Stock Price, Risk-free Rate and Learning

Stock Price, Risk-free Rate and Learning Tongbin Zhang Univeristat Autonoma de Barcelona and Barcelona GSE April 2016 Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 1 / 31

Introduction Research papers addressing the basic asset market facts often ignore the comovement between stock and short-term bond markets. For example: Campbell and Cochrane (1999), Bansal and Yaron (2004) and Adam, Marcet and Nicolini (2016) The comovement is important for investors asset allocation decision. And this comovement should also be well studied before exploring how to design monetary policy for stabilizing stock price uctuation. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 2 / 31

Roadmap In this paper In the US data, the comovement between stock and short-term bond markets is weak. Two consumption based asset pricing models with rational expectation generates counterfactual comovement. Model with learning, a relaxation of rational expectation assumption to allow "Internal Rationality" agents, can reproduce the weak comovement. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 3 / 31

Literature Review Campbell and Ammer (1993) adopt variance decomposition approach to argue that riskless rate is not important role in driving the volatility of excess stock return. Gali and Gambetti (2015) use the impulse response functions from time-varying VAR model to explore the response of stock price to exogenous monetary policy shock. Gali (2014) theoretically studies monetary policy and rational asset price bubbles. Shiller and Beltratti (1992) discover the puzzle about the comovement between stock price and nominal long-term bond yields. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 4 / 31

Illustrative Model A discreet time Gordon Model The representative risk-neutral agent uses risk-free rate to discount future dividend stream. P t = E t j=1 D t+j R t+j, Assume D t+1 /D t = aɛ d t, R t = R t P t = a R t a D t 1 + ɛ R t Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 5 / 31

Empirical Facts Basic stock market facts as Fact 0. The correlation between stock price-dividend ratio and risk-free rate as Fact 1. The statistics of variance decomposition as Fact 2. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 6 / 31

Fact 0 Statistics Estimate SE Quarterly mean stock return E rs 2.25 0.39 Mean PD ratio E PD 123.91 21.25 Std.dev. stock return σ rs 11.44 2.69 Std.dev. PD ratio σ PD 62.42 17.54 Autocorrel. PD ratio ρ PD, 1 0.97 0.02 Excess return reg. coe cient c5 2-0.0038 0.0013 R 2 of excess return regression R5 2 0.1772 0.0828 Mean risk-free rate E R 0.15 0.19 Std.dev. risk-free rate σ R 1.27 0.27 Mean dividend growth E D /D 0.41 0.18 Std. dev. dividend growth σ D /D 2.88 0.80 Table: The Statistics Regarding the Stock and Short-term Bond Markets (Sample Size: 1927:2-2007:1) Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 7 / 31

Fact 1 Statistics Estimate SE corr(pd, R) 0.069 0.12 Table: The Correlation between Risk-free Rate and Price-dividend Ratio (Sample Size: 1927:2-2007:1) Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 8 / 31

Fact 2 The method of variance decomposition follows Campbell and Ammer (1993) rs t = log( P t +D t P t 1 ) and e t = rs t r t e t+1 E t e t+1 = (E t+1 E t ) ( j=0 φ j d t+1+j e t+1 = e d,t+1 e r,t+1 e e,t+1 φ j r t+1+j j=0 ) φ j e t+1+j j=1 Statistics Estimate SE Var(e d ) 21.1% 0.242 Var(e r ) 4.4% 0.026 Var(e e ) 50.8% 0.257 Table: Variance Decomposition of Excess Stock Return Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 9 / 31

The External Habit Model U = E t=0 δ t (C t X t ) 1 γ 1 1 γ, where C t is consumption and X t is exogenous habit De ne surplus consumption ratio as S t = C t C t s t+1 = (1 φ)s + φs t + λ(s t )[ c t+1 E ( c t+1 )] Price-dividend ratio as P t D t (s t ) = E t [M t+1 D t+1 D t [1 + P t+1 D t+1 (s t )]] Risk-free rate as r f t = r f 0 B(s t s) X t Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 10 / 31

Simulated Moments for External Habit Model The parameter vector for MSM estimation Φ EH (δ,φ, g, σ) US Data External Habit Moment SE Moment t-stat corr(pd, R) 0.069 0.12-0.956 8.27* Var(e d ) 21.1% 0.242 18.8% 0.10 Var(e r ) 4.4% 0.026 1.1% 1.25 Var(e e ) 50.8% 0.257 154.5% -3.99* Table: Simulated Statistics of External Habit Model Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 11 / 31

The Long-Run Risk Model Recursive Preference V t = [(1 δ)c 1 γ θ t + δ(e t [V 1 γ t+1 ]) 1 θ ] 1 θ γ The dynamics for consumption and dividend c t+1 = µ c + x t + σ t η t+1 x t+1 = ρx t + ϕ e σ t e t+1 σ 2 t+1 = σ2 + ν(σ 2 t σ 2 ) + σ w w t+1 d t+1 = µ d + φx t + πσ t η t+1 + ϕσ t u d,t+1 log( P t D t ) = A 0,d + A 1,d x t + A 2,d σ 2 t r f t = A 0,f + A 1,f x t + A 2,f σ 2 t Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 12 / 31

Simulated Moments for Long-run Risk Model The parameter vector for MSM estimation Φ LRR (δ,ψ, µ d, ϕ d ) US Data LRR Moment SE Moment t-stat corr(pd, R) 0.069 0.12 0.608-4.35* Var(e d ) 21.1% 0.242 96.6% -3.12* Var(e r ) 4.4% 0.026 3.5% 0.33 Var(e e ) 50.8% 0.257 52.7% -0.08 Table: Simulated Statistics of Long-run Risk Model Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 13 / 31

My Model A simple model based on Adam, Marcet and Nicolini (2016, JF) Exogenous dividend D t and endowment growth process Y t D t D t 1 = aɛ d t, log ɛ d t iin( s 2 d 2, s2 d ) (1) Y t = aɛ y t, log ɛ y sy 2 t iin( Y t 1 2, s2 y ) (2) The representative agent i 2 [0, 1] needs to solve the maximization problem max E0 P δ t (Ct i ) 1 γ 1 γ t=0 s.t.ct i + R t 1 bt i 1 + P t St i = (P t + D t )St i 1 + bt i + Yt i bt i 5 θ E t P (P t+1 + D t+1 ) St i R t Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 14 / 31

Optimal Behaviors First-order conditions for three control variables: C i t, b i t and S i t C i t : (C i t ) γ λ t = 0 S i t : λ t P t + δe P t (λ t+1 (P t+1 + D t+1 )) + γ t θe P t (P t+1 + D t+1 ) = 0 b i t : λ t = δr t E P t λ t+1 + γ t R t & γ t (θe P t (P t+1 + D t+1 )S i t R t b i t) = 0 Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 15 / 31

Optimal Behaviors Approximation C t 6= Y t since C t = Y t + D t + b t R t 1 b t 1 in small open economy In general, E P t (C i t+1 ) 6= E P t (C t+1 ) for individual decision Assumption: We assume that Y t is su ciently large and that E P t (P t+1 + D t+1 ) < M for some M <. Then, income from holding stock should be su cient small give nite asset bounds S, S. Based on this assumption, collateral constraint can guarantee that b t is also small enough compared to Y t. Therefore, the above approximations hold with su cient accuracy. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 16 / 31

Optimal Behaviors Approximation Under the assumption, one can rely on approximations C t ' Y t Et P [( C t+1 i Ct i ) γ (P t+1 + D t+1 )] ' Et P [( C t+1 ) γ (P t+1 + D t+1 )] C t Et P [( C t+1 i Ct i ) γ ] ' Et P [( C t+1 ) γ ] C t Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 17 / 31

After algebraic computation, in the equilibrium we have P t = E P t η t (P t+1 + D t+1 ) (3) η t δ( Y t+1 Y t ) γ + θ( 1 R t ϕ) ϕ δe P t ( Y t+1 Y t ) γ (4) The process of exogenous risk-free rate is ( (1 ρr )R + ρ R t = r R t 1 + ɛ R t if R t < 1 ϕ if else 1 ϕ ) Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 18 / 31

Rational Expectation Equilibrium The Present-value expression for equilibrium stock price (E P t = E t ) is δa 1 γ ρ t = [ ɛ 1 δa 1 γ + E t ρ ɛ P RE j=1 θ j a j j 1 1 ( ϕ)]d t (5) R t+k k=0 E t [R t+k ] = (1 ρ k r )R + ρk r R t for any integer k Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 19 / 31

Quantitative Performance Simulation The model moments from simulated data Statistics US Data RE Estimate SE Statistics corr(pd, R) 0.069 0.12-1.000 Var( ee d ) 21.2% 0.242 96.2% Var( ee r ) 4.4% 0.026 17.0% Var( ee e ) 50.8% 0.257 5.0% Table: Simulated Moments of Rational Expectation Equilibrium Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 20 / 31

Dynamic Analysis with Learning If agents have subjective beliefs instead of objective ones, equation (5) doesn t hold. δa 1 γ ρ t = [ ɛ 1 δa 1 γ + E t ρ ɛ P RE j=1 θ j a j j 1 ( k=0 1 R t+k ϕ)]d t Only rst-order condition equation (3) holds. P t = E P t η t (P t+1 + D t+1 ). Agents should have their own beliefs on stock price behavior as β t E P t [( Y t+1 Y t ) γ P t+1 P t ] risk-adjusted price growth (6) m t E P t [ P t+1 P t ] non-adjusted price growth (7) Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 21 / 31

Dynamic Analysis with Learning The equation maps from perceived stock price to realized one as P t = δa1 γ ρ ɛ + θa( 1 R t ϕ) 1 δβ t θ( 1 R t ϕ)m t D t (8) Di erent from rational price as equation (5) the RE equilibrium, P t here is also driven by agent s beliefs β t and m t except R t. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 22 / 31

Agents Information Agents think that ( Y t+1 Y t ) γ P t+1 P t follows the process as ( Y t+1 Y t ) γ P t+1 P t = e β t + ɛ β t, ɛ β t iin(0, σ 2 ɛ,β ) e β t = e β t 1 + ξβ t, ξ β t iin(0, σ 2 ξ,β ) And agents can only observe the realizations of ( Y t+1 Y t ) γ P t+1 P t instead of transitory component ɛ β t and persistence component e β t separately. This setup encompasses the rational expectation equilibrium as a special case when agents believe σ 2 ξ,β = 0 and assign probability one to e β 0 = a1 γ ρ ɛ. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 23 / 31

Agents Information (Cont d) When we allow for a non-zero variance σ 2 ξ,β, the requirement to lter out the persistent component e β t calls for a ltering problem. The prior of agents initial belief could be e β 0 N(a1 the posterior will be e β t N(β t, σ 2 0,β ). The optimal updating rule implies that γ ρ ɛ, σ 2 0,β ), and β t = β t 1 + 1 α ((Y t 1 Y t 2 ) γ P t 1 P t 2 β t 1 ) And if agents think that endowment growth ( Y t+1 Y t ) γ and price growth P t+1 P t are not correlated, m t = β t /(a γ τ). Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 24 / 31

The Method of Simulated Moments (MSM) The parameter vector for estimation Φ (δ, α, a, σ D /D, σ R ) The moments chosen for matching [E rs, E PD, σ rs, σ PD, ρ PD, 1, c5 2, R2 5, E R, σ R, E D /D, σ D /D, corr(r, PD), var(e d,t+1 ), var(e r,t+1 ), var(e e,t+1 )] The MSM parameter estimate bφ T is de ned as bφ T arg min Ω [bs T es(φ)] 0 bσ S,T 1 [ bs T es(φ)] Under null hypothesis that the model is correct, cw T T [bs T es(φ)] 0 bσ S,T 1 [ bs T es(φ)] χ 2 s 5 as T! Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 25 / 31

Quantitative Performance The Fact 0 moments US data Model Moment SE Moment t-stat E rs 2.25 0.39 2.08 0.44 E PD 123.91 21.25 88.94 1.65 σ rs 11.44 2.69 12.30-0.32 σ PD 62.42 17.54 62.64-0.01 ρ PD, 1 0.97 0.02 0.93 1.72 c5 2-0.0038 0.0013-0.0060 1.72 R5 2 0.1772 0.0828 0.1108 0.80 E R 0.15 0.19 0.12 0.15 σ R 1.27 0.27 0.71 2.04 E D /D 0.41 0.18 0.03 2.10 σ D /D 2.88 0.80 2.22 0.82 Table: Moments from MSM Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 26 / 31

Quantitative Performance (Cont d ) The Fact 1 and Fact 2 moments US Data Model Moment SE Moment t-stat corr(pd, R) 0.069 0.12-0.170 1.92 Var(e d ) 21.1% 0.242 39.7% -0.77 Var(e r ) 4.4% 0.026 1.7% 1.01 Var(e e ) 50.8% 0.257 56.1% -0.21 Discount factor bδ T 0.9886 Gain coe cient 1/bα T 0.0085 p-value of cw T 0.000% Table: Moments from MSM Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 27 / 31

Vector-Autoregression Analysis Gali and Gambetti (2015) provide evidence about the response of real stock price to exogenous monetary policy shock using vector-autoregression (VAR) model. Being di erent from their paper we estimate the response of stock price to real risk-free shock instead of nominal risk-free rate shock. We de ne the state space The VAR model is x VAR t x VAR t [ y t, d t, r t, p t ] 0 = A 1 x VAR t 1 + A 2x VAR t 2 + A 3x VAR t 3 + A 4x VAR t 4 + u t Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 28 / 31

Vector-Autoregression Analysis-Impulse Response 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0 2 4 6 8 10 12 14 16 18 20 Figure: The Impulse Response of Stock Prices to Risk-free Rate Shock Using US Data. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 29 / 31

Vector-Autoregression Analysis-Impulse Response 0.08 0.06 0.04 0.02 0 0.02 0.04 0 2 4 6 8 10 12 14 16 18 20 Figure: The Impulse Response of Stock Prices to Risk-free Rate Shock Using Simulated Data. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 30 / 31

Concluding Remarks The empirical studies con rm that stock price is not correlated with risk-free rate, and the latter almost have no power in explaining the volatility of stock excess return. Two consumption based asset pricing models with rational expectation fails in reproducing these, but model with learning can match data. Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April 2016 31 / 31