2.1 Investor’s Problem

To illustrate the investor’s problem, let the reward of the investor be the constant relative risk averse utility used by Mehra and Prescott (1985, 2003).

(6)

where the consumption is a control variable for the investor. This reward is not a function of the state variable. Assume that the investor can purchase a risk-free bond with constant logarithmic return or a stock which pays a logarithmic return . The return on a portfolio of bonds and stocks, , is given by

(7)

where the control variable, , is the percentage of investor’s wealth invested in stocks, , and . The equation of motion for investor’s wealth is

(8)

where is the current wealth of the investor.

Since Kandel and Stambaugh (1996), the expectation of the logarithm of the return on stocks, , has been represented by a first order autoregressive process with long run value .

(9)

which is independent of and . is a random shock to the expected return on stocks. Consequently, the return on stocks is given by

(10)

Here, it is generally assumed that the correlation between and is negative based on the evaluation of the data of the return on stocks. In this example, the current wealth and the expected return on stocks are two state variables.

In this case, the first order conditions are

(11)

(12)

and

(13)

There is also a transversality or terminal condition at time

(14)

In the terminal period, the investor expects to consume all her wealth and any additional wealth in the last period is expected to provide no additional utility.

Combining (11) and (13) yields

This means that the marginal value of wealth to the investor is equal to the marginal utility of consumption. Substituting this into (13) results in the following equation which determines the consumption by the investor.

(15)

The investor compares her loss of utility from consumption today, , with her gain in utility from payment on the optimal portfolio at a future time, , since she saves part of her initial wealth. Thus, this is just a more sophisticated version of the comparison of present value with future value in which the stochastic discount factor, , is

Consequently, (15) may be written as

(16)

In addition, the choice between stocks and bonds in (12) is such that

(17)

This condition says that the investor chooses the allocation between stocks and bonds so that the expected marginal values of the payoffs from stocks and bonds are identical.

The functional Eqs. (15) and (17) determine the optimal consumption of the investor and the percentage of her wealth allocated to stocks. We seek the functions and in which is the current wealth of the investor and is the expected return on stocks. Once these functions are found, (8) tells us how the wealth of the investor evolves. The equation of motion for the investor’s wealth under this optimal behavior is

(18)

where refers to the value of wealth at the end of the next period. Here we have used the equations of motion for the return on stocks (10) and its expected value (9). This stochastic difference equation also needs to satisfy the terminal condition (14).

By assuming a particular cumulative probability distribution for , say , we end up with the following integral equation which satisfies the investor’s optimal decisions (15) and (17).

(19)

As a result, the stochastic discount factor is

The choice between stocks and bonds is such that

(20)

We are looking for the investor’s consumption and allocation to stocks, which satisfy Eqs. (19) and (20). These functions are independent of time so that the path of consumption and allocation to stocks evolves over time as different random shocks to the return on stocks and its expected value impact the stochastic difference equations for the state variables, wealth (18), and the expected return on stocks (9).

Except for certain simple examples, there are no known explicit solutions to these nonlinear integral equations. Altug and Labadie (1994, 2008) provide an analysis of the necessary conditions for the existence of the solution for consumption and allocation to stocks when the support of the probability density function is compact and a general utility function is specified. Researchers appeal to this analysis to suppose that a solution does exist, and immediately proceed to some approximation methods.

2.2 A Survey of Approximation Methods

Campbell (1993), Campbell and Viceira (1999, 2001), and Campbell et al. (2003) use a first order Taylor polynomial approximation of the logarithm of the wealth Eq. (8). In addition, they assume that the return on stocks and consumption growth have a joint log-normal distribution. This transforms the integral equation so that Campbell (1993) can use Euler conditions (15) and (17) to infer the properties of consumption growth. Subsequently, Campbell and Viceira (1999) guess and verify the solution to the approximated problem is log-linear in the state variable for the allocation to stocks and quadratic in the state variable for consumption growth. Campbell and Viceira (2002) use the same method to analyze the investment in long-term bonds. Campbell et al. (2003) extend this work to the case of more than two state variables.

Kandel and Stambaugh (1996) examine a more limited problem in which the investor cares only about the terminal wealth. They use Bayesian priors to specify the probability distribution, , for the return on stocks. Given the probability distribution, a search is constructed for the optima using a parabolic interpolation. The exact form of this numerical procedure is not specified. The reader is referred to Brent (1973) which is a standard reference on the use of polynomial methods to approximate integrals. Brandt (1999) estimates the first order conditions using the general method of moments.³ For a fixed number of states, he finds the optimal consumption and allocation to stocks in which the Euler conditions (19) and (20) are satisfied. The procedure for finding these fixed states is not specified. Balduzzi and Lynch (1999) and Lynch and Balduzzi (2000) use backward recursion on the value function to find the optimal investment in which the investor cares about only the terminal wealth. Brennan et al. (1997) use finite difference methods to approximate the PDE in the same case. These decisions are being made in the face of transaction cost and predictable return on stocks. The optimal investment choices are the solution to the Bellman equation at discrete points in the interval so that short selling and borrowing are not allowed. They increase the number of discrete points until there is virtually no change in the approximate solution. Barberis (2000) also looks at an investor interested in the terminal wealth. Given a posterior distribution for the return on stocks, he finds the percentage of wealth allocated to stocks among a finite number of points in the interval , which maximizes the expected utility of wealth.

A similar problem is analyzed in the life-cycle literature. In Carroll (1997) and Gourinchas and Parker (2002), the return on investment is constant and the focus is on permanent labor income so that the second state variable is expected labor income.⁴ Both papers use backward induction to calculate the optimal conditions. Carroll (1997) uses a discrete state space and a -point cumulative probability distribution function for . Gourinchas and Parker (2002) use Gauss-Hermite quadrature rules from Judd (1998) to form the expectations in (19). They then use a discrete state space to iterate backwards to obtain the optimal consumption rule. Both of these papers provide appendices to explain the algorithms used to find the solution.

Brandt et al. (2005) provide a more systematic algorithm for solving the optimal consumption and investment problem. First, they use a fourth order Taylor polynomial approximation of the investor’s utility function in the neighborhood of the wealth associated with investing in the risk-free bond and consumption in a deterministic problem.⁵ Under these circumstances optimal consumption and investment are approximated using the first four moments from the integrals in (19) and (20). The conditional probability distribution for the return on investment is then used to create sample paths for the return and the investor’s information set. Next, they use the sample paths to solve backwards from a terminal date for the optimal consumption and investment decisions of the investor. Finally, they use the parameterized expectations following Den Haan and Marcet (1990) to calculate all the moments in which they use monomials as basis to represent the information set of the investor. To check the accuracy of their method, they rely on the special cases in which the solutions are known. In these circumstances, they can check to see how close their approximate solutions are to the true solutions.

Chacko et al. (2005) use the second order Taylor polynomial approximation to solve the consumption and portfolio problem with CRRA utility and time-varying standard deviation of the return on stocks. They only use the second order Taylor polynomial approximation, since the error of their approximate solution is small for the case in which they know the true solution.

2.3 Polynomial Methods

The projection method of Judd (1992, 1996, 1998) can generalize the approximation method of Chacko et al. (2005) by using orthogonal polynomials. To apply this procedure, we need to assume a probability distribution on the shocks. Suppose that and are two log-normally distributed random variables, which are consistent with Campbell and Viceira (1999, 2001).⁶ We then replace the integration with double summation using the Gauss-Hermite quadrature procedure (see Judd, 1998, pp. 261–271). In addition, replace the unknown policy functions for consumption and investment with a linear combination of orthogonal polynomials such as the Chebyshev functions (see Judd, 1998, p. 206):

(21)

and

(22)

These two functions are only defined in the region . However, the probability distribution which determines and is defined over the entire real line. To account for this possibility we can follow Chen et al. (2008b) by extending and to .

By substituting these hypothesized functions (21) and (22) into the Euler conditions (19) and (20), we end up with two residual equations. These equations are represented by

(23)

These equations are evaluated at the nodes of Chebyshev polynomials. Newton’s method can then be used to find the vector of coefficients following

(24)

Each of the four partial derivatives in (24) are matrices, where is the order of the Chebyshev polynomials in (21) and (22). Fortunately, each of these partial derivatives has the same functional form, since they are made up of Chebyshev polynomials and their derivatives. Also, in the implementation of the projection method, the iteration in (24) can be more precise and quicker, since the actual derivatives can be found by using the derivatives of Chebyshev polynomials in (21) and (22).

It is frequently argued that the polynomial method cannot be used in circumstances with occasionally binding constraints such as the cost of security transactions. The Chebyshev parameterized expectation algorithm of Christiano and Fisher (2000) can be used to overcome this problem. Consequently, the above algorithm for solving the investor’s problem can be modified to handle transaction costs. However, the degree of accuracy of such an approximation is still an unresolved problem.

3.1 Analytic Method

Calin et al. (2005) introduce the analytic method to solve this problem. The benefit of this method is that it identifies when the solution may be approximated by a polynomial within a certain interval. In addition, the approximation errors can be estimated. In Chen et al. (2008a,b), they apply this method to the more general utility functions developed by Abel (1990) and Campbell and Cochrane (1999), respectively. The analytic method to solve these asset pricing problems may be illustrated with the Mehra and Prescott model.

Step 1: Simplify the integral equation and identify the solution space.

Use the change of variable, , to rewrite (30).

(31)

The advantage of this change of variable is that the properties of the probability density function are known. For example, Calin et al. (2005) use the Gaussian distribution with mean and standard deviation . The integral equation is

(32)

where , , and . The support of the Gaussian distributed probability density function is not compact, so the usual contraction mapping theorem (see Altug and Labadie, 2008) is not applicable. Yet, once the integral function is written in this form, the price-dividend function cannot grow faster than . As a result, the solution lies in the following vector space.

Step 2: Verify the existence and uniqueness of the solution to the integral equation (32) in .

Calin et al. construct a linear transformation given by

(33)

where and . The mapping (33) has a unique fixed point in the vector space , which turns out to be the solution to the integral equation (32), as long as a new combination of parameters satisfies

where . This condition is satisfied for the Mehra and Prescott parameters.

Step 3: Verify that the solution to the integral equation (32) is analytic.

Whether or not a particular asset pricing problem has an analytic solution relies on the properties of the pricing kernel, the equation of motion for the state variable, and the probability density function. Each of these contributes to the integrand in the integral equation (32) so that the analyticity may be established in some interval. First, a summary of analytic functions is provided.

Analytic functions: To understand analytic functions, a brief introduction to holomorphic functions in one complex variable follows. These properties are also valid in several complex variables with some additional technical complications. Let be a complex-valued function of defined in an open set (domain) of the complex plane . Suppose that this function is smooth and it can be expressed as

(34)

where denotes the order complex derivative of at (the usual calculus formulas for real differentiation hold true for complex differentiation, too). Separating this function into its real and imaginary parts , we obtain

and

Multiplying the second equation by and adding it to the first yields⁸

(35)

This equation is called the Cauchy-Riemann equation, which can also be written in the form:

This motivates the following definition.

In fact, it can be shown that is holomorphic in if in the sense of distribution theory (weak derivatives). A proof of this fact can be found in Hörmander (1983, p. 110, Theorem 4.4.1). The basic theory of holomorphic functions can be found in the classical text of Ahlfors (1979) or in Hörmander (1979), where it is presented in the more general context. There, it is proved that a holomorphic function has derivatives of any order, and therefore its Taylor series can be formed. Moreover, for each there is a (maximal) number , called the radius of convergence for at , such that can be represented by its Taylor series, like (34), for .

The “calculus” definition of an analytic function of a real variable in an open interval is that it is in and at each point the remainder of the degree Taylor polynomial approximation of tends to zero as when . By complexifying to , we obtain a holomorphic function in an open subset of , and we may think of as the restriction of from to .

Using complex variables allows us to determine the precise radius of convergence, .

Now we are ready to show that the solution to the integral equation (32) is analytic in . Suppose that the function in the vector space is the solution to this integral equation. It suffices to show that

(36)

is analytic at each . In this step, we complexify so that

(37)

is analytic if

(38)

Since is bounded on any compact set in the complex plane, there exists depending on such that

(39)

where and . By the Dominated Convergence Theorem (see Folland, 1984, pp. 53–54), has continuous partial derivatives with respect to and , and the partial derivatives can pass through the integral sign in (37). Consequently, we get for , since

(40)

Thus, the price-dividend function is analytic over .

Step 4: Find the numerical solution to the integral equation (32).

Having established the mathematical properties of the solution to the integral Eq. (32), one can proceed to the numerical approximation of the solution. It is easier to work with a transformed price-dividend function

(41)

since it reduces the number of summations necessary in the numerical algorithm. As a result, the integral equation (32) is changed to

(42)

The solution of the integral equation (42) can be approximated by the order Taylor polynomial around the point

(43)

Alternatively, one could use orthogonal polynomials such as (21) to construct the approximation, which have superior numerical properties. As in the projection method, the polynomial approximation (43) is substituted into the integral equation (42) to yield a system of equations in the undetermined coefficients , which is given in Calin et al. In this case, the system is linear, since the integral equation (42) is linear in the unknown function.

Step 5: Estimate the approximation error.

An important benefit of the analytic method is that one can estimate the difference between the polynomial approximation and the true solution. In particular, Cauchy’s integral formula can be used to bound all the derivatives of the transformed price-dividend function which is used to estimate the approximation error. Let be the circle of radius centered at in the complex plane. Figure 1 provides an example when and . Cauchy’s integral formula (see Corollary 5.9 in Conway, 1973) gives

(44)

is the contour integral on the path . Each point on satisfies and . The proof of the existence and uniqueness of provides an upper bound on where so that

(45)

Let be the order Taylor polynomial of about . Then the Taylor remainder, given by , satisfies

(46)

for , where . Thus, the approximation error is estimated by

(47)

Up to now the error in the second summand in (47) has not been calculated. This error occurs because the coefficients in could be different from the true values of . In practice, these differences are small when is small.

Having estimated the approximation error, we may examine other numerical schemes to learn the tradeoffs of these schemes. For example, Calin et al. solve the Mehra and Prescott model using both the analytic method and Tauchen and Hussey’s (1991) procedure. In the analytic method is increased to at least in (43) for the approximation error to be less than . Consequently, even for the basic Mehra and Prescott model, a lower degree ( or ) polynomial is not sufficient to get an accurate representation of the true solution. They also find that using for the interval of integration in Tauchen and Hussey’s procedure is not sufficient to obtain an accurate solution. In this case, the price-dividend ratio is about below the true price-dividend ratio, and the interval of integration must increase to to remove this error. Thus, Tauchen and Hussey’s procedure should use a larger support than what one would expect from intuition.

3.2 More Complicated Models

Chen et al. (2008a,b) have used the analytic method to study the solution to Abel’s (1990) and Campbell and Cochrane’s (1999) models, respectively. In Abel’s model, (6) is replaced with

(48)

where is the vector of state variables.⁹ Everything else stays the same as in the Mehra and Prescott (1985) model. represents what you consumed in the past period, while is the consumption of your neighbor. The equations of motion for these two state variables are trivial, i.e., and .¹⁰ If , then this reduces to the Mehra and Prescott model.

In this case, the integral equation (30) becomes

(49)

where , , , and . The stochastic discount factor is more complicated and is given by

While the integral equation (49) is more complicated, a new transformed price-dividend function is introduced.

(50)

so that this integral equation becomes

(51)

where

(52)

The transformed integral equation (49) is similar to (32) so that the analytic method applies to this model as well. Step 2 leads to an explicit formula for the transformed price-dividend function. In addition, Chen et al. (2008a) show that this explicit solution is an analytic function so that the analytic method is not necessary.

The main lesson from the analysis of the price-dividend function in the Abel model is the importance of the specification of the probability distribution for the future dividend growth.¹¹ Following the Mehra and Prescott model, the probability distribution in (49) is assumed to be a Gaussian distribution; however, the price-dividend function is only defined on the interval . This restriction occurs since the marginal utility of consumption is non-positive outside this interval. Consequently, one has to be careful to specify a probability distribution such that the economic properties of the model are sensible. Chen et al. (2008a) show that by restricting to such that , an error in the numerical approximation is introduced. This error is bounded by cents out of a million dollars worth of stock for dividend growth within , when the parameter values are chosen to match the equity premium.

Chen et al. (2008b) solve the model by Campbell and Cochrane (1999). Here the reward function (6) becomes

(53)

where is the individual’s consumption at time is the surplus consumption ratio at time is the average consumption of all individuals at time , and is their habitual level of consumption at time .¹²

The control variable is now and the state variable is . To maintain positive values for these variables, Campbell and Cochrane define and . The equation of motion for the state variable is given as follows: The surplus consumption ratio satisfies the AR(1) process with ,

(54)

Consumption growth follows a normal distribution such that

(55)

To introduce heteroscedasticity of the random shock to the consumption growth, Campbell and Cochrane introduce the sensitivity function:

(56)

Here,¹³

(57)

Campbell and Cochrane assume that the dividend growth follows

As a result, the dividend growth (9) is the same as consumption growth plus a random shock which is correlated with the random shock to consumption growth. The random shock to the dividend growth follows a log-normal distribution.

The integral equation for the price-dividend function is

(58)

where

(59)

Now, the state variable is and the constants are

(60)

In this case the stochastic discount factor is

Thus, each asset pricing model makes the stochastic discount factor more complex, so that it better represents the empirical properties of the price-dividend and asset returns.

The integral equation (58) is similar to that found in the Mehra and Prescott model and the Abel model. The main difference is that the sensitivity function is written as

(61)

where . This sensitivity function is designed to magnify uncertainty during bad times and to minimize it during good times. The purpose of this modification is to introduce volatility in the return on stocks, which is low (high) during good (bad) times. This mimics the empirical properties of stock returns. Chen et al. (2008b) show that this generates multiple problems in the implementation of the analytic method. On the other hand, it illustrates the ability of the analytic method to handle more complicated integral equations. In step 1 of the analytic method, the form of , which measures how much the investors discount the future, determines the conditions for the existence and uniqueness of the price-dividend function. has the property that for all , where

(62)

As a result, one would seek a solution in the following vector space:

Chen et al. (2008b) show that under the assumptions , and , the integral equation (58) has a unique solution in the space . The problem is that the parameters of Campbell and Cochrane do not satisfy the condition , so that the discounting factor is too big. Given that Campbell and Cochrane want to match the Sharpe ratio, which is the ratio of excess return on stocks relative to its standard deviation. It turns out that the persistence of the surplus consumption ratio must be almost a unit root, i.e., . The problem can be traced to the term in the exponent of . Campbell and Cochrane show that this term is approximately the Sharpe ratio at the steady-state value for the surplus consumption ratio, which is about per quarter. This value is enough to push the discounting factor above one so that a solution in cannot be found.

Chen et al. (2008b) presumed that Campbell and Cochrane did not want dividend growth to be unbounded both above and below. In fact, they explicitly ruled this out in their computer program. As a result, they restricted the price level outside a reasonable range for dividend growth. In this case, the transformed price-dividend function is

(63)

where , , , and

(64)

Here, .

In the new integral equation (63), one can see why the original integral equation fails to have a solution over the entire real line. In the calculation of the expected transformed price-dividend function, the probability distribution is a Gaussian distribution with mean and standard deviation . The functional form of implies that the standard deviation of the transformed price-dividend ratio tends to as the dividend growth approach . Consequently, the integral in (63) tends to for extreme negative dividend growth. To avoid this possibility, the new integral equation restricts the transformed price-dividend function to () for dividend growth less (more) than (). In the calibration of the model, was set at per month so that the model applies for all historically observed dividend growth.

This new integral equation for the Campbell and Cochrane model has a unique solution in the vector space of all continuous and bounded functions defined in when . In addition, the supremum norm of the transformed price-dividend function satisfies

(65)

Thus, step 2 can be successfully completed for the Campbell and Cochrane model. Consequently, the value of correctly identifying the vector space in step 2 is demonstrated in this model.

The new integral equation (63) does have a solution under the parameter values of Campbell and Cochrane. However, the supremum norm is too small so that the price-dividend function and Sharpe ratio cannot match their historic values. Chen et al. (2008b) demonstrate that a feasible solution, consistent with the Sharpe ratio, can occur for a coefficient of risk aversion and a persistence of the surplus consumption ratio .

Step 3 can also be completed for the Campbell and Cochrane model. In this case, the problem is to find the domain in the complex plane such that

(66)

where Re refers to real part. This domain is

As a result, the solution to the transformed price-dividend function is analytic within this domain, since the integral in (63) and its derivative are well defined. The final part of this step is to find the radius of convergence which is the minimum distance from the point on the real line to the boundary of . This problem is illustrated in Figure 2. This is a simple maximization problem which yields a radius of convergence around for . This value is close to the point at which the sensitivity function (64) becomes non-differentiable. Thus, the analytic method can be applied to this more complicated model.

Step 4 is also straightforward when one chooses to satisfy the additional condition:

(67)

Since the Taylor series of about the origin converges to for satisfies equation:

(68)

Chen et al. (2008b) derive a system of linear equations which solves for the coefficients in (68). This derivation includes all algebraic manipulations necessary to design an efficient computer program.

Finally, step 5 is completed for the Campbell and Cochrane model as in the Mehra and Prescott model. In this case, the upper bound in (45) depends on the supremum norm of the transformed price-dividend function (65) and the radius of convergence. Thus, the analytic method can be used to systematically approximate the solution to an asset pricing model as complicated as that posed by Campbell and Cochrane.

Step 2 shows that the persistence of the surplus consumption ratio must be close to a unit root. This means that a large number of coefficients need to be calculated for the solution to be stable. Unfortunately, Chen et al.’s (2008b) Fortran program does not allow for higher precision so that the calculation of the coefficients was not feasible in this case. Their Maple program does allow for higher precision but is too slow to calculate the required number of coefficients.

3.3 Alternative Asset Pricing Models

More recently, Bansal and Yaron (2004) use the recursive preferences of Epstein and Zin (1989, 1990, 1991) and functional form of Kreps and Porteus (1978) to examine the empirical behavior of stock returns. In this case, the reward becomes

(69)

Here, the utility today, , depends on the consumption today, , and the utility that the investor expects to receive in the future, . As a result, the control variable is and the state variable is . The intertemporal rate of substitution, , measures how much the investor is willing to substitute the consumption today for more utility in the future.

For the representative agent economy, in which consumption equals dividends, the equilibrium condition for the price-dividend function becomes

(74)

This model leads to the integral equation

(75)

This integral equation is no longer linear in the unknown price-divided function , since the stochastic discount factor is nonlinear.

As a result, the analytic method outlined for the Mehra and Prescott model cannot directly be applied to the Epstein and Zin model. In particular, the integral equation (74) is nonlinear as in the portfolio decision problem (19) and (20), so Judd’s (1992) projection method has been used by Bansal and Yaron (2004) to approximate the solution of this model.

In summary, the analytic method can be used to solve most one-dimensional discrete time asset pricing models. In all of these models, the price-dividend ratio and hence the return on stocks (26) can be accurately approximated by a high order polynomial. For example, the order of the polynomial is in the Mehra and Prescott case. The analytic method also points out the need to use a large support for the probability density function when the Gaussian quadrature procedure of Tauchen and Hussey (1991) is used. In particular, there is an error of in the price-dividend ratio when the support is even though the relative error is driven to zero. To minimize this error, one must consider the support of . Chen et al. (2008b) need to use even higher order polynomial approximations to accurately solve the Campbell and Cochrane (1999) model.

3.4 A Survey of Log-Linearized Approximations

The conclusion from using the analytic method to solve well-known asset pricing models begs the question: How many anomalies in financial markets are the result of low order polynomial approximations commonly used in the research on financial markets? For example, ever since Campbell and Shiller (1988a,b) it is common to explain the logarithm of the return on stocks in terms of unanticipated changes in the future dividends or discount rate. Campbell and Shiller start with (26) which may be written as

(76)

Campbell and Shiller do not have the solution to any particular asset pricing model so they do not know or . As a result, they calculate a first order Taylor’s polynomial approximation of the first term near the point , where is the stationary point of dividend growth (9).¹⁵

where and . Substitute this approximation into (76) to yield a first order stochastic difference equation in , which has forcing term , such that

(77)

If this difference equation is solved for and substituted back into (77), then one finds that the unanticipated changes in stock returns are given by

(78)

The first term is the news about future dividend growth, while the second term refers to the news about future rates of return. Campbell (1991), Campbell and Vuolteenaho (2004), and Campbell et al. (2010) among others find that most of the unanticipated changes in the return on stocks may be associated with the news on the future return on stocks rather than the news on the dividend growth. For example, Campbell finds that less than one half of the variation in the return on stocks is due to the news on the dividend growth. Yet, if one has the solution to one of the various models for the price-dividend function such as Campbell and Cochrane (1999) or Bansal and Yaron (2004), then it is clear from (76) that the unanticipated changes in the return on stocks should only be a function of the unanticipated changes in the state variables , i.e., . Thus, the unanticipated movement in the return on stocks from the news must be associated with a nonlinear relation between the future state variables and return on stocks. Future research should examine how the nonlinear relation between the state variables and the return on stocks color the understanding of what causes the return on stocks to move.

Bansal and Yaron (2004) also rely on the Campbell and Shiller (1988a,b) approximation to solve their model with Epstein and Zin preferences. In their model, they separate out the dividend growth from the consumption growth, which they assume is driven by the same state variable, . This state variable is assumed to be a first order autoregressive process such as (9) with . As a result, they hypothesize a function for the logarithm of the price-dividend function, which is linear in this state variable. Next, they log-linearize the return on stocks as in Campbell and Shiller. The guess for the price-dividend function and the approximation for the return on stocks are then substituted into the Euler condition (74). They then determine the two coefficients in the guess for the price-dividend function.¹⁶ Subsequently, Bansal and Yaron use the projection method to find the approximate solution of the model, which they say is “quite close to” the first order approximation. Yet, they do not say the order of the polynomial approximation used in the projection method.¹⁷ Given the close relation between the integral in (75) and that found in the Mehra and Prescott model (32), one would expect the need for at least a 9th order polynomial for an accurate approximation of the solution to the Bansal and Yaron model. Consequently, the long run risk models should also be examined to see whether or not the interpretations and approximation errors are influenced by low order polynomial approximations.

An alternative approach to developing log-linearized empirical models of asset returns is illustrated by Gabaix (2009). In this case, he identifies a linear generating process. Here, the stochastic process for dividend growth is found such that the price-dividend ratio is linear in the current state variable. A more sophisticated version of this approach is developed by Le et al. (2010) who incorporate the properties of continuous time affine term structure models into discrete time asset pricing models.¹⁸ The idea is to find a probability density function which yields an affine moment generating function under the risk neutral distribution

(79)

Here indicates the expectation is with respect to the risk neutral distribution and is now a vector of all state variables. They show that a gamma probability distribution in which the random parameters follow a Poisson probability distribution leads to this affine moment generating function. In this case the functions and have explicit formulas so that one can derive the affine bond prices using well-known recursive rules.¹⁹ Consequently, the probability distribution under the risk neutral measure, , has an explicit functional form. One then specifies a market price of risk such that the Radon-Nikodym derivative has the functional form

(80)

Thus, the physical distribution of the state vector is known and given by

(81)

As a result, the stochastic discount factor is given by

(82)

since . Also, is the risk-free interest rate which is assumed to be linear in the current state variables, . Given the stochastic discount factor one can proceed to price any financial asset.²⁰ To illustrate how this procedure can be applied to a particular asset pricing model Le, Singleton, and Dai specify a stochastic process for the inverse of the surplus consumption ratio in Campbell and Cochrane (1999) and Wachter (2006). This stochastic process has the risk neutral probability distribution with moment generating function (79). In addition, they derive a new sensitivity function for (56) which satisfies the local properties in Campbell and Cochrane’s original model.

This approach establishes a theoretical foundation for the log-linearized asset pricing models. However, the analytic method shows that even the Mehra and Prescott model needs a higher order Taylor polynomial to accurately represent asset returns in this model. Consequently, the stochastic discount factor from the original asset pricing models is not given by (82). Thus, an open question is how close the stochastic discount factor from the original asset pricing models matches the empirical model (82) developed by Le, Singleton, and Dai.

3.5 Heterogeneous Agents

Asset pricing models with heterogeneous characteristics for investors significantly increase the level of complexity of the computational algorithms. The heterogeneity takes many forms including differences in risk aversion, income, beliefs, and access to information. The complexity arises from the need to keep track of the differences among investors, as well as the coordination of their behavior with market equilibrium.

To illustrate these issues, consider the differences among investors’ incomes as in Lucas (1994) and Heaton and Lucas (1996). The budget constraint of the investors (8) now becomes

(83)

for each individual investor . The cash in hand for investing, , includes the initial wealth of the investor, , plus her labor income, .

Suppose that we still have only two assets as in the representative agent models. The return on the ith investor’s portfolio is found by combining (7), (9), (29), and (26) to yield

(84)

Here, the price-dividend ratio depends on the current growth rate of dividends, . In addition, the distribution of investors’ cash in hand, , as well as its future value, , influence investors’ demand for stocks, so that the price of stocks is a function of these distributions. To keep the analysis trackable it is also assumed that this probability distribution has the Markov property as in Heaton and Lucas (1996). The risk-free interest rate depends on the dividend growth and the distribution of investors’ current cash in hand, since they alter investors’ consumption.

The first order conditions (11)–(13), and the terminal condition (14) still hold under the constant relative risk averse utility function (6), where each variable has a subscript to represent investor .

For each agent, we seek functions and . As a result, the optimal portfolio return is , where the optimal investment in stocks, , is substituted into (84). These functions must satisfy the optimal investment decisions for each investor given by

(85)

and

(86)

Here, the future cash in hand available to the investor, , is endogenous and is determined by (83), and the optimal decision of investor is such that

(87)

where is future labor income.

Market prices coordinate the behavior of the agents. To illustrate the concept, suppose that there are two agents, i.e., . Also, assume that there is one share of stock so that the equilibrium in the stock market is given by

(88)

where is the initial level of dividends.²¹ Note that is now the percentage of the cash in hand invested in stocks rather than the percentage of wealth. The market for bonds must be cleared. Usually, it is assumed that these bonds are trading among the investors.

(89)

This relation means that one investor borrows, i.e., , and the other lends, i.e., .

Finally, total labor income each period is given by

(90)

To complete the model, one needs to specify how labor income and dividend growth move from time to time which determines the evolution of the distribution of the cash in hand from to following (87).

Asset pricing models with heterogeneous agents are more complicated for two reasons. First, both the optimal portfolio decisions and the equilibrium prices must be determined simultaneously. Second, one must keep track of the distribution of the cash in hand across agents. A standard way to circumvent the second issue is to assume a utility function which is an exponential function of consumption and that returns have a normal distribution. These assumptions make investors’ decisions independent of wealth. It generally leads to a linear pricing function for equity. Such models of financial markets are common in the study of the impact of information on stock prices (see Wang, 1994, Easley and O’Hara, 2004, Lo and Wang, 2006).

If one does not make these assumptions, then a tractable mechanism to keep track of investors’ cash in hand is essential in avoiding the curse of dimensionality. Constantinides and Duffie (1996) provide a clever way to reverse engineer the distribution of income such that the reduced form is mathematically the same as the representative agent model (30), albeit with different parameters. Consequently, it may be solved using the same procedure outlined above. Lucas (1994) and Heaton and Lucas (1996) assume finite states for aggregate income growth . In addition, investors’ share of labor income is given by so that following (90). As a result, the exogenous behavior of investors’ income may be represented by two state variables, . In addition, it is assumed that these two state variables follow a finite state Markov chain.

These assumptions make it easier to keep track of the distribution of investors’ cash in hand, . Given the possible finite realizations of , one can calculate the income levels of each individual so that (87) determines the realizations of cash in hand with probabilities of these realizations dictated by the Markov chain. Thus, the distribution of investors’ cash in hand is incorporated into the calculations of expectations in (85) and (86).

In principle, the same five steps used in the representative agent models would be followed for developing a computer algorithm for solving the heterogeneous agent models with incomplete markets. Kubler and Schmedders (2003) develop a proof of the existence of an equilibrium for a heterogeneous agent model with collateral constraints on debt. In their model, the exogenous Markov shocks determine both the wage income and dividend growth of assets. They develop a Markov equilibrium concept which modifies Duffie et al. (1994). Their state space includes both the exogenous Markov process and all possible endogenous variables such as cash in hand. The Markov equilibrium consists of a correspondence of expectations which determines the evolution of the state from one period to the next subject to all the conditions for equilibrium including (85), (86), and (88)–(90). In addition, there are correspondences of equilibrium policy which determine all the endogenous variables in the state space. They are able to prove the existence of an equilibrium; however, the properties, including the uniqueness, are not known.²³ Following tradition in computational economics, they proceed to define an -Markov equilibrium in which the optimal conditions (85) and (86) for all investors are satisfied within an arbitrarily small relative error. They are able to prove the existence and uniqueness of the -Markov equilibrium. Subsequently, Kubler and Schmedders (2005) show how to interpret such an approximate equilibrium as an equilibrium for an economy with close by endowments or preferences. As a result, they are able to complete steps 1 and 2 outlined in the representative agent models. Step 3 is not analyzed in the heterogeneous agent models, since the analysis of equilibrium has been limited to general utility functions rather than specific functional forms such as (6). Whether or not step 3 can be successfully undertaken in a heterogeneous agent model is an open question.

Heaton and Lucas (1996) consider a recursive equilibrium concept similar to Kubler and Schmedders. However, very little was known at that time about conditions which lead to the existence of such an equilibrium. Heaton and Lucas use Monte Carlo simulations to calculate the expected returns rather than the polynomial approximations. They then use what they call an auctioneer algorithm to search for the equilibrium. They start with an initial guess of the optimal consumption and investment decision for each agent. Given these choices, one can calculate the equilibrium price and interest rates, as well as each agent’s marginal value of the assets. If an agent’s marginal value of a security is not consistent with the equilibrium price and interest rates, then investors’ optimal decisions are modified such that each buys more of the assets which they value higher. This process is continued until there is no discrepancy between the market and investors’ valuation of securities.

Given the state of the literature, Judd et al. (2000) choose to find an -Markov equilibrium which determines the consumption, investment, and pricing functions that satisfy all the conditions of the model. Under the specification of the Markov stochastic process for the labor income and dividend growth, one can specify equilibrium functions for the price-dividend ratio and risk-free interest rates as a function of dividend growth and the state variable .

(91)

and

(92)

The distribution for investors’ wealth is incorporated into the unknown parameters and of these functions, since the parameters of the Markov chain are incorporated into the expectations found in (85) and (86). As a result, only the current state is relevant for the equilibrium prices. Given these price functions, the consumption and investment decision of each individual can be given by (21) and (22), respectively. Here, these functions are dependent on individuals’ cash in hand, , and the current state .²⁴

With regard to the polynomial approximations, Judd et al. do not use orthogonal polynomials, since the equilibriums exhibit high curvature on the boundary of the state space. To improve the accuracy of approximations they use the cubic splines in , which is a third order polynomial on each subinterval with . These subintervals partition the domain of the spline at the knots , where the function to be approximated is evaluated. A basis for the cubic splines can be constructed using B-splines (see Judd, 1998, p. 227 and Stoer and Bulirsch, 2002, pp. 97–121). In this case, the approximation functions (21), (22), (91), and (92) are constructed by replacing the Chebyshev polynomials with the B-splines. If the functions being approximated have a uniform bound on the interval for its continuous fourth order derivatives, then an accurate solution can be found by choosing a fine enough partition of the domain of the functions. Since step 3 has not been undertaken for the heterogeneous agent models, these conditions for a bound on the error have not been verified. Consequently, Judd et al. (2000) rely on the -Markov equilibrium concept to represent the accuracy of their algorithm.

Proceeding to step 4, the approximations (21), (22), (91), and (92) are substituted into each investor’s optimal conditions (85) and (86) using the equilibrium conditions (88) and (90). Judd et al. (2000) find that solving this system of equations for the coefficients of the approximation functions is problematic since the system is ill-conditioned. As a result, they adopt the following iterative scheme based on economic intuition. Start with an initial guess for the coefficients of the approximation functions. Use these approximate coefficients only for the future decisions and prices in the optimal conditions (85) and (86) subject to the equilibrium conditions (91) and (92). Then choose a new set of coefficients for the approximation functions so that the conditions (85) and (86) are satisfied. Proceeding in this fashion, the algorithm is continued until there are no further changes in the coefficients.

Each iteration in the algorithm in Judd et al. (2000) requires the solution of a system of nonlinear equations for the coefficients of the approximation functions. The numerical solution of this system using Newton’s method is sensitive to initial values for the coefficients of the approximation functions. Consequently, they use the homotopy method of Schmedders (1999) to locate reasonable initial values for Newton’s method. The homotopy method introduces a marginal penalty to the optimal conditions of each investor (21) and (22) which disappears if the homotopy parameter is set to zero. This procedure helps to mitigate the problems associated with possible discontinuities in the equilibrium solution. After running the homotopy procedure a few times, they switch back to Newton’s method to speed up the convergence to the solution. The procedure stops when the relative error in the optimal conditions for investors’ decisions is within some given tolerance level.

The approach for solving the heterogeneous agent models with incomplete asset markets should follow the same five steps outlined in the representative agent model. Up to now, we are not aware of a heterogeneous agent model which has completed all five steps. In particular, step 3 is bypassed. Our experience with the representative agent models suggests working out the details of the existence and uniqueness of equilibrium for specific examples can yield a better understanding of the equilibrium for model. Under these specific examples it is more likely that step 3 could be successful. Alternatively, enough information about the equilibrium can be found so that the conditions for a bound on the approximation error can be found.

4.1 Optimal Investment Decisions in Continuous Time

To illustrate the optimal investment decisions, we follow Schroder and Skiadas (1999, 2002, 2003, 2005, 2008), Wachter (2002b), Campbell et al. (2004), Liu (2007), Sangvinatsos and Wachter (2005), and Wachter (2010) who use the preferences of Duffie and Epstein (1992a,b) with the particular functional form of Kreps and Porteus (1978). These preferences are the continuous time version of Epstein and Zin (1989, 1990, 1991), (69). The investor chooses the percentage of wealth consumed, , and invested in stocks, , each period so as to maximize

(104)

where

(105)

Let . From now on we use the notation rather than to cut down on space. The measure of the investor’s aversion to risk is . The bequest function is . In particular, one could choose the of the form

(106)

The parameter measures how much weight the investor places on the welfare of some future person or institution, such as, the investor’s children or a charitable organization. If , then the weight is the same as if it is herself. If , then no weight is put on this future person.²⁸

The investor’s optimization of lifetime utility (104) is subject to the stochastic process for her wealth

(107)

The risk-free interest rate for bonds is assumed to be constant. The expected equity premium, , has a stationary value while the standard deviation for the equity premium is . The equity premium randomly changes following the standard Brownian motion .

The empirical characteristics of the equity premium are based on the work of Campbell and Viceira (1999, 2002) in which the expected excess return on stocks is dependent on the price-dividend ratio.

(108)

Here is the price of equity and is the price of the risk-free bond. The expected equity premium follows

(109)

The risk-free interest rate for bonds is assumed to be constant. is also a standard Brownian motion. The shocks to the expected equity premium, , and the equity premium, , are negatively correlated so that . The expected equity premium has the solution for any

(110)

where the integral has a normal probability distribution following Shreve (2004).²⁹

(111)

This specification captures the empirical properties demonstrated in Campbell and Cochrane (1999) and summarized in Lettau and Ludvigson (2010). In particular, when stock prices increase from a higher surplus consumption ratio in Figure 4, the expected return on stocks falls in Figure 5.

Following the analysis of Schroder and Skiadas (1999, 2002, 2003, 2005, 2008) one can show that the lifetime utility of the investor is separable between the current value of wealth and the current expected equity premium . This separation is given by

(112)

where with . One also needs and , so that . Finally, (112) is true at the terminal time , so that the terminal condition (106) has the same form as (112).

Using the dynamic programming procedure following Fleming and Soner (2006) one can derive the optimal consumption and investment decisions

(113)

For the solution to have a unique critical point, the restrictions , and must hold. Given that these critical points are unique, must satisfy the initial value problem

(114)

The constants are defined by³⁰

This initial value problem can be derived from the backward stochastic differential equation of Schroder and Skiadas (2003) for the investor’s problem (104)–(111). If there are multiple expected excess rates of returns and investment alternatives, becomes a vector. For the higher-dimensional problem and are replaced by positive definite matrices, since they correspond to variance-covariance matrices for the expected excess returns and actual returns.³¹

The solution to the initial value problem does not guarantee that the investor’s problem is solved. One must also show that the integrals in (104) are well defined. Proceeding formally apply Ito’s lemma to the utility of wealth from (112) using the stochastic process for wealth (107). Then use the optimal conditions (113) and the PDE (114) to obtain the following expression for the expected utility of wealth in the terminal period.

(115)

where .³² This expected utility of wealth is a stochastic exponential of a semi-martingale . For (115) to be well defined, Theorem (45) of Protter (2005, p.141) implies that the lifetime utility is a locally integrable martingale when

(116)

We can now discuss the procedure for assuring the lifetime utility (104) is well defined. From (110) and (111) the future expected equity premium increases (decreases) monotonically to for () as . In addition, the variance of increases monotonically to . Thus, one can establish a bound on the normal probability distribution used to calculate the expected value in (115) and (116) such that³³

(117)

where is the probability that at time the state becomes at time . In addition, the constant and is a locally bounded function of the current expected excess returns on stocks.

It is known that the normal probability distribution has the property

(118)

as long as . Since the integrands in (115) and (116) are the product of the transition density (117) and some combination of the first and second derivatives of the solution (114), , these combinations must grow more slowly than the bound on the transition density (117). This leads to sufficient conditions on the solution to the IVP (114) for the integrals in (115) and (116) to be bounded.³⁴ For example, one can explicitly calculate a bound for the expectation of the integral

(119)

in (115) as long as does not grow faster than in (117). In particular, suppose , then one can use (110) and (111) and recognize has a normal probability distribution with given mean and variance, as in Hugonnier et al. (2012 Lemma F.1), so that a bound for (119) can be determined.

The investor’s lifetime utility (104) is also related to the IVP solution, , through (112). From the application of Ito’s lemma to the lifetime utility function (112), the optimal conditions (113), and the solution to IVP (114) given by yields the integral equation

(120)

If one uses the change of variable , then the conditions for (120) to be well defined are the same as the growth conditions applied to (115). Thus, the solution to the IVP problem needs to satisfy growth conditions for this solution to satisfy the investor’s problem (104)–(111).

4.2 Solutions

Solutions to the investor’s problem have been published by Kim and Omberg (1996), Wachter (2002b), Schroder and Skiadas (2003), Sangvinatsos and Wachter (2005), and Liu (2007). Wachter (2002b) examines the case in which , while the others are in dimensions. These solution procedures work when the IVP (114) is linear, i.e., . One case, in which this occurs, is the constant relative risk averse case, , with and markets are complete . However, any other combination of parameters such that would also lead to the same mathematical problem. In addition, the solution applies for the case , as long as the solution satisfies bounds like (119). These conditions may not hold, since the constant in (119) switches sign as changes to .

In all these papers the procedure is the same. First assume the fundamental solution , when and , has the Gaussian functional with time-varying mean and variance:

(121)

Here, for are functions of the investment horizon for with initial conditions for , and the constant . These initial conditions assure that the solution converges to its terminal value as . is the endogenous discount rate for future values which assures the integrals over time in (115) and (116) imply a present value less than future payments. This along with the stationarity of the stochastic process for the expected equity premium (110) and (111), , is needed to assure that the integrals over time in (115), (116), and (120) exist even when the time horizon tends to infinity.

One then calculates the derivatives of the hypothesized solution (121) and substitute them into the IVP. By equating the coefficients on , and one obtains ordinary differential equations in the investment horizon for each of the coefficients. The remaining terms are used to obtain an ordinary differential equation for .³⁵ These equations are known as the Riccati equations and their solutions can be quickly and accurately calculated in computer programs.

After finding the fundamental solution to the IVP (114) one must make sure that conditions like (119) are true. In the case this condition is true, since is negative so that (118) can be used to calculate the expectation in (119).³⁶ In this case the solution has the form of a Gaussian probability density function so that the expected values in (115), (116), and (120) are well defined. One can see this by looking at Figure 3 which plots the solution (121) for the parameters in Table 1 with and . These parameters are updated following Campbell and Viceira (1999) and Campbell et al. (2004) for the US from 1947 to 2006.³⁷

When the sign of is positive so that (119) is not necessarily true. In this case Figure 3 would be the inverse of the previous case so that as for each . As a result, the convergence of the integrals over in (115), (116), and (120) becomes more delicate. In particular, the positive growth of must be less than the decline in the bound on the normal distribution (117) as . This case is important since Bansal and Yaron (2004), and Bansal et al. (2007, 2009) argue that the elasticity of intertemporal substitution is greater than one, which corresponds to the case of , and .³⁸

Once the fundamental solution is found, the non-homogeneous linear IVP, i.e., and , can be found using Duhamel’s principle:

(122)

so that the solution to the linear IVP (114) with is a combination of the fundamental solution (121) at different time to maturities.

The nonlinear case has not been solved and there is limited information about the solution. If one uses the transformation

(123)

then the fundamental solution can be found for the case in which there is no intermediate consumption. The solution has the same form as (121) with different constants in the IVP (114). In the general case and one cannot use Duhamel’s principle, since this principle does not hold for nonlinear differential equations. In addition, the Cauchy-Kovalevsky Theorem cannot be used, since the highest derivatives with respect to each variable must be the same. This condition is violated for the IVP (114).

For the infinite investor’s horizon case one would want to let once one obtains a solution to the finite horizon case. For example, the functions , and in Wachter’s (2002a) solution converge to constants , and . One can see this behavior of the solution in Figure 3 in that the curve in the dimension stabilizes as gets larger. In addition, the value of the fundamental solution (121) converges as because . Thus, one needs to make sure the final solution (122) converges for .

Some approximations for the nonlinear IVP Campbell and Viceira (2002), Campbell et al. (2004), and Chacko and Viceira (2005) approximate the PDE (114) in the case and by setting and replacing with a logarithm approximation around a known solution in which . This leads to a solution to an approximate PDE. However, it is not clear what the relation is with the solution to the original PDE as . Thus, the solution to the infinite horizon problem is not complete and is dependent on the characterization of the finite horizon problem.

For the case little is known about the solution to the IVP (114). One could use a traditional procedure, used by scientist and engineers, such as the perturbation method as in Bender and Orszag (1999). This method breaks the nonlinear PDE into an infinite number of linear PDEs which can be solved recursively. In these linear PDEs the nonlinear term in (114) is replaced by an independent function of the solutions to previous linear PDEs. The solution to the IVP (114) is then the sum of the solutions to all these linear PDEs. The mathematical problem is that one does not know whether or not this sum converges and whether it converges to the true solution of (114). Finally, one needs to check whether or not the condition (119) is true such that the integrals over in (115), (116), and (120) exist. Thus, there is still substantial work that needs to be done before the investor’s IVP (114) is fully characterized.

5.1 One-Dimensional Asset Pricing Model Under Infinite Horizon

Given the limited knowledge of the solution to the IVP problem (114), we initially focus on the infinite horizon case in which there is a single risky asset and a risk-free bond under a representative investor. This problem leads to a time independent differential equation, whose solution leads to the price-dividend function, the expected return on stocks, and the risk-free return. This problem can be quickly and accurately approximated using a high order Taylor polynomial using results from and extensions of the Cauchy-Kovalevsky Theorem. The main issue with this approach is that one must specify two conditions for the solution of the differential equation problem. Below we discuss several ways to establish these conditions using an empirical approach or approximations of the solution at the stationary point of the state variable to determine initial conditions. We then discuss the possibility of using boundary values rather than initial conditions to set the conditions on the differential equation. Which procedure best represents the economic problem is an open question, however the mathematical properties can be established once these conditions are known.

5.2 Multi-Dimensional Asset Pricing Models

The analytic method may be extended to higher-dimensional continuous time asset pricing models. However, up to now the error analysis for polynomial approximations is not done. To illustrate the issues, consider the two-dimensional extension of the Campbell and Cochrane model by Wachter (2002a, 2006).⁵² Wachter (2002a) changes the specification of consumption growth from a random walk (130) to

(139)

where is the long run growth of consumption, . As in Bansal and Yaron (2004) this long run growth follows a stochastic process like (109) in which would be the correlation of consumption growth with the long run risk variable, and is its standard deviation.

Chen et al. (2011) use the methods in Chen et al. (2010) to show that the external habit model subject to long run risk leads a price-dividend function which is the solution to the IVP:

(140)

(141)

where the coefficients are analytic functions near . In addition, they are similar to (133) in which in is replaced by . In addition, the initial conditions (141) must be analytic near . Chen et al. (2011) relate these initial conditions to the analytic solution for the Campbell and Cochrane model, since the stationary mean for the long run risk variable is . As a result, they use the Cauchy-Kovalevsky Theorem to prove that the solution to the PDE (140) subject to the initial conditions (141) is an analytic price-dividend function

(142)

Here, is the solution to the continuous time Campbell and Cochrane model (132). In addition, they provide a bound on the relative error between the Taylor polynomial approximation and the analytic solution (142) within a region of convergence. This region is and . Unlike the one-dimensional case, obtaining the maximum region of convergence in multiple dimensions is a well-known challenging mathematical problem. For our model we expect that the maximum region of convergence is larger than the one obtained rigorously, since our numerical simulations indicate a convergence within the rectangle and . For example, in this region a comparison of the Taylor polynomial of order with one of order yields a relative error less than 6 in 10 million.

The evidence from the Wachter model demonstrates that it is feasible to use the analytic method in multi-dimensional continuous time asset pricing models, although additional work still has to be done on the analysis of the approximation error. However, one must think carefully about how to choose appropriate initial conditions (141). In addition, these initial conditions will become more demanding as the dimension of the problem increases. For example, one could introduce learning about the underlying volatility of the dividend growth as in Wang (1993, 1994) and Lo et al. (2004). In this case, the variation in the conditional variance or under the continuous time Kalman filter is an ODE with analytic coefficients (see Eq. (A.4) of Wang, 1994). Consequently, the coefficients in the PDE (140) would still be analytic. Bhamra and Uppal (2010) take advantage of this property to analyze a dynamic general equilibrium model in which agents have heterogeneous beliefs and subjective rates of discount. The introduction of such a learning mechanism into Wachter’s model would add a third state variable. In this case, the initial conditions (141) would freeze one of the variables, presumably the conditional variance of dividend growth, so that these conditions would be an analytic function of two variables, the consumption growth and the surplus consumption ratio, i.e., the solution (142) of the Wachter model. Of course, one would also face the same curse of dimensionality problems observed in the discrete time asset pricing models, so that steps have to be taken to minimize the curse. Finally, one could use orthogonal polynomials rather than Taylor polynomials to improve the efficiency of the computer algorithm which implements the analytic method in continuous time asset pricing models.

5.3 Stochastic Differential Utility: Choosing Initial Conditions

To illustrate the second approach for setting the initial conditions Chen et al. (2013) consider the stochastic differential utility model, when consumption follows a stochastic process for consumption. This process mimics the long run growth model of Bansal and Yaron (2004). Chen et al. (2013) also consider a simpler model following Ai (2010) in which the standard deviation of the state variable is constant. To illustrate the initial value problem for infinite horizon problems we use the Ai model, which can be aligned with the investor’s problem in Section 4.

In the Ai (2010) model the long run risk variable of Bansal and Yaron is represented by investor’s estimates of the persistent component of technological shocks to production, , by firms in the economy. Ai uses a representative investor with an infinite investment horizon, so that the investor’s problem is given by (104)–(111) as and the state variable is rather than . Following Duffie and Lions (1992) the solution to this equation is time independent as a result, and the nonlinear differential equation (114) for Ai’s model is given by⁵³

(143)

As in the Campbell and Cochrane (1999) model one must specify two initial conditions for the existence of a solution to this ODE. The coefficients of this ODE are simpler than those found in the Campbell and Cochrane (1999) model. However, this ODE is nonlinear as long as , which corresponds to the case in which or in Ai’s model. From the Cauchy-Kovalevsky Theorem we know that the solution is analytic so that the solution can be represented as a power series within the radius of convergence. However, the error analysis for the Campbell and Cochrane model in Corollary 5.2 had to be modified to account for the nonlinear term in the ODE (143). Chen et al. (2013) carry out this analysis and establish a bound on the error between the solution to (143), , and its order Taylor polynomial approximation within a known radius of convergence. A bound is also established for the derivative of the solution within a fraction of the same radius of convergence. Thus, they obtain an accurate solution to the nonlinear ODE when the initial conditions are given.

Ai (2010) uses the procedure developed by Campbell and Viceira (2002), Campbell et al. (2004), and Chacko and Viceira (2005) which was discussed in Section 4.2. This approximation takes the form

(144)

where and are known functions of the parameters in the original ODE (143). This is the actual solution to the approximate ODE, not the original ODE from Ai (143), so there is some unknown approximation error in this procedure.

One way to think about this problem is to examine the case , which has the solution (121) and (122). Suppose all the unknown coefficients converge to constants as the investment horizon converges to infinity, . In this case, the first term in (122) approaches zero, as long as . Consequently, the terminal value in (122) provides no information about the solution to the ODE in the infinite horizon case. However, the integral does converge to

(145)

which corresponds to (144) with the appropriate definition of constants and . Thus, the Ai approximation and the solutions agree, when and the investor has an infinite horizon.

Given the approximation by Ai the initial conditions for the original ODE (143) can be taken as

(146)

To understand the impact of this choice of initial conditions, consider the perturbation method for and the state variable is . Also notice that the solution to the first ODE (143) when , corresponds to the Ai approximation (144). Thus the initial conditions (146) are ignoring the terms associated with the subsequent ODEs in the perturbation method which was discussed in Section 4.2. While the approximation (144) can be shown to satisfy the bounds (119), we do not know if the actual solution does, since the error in using (146) rather than their true values is unknown.

For the parameter values in Ai (2010) and Chen et al. (2013) approximate the function for the lifetime utility of the representative investor. The initial conditions are and where is the stationary point for the probability distribution for . Here, is the estimate for the radius of convergence. The relative error between the and polynomial approximation is less than one in a million within this radius of convergence. In Figure 6, the solution to the approximate ODE (144) is plotted along with the order polynomial approximation using the analytic method. Again this shows that high order approximations make a significant difference given the same initial conditions. This difference is more pronounced in Figure 7 in which the expected excess return on wealth (126) is plotted. The Ai approximation yields a constant expected excess return on wealth, since the elasticity of the exponential function is constant. On the other hand the order approximation yields an expected excess return on wealth which is low (high) when technological progress is high (low). This conforms with the empirical behavior found in the Campbell and Cochrane model (see Figure 5).⁵⁴

Using initial boundary value problem (I-BVP) rather than IVP An alternative way to place conditions on the differential equation (143) would be to establish boundary conditions on the solution to (143) as the state variable becomes arbitrarily large. An example of such an approach is the solution to the Black-Sholes-Merton European call option pricing problem.⁵⁵ Consider the case . Given the functional form of the fundamental solution to the investor’s problem (104) to (111), (121) and Figure 3, the solution should have the following property:

(147)

Given the parameter values this would be the case in Ai (2010) when and or and .⁵⁶ Given these properties of the fundamental solution, as well as the intuition from the Black-Scholes-Merton and heat diffusion one can specify the IVP as an I-BVP problem using (147) as boundary conditions on the PDE in (114) for the general case . One could then examine the asset pricing problem by letting the investor’s time horizon go to infinity.

The idea is that the investment horizon is far enough into the future so that the fundamental solution (121) does not change much for longer time horizons. In addition, the investor switches to an environment or preferences such that at the end of the investment horizon. This approach would be consistent with the strategy for identifying the terminal conditions in the standard neoclassical growth theory (see Acemoglu, 2009, Chapter 8). Once the I-BVP problem is solved, one can analyze how the solution behaves as becomes arbitrarily large. By using the tools of I-BVP one would study whether the mathematical problem is well posed. In particular, one would want to know how sensitive the solution is to the initial data in (145). In addition, numerical methods can be structured to accurately solve both the investor’s problem (104)–(111), and the asset pricing problem (124)–(126).