Solving Asset-Pricing Models with Recursive Preferences Walter Pohl University of Zurich Karl Schmedders University of Zurich and Swiss Finance Institute Ole Wilms University of Zurich July 5, Abstract We present a projection-based solution algorithm to solve asset pricing models with recursive preferences. The method outperforms common methods like discretization and loglinerization in terms of efficiency and accuracy. These improvements become particularly significant for the latest generation of asset-pricing models, such as the Bansal-Yaron long-run risk model. The increasing complexity of asset pricing models requires numerical solution methods, such as ours, that are robust to changes in model specification. Keywords: Asset pricing, discretization, log-linearization, projection methods. JEL codes: G, G. We are indebted to Lars Hansen, Ken Judd, and Martin Lettau for helpful discussions on the subject. We thank seminar audiences at the University of Zurich and the Becker Friedman Institute for comments. Karl Schmedders gratefully acknowledges financial support from the Swiss Finance Institute.
Introduction Asset pricing models have become increasingly complex over the last two decades. The first generation of equilibrium asset pricing models, developed in the 98s, (Grossman and Shiller (98), Hansen and Singleton (98), Mehra and Prescott (985)), proved unable to explain many financial market characteristics such the high equity premium and low risk-free rate. Modern asset pricing models not only feature highly nonlinear preference structures but also complex models of the underlying economy (see e.g. Bansal and Yaron () or Hansen, Heaton, and Li (8)). The methods that are commonly used to solve these models were first developed twenty years ago, and proved themselves adequate for the models of the time. A popular approach is to linearize the model around the steady state, as shown in Campbell and Shiller (988). Bansal and Yaron (), Bansal, Kiku, and Yaron (), Lettau and Wachter (7), and Beeler and Campbell () illustrate this approach. Loglinearization by its very nature misses higher-order approximation errors. For example Caldara, Villaverde, Ramirez, and Yao () show, that to compute DSGE models with recursive preferences and stochastic volatility accurately, higher order approximations are needed. An alternative procedure is to replace the continuous state space with a discrete approximation by a finite-state Markov chain and solve the discrete model. Popular discretization techniques were introduced by Tauchen (986) and Tauchen and Hussey (99), and have been applied by many subsequent works, such as Guvenen (9), Heaton and Lucas (996), Heaton and Lucas (), Hansen, Heaton, and Yaron (996), Campbell (99) and Judd, Kubler, and Schmedders (). In this paper, we show that these methods do not meet the requirements to efficiently and accurately solve the newest generation of asset pricing models. We propose a projection-based solution method that, unlike existing methods, is both efficient and accurate and robust to changes in model specification. We present a two-step solution algorithm to solve general asset pricing models with recursive preference structures using projection methods as in Judd (99). While linearization methods are very fast, and able to compute reasonable solutions for preferences close to log utility, the approximation error becomes large for more complex models. The method presented in this paper is nonlinear, and can provide arbitrary accuracy. For example in the recent version of the long run risk model of Bansal, Kiku, and Yaron () approximation errors are already with a -degree polynomial approximation about orders of magnitude smaller than the linearization solution, while it takes only slightly more computation time. Discretization methods, in contrast, can provide an accurate solution given sufficiently many nodes in the discrete approximation, but the number of nodes required is very sensitive to the particular model. Computation times increase dramatically with the dimension of the model, and the number of discretization nodes. The method presented in this paper is capable of arbitrary accuracy, but provides much higher degrees of accuracy for the same computation time. The differences between methods become particularly significant for the latest generation of asset pricing models, Bansal and Yaron () and Bansal, Kiku, and Yaron (). The paper is organized as follows. In Section we present the general solution algorithm and Section gives a short summary of the solution methods that are commonly used to solve asset pricing models. In Section we consider two recent asset pricing models, namely the model by
Tallarini () and Bansal and Yaron () to evaluate the performance of the different solution methods and Section 5 concludes. Solution Method We consider a standard asset-pricing model with a representative agent and recursive preferences as in Epstein and Zin (989) and Weil (989). Indirect utility at time t, V t, is given recursively as [ V t = ( δ)c γ θ t ( + δ [E t V γ t+ )] ] θ γ θ. () In this parametrization, C t is consumption, δ is the time discount factor, γ is the coefficient of relative risk aversion, ψ is the intertemporal elasticity of substitution, and θ = γ. For θ = ψ the agent has standard CRRA preferences and θ < indicates a preference for the early resolution of risk. Using the agent s Euler equation, Epstein and Zin (989) show that the gross return of asset i, R i,t+ = P i,t++d i,t+ P i,t must satisfy the following pricing equation, E t [ ( ) θ ] δ θ Ct+ ψ R θ w,t+ C R i,t+ t =. () The term R w,t+ denotes the return on a claim to aggregate consumption, R w,t+ = W t+ W t C t, where W t is the (unobserved) wealth level of the agent at time t. As Equation () has to hold for all assets i, it must also hold for the return of the aggregate consumption claim. So R w,t+ is determined by the wealth-euler equation given by E t [ ( ) θ ] δ θ Ct+ ψ R θ C w,t+ t =. () We present an algorithm using projection methods to solve the general asset pricing model given by equations () (). Projection methods are a general-purpose tool for solving function equations. They were first introduced by physicists and engineers to solve partial differential equations, but they can be used to solve the types of fixed-point equations that arise in economics, see Judd (99) for an introduction to projection methods for economists. In the following we first briefly describe the idea of projection methods and then apply it to the asset pricing model given above. Our description does not strive for maximal generality but instead is meant to simply convey the key steps of projection methods.. Projection Methods for Functional Equations Projection methods are a general tool to solve functional equations of the form G(z(x)) =, ()
where the variable x resides in a (state) space X R l, l, and z is an unknown solution function with domain X, so z : X R m. The given function G is a continuous mapping between two function spaces. Note that solving Equation () requires finding an element z in a function space, that is, in an infinite-dimensional vector space. The first central step of a projection method is to approximate the unknown function z on its domain X by a linear combination of basis functions. For the applications in this paper, it suffices to assume that the domain X is bounded and that the basis functions are polynomials. For a set {Λ k } k {,,...,n} of chosen basis functions the approximation ẑ of z is ẑ(x; α) = n α k Λ k (x), (5) k= where α = [α, α,..., α n ] are unknown coefficients. Replacing the function z in Equation () by its approximation ẑ, we can define the residual function ˆF (x; α) as the error in the original equation, ˆF (x; α) = G(ẑ(x; α)). (6) Instead of solving Equation () for the unknown function z, we now attempt to solve Equation (6) for the coefficients α in the approximation term ẑ(x; α) (which have to be, in some notion, optimal). Note that instead of finding an element in an infinite-dimensional vector space we are now looking for a vector in R n+. Obviously, this approximation step greatly simplifies the mathematical problem. The second central step of a projection method is to impose certain conditions on the residual function, the so-called projection conditions, to determine the unknown solution coefficients. Put differently, the purpose of the projection conditions is to establish a set of requirements that the coefficients α must satisfy. For a formulation of the projection conditions, define a weight function (term) w(x) and a set of test functions {g k (x)} n k=. We can then define an inner product between the residual function ˆF and the test function g k, X ˆF (x; α)g k (x)w(x)dx. This inner product induces a norm on the function space X. Natural restrictions for the coefficient vector α are now the projection conditions, X ˆF (x; α)g k (x)w(x)dx =, k =,,..., n. (7) Observe that this system of equations imposes n+ conditions on the (n+)-dimensional vector α. Different projection methods vary in the choice of the weight function and the set of test functions. In this paper we use two different projections, the collocation and the Galerkin method. A collocation method chooses n + distinct nodes in the domain, {x k } n k=, and defines the In addition to polynomial approximations, approximations using cubic splines or B-splines are often very useful.
test functions g k by g k (x) = { if x x k if x = x k. With a weight term w(x), the projection conditions (7) simplify to ˆF (x k ; α) =, k =,,..., n. (8) Simply put, the collocation method determines the coefficients in the approximation (5) by solving the square system (8) of nonlinear equations. The Galerkin method also sets the weight term w(x) but uses the basis functions as the test functions, g k (x) = Λ k (x). Put differently, the residual function F is multiplied by a basis function Λ k and the resulting product is integrated over the entire domain. The projection conditions (7) simplify to X ˆF (x; α)λ k (x)dx =, k =,,..., n. (9) In the following we describe how to apply the general projection method to the asset-pricing model given by Equations () ().. Projection Methods Applied to Asset-Pricing Models Before we can apply a projection method to an asset-pricing model, we need to express the equilibrium conditions as a functional equation of the type (). For this purpose, we need to choose an appropriate state space and perform the usual transformation from an equilibrium described by infinite sequences (with a time index t) to the equilibrium being described by functions on the state space. Again, we denote the state of the economy by x; and x denotes the subsequent state in the next period. (For example in the original model by Mehra and Prescott (985), the state x is given by log consumption growth; in the two-dimensional model of Bansal and Yaron (), the state x consists of the long-run mean of consumption growth (denoted by x t in that paper) and the variance of consumption growth (denoted by σ t ).) The state-dependent return of the aggregate consumption claim can be written as R w (x x) = W (x ) W (x) C(x) = Taking logarithms on both sides of this equation yields W (x ) C(x ) W (x) C(x) C(x ) C(x). () ( ) r w (x x) = w(x ) c(x ) log e w(x) c(x) + c(x x), () where lower case letters denote logs of variables and c(x x) = c(x ) c(x). Hence Equation () in state space representation becomes [ E e θ log δ θ ψ c(x x)+θr w(x x) ] x =. () 5
As a final step, we rewrite this equilibrium condition in the format of the functional equation (). For this purpose, we define the unknown solution function z as the logarithmic wealthconsumption ratio z w (x) = w(x) c(x) and obtain the functional equation ( E [e θ log δ+( ψ ) c(x x)+z w(x ) log(e zw(x) ) ) ] x =. () Using the approximation (5), ẑ w (x; α w ) = n k= α w,kλ k (x), we obtain the approximation of the log return function for the aggregate consumption claim, ( ) ˆr w (x x; α w ) = ẑ w (x ; α w ) log eẑw(x;αw) + c(x x). () Finally, the residual function for the wealth-euler equation (see Equation (6)) is given by ( ˆF w (x; α w ) = E [e θ log δ+( ψ ) c(x x)+ẑ w(x ;α w) log(eẑw(x;αw) ) ) ] x. (5) Applying one of the projections (collocation or Galerkin method) to the residual function ˆF w (x; α w ) yields the unknown solution coefficients α w. These determine the state-dependent wealth-consumption ratio ẑ w (x; α w ) which in turn leads to the (approximate) return function of the aggregate consumption claim, ˆr w (x x; α w ). With ˆr w (x x; α w ) at hand, we can now develop an approach to compute the return of any asset i using Equation (). Again, first define the state-dependent return of asset i as R i (x x) = P i(x ) + D i (x ) P i (x) = P i (x ) D i (x ) + P i (x) D i (x) D i(x ) D i (x) (6) or, equivalently, ( r i (x x; α) = log e p i(x ) d i (x ) ) + (p i (x) d i (x)) + d i (x x). (7) Performing the analogous step to the derivation of Equation (), we approximate the log pricedividend ratio z i (x) = p i (x) d i (x) by ẑ i (x; α i ) = n k= α i,kλ k (x) which leads to ( ) ˆr i (x x; α i ) = log eẑi(x ;α i ) + ẑ i (x; α i ) + d i (x x). (8) Equation () in state-space representation is then given by [ E e θ log δ θ ψ c(x x)+(θ )r w(x x)+r i (x x) ] x =. (9) Again, we define the residual function using the approximation ẑ i (x; α i ) as the deviation of the pricing equation from zero, [ ( ) ] ˆF i (x; α i ) = E e θ log δ θ ψ c(x x)+(θ )ˆr w(x x;α w)+log eẑi (x ;α i ) + ẑ i (x;α i )+ d i (x x) x. () 6
Recall that the coefficients α w and thus the function ˆr w (x x; α w ) have been computed previously. Therefore, we apply one of the projections only to solve for the unknown vector α i. In sum, we apply the projection method twice. In the first step, we approximate the log wealth-consumption ratio ẑ w (x; α w ) by applying the projections on the residual function of the wealth-euler equation (5). Once α w is known, the projections can be applied to Equation () to solve for the price-dividend ratio ẑ i (x; α i ) of any asset i. Formally the algorithm can be described as follows. Algorithm Solving Asset-Pricing Models with Recursive Preferences. Initialization. Define the state space X R l ; choose the functional forms for ẑ w (x; α w ) and ẑ i (x; α i ) as well as the projection method; choose the nodes for the projection method, X = {x j : j m} X. Step. Use the wealth-euler Equation () together with approximated log wealth consumption ratio ẑ w (x; α w ) and the definition of the return equation () to derive the residual function for the return on wealth ( ˆF w (x; α w ) = E [e θ log δ+( ψ ) c(x x)+ẑ w(x ;α w) log(eẑw(x;αw) ) ) ] x. () Compute the unknown solution coefficients α w by imposing the projections on Equation (). Step. Take the solutions for the wealth-consumption ratio ẑ w (x; α w ) as given and use the Euler equation () for asset i together with the approximated log price-dividend ratio ẑ i (x; α i ) and the definition of the return equation (8) to derive the residual function for asset i given by [ ( ) ] ˆF i (x; α i ) = E e θ log δ θ ψ c(x x)+(θ )ˆr w(x x;α w)+log eẑi (x ;α i ) + ẑ i (x;α i )+ d i (x x) x. Compute the unknown solution coefficients α i by imposing the projections on Equation (). () Evaluation. Choose a set of evaluation nodes X e = {x e j : j me } X and compute approximation errors in the residual function of the wealth portfolio and the residual function of asset i. If the errors don t fulfill a predefined error bound, start over at Initialization and change the number of approximation nodes or the degree of the basis functions. Before we can perform the individual steps of this algorithm, we need to specify more algorithmic details such as the choices for basis functions and the integration technique. 7
. Algorithmic Ingredients In the Initialization step, we need to choose a set of basis functions for the polynomial approximation, a projection method and a set of nodes. To simplify the presentation, we describe the necessary choices for a one-dimensional state space X = [x min, x max ]. We approximate the solution functions ẑ w (x; α w ) and ẑ i (x; α i ) by Chebyshev polynomials (of the first kind), see Judd (998). We obtain Chebyshev polynomials via the recursive relationship T (ξ) = () T (ξ) = ξ () T k+ (ξ) = ξt k (ξ) T k (ξ). with T k : [, ] R. Since we need to approximate functions on the domain X and the Chebyshev polynomials are defined on the interval [, ], we need to transform the argument for the polynomials. The basis functions for the approximate solution (5) are given by for k =,,..., n. ( ) ) x xmin Λ k (x) = T k ( x max x min For the set of nodes X for the projection method we use the m + zeros of the Chebyshev polynomial T m+. These points are called Chebyshev nodes, (5) ( ) j + ξ j = cos n + π, j =,,..., m. (6) Since all Chebyshev nodes are in the interval [, ], we need to transform them to obtain nodes in the state space X. This transformation is x j = x min + x max x min ( + ξ j ), j =,,..., m. (7) For the collocation projection the number of basis functions n+ must be identical to the number of approximation nodes m +, and so m = n. In Step (and Step, if applicable), we must solve the projection conditions involving the residual function. In the asset-pricing models, first an additional challenge arises when we solve the projection conditions. The residual functions defined in Equations (5) and () contain a conditional expectations operator. We need to approximate the integral arising by this operator. Here we apply a Gauss-Hermite quadrature method, a natural choice since many asset pricing models assume normally distributed shocks to the economy. For the collocation approach, we finally need to solve the square nonlinear system of equations (6) (with a standard solver). The Galerkin projection requires us to compute yet another integral, namely its projection condition (9). We approximate this integral by Gauss- Chebyshev quadrature using m quadrature nodes over the state space [x min, x max ]. And finally, we need to solve the resulting nonlinear system of equations. For the Evaluation step we use m e >> m equally spaced evaluation nodes in the state space 8
X to evaluate the errors in the residual function. In particular we compute the root mean squared errors (RMSE) and maximum absolute errors (MAE) in the residual function () given by RMSE = m e ˆF m e (x e j α) (8) j= MAE = max j=,,...,m e ˆF (x e j α) (9) with x e j = x min + x max x min m e (j ), j =,..., m e. () Alternative Solution Methods Several other methods exist for solving asset-pricing models with recursive preferences. The most prominent ones are discretization and linearization techniques. We provide a brief description of these methods because we compare their performance to that of our projection algorithm in our section on numerical results below. Specifically, we focus on the discretization methods by Tauchen (986) and Tauchen and Hussey (99) and the log-linearization approach that has been applied to asset pricing as early as [XXX CORRECTION NEEDED?] Campbell and Shiller (988). Similar to the projection algorithm, these three methods have to be conducted in two steps by first solving for the return on wealth, and then solving for the return of any individual asset.. Discretization The idea of discretization methods is to discretize the continuous state space by a finite number of discretization nodes and to design a Markov transition matrix for a Markov chain on the set of nodes. Put differently, these methods replace the continuous state space and conditional density functions by a discrete state space and transition probabilities, respectively. With the nodes and the Markov transition probabilities at hand, the pricing equation (9) becomes a square system of nonlinear equations. This nonlinear system has as many equations as nodes; the unknown variables are the log price-dividend ratio at each node. We can then solve this system with a standard nonlinear equation solver. Discretization methods differ in how they choose the discretization nodes and transition probabilities. For demonstration purposes, we consider the simple case of the discretization of an AR() process that is given by x t+ = ( ρ)µ + ρx t + ɛ t+, ɛ t N(, σe), () with persistence ρ < and the unconditional mean µ. The unconditional volatility of the process is given by σ x = σ e / ρ. 9
Tauchen (986) s method Tauchen (986) assumes a set of equally spaced nodes X nt = {x,..., x nt } for the discrete state space with x = µ m T σ y and x nt = µ + m T σ y. The factor m T is a positive real number and determines the range of the state space. (To the best of our knowledge there is no optimal rule for choosing m T even though its value strongly influences the approximation results.) Denote the step size between two adjacent grid points by h = x i x i. Then the elements π ij of the (n T n T )-transition probability matrix π are defined by π ij = Φ Φ ( xj +h/ ( ρ)µ ρx Φ i ) ( σ e xj +h/ ( ρ)µ ρx i ) σ e Φ ( ) xj h/ ( ρ)µ ρx i σ e ( ) xj h/ ( ρ)µ ρx i σ e for j =, for < j < n T, for j = n T. Tauchen and Hussey (99) s method and the extension by Floden (7) Tauchen and Hussey (99) s method is based on Gauss-Hermite quadrature. Let ξ i and ω i, i =,..., n T H, be the Gauss-Hermite nodes and weights on the interval [, + ], respectively. The approximation nodes are then given by x i = µ + σ e ξ i and the entries π ij of the transition probability matrix π can be computed by π ij = ˆω ij N j= ˆω ij () with ˆω ij = π.5 ω j f(x j x i ) f(x j µ) () where f( x i ) is the density function of N(( ρ)µ + ρx i, σ ). Tauchen and Hussey (99) simply choose σ = σ e. Floden (7) chooses σ = aσ e + ( a)σ x with a =.5 +.5ρ, so σ is a weighted average of σ x and σ e. He claims that his approach performs significantly better than the original Tauchen and Hussey (99) method, particularly for highly persistent processes.. Log-Linearization The idea of log-linearization is to solve the model by a linear approximation around the steadystate of the model. Assume that the log price-dividend ratio z i (x) of asset i is a linear function of the state variable x R l, that is, z i (x) = A,i + A,i x () where A,i is an unknown constant, x is an l vector representing the state of the economy in period t and A,i is a l vector of unknown slope coefficients. Again we denote the current state by x and the state in the subsequent periods by x. The return of asset i is given by (see equation (8)) ( r i (x x) = log e z i(x ) ) + z i (x) + d i (x x). (5)
Campbell and Shiller (988) use a first-order Taylor approximation around the average level of the log price-dividend ratio z i : r i (x x) κ i, + κ i, z i (x ) z i (x) + d i (x x), (6) where constants κ i, and κ i, depend only on the average level of the log price-dividend ratio z i, e z i κ i, = + e z i ( κ i, = log ( κ i, ) κ i, κ κ i, i, (7) ). (8) After substituting expressions () and () into the pricing equation (9), the unknown coefficients A,i and A,i can be derived analytically as a function of the linearization parameters κ i, and κ i,. The parameters A,i and A,i determine z while κ i, and κ i, are nonlinear functions of that mean. Hence, by iteratively computing a fixed point we can find both the mean of z i,t and the parameters. For the return of the wealth portfolio the approximation can be applied analogously. As the return is given by the approximation becomes ( r w (x x) = z i (x ) log e z i(x ) ) + c(x x). (9) with r w (x x) z i (x ) + κ i, κ i, z i (x) + d i (x x), () Numerical Evaluation e zw κ w, = e zw ( κ w, = log (κ w, ) κ w, κ κ w, w, () ). () In this section we compare the performance of our projection method using collocation or the Galerkin condition with the two discretization methods and the log-linearization approach. For this purpose we report numerical solutions, error measures, and running times for two well-known asset-pricing models with recursive preferences. The first asset-pricing model is the endowment economy of Tallarini (). Log consumption is modeled as AR() deviations from a deterministic linear trend. The deviation from trend is the only state variable in the model and so the state space is one-dimensional. As the model is rather simple and doesn t account for many empirical features found in the data, we rather view it as a technical analysis to understand the strengths and weaknesses of the different solution methods especially with regard to changes in the preference and model parameters. Hence, for the second example, we consider the long-run-risk model of Bansal and Yaron (), which has gathered much attention recently for its ability to match
many financial market characteristics. This model has a two-dimensional state space. For both models, we first compute Euler errors in the pricing equations on the continuous state space for the projection methods and the log-linearization approach. For this computation, we use m e = n σ equally spaced evaluation nodes (in each dimension) in the interval [x min, x max ] to evaluate the (absolute) error in the pricing equation. All results are computed in Matlab using the solver Fmincon. We use Fmincon s active-set algorithm with an error tolerance of 8. In addition to the Euler errors, we also report errors in economically meaningful variables such as the mean and standard deviation of the wealth-consumption and price-dividend ratio. To do so, we compute the true solution by using a very large number of nodes in the Tauchen and Hussey (99) procedure or using a Chebyshev approximation of extremely high degree. While these calculations take a very long time, they allow us to evaluate numerical results for discretizations with fewer nodes and projections with polynomials of smaller degree.. Tallarini () s Model We consider the endowment economy of Tallarini (). Log consumption c t is modeled as simple AR() deviations from a linear trend, c t = µt + x t () x t = ρx t + σ ɛ ɛ t, ɛ t N(, ), () where µ is the average net growth rate of consumption and ρ the degree of persistence. The state of the economy is given by x t and log consumption growth is given by c t+ = µ + x t+ x t. (5) The state space is one-dimensional. In the following analysis, we focus exclusively on the pricing of the wealth portfolio (Step in our solution algorithm). For the projection methods we approximate solution functions within ±n σ standard deviations around the steady state (E(x t ) = ) of the model. So, x min = E(x t ) n σ σ(x t ) and x max = E(x t ) + n σ σ(x t ) with E(x t ) = and σ(x t ) = σ ɛ / ρ. We use 8 quadrature nodes for the Gauss-Hermite quadrature to solve the integral in the pricing equation. Similarly, the integral that arises in the Galerkin projection is solved by an 8-node Gauss-Chebyshev quadrature. For the log-linearization approach the log wealth-consumption ratio is a linear function of the state x t of the economy, z w (x t ) = A,w + A,w x t. (6) The unknown solution coefficients A,w and A,w can be derived by substituting expression () and () into the pricing equation (9) and are given by A = log δ + ( ψ )µ +.5θ(( ψ ) + A ) σɛ + κ (7) κ ρ A = ( ψ )(ρ ). (8) κ ρ
.. Approximation Errors in the Euler Equations Tables and report Euler approximation errors for the pricing of the wealth portfolio for the two projection methods (for degrees, 6, and 9), and the log-linearization approach. We report Euler errors for six combinations of the preference parameters, γ and ψ. The first two combinations correspond to CRRA preferences with a low (γ =, ψ =.5) and a high (γ =, ψ =.) degree of risk aversion, respectively. The third combination are the parameter estimates from Bansal and Yaron (), γ = and ψ =.5. Table reports Euler errors for these three cases. Table depicts errors for three more cases of Epstein-Zin preferences, namely for γ =, ψ =.5, for γ = 7.5, ψ =.5, and for γ =, ψ =.5. The leftmost column indicates the state space by providing the number n σ of standard deviations around the steady state of the model. The parameters for the consumption process are taken from Pohl, Schmedders, and Wilms () and are given by σ ɛ =., µ =., ρ =.9. We set δ =.99. We later vary the parameters to analyze their influence on the performance of the different solution methods. [Table about here.] [Table about here.] For each value of n σ, the first row in Tables and shows the maximum absolute error and the second row the root mean squared error of m e = n σ uniformly distributed evaluation nodes within ±n σ standard deviations around the steady state. We observe that the Euler errors for the log-linearization approach depend strongly on the approximation range, n σ, and the preference parameters, γ and ψ. For the standard case of CRRA utility with γ = and ψ =.5 the maximum absolute approximation error is as small as. even for standard deviations around the steady state (n σ = ). The Euler errors increase dramatically for large values of risk aversion (γ = and ψ =.) with the maximum error being as large as 89 for n σ =. For the parameter set of Bansal and Yaron (), γ =, ψ =.5, the approximation errors become significantly smaller, so the log-linearization seems to give a good approximation of the model. Both the collocation and the Galerkin method produce about the same magnitude of errors for the same degree n of the polynomial approximation. Already for the -degree approximations, the Euler errors are at least orders of magnitude smaller than those of the log-linearization approach. Moreover, the errors for the projection methods decrease dramatically with larger n. Both projection methods are also very robust to changes in the preference parameters. The findings get confirmed for the second parameter set (see table )... Approximation Errors in the Wealth-Consumption Ratio Until now we only examined the Euler errors. In the following we compare the performance of the different solution methods in computing economically relevant moments. For this purpose, we report the absolute relative errors in the unconditional mean and the standard deviation of the log wealth-consumption ratio for the different methods in Tables (for the first set of Parameters are estimated from the Shiller dataset on annual all real consumption in the U.S. for the period from 889-9, see http://www.econ.yale.edu/ shiller/data.htm. (last accessed January 8, )
preference parameters) and (for the second set of preference parameters). We report the relative numerical errors as well as the computation times for different sets of preference parameters and approximation degrees. For the projection methods we set n σ = and for Tauchen (986) s method m T =. For the computation of the true solution we use Tauchen and Hussey (99) s method and let n T H until the change in the results is smaller than. [Table about here.] [Table about here.] We observe that the projection methods show very low approximation errors already for the -degree approximations and that the results are very robust across the different preference parameters. The log-linearization does a good job for all sets of parameters and only shows small problems for γ =, ψ =.. The discretization methods do a reasonable job in approximating first moments, but also show difficulties when it comes to the standard deviation of the wealth consumption ratio. Comparing the different discretization methods we find that the adjustment by Floden (7) (TH-F) slightly improves the original method (TH) as proposed in Tauchen and Hussey (99) especially with regard to the speed of convergence when the number of discretization nodes is increased. Tauchen (986) s method shows even smaller errors for the node discretization but doesn t converge as fast as TH-F. Compared to the projection methods, the discretizations perform significantly worse while needing about the same computation time. Log-linearization is the fastest method (usually about twice as fast as the -degree polynomial approximations) but as mentioned before, it is only able to achieve a reasonable accuracy of the model for preferences that are close to linear. In the following, we evaluate the robustness of the methods with regard to changes in the underlying parameter estimates. This exercise demonstrates the advantages and drawbacks of the different methods which will be essential to understand, why and in which cases the methods fail to compute accurate solutions. In particular it delivers interesting insights with regard to the evaluation of the long run risk model in section.... Robustness with Regard to Changes in the Input Parameters To evaluate the performance of the different methods with regard to different parameter specifications, Figure shows the approximation errors as in Table for different values of ρ. We slightly decrease σ ɛ so that the overall volatility of x t does not become too large: we set σ ɛ =., µ =., δ =.99 with the preference parameters being γ = and ψ =.5. For the projection methods a 9-degree approximation is used with n σ = and for the discretizations 9 nodes are used with m T =. We observe that the performance of the linearization and discretization methods strongly depends on the choice of model parameters. In particular the approximation errors increase Since Tauchen and Hussey (99) s method converges to the true distribution for n T H, this approach is a reliable but computationally rather slow method to calculate the correct solution. We compared the results to the solution from the projection methods with n σ = and a -degree approximation and observed that the results are essentially identical and that the projection methods are much faster.
dramatically with the serial correlation ρ c as well as with the standard deviation σ ɛ. The discretization method of Tauchen and Hussey (99) illustrates the fact (documented in Floden (7)) that the discretization becomes increasingly inaccurate, the larger the serial correlation. When looking at the standard deviation, we find that the approximation errors become very large with values up to 5%. The method of Floden (7) perform better but also shows large difficulties in approximating second order moments. The errors for the projection methods are hard to distinguish from zero and prove to be very robust (denote that the scales on the y-axes are 9 ). [Figure about here.] Figure shows a similar approximation error graph for different σ ɛ. Here we find, that the approximation errors for the log-linearization dramatically increase with the volatility in consumption. The discretization methods are robust to changes in σ ɛ with larger errors in the second moments that are independent of σ ɛ. Again, the errors for the projection methods are difficult to distinguish from zero and do not show any significant dependencies on changes in the volatility parameter (again denote the scale on the y-axes). [Figure about here.] In sum, the log-linearization is the fastest solution method and provides reasonably well approximations for this very simple model. The discretization methods show small errors for the original calibration but have difficulties when the underlying process shows large serial correlation patterns. The projection methods outperform all other methods and proves to be very robust with regard to different parameter specifications. Hence we find, that even in this very simple model, the robustness of the linearization and discretization methods strongly depend on the model specification for the underlying processes. The long run risk model has a rather complex system of processes to model consumption and dividend, with high serial correlations in the long run risk component as well as the conditional variance. In the following we show a complete evaluation of the model, comparing the different solution methods.. Bansal and Yaron () s Model For the second example we consider the long run risk model by Bansal and Yaron (). The main innovation in the model is that growth rates feature highly persistent but yet stochastic long run shocks. Additionally the conditional variance of the growth rates is stochastic. So the model has two states, the long run component x t and the variance level σt. Bansal and Yaron () specify the model as follows: c t+ = µ c + x t + σ t η t+ (9) x t+ = ρx t + φ e σ t e t+ σt+ = σ ( ν) + νσt + 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(, ). 5
We consider two sets of parameters for the model. The original set used in Bansal and Yaron () and the more recent version in Bansal, Kiku, and Yaron (). The parameters are calibrated to match annual financial market characteristics for the period from 9-8 while the decision interval of the representative agent is monthly. The parameter estimates are shown in table 5. The main difference between the two sets of parameters is that in Bansal, Kiku, and Yaron () the shock in consumption growth also effects the growth rate of dividends (π =.6), while this is not the case in Bansal and Yaron () (π = ). Also the persistence of the variance process is larger in Bansal, Kiku, and Yaron (). [Table 5 about here.] We solve the model for the return of the wealth portfolio z w, the market portfolio z m and the risk free rate z rf. As in the previous section we first look at Euler errors for the continous methods and evaluate all methods with respect to their ability to compute unconditional moments of the model variables. Afterwards we conduct a full evaluation of the different methods with respect to their ability to approximate different economically relevant moments on an annual basis. To compute the moments, we simulate.. years of artificial data and compute the annualized moments. A detailed description of how to compute the annual moments from the monthly observations can be found in Beeler and Campbell (). One critical issue in the model is, that the variance process σt can in fact become negative. To overcome this problem, Bansal and Yaron () replace all negative realizations with very small, but positive values. We proceed in the same way for all methods to achieve consistent results. For the approximation interval of the projection methods we choose x min =., x max =., σmin = and σ max =.. These are the minimum and maximum values that were reached within the simulations of the processes and therefore should provide a reasonable approximation range. For the collocation method we use the full tensor product of one-dimensional basis functions, which allows us to use Chebychev nodes in each dimension and still maintain the exactly identified system of equations ((n + ) unknown solution coefficients and (m + ) approximation nodes with n = m). For the Galerkin method we choose instead the set of complete polynomials, where the sum of the degrees of the two one-dimensional functions is smaller than n +. This choice reduces the number of unknown solution coefficients from (n + ) to (n + ) / + (n + )/ and thus lowers the computational costs without much loss of approximation quality. To further decrease computation times, we take the lower degree solutions as initial guesses for the higher degree approximations. For example to compute the 6-degree approximation we first compute the -degree approximation and take that as an initial guess. Computations times are always stated as total times to compute solutions including the computations for the initial guesses. For the log-linearization parameters, equations and are plugged in the pricing equation 9 for the wealth, market and risk free return to derive the linearization parameters for z w (x t, σ t ) = A,w + A,w x t + A,w σ t z m (x t, σ t ) = A,m + A,m x t + A,m σ t z rf (x t, σ t ) = A,rf + A,rf x t + A,rf σ t. 6
The solutions are given in Appendix A... Approximation Errors in the Euler Equations Table 6 shows Euler errors for the wealth, market and risk-free returns for uniformly distributed evaluation nodes in each dimension. The first row of each entry shows the maximum absolute error and the second row the root mean squared error. We find that already with the -degree polynomial approximation errors are at least digits smaller compared to the linear approximation. The collocation performs slightly better than Galerkin projection in most cases, which is a reason for the larger number of approximation nodes used. In general we find, that approximation errors are larger for the parameters in Bansal, Kiku, and Yaron () than in Bansal and Yaron (). [Table 6 about here.].. Approximation Errors in the Wealth-Consumption and Price-Dividend Ratio Table 7 shows absolute relative errors in the unconditional mean and standard deviation of the monthly log wealth-consumption and log price-dividend ratio as well as computation times. All results are compared to a 8 degree polynomial approximation using the collocation method. We find that the linearization does a reasonably good job for the parameters as in Bansal and Yaron () with a maximum error of.5% in the volatility of the price-dividend ratio. For the parameters set of Bansal, Kiku, and Yaron () the results get much worse. The linearization especially shows difficulties in approximating second order moments with a maximum error of 6.9% in the volatility of the price-dividend ratio. These large errors are a reason of the false assumption of the linear structure of the model. In particular the linearization approach assumes that first derivatives are constant and second derivatives zero. In section.. we show that this assumption is not true but leads to wrong implications especially regarding the volatility of the assets. The projection methods on the other hand show much smaller approximation errors already with the -degree approximation with very fast convergences for larger degrees. Denote that we use the 6-degree solutions as an initial guess for the 9 degree solutions, which is the reason why the approximation errors don t decrease further. Computing the higher order solution directly could further decrease the errors but increase computation time. Tauchen and Hussey (99) s method isn t able to produce any reliable results with a node discretization and shows very slow convergence properties especially for the calibration by Bansal, Kiku, and Yaron () with the large persistence of ν. With the extension of Floden (7) the method converges much faster at least for the parameter set of Bansal and Yaron (). The same is true for the method of Tauchen (986). Again we find that all discretization methods show especially difficulties in approximating the second order dynamics. The computation time of the discretization methods increases dramatically with the number of discretization nodes. For example for nodes, Tauchen and Hussey (99) s method takes about seconds to compute the optimal solution, while it takes We also computed the solution using Tauchen and Hussey (99) s method with a very large number of nodes in each dimension (). We got very close to the 8 degree solution but it takes considerably longer (about days using a standard laptop). 7
about 5 minutes to compute the 5 node approximation. The projection methods take less than a second to compute the -degree and about seconds for the 6-degree approximations, which already provides highly accurate results. In addition the Galerkin method is about 5% faster compared to collocation. [Table 7 about here.].. Understanding the Shortcomings of Log-Linearization Figures (calibration BKY ()) and 5 (calibration BY ()) show the wealth-consumption (left panel) and price-dividend (mid panel) ratio as a function of the state variables x and σ for the collocation projection and the log-linearization as well as the histogram of 7 simulations of the states x and σ. Corresponding first and second derivatives with respect to the state variables are shown in figures and 6. We find that in the BY () calibration the wealth-consumption ratio is almost linear and the log-linear approximation almost perfectly matches the higher order approximation, but the price-dividend ratio shows significantly larger non-linearities (see figure 6). This finding is reflected in the approximation errors in the first two moments of the two ratios (compare table 7). Looking at the calibration of BKY () we find that the wealth-consumption as well as the price-dividend ratio become strongly nonlinear (see figure ) which explains the larger approximation errors of the log-linearization for the second calibration especially for the second order moments. [Figure about here.] [Figure about here.] [Figure 5 about here.] [Figure 6 about here.].. Evaluation of Annualized Moments in the Long Run Risk Model Finally, in table 8 we show absolute relative errors in the mean and standard deviation of the annualized price-dividend ratio, the annualized market and risk free return as well as the equity premium for the recent version of the long run risk model in Bansal, Kiku, and Yaron (). The corresponding absolute values of the moments are shown in table 9. Again we find that maximum error for the projection methods are already very small for the three degree approximation with maximum errors around % and the convergence rates are quite fast for larger degrees. The linear approximation shows large approximation errors in the standard deviation of the annualized price-dividend ratio (up to %). Also the errors in expected returns are quite larger with errors up to % for the equity premium. For example for the equity premium this means, that the linearization overestimated the absolute premium by about %. When evaluating a consumption based asset pricing model, this difference might have a large influence on the calibration and hence economic implications of the model. Therefore in figures 7 and 8 we show the changes in the model parameters that are needed to adjust for the error induced by the linearization. In 8
particular we consider the parameters, to which there exist large uncertainty about their true magnitude, namely the risk aversion γ, the intertemporal elasticity of substitution ψ and the serial correlation in the long run risk ρ as well as the conditional variance ν. In figure 7 each plot shows the equity premium as a function of some parameter (top-left panel: ν, top-right panel: ρ, bottom-left panel: γ, bottom-right panel: ψ). The black dotted lines show the parameter of the calibration used by Bansal, Kiku, and Yaron () (x axis) and the true implied equity premium (y axis). The gray dashed line shows the equity premium implied by the linearization (y axis) and the parameter value that is needed to match this premium assuming that all other parameters are kept constant (x axis). The equity premium implied by the linearization is 5.8% while the true premium is only.7% (see table 9). Figure 7 shows that ν would have to increase from.999 to.9996 to match the equity premium implied by the linearization. ρ would have to increase from.975 to.979, γ from to. and ψ from.5 to.8. In figure 8 the corresponding results are shown for the volatility of the log price-dividend ratio. It shows, that ν would have to increase from.999 to.9996 to match the price-dividend ratio implied by the linearization. ρ would have to increase from.975 to.986 and the preference parameters γ and ψ have a negligible influence on the volatility of the price-dividend ratio. The long run risk model has in general difficulties matching the empirical volatility of the price dividend ratio (see Beeler and Campbell () who show, that the volatility implied by the model is much lower, compared to the volatility found in the data) and solving the model accurately, lowers the ratio even further. We can conclude, that the model is very sensitive to even small changes in the parameters of the underlying processes, so the false computations of the linearization have a large impact on the calibration and hence the economic implications of the model. [Table 8 about here.] [Table 9 about here.] [Figure 7 about here.] [Figure 8 about here.] 5 Conclusion The paper presents an efficient and robust method to solve asset pricing models with recursive preferences. The performance of standard methods for solving asset pricing models is highly dependent on the input parameters and the methods are not suitable for solving modern asset pricing models with non-linear preference structures. For example log-linearization provides a fast and easy solution method but the results are only accurate for preferences close to log utility. Recent findings in the asset pricing literature have shown, that a risk aversion around and an elasticity of intertemporal substitution of around.5 are reasonable values, so linearization techniques aren t appropriate for solving those models. While discretization can at least in theory converge to the true distribution of the underlying process, they show to be very inefficient and highly dependent on the choice of parameters. They show large difficulties when it comes to highly 9
persistent processes and hence require a large number of discretization nodes. But computation times dramatically increase with the number of nodes especially in higher dimensions. The solution method presented in this paper proves to be highly accurate and the performance does neither depend on the choice of preference parameters nor on the specification of the underlying processes. Already the -degree approximations give highly accurate solutions in the asset pricing examples under consideration while they take only slightly longer than the log-linearization approach. In low dimensions the difference between the Galerkin and the collocation projection is only marginal, but the Galerkin methods proves to be more efficient for higher dimensions as the approximation degree can be independently chosen from the number of approximation points. This suggest that the solution methods that have been used in the past don t meet the requirements to accurately solve modern asset pricing models while the method presented in this paper provides an efficient and robust alternative.