Why Surplus Consumption in the Habit Model May be Less Pe. May be Less Persistent than You Think

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1 Why Surplus Consumption in the Habit Model May be Less Persistent than You Think October 19th, 2009

2 Introduction: Habit Preferences Habit preferences: can generate a higher equity premium for a given curvature parameter, γ (Constantinides, 1990 and Sundaresan, 1989). Campbell and Cochrane (1999): an economy with i.i.d. consumption and a representative agent with external habit preferences. conditional volatility of the log consumption surplus (the price of risk) varies inversely with the log surplus. shock to the price of risk is close to perfectly negatively correlated with the shock to consumption growth. log surplus is a highly persistent process. Consistent with U.S. data: the price-dividend ratio is highly persistent. long-horizon stock returns are forecastable using the price-dividend ratio.

3 Introduction: Habit Preferences and the Value Premium U.S. data: On average value stocks have a higher mean return and higher CAPM α than growth stocks, but do not have a higher beta with the market. 1 Lettau and Wachter (2007) propose duration-based explanation: economy: state variable driving the price of risk is highly persistent. mean of consumption growth is slowly mean-reverting as in Bansal and Yaron (2004). vary the conditional correlation between price of risk and log consumption growth: large negative correlation generates growth premium for raw returns vs the value premium found in U.S. data. have to set correlation to zero to produce value premium. 2 Santos and Veronesi (2008) also show habit preferences deliver a growth premium, unless (excessive) cash flow heterogeneity across firms is introduced. Q: Can habit preferences generate value premium as in U.S. data? A: Yes! as long as surplus consumption is not too persistent.

4 Evidence against Persistent Habit in Micro Data Brunnermeier and Nagel (2006) find evidence against persistent habit in risky asset holdings: Test: in presence of persistent habit, risky asset holdings should increase in response to wealth increases. Reject persistent habit over CRRA. When habit is not persistent, this hypothesis no longer follows. Ravina (2007) finds evidence against persistent habit in credit card purchases: Internal habit depends on own consumption last quarter and external habit depends on current and last quarter s consumption in the city that the household lived. Estimates coefficient of lagged own consumption in internal habit as 0.5 Coefficient on current household city consumption in external habit is 0.29 These numbers are too high to be consistent with persistent habit assumed by CC, since persistent habit means that last period s consumption has very little effect on this period s habit.

5 Motivation for Less Persistent Surplus Consumption CC and LW: the 2 most recent years of consumption contribute a much smaller fraction (less than 26%) to the agent s habit level than all past consumption from more than 2 years ago. the 4 most recent years of consumption still contribute less to the agent s habit level than all past consumption from more than 4 years ago. Such high persistence seems counterintuitive. Our paper: assumes a less persistent state variable drives the price of risk, which would be implied by a less persistent (but more realistic) log surplus ratio. the 2 most recent years of consumption contribute more than 98% to the agent s habit level.

6 Our Main Results With less persistent surplus consumption: Can generate a value premium with habit preferences. Can still match high persistence of P/D ratio seen in U.S. data, since mean of log consumption growth is highly persistent as in BY. Can still match E[P/D] and Sharpe ratio of market as closely as LW. But: P/D s forecasting power for returns is low.

7 The Model Log dividend growth: d m t+1 = g m + z m t + ε m t+1 (1) Expected log consumption growth is g m + z m t where: zt+1 m = φ zzt m + ε z t+1 (2) Log dividend growth is levered version of log consumption growth: d t+1 = g + z t + ε d t+1 (3) where: g m δ m g zt m δ m z t ε m t+1 δ m ε d t+1 + ε u t+1 Price-of-risk driven by x t: x t+1 = (1 φ x) x + φ xx t + ε x t+1 (4) {ε d, ε z, ε x, ε u } multivariate normal, and independent over time.

8 SDF and Risk-free Rate We propose a reduced form of CC with persistent mean consumption growth, with stochastic discount factor: { Mt+1 LR = exp a + bz t 1 2 x t 2 x } t ε d t+1. (5) σ d Only the shock to dividend growth is priced. Risk-free rate: r f t ln (E t [M t+1 ]) = a bz t (6) To relate to LW and CC, who both assume constant interest rates, we assume this too and thus set b = 0.

9 Relation to CC (i.i.d. consumption growth) CC assume that a representative agent maximizes the utility function: [ ] E (δ cc ) t (Dt cc Ht cc ) 1 γcc 1 1 γ cc t=0 with i.i.d. consumption growth: d cc Log of surplus consumption ratio, s cc t ( D cc t Hcc t Dt cc (7) t+1 = g cc + ε d t+1 (8) ) log, has dynamics: st+1 cc = (1 φ cc s ) s cc + φ cc s st cc + λ cc (st cc )ε d t+1 (9) Specify sensitivity function λ cc (.) such that risk-free rate is constant and habit is predetermined at and near the steady state of st cc : { 1 S 1 2(s cc cc t s cc ) 1 st cc smax cc λ cc (s cc t ) = 0 s cc t s cc max (10) This implies the sdf: { } Mt+1 CC = exp log(δ cc ) γ cc g cc γ cc (φ cc s 1)(st cc s cc ) + (1 + λ cc (st cc ))ε d t+1 (11)

10 Relation to CC with persistent mean consumption growth So habit is predetermined by past consumption, specify dynamics for s: s t+1 = (1 φ s ) s + φ s s t + λ( s)z t + λ cc (s t )ε d t+1 (12) Implies sdf: t+1 = exp{ γ cc g cc + log(δ cc ) + γcc (1 φcc s ) +[ γ cc (1 + λ cc ( s)) z t ] }{{ 2 }}{{} b a }{{} rt f (13) M CC+ 1 2 γcc (1 φ cc s ) (1 2(st cc s cc )) γ cc (1 + λ cc (st cc )) }{{}}{{} xt 2 xt σ d d εt+1} x t = γ cc σ d (1 + λ cc (s cc t )) which we approximate as homoskedastic AR(1). As long as sensitivity function rarely zero, ρ d,x 1 and φ x φ cc s.

11 Relation to other models Nests LW, who don t distinguish between consumption and dividends: { Mt+1 LW = exp r f 1 2 x t 2 x } t ε d t+1 (14) σ d Setting σ x = 0, nests power utility with persistent mean consumption growth: M power t+1 = exp{ γg + log(δ) γz t γε d t+1} (15)

12 Relation between external habit and past consumption Now show log habit approximately moving average of lagged log consumption: Define h t log(h t), and apply a log-linear approximation to s t: ( ) [ ( )] ( ) s t log 1 e h d e h d + (h t d t) h d 1 e h d (16) Substitute in to law of motion for s, and set λ(s t) λ( s) = eh c, so that 1 eh d h t+1 is predetermined, depending on d t, d t 1,... but not d t+1, can show: h t+1 h d + (1 φ s) (φ s) j d t j + g (17) 1 φ j=0 s Can also show: Consumption Lag Habit Contribution (%) (yrs) LW Ours to > [ ] h t+1 h t g + (1 φ s) (d t h t) d h (18)

13 Zero-coupon Equity: P/D Ratio P m n,t time-t price of claim to zero-coupon market equity, paying off in n periods. Can show: P m n,t D m t = F (x t, z m t, n) = exp{a(n) + B x(n)x t + B z(n)z m t } (19) where: A(n) = A(n 1) + a + g m + B x(n 1) x(1 φ x) (20) (δm ) 2 σ 2 d + δ m Σ d,ε (G m n 1) G m n 1Σ ε,ε(g m n 1) B x(n) = φ xb x(n 1) δ m σ d 1 σ d Σ d,ε (G m n 1) (21) B z(n) = (1 + b)(1 φn z) 1 φ z (22) where Gn m [1 B x(n) B z(n)], Σ d,ε σ[ε d, ε], and Σ ε,ε σ 2 [ε] where ε [ε u ε x ε z ].

14 Market Equity: P/D Ratio and Returns By the law of one price: P m t = Pn,t m (23) Price-dividend ratio of market is sum of price-dividend ratios of zero-coupon claims: Pm t D m t = = n=1 n=1 Pn,t m (24) Dt m exp{a(n) + B x(n)x t + B z(n)zt m } (25) n=1 Returns are function of price-dividend ratios and dividend growth: R m t+1 Pm t+1 + D m t+1 = P m t ( ) ( ) P m t+1 /Dt+1 m + 1 D m t+1 P m t /D m t D m t (26) (27)

15 Sharing the Dividend: Forming Value/Growth Deciles Following LW, aggregate dividend split between 200 firms: share increases by 5.5% / quarter for 100 quarters, then declines at same rate for 100 quarters to original value: Firm 1's Share of Aggregate Dividend % Share of aggregate dividend Quarter At the start of each year firms are sorted into deciles from value to growth based on their annual price-dividend ratios.

16 Calibrating our Base case We follow LW by: Calibrating z m and d m processes, and r f, to the data. Assuming the aggregate consumption process is the same as the aggregate dividend process (we relax this later). We depart from LW in: Also: Setting ρ[ε d, ε x ] = 0.99 in the spirit of CC (LW set ρ[ε d, ε x ] = 0). Choosing φ x be low (0.14 annually or 0.61 quarterly). Choosing x and σ x so that E[P/D] and market Sharpe ratio are as close to the data as LW. Choosing ρ[ε x, ε z ] to ensure covariance matrix of shocks is positive definite.

17 Wedge cases: Distinguishing Consumption from Dividends Now consider log dividend growth to be a levered version of log consumption growth, as in Abel (1999). Keep σ m and σ m,z same as base case, to match the data moments. Annual correlation of log consumption and dividend growth is 0.55 in Bansal-Yaron s sample period. Requires x < 0 for P. Since x is positive in D CC, instead set correlation to 0.82, so P/D converges for range of x > 0. Joint {z m, d m } same: φ z, σ z, σ m, g m, ρ m,z unchanged from base case. Choose δ m = σm σ d (typical in literature). Also setting ρ[ε d, ε z ] = ρ[ε m, ε z ]: 1 There is an asset in the wedge cases with the same cash-flows and price as the market dividend in the base case. 2 Given σ z and σ m fixed, as ρ[ε d, ε m ] tends to 1, pricing implications for wedge cases converge to those for base case. Given δ m and ρ[ε d, ε z ] = ρ[ε m, ε z ], system of equations yields σ d, σ u, σ d,u and σ z,u. Iterate until obtained σm σ d value equals the δ m used to obtain it. σ[ε x, ε z ] and σ[ε x, ε u ] set so covariance matrix of shocks positive definite 2 Wedge cases differ only in x and σ x.

18 Model Parameters Variable Frequency LW Base Wedge 1 2 g m annual 2.28% 2.28% 2.28% 2.28% r f = a annual 1.93% 1.93% 1.93% 1.93% b x quarterly φ z annual φ x annual σ m quarterly σ d quarterly σ z quarterly σ x quarterly σ u quarterly ρ m,z = ρ d,z quarterly ρ d,x quarterly ρ z,x quarterly ρ d,u quarterly ρ z,u quarterly ρ x,u quarterly δ m quarterly

19 Aggregate Moments Data same as Lettau-Wachter, annual Model simulated for 4 million quarters, by which point moments have converged, i.e. first and second halves of simulation yield similar moments. Returns, dividends and P/D ratios aggregated to annual frequencies. Moment Data LW Base Wedge 1 2 E[P m /D m ] σ[p m d m ] AC[p m d m ] E[R m R f ] σ[r m R f ] AC[R m R f ] Sharpe m AC[ d m ] σ[ d m ] p m d m still persistent when φ x = 0.14, since φ z = 0.91.

20 Value vs Growth Portfolios Portfolio E/P C/P B/M LW Base Wedge 1 2 Expected Excess Return: E[R i ] R f E[R i ] R f HML σ[r i R f ] HML Sharpe i HML CAPM α: R i t Rf t = α i + β i (R m t R f t ) + ɛ it α i HML β i HML Ri HML

21 Zero-Coupon Equity: Returns Zero-coupon returns: R m n,t time-t price of claim to zero-coupon market equity, paying off in n periods. r m n,t log(r m n,t). It can be shown that: r m n,t+1 = E t[r m n,t+1] + δ m ε d t+1 + ε u t+1 + B z(n 1)ε z t+1 (28) +B x(n 1)ε x t+1 σt 2 [rn,t+1] m = Cn 1Σ(C m n 1) m (29) where C m n 1 [δ m, 1, B z(n 1), B x(n 1)] and Σ = σ 2 [ε d ε u ε z ε x ]. Zero-coupon risk premium: ( [ R m ]) ln E n,t+1 t Rt f = E t[rn,t+1 m rt f ] σ2 t [rn,t+1] m (30) ( ) δ m σd 2 + σ d,u + B z(n 1)σ z,d + B x(n 1)σ x,d = x t (31) σ d

22 Intuition: LW s Value Premium when ρ[ε d, ε x ] = 0 ( σ RPn,t m 2 = d + B z (n 1)σ z,d σ d ) x t Given: Then: B z (n 1) positive for all n and increasing in n. σ z,d < 0. B z (n 1)σ z,d negative for all n and decreasing in n. E[RP m n,t] is declining in n. Value premium for expected excess returns, since value stocks have shorter duration cash flows than growth stocks (LW).

23 Zero-coupon Aggregate Equity: B z (n) and risk premium for LW B z(1 φz) LW Base Wedge 1 Wedge 2 risk premium LW Maturity (years) Maturity (years)

24 Intuition: LW s Growth Premium when ρ[ε d, ε x ] = 0.99 and x Very Persistent Given: Then: ( σ RPn,t m 2 = d + B x(n 1)σ x,d + B z(n 1)σ z,d B z(n)σ z,d negative for all n and decreasing in n. ρ x,d = x very persistent. σ d ) x t shock to x today impacts value of x for many periods in the future. B x(n 1) negative for all n and decreasing in n out to very large n. B x(n 1)σ x,d positive for all n and increasing in n out to very large n. Net effect: E[RP m n,t] increasing in n out to very large n: B x(n 1)σ x,d outweighs B z(n 1)σ z,d. Growth premium for expected excess returns (as LW found).

25 Zero-coupon Aggregate Equity: B x (n) and risk premium for high φ x B x LW LW ρ dx risk premium LW ρ dx LW Maturity (years) Maturity (years)

26 Intuition: Our Base case Value Premium when ρ[ε d, ε x ] = 0.99 and x Not Very Persistent Given: Then: ( σ RPn,t m 2 = d + B x(n 1)σ x,d + B z(n 1)σ z,d B z(n)σ z,d negative for all n and decreasing in n. ρ x,d = x not very persistent. σ d ) x t shock to x today impacts value of x for only a few periods in the future. B x(n 1) negative for all n and inverted hump-shaped in n with min. at small n. B x(n 1)σ x,d positive for all n and hump-shaped in n with max. at small n. Net effect: E[RP m n,t] hump-shaped with the max at a quite small n. Value premium for expected excess returns if the max is at a sufficiently small n.

27 Zero-coupon Aggregate Equity: B x (n) and risk premium B x LW Base Wedge 1 Wedge 2 risk premium LW Base Wedge 1 Wedge Maturity (years) Maturity (years)

28 Predictive Regressions Horizon (yrs) Data LW Base Wedge 1 2 H i=1 (r m t+i r f t+i ) = β 0 + β 1 (p m t d m t ) + ɛt β R H i=1 dm t+i = β 0 + β 1 (p m t d m t ) + ɛt β R H i=1 dm t+i = β 0 + β 1 z m t + ɛ t β R

29 Discussion Add second persistent state variable to increase predictability, e.g. persistent consumption growth volatility as in BY. Cointegrating relationship between log dividend growth and consumption growth, instead of levered. Make firms share process stochastic, instead of deterministic. Non-constant interest rates. CC + long-run risk directly: no closed form for P/D, and computationally more intensive.