Intertemporally dependent preferences: the link between asset pricing, the term structure and the market portfolio

Rallis, Nicholas (2004) Intertemporally dependent preferences: the link between asset pricing, the term structure and the market portfolio. Doctoral thesis, University of London: London Business School.

Abstract

This thesis derives a two-factor asset pricing model based on either (i) the term structure of interest rates and aggregate consumption or (ii) the market portfolio and a proxy for aggregate consumption in a multi-agent, continuous time setting where each individual's utility function is a concave function of current consumption and the historical consumption path. Markets are generally assumed to be dynamically incomplete and no assumption is made about the existence of a representative agent. In addition, there are no assumptions regarding the dynamics of either the asset prices or the risk-free rate, both of which are endogenously determined at equilibrium. The dynamics of the habit variable have a non-linear form, that includes as a special case the standard linear index formulation. Depending on whether utility is derived from (a) surplus consumption over habit or, alternatively, (b) the ratio of consumption to habit, it is shown that an asset's risk premium is determined by its instantaneous covariances with (a) changes in aggregate consumption and the price of a "pivotal" bond paying a continuous coupon that is a simple exponential function of time or (b) changes in individual consumption and individual wealth. After aggregation, the latter leads to a weighted average of individual consumption changes (a proxy for aggregate consumption changes) and changes in the market portfolio. In the first case the model utilises the observability of the term structure to capture the impact on instantaneous risk premia of term (as distinct from instantaneous) uncertainty in the aggregate consumption process. For sufficiently strong intertemporal dependence, the term structure related component of the risk premium is shown to dominate the consumption driven term; moreover, it is shown to be counter-cyclical with regard to consumption growth prospects. The two factors are positively correlated if rising consumption is associated with a slowdown in consumption growth prospects and vice versa. With linear habit dynamics, this bond driven term is expressed independently of the utility functions of individuals, while the non-linearity in the habit dynamics can increase substantially the coefficients of both the consumption and the bond related term. We also provide conditions for a constant risk-free rate, under which we focus on the effect of habit on the consumption term. For log utility, we achieve a transition from consumption history to price history and provide explicit expressions for the risk premium, the dividend yield and the volatility of the market portfolio in terms of the ratio of the price of the market portfolio to an exponentially weighted "moving average" of past prices. This ratio exhibits mean reversion while the three aforementioned variables are counter-cyclical with regard to market performance. Only in the special case of linear habit dynamics coupled with dynamic completeness, individual consumptions are perfectly correlated and the existence of a representative agent is guaranteed. If aggregate consumption is assumed to be locally deterministic, yet discretely stochastic, which is broadly consistent with empirical evidence, the general result simplifies to a simple single-beta form, with risk premia on all assets proportional to their "betas" relative to the pivotal bond. This may provide a resolution of the "equity premium puzzle", as substantial risk premia can arise even in the absence of instantaneous uncertainty in aggregate consumption. In the second case, the market portfolio related term of the risk premium is larger the smaller the aggregate habit is relative to aggregate wealth. Again, for linear habit dynamics, this term is expressed independently of utility functions, while non-linearity in the habit dynamics can increase substantially the coefficient on the consumption related term. If aggregate consumption is assumed instantaneously deterministic, we recover the classical beta form of the CAPM. This formulation may therefore provide an alternative resolution of the "equity premium puzzle". Finally, as an application, we employ the pivotal bond principle, appropriately extended to the discrete-time habit and durability setting of Ferson and Constantinides (1990), to obtain a tractable one-lag form for the Euler equation and thus avoid its truncation. In this case, the simultaneous presence of habit persistence and durability makes the coupon function of the pivotal bond a more complex sum of two exponential terms.