Elsevier

European Economic Review

Volume 52, Issue 6, August 2008, Pages 1072-1096
European Economic Review

Expected life-time utility and hedging demands in a partially observable economy

https://doi.org/10.1016/j.euroecorev.2007.09.003Get rights and content

Abstract

This paper analyzes the expected life-time utility and the hedging demands in an exchange only, representative agent general equilibrium under incomplete information. We derive an expression for the investor's expected life-time utility, and analyze his hedging demands for intertemporal changes in the stochastic unobservable growth of the endowment process and the changing quality of information regarding these changes. The hedging demands consist of two components, which could work in opposite directions so that a conservative consumer may end up having positive hedging demands. Our results are qualitatively different from those prevailing under constant growth (cf. [Brennan, M.J., 1998. The role of learning in dynamic portfolio decisions. European Finance Review, 1, 295–306; Ziegler, A., 2003. Incomplete Information and Heterogeneous Beliefs in Continuous-Time Finance. Springer, Berlin, Chapter 2].

Introduction

A recent strand of literature studies the effects of incomplete information on various aspects of the economy. While traditional models assume that investors already know all the relevant quantities in the economy, some more recently developed models incorporate the fact that investors do not observe, thus have to learn the growth of the economy. They observe the realizations of different variables in the economy, and use these to make assessments of the economy's growth.

Because the moments of the returns’ distributions are unobservable to both real-world investors and empiricists, implementing and testing traditional, complete information models raises a number of issues. First of all, one might question whether the econometric assumptions of the estimation procedure are consistent with the structure and assumptions of the original model. Second, one might ask whether the “responsibility” of implementation shortcomings should be attributed to the original model or to the estimation procedure. Third, the attribution of “responsibility” for rejection or non-rejection of statistical hypotheses to the original model or to the estimation procedure is ambiguous (Feldman, 2007). To resolve these issues and to explain real-world hedging demands that hedge not only fundamental changes but also changes in the quality of information, this paper allows for unobservability and endogenously models the estimation/learning process.

The purpose of this paper is to analyze the expected life-time utility and the hedging demands in a Lucas (1978) economy, but where the growth rate of the stochastic endowment process is unobservable. The main contribution of this paper is the analysis of the expected life-time utility and hedging demands in a partially observable economy with a stochastic mean-reverting endowment growth rate. We derive analytical expressions for the equilibrium value function and hedging demands.

Consumers have a constant relative risk aversion (γ) and maximize expected life-time utility of consumption. Empirical evidence suggests that the coefficient of relative risk aversion should be greater than unity. We show that conservative (γ>1) consumers dislike both variability in estimates and local covariation between estimates and the endowment process.1

We also analyze the individual consumer's hedging demands derived in this paper. We show that the hedging demands consist of two components. The first is a hedging component that arises because the true growth rate is stochastic and there is a correlation between the endowment flow and changes in the true growth rate. The second is a hedging component that arises because the consumer has to consider the fact that there is a difference between his estimate and the true growth rate, i.e., he has to take into account the presence of an estimation error. In the case of a negative local correlation between the endowment flow and changes in the true growth rate, the two hedging components work in opposite directions, and a conservative (γ>1) consumer can end up having non-negative hedging demands, even if the equity premium is positive.

A negative equity premium can occur naturally in this economy. The explanation is that, with a negative equity premium, the stock returns and innovations in aggregate consumption are perfectly negatively correlated. The representative consumer accepts a negative equity premium, since the stock acts as a hedge against low future consumption. Because a negative equity premium can occur naturally in this economy, we analyze the implications of a negative equity premium for hedging demands.

Further, we show how the hedging demands, which consist of a sum of two components, are related to the equity premium. Assuming a positive equity premium and conservative consumers, we show that if the instantaneous covariance between the endowment flow and changes in the true growth rate is sufficiently high (low), then the hedging demands are positive (negative) and the market price of risk is relatively low (high). If this instantaneous covariance happens to be equal to the negative of the estimation error, then the consumers’ hedging demands are zero and the market price of risk is equal to the relative risk aversion of the consumers.

In addition, we calculate numerical values on the two hedging components, the total hedging demands and the market price of risk, allowing the length of the time horizon, the volatility of the true growth rate and the local correlation between the endowment process and changes in the true growth rate to vary, while keeping the values of the other parameters constant. We find that when the estimation error has not reached its steady state, the magnitude of the first hedging component is increasing in the volatility of the true growth rate and U-shaped with respect to the local correlation between the endowment flow and changes in the true growth rate. However, the magnitude of the second hedging component is increasing with respect to this local correlation and its reaction with regard to the volatility of the true growth rate depends on the aforementioned local correlation. Moreover, we find that the magnitudes of the first and second hedging components increase with the length of the horizon. We also calculate numerical values for the steady state. The results for the steady state are similar. However, in steady state, the magnitude of the second hedging component is inversely U-shaped with respect to the local correlation between the endowment flow and changes in the true growth rate, and increasing in the volatility of the true growth rate. We argue that these effects occur because the second hedging component is largely determined by the estimation error, whose steady state value is inversely U-shaped with respect to the local correlation between the endowment flow and changes in the true growth rate, and increasing in the volatility of the true growth rate.

Early contributions within the field of incomplete information are Williams (1977), Detemple, 1986, Detemple, 1991, Dothan and Feldman (1986) and Feldman (1989). In a partial equilibrium framework, Williams (1977) studies the CAPM under heterogeneous Gaussian distributions of consumer beliefs consistent with available information. Dothan and Feldman (1986) and Detemple (1986) solve the general equilibrium problem under incomplete information. Feldman (1989) analyzes equilibrium interest rates and multiperiod bonds under partial information. Detemple (1991) allows for non-Gaussian distributions of investors’ prior beliefs.

More recent contributions include Brennan (1998), Yan (2000), Ziegler (2000), Riedel (2000), Brennan and Xia (2001), Xia (2001), Feldman, 2002, Feldman, 2003 and Cvitanić et al. (2006). Brennan (1998) studies the portfolio optimization problem when the consumer learns about a constant growth in exogenously given stock returns. Yan (2000) solves for the equilibrium asset prices and analyzes the volatility of stock index options. In contrast to Brennan (1998), Yan (2000) develops an equilibrium model, in which agents learn about a mean-reverting endowment growth rate. Goldstein and Zapatero (1996) analyze asset prices and stock index options in a similar full information economy where there is a perfect positive local correlation between the endowment flow and changes in the true growth rate. Ziegler (2000) studies the portfolio choice problem of an agent whose beliefs about the dividend growth rate differ from the market beliefs. Riedel (2000) shows that in a Lucas (1978) economy in which there is an unobservable constant growth rate, the term structure of interest rates decreases to negative infinity. The result in Riedel (2000) is in contrast to the result in Feldman (1989), who shows that in a similar economy, where the unobservable productivity factors are stochastic, equilibrium term structures are bounded. Feldman (2003) resolves the apparent contradiction between Feldman (1989) and Riedel (2000). Feldman (2002) provides a theoretical framework for empirical tests of asset-pricing models with unobservable productivity factors. Brennan and Xia (2001) analyze the optimal portfolio strategy for an investor who has discovered an asset-pricing anomaly but is not certain whether the anomaly is genuine. Xia (2001) investigates the horizon effect in optimal portfolio choice when there is uncertainty about stock return predictability. Cvitanić et al. (2006) use the incomplete information framework in order to assess the economic value of analysts’ recommendations.

There is also a complete information literature, including e.g., Merton, 1971, Merton, 1973, Kim and Omberg (1996) and Wachter (2002), which solves portfolio choice problems/equilibria that are similar to the incomplete information portfolio choice problems/equilibria discussed above. Since it is possible to re-represent the non-Markovian incomplete information problem as a Markovian problem that is mathematically identical to a complete information problem (Feldman, 2007), the solution of incomplete and complete information equilibria share the same techniques. However, the economic intuition is totally different, because in complete information equilibria, agents respond to unanticipated changes in the true economic environment, whereas in incomplete information equilibria, agents respond to unanticipated changes in the perceived economic environment.

Korn and Kraft (2004) and Kraft (2004) document some inconsistencies in the formulations of some of the problems studied in the previous complete information literature, which are related to the so-called nirvana solutions in Kim and Omberg (1996). Since we are concentrating on the most relevant case in which the coefficient of relative risk aversion is greater than 1, and since the time horizon is finite, we can be assured that our consumption/investment problem is well-defined. This is because, with a coefficient of relative risk aversion greater than 1, the elementary utility function is bounded from above by zero.

We consider an economy that is similar to that in Yan (2000) but, instead of focusing on asset prices and option volatility, we focus on the value function and the hedging demands. The results in this paper, which explores a stochastic growth rate economy, are both qualitatively and quantitatively different from those in Brennan (1998), who examines a constant growth rate economy.2 With a constant growth rate as in Brennan (1998), the filtering error will vanish as time goes to infinity. Our finding that there are two hedging components which can work in opposite directions also differs from the findings in Brennan (1998). He shows that, assuming a positive equity premium, a conservative agent will always have negative hedging demands. This paper demonstrates that this is generally not true in an equilibrium model where consumers learn about a mean-reverting endowment growth rate, and shows under what conditions the hedging components will work in the same and opposite directions, respectively. As opposed to Xia (2001), who studies horizon effects on optimal portfolio choice in a model in which stock returns are exogenously given, we analyze the expected life-time utility and the hedging demands in an equilibrium model. In contrast to Brennan and Xia (2001), the growth rate of the entire economy is unobservable in our model. In Brennan and Xia (2001) the endowment flow is separated from the dividend flow, and the endowment growth rate (and hence the economy's growth rate) is observable, whereas the dividend growth rate is unobservable. The focus of Brennan and Xia (2001) is also different: while we provide a theoretical analysis of the expected life-time utility and the hedging demands, they focus on generating stock price volatilities and equity premia that are close to historical values.

The organization of the rest of the paper is as follows. In Section 2, we describe the nature of the economy that we consider. In Section 3, we derive the theoretical results: first, in Section 3.1, we examine the filtering problem of the partially informed agents, then in Section 3.2, we analyze the equilibrium expected life-time utility under partial information, and we examine its relation to the dynamics of stock prices. In Section 3.3, we analyze the individual investors’ hedging demands. Finally, Section 4 concludes the paper.

Section snippets

The economy

We consider a Lucas (1978)-type exchange economy. In this economy, there is an exogenous aggregate endowment process, Dt. The consumption good is perishable, so that in each period, the entire endowment is consumed. There is a complete probability space (Ω,F,P). The flow of aggregate endowments follows the process:dDtDt=μtdt+σDdBt,where the endowment growth rate (μt) is stochastic and evolves according to a mean-reverting Ornstein–Uhlenbeck process,dμt=κ(μ¯-μt)dt+σμdZtwith σD, κ and σμ being

The equilibrium

In this section, we will first analyze the filtering problem of the consumers. Then, we will examine the equilibrium properties of the interest rate, the stock price and the value function. Finally, we will analyze the consumers’ portfolio choice problem. The filtering problem is analyzed in Section 3.1 below. In Section 3.2, we examine the equilibrium properties of the interest rate, the stock price and the value function. The consumers’ portfolio choice problem is analyzed in Section 3.3.

Conclusions

We analyze an exchange economy in which consumers learn about a mean-reverting endowment growth rate. Consumers have power utility and maximize expected utility of life-time consumption. First, we derive the expected life-time utility of the representative consumer and analyze its properties. Then, we derive an expression for the value function of an individual consumer in equilibrium, and analyze his hedging demands. We show how the value of the instantaneous covariance between the endowment

Acknowledgments

I thank Gerard Pfann (the editor), two anonymous referees, David Feldman, Michael Brennan, Yihong Xia, Carsten Sørensen, Erik Lindström and Björn Hansson for helpful comments and suggestions, and conference participants at the meetings of The Asian Finance Association in Taipei 2004 and The Econometric Society in Madrid 2004. Financial support from the Jan Wallander and Tom Hedelius Foundation (Grant W2005-0365:1), Bankforskningsinstitutet and the Swiss National Science Foundation (NCCR FINRISK

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