Term Structure Models with Differences in Beliefs. Imperial College and University of Chicago

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1 Term Structure Models with Differences in Beliefs Andrea Buraschi Imperial College and University of Chicago Paul Whelan Imperial College

2 Term Structure Puzzles Several properties of bond prices are difficult to reconcile with traditional models: 1. Predictability: violations of the EH: (a) Fama and Bliss (1987), Cochrane and Piazzesi (2005) for excess returns; (b) Campbell and Schiller (1991) neg regression coefficients; (c) also predictability in high-freq [Mueller, Vedolin, Yen (2011)] 2. Volatility anomalies: (a) long term bond yields appear too volatile to accord with traditional rational expectations models; (b) the term structure of volatility is itself a complex object. [Shiller (1979), Piazzesi (2005), and Piazzesi and Schneider (2006)]. 3. Statistical properties of bonds: (a) essentially affine models cannot simultaneously match conditional first and second moments of yields; (b) traditional affine models imply perfect spanning which is strongly rejected in the data. [Duffee(2002), Duffee(2008), Joslin, Priebsch, and Singleton (2011)]. 4. Unspanned stochastic volatility: Bonds do not span interest rate derivatives [Collin-Dufresne and Goldstein (2002a), Heidari and Wu (2003a), Li and Zhao (2006)].

3 Our Contribution We investigate empirically a specific economic channel: The properties of the SDF change when one moves from single agent to multiple agent economy; In particular: 1. Bond Risk Premia 2. Bond Volatility 3. Bond Trading Activity Builds on earlier work in the Literature on Differences in Beliefs: Theory: Harris and Raviv (1993), Detemple and Murthy (1994) and Zapatero (1998), Buraschi and Jiltsov (2006) Scheinkman and Xiong (2003), Dumas, Kurshev and Uppal (2009), Bhamra and Uppal (2009). Empirical: [Equity] Diether, Malloy, and Scherbina (2002); [Correlation Risk Premium] Buraschi, Trojani and Vedolin (2009), [FX] Buraschi, Beber, Breeden (2010), [Credit] Buraschi, Trojani, Vedolin (2009). [Bonds]: Xiong and Yan (2010) who study speculative trading regarding the central banks inflation target.

4 Heterogeneous Belief Structure Consider an economy with two agents with different subjective conditional probability measures dq a t and dq b t. Q a u e u e u Q b u Q b u > Q a u Pessimist Optimist Q a d e d Q b d e d Q b d < Qa d If absolute continuous, a convenient way to summarize difference in subjective measures (disagreement) is the Radon-Nikodym derivative process η = dqb dq a ; if X t be an I t-measurable random variable then «E b (X T I t) = E a ηt X T I t. η t

5 The Homogeneous Benchmark Economy Assume u (c t ) = e ρt c γ t and dd t /D t = β g t dt + σ D dw D t dg t = κ g (g t θ)dt + σ g dw g t. Common beliefs: dm /M = r t dt κ dw t, since M t = u (D t ): r t = δ + γβ g t 1 2 γ(1 + γ)σ2 D. If growth rates are constant, so are interest rates and the term structure is flat.

6 The Homogeneous Benchmark Economy Assume u (c t ) = e ρt c γ t and dd t /D t = β g t dt + σ D dw D t dg t = κ g (g t θ)dt + σ g dw g t. Common beliefs: dm /M = r t dt κ dw t, since M t = u (D t ): r t = δ + γβ g t 1 2 γ(1 + γ)σ2 D. If growth rates are constant, so are interest rates and the term structure is flat. When gt is stochastic: Excess returns: (T t) B t = Et rx (T ) [ ] M T = exp [A h (t, T ) + G h (t, T )g t ] M t t,t+dt = γg(t, T )σ Dσ g E ( dwt D dwt g ). Very tight connection between volatility and risk compensation

7 Introduce Heterogeneity Growth rates are unobservable; Assume two factors: g t = [g 1 t, g 2 t ] dg i t = κ i g i (g i t θ i g )dt + σ i g dŵ g i t. suppose agents agree on σ i g and θ i g but disagree on κ i g i. Since gt is unobservable it is (a) hard to agree on it s dynamics; and (b) the relative importance of long-run versus business cycle components [see Hansen, Heaton and Li (2008); Pastor and Stambaugh (2006)]. Difference in beliefs summarized by η t = dpb t, so that dp b «E b (X T I t) = E a ηt X T I t. η t

8 Immediate Implications Example. (Xiong and Yan (2010)): Agents have ex-ante incentives to trade with each others. If, u(c i ) = e ρ(t t) ln c i t, agents trade until their wealth ratio is equal to W b T /W a T = η T Complete markets imply perfect risk sharing only if dq a t = dq b t. Furthermore, in equilibrium (T t) B t = 1 (T t),a B t + η t B 1 + η t 1 + η t (T t),b t Even if ηt were constant, bond prices in the heterogeneous economy would not be affine! Sketch Consider tradable asset with terminal payoff B T. In equilibrium, agents must agree on its value. Under logarithmic preferences, this requires E b t ( cb t c b T B T ) = Et a ( ca t c T a " Define W t b W T b B T = B T, then Et b ( B T ) = Et a B T ). From Merton, ct i = ρw t i, so that E t b ( W t b W T b B T ) = Et a ( W t a W T a B T ). ( W T b /W T a W t b /W t a ) B T #, so that η T η t = W T b /W T a W t b /W t a.

9 A General Model Allow for Learning. Two types of signals in the literature: First-order signals: a process S i t whose drift is correlated with the conditional first moments of ĝ t or E t [dd t ] [Detemple (1986), Veronesi (2000), Buraschi and Jiltsov, 2006]. dg t = κ g (g t θ)dt + σ g dw g t, dst i = φg t + (1 φ)ε i t dt + σsdw i t Si. Second-order signals: a process S j t whose Brownians are correlated with the second moments of ĝ t or dd t 2 [Scheinman and Xiong, 2003, Dumas, Krushev and Uppal, 2009, Xiong and Yan, 2010]. dg t = κ g (g t θ)dt + σ g dw g ds j t = σ j sφdw g t + σ j s t, p 1 φ2 dw Sj t. Filtering using first versus second order filtering has important pricing implications.

10 Learning Conditional on priors agents are rational. State-space representation: X t = [log D t, S 1 t, S 2 t ] and µ t = [g 1 t, g 2 t, ε], dx t = (A 0 + A 1 µ t ) dt + BdW X t dµ t = (a 0 + a 1 µ t ) dt + bdw µ t, Lemma. (Beliefs) The posteriors solve: dm n t = (a 0 + a 1 m t )dt + υ t A 1B 1 dŵ X,n t υ t = a 1 v t + υ t a 1 + bb υ t A 1(BB ) 1 A 1 υ t where dŵ X t = B 1 [dx t (A 0 + A 1 µ t ) dt]. Agents have different subjective Brownians dŵ t X,b = dŵ t X,a + B 1 A 1 (mt a mt b )dt. Define Ψ t B 1 A 1 (m a t m b t ) as the standardized difference in beliefs.

11 Equilibrium Problem max {ci,m i } E i o 0 ϱ t u(c i t)dt s.t. E i [ 0 M i 0 t c i t et] i dt 0; i ci t = D t for t. Solution: η t = α b u a(c t ) α a u b (c t) = Ma t M b. t dη t /η t = Ψ t dŵ X,a t Disagreement Ψt affects the M t process, thus directly bond prices.

12 Bond Market Implications Stochastic discount factor dm t /M t = r f (t, Ψ t )dt κ (t, Ψ t )dŵ X, t (T t) B t = Et (M T /M t ) M t = ϱ t Dt γ 1 + (αη }{{} t ) 1/γ }{{} Lucas term Disagreement γ

13 Interest Rate From the drift of dm t : risk free rate r f = ρ + γβ (ω a (t)ĝt a + ω b (t)ĝt b ) 1 }{{} 2 γ(γ + 1)σ2 D + γ 1 }{{} 2γ ω a(t)ω b (t)ψ t Ψ t, Consensus Aggregation Bias }{{} Precautionary Savings Differences in Beliefs where ω i (t) = c i t/d t is investor s i total consumption share. Standard Vasicek economy: rf = ρ + γβ ĝ t 1 2 γ(γ + 1)σ2 D. When γ > 1 interest rates are increasing in Ψt.

14 Risk Premia Vasicek: only priced shocks are dw D (t) and since dd t /D t is homoskedastic, the price of risk is constant. Here, more interestingly, the dynamics of dm t also depend on dη t. Relative consumption is stochastic: Optimistic (pessimistic) investors consume more (less) in states of high aggregate cash flows, at a lower (higher) marginal utility, because they perceive those states as more (less) likely. This also implies that the consumption volatility of the optimist is higher than the pessimist. Or... κ a(t) = γσ D + ω b (t)ψ t and κ b(t) = γσ D ω a (t)ψ t. M i t = M i t ω i (t) γ, with M i t 1 α i ϱ t D γ t Risk does not cancel out at the level of representative agent:

15 Disagreement drives price of risk Theorem The price of risk is affected directly by disagreement: κ g (t) = γσ D + ω b (t)ψ g (t) κ s (t) = ω b (t)ψ s (t), which implies that bond excess returns explicitly depend on the dynamics of Ψ t. This channels are absent in Jouini and Napp (2010): constant beliefs and no learning; or in Xiong and Yan (2010) with γ = 1. Different than in the Vasicek benchmark economy the price of risk is time varying.

16 ... and of signals Corollary [Dimensionality of the State Space] Suppose that agents use N + M signals for inference. Let N be the number of signals correlated with expected innovations (first-order signals) and M being the number of signals correlated with unexpected innovations of the fundamentals (second-order signals). Then, there are 1 + N priced state variables. Disagreements on first-order signals, in addition to disagreement on fundamentals, are priced. Second order signals are not priced instantaneously. Signals that affect the conditional first moment of the growth rates enter directly as state variables in agents belief-dependent optimal consumption plans. Thus, this channel is absent in the economies studied in Dumas, Krushev, Uppal (2010) where κ s (t) = 0. Intuition: Second order filtering requires no change of measure; thus, no risk sharing on information flow.

17 The Term Structure of Bond Prices Theorem [The Term Structure of Bond Prices] Exponential linear-quadratic solution in m D and Ψ. Bond prices are equal to the product of two deterministic functions: 1. The first is exponentially affine in the posterior growth rate of the endowment; 2. the second is exponentially quadratic in the level of differences in beliefs: B(t, T ) = ϱ T t F m a D (m a D, t, T ; γ)g(t, T, γ; ψ g ; ψ s), F m a D (m a D, τ, ɛ) = exp(a m a D (ɛ, τ) + B m a D (ɛ, τ)m a D), F ψg,ψ S = exp(a Ψ (τ) + B Ψ (τ)ψ t + Ψ tc(τ)ψ t).

18 Testable Implications Hypothesis 1: Expected Returns rx τ t,t+dt = γb m a D (τ)σ D σ g E dŵ D t dŵ g t {z } Learning 1. First term homoskedastic. + ω b (t)ψ g (t)σb,d(t) T {z } DiB fundamentals 2. Second term: disagreement on fundamentals (on g t and potentially π t) + ω b (t)ψ s(t)σ T B,S(t) {z } DiB signals 3. Third term: disagreement on first-order signals (which we proxy using with disagreement about future bond yields).

19 Testable Implications Hypothesis 2: Disagreement about (future) prices By no-arbitrage all agents must agree on B(t, T ) a time t: B a (t, T ) = B b (t, T ). But for t < τ < T disagreement can extend to future bond prices: E a t [B(τ, T )] E b t [B(τ, T )]. Dispersion in price forecasts is due to η t(ψ g t, ψt S ) which depends h on disagreement i about both fundamentals and signals: Et a [B(τ, T )] = Et b ητ η t B(τ, T ).

20 Testable Implications Hypothesis 3: Stochastic Volatility Disagreement generate endogenous stochastic volatility of bond yields even if the endowment process is homoskedastic. When γ > 1, changes in disagreement directly affect both the level and slope of the term structure of volatility of (changes in) bond yields: Hypothesis 4: Spanning σ T B,D(t) = α 0,D (τ) + α 1,D (τ)ψ g + α 2,D (τ)ψ s σ T B,S(t) = α 0,S (τ) + α 1,S (τ)ψ g + α 2,S (τ)ψ s Heterogeneous beliefs generate dynamics in bond prices suggestive of unspanned factors. Hypothesis 5: Trade In a heterogeneous agent economy with difference in belief agents risk share by trading state contingent claims; thus, shocks to disagreement induce an increase in trading activity.

21 Data Description

22 A Unique Dataset on the Formation of Expectations BlueChip Economic Indicators: Single biggest data set of agent s expectations. Unfortunately, digital copies of BCEI are only available since We sourced the paper archive directly from Wolters Kluwer and digitize the data: it now includes 350,000 data points of individual observations on: 1. Real: Real GDP, Personal Income Expenditure, Unemployment, Industrial Production, Housing Starts, Non-residential Investment, Corporate Profits, Auto and Truck Sales 2. Nominal: Consumer Price Inflation, GDP Price index. 3. Monetary: 3 Month bank discount rate; 10Y Treasury Rate. We have data on two sets of forecasts for January December 2011: 1. Short-term: an average for the remaining period of the current calendar year. 2. Long-term: an average for the following year.

23 Modelling constant maturity DiB Important advantages in new dataset: 1. Monthly frequency (vs. Quarterly frequency of SPF) 2. Large Cross-section Agents (never less than 40 vs. 9 of SPF) 3. One agency since SPF changed questions and/or the forecast horizon Can use individual short & long forecast to obtain implied constant maturity forecast with no seasonality (lots of seasonality in SPF, which people filter using X-12 ARIMA - TRAMO/SEATS filters). short term forecast long term forecast January April December

24 CPI GDPI 0.6 Ψt date

25 RGDP IP 1 Ψt date

26 0.5 SR LR 0.4 Ψt date

27 Hypothesis 3: Return Predictability rx (5) t,t+12 = const + 3X β i ψ t( ) + i=1 4X γ i E t( ) + i=1 11X i=1 φ i Macro t( ) + δslope t + ε t,t+12, Ψ π Ψ g Ψ s Slope t E(π) E(g) E(LR) E(SR) F 1 F 8 σ π σ g ρ π,g R 2 (i) (ii) (iii) (iv) (v) (vi) (vii)

28 Hypothesis 3: Price Dispersion Hypothesis 2 questions the relative importance of disagreement on fundamentals versus disagreement on future prices. This is important since disagreement on future prices cannot exist in ambiguity models even if disagreement on fundamentals serves as a proxy for Knightian uncertainty: ψt s = 0.06 ψπ,t ψ g,t (ψ π,t) ( 0.49) (1.22) (0.01) (1.90) (ψg,t) ( 2.44) (ψπ,t ψg,t) + εs t R 2 = 0.14 ψt s = 0.02E t[π] 0.00 E t[g] E t[lr] E t[sr] + ε s t (1.17) ( 0.11) ( 0.65) (0.58) R 2 = 0.07 This results is difficult to reconcile with single agent ambiguity models. Moreover, within the family of heterogeneous beliefs models, we learn that the cross-section of beliefs on asset prices is richer than what can be explained by disagreement on fundamentals alone, highlighting the potential importance of studying learning models with first order signals.

29 Hypothesis 3: Volatility σ slope t,t+1 = const + ασslope t 1,t + 3X β i ψ t( ) + i=1 2X γ i E t( ) + i=1 3X φ i Macro t( ) + ε t+1, i=1 Ψ π Ψ g Ψ s E(π) E(g) E(LR) E(SR) F 1 F 8 σ π σ g ρ π,g σ slope t 1,t R 2 (i) (ii) (iii) (iv) (v) (vi)

30 Hypothesis 4: Spanning 5X ψt i = const + β i PCt i + ε i t, i=1 PC 1 PC 2 PC 3 PC 4 PC 5 R 2 Ψ π Ψ g Ψ s

31 Hypothesis 4: Spanning rx (n) t,t+12 = const(n) + 3X i=1 β (n) i ψun + ε (n) t+12, regressor rx (2) rx (3) rx (4) rx (5) ψun π (0.25) (0.22) ( ) ψ g UN (2.47) (2.65) (2.39) (2.05) ψun s (5.71) (5.42) (5.04) (4.69) R

32 Hypothesis 4: Spanning 4X Hidden t = const + β i ψun i + ε i t, i=1 ψ π UN ψ g UN ψ s UN R 2 Hidden t (-0.23) (-0.14) (2.24)

33 Hypothesis 4: Spanning rx (n) t,t+12 = const(n) + 3X i=1 β (n) i ψab + ε (n) t+12, regressor rx (2) rx (3) rx (4) rx (5) ψab π (0.39) (0.27) (0.43) (0.21) ψ g AB (2.05) (1.96) (1.55) (1.19) ψab s (3.48) (3.60) (3.42) (3.29) R

34 Hypothesis 5: Trade 3X Vol t,t+1 = const + β i ψ t( ) + γvol t,t 1 + ε t,t+1., i=1 regressor (i) (ii) (iii) (iv) (v) (vi) ψ π (8.88) (4.07) (3.83) (2.89) ψ g (3.62) (2.69) (2.95) (2.10) ψ s (-1.14) (-0.89) (0.05) Vol t,t (8.98) R Vol t,t+1 is the log ratio of the number of monthly transaction volumes between primary dealers and secondary customers for Treasury Bills versus coupon paying securities due in more than 6 years but less than or equal to 11 years.

35 Conclusions Existence of time-varying bond risk premia is one of the most interesting and challenging topic in fixed income. The weak empirical link between observable macro variables and bond returns has been a long standing puzzle. In this study, we learn that: 1. A simple parsimonious specification with Ψ π t, Ψ g t and Ψ s t helps explain a lot of bond risk premia. 2. Ψ π,g t is consistent with simple RE-heterogeneous model with risk sharing (also affects vol and trade) and explains CP factor. 3. Ψ s t explain predictability and slope effects to the term structure of vol but it is not spanned or linked to trade: proxy monetary policy? 4. Very large R-square for excess bond returns ( between 35% and 40% ) and economically significant. 5. Robust to: (a) H1B errors; (b); reverse regressions; (c) lagged disagreement; (e) a host of alternatives ; and (d) stable link over time.