This paper has considered Merton's consumption and investment choice prob-lem in the case of non-addictive habits. As shown by Detemple and Karatzas (2003), for the case in which habit is non-addictive there is a signicant re-lationship between the solution with and without the non-negativity con-straint. By using the martingale representation theorem, Detemple and Za-patero (1992, p. 257-259) have proved the existence and uniqueness of the optimal portfolio of an agent with habit utilities. However, they characterize this solution only as an unknown process. Optimal consumption and portfo-lio choice of the agent with habit utility function can also be found via the Hamilton-Jacobi-Bellman equation following Egglezos (2007, p.48-50). Then
the r(t), λ(t), σ(t) have to follow Markov processes. The consideration of the previous section allows a non-Markovian framework.
The so-called habit factors a(t) and b(t) are deterministic. The optimal consumption and investment choices were shown in the general case as well as in the case of the HARA utility function. The optimal portfolio solution was found using results on isomorphism in Schroder and Skidas (2002).
Throughout this paper, I have assumed complete markets, in line with the bulk of the literature. This makes it easier to nd solutions. Obviously, this assumption is restrictive and rules out many interesting aspects of nancial markets. By using numerical methods, it is possible to relax this assumption and consider more complicated and realistic situations. Future research could consider these alternative scenarios.
Appendix
.1 Isomorphism between the separable and habit case
Schroder and Skiadas (2002) also consider the isomorphism in the case of more general market processes where stock prices and money markets can be non-Markovian processes.
The isomorphism is useful,if a primal problem is easier to solve and the solution can be reformulated to obtain the habit solution. The duality results are symmetric and the roles of primal and dual qualities are interchangeable.
The process in equation (2.4.15): ρ(t)=. Et are material for the solution. BsT is a unit discount bond that matures at time T andσBis the volatility of that bond. The solution of a primal optimization problem can be found from the dual problem
supcˆ(t)E If the dual agent's problem has been solved and the optimal consumption plan ˆc∗ is the corresponding optimal consumption plan,the price of risk, wealth process and portfolio choice are
ˆ
and
.2 Solutions in the time separable case and in the habit in case
This derivation is a straightforward extension of the proof of Theorem 1 in appendix of Munk (2008) and proposition 1 in the appendix of Munk and Sørensen (2000). whereKand(1−K)are weights between utility in periods[t, T]and terminal utility.
According to the martingale approach, the problem can be solved as a static problem by using the Lagrange method. The rst order conditions are
Ke−υ(s−t)(βc(s)
and
Inserting equations (.2.5) and (.2.6) into the budget constraint gives w(t) =λ−1μ Et
A positive stochastic process O(t) adapted to the ltration generated by dz(t) is dened by:
The dynamics of O(t) can be described by
dO(t) =O(t)[μOtdt+σOtdz(t)]. (.2.9) Then, optimal consumption plan is
c(s) =Kμ1w(t)−L(t) and optimal terminal utility is
w(T) = w(t)−L(t)
This optimal plan for the terminal wealth in (.2.12) is valid for any t,
Following the appendix of Munk and Sørensen (2000) and using Itˆo's rule we obtain a stochastic dierential equation of an optimal investment policy
dw(t)
w(t) = [...]dt+ (1− L(t) w(t))1
μλ(t)dz+σOtdz. (.2.13) Inserting a testing equation for optimal portfolio choice
π(t) = (1− L(t)
w(t))(σ(t))−1(1
μλ(t) +σOt) (.2.14) into budget constraint equation (2.3.10)we obtain
dw(t)
w(t) = [...]dt+ [1
μλ(t) +σOt]dz. (.2.15) We can see that using the strategy (.2.14)the diusion coecients of(2.3.10) and (.2.15)match and it gives the optimal portfolio choice in the non-habit case. Under the martingale solution approach, the drift term will be matched if the diusion coecients are matched.
By next using Schroder and Skiadas's rule for optimal portfolio choice (2.5.2), we obtain the optimal choice in the habit case.
Let consider equation (.2.8) in the caseK = 1:
Using the relation between state-price deators in separable and in habit markets: ζˆt =ζt(1 +ρ(t)), we get
⇔O1(t) =M(t)(1 +ρ(t))1−μμ . (.2.18) By using Itˆo's rule and (.2.18), we obtain
dO(t)
O(t) = [...]dt+
σM(t)+ (1
μ −1)(μ1 −1)ρ(t) 1 +ρ(t) σρ(t)
dz. (.2.19) We can see that the percentage diusion process of O1(t)is
σO
1(t)=σM(t)+(1μ−1)ρ(t)
1 +ρ(t) σρ(t). (.2.20) Inserting(.2.14), (.2.20) and (.1.5) into π in equation(2.5.2) we get
π(t) = (1−bρ(t)h(t)
w(t))[σM(t)+(1μ−1)ρ(t)
1 +ρ(t) ]σρ(t)+ (1 + h(t)
w(t))(σ(t))−1φ(t) (.2.21) which gives equation (2.5.7) after reformulation.
.3 Symbol description
yx= dydx derivative of y with respect to x x the transpose of matrix x
x(t) =xt the value of process xat time t
x∗ the optimal value of xrelated to dened optimization problem St the price process of risky assets
αi(·) the drift of a risky asset i σi(·) the volatility of a risky asset i zt standard Brownian motion
Et expectation given information available at time t P, Q probability measures
Ωprobability set rt riskless interest rate
λt the market price of risk process ζ(t)state-price density
ζˆ(t) the state price density in the habit case λt the market price of risk process
ΓTt the price at time t of a zero-coupon bond which pays a principal of one at time T
U(·) utility function V value function
υ subjective discount rate ct,0≤t≤T consumption ˆ
ct subsistence consumption dened in page 30 c∗ consumption in a constrained problem Z(·)the bequest function
h(t) habit factor
a(t) andb(t)habit coecient
wt,0≤t≤T the value of wealth process at time t μ the degree of risk aversion
w0 initial wealth
I(·) inverse function of marginal utility
π proportional amount which agent invests to risky object L(·) a Lagrange function
y a Lagrange multiplier
q marginal cost of subsistence consumption per unit of standard of living ρ(t) process is dened in page 42
G(t) process is dened in page 44 φ(t)process is dened in page 49 O(t) process is dened in page 46
Other necessary denitions are in the text.
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Chapter 3
Monte Carlo Computation of Optimal Portfolio Choice with Habit Formation
Abstract
This paper studies optimal consumption and portfolio choice of an investor with habit formation in preferences. The Monte Carlo co-variation (MCC) method is used for optimal portfolio selection when an investor's preferences are time-separable. This paper works on the MCC method so that it is applicable to the case of more general utili-ties. As an example,I solve the optimal portfolio problem in the case where the interest rate adheres to Cox-Ingersoll-Ross dynamics and stock prices are mean reverting using the method and compare my results to the time-separable case.
3.1 Introduction
Time separability of consumption utilities is a common assumption in the theory of nancial economics. Various empirical studies suggest that there are problems with this assumption. Sometimes these problems are solvable by applying a more general utility formulation.
Applying time separable utilities, rational expectation models often gen-erate results that are empirically valid only if we assume a very risk-averse investor. If the risk aversion coecient is plausible, then the representative investor in the models puts much more money in the risky investment than has been observed empirically. Between 1889-2005, the average return on equity in the U.S stock market was seven percent and the average yield on short-term debt was less than one percent. Mehra and Precott (1985) and Mehra (2006) show that the common general equilibrium model with separa-ble utilities cannot explain why the rst rate is so low and the second rate is so high. That so-called equity premium puzzle is solvable using a more gen-eral form of the utility function. In this paper, I reject the time-separability assumption and assume that an agent's utilities adhere to a more general function, the habit utility function.
Merton (1971) examines the continuous-time consumption-portfolio prob-lem for an individual whose income is generated by capital gains on invest-ments in assets with prices that satisfy geometric Brownian motion. For the solution of an individual's optimization problem, Merton uses Itô's lemma and stochastic analysis. A few papers have studied the consumption and investment problem of an agent with habit utilities either in the general equilibrium or in the partial equilibrium model (e.g. Sundaresan (1989), Constantinides (1990), Ingersoll (1992), Munk (2008)).
Constantinides (1990) and Sundaresan (1989) present a solution to the equity premium puzzle by applying habit utilities. Constantidines's (1990) reason for including habit function in preferences form is simply to nd a the-oretical model that can explain the equity premium puzzle. In the literature, an intuitive interpretation is also given for the habit utility function. There
are temporal dependencies in the sense that utility in period t depends not just on consumption in the same period but also on the level of consumption in previous periods. An individual who consumes a lot in period (t-1) will get used to that high level of consumption, and will more strongly desire consumption in period t (Kocherlakota (1996)).
If the assumption of time separability is rejected, two types of eects are possible: intertemporal substitution or intertemporal complementarity.
In the case of intertemporal substitutes a consumer buys a durable good in period t, but receives the utility of this good in periods t+i, i > 0 without spending any money.
Ferson and Constantinides (1991) empirically study habit persistence in preferences and the durability of consumption goods which both imply time-nonseparability of the derived utility for consumption expenditures. They investigate which eect dominates and nd evidence in monthly, quarterly and annual data that habit persistence dominates over durability. Obviously, nondurables are more habit" than durables. Detemple and Zapatero (1992) and Egglezos (2007) solve the optimal consumption when an investor has habit utilities, but they do not nd an exact solution for an optimal portfolio choice.
Munk (2008) nds a closed-form solution to the optimal consumption and portfolio choice with habit utilities and mean-reverting stock returns.
He also solves numerically the problem in the habit case when the interest rate is stochastic and stock prices are mean-reverting. Munk uses a Monte Carlo simulation to solve the relevant partial dierential equation.
Cvitanic et al. (2003) propose a numerical method for an optimal port-folio choice in the case where the interest rate follows Cox-Ingersoll-Ross dynamics and the stock prices are mean-reverting. This is a very exible method and, by exploiting it, it is possible to solve the optimal portfolio problem in the habit case, making dierent assumptions about nancial as-sets. The only requirements are that markets have to be complete and the expanded opportunity set has to be Markovian, i.e. all parameters of
mar-ket processes depend on the n-dimensional Brownian motion process which describes uncertainty in the economy. In this paper, I extend this method to the problem of an investor with habit utilities.
The rest of the paper is structured as follows. Section 2 gives some set-ups and denes utilities. Sections 3 and 4 present the assumptions related to nancial markets and dene a precise optimization problem. Section 5 shows how to nd the optimal consumption in the case of habit utilities by using the martingale method solution. Section 6 presents the extension of Cvitanic's (2003) Monte Carlo covariation method in the habit case. Section 7 shows the results for the optimal portfolio choice problem and nally, section 8 concludes.