 75% 95% I it /K it W it /W it Y it+1 /K it Y it+1 /W it Y it+1 /(K it+1 + W it+1 ) K it+1 /(K it+1 + W it+1 ) wit B δ it rit+1 B

26 Data Correlation matrix, rm-level variables in the fundamental returns, June 1967December 216 I it+1 W it K it+1 W it W it+1 W it+1 Y it+1 K it+1 Y it+1 W it+1 Y it+1 K it+1 +W it+1 K it+1 K it+1 +W it+1 w B it δ it+1 r B it+1 I it /K it I it+1 /K it W it /W it W it+1 /W it Y it+1 /K it Y it+1 /W it Y it+1 /(K it+1 + W it+1 ) K it+1 /(K it+1 + W it+1 ) wit B.33.4 δ it+1.6

27 Data Histograms, rm versus portfolio I it /K it

28 Data Histograms, rm versus portfolio K it+1 /(K it+1 + W it+1 )

29 Data Histograms, rm versus portfolio Y it+1 /(K it+1 + W it+1 )

30 Data Histograms, rm versus portfolio Y it+1 /K it

31 Data Histograms, rm versus portfolio r B it

32 Outline 1 The Model of the Firms 2 Econometric Methods 3 Data 4 GMM Estimation and Tests 5 Diagnostics: Dynamics of Factor Premiums 6 Out-of-sample Tests

33 GMM Estimation and Tests Replication: The physical capital model estimated at the portfolio level d.f. γ K [γ K ] a [a] α α H L p Bm R I/A Roe Bm-R I/A-Roe Bm-R 11 -I/A-Roe

34 GMM Estimation and Tests Average predicted versus realized stock returns, Bm-R 11, Bm-R 11 -I/A-Roe, the physical capital model at the portfolio level L H 15 1 L H

35 GMM Estimation and Tests The 2-capital model estimated at the rm level d.f. γ [γ] a [a] α α H L p Bm R I/A Roe Bm-R I/A-Roe Bm-R 11 -I/A-Roe

36 GMM Estimation and Tests Average predicted versus realized stock returns, Bm-R 11, Bm-R 11 -I/A-Roe, the 2-capital model at the rm level H 15 H 1 L 1 L

37 GMM Estimation and Tests Intuition The physical capital investment return: [ Y (1 τ t+1 ) γ it+1 K r I it+1 = +(1 δ it+1 ) K it+1 + a 2 ( Iit+1 K it+1 ) 2 ] + τ t+1 δ it+1 [ 1 + (1 τ t+1 )a ( Iit+1 K it+1 )] 1 + (1 τ t )a ( Iit K it ) The tug of war between current investment-to-physical capital and expected investment-to-physical capital

38 GMM Estimation and Tests Comparative statics on the high-minus-low investment CAPM alphas Bm R 11 I/A Roe Benchmark I it /K it I it+1 /K it Y it+1 /(K it+1 + W it+1 )

39 GMM Estimation and Tests The impact of aggregation, Bm-R 11, Bm-R 11 -I/A-Roe, the 2-capital model at the portfolio level L H 15 L H

40 GMM Estimation and Tests The impact of capital heterogeneity, Bm-R 11, Bm-R 11 -I/A-Roe, the physical capital model at the rm level L H 15 1 L H

41 Outline 1 The Model of the Firms 2 Econometric Methods 3 Data 4 GMM Estimation and Tests 5 Diagnostics: Dynamics of Factor Premiums 6 Out-of-sample Tests

42 Diagnostics Correlations between the stock returns of various leads and lags and the contemporaneous fundamental return, r F it r S it 6 Correlations of the stock returns with the fundamental returns, r F it r S it 36 r S it 24 r S it 12 r S it 3 r S it r S it+3 r S it+12 r S it+24 r S it+36 r S it+6 Firms.2 c.3 c.3 c.2 c.1 c.11 c.12 c.5 c..1.1 a Port.5 a.9 c.5 a.9 c.17 c.19 c.2 c.12 c.8 c.12 c.11 c Correlations between the stock and fundamental returns across the testing deciles L H H L Bm b.2 a c R 11.2 b c.14 a I/A.19 b c.42 c Roe a

43 Diagnostics The long-term dynamics of the value premium, the stock and fundamental returns in event-time

44 Diagnostics The short-term dynamics of the momentum premium, the stock and fundamental returns in event-time

45 Diagnostics The long-term dynamics of the investment premium, the stock and fundamental returns in event-time

46 Diagnostics The short-term dynamics of the Roe premium, the stock and fundamental returns in event-time

47 Diagnostics Time series, the stock and fundamental value premium, corr = , 21/

48 Diagnostics Time series, the stock and fundamental momentum premium, corr = , 1979/8 8, 1998/7 85, 1999/8 6 82, 23/3 168, 29/

49 Diagnostics Time series, the stock and fundamental investment premium, corr =

50 Diagnostics Time series, the stock and fundamental Roe premium, corr = , 1999/

51 Diagnostics Higher moments Bm L H H L σ r S c r F c S k r S r F K u r S r F R 11 L H H L σ r S c r F c S k r S a r F a K u r S c r F b

52 Diagnostics Higher moments I/A L H H L σ r S c r F c S k r S r F K u r S r F Roe L H H L σ r S c r F c S k r S a r F K u r S c r F c

53 Outline 1 The Model of the Firms 2 Econometric Methods 3 Data 4 GMM Estimation and Tests 5 Diagnostics: Dynamics of Factor Premiums 6 Out-of-sample Tests

54 Out-of-sample Tests Recursive estimation: The 1st window from 1967/6 to 1985/7, add 1 month at a time until 217/12, time series of γ and a estimates

55 Out-of-sample Tests 1-month-ahead ts, stock versus fundamental returns, the 2-capital model at the rm level, the physical capital model at the portfolio level, the q-factor model L L H 15 1 H 15 1 L H

56 Out-of-sample Tests Expected return estimates, pooled panel forecasting regressions, WLS, June 1967December 217 Investment-to-capital, I t+1/k t+1 Annual sales growth, Y t+1/y t Slope [t] R 2 Slope [t] R 2 log(q t) gq 1 Y Y t/(k t + W t) gq 2 Y I t/k t gq 3 Y gq 4 Y Industry xed eects Yes Yes

57 Out-of-sample Tests Deciles formed on the expected return estimates, July 1985December 217 h L H H L The two-capital model estimated at the rm level

58 Out-of-sample Tests Deciles formed on the expected return estimates, July 1985December 217 h L H H L The physical capital model estimated at the portfolio level

59 Out-of-sample Tests Deciles formed on the expected return estimates, July 1985December 217, evidence consistent with Fama and French (1997) h L H H L The q-factor model

60 Conclusion Gonçalves, Xue, and Zhang (218) A detailed treatment of aggregation and capital heterogeneity substantially improves the performance of the investment CAPM Future work: The fundamental cost of capital