Habit formation has been reported in the literature to play important roles in individual consumers’ intertemporal decisions and macroeconomic phenomena [2,3]. One of the seminal papers by Sundaresan [4] shows that adjacent complementarity in consumption due to habits lowers the intertemporal elasticity of substitution and, consistently with the empirical fact [5], thereby making consumption less volatile. Constantinides [6] derives implications of this effect on the equity premium puzzle.1 The restricted assumption that has been put commonly in the existing literature is that the process of habit formation is deterministic so that consumers can predict perfectly their future habits once their plans of the future consumption stream are set. In reality, however, even with a given consumption stream, it is quite uncertain how deep consumers would become in the habit of consuming in future. The resulting consumers’ choices and hence asset price dynamics would be affected by the risk associate with habit formation.

By extending the habit model by incorporating uncertainty over the habit formation process, the purpose of this paper is to explore the implications of the habit shifting risk for optimal consumption/portfolio choices and for asset pricing.2 As my main conclusions, it is shown that 1) by extending [1], the optimization problem with stochastic habit forming preferences can be transformed into that with simple time-additive preferences; that 2) consumers’ optimal portfolios are composed of tangency portfolio, which has minimum variance of returns, and habit hedging portfolio, duplicating stochastic habit formation; and that 3) risk premia of asset returns are characterized by consumption beta, which is proportionate to the covariance with consumption changes, and habit beta, defined by using the covariance with fluctuations in the habitual subsistence level.

The key relation underlying these results is the optimal condition that the marginal utility of wealth equals that of contemporaneous utility of consumption plus the shadow price of the uncertain future habit stream.

As in [6], the optimal surplus consumption, defined as consumption in excess of subsistence level, is expressed as proportionate to surplus wealth defined as wealth in excess of the capitalized value of future uncertain habitual subsistence level (say, the habitual subsistence wealth). A distinct feature in my paper is that the marginal propensity to consume and habitual subsistence wealth both depend on risk associate with habit formation.

The optimal portfolios are composed of riskless bond, the tangency portfolio and habit hedging portfolio, that is, three-fund separation holds. Wealth is divided into two parts; surplus wealth and habitual subsistence wealth. The amount of habitual subsistence wealth is invested in habit hedging portfolio and riskless bond to finance future uncertain habitual subsistence level. While, the residual wealth: surplus wealth is invested in tangency portfolio and riskless bond to finance surplus consumption.

From the optimal condition, stochastic movement of the marginal utility of wealth is duplicated by fluctuations in consumption and that in habitual subsistence level. As a result, risk premia are characterized by consumption beta and habit beta. One important implication of this for the risk premium puzzle is that neglecting the effect of the habit beta might lead to underestimation of the risk underlying security returns, and hence to overestimation of risk aversion parameters.

By using an endowment economy model with deterministic habit formation, Detemple and Zapatero [10] derive a two-factor pricing formula with consumption beta and habit beta which is perfectly correlated to changes in shadow price of habit.3 However, in the deterministic opportunity set, shadow price of habit is perfeclty correlated to changes of consumption and it turns to be single consumption beta model. In contrast, by assuming a constant opportunity set, we show that the stochastic habit formation preference results in a two-factor model with habit and consumption betas.

The remainder of this paper proceeds as follows. The model is presented in Section 2. Section 3 provides the optimal solution under linear stochastic habit formation by applying the linear transformation procedure developed by [1]. Section 4 characterizes risk premia and riskfree interest rates. Section 5 concludes the paper.

Suppose that a representative consumer endowed with an initial wealth W(0) faces D + 1 investment opportunities: one riskless bond and D risky assets. Underlying the model is a complete probability space where Ω is the set of states of nature, F is the σ-field of events, and P is a probability measure on. A D-dimensional standard Wiener process B is defined on.

The riskless bond yields a constant rate of return, r. The price of i-th risky asset (i = 1, ···, D) is given by

where (j = 1, ···, D) are independent of each other:

⁴This specification of habit dynamics is no longer valid in incomplete market economy because consumption surplus can be negative with positive probabilities.

if; and expected returns and diffusion coefficients are assumed to be constant. I assume that the markets are complete in which has full rank D. The market-price-of-risk process η i.e., risk premium on portfolios that duplicate is thus determined uniquely as follows,

The representative consumer determines the optimal portfolio and consumption processes in order to maximize his expected lifetime utility specified as follows,

where is subjective discount rate; is a parameter related with relative risk aversion; c and z are consumption and habitual living standard, respectively.

The key assumption in this paper is that the habitual living standard grows stochastically as follows,

or integral representation

where the parameters and measure intensity of consumption and the depreciation rate of the past habit, respectively. In (3) or (4), his habitual living standard depends not only on his past consumption history but also on states of nature. From the assumption of complete markets, the habit shifting shocks are given by the same Brownian motions as in (1). The stochastic property of the shocks is thus captured by diffusion term.4

With an initial wealth W(0) and an initial habitual living standard z(0), the representative consumer determines the optimal policies for consumption c and proportion of risky asset portfolio subject to the following dynamic budget constraint,

and the habit formation process (3).

Thus, the optimization problem is summarized as

subject to (3) and (5).

Previous section derives optimal consumption and portfolio rules in the production economy where risk-free rate is constant and risky production technologies follow (1). This section characterizes risk premium and risk-free interest rate to make sure of the relationship between consumption and asset pricing.

As discussed in (11), due to linear structure of habit formation, the present model can duplicate the shadow price of habit by contemporaneous marginal utility. Applying Ito lemma to (11) yields

Dividing both sides by (11) yields

Equating both sides of the deterministic and stochastic parts provides as follows,

where and are diffusion coefficient and expected rate of surplus consumption growth, respectively.

From the definition of, (20) reduces to

where is instantaneous covariance of surplus consumption growth rate with i-th stock return.6 Substituting into (22) provides two-factor asset pricing formula as follows,⁷

Note that risk premium is give by weighted average of covariance of consumption and that of habitual living standard. This implies that economic risks over the assets are captured not only by consumption risk but also habit risk.8 In the case that risky assets are negatively correlated to habit, observers who lack evaluating risk over habit tend to underestimate risk premium.

By using definitions of and, (21) reduces to

where is instantaneous variance of surplus consumption growth rate.

⁹In the case of time-additive utility, Gollier [17] provides three basic determinants: the pure time preference for the present, consumption smoothing effect, and the precautionary effect.

Substituting into (23) provides risk-free interest rate as follows:

This implies that the risk-free interest rate is characterized by 6 determinants.9 The second and third terms correspond to surplus consumption smoothing effect. The second term is positive because the growth rate of consumption induces the present consumption thereby requiring a higher return on the saving. The third term is negative because the high growth rate of habitual living standard induces the reserve thereby increasing the investment in risk-free asset.

Last three terms capture the precautionary effect caused by the risk over surplus consumption. Provided that the consumer is prudent, volatilities of consumption and habit are negatively correlated to the risk-free interest rate.

This paper examines optimal consumption/portfolio choices under stochastic habit formation. Due to introducing surplus consumption and single lifetime budget constraint with respect to surplus wealth, this optimization problem with stochastic habit forming preferences is transformed into that with simple time-additive preferences.

Asset pricing implications are provided as follows. From the optimal condition that the marginal utility of wealth equal that of contemporaneous utility of consumption plus the shadow price of habit, the marginal utility of wealth is driven by the fluctuations in the habitual subsistence level and those in the consumption. Therefore, for stochastic habit forming consumers, risks are measured both by covariance with changes in consumption and by covariance with fluctuations in the habitual subsistence level. One empirical implication of this is that provided that risky assets are negatively correlated to habit, observers who lack evaluating risk over habit tend to overestimate risk aversion parameters.

The author is grateful to Shinsuke Ikeda for encouragement and valuable comments. I would like to thank Yuichi Fukuta, Kazuhiko Hashimoto, Keichi Hori, Yoshiyasu Ono, Akiko Yamane and the participants at the annual meeting of Japanese Economic Association at Osaka Gakuin University.