Relative Risk Aversion

Relative Risk Aversion



September 1999

Donald J. MeyerJack Meyer

Department of EconomicsDepartment of Economics

Western Michigan UniversityMichigan State University

Kalamazoo, MI 49008East Lansing, MI 48824

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 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 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 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 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. 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) + Ev(Cn)], where Cn = (Wn-1 + y - Cn-1) [n-1Rs+(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-1Rs+(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 theorem[6], 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 slope at the optimum; that is un-1(Wn-1) = v(Cn-1(Wn-1)). In fact, with this additive form for utility, this same relationship between marginal utilities, ut(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 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 portfolio decisions. This important point is summarized in the following proposition.