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Optimization of investor's intertemporal consumption strategy in a continuous-time framework.

Introduction

We consider the intertemporal consumption optimization problem of an investor. We assume that the investor invest his resources into a risk-free asset and N risky assets. The investor will be encourage to continuously invest into the financial markets if the market is growing positively. Our aim is to determine the behavoiur of the investor towards consumption. In the related literature, Mason and Wright (2001), Chan and Viceira (2000) and Chamberlain and Wilson (2000) provide and approximate solution of investor intertemporal consumption strategy with some restrictive assumption in other to obtain analytical results. Blanchard and Fischer (1989) considered investor's intertemporal consumption strategy with the assumption that the asset return is non-stochastic. They adopted the quadratic utility function that has the characterization of linear marginal utility. This quadratic utility has an unattractive description of investor attitude towards risk since it involve increasing absolute risk aversion. Bodie, Merton and Samuelson (1992) considered the objective of maximizing expected discounted lifetime utility and assume that the utility function has two arguments (consumption and Labour/leisure). Zhang (2007) considered a closed-form solution to the intertemporal consumption problem, in which both asset return and labour income uncertainty are considered simultaneously with the assumption that the investor has no initial capital. Toche (2005) assumed that the uncertainty is about the timing of the income loss as well as the assumption of non-stochastic asset return. Pitchford (1991) similarly, takes the form of uncertainty as the timing is the reversal of an income shock. Zariphopoupou (1999) found a closed-form solution when the investor maximizes its terminal wealth and invests in a bond and a risky security whose price process has non-linear coefficients in the shock level. When there is intermediate consumption, optimal portfolio composition is expressed via the solution of a non-homogeneous linear parabolic function. She uses viscosity solution theory in the case when the HJB equation has only a weak solution.

In this paper, we consider the optimal intertemporal consumption problem by assuming that the asset prices is driven by geometric Brownian motion. We also remove the assumption made by Zhang (2007) that the investor has no initial capital. We consider a situation in which the investor has initial capital by adopting the Martingale method.

The Financial Market

In this section, we consider the financial market M that consists of n +1 assets being open and traded upon continuously. These assets include financial stocks and bond, respectively.

Risk-free Bond

The risk-free bond with price process, B(t) is given by the dynamics

dB(t)/B(t) = r(t)dt, B(0) = 1, (1)

where r(t) represents the short term interest rate at time t.

Financial stock price dynamics

The n assets are the risky financial assets, whose prices are denoted by [S.sub.i](t), i = 1,2,..., n. The dynamics of [S.sub.i](t) given by

d[S.sub.i](t)/[S.sub.i](t) = [[mu].sub.i](t)dt + [n.summation over (j=1)][[sigma].sub.i,j](t)d[W.sub.j](t), 1 = 1,2,...,n, (2)

[S.sub.i](0) = [s.sub.i] > 0, i = 1,2,...,n, (2)

where the randomness W(t) = [{[W.sub.1](t),...,[W.sub.n](t)}.sup.T]; t [member of] [0,T] is n-dimensional Brownian motion defined on a complete probability space ([OMEGA],F,P), where P is the real world probability measure and [[sigma].sub.i,j](t) is the volatility of asset i at time t with respect to changes in [W.sub.j](t). [mu](t) := {[[mu].sub.1](t),...,[[mu].sub.n](t)} is the appreciation rate vector. Moreover, [sigma](t) = [{[[sigma].sub.i,j](t)}.sup.n.sub.i,j] is the volatility matrix referred to as the coefficients of the market. The volatility matrix {[sigma](t) = [{[[sigma].sub.i,j](t)}.sub.1[less than or equal to]i, j[less than or equal to]n], 0 [less than or equal to] t [less than or equal to] T} are progressively measurable with respect to the filtration [F.sup.S] and satisfy the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

where [parallel]*[parallel] denotes the Euclidean norm in [[Real part].sup.n] and where [[sigma].sub.i](t) denotes the i-th row of [sigma](t). The filtration F = [(F(t)).sub.t[greater than or equal to]0], represents the information structure generated by the Brownian motion and is assume to satisfy equation (3).

We assume that the financial market is arbitrage-free, complete and continuously open between time 0 and T, i.e., there is only one process [theta](t) satisfying

[theta](t) = [([[sigma].sub.i]).sup.-1](t)([[mu].sub.i](t) - r(t)[1.sub.n]), 0 [less than or equal to] t [less than or equal to] T, 1 [less than or equal to] i [less than or equal to] n, (4)

with [1.sub.n] = [(1,...,1).sup.T] [member of] [[Real part].sup.n] where [sigma](t,w) is non-singular, for ([lambda] [cross product] P) almost everywhere and (t,w) [member of] [0,T] x [OMEGA]. The exponential process

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

is assumed to be a martingale, and the risk-neutral equivalent martingale measure, denoted by

[??](A) = E[Z(T)[1.sub.A]], A [member of] F(T). (6)

We further defined the state-price density process by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[LAMBDA](t) is referred to as the stochastic discount factor.

Using Ito lemma on Eq.(1) and (2), we respectively obtain the following solutions

B(t) = B(0)exp[[[integral].sup.t.sub.0]r(u)du]

= exp [[[integral].sup.t.sub.0]r(u)du] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Definition 3.1: Let [DELTA] consists of portfolio [[DELTA].sub.i] (t) the proportion of wealth invested in financial stock i, at time t. Again, let C(t) be the consumption process at time t, then the pair ([DELTA], C) is said to be self-financing if the corresponding wealth process [X.sup.[DELTA],C](t), t [member of] [0, T], satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Where, 1 - [n.summation over (i=1)][[DELTA].sub.i](t) is the proportion of wealth invested in the bond market.

The requirement of being self-financing states that the change in wealth must equal the different of the capital gains and infinitesimal consumption. Substituting the assets returns in Eq. (1) and (2) into Eq (10), we obtain the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where, [DELTA](t) [equivalent to] [([[DELTA].sub.1](t),...,[[DELTA].sub.n](t)).sup.T], [1.sub.n] [equivalent to] [(1,...,1).sup.T] [member of] [[Real part].sup.n].

The terms appearing in the braces in Eq. (11) are referred to as the percentage of the capital gains, during a time interval of dt and are made up of three parts:

The percentage of an average underlying gross return on the n+1 assets, which is reflected by the term r(t)dt.

The percentage of a risk premium for investing in n financial stocks, which is reflected by the term ([DELTA])'(t)([mu](t) - r(t)[1.sub.n])dt;

The volatility term proportional to the amount of the investment in the financial stocks, which is the term ([DELTA])'(t)[sigma](t)dW(t);

The proposition below tells us that the sum of an accumulated discounted consumption process and its corresponding discount wealth process can be expressed as a stochastic integral with respect to the Brownian motion.

Theorem 1: Let [X.sup.[DELTA],C](t) be the investor's wealth process of a portfolio process [DELTA], then the process

[LAMBDA](t)[X.sup.[DELTA],C](t) + [[integral].sup.t.sub.0][LAMBDA](u)C(u)du is a P-local martingale.

Proof: (see Zhang, 2007)

Corollary 1: Let [X.sup.[DELTA]](t) be the portfolio value process of a portfolio [DELTA] for the investor, then we have that

[X.sup.[DELTA]](t)[LAMBDA](t) = x + [[integral].sup.t.sub.0][X.sup.[DELTA]](u)[LAMBDA](u)[([DELTA])'(u)[sigma]- [theta]]dW(u) (12)

is a P-local martingale.

Corollary 2: Let [X.sup.[DELTA],C](t) be the investor's wealth process of a portfolio [DELTA], then the process

E[[LAMBDA](t)[X.sup.[DELTA],C](t) + [[integral].sup.t.sub.0][LAMBDA](u)C(u)du = x (13)

Corollary 3: Let [X.sup.[DELTA]](t) be the portfolio value process of a portfolio [DELTA] for the investor's, then we have that

E[[X.sup.[DELTA]](t)[LAMBDA](t)] = x. (14)

Optimization program of investor's consumption strategy

In this section, we consider the optimization process using Lagrangian function.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

subject to:

d[X.sup.[DELTA],C](t) = [X.sup.[DELTA],C](t)[r(t)dt + [pi]'(t)[sigma](t)([theta](t)dt + dW(t))] - C(t)dt (16)

[X.sup.[DELTA],C](0) = x

With

Q(x) [equivalent to] {([pi],C) [member of] Q(x): E[([[integral].sup.T.sub.0]U(C(t))exp(-[rho]t)dt < [infinity]} (17)

U(x) = [x.sup.1-[gamma]]/[1-[gamma]], [gamma] > 0, [gamma] [not equal to] 1 (18)

Eq. (15)-Eq. (18) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Subject to:

E[[[integral].sup.T.sub.0][LAMBDA](t)C(t)dt] = x (20)

(see Zhang, 2007).

The Lagrandian function of the problem is written as follows:

L(C(t),[lambda]) [equivalent to] E[[[integral].sup.T.sub.0]U(C(t))exp[-[rho]t]dt + [lambda](x - [[integral].sup.T.sub.0][LAMBDA](t)C(t)dt]

Finding partial derivative with respect to C(t) and equate to zero, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

[C.sup.*](t) = [([lambda][LAMBDA](t)exp[[rho]t]).sup.-1/[gamma]]

= [[lambda].sup.-1/[gamma]][[LAMBDA].sup.-1/[gamma]]exp[-[rho]t/[gamma]] (21)

Finding partial derivative with respect to [lambda] and equate to zero, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

From Eq.(17), we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that exp exp[-[[[gamma]-1]/[gamma]][theta]W(t) - [1/2][([[gamma]-1]/[gamma]).sup.2][[theta].sup.2]t] is a martingale and thus has unit expectation, we obtain

E[[[LAMBDA](t).sup.[[gamma]-1]/[gamma]]] = exp[-[[[gamma]-1]/[gamma]](r + [[theta].sup.2]/2[gamma])t] (23)

Now Eq. (22) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Setting g = [[[gamma]-1]/[gamma]](r + [[theta].sup.2]/2[gamma]) + [rho]/[gamma]

Eq.(24) becomes

[[lambda].sup.-1/[gamma]][[integral].sup.T.sub.0]exp[-gt]dt = x (25)

Finding the value of the integral, we obtain

[[lambda].sup.-1/[gamma]][[1/g](1 - exp[-gT])]= x

[[lambda].sup.-1/[gamma]] = gx/(1 - exp[-gT]) (26)

Therefore, the expected optimal consumption is express as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

The Rate of Growth of the Investor's Wealth and Economic Interpretation

The growth rate of the expected optimal consumption of the investor's wealth as well as its economic interpretation are given in this section. Hence, the growth rate of the expected optimal consumption of the investor's wealth is given by

[1/[[??] x (t)]][[d[??] x (t)]/dt] = [1/[gamma]](-[rho] + r + [[[gamma]+1]/2[gamma]][[theta].sup.2]) (28)

Eq. (29) is referred to as the Euler equation for the intertemporal maximization under uncertainty. The positive term 62 captures the uncertainty of the financial market. When the financial market is risky, it will induces the investor to shift consumption overtime. Obviously, the growth rate is decreasing in [gamma] or increasing in the elasticity of the investor's substitution between consumption 1/[gamma]. If [gamma] is smaller, the smaller the marginal utility changes as consumption changes, the more the investor is willing to subtitute consumption between periods. If r - [rho] is held fixed, a higher market price of risk [theta] will bring about sharp gradient of the expected consumption, thus a more prudent the investor will be. If the risk premium [mu] - r equals zero, then the market price of risk becomes zero as well. In that case, all the wealth will be optimally be invested into the risk-free asset. In that case, the Euler equation under uncertainty will be equal to the well known Euler equation under certainty, that is,

[1/[[??] x (t)]][[d[??] x (t)]/dt] = [1/[gamma]](r - [rho]) (29)

Eq. (29) states that, when the nominal interest rate exceeds the discount rate, the expected consumption of the investor with a CRRA with respect to consumption is rising and falling if the reverse holds. From Eq. (28), the growth rate of the expected consumption is strictly positive if [rho] < r + [[[gamma]+1]/2[gamma]][[theta].sup.2], strictly negative if [rho] > r + [[[gamma]+1]/2[gamma]][[theta].sup.2]and constant if [rho] = r + [[[gamma]+1]/2[gamma]][[theta].sup.2].

Conclusion

We derived the optimal consumption strategy of an investor with initial capital using Martingale approach. We derived analytical closed-form solution to the intertemporal consumption strategy of the investor's problem. We further established the Euler equation under both certainty and uncertainty. We found that when the financial market is doing well, investors will be more prudent. We also found that if the expected rate of return from the risky assets equals the short term interest rate, the investor will prefer to leave his/her resources in the risk-free asset alone. This to a great extent will leads to high level of consumption especially when the investor's preference rate is greater than the short term interest rate.

Reference

[1] Blanchard, O.J. and Fischer, S. (1989), "Lectures on Macroeconomics", The MIT press.

[2] Bodie, Z., Merton, R.C., and Samuelson, W.F. (1992), "Labour supply flexibility and portfolio choice in a life cycle model", Journal of Economic Dynamics and Control 16, 427-449.

[3] Champbell, J., and Viceira, L.M. (2000), "Optimal intertemporal consumption under uncertainty", Review of Economic Dynamics 3, 365-395.

[4] Chan, Y.L. and Viceira, L.M. (2000), "Asset allocation with endogeneous labour income", Preliminary.

[5] Hamada, M. (2001), "Dynamic portfolio optimization and asset pricing: Martingale methods and probability distortion functions", Ph.D thesis, Department of Mathematics, The University of New South Wales, Britain.

[6] Mason R. and Wright, S. (2001), "The effects of uncertainty on optimal consumption", Journal of Economic Dynamics and Control 25, 185-212.

[7] Pitchford, J. (1991), "Optimum responses of the current account when income is uncertain", Journal of Economic Dynamics and Control, Vol. 15, pp. 285-296.

[8] Toche, P. (2005), "A tractable model of precautionary saving in continuous time", Economics letters 87, 267-272.

[9] Zariphopoulou, T. (1999), "Optimal investment and consumption models with non-linear stochastic dynamics", Mathematical Methods of Operation Research, Vol. 50, pp. 271-296.

[10] Zhang, A. (2007), "Stochastic optimization in finance and life insurance: Application of the Martingale method", Ph. D thesis, Department of Mathematics, University of Kaiserslautern, Germany.

Chukwuma Raphael Nwozo (1) and Iwebuke Charles Nkeki (2)

(1) Department of Mathematics, University of Ibadan, Ibadan, Oyo State, Nigeria Email: crnwozo@yahoo.com

(2) Department of Mathematics, University of Benin, P. M. B. 1154, Benin City, Edo State, Nigeria Email: nkekicharles2003@yahoo.com
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Author:Nwozo, Chukwuma Raphael; Nkeki, Iwebuke Charles
Publication:International Journal of Computational and Applied Mathematics
Date:Mar 1, 2010
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