External Habit Formation and Asset Prices

Table of Contents

Table of Contents

List of Figures

List of Tables

1. Introduction

2. “Catching up with the Joneses” Preferences

3. Conceptual Framework
3.1 The Model
3.2 The Competitive Equilibrium

4. The Results
4.1 Theoretical Implications
4.2 Simulated Outcome

5. Discussion

6. Conclusion

Appendix

References

List of Figures

Figure 1: Aggregate wealth distribution as function of individual risk aversion

Figure 2: Conditional volatility of returns in 3 different homogeneous economies

List of Tables

Table 1: Moments of Historical and Simulated Data

Table 2: Input Parameters for the Heterogeneous and Homogeneous Models

Table 3: Autocorrelation and Cross Correlation of Historical and Simulated Data

1. Introduction

One of the most studied topics in modern economics are the market mechanisms that lead to the determination of asset prices in an economy. The empirical research indicates that there is a link between the historically observed asset prices and macroeconomic developments. One of the most important observations are the countercyclical behavior of the equity risk premium and the stock return volatility, implying that the excess return of common stocks over the risk-free rate during business cycle troughs is significantly higher than during expansions (Fama & French, 1989; Cochrane, 1991). This paper aims to explain this countercyclical behavior by introducing an external habit formation feature in the standard representative-agent consumption-based asset pricing model, in form of the so called “catching up with the Joneses” preferences (Abel, 1990; Galí, 1994; Chan & Kogan, 2002). These preferences imply that the relative risk aversion of the agents in the economy is constant over time and varies across the agents, which generates an endogenous wealth process, that in turn creates a countercyclical behavior in the risk premium and the conditional stock return volatility (Chan & Kogan, 2002). As the agents with lower risk aversion distribute a greater fraction of their wealth to risky assets, their wealth decreases relatively more in reaction to cyclical downturns, shifting the aggregate wealth towards more risk averse individuals. These more risk averse agents, however, demand a higher compensation for risk, leading to an increase of the aggregate equity risk premium in response to a fall in stock prices (ibid.).

Studying the excess return of common stocks over the period from 1889 to 1978, Mehra and Prescott (1985) find that the historically observed equity risk premiums cannot be reconciled with a standard general equilibrium model for reasonable levels of risk aversion. In the literature, one can find several approaches to explain this “equity premium puzzle” (Abel, 1990; Constantinides, 1990; Campbell & Cochrane, 1999). Campbell and Cochrane (1999) use a consumption-based representative agent model with external habit formation to explain the countercyclicality of excess returns and stock market volatility and thereby, the “equity premium puzzle”. However, they use a power utility function with a countercyclical variation in the risk aversion. This implies that investors demand risk premia for holding stocks, not because stock returns are correlated with declines in wealth or consumption, but because investors fear that stocks perform badly in recessions, when individuals need the surplus from stocks the most (ibid.). Abel (1990) demonstrates that by incorporating “catching up with the Joneses” preferences in the utility function of an agent, one can generate equity premia and risk-free rates close to the historically observed averages. However, the volatility of both, equity premium and risk-free rate predicted by the model exceeds the empirically observed values (ibid.). Therefore, the following paper builds up on this approach, while overcoming the flaws of the model by Abel (1990).

The paper is structured as follows. Section 2 explains the origin and the impact of the “catching up with the Joneses” preferences. Section 3 develops the model and its competitive equilibrium. Section 4 presents the theoretical and simulated results. Section 5 and 6 discuss the results and conclude.

2. “Catching up with the Joneses” Preferences

To understand what distinguishes the following model from other models trying to explain the determination of asset prices, it is useful to understand what are “catching up with the Joneses” preferences and in which way they affect the determination of asset prices.

Gali (1994)¹ studies the effect of consumption externalities on portfolio decisions and asset pricing models. To do so, he develops a utility function with “keeping up with the Joneses”-preferences, meaning that an agent’s consumption preferences are not only defined by his own consumption, but also by the average consumption in the economy, which incorporates the belief that households care about their relative standard of living. The author therefore develops the following utility function:

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One can see that the utility of an agent does not only depend on his own consumption c, but also on the average consumption level in the economy C, which describes the standard of living. The agent’s risk aversion and the effect of consumption externalities are incorporated by and g, respectively. Gali (1994) explains that for the case that g is greater than zero, the marginal utility of an agent increases with the average level of consumption. This describes the “keeping up with the Joneses” preferences, as the agent will increase his consumption in function of the average consumption, to maintain his relative standard of living (ibid.). The author finds that this preference type leads to increases of the optimal risky portfolio in the capital asset pricing model (CAPM). Moreover, in a multiperiod asset pricing model, the asset prices and returns in equilibrium are identical to those of an economy with a properly adjusted degree of risk aversion, without consumption externalities (ibid.).

However, to emphasize that consumers care about the lagged value of consumption, Abel (1990) introduces the term “catching up with the Joneses”. In his attempt to explain the “equity premium puzzle” (Mehra & Prescott, 1985), using this type of preferences, the author is able to create an equity premium and a risk-free rate close to historical averages. Nevertheless, the expected returns vary more than what is empirically observed.

3. Conceptual Framework

3.1 The Model

To develop their model, Chan and Kogan (2002) define an exchange economy with complete financial markets and a single consumption good, that is not intertemporally transferable. They assume that there is only one source of uncertainty and that this uncertainty can be hedged in the financial markets. To ensure that the risk aversion of the agents does not change over time i.e., that they have constant relative risk aversion, the individual utility function is set as a power function of the relative standard of living (Chan & Kogan, 2002). This feature introduces the above explained “catching up with the Joneses” preferences in the model, which creates a nondegenerate stationary cross-sectional distribution of wealth, as well as relatively low equilibrium interest rates (ibid.).

The aggregate endowment process Yt follows a standard Brownian motion with lognormality and nonnegativity (Chan & Kogan, 2002). In the economy, there is one risky asset with dividend stream Yt and price Pt and a risk-free asset with the instantaneous interest rate rt (ibid.). The authors define the consumption rate and the standard of living at time point t as Ct and Xt, respectively. From these assumptions, the authors derive the expected utility function with the common time discount factor

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The individual risk aversion of an investor for consumption, as well as wealth gambles is given by (Chan & Kogan, 2002). Considering the utility function for one individual agent, it is possible to show that the relative risk aversion coefficient, corresponding to wealth gambles, is also equal to and stable over time (ibid.). However, the authors show that this does not hold for the aggregate wealth, as the aggregate utility function is derived from the individual preferences and is only homothetic for the case that all agents have identical preferences. Therefore, any changes in the risk aversion of the representative agent must be due to the differences in the aggregate relative risk aversion of the individuals (ibid.).

The complementary effect of the standard of living Xt on an agent’s utility presupposes that the marginal utility of the agent increases with the average consumption per capita. Therefore, the derivative of the utility function with respect to Xt must be greater than zero, which requires a relative risk aversion . Thus, Chan and Kogan (2002) decide to analyze a continuum of preference types, defined over .

The authors define the standard of living as a weighted geometric average of past realizations of the aggregate endowment process

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with xt and yt defined as the logarithm of the standard of living Xt and the aggregate endowment Yt, respectively. l defines the degree of history dependence of the standard of living Xt. If l is close to zero, the past history has a strong effect on the current standard of living, while with an increasing l, the most recent realizations of the aggregate endowment have a stronger influence (Chan & Kogan, 2002). The authors clarify that the infinite moving average gives rise to a more stable expected growth rate of the marginal utility of consumption, which reduces the volatility of the interest rate in equilibrium, compared to the model by Abel (1990). Chan and Kogan (2002) further demonstrate that the process Xt allows for shocks in the aggregate endowment to be incorporated in the relative (log) consumption t = yt - xt instantaneously. However, as the standard of living process slowly adapts to the new level of endowment, the effect of the shock decays exponentially at the rate l , which indicates lower persistence and reduced steady-state variance of relative consumption, with increasing degree of history dependence of the standard of living Xt (ibid.).

3.2 The Competitive Equilibrium

To analyze the model’s equilibrium, Chan and Kogan (2002) develop the optimal consumption sharing rule, making use of the social planner’s problem, which they support with an Arrow-Debreu economy. The developed equilibrium is then implemented as a sequential-trade economy (ibid.).

A social planner is a decision-maker that tries to achieve a pareto optimal allocation of the aggregate endowment. Making the assumptions that risk aversions are unique, and that the social planner assigns the social weight to each type of , Chan and Kogan (2002) explain that the social planner solves the static optimization problem

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as resources are not intertemporally transferable. To develop the optimal consumption sharing rule, they use the first-order condition for consumption. By doing so, the authors develop the following optimal consumption sharing rule

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with t) as the logarithm of the shadow price of the resource constraint. In equation (7) one can see that the individual consumption can be separated into the aggregate endowment Yt and the proportion assigned to the individual. This proportion is a function of the stationary relative consumption wealth, which implies that the consumption of all investors grows at the same rate, due to the dependence of the individual consumption on the relative consumption (Chan & Kogan, 2002). This results from an equalizing effect of the “catching up with the Joneses” preferences on the marginal utilities of the investors and guarantees that the economy will not be dominated by a single investor in the long run (ibid.). Therefore, the share of the aggregate endowment for each investor remains stationary over time and depends on the ratio of the aggregate endowment to the standard of living (ibid.).

In an Arrow-Debreu economy, investors trade on an Arrow-Debreu security which pays one unit of consumption in a certain state of the economy and zero otherwise. The arbitrary payoff stream of the security is discounted with a stochastic discount factor, which Chan and Kogan (2002) define as function of the shadow price z, the change in the logarithm of the standard of living x and the common discount rate

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Using this stochastic discount factor in a sequential-trade economy, the authors set the stock prices of long-lived securities in equilibrium as the expected infinite payoff stream discounted with the stochastic discount factor and show that the instantaneous interest rate is given by

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4. The Results

4.1 Theoretical Implications

To facilitate the analysis of the asset price behavior in the model, Chan and Kogan (2002) show that the equilibrium price-dividend ratio, which summarizes the conditional expectations of future discount and dividend growth rates, only depends on the relative consumption wt. Resulting from the specified aggregate endowment process, the dividend is independent of the current state of the economy and is inversely related to the distribution of future discount rates (ibid.).

The authors show that in an economy populated by more than one type of agents, the instantaneous Sharpe ratio is a monotonically decreasing function of relative consumption, which results from the endogenous redistribution of wealth in the modeled economy. A decline in asset prices would reduce the wealth of the agents, and therefore their relative consumption. Hence, to induce them to continue to hold the entire stock market in the aggregate, the return on risk must increase (ibid.). This shows that the preference heterogeneity implemented in the model allows a countercyclical behavior of the Sharpe ratio, which generates empirically plausible patterns of predictability of asset prices.

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¹ Jordi Gali’s paper "Keeping up with the Joneses Consumption Externalities, Portfolio Choice and Asset Prices," was first published in 1989. The paper published in 1994 is a revised version.