Habits and Leverage. Tano Santos. Columbia Business School Columbia University. Pietro Veronesi. University of Chicago Booth School of Business

Transcription

1 Habits and Leverage Tano Santos Columbia Business School Columbia University Pietro Veronesi University of Chicago Booth School of Business 1st Annual Research Conference of Banco de España Bank of Spain - Madrid - August 31st September 1st 217

2 Motivation What determines the amount of aggregate credit and leverage (debt over GDP)? How does leverage relate to aggregate economic conditions? Is it procyclical or countercyclical? Adrian and Shin (21) How does leverage (and deleverage) affect individual household consumption and who levers up? What determines the cross section of leverage? Mian and Sufi (215) and Justiniano, Primiceri and Tambalotti (213) How does leverage relate to current and future financial conditions? Jordà, Schularick and Taylor (211) Does deleveraging lead to fire sales and asset deflation? Impact on aggregate economic activity? Shleifer and Vishny (211)

3 What we do Study a frictionless dynamic general equilibrium model featuring Heterogeneous agents External habit preferences Heterogeneous time varying risk-bearing capacity = leverage dynamics We focus on The dynamics of aggregate debt and leverage (debt over aggregate income) The relation between individual consumption and leverage Leverage patterns in the cross section. Leverage growth and future stock retuns Deleveraging and asset prices The stationary dynamics of the wealth distribution (not today!)

4 What we find - 1 Our model predicts: 1. Leverage in good times when prices and volatility 2. Leveraged agents enjoy a consumption boom in good times, followed by a consumption slump 3. High aggregate leverage should predict low future returns 4. Negative shocks = leveraged agents delever by fire-selling stocks 5. While leverage, debt/wealth ratio due to strong discount effects. 6. Wealth dispersion in good times

5 What we find - 2 Asset pricing Model aggregates to standard representative agent models with external habit, such (Campbell and Cochrane (1999) and Menzly, Santos, and Veronesi (24)). It can be calibrated to yield reasonable quantitative implications. Caveat: Cash-flow shocks vs. discount shocks The asset pricing implications are orthogonal to the cross sectional implications A peculiar feature of our model: The wealth distribution plays no role in the determination of asset prices Longstaff and Wang (212) and Chan and Kogan (24) But obviously asset prices determine the wealth distribution.

6 Literature Risk sharing Borch (1962), Dumas (1989), Wang (1996), Chan and Kogan (22), Bolton and Harris (213), Longstaff and Wang (213). Supply of safe assets Barro, Fdez-Villaverde,Levintal and Mollerus (216), Gorton and Ordoñez (213), Gennaioli, Shleifer and Vishny (212), Caballero and Fahri (216),... Household and leverage Justiniano, Primiceri and Tambalotti (213) and Mian and Sufi (215). Inequality, leverage and asset pricing Rajan (29), Johnson (212)

7 A word on terminology What is leverage? Leverage: L Debt Output Debt-to-assets: D Debt Assets Each measure has different properties Discount shocks, cash-flow shocks and leverage

8 Preferences and endowments Continuum of agents with external habit preferences: u (C i,t, X i,t, t) = e ρt log (C it X it ) Agents are heterogeneous in habit indices X it X it = g it (D t ) X jt dj (1) External Habit in Utility: Envy-the-Joneses Habits loadings g it are heterogeneous and time varying: g it = a i Y t + b i a i > and a i di = 1 (2) Y t : Recession Indicator Habits matter more in bad times. Each agents is endowed with a share w i of total wealth: w i di = 1

9 Aggregate output Aggregate output follows: dd t D t = µ D dt + σ D (Y t )dz t (3) σ D (Y t ) : Economic Uncertainty Stock and Watson (22) and Jurado, Ludvigson and Ng (215). Recession indicator Y t follows: dy t = k(y Y t )dt v Y t [ ddt D t E t ( )] ddt D t (4) Recession indicator: Y t after bad economic shocks, i.e. dd t D t < E t ( ddt D t ). v constant but we can handle more flexible specifications

10 Restrictions: (a) On Y t Y t > λ 1 for all t: σ D (Y t ) as Y t λ. Otherwise σ D (Y t ) general. (b) On endowments Endowment satisfy w i > a i(y λ) + λ 1 Y Ensures agents have enough wealth to ensure positive consumption over excess habit. But it induces a positive correlation between a i and w i.

11 Optimal risk sharing No consumption externalities = solve planner s problem U (D t, {X it }, t) = max C it φ i u (C it, X it, t) di subject to C it di = D t Pareto weights φ i > such that φ i di = 1. First order conditions: u C (C it, X it, t) = φ ie ρt C it X it = M t M t = Lagrange multiplier of resource constraint. Solve for C it, integrate across i, and obtain ( C it = (g it + φ i ) D t ) X jt dj C it if (i) excess output, or (ii) habit loading g it

12 An aside: An alternative representation of preferences Aggregate agents consumption C it di = D t and solve for excess output D t X it di = D t git di + 1 > Thus substituting in preferences u (C i,t, X i,t, t) = e ρt log (C it ψ it D t ) with ψ it g it git di + 1.

13 Optimal allocation Proposition. Initial conditions: Y = Y and normalize D = ρ. Then: (a) Pareto weights: φ i = w i Y a i (Y λ) + 1 λ (b) Consumption shares: s it C it D t = a i + (w i a i ) Y Y t (c) Planner s transfers to/from agents relative to endowments w i D t : ( ) Y τ it = (w i a i ) 1 D t Y t Thus w i a i > : High wealth w i or low habit loading a i = s it when Y t w i a i < : Low wealth w i or high habit loading a i = s it when Y t If v = (Y t constant) = s it = w i = usual log-utility case

14 Risk aversion (curvature): Risk aversion and risk sharing Curv it = C itu cc (C it, X it, t) u c (C it, X it, t) = 1 + a i (Y t λ) + λ 1 w i Y a i (Y λ) λ + 1 Cross-section: risk aversion if w i or a i Time-series: 1. All agents risk aversion if Y t 2. Risk aversion of i more if w i is low or a i is high Recall τ it = (w i a i ) ( ) Y 1 D t Y t Those agents with w i a i > insure agents with w i a i <

15 Decentralization - 1 Market structure I. Complete Arrow-Debreu markets P it = E t [ t ( Mτ M t ) ] C iτ dτ = ( ) ρai + [ρ (w i a i ) + kw i ] Y S t ρ (ρ + k) D t II. Dynamic replication with stocks and bonds III. Other market structures Dynamically complete markets N it P t + N itb t = P it

16 Decentralization - 2 Given price processes {P t, r t }, agents solve [ ] max {C it,n it,nit} E e ρt log (C it X it ) dt subject to dw it = N it (dp t + D t dt) + N itb t r t dt C it dt with W i, = w i P A competitive equilibrium is a set of stochastic processes for prices { P t, r t } and allocations {Ĉit, N it, N it } such that agents maximize their utilities, and good and financial markets clear Ĉ it di = D t, Nit di = 1, N it =.

17 Decentralization - 3 Proposition. Consider the following prices and allocations ( ρ + ky Y 1) t (Stock price) Pt = D t ρ (ρ + k) (Risk-free rate) r t = ρ + µ D (1 v)σ D (Y t ) 2 + k ( 1 Y Y 1 t (Stock position) Nit = a i + (ρ + k)(1 + v)(w i a i )H(Y t ) ) (Bond position) N it B t = v(w i a i )H(Y t )D t where H(Y t ) = Y Y 1 t ρ + k(1 + v)y Y 1 t > Then { P t, r t, N it, N it } is an equilibrium supporting the efficient allocation.

18 Aggregation and state price density Our model aggregates to Menzly, Santos, and Veronesi (24): Proposition. The equilibrium state price density which follows M t = e ρt Dt 1 Y t dm t /M t = r t dt σ M,t dz t with σ M,t = (1 + v)σ D (Y t ) As in Campbell and Cochrane (1999), define Surplus consumption ratio = S t = D t X it di D t = 1 Y t Y t = inverse surplus in MSV = local curvature of the utility function We use S t as state variable for notational convenience.

19 Stock prices and returns The price of a claim to aggregate output is P t = E t [ t ] M τ D τ dτ M t = ( ) ρ + ky St D t ρ (ρ + k) Double kick of economic shocks: D t = S t = P t Return volatility: σ P (S t ) = σ D (S t ) ( 1 + vky S ) t ρ + ky S t Risk premium: E t [dr P r(s t )dt] = σ M (S t )σ P (S t )dt Corollary. Asset prices are independent of the distribution of endowments and preferences.

20 Individual portfolios The position in bonds Nit B t and stocks N it are NitB t = v (w i a i ) H (S t ) D t ; N it = a i + (ρ + k)(1 + v) (w i a i ) H (S t ) where H (S t ) = Y S t ρ + k(1 + v)y S t > with S t H (S t ) Define Individual leverage: L i (S t ) N it B t D t = v (w i a i ) H (S t ) Debt-to-assets: D i (S t ) N it B t N it P t

21 Implications: Individual leverage, consumption, and the business cycle Results: Agents with w i a i > : 1. Take on leverage: N it B t < 2. Over-invest in stocks: N itp t W it > 1 3. Increase leverage, L i (S t ), in good times (as H (S t ) > ) when S t, their risk aversion, take on more aggregate risk 4. Enjoy high consumption share s it when leverage is high Leverage = higher return = higher consumption in good times Lower risk aversion = even higher leverage in good times 5. Suffer consumption decline after consumption boom Spatial interpretation: e.g. counties with high wealth w i or low habit a i Good times = leverage and consumption = but lower future growth. Identification: Debt overhang versus risk sharing

22 Implications: Active Trading Results (cntd.). Agents with w i a i > : 6. Increase stock holdings in good times (trend chasers) 7. Decrease stock holdings drastically in very bad times (H (S t ) concave) 8. Agents with high w i a i trade more in response to changes to S t than those with low w i a i. As P t agents deleverage by selling stocks Gives the appearance of selling pressure that makes stock price decline, especially in deep crisis. But no causal link between trading activity and prices

23 The supply of safe assets: Aggregate leverage dynamics Aggregate leverage: L(S t ) i:w i a i > L i (S t ) di = i:w i a i > N it B tdi D t = vk 1 H(S t ) Corollary: H (S t ) > = Aggregate leverage is pro-cyclical. When S t is high, agents with high w i and low a i have lower risk aversion = more willing to take on more of the aggregate risk = issue more risk-free debt to agents with higher risk aversion.

24 1 A. Aggregate Leverage B. Aggregate Stock Holdings of Levered Agents Leverage.6 Leverage Surplus Surplus

25 Financial intermediation Define l t = Q (S t ), where Q is some monotone transformation of S t. Let S t = q (l t ) Q 1 (l t ). Then, trivially, M t = e ρt Dt 1 St 1 = e ρt D 1 t q (l t ) 1 σ M,t = σ D,t + q (l t ) q (l t ) σ l,t Premia ( E t [dr i,t r t dt] = cov t dr i,t, dd ) t D t + q (l t ) q (l t ) cov t (dr i,t, dl t ) Caution: The consumption CAPM holds in our framework!

26 Define l t = L i (S t ) N it B t D t = v (ω i a i ) H (S t ) l t = D i (S t ) N it B t = σ W i (S t ) W i,t σ P (S t ) Corollary. The market price of risk λ L t = ql (l t ) q L (l t ) > and λd t = qd (l t ) q D (l t ) < Assets that pay when L t rises pay up in good times, when marginal utility is low high premium Assets that pay when D t rises pay up in bad times, when marginal utility is high low premium

27 Quantitative Predictions Table 1. Parameters and Moments Panel A. Parameters (MSV) ρ k Y λ v µ σ σ max Panel B. Moments ( ) E[R] Std(R) E[r f ] Std(r f ) E[P/D] Std[P/D] SR E[σ t ] Std(σ t ) Data 7.13% 16.55% 1.% 1.% % 1.41%.52% Model 7.% 24.58% 1.68% 5.84% % 1.42% 1.2% Panel C. P/D Predictability R 2 1 year 2 year 3 year 4 year 5 year Data 5.12% 8.25% 9.22% 9.59% 12.45% Model 9.22% 14.47% 17.65% 2.13% 21.87% We assume a specific functional form for the volatility of aggregate output. σ D (Y t ) = σ max (1 λy 1 t )

28 The Cross-Section of Agents Behavior: Who Borrows? Need to ensure preferences a i and endowments w i are such to satisfy constraint: Assume first a i U[, 2]. w i > a i(y λ) λ + 1 Y To ensure constraint is satisfied, simulate Pareto weights. We assume φ i are log-normal to obtain skewed distribution φ i = e σ wε i 1 2 σ2 w with ε i N(, 1) and σ w = 2. Thus, i φ idi = E CS [φ i ] = 1. Endowment is then: w i = g i + φ i Y

29 Endowment and Preference Distribution A x 1 Distribution of Habit Loadings ai B. Distribution of Endowments wi Habit Loadings ai 2 1 C. Relation between wi and ai 2 3 Endowment wi 4 5 D. Leveraged Agents 2 5 LEVERAGED AGENTS wi ai Habit Loadings ai UNLEVERAGED AGENTS Endowment wi Endowment wi 4 5

30 The cross section of D i (S t )

31 The cross section of D i (S t ): Not just the US, also in Spain

32 Consumption of Levered Agents.9 A. Consumption Shares Leverage No Leverage 8 B. Expected Consumption Growth Leverage No Leverage Consumption Share.6.5 Percent Surplus Surplus Consumption boom of levered agents during good times But expected negative consumption growth going forward

33 The cross section of individual consumption

34 Fire Sales in a Simulation Run.6 A. Surplus Consumption Ratio 8 B. Economic Uncertainty.4.2 Percent Calendar Time (Quarters) C. Price / Dividend Ratio Calendar Time (Quarters) E. Leverage and Stock Holdings Leverage Stocks Calendar Time (Quarters) Percent Calendar Time (Quarters) D. Return Volatility Calendar Time (Quarters) F. Aggregate Debt/Wealth Calendar Time (Quarters)

35 Proposition. Wealth-ouput ratios: The dynamics of the wealth distribution - 1 W it D t W Lev. it D t = 1 ρ = 1 ρ [ ] ρ ρ + k a ( ) i 1 Y St + wi Y S t [ ] ρ ρ + k K ( ) 1 Y St + (K + K 1 ) Y S t Agents with higher w i or lower a i see their wealth increase in good times because of higher investment in stocks Proposition. Let w i and a i be independent. Then the cross-sectional dispersion of wealth/output ratio is V ar CS t ( Wit D t ) = V arcs (a i ) (ρ + k) 2 ( 1 Y St ) 2 + V ar CS (w i ) ρ 2 ( Y St ) 2

36 The dynamics of the wealth distribution - 2: Leveraged vs. unleveraged agents Wealth/Output A. Wealth/Output Ratios Leverage No Leverage Wealth Shares B. Wealth Shares Leverage No Leverage Surplus Surplus 8 6 C. Expected Return Leverage No leverage 8 6 D. Volatility Leverage No Leverage Percent 4 Percent Surplus Surplus

37 The dynamics of the wealth distribution - 3: Preference heterogeneity 9 A. Wealth/Output Dispersion (St.Dev.).29 B. Wealth Share Dispersion (St.Dev.) Standard Deviation Standard Deviation Surplus Surplus

38 The dynamics of wealth and of the wealth distribution - 4: No Preference heterogeneity 7 A. Wealth/Output Dispersion (St.Dev.).2 B. Wealth/Stock Dispersion (St.Dev.) Dispersion/Output 4 3 Dispersion/Stock Surplus Surplus

39 Conclusions A frictionless dynamic general equilibrium model with heterogeneous agents and external habits seem consistent with many stylized facts. Risk sharing motives generate endogenous leverage dynamics Our model predicts: I. Leverage increases in good times when prices are high and volatility is low II. Leverage should predict low future return III. Leveraged agents enjoy a consumption boom in good times, followed by a consumption slump IV. Negative shocks = leveraged agents delever by fire-selling stocks V. While leverage, debt/wealth ratio due to strong discount effects. VI. Level wealth dispersion increases in good times, while relative wealth dispersion depends on preference heterogeneity Leverage dynamics is due to the differential impact of aggregate shocks on agents risk aversion.

40 Figure 1: Household total debt to disposable income and to real estate assets at market value. Annual: ; Quarterly: 26Q1-212Q1. Data Source: Household total debt and Real Estate at market value are from the Flow of Funds (Federal Reserve Board); Disposable Income is from the BEA and Real Estate at market values obtained by subtracting from Households and nonprofit organizations the corresponding entry for nonprofits Q1 26Q2 26Q3 26Q4 27Q1 27Q2 27Q3 27Q4 28Q1 28Q2 28Q3 28Q4 29Q1 29Q2 29Q3 29Q4 21Q1 21Q2 21Q3 21Q4 211Q1 211Q2 211Q3 211Q4 212Q1 Household Debt/Disposable Income Household Debt/Real Estate