solving asset-pricing models with recursive preferences · solving asset-pricing models with...

Solving Asset-Pricing Models with

Recursive Preferences∗

Walter Pohl

University of Zurich

Karl Schmedders


and

Swiss Finance Institute

Ole Wilms


July 25, 2014

Abstract

We present a projection-based solution algorithm to solve asset pricing models with re-

cursive preferences. The method outperforms common methods like discretization and log-

linerization 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: G11, G12.

∗We are indebted to Lars Hansen, Ken Judd, and Martin Lettau for helpful discussions on the subject. We thankseminar audiences at the University of Zurich and the Becker Friedman Institute for comments. Karl Schmeddersgratefully acknowledges financial support from the Swiss Finance Institute.

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1 Introduction

Asset pricing models have become increasingly complex over the last two decades. The first

generation of equilibrium asset pricing models, developed in the 1980s, (Grossman and Shiller

(1981), Hansen and Singleton (1982), Mehra and Prescott (1985)), 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 (2004) or Hansen, Heaton, and Li (2008)).

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 (1988). Bansal

and Yaron (2004), Bansal, Kiku, and Yaron (2012), Lettau and Wachter (2007), and Beeler and

Campbell (2012) illustrate this approach. Loglinearization by its very nature misses higher-order

approximation errors. For example Caldara, Villaverde, Ramirez, and Yao (2012) 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 (1986) and Tauchen and Hussey

(1991), and have been applied by many subsequent works, such as Guvenen (2009), Heaton and

Lucas (1996), Heaton and Lucas (2000), Hansen, Heaton, and Yaron (1996), Campbell (1993) and

Judd, Kubler, and Schmedders (2011). 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 (1992).

While linearization methods are very fast, and able to compute reasonable solutions for pref-

erences 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 (2012) approximation

errors are already with a 3-degree polynomial approximation about 2 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 (2004) and Bansal, Kiku, and Yaron (2012).

The paper is organized as follows. In Section 2 we present the general solution algorithm and

Section 3 gives a short summary of the solution methods that are commonly used to solve asset

pricing models. In Section 4 we consider two recent asset pricing models, namely the model by

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Tallarini (2000) and Bansal and Yaron (2004) to evaluate the performance of the different solution

methods and Section 5 concludes.

2 Solution Method

We consider a standard asset-pricing model with a representative agent and recursive preferences

as in Epstein and Zin (1989) and Weil (1989). Indirect utility at time t, Vt, is given recursively as

Vt =

[(1− δ)C

1−γθ

t + δ[Et

(V 1−γt+1

)] 1θ

] θ1−γ

. (1)

In this parametrization, Ct 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 = 1−γ1− 1

ψ

. For θ = 1

the agent has standard CRRA preferences and θ < 1 indicates a preference for the early resolution

of risk. Using the agent’s Euler equation, Epstein and Zin (1989) show that the gross return of

asset i, Ri,t+1 =Pi,t+1+Di,t+1

Pi,tmust satisfy the following pricing equation,

Et

[δθ(Ct+1

Ct

)− θψ

Rθ−1w,t+1Ri,t+1

]= 1. (2)

The term Rw,t+1 denotes the return on a claim to aggregate consumption, Rw,t+1 = Wt+1

Wt−Ct , where

Wt is the (unobserved) wealth level of the agent at time t. As Equation (2) has to hold for

all assets i, it must also hold for the return of the aggregate consumption claim. So Rw,t+1 is

determined by the wealth-euler equation given by

Et

[δθ(Ct+1

Ct

)− θψ

Rθw,t+1

]= 1. (3)

We present an algorithm using projection methods to solve the general asset pricing model

given by equations (1)–(3). 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 (1992) 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.

2.1 Projection Methods for Functional Equations

Projection methods are a general tool to solve functional equations of the form

G(z(x)) = 0, (4)

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where the variable x resides in a (state) space X ⊂ Rl, l ≥ 1, and z is an unknown solution

function with domain X, so z : X → Rm. The given function G is a continuous mapping between

two function spaces. Note that solving Equation (4) 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.1

For a set {Λk}k∈{0,2,...,n} of chosen basis functions the approximation z of z is

z(x;α) =n∑k=0

αkΛk(x), (5)

where α = [α0, α1, . . . , αn] are unknown coefficients. Replacing the function z in Equation (4)

by its approximation z, we can define the residual function F (x;α) as the error in the original

equation,

F (x;α) = G(z(x;α)). (6)

Instead of solving Equation (4) for the unknown function z, we now attempt to solve Equation (6)

for the coefficients α in the approximation term z(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 Rn+1. 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 {gk(x)}nk=0. We can then define an inner

product between the residual function F and the test function gk,∫XF (x;α)gk(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,∫XF (x;α)gk(x)w(x)dx = 0, k = 0, 1, . . . , n. (7)

Observe that this system of equations imposes n+1 conditions on the (n+1)-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 + 1 distinct nodes in the domain, {xk}nk=0, and defines the

1In addition to polynomial approximations, approximations using cubic splines or B-splines are often very useful.

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test functions gk by

gk(x) =

{0 if x 6= xk

1 if x = xk.

With a weight term w(x) ≡ 1, the projection conditions (7) simplify to

F (xk;α) = 0, k = 0, 1, . . . , 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) ≡ 1 but uses the basis functions as

the test functions, gk(x) = Λk(x). Put differently, the residual function F is multiplied by a

basis function Λk and the resulting product is integrated over the entire domain. The projection

conditions (7) simplify to ∫XF (x;α)Λk(x)dx = 0, k = 0, 1, . . . , n. (9)

In the following we describe how to apply the general projection method to the asset-pricing

model given by Equations (1)–(3).

2.2 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 (4). 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 (1985), the

state x is given by log consumption growth; in the two-dimensional model of Bansal and Yaron

(2004), the state x consists of the long-run mean of consumption growth (denoted by xt in that

paper) and the variance of consumption growth (denoted by σ2t ).) The state-dependent return of

the aggregate consumption claim can be written as

Rw(x′|x) =W (x′)

W (x)− C(x)=

W (x′)C(x′)

W (x)C(x) − 1

× C(x′)

C(x). (10)

Taking logarithms on both sides of this equation yields

rw(x′|x) = w(x′)− c(x′)− log(ew(x)−c(x) − 1

)+ ∆c(x′|x), (11)

where lower case letters denote logs of variables and ∆c(x′|x) = c(x′)− c(x). Hence Equation (3)

in state space representation becomes

E

[eθ log δ− θ

ψ∆c(x′|x)+θrw(x′|x)

∣∣∣∣x] = 1. (12)

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As a final step, we rewrite this equilibrium condition in the format of the functional equa-

tion (4). For this purpose, we define the unknown solution function z as the logarithmic wealth-

consumption ratio zw(x) = w(x)− c(x) and obtain the functional equation

E

[eθ(

log δ+(1− 1ψ

)∆c(x′|x)+zw(x′)−log(ezw(x)−1))− 1

∣∣∣∣x] = 0. (13)

Using the approximation (5), zw(x;αw) =∑n

k=0 αw,kΛk(x), we obtain the approximation of the

log return function for the aggregate consumption claim,

rw(x′|x;αw) = zw(x′;αw)− log(ezw(x;αw) − 1

)+ ∆c(x′|x). (14)

Finally, the residual function for the wealth-Euler equation (see Equation (6)) is given by

Fw(x;αw) = E

[eθ(

log δ+(1− 1ψ

)∆c(x′|x)+zw(x′;αw)−log(ezw(x;αw)−1))− 1

∣∣∣∣x] . (15)

Applying one of the projections (collocation or Galerkin method) to the residual function Fw(x;αw)

yields the unknown solution coefficients αw. These determine the state-dependent wealth-con-

sumption ratio zw(x;αw) which in turn leads to the (approximate) return function of the aggregate

consumption claim, rw(x′|x;αw).

With rw(x′|x;αw) at hand, we can now develop an approach to compute the return of any

asset i using Equation (2). Again, first define the state-dependent return of asset i as

Ri(x′|x) =

Pi(x′) +Di(x

′)

Pi(x)=

Pi(x′)

Di(x′)+ 1

Pi(x)Di(x)

× Di(x′)

Di(x)(16)

or, equivalently,

ri(x′|x;α) = log

(epi(x

′)−di(x′) + 1)− (pi(x)− di(x)) + ∆di(x

′|x). (17)

Performing the analogous step to the derivation of Equation (14), we approximate the log price-

dividend ratio zi(x) = pi(x)− di(x) by zi(x;αi) =∑n

k=0 αi,kΛk(x) which leads to

ri(x′|x;αi) = log

(ezi(x

′;αi) + 1)− zi(x;αi) + ∆di(x

′|x). (18)

Equation (2) in state-space representation is then given by

E

[eθ log δ− θ

ψ∆c(x′|x)+(θ−1)rw(x′|x)+ri(x

′|x)

∣∣∣∣x] = 1. (19)

Again, we define the residual function using the approximation zi(x;αi) as the deviation of the

pricing equation from zero,

Fi(x;αi) = E

[eθ log δ− θ

ψ∆c(x′|x)+(θ−1)rw(x′|x;αw)+log

(ezi(x

′;αi)+1)−zi(x;αi)+∆di(x

′|x) − 1

∣∣∣∣x] . (20)

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Recall that the coefficients αw and thus the function rw(x′|x;αw) have been computed previously.

Therefore, we apply one of the projections only to solve for the unknown vector αi.

In sum, we apply the projection method twice. In the first step, we approximate the log

wealth-consumption ratio zw(x;αw) by applying the projections on the residual function of the

wealth-Euler equation (15). Once αw is known, the projections can be applied to Equation (20) to

solve for the price-dividend ratio zi(x;αi) 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 ⊂ Rl; choose the functional forms for zw(x;αw)

and zi(x;αi) as well as the projection method; choose the nodes for the projection method,

X = {xj : 0 ≤ j ≤ m} ⊂ X.

Step 1. Use the wealth-Euler Equation (3) together with approximated log wealth consumption

ratio zw(x;αw) and the definition of the return equation (14) to derive the residual function

for the return on wealth

Fw(x;αw) = E

[eθ(

log δ+(1− 1ψ

)∆c(x′|x)+zw(x′;αw)−log(ezw(x;αw)−1))− 1

∣∣∣∣x] . (21)

Compute the unknown solution coefficients αw by imposing the projections on Equation (21).

Step 2. Take the solutions for the wealth-consumption ratio zw(x;αw) as given and use the

Euler equation (2) for asset i together with the approximated log price-dividend ratio zi(x;αi)

and the definition of the return equation (18) to derive the residual function for asset i given

by

Fi(x;αi) = E

[eθ log δ− θ

ψ∆c(x′|x)+(θ−1)rw(x′|x;αw)+log

(ezi(x

′;αi)+1)−zi(x;αi)+∆di(x

′|x) − 1

∣∣∣∣x] .(22)

Compute the unknown solution coefficients αi by imposing the projections on Equation (22).

Evaluation. Choose a set of evaluation nodes Xe = {xej : 1 ≤ 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 algo-

rithmic details such as the choices for basis functions and the integration technique.

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2.3 Algorithmic Ingredients

In the Initialization step, we need to choose a set of basis functions for the polynomial ap-

proximation, a projection method and a set of nodes. To simplify the presentation, we describe

the necessary choices for a one-dimensional state space X = [xmin, xmax]. We approximate the

solution functions zw(x;αw) and zi(x;αi) by Chebyshev polynomials (of the first kind), see Judd

(1998). We obtain Chebyshev polynomials via the recursive relationship

T0(ξ) = 1 (23)

T1(ξ) = ξ (24)

Tk+1(ξ) = 2ξTk(ξ)− Tk−1(ξ).

with Tk : [−1, 1] → R. Since we need to approximate functions on the domain X and the

Chebyshev polynomials are defined on the interval [−1, 1], we need to transform the argument for

the polynomials. The basis functions for the approximate solution (5) are given by

Λk(x) = Tk

(2

(x− xmin

xmax − xmin

)− 1

)(25)

for k = 0, 1, . . . , n.

For the set of nodes X for the projection method we use the m + 1 zeros of the Chebyshev

polynomial Tm+1. These points are called Chebyshev nodes,

ξj = cos

(2j + 1

2n+ 2π

), j = 0, 1, . . . ,m. (26)

Since all Chebyshev nodes are in the interval [−1, 1], we need to transform them to obtain nodes

in the state space X. This transformation is

xj = xmin +xmax − xmin

2(1 + ξj), j = 0, 1, . . . ,m. (27)

For the collocation projection the number of basis functions n+1 must be identical to the number

of approximation nodes m + 1, and so m = n. In Step 1 (and Step 2, 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 (15) and (20) 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 [xmin, xmax]. And finally,

we need to solve the resulting nonlinear system of equations.

For the Evaluation step we use me >> m equally spaced evaluation nodes in the state space

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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 (20) given by

RMSE =

√√√√ 1

me

me∑j=1

F (xej |α)2 (28)

MAE = maxj=1,2,...,me

|F (xej |α)| (29)

with

xej = xmin +xmax − xminme − 1

(j − 1), j = 1, . . . ,me. (30)

3 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 (1986) and Tauchen and Hussey (1991) and the log-linearization approach that has

been applied to asset pricing as early as [XXX CORRECTION NEEDED?] Campbell and Shiller

(1988). 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.

3.1 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 (19) 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 prob-

abilities. For demonstration purposes, we consider the simple case of the discretization of an

AR(1) process that is given by

xt+1 = (1− ρ)µ+ ρxt + εt+1, εt ∼ N(0, σ2e), (31)

with persistence |ρ| < 1 and the unconditional mean µ. The unconditional volatility of the process

is given by σx = σe/√

1− ρ2.

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Tauchen (1986)’s method

Tauchen (1986) assumes a set of equally spaced nodes XnT = {x1, . . . , xnT } for the discrete state

space with x1 = µ −mTσy and xnT = µ + mTσy. The factor mT 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 mT even though its value strongly influences the approximation results.) Denote

the step size between two adjacent grid points by h = xi − xi−1. Then the elements πij of the

(nT × nT )-transition probability matrix π are defined by

πij =

Φ(xj+h/2−(1−ρ)µ−ρxi

σe

)Φ(xj+h/2−(1−ρ)µ−ρxi

σe

)− Φ

(xj−h/2−(1−ρ)µ−ρxi

σe

)1− Φ

(xj−h/2−(1−ρ)µ−ρxi

σe

)for j = 1,

for 1 < j < nT ,

for j = nT .

Tauchen and Hussey (1991)’s method and the extension by Floden (2007)

Tauchen and Hussey (1991)’s method is based on Gauss-Hermite quadrature. Let ξi and ωi,

i = 1, . . . , nTH , be the Gauss-Hermite nodes and weights on the interval [−∞,+∞], respectively.

The approximation nodes are then given by xi = µ+√

2σeξi and the entries πij of the transition

probability matrix π can be computed by

πij =ωij∑Nj=1 ωij

(32)

with

ωij = π−0.5ωjf(xj |xi)f(xj |µ)

(33)

where f(·|xi) is the density function of N((1− ρ)µ+ ρxi, σ2). Tauchen and Hussey (1991) simply

choose σ = σe. Floden (2007) chooses σ = aσe+(1−a)σx with a = 0.5+0.25ρ, so σ is a weighted

average of σx and σe. He claims that his approach performs significantly better than the original

Tauchen and Hussey (1991) method, particularly for highly persistent processes.

3.2 Log-Linearization

The idea of log-linearization is to solve the model by a linear approximation around the steady-

state of the model. Assume that the log price-dividend ratio zi(x) of asset i is a linear function

of the state variable x ∈ Rl, that is,

zi(x) = A0,i +A1,ix (34)

where A0,i is an unknown constant, x is an l × 1 vector representing the state of the economy

in period t and A1,i is a 1× 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 (18))

ri(x′|x) = log

(ezi(x

′) + 1)− zi(x) + ∆di(x

′|x). (35)

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Campbell and Shiller (1988) use a first-order Taylor approximation around the average level of

the log price-dividend ratio zi:

ri(x′|x) ≈ κi,0 + κi,1zi(x

′)− zi(x) + ∆di(x′|x), (36)

where constants κi,0 and κi,1 depend only on the average level of the log price-dividend ratio zi,

κi,1 =ezi

1 + ezi(37)

κi,0 = − log(

(1− κi,1)1−κi,1κκi,1i,1

). (38)

After substituting expressions (34) and (40) into the pricing equation (19), the unknown coeffi-

cients A0,i and A1,i can be derived analytically as a function of the linearization parameters κi,0

and κi,1. The parameters A0,i and A1,i determine z while κi,0 and κi,1 are nonlinear functions

of that mean. Hence, by iteratively computing a fixed point we can find both the mean of zi,t

and the parameters. For the return of the wealth portfolio the approximation can be applied

analogously. As the return is given by

rw(x′|x) = zi(x′)− log

(ezi(x

′) − 1)

+ ∆c(x′|x). (39)

the approximation becomes

rw(x′|x) ≈ zi(x′) + κi,0 − κi,1zi(x) + ∆di(x′|x), (40)

with

κw,1 =ezw

ezw − 1(41)

κw,0 = log(

(κw,1 − 1)1−κw,1κκw,1w,1

). (42)

4 Numerical Evaluation

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 (2000). Log consumption is modeled as AR(1) 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 (2004), which has gathered much attention recently for its ability to match

11

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 me = 100nσ equally spaced evaluation nodes (in each dimension) in the interval [xmin, xmax]

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 10−8.

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

(1991) 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.

4.1 Tallarini (2000)’s Model

We consider the endowment economy of Tallarini (2000). Log consumption ct is modeled as simple

AR(1) deviations from a linear trend,

ct = µt+ xt (43)

xt = ρxt−1 + σεεt, εt ∼ N(0, 1), (44)

where µ is the average net growth rate of consumption and ρ the degree of persistence. The state

of the economy is given by xt and log consumption growth is given by

∆ct+1 = µ+ xt+1 − xt. (45)

The state space is one-dimensional. In the following analysis, we focus exclusively on the pricing

of the wealth portfolio (Step 1 in our solution algorithm).

For the projection methods we approximate solution functions within ±nσ standard deviations

around the steady state (E(xt) = 0) of the model. So, xmin = E(xt) − nσσ(xt) and xmax =

E(xt) + nσσ(xt) with E(xt) = 0 and σ(xt) = σε/√

1− ρ2. 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

xt of the economy,

zw(xt) = A0,w +A1,wxt. (46)

The unknown solution coefficients A0,w and A1,w can be derived by substituting expression (34)

and (40) into the pricing equation (19) and are given by

A0 =log δ + (1− 1

ψ )µ+ 0.5θ((1− 1ψ ) +A1)2σ2

ε + κ0

κ1 − ρ(47)

A1 =(1− 1

ψ )(ρ− 1)

1− κ1ρ. (48)

12

4.1.1 Approximation Errors in the Euler Equations

Tables 1 and 2 report Euler approximation errors for the pricing of the wealth portfolio for the two

projection methods (for degrees 3, 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 = 2, ψ = 0.5) and a high (γ = 10, ψ = 0.1)

degree of risk aversion, respectively. The third combination are the parameter estimates from

Bansal and Yaron (2004), γ = 10 and ψ = 1.5. Table 1 reports Euler errors for these three cases.

Table 2 depicts errors for three more cases of Epstein-Zin preferences, namely for γ = 10, ψ = 0.5,

for γ = 7.5, ψ = 1.5, and for γ = 2, ψ = 1.5. The leftmost column indicates the state space

by providing the number nσ of standard deviations around the steady state of the model. The

parameters for the consumption process are taken from Pohl, Schmedders, and Wilms (2014) and

are given by σε = 0.0343, µ = 0.02, ρ = 0.91.2 We set δ = 0.99. We later vary the parameters to

analyze their influence on the performance of the different solution methods.

[Table 1 about here.]


For each value of nσ, the first row in Tables 1 and 2 shows the maximum absolute error and

the second row the root mean squared error of me = 100nσ uniformly distributed evaluation nodes

within ±nσ standard deviations around the steady state. We observe that the Euler errors for the

log-linearization approach depend strongly on the approximation range, nσ, and the preference

parameters, γ and ψ. For the standard case of CRRA utility with γ = 2 and ψ = 0.5 the maximum

absolute approximation error is as small as 0.011 even for 10 standard deviations around the steady

state (nσ = 10). The Euler errors increase dramatically for large values of risk aversion (γ = 10

and ψ = 0.1) with the maximum error being as large as 289 for nσ = 10. For the parameter set of

Bansal and Yaron (2004), γ = 10, ψ = 1.5, 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 3-degree approximations, the Euler errors are at

least 2 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 2).

4.1.2 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 3 (for the first set of

2Parameters are estimated from the Shiller dataset on annual all real consumption in the U.S. for the period from1889-2009, see http://www.econ.yale.edu/ shiller/data.htm. (last accessed January 28, 2014)

13

preference parameters) and 4 (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 nσ = 3 and for Tauchen (1986)’s

method mT = 2. For the computation of the “true” solution we use Tauchen and Hussey (1991)’s

method and let nTH →∞ until the change in the results3 is smaller than 10−10.



We observe that the projection methods show very low approximation errors already for the

3-degree approximations and that the results are very robust across the different preference pa-

rameters. The log-linearization does a good job for all sets of parameters and only shows small

problems for γ = 10, ψ = 0.1. 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 (2007) (TH-F) slightly improves the original method (TH) as proposed in Tauchen

and Hussey (1991) especially with regard to the speed of convergence when the number of dis-

cretization nodes is increased. Tauchen (1986)’s method shows even smaller errors for the 3

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 3-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 4.2.

4.1.3 Robustness with Regard to Changes in the Input Parameters

To evaluate the performance of the different methods with regard to different parameter specifica-

tions, Figure 1 shows the approximation errors as in Table 3 for different values of ρ. We slightly

decrease σε so that the overall volatility of xt does not become too large: we set σε = 0.03, µ =

0.02, δ = 0.99 with the preference parameters being γ = 10 and ψ = 1.5. For the projection

methods a 9-degree approximation is used with nσ = 3 and for the discretizations 9 nodes are

used with mT = 2.

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

3Since Tauchen and Hussey (1991)’s method converges to the true distribution for nTH → ∞, this approach is areliable but computationally rather slow method to calculate the correct solution. We compared the results to thesolution from the projection methods with nσ = 12 and a 33-degree approximation and observed that the resultsare essentially identical and that the projection methods are much faster.

14

dramatically with the serial correlation ρc as well as with the standard deviation σε. The dis-

cretization method of Tauchen and Hussey (1991) illustrates the fact (documented in Floden

(2007)) 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 50%. The method of Floden (2007) perform better but also shows large diffi-

culties 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

10−9).

[Figure 1 about here.]

Figure 2 shows a similar approximation error graph for different σε. 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 σε with larger errors in the

second moments that are independent of σε. 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).


In sum, the log-linearization is the fastest solution method and provides reasonably well ap-

proximations 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.

4.2 Bansal and Yaron (2004)’s Model

For the second example we consider the long run risk model by Bansal and Yaron (2004). 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 xt and the variance level σ2t . Bansal and Yaron

(2004) specify the model as follows:

∆ct+1 = µc + xt + σtηt+1 (49)

xt+1 = ρxt + φeσtet+1

σ2t+1 = σ2(1− ν) + νσ2

t + σωωt+1

∆dt+1 = µd + Φxt + φσtut+1 + πσtηt+1

ηt+1, et+1, ωt+1, ut+1 ∼ i.i.d.N(0, 1).

15

We consider two sets of parameters for the model. The original set used in Bansal and Yaron

(2004) and the more recent version in Bansal, Kiku, and Yaron (2012). The parameters are

calibrated to match annual financial market characteristics for the period from 1930-2008 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 (2012) the shock in consumption growth also effects the growth rate of dividends (π = 2.6),

while this is not the case in Bansal and Yaron (2004) (π = 0). Also the persistence of the variance

process is larger in Bansal, Kiku, and Yaron (2012).


We solve the model for the return of the wealth portfolio zw, the market portfolio zm and the

risk free rate zrf . 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 1.000.000 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 (2012). One critical issue in the model is,

that the variance process σ2t can in fact become negative. To overcome this problem, Bansal and

Yaron (2004) 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 xmin = −0.013, xmax = 0.013, σ2min = 0 and σ2

max = 0.00032. 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 + 1)2

unknown solution coefficients and (m+ 1)2 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+ 1. This choice reduces the number of unknown

solution coefficients from (n + 1)2 to (n + 1)2/2 + (n + 1)/2 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 3-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 34 and 40 are plugged in the pricing equation

19 for the wealth, market and risk free return to derive the linearization parameters for

zw(xt, σ2t ) = A0,w +A1,wxt +A2,wσ

2t

zm(xt, σ2t ) = A0,m +A1,mxt +A2,mσ

2t

zrf (xt, σ2t ) = A0,rf +A1,rfxt +A2,rfσ

2t .

16

The solutions are given in Appendix A.

4.2.1 Approximation Errors in the Euler Equations

Table 6 shows Euler errors for the wealth, market and risk-free returns for 200 uniformly dis-

tributed 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

3-degree polynomial approximation errors are at least 2 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 (2012) than in

Bansal and Yaron (2004).


4.2.2 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 18 degree polynomial approximation using the collocation method.4

We find that the linearization does a reasonably good job for the parameters as in Bansal and

Yaron (2004) with a maximum error of 1.52% in the volatility of the price-dividend ratio. For the

parameters set of Bansal, Kiku, and Yaron (2012) the results get much worse. The linearization

especially shows difficulties in approximating second order moments with a maximum error of

26.9% 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 4.2.3 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 3-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

(1991)’s method isn’t able to produce any reliable results with a 10 node discretization and shows

very slow convergence properties especially for the calibration by Bansal, Kiku, and Yaron (2012)

with the large persistence of ν. With the extension of Floden (2007) the method converges much

faster at least for the parameter set of Bansal and Yaron (2004). The same is true for the method

of Tauchen (1986). 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 10 nodes, Tauchen

and Hussey (1991)’s method takes about 3 seconds to compute the optimal solution, while it takes

4We also computed the solution using Tauchen and Hussey (1991)’s method with a very large number of nodes ineach dimension (200). We got very close to the 18 degree solution but it takes considerably longer (about 2 daysusing a standard laptop).

17

about 50 minutes to compute the 50 node approximation. The projection methods take less than

a second to compute the 3-degree and about 4 seconds for the 6-degree approximations, which

already provides highly accurate results. In addition the Galerkin method is about 25% faster

compared to collocation.


4.2.3 Understanding the Shortcomings of Log-Linearization

Figures 3 (calibration BKY (2012)) and 5 (calibration BY (2004)) show the wealth-consumption

(left panel) and price-dividend (mid panel) ratio as a function of the state variables x and σ2 for

the collocation projection and the log-linearization as well as the histogram of 107 simulations of

the states x and σ2. Corresponding first and second derivatives with respect to the state variables

are shown in figures 4 and 6. We find that in the BY (2004) 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 (2012) we find that the wealth-consumption

as well as the price-dividend ratio become strongly nonlinear (see figure 4) which explains the

larger approximation errors of the log-linearization for the second calibration especially for the

second order moments.





4.2.4 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 (2012). 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 1% 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 22%). Also the errors in expected returns are quite larger with errors

up to 22% for the equity premium. For example for the equity premium this means, that the

linearization overestimated the absolute premium by about 1%. 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

18

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 (2012) (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 5.81% while the true

premium is only 4.74% (see table 9). Figure 7 shows that ν would have to increase from 0.999 to

0.9996 to match the equity premium implied by the linearization. ρ would have to increase from

0.975 to 0.979, γ from 10 to 11.4 and ψ from 1.5 to 4.8. 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 0.999 to 0.9996 to match the price-dividend ratio implied by the linearization. ρ would have

to increase from 0.975 to 0.986 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 (2012) 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.





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 10 and

an elasticity of intertemporal substitution of around 1.5 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

19

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 3-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.

20

Appendix

A Log-Linearization Coefficients for the Long Run Risk Model

A0,w =log δ + (1− 1

ψ )µ+A2,wσ2(1− ν) + κ0,w + 0.5θ(A2,wσw)2

κ1,w − 1

A1,w =(1− 1

ψ )

κ1,w − ρ

A2,w =0.5θ

((1− 1

ψ )2 + (A1,wφe)2))

κ1,w − ν

A0,m =

[θ log δ − γµ+ µd + (θ − 1)(κ0,w +A0.w(1− κ1,w)) + (θ − 1)A2,wσ

2(1− ν)

+ κ0,m + κ1,mA2,mσ2(1− ν) + 0.5 ((θ − 1)A2,wσω + κ1,mA2,mσω)2

]/(1− κ1,m)

A1,m =(Φ− 1

ψ )

1− κ1,mρ

A2,m =0.5((π − γ)2 + φ2) + 0.5((θ − 1)A1,wφe + κ1,mA1,mφe)

2 + (θ − 1)A2,w(ν − κ1,w)

1− κ1,mν

A0,rf = θ log δ − γµ+ (θ − 1)(κ0,w +A0,w(1− κ1,w) +A2,wσ2(1− ν)) + 0.5((θ − 1)A2,wσω)2

A1,rf = −γ + (θ − 1)A1,w(ρ− κ1,w)

A2,rf = 0.5(γ2 + ((θ − 1)A1,wφe)2) + (θ − 1)A2,w(ν − κ1,w)

21

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23

List of Figures

1 Absolute Relative Errors in the Pricing of the Mean and Standard Deviation of theLog Wealth-Consumption Ratio as a Function of ρ . . . . . . . . . . . . . . . . . . 25

2 Absolute Relative Errors in the Pricing of the Mean and Standard Deviation of theLog Wealth-Consumption Ratio as a Function of σε . . . . . . . . . . . . . . . . . . 26

3 Log Wealth-Consumption and Price-Dividend Ratios as a Function of the States xand σ2 in the Model of Bansal, Kiku, and Yaron (2012) . . . . . . . . . . . . . . . 27

4 First and Second Derivatives of the Log Wealth-Consumption and Price-DividendRatios as a Function of the States x and σ2 in the Model of Bansal, Kiku, andYaron (2012) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Log Wealth-Consumption and Price-Dividend Ratios as a Function of the States xand σ2 in the Model of Bansal and Yaron (2004) . . . . . . . . . . . . . . . . . . . 29

6 First and Second Derivatives of the Log Wealth-Consumption and Price-DividendRatios as a Function of the States x and σ2 in the Model of Bansal and Yaron (2004) 30

7 Implications of the Errors of the Linearization . . . . . . . . . . . . . . . . . . . . . 318 Implications of the Errors of the Linearization . . . . . . . . . . . . . . . . . . . . . 32

24

Figure 1: Absolute Relative Errors in the Pricing of the Mean and Standard Deviation of the LogWealth-Consumption Ratio as a Function of ρ

0.9 0.92 0.94 0.96 0.980

1

2

3

4x 10

−4

ρc

Err

or(

E(w

t−c

t))

0.9 0.92 0.94 0.96 0.98 10

1

2

3

4

5

6x 10

−3

ρc

Err

or(

σ(w

t − c

t))

0.9 0.92 0.94 0.96 0.980

1

2

3

4

5x 10

−3

ρc

Err

or(

E(w

t−c

t))

0.9 0.92 0.94 0.96 0.980

0.1

0.2

0.3

0.4

0.5

ρc

Err

or(

σ(w

t − c

t))

0.9 0.92 0.94 0.96 0.980

1

2x 10

−11

ρc

Err

or(

E(w

t−c

t))

0.9 0.92 0.94 0.96 0.980

0.5

1

1.5

2

2.5

3x 10

−10

ρc

Err

or(

σ(w

t − c

t))

LogLin LogLin

TH

TH−F

T

TH

TH−F

T

Col

Gal

Col

Gal

The graph shows absolute relative errors in the unconditional mean (left panel) and standard deviation(right panel) of the wealth-consumption ratio for different values of ρ. For the projection methods a 9-degree approximation is used with nσ = 3. For the discretizations 9 nodes are used and mT = 2. Tocompute the true solution we use Tauchen and Hussey (1991)’s method and let nTH →∞ until the changein the results < 1.0e−10. The parameters for the consumption process of the model and the preferenceparameters are µ = 0.02, δ = 0.99, σε = 0.03, γ = 10 and ψ = 1.5.

25

Figure 2: Absolute Relative Errors in the Pricing of the Mean and Standard Deviation of the LogWealth-Consumption Ratio as a Function of σε

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

2

4

6

8x 10

−5

σε

Err

or(

E(w

t−c

t))

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.5

1

1.5

2

2.5x 10

−5

σε

Err

or(

σ(w

t − c

t))

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.5

1

1.5

2

2.5

3x 10

−5

σε

Err

or(

E(w

t−c

t))

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.02

0.04

0.06

0.08

0.1

σε

Err

or(

σ(w

t − c

t))

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

1

2

3

4x 10

−11

σε

Err

or(

E(w

t−c

t))

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.2

0.4

0.6

0.8

1x 10

−7

σε

Err

or(

σ(w

t − c

t))

LogLin LogLin

TH

TH−F

T

TH

TH−F

T

Col

Gal

Col

Gal

The graph shows absolute relative errors in the unconditional mean (left panel) and standard deviation(right panel) of the wealth-consumption ratio for different values of σε. For the projection methods a9-degree approximation is used with nσ = 3. For the discretizations 9 nodes are used and mT = 2. Tocompute the true solution we use Tauchen and Hussey (1991)’s method and let nTH →∞ until the changein the results < 1.0e−10. The parameters for the consumption process of the model and the preferenceparameters are µ = 0.02, δ = 0.99, ρ = 0.91, γ = 10 and ψ = 1.5.

26

Figure 3: Log Wealth-Consumption and Price-Dividend Ratios as a Function of the States x andσ2 in the Model of Bansal, Kiku, and Yaron (2012)

The graph shows the wealth-consumption (left panel) and price-dividend (mid panel) ratio as a function ofthe state variables x and σ2 as well as the histogram of 107 simulations of the states x and σ2. The lightgray layer shows the result of the log-linearization and the dark grey the result of the collocation methodwith a 9 degree polynomial approximation in xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max = 0.00032.

Calibration BKY (2012).

27

Figure 4: First and Second Derivatives of the Log Wealth-Consumption and Price-Dividend Ra-tios as a Function of the States x and σ2 in the Model of Bansal, Kiku, and Yaron(2012)

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

12

12.5

13

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

50

60

70

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−3000

−2000

−1000

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−10000

−5000

0

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

5

10

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

500

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

1

2

x 106

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

1

2

x 107

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−2000

−1000

0

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−4

−2

0

x 104

xσ2

∂ (p t−d t)∂σ 2

∂ (p t−d t)∂x

∂ 2(p t−d t)∂x2

∂ 2(p t−d t)∂ (σ 2)2

∂ 2(p t−d t)∂x∂σ 2

∂ 2(wt−ct)∂x∂σ 2

∂ 2(wt−ct)∂ (σ 2)2

∂ (wt−ct)∂σ 2

∂ 2(wt−ct)∂x2

∂ (wt−ct)∂x

The graph shows the first and second derivatives of the wealth-consumption (left panel) and price-dividend(mid panel) ratio as a function of the state variables x and σ2. The light gray layer shows the result ofthe log-linearization and the dark grey the result of the collocation method with a 9 degree polynomialapproximation in xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max = 0.00032. Calibration BKY (2012).

28

Figure 5: Log Wealth-Consumption and Price-Dividend Ratios as a Function of the States x andσ2 in the Model of Bansal and Yaron (2004)

The graph shows the wealth-consumption (left panel) and price-dividend (mid panel) ratio as a function ofthe state variables x and σ2 as well as the histogram of 107 simulations of the states x and σ2. The lightgrey layer shows the result of the log-linearization and the dark grey the result of the collocation methodwith a 9 degree polynomial approximation in xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max = 0.00032.

Calibration BY (2004).

29

Figure 6: First and Second Derivatives of the Log Wealth-Consumption and Price-Dividend Ra-tios as a Function of the States x and σ2 in the Model of Bansal and Yaron (2004)

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

14.5

15

15.5

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

50

100

150

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−540

−520

−500

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−4000

−2000

0

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

5

10

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

1000

2000

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

2

4

x 104

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

0

2

4

x 106

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−400

−200

0

xσ2

−0.015 −0.01 −0.005 0 0.005 0.01

01

23

4

x 10−4

−10

−5

0

x 104

xσ2

∂ (wt−ct)∂x

∂ (wt−ct)∂σ 2

∂ 2(wt−ct)∂x2

∂ 2(wt−ct)∂ (σ 2)2

∂ 2(wt−ct)∂x∂σ 2

∂ (p t−d t)∂x

∂ (p t−d t)∂σ 2

∂ 2(p t−d t)∂x2

∂ 2(p t−d t)∂ (σ 2)2

∂ 2(p t−d t)∂x∂σ 2

The graph shows the first and second derivatives of the wealth-consumption (left panel) and price-dividend(mid panel) ratio as a function of the state variables x and σ2. The light gray layer shows the result ofthe log-linearization and the dark grey the result of the collocation method with a 9 degree polynomialapproximation in xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max = 0.00032. Calibration BY (2004).

30

Figure 7: Implications of the Errors of the Linearization

0.997 0.9975 0.998 0.9985 0.999 0.99950

0.02

0.04

0.06

0.08

0.1

ν

EP

0.96 0.965 0.97 0.975 0.980

0.02

0.04

0.06

0.08

ρ

EP

8 9 10 11 12 130

0.02

0.04

0.06

0.08

γ

EP

2 3 4 5 60

0.02

0.04

0.06

0.08

ψ

EP

Each panel 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) for the model of Bansal, Kiku, and Yaron (2012). Theblack dotted lines show the parameter of the calibration used by Bansal, Kiku, and Yaron (2012) (x axis)and the true implied equity premium (y axis). The gray dashed line shows the equity premium impliedby the linearization (y axis) and the parameter value that is needed to match this premium assuming thatall other parameters are kept constant (x axis). The equity premium implied by the linearization is 5.81%while the true premium is only 4.74% (see table 9).

31

Figure 8: Implications of the Errors of the Linearization

0.997 0.9975 0.998 0.9985 0.999 0.99950

0,1

0,2

0,3

0,4

0.5

ν

σ(p

t − d

t)

0.96 0.965 0.97 0.975 0.98 0.985 0.990

0.1

0.2

0.3

0.4

ρ

σ(p

t − d

t)

8 9 10 11 120

0.1

0.2

0.3

0.4

γ

σ(p

t − d

t)

1.5 2 2.5 30

0.1

0.2

0.3

0.4

ψ

σ(p

t − d

t)

Each panel 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) for the model of Bansal, Kiku, and Yaron (2012). Theblack dotted lines show the parameter of the calibration used by Bansal, Kiku, and Yaron (2012) (x axis)and the true implied equity premium (y axis). The gray dashed line shows the equity premium impliedby the linearization (y axis) and the parameter value that is needed to match this premium assuming thatall other parameters are kept constant (x axis). The equity premium implied by the linearization is 5.81%while the true premium is only 4.74% (see table 9).

32

List of Tables

1 Euler Approximation Errors for the Log-Linearization and Projection Methods inthe Model of Tallarini (2000) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Euler Approximation Errors for the Log-Linearization and Projection Methods inthe Model of Tallarini (2000) - Second Parameter Set . . . . . . . . . . . . . . . . 35

3 Absolute Relative Errors in the Unconditional Mean and Standard Deviation ofthe Log Wealth-Consumption Ratio in the Model of Tallarini (2000) . . . . . . . . 36

4 Absolute Relative Errors in the Unconditional Mean and Standard Deviation ofthe Log Wealth-Consumption Ratio in the Model of Tallarini (2000) - Second Pa-rameter Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Parameters for the Long Run Risk Model . . . . . . . . . . . . . . . . . . . . . . . 386 Euler Approximation Errors for the Log-Linearization and Projection Methods in

the Long Run Risk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Absolute Relative Errors in the Unconditional Mean and Standard Deviation of

the Log Wealth-Consumption and Log Price-Dividend Ratio in the Long Run RiskModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8 Absolute Relative Errors of the Annualized Moments in the Long Run Risk Model 419 Annualized Moments in the Long Run Risk Model . . . . . . . . . . . . . . . . . . 42

33

Table 1: Euler Approximation Errors for the Log-Linearization and Projection Methods in theModel of Tallarini (2000)

Log-Lin Collocation Galerkin

n = 3 n = 6 n = 9 n = 3 n = 6 n = 9

γ = 2, ψ = 0.5

nσ = 22.5e-4 1.5e-6 3.9e-11 3.9e-11 1.5e-6 5.4e-11 5.1e-111.1e-4 1.0e-6 1.4e-11 1.4e-11 1.0e-6 1.9e-11 1.8e-11

nσ = 100.0076 1.9e-4 1.7e-7 1.7e-7 1.9e-4 1.1e-7 1.0e-070.0028 1.2e-4 7.5e-8 7.5e-8 1.2e-4 5.3e-8 4.8e-8

γ = 10, ψ = 0.1

nσ = 20.0340 6.3e-4 1.5e-7 1.4e-7 6.3e-4 1.5e-7 2.4e-70.0125 3.2e-4 6.8e-8 6.8e-8 3.2e-4 6.8e-8 6.1e-8

nσ = 10289 0.1266 0.0018 2.4e-4 0.1266 0.0018 2.4e-4

12.52 0.0451 0.0011 1.2e-4 0.0451 0.0011 1.2e-4

γ = 10, ψ = 1.5


nσ = 100.0035 7.3e-5 5.6e-10 2.1e-12 7.2e-5 1.8e-6 2.2e-60.0015 4.8e-5 3.9e-10 2.7e-13 4.8e-5 3.3e-7 3.1e-7

The table shows Euler approximation errors for the pricing of the wealth return in the model of Tal-larini (2000) for the log-linearization and the projections methods for different degrees of polynomialapproximation n. Errors are reported for different sets of preference parameters γ and ψ. The firstrow of each pair of parameters shows the maximum absolute error and the second row the root meansquared error of me = 100nσ uniformly distributed evaluation nodes, nσ standard deviations aroundthe steady state of the model. The parameters for the consumption process of the model are given byσε = 0.0343, µ = 0.02, δ = 0.99, ρ = 0.91.

34

Table 2: Euler Approximation Errors for the Log-Linearization and Projection Methods in theModel of Tallarini (2000) - Second Parameter Set


n = 3 n = 6 n = 9 n = 3 n = 6 n = 9

γ = 2, ψ = 1.5

nσ = 21.5e-5 6.6e-8 5.2e-9 5.1e-9 6.6-8 5.1e-9 5.1e-96.5e-6 4.3e-8 4.9e-9 4.9e-9 4.2e-8 4.9e-9 4.9e-9


γ = 10, ψ = 0.5

nσ = 20.0022 1.3e-5 5.9e-9 6.5e-9 1.4e-5 1.1e-4 6.5e-99.7e-4 9.2e-6 3.0e-9 3.0e-9 9.2e-6 4.3e-6 3.2e-9

nσ = 100.0702 0.0017 4.7e-7 4.5e-7 1.3e-10 4.7e-7 1.4e-100.0257 0.0011 3.3e-7 3.3e-7 0.0011 3.3e-7 8.5e-11

γ = 7.5, ψ = 1.5


nσ = 100.0026 5.2e-5 3.2e-9 4.5e-9 5.3e-5 3.2e-9 4.5e-90.0011 3.4e-5 7.9e-10 6.3e-10 3.4-5 7.9e-10 6.3e-10

The table shows Euler approximation errors for the pricing of the wealth return in the model of Tal-larini (2000) for the log-linearization and the projections methods for different degrees of polynomialapproximation n. Errors are reported for different sets of preference parameters γ and ψ. The firstrow of each pair of parameters shows the maximum absolute error and the second row the root meansquared error of me = 100nσ uniformly distributed evaluation nodes, nσ standard deviations aroundthe steady state of the model. The parameters for the consumption process of the model are given byσε = 0.0343, µ = 0.02, δ = 0.99, ρ = 0.91.

35

Table 3: Absolute Relative Errors in the Unconditional Mean and Standard Deviation of the LogWealth-Consumption Ratio in the Model of Tallarini (2000)

γ = 2, ψ = 0.5

Collocation Galerkin Log-Lin3 6 9 3 6 9

E (wt − ct) 5.3e-7 6.9e-10 6.9e-10 5.3e-07 4.0e-10 3.5e-10 0.0020σ (wt − ct) 3.7e-4 4.0e-10 4.0e-10 3.7e-4 3.3e-10 2.8e-10 0.0028

Time 0.0217 0.0334 0.0425 0.0212 0.0375 0.0549 0.0149

TH TH-F Tauchen3 6 9 3 6 9 3 6 9

E (wt − ct) 6.0e-4 3.1e-4 1.5e-4 3.9e-4 7.5e-5 1.3e-5 1.5e-4 7.4e-5 4.8e-5σ (wt − ct) 0.3390 0.1326 0.0561 0.2649 0.0396 0.0059 0.2548 0.0703 4.5e-4

Time 0.0248 0.0303 0.0354 0.0277 0.0356 0.0413 0.0279 0.0337 0.0395

γ = 10, ψ = 0.1


E (wt − ct) 7.2e-4 4.5e-7 4.5e-9 7.2e-4 4.5e-7 4.5e-9 0.0361σ (wt − ct) 0.0106 2.6e-6 1.5e-7 0.0106 2.6e-6 1.5e-7 0.0775

Time 0.0242 0.0321 0.0424 0.0236 0.0324 0.0525 0.0120

TH TH-F Tauchen3 6 9 3 6 9 3 6 9

E (wt − ct) 0.0419 0.0185 0.0083 0.0361 0.0064 0.0010 0.0248 0.0115 1.3e-4σ (wt − ct) 0.0979 0.0011 0.0072 0.3085 0.0308 0.0016 0.4572 0.1615 0.0695

Time 0.0270 0.0297 0.0377 0.0252 0.0293 0.0376 0.0224 0.0293 0.0364

γ = 10, ψ = 1.5


E (wt − ct) 8.7e-8 2.4e-11 2.4e-11 8.7e-8 1.2e-11 1.2e-11 3.5e-4σ (wt − ct) 1.5e-5 1.0e-11 1.0e-11 1.5e-5 1.0e-11 1.0e-11 4.0e-4

Time 0.0224 0.033 0.0379 0.0217 0.036 0.0494 0.0141

TH TH-F Tauchen3 6 9 3 6 9 3 6 9

E (wt − ct) 6.3e-5 4.5e-5 2.5e-5 1.4e-5 6.2e-6 1.7e-6 4.7e-4 1.4e-5 1.2e-5σ (wt − ct) 0.4234 0.1901 0.0862 0.2403 0.0423 0.0074 0.1477 0.0380 0.0284

Time 0.0277 0.0294 0.0406 0.0237 0.0289 0.0350 0.0226 0.0292 0.0354

The table shows absolute relative errors in the unconditional mean and standard deviation of the log wealth-consumption ratio as well as computation times for different sets of preference parameters. For the projec-tion methods we set nσ = 3 and for Tauchen (1986)’s method mT = 2. To compute the true solution we useTauchen and Hussey (1991)’s method and let nTH → ∞ until the change in the results falls below 10−10.The parameters for the consumption process of the model are σε = 0.0343, µ = 0.02, ρ = 0.91. (δ = 0.99)

36

Table 4: Absolute Relative Errors in the Unconditional Mean and Standard Deviation of the LogWealth-Consumption Ratio in the Model of Tallarini (2000) - Second Parameter Set

γ = 2, ψ = 1.5


E (wt − ct) 8.5e-8 8.5e-8 8.5e-8 8.5e-8 1.0e-7 1.0e-7 3.5e-4σ (wt − ct) 1.5-5 1.5-5 1.5e-5 1.5e-5 1.0e-5 9.9e-6 3.8e-4

Time 0.0225 0.0304 0.0393 0.0222 0.0340 0.0525 0.0147

TH TH-F Tauchen3 6 9 3 6 9 3 6 9

E (wt − ct) 3.0e-5 1.4-5 6.4e-6 2.2e-5 3.9e-6 7.1e-7 4.4e-5 5.6e-6 1.1e-6σ (wt − ct) 0.4235 0.1902 0.0862 0.2403 0.0423 0.0074 0.1475 0.0382 0.0284

Time 0.0246 0.0295 0.0380 0.0203 0.0255 0.0325 0.0222 0.0274 0.0361

γ = 10, ψ = 0.5


E (wt − ct) 5.9e-6 4.8e-9 4.8e-9 6.3e-6 2.8e-9 2.6e-9 0.0021σ (wt − ct) 3.5e-4 8.8e-9 8.8e-9 3.5e-4 1.6e-9 9.0e-10 9.0e-4

Time 0.0234 0.0335 0.0419 0.0236 0.0388 0.0595 0.0150

TH TH-F Tauchen3 6 9 3 6 9 3 6 9

E (wt − ct) 0.0029 0.0017 8.6e-4 0.0015 3.5e-4 7.7e-5 0.0033 2.3e-5 4.7e-4σ (wt − ct) 0.3392 0.1320 0.0550 0.2625 0.0384 0.0054 0.2203 0.0722 0.0018

Time 0.0261 0.0311 0.0395 0.0216 0.0259 0.0326 0.0244 0.0285 0.0360

γ = 7.5, ψ = 1.5


E (wt − ct) 1.7e-6 9.5e-12 9.5e-12 6.0e-8 9.4e-12 8.6e-12 3.5e-4σ (wt − ct) 1.5e-5 5.0e-12 5.0e-12 1.5e-5 5.0e-12 1.0e-11 3.9e-4

Time 0.0229 0.0336 0.0410 0.0225 0.0386 0.0567 0.0135

TH TH-F Tauchen3 6 9 3 6 9 3 6 9

E (wt − ct) 3.4e-5 2.6e-5 1.4e-5 2.3e-6 2.6e-6 5.8e-7 3.3e-4 1.1e-5 1.3e-5σ (wt − ct) 0.4235 0.1902 0.0861 0.2404 0.0424 0.0073 0.1475 0.0381 0.0285

Time 0.0248 0.0295 0.0347 0.0208 0.0260 0.0305 0.0236 0.0291 0.0356

The table shows absolute relative errors in the unconditional mean and standard deviation of the wealth-consumption ratio as well as computation times for different sets of preference parameters. For the projec-tion methods we set nσ = 3 and for Tauchen (1986)’s method mT = 2. To compute the true solution we useTauchen and Hussey (1991)’s method and let nTH → ∞ until the change in the results falls below 10−10.The parameters for the consumption process of the model are σε = 0.0343, µ = 0.02, ρ = 0.91. (δ = 0.99)

37

Table 5: Parameters for the Long Run Risk Model

µc ρ φe σ ν σω µd Φ φ π γ ψ δ

BKY (2012) 1.5e-3 0.975 0.038 7.2e-3 0.999 2.8e-6 1.5e-3 2.5 5.96 2.6 10 1.5 0.9989

BY (2004) 1.5e-3 0.979 0.044 7.8e-3 0.987 2.3e-6 1.5e-3 3.0 4.5 0 10 1.5 0.998

38

Table 6: Euler Approximation Errors for the Log-Linearization and Projection Methods in theLong Run Risk Model


Calibration BKY (2012)

n = 3 n = 6 n = 9 n = 3 n = 6 n = 9

Wealth-Euler0.0112 2.4e-5 3.8e-7 3.8e-7 1.2e-4 2.5e-7 2.5e-70.0029 9.2e-6 9.7e-8 9.7e-8 2.9e-5 5.2e-8 5.3e-8

Market-Euler0.0091 1.6e-4 2.0e-7 2.0e-7 2.7e-4 1.5e-6 9.8e-90.0014 5.3e-5 7.1e-8 7.1e-8 7.7e-5 3.2e-7 1.6e-9

RiskFree-Euler0.0117 2.5e-5 3.1e-9 3.1e-9 1.3e-4 1.6e-8 1.6e-90.0030 9.5e-6 1.8e-9 1.8e-9 3.0e-5 3.9e-9 3.9e-9

Calibration BY (2004)

n = 3 n = 6 n = 9 n = 3 n = 6 n = 9

Wealth-Euler0.0018 1.2e-5 1.3e-8 1.3e-8 4.7e-5 3.3e-8 3.2e-84.3e-4 6.3e-6 1.2e-9 1.2e-9 1.0e-5 2.7e-9 2.7e-9

Market-Euler0.0119 2.3e-4 3.8e-7 1.0e-9 5.0e-4 3.0e-6 2.6e-80.0023 1.3e-4 2.1e-7 2.1e-7 1.5e-4 5.2e-7 2.5e-9

RiskFree-Euler0.0019 1.2e-5 2.2e-11 3.9e-12 4.9e-5 1.2e-9 2.9e-124.5e-4 6.6e-6 8.9e-12 8.9e-12 1.1e-5 1.1e-10 1.1e-12

The table shows Euler approximation errors for the pricing of the wealth return, the market return andthe risk free rate for the log-linearization and the projections methods for different degrees of polynomialapproximation n. Errors are reported for the parameter specification as in Bansal and Yaron (2004) andBansal, Kiku, and Yaron (2012). The first row of each pair of parameters shows the maximum absoluteerror and the second row the root mean squared error of 200 uniformly distributed evaluation nodes in eachdimension with xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max = 0.00032.

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Table 7: Absolute Relative Errors in the Unconditional Mean and Standard Deviation of the LogWealth-Consumption and Log Price-Dividend Ratio in the Long Run Risk Model

Calibration BKY (2012)


E (wt − ct) 2.7e-5 2.6e-7 2.6e-7 1.6e-5 4.2e-7 4.2e-7 0.0105σ (wt − ct) 9.2e-6 3.6e-6 3.6e-6 9.9e-4 6.9e-6 6.9e-6 0.1225E (pt − dt) 0.0015 3.8e-7 3.8e-7 3.6e-4 2.6e-7 8.3e-8 0.0315σ (pt − dt) 0.0059 1.9e-6 1.9e-6 0.0292 2.1e-6 1.0e-6 0.2690

Time 1.01 4.76 12.14 0.59 3.18 9.99 0.023

TH TH-F Tauchen10 30 50 10 30 50 10 30 50

E (wt − ct) 0.0794 0.0608 0.0533 0.0556 0.0432 0.0203 0.0378 0.0121 0.0056σ (wt − ct) 0.9521 0.8627 0.8240 > 1 > 1 0.7682 > 1 0.7672 0.0999E (pt − dt) 0.2533 0.1223 0.0877 0.1171 0.0816 0.0334 0.0773 0.0249 0.0060σ (pt − dt) 0.888 0.6981 0.6267 > 1 > 1 0.6992 > 1 0.6123 0.1312

Time 2.35 228.8 2944 1.67 354.7 4664 2.16 293.5 3697

Calibration BY (2004)


E (wt − ct) 1.1e-5 3.1e-11 3.1e-11 2.1e-5 3.9e-9 3.9e-9 1.7e-4σ (wt − ct) 1.9e-4 5.9e-10 5.9e-10 2.9e-4 1.0e-8 1.0e-8 0.0013E (pt − dt) 0.0028 7.0e-6 7.0e-6 0.0017 2.2e-5 2.9e-8 0.0054σ (pt − dt) 0.0145 2.3e-5 2.3e-5 0.0207 2.8e-5 4.9e-7 0.0152

Time 0.67 4.27 12.18 0.53 2.52 9.33 0.023

TH TH-F Tauchen10 30 50 10 30 50 10 30 50

E (wt − ct) 0.0538 0.0238 0.0123 0.0015 2.8e-4 9.8e-5 0.0318 0.0019 0.0038σ (wt − ct) 0.7739 0.3227 0.1634 0.0758 0.0123 0.0105 0.5886 0.0386 0.0560E (pt − dt) 0.6509 0.1085 0.0518 0.0164 3.9e-4 5.6e-4 0.0635 0.0104 0.0169σ (pt − dt) 0.7427 0.2968 0.15472 0.0074 0.0063 0.0033 0.2479 0.0512 0.0611

Time 2.87 215.8 2867 1.44 285.0 3792 1.87 234.9 2893

The table shows absolute relative errors in the unconditional mean and standard deviation of the monthly logwealth-consumption and log price-dividend ratio as well as compuation times. For the projection methodsthe approximation interval is given by xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max = 0.00032 and

for Tauchen (1986)’s method mT = 2. All results are compared to the 18 degree polynomial approximationusing the collocation method.

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Table 8: Absolute Relative Errors of the Annualized Moments in the Long Run Risk Model

n E (pt − dt) σ (pt − dt) E (rm,t) σ (rm,t) E (rrf,t) σ (rrf,t) EP

Collocation

3 0.0026 0.0068 0.0056 4.6e-4 2.3e-4 8.2e-5 0.00696 1.8e-5 1.3e-5 4.2e-5 8.3e-7 2.2e-4 4.3e-5 1.0e-49 1.8e-5 1.3e-5 4.2e-5 8.3e-7 2.2e-4 4.3e-5 1.0e-4

Galerkin

3 2.5e-4 0.0166 0.0012 0.0017 8.1e-4 2.7e-4 0.00166 1.9e-5 6.4e-6 4.3e-5 5.5e-7 2.4e-4 3.7e-5 1.1e-49 1.8e-5 1.0e-6 4.2e-5 4.6e-7 2.4e-4 3.6e-5 1.0e-4

Log-Linearization

0.0607 0.2250 0.1636 0.0129 0.1050 0.0148 0.2252

Tauchen-Hussey

10 0.4729 0.6118 0.5460 0.2206 0.8253 0.5614 0.860730 0.2390 0.5068 0.3787 0.1863 0.7167 0.3179 0.630150 0.1771 0.4584 0.3079 0.1725 0.6690 0.2573 0.5321

Tauchen-Hussey-Floden

10 0.2116 0.4959 0.3468 0.2455 0.7809 0.3564 0.605730 0.1757 1.3343 0.2538 0.1509 0.5624 0.0985 0.441150 0.0789 0.5901 0.1315 0.1062 0.3480 0.0445 0.2416

Tauchen

10 0.1663 0.4901 0.2924 0.3328 0.8518 0.3183 0.555030 0.0645 0.5495 0.1072 0.0768 0.2318 0.0214 0.185050 0.0302 0.1410 0.0563 0.0773 0.2133 0.0013 0.1183

The table shows absolute relative errors in the mean and standard deviation of the annualized price-dividendratio, the annualized market and risk free return as well as the equity premium for the different solutionmethods and the parameter set as in Bansal, Kiku, and Yaron (2012). To compute the moments, 1.000.000years of each 12 monthly observations are simulated and aggregated to annual frequencies. For the projectionmethods the approximation interval is given by xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max =

0.00032 and for Tauchen (1986)’s method mT = 2. All results are compared to the 12 degree polynomialapproximation using the collocation method.

41

Table 9: Annualized Moments in the Long Run Risk Model

n E (pt − dt) σ (pt − dt) E (rt) σ (rt) E(rft

)σ(rft

)EP

Collocation

3 3.24 0.24 5.78 21.20 1.09 1.27 4.716 3.24 0.24 5.83 21.21 1.09 1.27 4.749 3.24 0.24 5.83 21.21 1.09 1.27 4.74

Galerkin

3 3.23 0.24 5.82 21.18 1.09 1.27 4.736 3.24 0.24 5.83 21.21 1.09 1.27 4.749 3.24 0.24 5.83 21.21 1.09 1.27 4.74

Log-Linearization

3.04 0.30 6.78 21.49 0.97 1.29 5.81


10 4.77 0.09 2.65 16.53 1.98 0.56 0.6630 4.01 0.12 3.62 17.25 1.87 0.87 1.7550 3.80 0.13 4.03 17.55 1.81 0.94 2.21


10 3.92 0.12 3.81 16.00 1.93 0.82 1.8730 3.80 0.56 4.34 18.01 1.69 1.14 2.6450 3.49 0.38 5.06 18.95 1.46 1.21 3.59

Tauchen

10 3.77 0.12 4.12 14.15 2.01 0.86 2.1130 3.44 0.37 5.21 19.58 1.34 1.30 3.8650 3.33 0.27 5.50 19.57 1.32 1.27 4.18

The table shows the mean and standard deviation of the annualized log price-dividend ratio, the annualizedmarket and risk free return as well as the equity premium for the different solution methods and theparameter set as in Bansal, Kiku, and Yaron (2012). All returns are shown in percent, so a value of 1.5 isa 1.5% annualized return.To compute the moments, 1.000.000 years of each 12 monthly observations aresimulated and aggregated to annual frequencies. For the projection methods the approximation interval isgiven by xmin = −0.013, xmax = 0.013, σ2

min = 0 and σ2max = 0.00032 and for Tauchen (1986)’s method

mT = 2. All results are compared to the 12 degree polynomial approximation using the collocation method.

42

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