Higher-Order Dynamics in Asset-Pricing Models with Recursive Preferences

Transcription

1 Higher-Order Dynamics in Asset-Pricing Models with Recursive Preferences Walt Pohl Karl Schmedders Ole Wilms Dept. of Business Administration, University of Zurich Becker Friedman Institute Computational Economics Colloquium April 0, 205 / 6

2 Asset-Pricing Models Asset pricing models have become increasingly complex over the last three decades First generation: Grossman and Shiller (98), Hansen and Singleton (982), Mehra and Prescott (985) Modern generation: Bansal and Yaron (2004) or Hansen, Heaton, and Li (2008) Highly nonlinear preferences structures, complex exogenous specifications Complexity requires numerical approximations for tractability 2 / 6

3 Standard Solution Methods Discretization methods: Tauchen (986), Tauchen Hussey (99) Finite-state Markov chain replaces continuous state space Heaton and Lucas (996, 2000), Hansen, Heaton, Yaron (2009) Guvenen (2009) and many others Log-linearization methods: Judd (996), Bansal and Yaron (2004) Linear approximation of policy and price functions Bansal, Kiku, Yaron (200), Constantinides and Ghosh (20), Bansal et al. (204), Beeler and Campbell (202) Methods are well suited for the early models and provide (more or less) accurate and robust solutions / 6

4 Motivation But: Models are now so much more complex, still the same solution methods are applied By construction: Log-linear approximation misses higher-order effects Question: Does it matter? Purpose: Solve models accurately using projection methods, analyze the influence of higher order dynamics 4 / 6

5 Outline Motivation Introductory Example Projection Method Long-Run Risks Model Conclusion 5 / 6

6 Standard Asset Pricing Framework Euler equation for asset with price P t that pays dividend D t ( Ct+ ) } γ P t = E t {β (Pt+ + D t+ ) C t Assumption: dividends are the only source of income (C t = D t ) { P ( t Ct+ ) ( )} γ Pt+ = E t β + C t C t+ C t Log-consumption growth c t+ = log( c t+ c t ) is an AR() process c t+ x t+ = ( ρ)µ + ρx t + ɛ t+, ɛ t+ N(0, σ 2 ) 6 / 6

7 State-Space Representation Log-consumption growth c = x is state variable Assumption: p-c ratio Pt C t y(x) = E is a function of the state x, y(x), { ( β e x ) γ ( y(x ) + ) } x = E { h(x ) x } with x denoting the state in the next period Projection, Step I Approximation of the p-c ratio y(x) by a linear combination of basis functions Example: y(x) = α 0 + α x 7 / 6

8 Gaussian Quadrature Gauss-Hermite Quadrature + e z2 f (z)dz n ω i f (z i ) i= with the Gauss-Hermite quadrature nodes z i and weights ω i Change of variable for normally distributed variables z N(a, b 2 ) E(h(z)) π n ω i h( 2bz i + a) i= In our example: x x N(( ρ)µ + ρx, σ 2 ) E { h(x ) x } π n ω i h( 2σz i + ( ρ)µ + ρx) i= 8 / 6

9 Solving the Integral Define x i (x) 2σz i + ( ρ)µ + ρx, so for the linear example α 0 + α x = π n ( ω i βe ( γ)x (x)) ( i α0 + α x i (x) + ) }{{} :=A i (x) i= Note that x i (x) and hence A i(x) are known for a given value of x Define the residual function as ˆF (x, α) = α 0 + α x π n A i (x) ( α 0 + α x i (x) + ) i= which is linear in the coefficients α = [α 0, α ] 9 / 6

10 Applying the Projection Projection, Step II: Collocation conditions: ˆF (x j, α) = 0 for j =, 2 Conditions lead to 2 linear equation and 2 unknowns π i π i A i (x ) x A i (x 2 ) x 2 = π i π π i π i i A i (x ) A i (x 2 ) A i (x )x i (x ) A i (x 2 )x i (x 2) ( α0 α ) 0 / 6

11 Example Parameter values γ = 5, β = 0.95, µ = 0.02, ρ = 0.8, σ = Approximate solution function within ±m standard deviations around the unconditional mean of the underlying process With m = 2 we obtain x min = and x max = For the degree approximation with 2 nodes, x = x min and x 2 = x max and the system of equations becomes ( ) ( ) ( ) α = α.60 / 6

12 Results Degree Degree Approximation, Maximum Absolute Error: Closed Form Solution Collocation Method 0 9 Price Consumption Ratio log Consumption Growth Absolute Error log Consumption Growth Closed-form solution by Burnside (998) 2 / 6

13 Results Degree 2 Degree 2 Approximation, Maximum Absolute Error: Price Consumption Ratio Closed Form Solution Collocation Method Absolute Error log Consumption Growth log Consumption Growth / 6

14 Results Degree 5 Degree 5 Approximation, Maximum Absolute Error: Price Consumption Ratio Closed Form Solution Collocation Method Absolute Error log Consumption Growth log Consumption Growth 4 / 6

15 Results Degree 0 Degree 0 Approximation, Maximum Absolute Error: 5.e-5 Price Consumption Ratio Closed Form Solution Collocation Method Absolute Error log Consumption Growth log Consumption Growth 5 / 6

16 General Projection Approach Projection methods in economics, Judd (992), Chen, Cosimano and Himonas (204) Functional equation (Gz) (x) = 0, with the variable x in a (state) space X R l, l an unknown solution function, z : X R m the given function G is a continuous mapping between two function spaces Projection Method, Step I: Approximate term z(x) on its domain X by a linear combination of basis functions n ẑ(x; α) = α k Λ k (x) k=0 6 / 6

17 Residual Function Residual function ˆF (x; α) is the error in the original equation, ˆF (x; α) = (Gẑ) (x; α) Projection Method, Step II: Coefficients α are chosen by imposing certain conditions on the residual function (projection conditions), so that ˆF (x; α) close to zero Different projection methods differ in how close to zero is defined This paper: Collocation and Galerkin Projection 7 / 6

18 Collocation Method: Projection Conditions ˆF (x k ; α) = 0, k = 0,,..., n with n + distinct nodes {x k } n k=0 Collocation requires solving a nonlinear system of equations Galerkin Method: ˆF (x; α)λ k (x) dx = 0, k = 0,,..., n x 2 X for domain [, ] Galerkin projection requires numerical integration and solving a nonlinear system of equations 8 / 6

19 Application to Asset Pricing Asset pricing model with recursive preferences Wealth-Euler equation [ ( ) θ ] E t δ θ Ct+ ψ R θ C w,t+ t = with R w,t+ return on claim to aggregate consumption In state-space representation (lower case letters = log of variables) [ ( E exp θ log δ θ ] ψ c(x x) + θr w (x x)) x =, x with c(x x) = c(x ) c(x) 9 / 6

20 Functional Equation Unknown solution function: ( ) log wealth-consumption ratio z w (x) = log W (x) C(x) State-dependent log return of aggregate consumption claim ( ) r w (x x) = z w (x ) log e zw (x) + c(x x) Functional equation 0 = X [ exp ( ( θ log δ + ( ψ ) c(x x)+ ( ))) ] z w (x ) log e zw (x) df x 20 / 6

21 Residual function ˆF w (x; α w ) = Approximate Solution Functions X [ ( ( exp θ log δ + ( ψ ) c(x x)+ ( ẑ w (x ; α w ) log eẑw (x ;α w) ))) ] df x Projection methods determine unknown solution coefficients α w Approximate state-dependent wealth-consumption ratio ẑ w (x; α w ) Approximation of return of aggregate consumption claim ( ) ˆr w (x x; α w ) = ẑ w (x ; α w ) log (x;αw) eẑw + c(x x) 2 / 6

22 Return of any Asset Euler equation E t [ ( ) θ ] δ θ Ct+ ψ R θ w,t+ C R i,t+ t with R i,t+ (gross) return of asset i = In state-space representation and with logs [ ( E exp θ log δ θ ] ψ c(x x) + (θ )r w (x x) + r i (x x)) x = 22 / 6

23 Residual for Asset i Unknown solution function: ( ) log price dividend ratio z i (x) = log P(x) D(x) State-dependent log return of asset i ( r i (x x) = log e z i (x ) ) + z i (x) + d i (x x) Residual function [ ( ˆF i (x; α i ) = exp θ log δ θ X ψ c(x x) + (θ )ˆr w (x x; α w ) ( ) ) ] (x + log ;α i ) eẑi + ẑ i (x; α i ) + d i (x x) df x 2 / 6

24 Projection Methods... Again Coefficients α w and function ˆr w (x x; α w ) previously computed Projection methods determine unknown solution coefficients α i Approximate state-dependent price-dividend ratio ẑ i (x; α i ) Approximation of log return of asset i, ( ) ˆr i (x (x;α x; α w ) = log i ) eẑi + ẑ i (x; α i ) + d i (x x) 24 / 6

25 Risks for the Long Run Highly influential asset pricing model by Bansal and Yaron (2004) Growth rates have random but highly persistent long-run shocks Conditional variance of growth rates is stochastic Model has two state variables long-run component of growth, x t conditional variance level, σ 2 t Calibration of Bansal, Kiku, and Yaron (202) increases influence of volatility channel versus the long-run risks channel compared to original Bansal Yaron (2004) calibration 25 / 6

26 Model Specification c t+ = µ c + x t + σ t η t+ x t+ = ρx t + φ e σ t e t+ σ 2 t+ = σ 2 ( ν) + νσ 2 t + σ ω ω t+ d t+ = µ d + Φx t + φσ t u t+ + πσ t η t+ η t+, e t+, ω t+, u t+ i.i.d. N(0, ). µ c ρ φ e σ ν σ ω µ d BKY (202).5e e e-6.5e- BY (2004).5e e e-6.5e- Φ φ π γ ψ δ BKY (202) BY (2004) / 6

27 Relative Errors and Running Times Collocation BY Log-Lin 6 9 E (w t c t ).5e-5 2.0e-8 2.0e σ (w t c t ) 4.6e-4 4.9e-7 4.9e E (p t d t ) e-8 2.9e σ (p t d t ) e-6 9.4e Time (sec) Tauchen-Hussey Tauchen E (w t c t ) σ (w t c t ) > E (p t d t ) σ (p t d t ) > Time (sec) / 6

28 Relative Errors of Annualized Moments n E (p t d t ) σ (p t d t ) E (rt m ) σ (rt m ) E ( ) rt f σ ( ) rt f EP Collocation e-4.8e-4 2.e e-8.e-5 6.2e-7 2.0e-7.4e-6.e-6.6e-8 9.2e-8.e-5 6.2e-7 2.0e-7.4e-6.e-6.6e-8 BY Log-Linearization Tauchen-Hussey Tauchen / 6

29 Annualized Moments (in %) n E (p t d t ) σ (p t d t ) E (rt m ) σ (rt m ) E ( ) rt f σ ( ) rt f EP Collocation BY Log-Linearization Tauchen-Hussey Tauchen / 6

30 Approximation Errors x 0 4 w t c t x 0 4 p t d t σ σ x x 0 / 6

31 Errors: First Derivative x 0 4 (wt ct) x x 0 4 (pt dt) x σ σ x x x 0 4 (wt ct) σ 2 x 0 4 (pt dt) σ σ x σ x / 6

32 Errors: Second Derivative x (wt ct) x 2 x (pt dt) x 2 σ x 2 (wt ct) x 0 4 (σ 2 ) 2 σ x 2 (pt dt) x 0 4 (σ 2 ) 2 00 σ 2 2.e+06.45e+06.4e+06.5e+06 σ e+07 e e+06 5e x 2 (wt ct) x 0 4 x σ x 2 (pt dt) x 0 4 x σ σ σ x x 2 / 6

33 Sensitivity of the Approximation Errors Collocation BY Log Lin EP 0. EP γ ψ EP 0.06 EP ρ ν / 6

34 Sensitivity of the Approximation Errors Collocation BY Log Lin σ(p t d t ) 0.25 σ(p t d t ) γ ψ σ(p t d t ) 0.25 σ(p t d t ) ρ ν 4 / 6

35 Conclusion: Log-linearization Recent asset pricing models feature interplay of long run risks, stochastic volatility, and recursive preferences Features can result in highly nonlinear relationships between equilibrium quantities and the state variables Log-linearization, common solution approach, cannot capture nonlinear relationships Overestimation of equity premium by more than 00 basis points 5 / 6

36 Conclusion: Projection Methods Projection methods prove to be highly accurate even with low degree approximation while requiring only slightly longer computation times Increasing complexity of asset-pricing models requires numerical solution techniques, such as projection methods, that can capture nonlinear dynamics and are robust to changes in model specification 6 / 6