Asset Prices and Time-Varying Risk

Observers have often characterized asset markets as being subject to periods of tranquility and periods of turbulence. Until recently, however, researchers were unable to produce closed-form asset pricing formulas in a model environment of time-varying risk. Some work by Abel provided us with the insights needed to produce such formulas. This paper gives a exposition of how to develop the formulas in an environment where the formulas may by obtained using a simple extension of standard tools. While the paper is intended mainly as an exposition of new work, it also contains a report on the asset market effect of fiscal reform. It is found that entering a period of weak coordination between government spending and taxing (tax rate) policy is good for stock prices.

Abstract

Observers have often characterized asset markets as being subject to periods of tranquility and periods of turbulence. Until recently, however, researchers were unable to produce closed-form asset pricing formulas in a model environment of time-varying risk. Some work by Abel provided us with the insights needed to produce such formulas. This paper gives a exposition of how to develop the formulas in an environment where the formulas may by obtained using a simple extension of standard tools. While the paper is intended mainly as an exposition of new work, it also contains a report on the asset market effect of fiscal reform. It is found that entering a period of weak coordination between government spending and taxing (tax rate) policy is good for stock prices.

I. Introduction

The purpose of this paper is to provide an exposition in a familiar setting of some new methods in asset pricing and to indicate some aspects of the extension of these methods to the open economy. While the paper is designed as an exposition of these new ideas, the fiscal policies studied in Section III may be applicable to discussions of the asset-market effects of recent U.S. fiscal policy efforts.

The paper builds on some work by Andrew Abel (1986) on obtaining closed-form asset pricing formulas for a model environment where agents understand correctly that the nature of risk is time-varying. 2/ In particular, Abel’s work was the first that allowed time-varying dividend risk and obtained explicit closed-form pricing formulas for a representative-agent asset-pricing model where agent preferences display constant relative risk aversion.

Researchers have often described asset markets as being characterized by periods of turbulence and periods of tranquility. Before Abel’s work, however, we were unable to study explicitly agents’ reactions to new information about the riskiness of the asset pricing environment in a framework of a well-specified time-series model of tranquility and turbulence. Indeed, typical in asset pricing work is the examination of agents’ reactions to inconceivable once-and-for-all shifts in higher moments. These shifts are inconceivable because the models are solved typically conditionally on the agents supposing that relevent higher moments are assuredly constant through time. Researchers then study the effects of risk changes by changing moments (apparently forever) in a pricing formula constructed under the assumption that such moments would never change. The asset market applicability of the study of such miraculous events is unclear.

If risk, in some sense, changes through time then one way to model explicitly intertemporal variations in the risk environment is by making risk the outcome of a stochastic process. The advantage to adopting an explicit stochastic process for risk is that a stochastic process is useful for separating shocks to the risk environment from predictable changes in the risk environment. Since it is the agents’ reactions to their understanding of the time variation in risk that provides the link between the risk environment and asset prices we should model carefully the formation of agents’ beliefs about the current and future riskiness of their environment.

Abel studies risk as a time-series process assuming representative agents with rational expectations, constant-relative-risk-aversion period utility functions and assuming a particular functional form for distribution functions generating the underlying shocks. This paper will study agents’ reactions to shocks in the time-series process generating their stochastic environment without assuming a representative agent and without making particular functional form distributional assumptions. In fact, in this model the stochastic environment enters the rational expectations problem in a very familiar way and obtaining explicit closed-form solutions requires only a minor extension of the now well-know method of linear undetetermined coefficients.

In order to simplify the analysis some of Abie’s assumptions are changed. For clarity, the simplest assumption set has been chosen. In making the assumption substitutions, however, a possibly important aspect of decision making may be lost. In particular, it is assumed that time-separable quadratic preferences are capable of capturing adequately the consumption-saving decision facing agents. Given this approximation it is simple to obtain closed-form asset pricing functions for a wide range of distribution functions for the exogenous variables without assuming that individual agents are very similar aside from the functional form of preferences. While the quadratic preferences assumption buys a lot of simplicity it also costs something in terms of a possibly important dimension of risk aversion. With quadratic period utility, agents are risk averse with respect the risks involving the returns on their asset holding but they are risk neutral with respect to the riskiness of the underlying environment. Quadratic preferences induce linearity in the second moments (or variances) of returns. This linearity is the source of much of the simplicity of the present approach.

The paper is divided into four additional sections. In Section II, the basic asset pricing model is developed for an infinitely lived representative agent inhabiting a world with time-varying risk. Two simple experiments (”warm-ups”) are conducted in this section to familiarize the reader with the asset-price solution algorithm. The warm-ups are chosen to illustrate simply environments that might be encountered in practice. In the first example, dividend risk moves exogenously and is unconnected to any other aspect of market fundamentals. In the second example the dividend risk is modelled as a function of past disturbances to dividends. In Section III the model is subjected to two kinds of fiscal policy experiments. In the first fiscal experiment we study how fiscal feedback policy (from the level of output) influences the asset market impact of a shift in the riskiness of the environment. In the second fiscal experiment we study the asset market effect of entering a period of “fiscal reform.” Section IV contains some concluding remarks. The text of the paper is followed by three appendices, the first listing some text-referenced coefficients, the second discussing some issues concerning nominal asset pricing and the third explaining a heterogenous-agent multi-country reinterpretation of the model.

II. A Real Asset Pricing Model

The model to be developed in this section is suitable for pricing assets in a hypothetical world that functions without money. Initially it is assumed that a set of representative agents inhabits a closed economy where output is exogenous, government is absent. Preferences are of the representative agent are represented by a quadratic period utility function with lifetime utility being the discounted sum of the period utilities. Some of these assumptions will be relaxed later. The agent’s problem is:

max EtΣi=0 u(ct+i)βi, 0<β<1

subject to:

ct+i + qt+ikt+i = qt+ikt+i + dt+ikt+i1, i = 0,1,2,...,

where

Et = the mathematical expectation operator conditional on complete time t information,

u() = the period utility function,

ct+i = per capita consumption in period t+i,

qt+i = price of equity in terms of consumption goods at time t+i,

kt+i = number of equity shares held by the representative agent at time t+i,

dt+i = dividend paid at the beginning of t+i to the holder of one equity share during t+i-1.

The agents’ first order condition for maximizing expected lifetime utility at time t is:

(1)u(ct)qt = βEt(u(ct+1)[qt+1 + dt+1]).

Define zt ≡ u’(ct)qt; at ≡ u’ (ct)dt. Equation (1) now becomes:

(2)zt = βEtzt+1 + βEtat+1

To simplify, assume that a fixed capital stock is the only factor of production, that all output is paid to equity holders as dividends and that there is one equity share per representative agent. Therefore, dt = yt, where yt is per capita output at time t. Output depreciates fully in one period so that storage is futile. In equilibrium, therefore, ct = yt.

The solution of (2) is:

zt = Σi = 1 Etat+iβi

The solution excludes dynamically-based inteterminacies and is the type of solution that will be dealt with throughout the paper.

Now, assume a particular functional form for period utility:

(3)u(ct) = αct - (1/2)ct2

which is quadratic utility. 3/ Since choices are invariant to linear transformations of the period utility function, the constant has been suppressed in (3), which has been written to highlight the fact that marginal period utility for quadratic preferences is a one parameter family of functions.

Using (3) and ct = yt = dt, equation (2) becomes

(3a)zt = βEtzt+1 + βEt(α - yt+1)yt+1,

which may also be written as

(3b)zt = βEtzt+1 + β[αEtyt+1 - (Etyt+1)2 - ht],

where ht is the conditional variance of yt+1

htEt[(yt+1 - Etyt+1)2].

Like yt, ht is an exogenous variable and it will provide most of the “action” presently. In particular, the time series process for h will be the vehicle for modelling tranquil and turbulent periods. Notice that the t subscript on ht refers to the period of the information set relevant to the ht calculation.

Before moving on to the “warm-ups” notice that equation (3b) is a first-order linear stochastic difference equation in the variable zt with the composite forcing process given by the right-hand-most term in (3a) or (3b). The experiments to be performed here all involve altering the composite forcing term. If we had simply adopted a linear time series process for the composite forcing term then we would solve (3b) directly by the methods of Hansen and Sargent (1981a,b). This would be an expedient method of solving for zt, but it would not allow us to study the riskiness of the environment in isolation from other aspects of the forcing process.

Our choice of period utility function implies the moments of yt that will appear in the forcing term in (3b). With quadratic utility the first two moments of yt will appear. If a different period utility function had been adopted different moments would appear in the forcing process.

The variable zt is the marginal utility attached to a unit of the equity. It is equal to period marginal utility of goods times the goods price of the equity. It turns out that it is convenient to work with z trather that qt and so most of the discussion centers on zt. Usually this is a harmless simplification but not always; in example 2 in this section we study a situation where the simplification is not harmless.

The exposition begins with some pedantic examples to make clear the aspects of the this type of problem and solution method that are familiar from previous work and those that are novel to this approach.

Example 1: Constant first moment, first order auto-regressive conditional variance

Assume the following time series processes for yt and ht:

(4a)yt = y + wt
(4b)Etht+1 = øht + h, -1<øβ<1,

where wt is mean zero and serially uncorrelated. The stochastic process for yt and ht are assumed to be such that both are always positive. Further, α is large enough compared to any possible value of yt that period marginal utility of consumption is always positive, which requires yt< α. Imposing an upper bound of α on yt implies boundaries on h t. 4/

Innovations in ht+1 can come from a variety of sources and maybe correlated with innovations in the yt process or not. 5/

An explicit example where yt shocks drive ht is given later. For now, we need not be concerned with how conditional variance shocks enter the model or how they might be correlated with shocks to other variables. Since Etyt+1 is constant, recognize from (3b) that the only variable entering the forcing process is ht. This equation can be solved using the method of linear undetermined coefficients by “guessing” that the solution is linear in ht, substituting the trial solution into the equilibrium condition and treating the resulting equation as an identity. In particular, consider the trial solution

(5)zt = λ10 + λ11ht.

The standard undetermined-coefficients algorithm reveals that (5) is a solution only for

(5a)λ10 = β1β{θ - βh1βø}
(5b)λ11 = -β1βø,

where θ ≡ (α - y)y > 0. Now divide each side of (5) by (α - yt) to obtain:

(6)qt = λ10 + λ11htα - yt

In this example, two things are responsible for the stochastic behavior of the price of equity, qt. First, real output disturbances will influence y and thereby u’ (yt) = u’(ct). The influence of current output shocks therefore shows up in the denominator of (6) with an increase in yt (through an increase in wt) increasing qt. Second, new information altering agents’ beliefs about the variance of future real output shocks shows up in ht. Information that makes agents think that future output variance will be higher, will lower qt at the rate λ11/u’(ct). Notice that depends on ø which is one of the parameters in the ht time series process. If ø is large then a shock to ht is relatively persistent and has a large effect on qt as compared to a case where ø is small and shocks to ht are transitory. Notice that the framework can handle very persistent shocks to ht in that it can handle any ø such that |ø|< 1/β.

Example 2: Arch

The methods of Engle (1982) and Bollerslev and Engel (1986) suggest an alternative formulation of the behavior of the conditional variance, ht. Retain equations (3b) and (4a), the equilibrium Euler equation and the yt process respectively, but alter the structure of the conditional variance equation to:

(7)ht = h + δwt2.

This example differs from the previous one in that output shocks, wt, now alter agents’ beliefs about the distribution of future output shocks. If an output shock is large (in absolute value) the agents will raise their current beliefs about the variance of future output shocks. Here, make use of the fact ht = Et [wt+12] and recognize that (7) is structurally very similar to (4b). Consequently the trial solution for zt retains its previous functional form but with δwt2 replacing øht. In particular:

(8)zt = λ20 + λ21wt2,

where

(8a)λ20 = λ10(given in (5a)),
(8b)λ21 = βδ1βδ

and

(8c)qt = λ20 - λ21wt2α - (y+tw)t

In this example, the effect of a positive output shock, wt, on equity price, qt (compared to wt = 0) depends on specific parameter magnitudes since positive wt both decreases current marginal utility of consumption acting toward increasing equity price and increases agents perception of future variance acting toward decreasing equity price. The net result will depend on specific parameters.

A negative wt, on the other hand, lowers qt (compared to wt = 0). 6/ This example presents an interesting possible asymmetry in the reaction of equity price to positive versus negative real shocks.

III. Government Policy and Asset Prices

In this section the model developed above will be used to study two issues. First, how does government expenditure policy influence the way changes in an the economy’s stochastic environment impact the asset market? Second, how does the asset market react to entering a period of likely fiscal reform? The concept of likely fiscal reform is characterized here by an upward shift in the covariance of innovations in government spending and innovations in the income tax rate. This definition of fiscal reform does not take a stand on the direction of movement of tax rates or government expenditure; it considers only the association between government spending changes and tax rate changes. While the questions studied in this section have some independent interest they were chosen to illustrate some aspects of the model. The first question is chosen to illustrate the asset pricing effect of the interaction of first moment and variance processes and the second is chosen to illustrate the adaptability of this kind of model to studying time series processes involving the covariance.

1. Introducing the government

Introducing the government requires that the budget constraints and equilibrium conditions be modified. It is assumed that the government purchases goods in the goods market, collects a proportional income tax and collects a lump-sum tax. The government’s budget constraint: is:

(9)gt = τtyt + Tt

where gt is per-capita government consumption τt is the income tax rate and Tt is the per-capita lump-sum tax. The after-income-tax individual budget constraint is now:

(9a)ct + qtkt + Tt = (qt + (1τt)dt)kt1.

Goods market equilibrium requires:

(9b)ct + gt = yt

For the issues studied in this section it is assumed that government purchases do not influence the marginal utility of private consumption. The Euler equation for equities, written for equilibrium is

(9c)zt = βEtzt+1 + βEt{(α - yt+1 + gt+1)(1τt+1)yt+1},

where zt is the product of the market price of equity and marginal utility of consumption.

In writing (9c), equilibrium average consumption, ct = yt - gt, has been used in place of ct and private after-tax income becomes the pay-out to equity holders. Lump-sum taxes are present only as a government-budget balancing item.

2. Government feedback policy, volatility, and the equity market

The question to be investigated here is: “How does government expenditure policy influence the way in which volatility impacts the equity market?” To investigate this question, we adopt the following settings.

(10a)yt = y + ρ1yt1 + wt  |ρ1|<1
(10b)Etht+1 = øht + h
(10c)gt = g + ρ2gt1 + ηyt + ϵt,|η|<1,|ρ2|<1.

Further, to focus on expenditure policy set τ rt = 0 all t. The important difference between the set up currently and that used previously is that now a government expenditure policy rule has been adopted. According to this rule, equation (10c), government spending is determined by a constant, past government spending, the amount of current output and a white noise disturbance, εt.

The short cut of examining the influence of policy on zt rather than on qt is adopted. As before, the transformation between zt and qt can be made by dividing zt by marginal utility. Keep in mind that by concentrating on zt we are looking only into the future. This is reflected in equation (9c) by the fact that the entire right-hand-side of the zt equation gives zt as the current expectation of future events.

Following the methods of the previous section, substitute (10a), (10b), and (10c) into (9c) with τt+1 = 0 and find that the forcing process (for zt) is linear in yt, ht, gt, and gt yt. The following trial solution is therefore attempted:

(12)zt = λ40 + λ41yt + λ42yt2 + λ43ht + λ44gt + λ45gtyt,

which yields:

(12a)λ42=[βρ12(1-βρ12)1(ηλ45+η-1)
(12b)λ43 = [ρ12(1 - βø)]1λ42
(12c)λ44 = yβρ2[(1 - ρ1ρ2)(1βρ22)]1
(12d)λ45 = βρ1ρ2(1βρ1ρ2)1

The coefficients λ40 and λ41are reported in the appendix.

The equity-market impact of ht is given by λ43, which is proportional to λ42, the coefficient attached to yt2. This occurs because ht enters the problem only through agents’ forecasts of future squared output.

Government expenditure policy can influence the asset market impact of changes in risk. In particular, the asset market impact of risk is proportional to λ45 + η - 1]. The size and sign of this term depends on η, which can assume any value between -1 and 1. 7/ When η < 0 government spending policy is counter-cyclical and the spending feedback exacerbates the effect of risk on the market. When n > 0 government spending is pro-cyclical and dampens the effect of risk on the market. What is going on is that counter-cyclical government spending policy, η < 0, induces a negative covariance of gt and yt. A larger absolute negative covariance (as would be induced by a positive ht shock) acts toward lowering the marginal utility of future dividends. This acts in conjunction with the standard effect of ht to give larger variations in the equity market as a result of ht disturbances. On the other hand, when η > 0 the above covariance is positive and its effect is to act toward offsetting ht ’s standard effect.

The effect isolated here is conceptually distinct from the way the equity market reacts to an output shock. In particular, λ43 gives the partial derivative of zt with respect to ht. It is therefore answering a question about the equity market impact of a risk shock as distinct from an output shock.

3. Fiscal reform and the equity market

The question to be studied here is: “What is the equity market effect of entering a period of likely fiscal reform?” To confront the question with our tools, “fiscal reform” must be a recurring event with agents understanding the time series process of fiscal reforms. We can then come-to-grips with the narrower question: “What is the equity market effect of a disturbance to the “fiscal reform” time series process?” The following settings are adopted:

(13a)yt = y,
(13b)gt = g + vt,
(13c)τt = τ + xt,
(13d)Etxt+1vt+1δt,
(13e)Etδt+1 = øδt + δ.

In equation (13a), output has been set at a constant allowing us to concentrate attention on fiscal policy disturbances. In (13b) and (13c) government spending and the tax rate respectively are assumed to follow stochastic processes around fixed means. The disturbance terms in (13b) and (13c), vt and xt, are white noises. The conditional covariance of vt and xt is given by δt-1 and the conditional expectation of the time-series process of this covariance is given by equation (13e). The conditional covariance, δt, is playing the same type of role played previously by ht.

A period of “fiscal reform” is defined presently to be a period when δt is high. For an economy to move into a surprising period of fiscal reform is to have the economy receive a high δt. The fiscal reform model used here is cynical in that nothing ever happens to the means of per-capita government spending and the income tax rate. This is intended only as an analytical convenience allowing separation of the riskiness of a fiscal reform period from the type of reform expected.

Notice that the present idea of fiscal reform deals only with the coordination of tax and spending policies. It is eclectic with respect to the direction of change of both spending and the tax rate. A period of fiscal reform is therefore a period when the tax rate and per-capita spending are likely to move in the same direction.

For this issue we follow previous practice by substituting from equations (13a)-(13e) into equation (10c) and noticing that the forcing process (for zt) is linear in δt. The trial solution therefore is:

(16)zt = λ50 + λ51δt,

where

(16a)λ51 = -βy1βø,

The constant term, λ50 is given in the appendix. Since λ51 < 0, we find that entering a period of likely fiscal reform, i.e., entering a period of close coordination of tax and spending policy, decreases zt. Conversely, entering a period of weak coordination of tax rates and per capita spending increases zt.

The intuition for this result is that (roughly) things that are good for zt are things that increase the expected future marginal utility of after tax dividends. The tax rate enters after tax dividends negatively while government spending enters marginal utility positively. The covariance of the tax rate and spending therefore enters negatively.

IV. Concluding Remarks

The purpose of this paper was to exposit and apply in a simple and familiar framework some ideas developed recently by Abel. This line of research allows us for the first time to ask in a consistent way questions concerning agents’ reactions to predictable and unpredictable alterations in the riskiness of their environment. The framework seems to have wide potential applicability to problems such as options pricing and optimal government stabilization policy where the nature of and reactions to time-varying risk are at the center of the problem.

APPENDIX I Text Coefficients

This appendix records several coefficients mentioned in the text.

(A1.1)λ40 = (1β)1{β[λ40 + λ41y + λ42y2 + λ43(1ø)h + αy - y2 + ηy2]}+λ44 g + λ45(ηy2 + yg)
A1.2λ41 = β(1ρ1β)1{(λ422ρ1 + λ45[η + ηρ1]2ρ1 + η + ρ1η)y+λ44ηρ1 + (λ55ρ1 + ρ1)g}
(A1.3)λ50 = (1β)1[βλ51δ + βy(α - y - g)(1 - τ)]

APPENDIX II Some Aspects of Nominal Asset Pricing

The model developed in the text was designed to allow pricing of assets in real terms in a hypothetical world that functions without money. One is interested, however, in nominal asset pricing since nominal units are the typical units of real-world asset pricing. While it is outside the direct realm of this paper, this appendix gives an example of asset pricing in a nominal environment where risk may be time-varying.

Money is introduced via the simplest mechanism available in the current literature, strong cash-in-advance. In this monetary introduction scheme, also known as a Clower constraint, it is assumed that agents must use money to buy goods. It is also assumed that when the money is acquired by agents the agents know exactly the quantity of goods that they will purchase with the money. Later a somewhat looser version of a Clower-constrained model will be discussed.

Accommodating money in our model requires that we reset budget constraints. In particular, it is assumed that all money enters (leaves) the model via lump-sum transfers (taxes) from (to) the government. The government, which takes no role in this economy other than provider of money, has a budget constraint given by PtTt = Mt - Mt-1, where Pt is the money price of goods and Mt is the per-capita money stock.

The private individual now must obey a pair of constraints, the first constraining his purchases of asset and the second constraining his purchases of goods. This is natural in a cash-in-advance economy since only goods carry the special condition that they must be purchased with money. The constraints are:

(A2.1)Mt + Ptqtkt + Bt = Pt(qt + dt)kt1 + (1+it1) Bt1 + PtTt,
(A2.2)Mt = Ptct,

with (A2.1) and (A2.2) constraining asset and goods purchases respectively. Previously defined symbols retain their meaning. Bt is per-capita nominal bonds and it is the nominal rate of interest. The Euler equation for bond holding is

(A2.3)u(ct)/Pt = β(1+it)Et[u(ct+1)/Pt+1].

Imposing ct = yt and Mt = Ptyt and rearranging obtain

(A2.4)(1+i)1 = βEt(gt+1nt+1),

where gt+1 ≡ u’(yt+1)yt+1/u’ (yt)yt and nt+1 = Mt/Mt+1. Rewrite (A2.4) as

(A2.5)(1+it)1 = βEtgt+1Etnt+1 + ωt,

where ωt is the covariance of gt+1, and nt+1 conditional on time t information.

Leaving aside ωt for now, we see that the moments of yt+1 will enter Etgt+1 as directed by utility function choice. If quadratic utility is chosen, the first two moments of the yt+1 distribution will be important in determining the nominal interest rate. For other utility function choices other moments will enter the pricing function.

To the extent that money is exogenous to the private sector we have much less economic guidance when calculating Etnt+1. The problem is that we have no way of knowing offhand what mathematical function of money is being given a distribution function by the policy authorities. For example, suppose policy authorities choose a rule like Mt = M + ut where ut is white and has some known distribution. Since agents need to calculate Etnt+1 they must calculate MtEt(1/Mt+1) and if a distribution is given for ut+1 then aspects of that distribution function must be invoked in the calculation and, in general, the calculation will be a function of the entire shape of the distribution function.

For the above reason, moments of the money supply rule other than those typically considered important for nominal interest rates might enter a reduced-form interest rate function. Of course there is no good reason for the authorities to adopt a distribution pertaining to the level of money. The monetary authority could make agents’ calculations a lot easier if they were to adopt a rule on gt+1, e.g., gt+1 = g + wt+1, where wt+1 is white noise. For a monetary authority rule on gt+1 no moments of wt+1’s distribution other than the mean, zero, would enter the nominal interest rate reduced form.

The reader may have noticed that the above methods for introducing unusual moments of the monetary disturbance into the interest rate reduced form are essentially unrelated to the asset pricing framework. These moments are being introduced by the model’s requiring agents to form expectations of nonlinear functions of the policy variables. For the same reasons these moments could turn up in standard descriptive macro-models. Suppose, for example, that agents are following a log-linear macro-model. The solution of the model (e.g., for the interest rate) might require the model’s agents to form expectations of future logs of money. If the policy authority is following money-supply a rule on the level of money with an additive shock then descriptions of the distribution of that shock (possibly including higher moments of the distribution) will appear in the reduced form expressions for the endogenous variables.

A much more interesting area of study is suggested by the term in ω1(A2.5). Through this term the covariance of a monetary variable, nt+1, and a real variable, gt+1, can influence the nominal rate of interest. Similar covariance terms have been an important topic in macroeconomics since its inception.

The model used in this appendix was adopted for ease of discussion. Essentially the same points apply in the cash-in-advance models of the type used by Hodrick (1987) and Giovannini (1987) where money must be gathered before the resolution of other aspects of the period’s uncertainty. This alteration in the timing of the resolution of uncertainty simply alters the timing of variables entering the interest rate reduced form.

APPENDIX III Heterogenous Agent. Multi-Country Reinterpretation

The analysis in the text was based on a representative-agent closed economy model to facilitate comparisons with other work in that setting. The purpose of this appendix is to show that the model we have used has a much broader interpretation than we have placed on it. Indeed we will find that once quadratic period utility has been assumed, aggregation in a world of heterogenous agents is straightforward.

In a world where agents from different countries are endowed with different wealth levels and have different period utility functions but where each agent has quadratic period utility, each agent faces the following problem:

max EtΣi=0(αj - [1/2]{ct+ij}2βi), 0 <β<1

subject to:

ct+ij + qt+ikt+i - 1j = qt+ikt+i - 1j + dt+ikt+i1j.

The symbols in the above problem retain most of their previous meaning, but now symbols with a j superscript apply only to individual j, where j = 1,2,3,...J.

It is not necessary to make any special assumptions about the country of location of the individuals for asset pricing. 8/ However, certain assumptions are required for simple aggregation. First, all agents use the same expectation operator; information is therefore homogenous across agents. Second, the subjective discount rate, β, is identical for all agents. Third, all agents face identical prices for goods and assets.

The parameters in the period utility function and individual wealths can differ across individuals. As before, however, constant terms in the period utility functions have been ignored and each individual’s period utility function has been normalized by dividing the function by 1/2 of the parameter attached to(ctj)2.. Individual period-utility-function differences are therefore contained entirely in the parameters αj.

Each individual’s optimization problem will imply an Euler equation like equation (1) in the text. Average these equation across individuals to obtain:

(A3.1)(Σi=1J[αj - ctj]/J)qt = Etβ(Σi = 1J[αj - ct+1j]/J)(qt+1 + dt+1).

Retaining the assumption of no government spending or taxation, equilibrium requires

(1/J)Σj=1J ctj = yt.

If all output is paid out as dividends we obtain an Euler equation very similar to (3a) in the text. The only difference between (A3.1) and (3a) is that in (A3.1), the Euler equation parameter derived from the period utility functions is the average across j of the constant terms in individual period marginal utility while in α(3a) is the representative agent’s constant term in period marginal utility.

This implies that the results in the text are not very sensitive to the closed economy representative agent assumption. They would, however, be sensitive to different discount rates across agents or agents facing different effective prices as would happen in a multi-country setting with goods or asset market distortions.

References

  • Abel, Andrew B., “Stock Prices Under Time-Varying Dividend Risk: in Exact Solution in an Infinite Horizon General Equilibrium Model,” mimeo, The Wharton School (University of Pennsylvania, Philadelphia, 1986).

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1/

The author is on leave from Northwestern University.

2/

Others who have adopted and extended the Able model include Giovannini (1987) and Hodrick (1987) both of whom include money in their models and can therefore price assets in nominal terms. In Appendix II, some aspects of nominal asset pricing are discussed.

3/

Quadratic utility is objectionable to some--apparently on aesthetic grounds. It is sometimes mentioned that any fixed-parameter quadratic utility function will eventually be “out grown” by an economy growing without bound. Another way of phrasing this objection is to say that attention should be restricted to model environments where per-capita consumption can grow without bound. This seems to me to be a singularly unworldly restriction.

Another line of attack on quadratic utility is that other utility functions such as constant relative (or constant absolute) risk aversion seem a priori “more plausible”. Such a point is, of course, unarguable. One can argue, however, that since empirical work on the stock market (e.g., Flood, Hodrick, and Kaplan (1987)) rejects many popular period utility functions including constant relative risk aversion, constant absolute risk aversion, risk neutrality and quadratic utility there is no way to choose among the popular forms on the basis of conformity with the data. Consequently, for studies like the current one it seems sensible to adopt the utility function which makes exposition as transparent as possible--quadratic utility.

One objection to quadratic period utility which should not be dismissed is that quadratic period utility implies risk neutrality with respect to the riskiness of the environment. As was mentioned in the text, this is the technical source of this paper’s simplicity.

4/

One way to keep variables positive is to introduce stochastic processes of the form:xt+1 = x + vt+1xt,vt+10.

where vt+1 is a positive stochastic coefficient with Et vt+1 = v and xt and is any variable in the model. If written in the standard linear form, the additive disturbance becomes proportional to xt. It is convenient to think about many of the variance and covariance processes in the paper in these terms. Further, when stochastic processes are assumed for variables like yt, one can think of the disturbances as being generated as above.

The requirement zt > 0 implies that ht is bounded above as well as below would be necessary, therefore, for the variance process, to impose conditions such as 0 < v < 1 with a moving upper bound on vtt+1 such that ht+1 meets the variance upper bound. Phillipe Weil pointed out to me the upper bound implied by the condition zt > 0.

In applied work one would work with noncentral higher moments (here the second moment) rather than with variances and covariances so that variance conditions could be ignored. On using this type of model in practice (see footnote 1, page 7).

5/

What is appearing here is not a dynamically-based indeterminacy similar to “bubbles” or “sunspots.” What is going on is the required modeling (and identification and estimation in applied work) of an unfamiliar (possibly nonlinear) forcing process. In fact, the nonlinear rational expectations model in this paper is even more tightly constrained than the typical linear rational expectations model in the sense that more exogenous variable equations need to be estimated along with the pricing equation than in the linear case for fully efficient estimation.

The choice of exogenous variable processes used in the examples was guided by the goal of exposition--not estimation--and may not be a good approach in applied work. For instance, to estimate parameters in example 3 (below in the text), I would estimate simultaneously: (i) equation (10a)--the reduced-form pricing function; (ii) equation (9a)--the y process; and (iii) a process on y. The y process will place additional overidentifying restrictions on the model (compared to estimating only the reduced-form pricing function and a yt2 process). From discussions with Robert Hodrick, and Adrian Pagan it seems to me that it might be a good idea in most applied work in this area to keep the estimated model in terms of yt and yt2 (or their analogs) as in equation (3a) rather than invoking the decomposition allowing the transition to equation (3b). It would be simple in example 3 to reformulate the pricing function to exclude ht and put the agents’ forecasting problem and the reduced-form pricing function in terms of a yt2 forcing process in addition to the y forcing process. The term ht2 was used in the text for ease of exposition and could be reintroduced for experiments once one has obtained consistent estimates of the behavioral parameters.

In practice, ht is difficult to model because the researcher has no direct observations on ht and because constructing ht requires that the researcher correctly model the agents’ Etyt+1. This point was made by Pagan (1984) and is very closely related to Flavin’s (1983) critique of some of the empirical stock-price variance bounds literature. At the time of Flavin’s paper it was pointed out to me by Richard Porter that the variance bounds literature could be consistently recast in terms of first and second moments and that message seems to extend here.

6/

Examining equation (6), it is clear that while the present approach is helpful for examining the impact of underlying risk shifts on asset price, it is no big help in examining the effects of underlying volatility shifts on asset-price volatility. The problem is that to convert to asset price one must divide zt by the marginal utility of consumption. Since this marginal utility will in equilibrium involve the level of output, the conversion to asset price from zt involves dividing by a linear function of a stochastic disturbance. Asset price is therefore, in this set up, a nonlinear function of the underlying disturbances. Studying the volatility of asset price usually requires that the expectation of asset price be calculated. Such calculations for nonlinear functions are distribution-function specific and will be avoided here.

7/

A condition that an increase in long-run output of one unit should bring about a less than unitary increase in long-run government spending,

η/(1 - ρ1) < 1, would ensure λ43< 0. It is not clear, however, why in this model such a condition should be imposed since long-run output is not time-varying.

8/

Of course locating agents by country would be crucial for country-bookeeping such as current account, capital accounts, etc.