RISK PREFERENCES IN THE ASSET PRICING MODEL1

September 1999

Donald J. Meyer Jack MeyerDepartment of Economics Department of EconomicsWestern Michigan University Michigan State UniversityKalamazoo, MI 49008 East Lansing, MI 48824donald.meyer@wmich.edu jmeyer@pilot.msu.edu

Abstract: The asset pricing model (APM) assumes that wealth is allocated among a portfolio of assets and that the returns from those assets are consumed over time. The representative consumer maximizes expected utility from consumption when making these decisions. In the APM there are two related but different utility functions. One represents direct utility from consumption, and the second gives the indirect utility from wealth. This research focuses on the relationship between the risk aversion properties of these two utility functions. It is argued that the indirect utility for wealth has received insufficient attention, and in fact, its properties should be an important consideration when choosing the form for utility from consumption. This theoretical discussion is supported with empirical analysis which shows that the equity premium and risk free rate puzzle need not arise when the utility function from consumption is chosen so that utility from wealth displays reasonable levels of constant relative risk aversion.

1. Introduction

Twenty years ago, Lucas [1978] included both portfolio selection and multiperiod

consumption decisions in a model which has been extensively used since that time to determine

equilibrium prices for financial assets. In this asset pricing model (APM), it is assumed that

wealth is allocated across a portfolio of assets whose returns are then consumed over time. The

consumer is represented as maximizing expected utility from consumption, where utility is

additive separable. Equilibrium conditions in the model reflect both the portfolio and

consumption decisions. Those reflecting optimal portfolio composition ensure that decreasing

the holding of one asset and increasing the holding of another does not increase expected utility.

Optimal time allocation requires that expected marginal utility from consumption be equalized

across time periods.

The first segment of this paper focuses on theoretical discussion of risk preferences

within the APM multi-period consumption framework. In multi-period consumption models of

this sort there are two related but different utility functions that one can examine, and in general,

these functions display different risk aversion characteristics. The first and most obvious such

function is the direct utility function from consumption in each period, denoted v(C). The

second utility function is denoted u(W), and is the maximum utility that can be obtained from a

given wealth level. It is derived under the assumption that wealth is allocated optimally among

assets, and that consumption is chosen optimally as well.2 Of course, since u(W) is derived from

v(C), the two functions are related, but as is demonstrated below, their functional forms and risk

taking properties generally are not the same.

When evaluating or analyzing the portfolio decisions of the consumer, u(W) rather than

v(C) is the utility function whose characteristics best reflect the consumer's attitude toward the

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risk involved. There are two reasons for this. First, the properties of u(W) are relevant because

each period's portfolio decision maximizes expected indirect utility from next period's wealth.

That is, even though the consumer chooses a portfolio to maximize expected utility from future

consumption, it is necessarily the case that expected indirect utility from next period's wealth is

also maximized. Second, the well-known Pratt [1964] and Arrow [1971] definitions of relative

and absolute risk aversion are formulated for utility from wealth. This has caused much of the

theoretical and empirical analysis of portfolio decisions in the past thirty five years to deal with

the risk aversion properties for u(W). Hence, this research examines the risk taking

characteristics of the consumer in the APM by looking at u(W) rather than v(C).

The second segment of this research supports this theoretical argument by showing that

choosing v(C) so that u(W) displays constant relative risk aversion (CRRA), can eliminate a well

known and important paradox in the APM literature, the equity premium and risk free rate

puzzle. The equity premium and risk free rate puzzle results from the fact that risk preferences

and time preferences are not specified independently in this model. Mehra and Prescott [1985]

exploit this characteristic, and show that the observed returns to risky and riskless assets are

inconsistent with the equilibrium conditions of the APM. The risk aversion level necessary to

explain the premium earned from assuming risk is in conflict with the rate of time preference

needed to explain the risk free return and the growth of consumption over time. In addition,

Mehra and Prescott note that the risk aversion level needed to explain the observed risk premium

seems to be unrealistically large. The risk aversion properties that Mehra and Prescott evaluate

and reject are those for utility from consumption, v(C).

This violation of the equilibrium conditions of the APM is referred to as the equity

premium and risk free rate puzzle, and has generated many attempts at resolution during the last

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fifteen years. A recent and excellent review of this large body of work by Kocherlakota [1996]

concludes that the puzzle is very robust, that the phenomenon is widespread, even across

countries, and that it has persisted over time.

The empirical analysis here demonstrates that for a v(C) chosen so that u(W) displays

CRRA, the equity premium and risk free rate puzzle can be made to disappear. A single

parameter can represent both risk and time preferences without implying that the observed

values for the risk premium and risk free rate are inconsistent with the equilibrium conditions of

the APM. Moreover, the relative risk aversion level for u(W) that is required for this

consistency is much smaller than that determined by Mehra and Prescott for v(C).

This claim is supported in two ways. First and primarily, the Kocherlakota empirical

analysis is replicated with the different functional form for v(C). A range of relative risk

aversion levels for the implied u(W) is found to be consistent with (to not reject) the asset

pricing equations, and the lowest magnitudes not leading to rejection are near one. Second and

more indirectly, the claim is also supported by observing that the form for v(C) in the habit

formation model of Constantinides [1990] is very similar to that which leads to a u(W)

displaying CRRA in this model. Thus, the evidence that the form for v(C) under habit formation

can resolve the equity premium puzzle is also indirect evidence for the risk preference

specification proposed here. Conversely, and perhaps more importantly, this research provides

an alternate explanation and justification for that functional form for v(C).

The paper is organized as follows. In the next section, the theory supporting the thesis of

the research is presented. First, the general relationship between v(C) and u(W) is analyzed

without specifying the form for utility or explicitly solving for the optimal portfolio or

consumption levels. That analysis allows several general statements to be made concerning the

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relationship between the risk taking properties of these two utility functions. The envelope

theorem is used to show that, in general, u(W) is less risk averse than v(C). It is also used to

derive expressions relating the absolute and relative risk aversion measures for these two utility

functions.3 How these risk aversion measures compare with one another depends on the

marginal and average propensities to consume from wealth.

The exact risk taking properties for v(C) and u(W) cannot be determined until a form for

v(C) is specified and optimal consumption and portfolio composition are determined for that

consumer. This calculation is often difficult to carry out, but it is presented for one particular

utility function. This is accomplished in a multi-period consumption model similar to that used

by Kimball and Mankiw [1989]. Kimball and Mankiw's model generalizes that of Lucas by

considering sources for consumption in addition to returns from assets. Such things as labor

income and government transfers are included in the budget constraint of the consumer, but

these payments are not capitalized and included with wealth. The form for v(C) that is used in

the analysis is shown to imply the CRRA form for u(W). This v(C) function itself does not

display constant relative risk aversion.

Sections 3 and 4 present empirical evidence concerning the impact of the theoretical

discussion. Using the APM and utility function from section 2, the equity premium and risk free

rate puzzle is reexamined. First, in section 3, one of the many statements of the paradox, that

given by Kocherlakota in his recent review of the literature, is presented. This analysis is

described so that it can be replicated in section 4 for the different functional form for v(C). The

replication confirms the fact that the equity premium and risk free rate puzzle need not arise,

even with a single risk aversion and time preference parameter, when v(C) takes the particular

form chosen to give CRRA risk preferences for u(W). Moreover, the risk aversion level for

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u(W) need not be unreasonably large.

Section 5 provides limited discussion of empirical evidence concerning the relationship

between consumption and wealth. In addition, the utility function for consumption resulting

from habit formation is pointed out as being similar to that leading to CRRA preferences for

wealth in the Kimball and Mankiw model. Finally, the section concludes with discussion of

extensions of this research and mentions questions generated by the analysis.

2. Risk Preferences in the APM

The asset pricing model employed here is an n-period consumption model with saving,

and is patterned after one used by Kimball and Mankiw [1989] to explain the effects of taxes on

saving. In this model, the consumer begins with an initial wealth, and in addition, receives

nonrandom income, y, in each period.4 This income can be thought of as labor income,

government transfers, or any other payment which occurs regularly, and whose future stream of

payments cannot be capitalized and purchased or sold as an asset.5 In each time period, the

consumer chooses two things, the amount to consume, and the portfolio of assets in which to

invest the wealth that is saved for future consumption. Consumption in each period can be no

larger than beginning of period wealth plus that period's income.

To simplify the notation, the consumer's portfolio is assumed to contain only two assets.

They are called stocks and bonds, and their returns are random and denoted Rs and Rb,

respectively. Future consumption levels are not known for certain because of the randomness of

the return on assets. The objective of the consumer is the maximization of expected utility from

consumption of the additive separable form with discounting. Thus, the consumer's goal is to

choose consumption levels to maximize EU = i Ev(Ci).

The presentation of the solution to this maximization problem proceeds at two levels.

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First, the conditions defining the expected utility maximizing value for consumption and asset

allocation for an arbitrary utility function v(C) are given. These conditions are then used to

determine several general relationships between the risk taking properties of v(C) and u(W), and

also to show why the risk taking properties of u(W) are relevant. Following this general

discussion, the second level of the analysis solves the APM for a particular v(C).

The optimal values for consumption and asset allocation in the last two time periods

depend on the level of wealth at the beginning of those periods. As usual, substituting these

optimal values into the utility function being maximized gives an identity defining indirect

utility from this initial wealth. Assume that there are two time periods remaining in which to

consume, and that wealth Wn-1 is available at the beginning of the first of these two periods.

Nonrandom income y occurs in each period. The general form for expected utility from

consumption in the last two periods is (n-1)[v(Cn-1) + Ev(Cn)], where Cn = (Wn-1 + y - Cn-1) [n-

1Rs+(1-n-1)Rb] + y. The consumer chooses Cn-1 and n-1 to maximize this sum of expected

utilities from consumption.

The optimal levels of Cn-1 and n-1 satisfy two first order conditions that reduce to the

following familiar expressions. First is Ev(Cn)(Rs - Rb) = 0, indicating portfolio equilibrium for

these two assets, and second is v(Cn-1) = Ev(Cn) [n-1Rs+(1-n-1)Rb], representing optimal

allocation of consumption across the two time periods. Utility from Wn-1 is the utility obtained

when the consumer has chosen the optimal values for Cn-1 and n-1. Writing this out formally,

un-1(Wn-1) = (n-1)[v(Cn-1(Wn-1)) + E[v((Wn-1+ y - Cn-1(Wn-1))(n-1(Wn-1)Rs+(1- n-1(Wn-1))Rb) + y)],

where Cn-1(Wn-1) and n-1(Wn-1) are the optimal values for these decision variables. This equation

is an identity holding for all Wn-1. The envelope theorem6, or direct derivation and use of the

two first order conditions, shows that the two utility functions un-1(W) and v(C) have a common

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slope at the optimum; that is un-1(Wn-1) = v(Cn-1(Wn-1)). In fact, with this additive form for

utility, this same relationship between marginal utilities, ut(Wt) = v(Ct(Wt)), holds as an

identity in each time period no matter how many time periods remain.

This identity relating marginal utilities can be used to calculate the relationship between

second and higher derivatives for the functions ut(Wt) and v(Ct). To reduce cumbersome

notation, the t subscripts are dropped until the model is solved for a specific utility function.7

Differentiating the identity with respect to W yields u(W) = v(C)[dC/dW]. Next, let

Au(W) = -u(W)/u(W)and Ru(W) = Au(W)W denote the Pratt-Arrow absolute and relative risk

aversion measures for u(W), respectively. Similarly, let Av(C) and Rv(C) denote the same ratio

of derivatives for utility function v(C). Then the following is trivial to establish and relates what

are frequently referred to as risk aversion measures for these two utility functions.

Proposition 1: Au(W) = Av(C)[dC/dW] and Ru(W) = Rv(C)[dC/dW][W/C].

Notice that when dC/dW is less than one, as is the case for all concave v(C) and finite ,

then u(W) is less risk averse than v(C). This reduced concavity or increased convexity of the

"envelope function" is the usual consequence of maximization. It is also the case that u(W)

displays risk aversion, Au (W) > 0, whenever v(C) is concave. This fact is easily demonstrated,

since for the additive separable form for utility, dC/dW is always positive. Hence, in the APM,

u(W) displays less risk aversion than v(C), but remains risk averse as long as v(C) is concave.

The relationship between absolute risk aversion measures for u(W) and v(C) given in

Proposition 1 indicates that each is a constant whenever a constant absolute risk averse v(C)

implies a linear consumption and wealth relationship. Kimball and Mankiw demonstrate that

this is indeed the case for their model. They do use the constant absolute risk averse form for

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utility in their analysis. For portfolio decisions in general, and for the APM specifically,

however, the assumption of constant absolute risk aversion, is not a very acceptable one.

Instead, decreasing absolute risk aversion, in the form of constant relative risk aversion (CRRA),

is often assumed.8

Proposition 1 indicates that both Rv(C) and Ru(W) are constant only when optimal

consumption is such that [dC/dW][W/C] is a constant. Indirect utility for wealth, u(W),

however, can also display CRRA if the slopes of [dC/dW][W/C] and Rv(C) are offsetting. Such

is the case for the specific utility function analyzed below. Also, note that the critical expression

relating relative risk aversion measures, [dC/dW][W/C], is the ratio of the marginal (MPC) and

average (APC) propensities to consume from saving or wealth. One line of possible research,

not explored in any detail here, is to gather evidence concerning the MPC and APC for the

representative consumer. Such evidence would allow the risk taking properties of these two

utility functions to be compared without ever facing the difficult task of solving the details of the

APM.9

The feature of this APM that explains why the risk aversion properties of u(W) are

relevant is the fact that expected utility from next period's wealth, by its definition, equals the

expected utility from future consumption beginning next period. The single term, utility from

wealth, equals the sum of the several utility from consumption terms. Hence, the portfolio

decision made in any period, which is assumed to maximize expected utility from future

consumption, also maximizes expected (indirect) utility from next period's wealth. That is,

when evaluating portfolio alternatives, the consumer's explicit goal is exactly the same as

maximizing expected utility from next period's wealth. Consequently, the risk aversion measure

for u(W) is a relevant and simple measure to use when evaluating or analyzing the consumer's

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portfolio decisions. This important point is summarized in the following proposition.

Proposition 2: In the n-period APM, the optimal portfolio in any period (t) maximizes expected

(indirect) utility for wealth in period (t+1).

Finally, the detailed n-period analysis is carried out for a specific utility function.

1 The authors thank Larry Martin, Jonathan Parker and Robert Rasche for helpful comments and discussion.

2 Dreze and Modigliani [1972] and Mossin [1969] are among those to have discussed this function in earlier research concerning multi-period consumption models. This function is sometimes referred to as the value function.

3 Both Arrow [1971] and Pratt [1964] define absolute and relative risk aversion measures as properties of utility from wealth. The profession seems to have extended these definitions to apply to all utility functions no matter what their argument, including utility from consumption. We reluctantly follow this broader use of the terminology, but recognize that this can lead to confusion, including that pointed out in this paper.

4 The Kimball and Mankiw model is a continuous time model. Also, in their formulation, y is random, and return on saving or wealth is not random. For the portfolio decision that is the basis for asset pricing, returns must be random. Therefore, to keep the model tractable, we reverse this assuming that y is not random and that return to wealth is random.

5 This assumes a lack of complete markets in that consumers are unable to capitalize these income payments. We thank Jonathan Parker for pointing this out.

6 Varian [1992, p. 490] discusses the envelope theorem and the properties of value functions.

7 It is important to recognize that utility from wealth, ut(Wt) is not necessarily the same in each time period. For the specific utility function used in this analysis it is, however.

8 One unacceptable feature of constant absolute risk aversion is that it implies that the amount of the risky asset held in a portfolio is independent of the size of the portfolio. Kimball and Mankiw do not deal with portfolio issues in their analysis so this is less important in their research.

9 For example, suppose consumption is estimated to be linear in wealth and given by C = a + bW. Also assume that a > 0 and b < 1 as would be the case if consumption is smoothed or less variable than wealth. In this instance, the ratio of the marginal to average propensity to consume is bW/(a + bW) which is less than one and is an increasing function of W. Thus, for this consumption and wealth relationship, u(W) displays a lower level of relative risk aversion than

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Assume that v(C) takes the form v(C) = (C-y)(1-)/(1-). For this function, the analysis below

shows that indirect utility from wealth in each of the n periods is of the constant relative risk

averse form, with as the relative risk aversion measure.10 For this utility function, the

conditions defining the optimal n-1 and Cn-1 are: E(Wn-1 + y - Cn-1)- (Rs - Rb) = 0 and

(Cn-1 - y)- = E(Wn-1 + y - Cn-1)- [n-1Rs+(1-n-1)Rb] (1-). The notation is simplified when

c = (Cn-1 - y) is substituted, and since y is exogenously given, solving for c is sufficient.

First, c is determined as a function of the portfolio selected. Let the return on the

portfolio be denoted R = [n-1Rs+(1-n-1)Rb]. With this notational simplification, the relevant

equation to solve becomes (c)- = E(W - c)- [R] (1-). In this equation only the return variable is

random so the equation simplifies to (c)- = (W - c)- E[R (1-)], and solving yields

c = [[E(R1-)]-1//(1 + [E(R1-)]-1/]W. This solution for c is proportional to W. The

proportionality factor is of the form [/(1+)], where = [E(R1-)]1/]. Notice that the

magnitude of this proportionality factor does depend on the return to wealth R = [n-1Rs+(1-n-

1)Rb], and hence on the portfolio selected, as well as on the levels of and describing the

consumer's preferences. The fact that c is proportional to W, however, is independent of the

particular portfolio selected or these preference parameters.

Since total consumption in period (n-1) is c + y, it is the case that Cn-1 is a linear function

of beginning of period wealth given by Cn-1 = y + Wn-1/(1+). This implies that consumption in

the last period is Cn = y + RWn-1/(1+), which is random because R is random. This is also

v(C), and its relative risk averse measure has a larger slope. If v(C) displays CRRA then u(W) is increasing relative risk averse.

10 This form for v(C) is selected for exactly this reason. In the APM, the usual form chosen for v(C) is v(C) = (C)(1-) /(1-), and the form selected here reduces to this when y = 0. So far we have been unable to solve for the indirect utility function associated with the standard form when y is not zero.

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linear in Wn-1. Indirect utility for Wn-1 is obtained by inserting these two optimal consumption

levels into the total utility expression un-1(Wn-1) = (n-1)[v(Cn-1) + Ev(Cn)]. Doing so yields un-

1(Wn-1) =

(n-1){[(/(1+)](1-) + [(1/(1+)](1-)[E(R(1-))]}(Wn-1)(1-)/(1-). The most important feature of

this un-1(Wn-1) is that it is a scale factor times (Wn-1)(1-)/(1-). This form, of course, is the

functional form representing CRRA preferences with risk aversion level for wealth Wn-1.

An alternate way of determining the relative risk aversion property of un-1(Wn-1) is to use

the relationship given in Proposition 1. Since consumption is of the form C = y + bW, the ratio

[dC/dW][W/C] = bW/(y + bW). Furthermore, the relative risk aversion measure for the selected

v(C) function is Rv(C) = C/(C-y). Using the relationship Ru(W) = Rv(C) [dC/dW][W/C] from

Proposition 1, together with these facts, also shows that Ru(W) = .

Determining indirect utility from wealth in periods before period (n-1) is accomplished

by extending backward one period at a time. Consider first the utility for the last three periods

n-2[v(Cn-2)] + Eun-1[(Wn-1 +y - Cn-1)[n-2Rs+(1-n-2)Rb]], where Cn-2 and n-2 are the consumption

and portfolio values selected in time period (n-2), and un-1() is the indirect utility for Wn-1. The

first term in this expression is utility from current period consumption, the second term is the

expected indirect utility from having random wealth [(Wn-1 +y - Cn-1)[n-2Rs+(1-n-2)Rb]], the

return on the amount saved, available at the beginning of the next period.

Choosing Cn-2 to maximize this sum of expected utilities requires that

v(Cn-2) = Eun-1[(Wn-1 +y - Cn-1)[n-2Rs+(1-n-2)Rb]] [n-2Rs+(1-n-2)Rb]. Using the assumed

form for v(C), the derived form for un-1(), and letting R = [n-2Rs+(1-n-2)Rb], this equation

becomes (Cn-2 - y)- = KE[(Wn-1 + y - Cn-2)]-R1- where K is a constant. This equation is

identical in structure to that solved when determining optimal Cn-1, and the solution possesses the

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property that indirect utility from Wn-2 is of the CRRA form. This same argument can be applied

any number of times and hence the following has been demonstrated.

Proposition 3: In this n-period APM, the consumer maximizing EU = i Ev(Ci) where v(C)

takes the form v(C) = (C-y)(1-)/(1-), chooses in each period so as to maximize expected

utility of the form u(W) = (W)(1-)/(1-), where W = W0[Rs + (1-)Rb] for some W0.

It is the case that the two utility functions, u(W) and v(C), are each of the CRRA

functional form only when income is zero. The Lucas formulation of the APM excludes income

and assumes the CRRA form for v(C). Hence risk preferences for u(W) in that model are also

CRRA. Omitting income from the model eliminates any need to decide which of the two utility

functions, if either, should display CRRA. When y is not zero, however, this question of which

should display CRRA does arise. It is beyond the scope of this paper to thoroughly review

support for the assumption of CRRA for u(W), but a brief summary is given below.

For more than thirty years, economic theory has examined the reasonableness of various

properties of u(W) when making portfolio decisions. The simple model used in that analysis

assumes that an investor has available one risky and one riskless asset whose returns are Rs and

Rb, respectively. The investor chooses to maximize Eu(W) where W = W0(Rs + (1-)Rb).

The feature of this simple portfolio model that we want to emphasize is that this form for

W is exactly the same as that identified in Proposition 3. The argument of utility in both the

portfolio model from theory, and for indirect utility for the APM, includes as wealth only assets

that can be purchased and sold. Thus, information concerning that utility function from the

literature is relevant here in determining acceptable properties for the indirect utility function in

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the APM.

There are many results published for the simple portfolio model, but we focus on just

one. Pratt, already in 1964, describes analysis of Arrow showing that the optimal is an

increasing, constant, or decreasing function of W0 depending on whether relative risk aversion,

as a property of u(W), is a decreasing, constant, or increasing function, respectively. Constant

relative risk aversion for u(W), when W is of the form W = W0(Rs + (1-)Rb), implies that the

share of wealth allocated to risky and riskless assets does not vary with changes in initial wealth.

This is considered a positive property of assuming CRRA, especially for a representative agent.

Empirically, one can attempt to determine the sign of the slope of the relative risk

aversion function for u(W) by determining how changes with changes in W0. What to include

as wealth, and which assets to consider risky and riskless are important decisions in this analysis

of this sort. Many studies have addressed this question and typically either support CRRA as a

property of u(W), or, if not, find decreasing relative risk aversion instead. The exact result is

often sensitive to which assets are included as risky and riskless assets.11

The conclusion we draw from this vast literature is that there is at least modest support

for the assumption of CRRA for u(W), when W is defined as in Proposition 3. To augment this,

the positive implications of choosing a v(C) so that u(W) displays CRRA for the equity premium

and risk free rate puzzle are pointed out in sections 3 and 4. Before turning to this topic,

however, one additional comment of a theoretical nature is given.

When utility from consumption takes the form v(C) = (C-y)(1-)/(1-), its relative risk

aversion measure is Rv(C) = C/(C-y). This expression is decreasing in C, and in addition, its

magnitude is larger than by factor C/(C-y) > 1. There is some evidence indicating that return

11 Friend and Blume [1975] is an early study in this area.

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to financial assets make up no more than 20% of consumption. For this model, this means that y

is on the order of 80% of C. If this is the case, the relative risk aversion measure for v(C) is

approximately five times larger than that for u(W).12 This fact may explain why Mehra and

Prescott and others find that the required magnitudes for relative risk aversion for v(C) seem

unreasonably large. The corresponding level for u(W) may well be much smaller, and in fact, of

a reasonable magnitude.

3. The Equity Premium and Risk Free Rate Puzzle: A Review

In this section, the theoretical discussion and the empirical facts pertaining to one

particular statement of the equity premium and risk free rate puzzle are briefly reviewed. The

material presented borrows heavily from the discussion and review of the literature on this

general topic by Kocherlakota. The specific equations that are presented, and the tests and the

data, are those employed in his summary of that literature. The purpose of this and the

following section is to illustrate the impact of choosing a v(C) so that u(W) displays CRRA.

The impact that is highlighted is the elimination of the equity premium and risk free rate puzzle.

We begin with Kocherlakota's description of the equity premium and risk free rate puzzle.

Over a long period of time, 1890-1979, the real rate of return from investing in equity

securities has averaged 7% with a standard deviation of 17%, while that from investing in short

term bills has averaged 1% with a standard deviation of only 5%. The 6% difference between

these mean returns is interpreted as a premium paid for investing in the riskier equity securities.

This equity premium is assumed to result from risk aversion on the part of the consumer. An

equity premium of this magnitude can persist only if consumers exhibit sufficient risk aversion.12 Rv(C) is not simply five times Ru(W) because the one is decreasing when the other is constant. Thus, for low wealth and consumption levels, Rv(C) is more than five times Ru(W), but for high wealth and consumption levels the ratio is less than five.

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It is also the case that over this same time period, real consumption has grown on average

about 1.8% per year. In a multi-period consumption model with impatience and a 1% real

riskless rate of return, this time pattern of consumption is optimal only if the consumer is not too

averse to consumption levels that differ across time periods. With strong aversion to

consumption disparity, the relatively small return earned from saving for future consumption is

not sufficient to generate this large a growth rate in consumption. The parameter in the CRRA

utility function v(C) represents both aversion to risk and aversion to differing levels of

consumption. Hence must be large enough to explain the observed equity premium, and must

be small enough to explain the time pattern of consumption. It is possible that explaining these

two historical facts using this single parameter is not possible. The equity premium and risk free

rate puzzle is that this is the case when v(C) takes the form v(C) = (C)(1-)/(1-).

Kocherlakota confirms the existence of the equity premium and risk free rate puzzle for

this utility function by testing for and rejecting two equilibrium conditions that arise in the asset

pricing model. One condition ensures that the expected utility maximizing allocation of wealth

between equities and bills prevails, and the other reflects optimal allocation of consumption over

time. Because these same two equations are used in the next section for the modified utility

function, they are presented first for an arbitrary v(C) and then specialized to Kocherlakota’s

form for v(C).

Optimal portfolio allocation between equities and bills at time t requires that expected

utility cannot be increased by shifting a portion of wealth from the one asset to the other. This is

expressed by the equation Et[v(Ct+1)(Rs,t+1 - Rb,t+1)] = 0 at each time t. In this equation, Et

denotes expectation at time t, and Ct+1, Rs,t+1 and Rb,t+1 denote the next period's optimal

consumption, return to equities and return to bills, respectively. The equation indicates that at

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each t, there is no expected utility gain from altering the portfolio allocation between equities

and bills.

Optimal allocation of consumption over time requires that at each time t, marginal utility

from consumption now equals expected marginal utility from consumption next period. This is

more conveniently written as [Etv(Ct+1)Rb,t+1/ v(Ct)] - 1 = 0 so that this equilibrium condition

also involves a particular expression having expectation zero at time t.

Given observed values for Rs, Rb and C, and a particular utility function, one can test to

determine if these two equations or equilibrium conditions have held in the past.13 Since neither

is likely to have been identically zero throughout the time period, the problem is one of

determining whether or not the observed values for the two expressions are significantly

different from zero. In conducting this test, Kocherlakota uses the data employed earlier by

Mehra and Prescott. These are annual time series for the various variables for the period 1890-

1979. Equities are represented by the S&P 500, and the bill rate is that associated with short

term government securities. A more complete description of the data and its sources is in

Kocherlakota.

Now for the specific equations tested by Kocherlakota. When utility is of the form

v(Ct) = t(Ct)(1-)/(1-), marginal utility is given by v(Ct) = t(Ct)-. In addition, because v(Ct)

is known at time t, it can be introduced into the denominator of the portfolio equilibrium

condition without effecting the expected value which is zero.14 Thus, the two expressions tested

13 is assumed to be .99 in both Kocherlakota's analysis and in ours.

14 Modifying the portfolio equilibrium condition in this way makes it depend on the growth rate of consumption rather than the consumption level. This makes the values less dependent on time. Kocherlakota indicates that Rs, Rb and the growth rate in consumption are each stationary and ergodic. Consequently it is reasonable to use a t-test to determine if either Expression 1 or Expression 2 is significantly different from zero.

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as significantly different from zero or not are: Expression 1: Et(Ct+1/Ct)-(Rs,t+1-Rb,t+1) and

Expression 2: Et(Ct+1/Ct)- (Rb,t+1) - 1.

Kocherlakota asks, for each expression, the following question. For which values for

do the data reject the hypothesis that the mean value of the observations for the ninety year

period is zero? These findings for the two expressions are then combined to ask if there is any

value, or set of values, for such that neither expression is significantly different from zero.

That is, is there any utility representation of this functional form, which is not rejected by the

historical evidence? Kocherlakota reports that Expression 1, the equilibrium condition

associated with the holding of assets, is rejected for all less than 8.5. Expression 2, on the

other hand, is significantly different from zero for all greater than or equal to 1.0. Thus, for

no value for is the observed equity premium and risk free rate consistent with the fundamental

equilibrium conditions in the APM. The 95% rejection level is used in stating this conclusion.

Kocherlakota’s test is only one of many giving evidence for what is referred to as the

equity premium and risk free rate puzzle. Different time periods, different countries and

different model formulations and a variety of empirical test procedures have been used. This

particular test is selected as the base case for our analysis because of its straightforward nature,

the fact that it is easily extendable to another functional form for utility, and because it was

recently published in a widely read outlet. In the next section, the same data15 and testing

methodology are used assuming the form for v(C) implies that u(W) is CRRA even when y > 0.

As shown in section 2, this form for v(C) is v(C) = (C-y)(1-) /(1-). The case where y = 0 is

exactly the Kocherlakota analysis. Thus, our findings when y = 0 must and do match his. When

y is greater than zero, however, these preferences are shown to allow resolution of the equity

premium and risk free rate puzzle. 15 Kocherlakota generously provided his data to us in electronic form and we thank him.

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4. A Resolution of the Equity Premium Puzzle

The analysis in section 2 indicates that when income and wealth are each sources for

consumption, then CRRA preferences for wealth result from utility from consumption of the

form v(C) = (C-y)(1-)/(1-), where y is income. In this section, the impact of y > 0 on the

Kocherlakota test for the equity premium puzzle is determined. The same two equilibrium

conditions examined by Kocherlakota are tested here. We find that when y is a sufficiently large

fraction of consumption, ranges of values for exist which are consistent with the asset pricing

conditions. Adjusting the form for v(C) so that u(W) displays CRRA can cause the equity

premium and risk free rate puzzle to disappear.

Expressions (1) and (2) listed below, are analogous to those formulated and tested by

Kocherlakota, and are for the more general v(Ct) = (Ct-y)(1-) /(1-). Expression (1) is now

[(Ct+1- yt+1)/(Ct- yt)]-(Rs,t+1-Rb,t+1) and Expression (2) is [(Ct+1 - yt+1)/(Ct - yt)]-[Rb,t+1] - 1. As

before, it is the case that each expression should not be significantly different from zero.

The focus here is on the impact of including yt in the analysis and therefore the

magnitude of yt is entered at several levels, including yt = 0. Before reporting the impact on the

range of values for for which the Expressions (1) and (2) are not significantly different from

zero, two details must be clarified. First, since Kocherlakota’s test only involves growth rates

for consumption, and the test here uses consumption levels, these levels must be determined.

This is done by choosing the consumption level in 1889 as the numeraire, and then using the

growth rate data for 1890 to 1979 to calculate the consumption levels in those years.

Consumption in each year from 1890 to 1979 is obtained by multiplying consumption in the

previous year by the growth rate for the year.

This analysis also requires values for yt. Now yt is assumed to be nonrandom and

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constant in the theoretical discussion. In this application, however, it makes more sense to

maintain the assumption that yt is nonrandom, but to assume that it is known to grow at the

average growth rate of consumption. Thus, in each period, yt is a constant fraction of trend

consumption. Trend consumption, Ct*, is calculated by choosing consumption in 1889 as the

numeraire, with a constant annual growth rate of 1.77% throughout the period. This 1.77%

growth rate differs by .03% from that reported by Kocherlakota, but is chosen because it yields

consumption in the last year of the sample at its observed value. In summary, yt is assumed to

take the form yt = Ct*, where Ct

* grows at a constant rate of 1.77% per year. The analysis

considers values for ranging from 0 to .8 in increments of .1.

With these preliminaries taken care of, the data are used to determine whether

Expressions (1) and (2) are significantly different from zero for various values for and using

the same test procedure as Kocherlakota. Table 1 summarizes these results and an appendix

contains the detailed summary statistics supporting these findings. Table 1 contains three

columns. The first two columns list the values for for which Expression (1) and Expression

(2) are not rejected, respectively. The third column combines this information to list those

values for such that neither expression is rejected. To match Kocherlakota, values for

ranging from 0 to 10 in increments of .5 are considered in conducting the tests. The parameter

is varied from 0 to .8 in increments of .1. At = .8, income is 80% of trend consumption and

only 20% of consumption on average comes from saving.

Table 1 indicates that as income is a larger and larger fraction of consumption, the set of

values for which does not lead to rejection of Expression (1) gets larger. That is, as saving is a

smaller and smaller source for consumption, the equity premium is consistent with lower levels

of relative risk aversion. Similarly, as gets larger, the values for which do not lead to

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rejection of Expression (2) also gets larger, including more values at both the low and high end

of the values considered. For large enough , Expression (2) is not rejected for any in the

range considered. For example, when consumption on average is half from wealth and half from

nonwealth sources ( = .5), an level of 4 or larger does not lead to rejection of either the

portfolio or the time allocation equilibrium condition. When is as large as .8, so that on

average only one fifth of consumption comes from portfolio wealth, as small as one does not

lead to rejection of either of these equilibrium conditions.

The broad conclusion we wish to draw from Table 1 is that when return on assets is not

the only source for consumption, so that u(W) and v(C) are of a different functional form,

choosing v(C) so that u(W) displays CRRA significantly alters equilibrium conditions in the

APM. Furthermore, at least for the equity premium puzzle, this change appears to be a positive

one in that the puzzle need not arise.

One way to interpret this empirical finding is that when large values for relative risk

aversion for v(C) are associated with much smaller values for u(W), then the equity premium

puzzle need not arise. For instance, as mentioned earlier, for = .8, relative risk aversion for

v(C) is approximately five times larger than for u(W). Hence, the magnitudes of near one for

u(W) that resolve the equity premium puzzle, do so while maintaining large relative risk

aversion levels for v(C). The findings here are consistent with those of Mehra and Prescott and

others.

5. Conclusions

This concluding section does several things. First, a very limited amount of empirical

evidence concerning issues raised here is presented. Primarily this is discussion of the

magnitude of y as compared with total consumption. Following this, the utility function derived

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under the assumption of habit formation by Constantinides is discussed. This habit formation

v(C) is quite similar to the one used to derive a u(W) with CRRA preferences. Finally, a number

of questions that arise because of this analysis are briefly discussed.

Some evidence concerning the relative magnitudes of income, wealth, and rate of return

on wealth for this time period does exist, and can be used to partially determine appropriate

values for . Ando and Modigliani [1963], for instance, indicate that rate of return from saving

represents only a small portion, on the order of 10%, of total consumption. Kuznets [1946],

when discussing sources of income, claims that income is roughly 4/5 from personal effort and

1/5 from invested capital. While these are certainly crude estimates, the information does

support values for near the upper end of the range examined in section 4.

Another way to attempt to determine the magnitude of is to compare the relative

variability of consumption with that of portfolio wealth. When consumption is linear in wealth,

C = y + bW, the variables C and W display the same coefficient of variation (/) only when

y = 0. C has a smaller coefficient of variation than W when y > 0. Hence, evidence from the

permanent consumption literature which indicates that the marginal propensity is smaller than

the average propensity to consume, also supports values for y > 0. This propensity to consume

information can also be used with Proposition 1 to indicate that relative risk aversion measures

for u(W) are smaller than those for v(C).

Turn now to the habit formation model. Constantinides proposes a model where utility

from consumption depends both on current and past consumption levels. Past consumption

determines a "subsistence" or habit level of consumption that diminishes the valuation of current

consumption. This assumption violates the time separability assumption typically employed in

the APM. Specifically, Constantinides proposes that current consumption Ct, reduced by a

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weighted average of past consumption levels, xt, is the argument of utility. Furthermore, the

exponential form associated with CRRA is used. Hence v(Ct) = (Ct - xt)(1-) /(1-) is the

specification for utility in the habit formation model.

Obviously, income, yt, in our analysis, and subsistence consumption, xt, in the habit

formation model play similar roles. Each enters utility for consumption in the same way, and

each is significantly less variable than consumption itself. Thus the effects on the risk attitudes

of the consumer are very similar. This is why the two utility functions for consumption resolve

the equity premium puzzle. Even though the two utility functions are technically quite similar, it

is important to recognize that they are derived from quite different assumptions. The one

assumes a lack of time separability and the other requires CRRA as a property of utility for

wealth. The research provided here gives another interpretation for what appears to be a useful

functional form for utility from consumption.

The derivation methods used here rely on being able to determine the relationship

between consumption and wealth so that the u(W) associated with v(C) function can be

calculated. Finding this optimal consumption and wealth relationship is difficult, and not likely

even possible for any more than a handful of simple functional forms for utility. In addition,

when y is more realistically assumed to also be random, optimal consumption and the

relationship between u(W) and v(C) may well be impossible to determine. Fortunately, certain

aspects of the analysis do not require knowledge of the optimal consumption and wealth

relationship.

The fact that u(W) = v(C(W)) holds as an identity implies that the equilibrium

conditions in the asset pricing model can be equivalently posed using marginal utility of wealth

rather than marginal utility of consumption. For instance, rather than writing the portfolio

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equilibrium condition as E[v(C)(Rs - Rb)] = 0 as is usual, E[u(W)(Rs - Rb)] = 0 can be used

instead. This implies that if the form for u(W), rather than that for v(C), is taken as known or

given, the equilibrium conditions for the pricing of assets can be tested using wealth rather than

consumption data. Work is underway to collect such data.

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TABLE 1

Expression (1) Expression (2) Both

= 0 8.5 .5

= .1 7.5 .5

= .2 6.5 .5, 7.5

7.5

= .3 6.0 .5, 4.5

6.0

= .4 5.0 1.0, 2.5 5.0

= .5 4.0 all 4.0

= .6 3.0 all 3.0

= .7 2.0 all 2.0

= .8 1.0 all 1.0

Values for for which Expression (1), (2) and both expressions are not rejected.

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