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This paper examines optimal consumption/portfolio choices under stochastic habit formation in which it is uncertain how deep consumers would become in the habit of consuming in future. By extending Shroder and Skiadas [1] to stochastic habit formation, the optimization problem with stochastic habit forming preferences is transformed into that with simple time-additive preferences. Optimal portfolios are composed of the tangency portfolio and habit hedging portfolio. Resulting risk premia are characterized by consumption beta, which is proportionate to the covariance with consumption changes, and habit beta, defined by using the covariance with habit.

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 [

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 [

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 [

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 [

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 [

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:

^{4}This 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).

To solve the problem, I first obtain the capitalized value of habit to finance the future uncertain habitual living standard, I refer it the value as the habitual subsistence wealth. Following [

Letting H(t) be state price deflator as

the habitual subsistence wealth level is obtained by capitalizing the subsistence consumption stream with H as

where

which represents the rate of return on the habitual subsistence wealth. I impose a restriction A > 0 for convergence.

Unlike in the case of deterministic habit formation [

Due to linear structure of habit formation, I can follow [

Define surplus wealth and surplus consumption. Two stochastic differential equations (3) and (5) can be combined into single equation with respect to surplus wealth deflated by H(t),

which can be integrated as

This lifetime budget constraint requires that the present value of the surplus consumption stream equals the initial value of surplus wealth, where surplus consumption at each instant is evaluated by. A surplus consumption deepens the future habit stocks by rate, which in turn increases the required value of the subsistence wealth and thereby decreases available surplus wealth. represents these additional costs of surplus consumption. To ensure the existence of optimal policies, I impose the restriction on the initial condition that.

Using surplus consumption and surplus wealth, the optimization problem (6) is reduced to that with simple time-additive preferences.

Letting y be the present value of Lagrange multiplier associated with the lifetime budget constraint (9), the necessary condition for optimality is

which requires that the marginal utility of consumption be equal to the marginal utility of wealth. Note that due to linear structure of habit formation, shadow price of habit can be represented by contemporaneous marginal utility.

The optimal policy for surplus consumption can be obtained by substituting solutions y and H(t) into this condition. The Lagrange multiplier y can be obtained by substituting (11) into (9),

where is given by

Substituting (12) into the first order condition (11) gives optimal surplus consumption process

From (9), surplus wealth is given as follows,

Substituting (13) into (14) gives optimal surplus wealth process

Substituting (15) into (13) provides optimal consumption process as follows,

or

Applying Ito lemma to (15) gives

As shown in Karatzas and Shreve [

This relation provides optimal portfolio as follows,

where and denote transpose and inverse, respectively; and denotes the D-dimensional vector with each component equal to one. Finally, substituting (16) into (10) provides the value function as follows,

As in the literature [6,13], (16) implies that the optimal surplus consumption is determined as the value of surplus wealth multiplied by the marginal propensity to consume. A unique property of the present model is that the discount rate and hence the marginal propensity to consume depend on the degree of riskiness of habit formation.

Wealth W(t) is composed of surplus wealth W(t)-z(t)/A and subsistence wealth z(t)/A. As for the surplus wealth, the usual two-fund separation theorem holds, so that it is held in the form of the tangency portfolio and the riskless bond. The subsistence wealth is held in the form of the habit hedging portfolio that duplicates random parts of habit formation and the riskless bond that is used to duplicate the drift part.

Note that in the case of deterministic habit formation [^{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.

^{9}In the case of time-additive utility, Gollier [

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.