Stock Price, Risk-free Rate and Learning

Transcription

1 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 / 31

2 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 / 31

3 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 / 31

4 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 / 31

5 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 / 31

6 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 / 31

7 Fact 0 Statistics Estimate SE Quarterly mean stock return E rs Mean PD ratio E PD Std.dev. stock return σ rs Std.dev. PD ratio σ PD Autocorrel. PD ratio ρ PD, Excess return reg. coe cient c R 2 of excess return regression R Mean risk-free rate E R Std.dev. risk-free rate σ R Mean dividend growth E D /D Std. dev. dividend growth σ D /D 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 / 31

8 Fact 1 Statistics Estimate SE corr(pd, R) 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 / 31

9 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% Var(e r ) 4.4% Var(e e ) 50.8% Table: Variance Decomposition of Excess Stock Return Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April / 31

10 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 / 31

11 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) * Var(e d ) 21.1% % 0.10 Var(e r ) 4.4% % 1.25 Var(e e ) 50.8% % -3.99* Table: Simulated Statistics of External Habit Model Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April / 31

12 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 / 31

13 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) * Var(e d ) 21.1% % -3.12* Var(e r ) 4.4% % 0.33 Var(e e ) 50.8% % Table: Simulated Statistics of Long-run Risk Model Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April / 31

14 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 / 31

15 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 / 31

16 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 / 31

17 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 / 31

18 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 / 31

19 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 / 31

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

21 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 / 31

22 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 / 31

23 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 / 31

24 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 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 / 31

25 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 / 31

26 Quantitative Performance The Fact 0 moments US data Model Moment SE Moment t-stat E rs E PD σ rs σ PD ρ PD, c R E R σ R E D /D σ D /D Table: Moments from MSM Tongbin Zhang (Institute) Stock Price, Risk-free Rate and Learning April / 31

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

28 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 / 31

29 Vector-Autoregression Analysis-Impulse Response 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 / 31

30 Vector-Autoregression Analysis-Impulse Response 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 / 31

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 / 31