optimal consumption/investment policies with undiversifiable income risk and liquidity constraints

E-mail address: [email protected] (C. Munk).

Journal of Economic Dynamics & Control24 (2000) 1315}1343

Optimal consumption/investment policieswith undiversi"able income risk

and liquidity constraints

Claus Munk

Department of Management, Odense University, DK-5230 Odense M, Denmark

Received 1 February 1997; accepted 25 March 1999

Abstract

This paper examines the continuous time optimal consumption and portfolio choiceof an investor having an initial wealth endowment and an uncertain stream of incomefrom non-traded assets. The income stream is not spanned by traded assets andthe investor is not allowed to borrow against future income, so the "nancial marketis incomplete. We solve the corresponding stochastic control problem numericallywith the Markov chain approximation method, prove convergence of the method,and study the optimal policies. In particular, we "nd that the implicit value theagent attaches to an uncertain income stream typically is much smaller in this incompletemarket than it is in the otherwise identical complete market. Our results suggest thatthis is mainly due to the presence of liquidity constraints. ( 2000 Elsevier Science B.V.All rights reserved.

JEL classixcation: C61; G11

Keywords: Optimal consumption and portfolio policies; Undiversi"able income risk;Liquidity constraints; Wealth equivalent of income; Markov chain approximation

0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 1 9 - 6

1Cox and Huang (1989) demonstrated a similar result for the no-income case.

1. Introduction

The consumption/investment choice of a price-taking investor is a classicalproblem of "nancial economics. In two pioneering papers, Merton (1969, 1971)introduced stochastic control techniques to analyze the continuous-time versionof the problem for an investor with an additively time-separable utility function.In particular, Merton studied the case where the investor has access to a com-plete "nancial market in which risky asset prices follow geometric Brownianmotions and the investor's utility function for consumption is of the constantrelative risk aversion type. He was able to solve analytically the Hamilton}Jacobi}Bellman (HJB) equation associated with the problem and hence toobtain closed-form expressions for the optimal control policies in feedback form,both for a "nite horizon and an in"nite horizon.

One of many interesting generalizations of Merton's setting appears when theinvestor besides having an initial endowment of wealth also receives a stream ofincome throughout her planning horizon. Merton (1971, Section 7) stated thatthe optimal policies when the agent has a deterministic stream of income areas if the agent has no income stream but instead adds the capitalized lifetimeincome #ow discounted at the risk-free rate to her initial wealth. However,it is easy to show, see e.g. He and Pages (1993, Example 1), that underthe policies derived this way the wealth process may go below zero. Due tomoral hazard and adverse selection problems, it may be impossible for theinvestor to borrow against future income, so that the investor can onlychoose her consumption/investment policy among those that keep her "nancialwealth non-negative.

He and Pages (1993) study a model where the income rate is spanned, so thatthe only source of incompleteness is that liquid wealth has to stay non-negative.Using the martingale techniques of Cox and Huang (1989) they "nd that thepresence of liquidity constraints has a smoothing e!ect on the optimal consump-tion across time. If the investor expects her income to rise, she will increase herconsumption at a smaller rate than if she was not subjected to liquidityconstraints. In a similar set-up, El Karoui and Jeanblanc-PicqueH (1996) demon-strate that the optimal trading strategy is to invest part of the wealth in thestrategy which is optimal in the corresponding unconstrained case, and theremainder in an American put option written on the optimal wealth process inthe unconstrained case.1 They also derive a formula linking the optimal con-sumption rate to current wealth and current income, and they show that for zerowealth the optimal consumption rate is a smaller fraction of current income inthe liquidity constrained case than in the unconstrained case.

1316 C. Munk / Journal of Economic Dynamics & Control 24 (2000) 1315}1343

Maintaining the liquidity constraints of He and Pages, but dropping thespanning assumption, Du$e and Zariphopoulou (1993) study an in"nitehorizon model with a single risky asset whose price follows a geometricBrownian motion and a non-negative income rate given by an Ito( processdriven by a Brownian motion imperfectly correlated with the risky asset price.In this general setting we cannot be sure that the value function, also known asthe indirect utility function, is a solution of the associated HJB equation. In factthe value function may not even be smooth. Du$e and Zariphopoulou are ableto show that the value function is the unique constrained viscosity solution in theclass of concave functions of the HJB equation.

The model of Du$e and Zariphopoulou is specialized in Du$e et al. (1997),henceforth abbreviated DFSZ. For power utility of consumption they reducethe control problem from a two-variable (wealth and income) to a one-variable(wealth divided by income) problem and show that the HJB equation for thereduced problem has a unique smooth solution. The optimal policies and thevalue function of the original problem can easily be restored from the solution tothe reduced problem. With analytical derivations DFSZ are able to "nd somecharacteristics of the solution, but they cannot solve the problem completely. Inan almost identical model Koo (1998) expresses the optimal policies in terms ofthe current liquid wealth and a measure of the agent's implicit value of the futureuncertain income stream and is able to analytically derive some general proper-ties of these policies. In Section 2 below, we shall review the main "ndings ofDFSZ and Koo, and we introduce an intuitively more appealing measure of theimplicit value of future income and relate it to Koo's measure. For othercontinuous-time models with stochastic income, see Andersson et al. (1995),Cuoco (1997), Detemple and Serrat (1997), and Svensson and Werner(1993).

The main contributions of this paper are to demonstrate how the reducedproblem of DFSZ can be solved by a relatively simple converging numericalscheme and to study the properties of optimal policies in the set-up of DFSZ indetail. The numerical method adopted is the Markov chain approximationapproach which basically approximates the continuous time, continuous statestochastic control problem with a discrete time, discrete state stochastic controlproblem that is easily solved numerically. The Markov chain approximationapproach is described in Kushner (1990) and more detailed in Kushner andDupuis (1992). See also Fleming and Soner (1993, Chap. IX). The method haspreviously been applied to consumption/portfolio problems by Fitzpatrick andFleming (1991) and Hindy et al. (1997).

We contrast the numerically computed value function and optimal controlsto the complete market case where the income rate is spanned by traded assetsand the investor is not liquidity constrained. In particular, we "nd that theimplicit value the investor attaches to the uncertain income stream is muchsmaller in the non-spanned, liquidity constrained case than in the complete

C. Munk / Journal of Economic Dynamics & Control 24 (2000) 1315}1343 1317

market case, even for high ratios of initial wealth to initial income. We "nd thatthis implicit value of income is very insensitive to the correlation betweenchanges in the risky asset price and changes in the income rate. Since a perfectlypositive correlation corresponds to spanning, this suggests that the large di!er-ence between the complete markets case and the non-spanned, liquidity con-strained case is to be attributed to the liquidity constraints. We study thesensitivity of both the optimal policies and the implicit value of the incomestream with respect to various parameters. Among other things we "nd that,ceteris paribus, liquidity constraints as modeled in this paper are most restrictivefor agents with a low "nancial wealth relative to income, an income ratepositively correlated with changes in the risky asset price, and a high timepreference for consumption.

While this paper apparently contains the "rst quantitative study of theproperties of the optimal consumption and portfolio policies in a continuous-time set-up with undiversi"able income risk and liquidity constraints, relateddiscrete-time models have been considered by other authors. Koo (1995) alsohas a non-spanned income process and a liquidity constraint, but in a discrete-time set-up the liquidity constraint explicitly implies bounds on the riskyinvestment in each period, which is not the case with continuous trading.Furthermore, Koo imposes even tighter bounds on the investment in the riskyasset than those implied by his liquidity constraint. Hence, it is di$cult toevaluate the implications of the liquidity constraint from his numerical results.Koo only considers the case where the shocks to the income rate process and theshocks to the risky asset price are uncorrelated, while we allow for a non-zerocorrelation and provide a detailed comparative statics analysis. For otherdiscrete-time models of optimal consumption and investment choice withincome, see, e.g., Cocco et al. (1998), Deaton (1991), and Heaton and Lucas(1997).

The outline of the rest of the paper is as follows. The problem is formalized inSection 2 which also reviews the analytical "ndings of DFSZ and discusses twoimplicit measures of the value of an uncertain non-spanned stream of income. InSection 3, we implement the Markov chain approximation method and prove itsconvergence. Numerical results are presented and discussed in Section 4. InSection 5, we study the sensitivity of the results with respect to selected para-meters. Finally, Section 6 brie#y summarizes the paper.

2. The problem

2.1. Statement of the problem

Let="(=1,=

2) be a standard two-dimensional Brownian motion on some

probability space (X,F,P), and let F"MFt: t50N be the "ltration generated by


2R`

denotes [0,R).

=. Consider an investor who wants to maximize her expected life-time utilityfrom consumption E[:=

0e~btu(c(t)) dt]. The investor can invest in a risky asset

with a price process S( ) ) given by

dS(t)"S(t)(bdt#pd=1(t)), S(0)'0,

and in a riskless asset with a constant continuously compounded rate of return r.The investor has an initial wealth endowment x50 and receives income fromnon-traded assets at a rate given by the process >( ) ), where

d>(t)">(t)(kdt#dod=1(t)#dJ1!o2 d=

2(t)), >(0)"y'0.

Here b, p, k, and d are positive constants, and o3(!1,1) is the correlationbetween changes in the risky asset price S and changes in the income rate >.Hence, the income rate process is not spanned by traded assets, i.e. the investorfaces undiversi"able income risk.

A consumption process is de"ned to be an F-progressively measurable processc:X]R

`PR

`with E[:T

0c(t) dt](R for all ¹50 and E[:=

0e~btu(c(t)) dt](R.2

c(t) is the rate of consumption at time t. The set of consumption processes isdenoted by C. A portfolio process is an F-progressively measurable processn:X]R

`PR satisfying :T

0n(t)2dt(R (a.s.) for all ¹50. n(t) is the dollar

amount invested at time t in the risky asset. The set of portfolio processes isdenoted by P.

De"ne X(t) as the time t value of the agent's portfolio of "nancialassets. Hence, X(t) is the liquid wealth of the investor at time t and does nottake into account the value of her future uncertain income stream. Givena consumption process c3C and a portfolio process n3P, X( ) )"Xc,n( ) )evolves as

dX(t)"(rX(t)#n(t)(b!r)!c(t)#>(t)) dt#n(t)pd=1(t), X(0)"x.

(1)

The investor can choose c and n from the set A(x,y) of admissible controls,where

A(x,y)"M(c,n)3C]P: Xc,n(t)50 (a.s.), ∀t50N.

The admissible control policies have the property that the liquid wealth isnon-negative (with probability one) at all points in time. The investor is notallowed to borrow funds against her future uncertain income.

De"ne the value function of the investor's problem by

v(x,y)" sup(c,n)|A(x,y)

ECP=

0

e~btu(c(t)) dtD. (2)


3 It is not clear a priori that the value function v is twice di!erentiable and hence that the HJBequation (3) has a solution in the classical sense. In fact, the HJB equation is of the degenerate elliptictype, so one cannot expect v to be a smooth solution to Eq. (3); see, e.g., the discussion in Flemingand Soner (1993, Section IV.5).

4The reduction idea was suggested by Davis and Norman (1990). A reduction is also possible inthe case of logarithmic utility, u(c)"log c, and for the "nite horizon case, a reduction of theproblem's dimension is possible, e.g., when the utility function for terminal wealth is identically equalto zero or if it is identical to the utility function of consumption. The homogeneity property is alsosustained if the amount invested in the risky asset is constrained to a convex cone, a speci"cationwhich contains many portfolio constraints of interest; see, e.g., Cuoco (1997). There can also bemultiple risky assets as long as their drift and noise terms are constant. Of course, this will increasethe number of controls in the reduced problem, but not the dimension of the state variable. Indeed,the extension to multiple risky assets is a real extension of the model in the sense that portfolioseparation does not obtain.

The HJB equation3 associated with this control problem is

bv(x,y)"1

2

L2vLy2

(x, y)d2y2#Lv

Lx(x, y)[rx#y]

#

Lv

Ly(x, y)ky#sup

c6 |R`Gu(c6 )!

Lv

Ly(x, y)c6 H

#supn6 |R G

1

2

L2vLx2

(x, y)n6 2p2#L2v

LxLy(x, y)n6 pdoy#

Lv

Lx(x, y)n6 (b!r)H.

(3)

2.2. Reduction of the problem

Now assume that the investor has a CRRA (constant relative risk aversion)utility function

u(c)"cc, 0(c(1,

where 1!c is the coe$cient of relative risk aversion. The subsequent stepsmeant to reduce the dimension of the problem from two to one follow DFSZ.For any positive constant k, it follows from the linearity of the wealth dynamicsin Eq. (1) that (c,n)3A(x, y) if and only if (kc, kn)3A(kx,ky) and hence thatv(kx, ky)"kcv(x, y), i.e., v is homogeneous of degree c. As demonstrated below,this allows a reduction in the dimension of the problem from two to one.4

De"ne F: R`PR

`by F(z)"v(z,1). Then v can be recovered from F since, for

yO0, v(x,y)"ycv(x/y,1)"ycF(x/y). For y"0, the value function v(x,0) isknown from Merton's work. De"ne

A"

b!rc1!c

!

c(b!r)2

2(1!c)2p2.


Under the assumption A'0, Merton showed that v(x, 0)"Ac~1xc withoptimal policies c(t)"C(X(t), 0) and n(t)"P(X(t), 0), where

C(x, 0)"Ax, P(x, 0)"b!r

p2(1!c)x. (4)

Exploiting v(x,y)"ycF(x/y), we get from Eq. (3) that

bK F(z)"1

2d2z2FA(z)#k

2zF@(z)# sup

fz~1

M!fF@(z)#(1#f)cN

#supt|R GA

1

2p2t2!pdoztBFA(z)#k

1tF@(z)H , (5)

where

bK "b!kc#12d2c(1!c),

k1"b!r!(1!c)pdo, k

2"d2(1!c)#r!k,

and the variables f"c6 /y!1 and t"n6 /y have been introduced. The maxi-mizers of the two sup-terms are

fH(z)"(F@(z)/c)1@(c~1)!1, tH(z)"doz

p!

k1

p2

F@(z)FA(z)

,

so for x, y'0 the candidate optimal consumption and investment policies infeedback form are c(t)"C(X(t),>(t)), n(t)"P(X(t),>(t)), where

C(x,y)"yAF@(x/y)

c B1@(c~1)

, P(x,y)"dop

x!k1y

p2

F@(x/y)

FA(x/y). (6)

By direct computations it can be veri"ed that

supt|R GA

1

2p2t2!pdoztBFA(z)#k

1tF@(z)H

"!

1

2

k21

p2

F@(z)2FA(z)

!

1

2d2o2z2FA(z)#

k1doz

pF@(z)

"supr|R G

1

2p2u2FA(z)#k

1uF@(z)H#

ok1dz

pF@(z)!

1

2d2o2z2FA(z),

where the maximizers tH and uH of the two sup-terms are related by theequation tH"doz/p#uH, and hence Eq. (5) may be rewritten as

bK F(z)"1

2d2(1!o2)z2FA(z)#kzF@(z)# sup

fz~1

M!fF@(z)#(1#f)cN

#supr|R G

1

2p2u2FA(z)#k

1uF@(z)H, (7)

where k"k2#ok

1d/p.


Eq. (7) is actually the HJB equation associated with the control problem

F(z)" sup(f,r)|AK (z)

ECP=

0

e~bK t(1#f(t))cdtD, (8)

wheredZ(t)"(kZ(t)#k

1u(t)!f(t)) dt#pu(t) d=

1(t)

#dZ(t)J1!o2 d=2(t), Z(0)"z, (9)

AK (z)"M(f,u)3C~1

]P: Z(t)50 (a.s.), ∀t50N.

Here C~1

is the set of progressively measurable processes f withE[:t

0f(s) ds](Rand f(t)5!1 for all t50, which is as the set C except that the

non-negativity constraint is replaced with the greater than or equal to !1constraint. Note that the HJB equation (7) can be written as

bK F(z)" supfz~1,r|R

M(1#f)c#Lf,rF(z)N, (10)

where the operator Lf,r is given by

Lf,rF(z)"12(d2(1!o2)z2#p2u2)FA(z)#(kz#k

1u!f)F@(z).

2.3. Review of analytical results

Applying viscosity solution techniques DFSZ are able to show that, under theparameter restrictions bK '0, A'0, and r'k, the reduced HJB equation (7) hasa unique C(R

`)WC2((0,R)) solution F in the class of concave functions; cf.

Theorem 1 of DFSZ. Furthermore, the unique optimal feedback policy (C,P) forthe original problem (2) is given by Eq. (6) for x,y'0, by Eq. (4) for y"0, and by

C(0,y)"AF@(0)

c B1@(c~1)

y, P(0, y)"0,

for x"0 and y'0, and DFSZ provide some results on the behavior of F( ) )near zero; cf. their Proposition 1. DFSZ also show that as the ratio of wealth toincome goes to in"nity, the value function, respectively the optimal policies, areasymptotically equal to the value function, respectively the optimal policies, inMerton's problem.

Koo (1998) shows that, for x'0 and y'0, the homogeneity property of thevalue function implies that the following two mappings B( ) ) and A( ) ) arewell-de"ned continuous functions of z"x/y :

B(x/y)"

Lv

Ly(x, y)

Lv

Lx( x, y)

, A(x/y)"Av(x, y)

(x#B(x/y)y)cB1@(c~1)

.


5 In the case of a certain income stream where d"0, we see that the value function and theoptimal policies are exactly as suggested by Merton (1971, Section 7), cf. our introduction.

Indeed,

B(z)"cF(z)

F@(z)!z, (11)

A(z)"Ac

F@(z)Bc@(1~c)

/F(z).

The value function can be written as

v(x,y)"A(z)c~1(x#B(z)y)c, x'0, y'0,

and hence

F(z)"A(z)c~1(z#B(z))c. (12)

In the complete market where there are no liquidity constraints and theincome rate is spanned, the certainty wealth equivalent of lifetime income isE[:=

0p(t)>(t) dt] where

p(t)"expG!rt!1

2 Ab!r

p B2t!

b!r

p=

1(t)H

is the unique state-price density. De"ne the constant j"r!k#d(b!r)/p.Then a simple computation yields that, if j'0, the certainty wealth equivalentof lifetime income is equal to y/j. If the constants A and j are both positive, wetherefore have that, in the complete market, the value function is given by

v#0.

(x, y)"Ac~1Ax#y

jBc

with optimal consumption policy

C#0.

(x, y)"AAx#y

jB.The optimal risky investment is5

P#0.

(x, y)"b!r

p2(1!c)Ax#y

jB!dpy

j. (13)

Thus, in the complete market, B(z)"1/j and A(z)"A for all z'0.Koo shows that, if bK '0, A'0, r'k, and DoDO1, then, for x'0 and y'0,

optimal consumption is given by

C(x,y)"A(z)(x#B(z)y), (14)


and the optimal amount invested in the risky asset is given by

P(x,y)"b!r

p2

x#B(z)y

1!c#B@(z)#

dop Ax!

1!c1!c#B@(z)

(x#B(z)y)B,where z"x/y; cf. his Theorem 4.1. If, furthermore, j'0, then B(z) is strictlyincreasing and

limz?=

B(z)"1

j, lim

z?=

A(z)"A.

In particular, the ratio of optimal consumption to the accounting total wealthx#y/j is strictly increasing in z.

Based on Eq. (14), Koo interprets the term B(z)y as the implicit value the agentassociates with her future uncertain income stream in the presence of liquidityconstraints and undiversi"able income risk. However, B(z) is by de"nitiona marginal rate of substitution between income and liquid wealth. An alterna-tive, and perhaps more natural, measure of the value of the entire income streamis the wealth equivalent

W(x, y)"infMq50: v(x#q,0)5v(x,y)N,

i.e. the least increase in initial wealth the investor would accept in exchange forthe entire stream of income. Since v(x#q,0)"Ac~1(x#q)c, we get

W(x, y)"A(1~c)@cv(x,y)1@c!x"A(1~c)@cyF(x/y)1@c!x"BH(z)y,

where

BH(z)"A(1~c)@cF(z)1@c!z (15)

and, as before, z"x/y. Notice that from Eq. (12) we have

B(z)"A(z)(1~c)@cF(z)1@c!z.

Since limz?=

A(z)"A, we see that for large z the two income multipliers B(z)and BH(z) will be approximately equal.

3. The numerical method

3.1. An approximating Markov chain

The reduced control problem (8) is solved with the Markov chain approxima-tion approach. Similar approximations have been used to solve other consump-tion/investment problems by Fitzpatrick and Fleming (1991) and Hindy et al.(1997). We approximate the controlled state variable process Z"(Z(t))

t|R`by

a controlled discrete-time Markov chain mh"(mhn)n|Z

`on the discrete state space


6The plus and minus superscripts indicate the positive and negative part, respectively, i.e.x`"maxMx, 0N and x~"maxM!x, 0N.

Rh"M0, h, 2h,2, IhN. Here, z6 ,Ih is an arti"cially imposed upper boundary.A control is a sequence (fh, uh)"(fh

n, uh

n)n|Z

`where fh

n"fh(mh

n) and uh

n"uh(mh

n).

The controls are bounded by the requirements

!14fh(z)4Kfz and Duh(z) D4Krz, (16)

where Kf and Kr are positive constants. The stochastic evolution of the control-led Markov chain mh is given by the transition probabilities6

ph (z, z!h D f,u)"12(p2u2#d2(1!o2)z2)#h(k~z#(k

1u)~#f`)

Qh(z),

(17a)

ph (z, z#h D f,u)"12(p2u2#d2(1!o2)z2)#h(k`z#(k

1u)`#f~)

Qh(z),

(17b)

ph(z, z D f,u)"1!ph(z, z!h D f,u)!ph(z, z#h D f,u) (17c)

for z3Mh,2h,2,(I!1)hN, where

Qh(z)"p2K2rz2#d2(1!o2)z2#h( D k D z# D k1D Krz#maxM1,KfzN).

Here, ph(z, z@ D f, u) denotes the probability of the state changing from z to z@ inone `time stepa when the control (f,u) is currently applied. At the upperboundary we can take

ph(z6 , z6 !h D f,u)"12(p2u2#d2(1!o2)z6 2)#h(k~z6 #(k

1u)~#f`)

Qh(z6 ), (17d)

ph(z6 , z6 D f,u)"1!ph(z6 , z6 !h D f,u). (17e)

Since Z must stay non-negative, we conclude from Eq. (9) that u at Z"0 must bezero and subsequently that f has to be non-positive at Z"0. Therefore, we take

ph (0, h D f,u)"hf~Qh(0)

"f~, (17f)

ph(0, 0 D f,u)"1!ph (0, h D f,u)"1!f~. (17g)

All other transition probabilities are zero.The control (fh,uh)"(fh

n,uh

n)n|Z`

is called an admissible control for mh ifEq. (16) is satis"ed for all z3Rh and mh is a Markov chain when it is controlledby (fh,uh). The set of admissible controls for mh given mh

0"z is denoted Ah(z).


De"ne the interpolation interval function *th(z)"h2/Qh(z) and let *thn"

*th(mhn) and th

n"+n~1

m/0*th

m. De"ne

Fh(z)" sup(fh,rh)|Ah(z)

EC=+n/0

e~bK thn(1#fhn)c*th

nDmh

0"zD, z3Rh. (18)

The dynamic programming equation for the discrete-time Markov chain con-trol problem (18) is

Fh(z)" sup~1yfyKfz, @ r @ yKrz

G*th(z)(1#f)c#e~bK *th(z) +z{|Rh

ph(z, z@ D f,u)Fh(z@)H.(19)

We note that the Markov chain mh is locally consistent with the process Z( ) ),since

E[mhn`1

!mhnD mh

n"z,fh

n"f,uh

n"u]"*th (z)(kz#k

1u!f), (20)

Var[mhn`1

!mhnD mh

n"z,fh

n"f,uh

n"u]"*th (z) (p2u2#d2(1!o2)z2)

#o(h*th(z)), (21)

for every z3Rh, !14f4Kfz, and D u D4Krz. Eq. (20) says that the expectedchange in the Markov chain mh divided by the length of the time step is equal tothe drift of the process Z( ) ), cf. Eq. (9). Similarly, Eq. (21) says that the varianceof the change in mh divided by the length of the time step is approximately equalto the squared volatility of Z( ) ).

3.2. Convergence of the numerical method

We shall argue that the discrete-time value function Fh( ) ) converges to thecontinuous-time value function F( ) ) as hP0, Kf,KrPR, and z6 "IhPR. Theconvergence of the Markov chain approximation approach has been proved forvarious stochastic control problems by, e.g., Kushner and Dupuis (1992, Chap.14), Fleming and Soner (1993, Chap. IX), Fitzpatrick and Fleming (1991), andHindy et al. (1997) using viscosity solution techniques. While these proofs onlyrequire that the value function of the continuous-time control problem isa viscosity solution of the associated HJB equation, we know from DFSZ thatthe value function for the problem (8) is actually a classical solution of the HJBequation (7). Therefore, convergence will follow immediately from the stabilityand consistency of the Markov chain approximation approach which we shallprove below.

De"ne the operator Th on the space B of real functions on Rh by

ThF(z)"e~bK *th(z) +z{|Rh

ph(z,z@ D f,u)F(z@).


Obviously,

DThF1(z)!ThF

2(z)D4e~bK *th(z) +

z{|Rh

ph(z, z@ D f,u)DF1(z@)!F

2(z@)D

4e~bK *h maxz{|Rh

DF1(z@)!F

2(z@)D,

where *h"min

z|Rh *th(z)'0. Hence, Th is a strict contraction in terms of theoperator norm DDTDD"sup

FDDTFDD

=/DDFDD

=where DD ) DD

=is the sup-norm on B.

This implies the stability of the method.Note that with the transition probabilities given by Eq. (17) we have

+z{|Rh

ph(z, z@ D f,u)Fh(z@)"*th(z)Lhf,rFh(z)#Fh(z),

where the operator Lhf,r is given by

Lhf,rFh(z)"12(d2(1!o2)z2#p2u2)D2Fh(z)

#(k`z#(k1u)`#f~)D`Fh(z)!(k~z#(k

1u)~#f`)D~Fh(z),

D`Fh(z)"Fh(z#h)!Fh(z)

h, D~Fh(z)"

Fh(z)!Fh(z!h)

h,

D2Fh(z)"Fh(z#h)!2Fh(z)#Fh(z!h)

h2.

Hence, the discrete dynamic programming Eq. (19) can be rewritten as

1!e~bK *th(z)*th(z)

Fh(z)" sup~1yfyKfz, @ r @ yKrz

M(1#f)c#e~bK *th(z)Lhf,rFh(z)N. (22)

Comparing Eq. (22) with the HJB equation (10) we see that consistency of themethod follows from the facts that (1!e~bK *th(z))/*th(z)PbK , e~bK *th(z)P1, andLhf,rPLf,r as hP0 and Kf,Kr,z6 PR. Note the link between the consistency ofthe method and the local consistency property of the approximating Markov chain.

The optimal control policies can be expressed in terms of the value functionand its derivatives (for the continuous-time, continuous-state problem) or itsdi!erences (for the approximating problem). Given the convergence of the valuefunction and the convergence of "nite di!erences to derivatives it follows thatthe optimal policies for the approximating Markov control problem convergesto the optimal control policies of the continuous-time, continuous-state prob-lem. For details the reader is referred to Hindy et al. (1997, Theorem 5) andFitzpatrick and Fleming (1991, Section 4).

As noted by, e.g., Fleming and Soner (1993, Chap. IX) and Hindy et al. (1997), thespeci"cation of the Markov chain at the arti"cial upper bound is not important forthe convergence of the method, but will, of course, a!ect the quality of the numericalresults. Also note the importance of the reduction of the problem form a


two-variable to a one-variable problem. Due to the non-trivial control-dependenceof the variance}covariance matrix of the original two-dimensional state-variable(X,>) it is not possible to approximate it by a locally consistent Markov chain.

3.3. Solving the approximating Markov chain control problem

The DPE (19) is solved with the policy iteration algorithm. Given an arbitraryadmissible control policy (fh

0, uh

0), i.e. given (fh

0(z),uh

0(z)) for all z3Rh, perform

a policy evaluation by solving the linear system of equations

Fh0(z)"*th(z)(1#fh

0(z))c

#e~bK *th(z) +z{|Rh

ph(z,z@ D fh0(z),uh

0(z))Fh

0(z@), z3Rh. (23)

Next, a policy improvement is computed by

(fh1(z),uh

1(z))

" argmax~1yfyKfz, @ r @ yKrz

G*th(z)(1#f)c#e~bK *th(z) +z{|Rh

ph(z, z@ D f,u)Fh0(z@)H.

Then a new guess Fh1

on the value function is computed on the basis of (fh1,uh

1)

similarly to the computation of Fh0, etc. Note that each policy iteration step

always generates an improvement so that Fhm

converges to Fh from below.A simple criterion for stopping the algorithm is to stop the "rst timesup

z|Rh DFhm(z)!Fh

m~1(z) D(e for some tolerance e'0.

Since the transitions of the approximating Markov chain are to only &nearestneighbors', the equation system Eq. (23) has a tri-diagonal matrix structurewhich makes it very fast to solve. For interior states z, the jth policy improve-ment step amounts to computing

fhj(z)" argmax

~1yfyKfz

M(1#f)c#e~bK *th(z)(!f`D~Fhj~1

(z)#f~D`Fhj~1

(z))N,

uhj(z)"argmax

@r@ yKrzG1

2p2u2D2Fh

j~1(z)!(k

1u)~D~Fh

j~1(z)

#(k1u)`D`Fh

j~1(z)H.

At the upper bound z6 the corresponding equations are

fhj(z6 )" argmax

~1yfyKfz6M(1#f)c!e~bK *th(z6 )f`D~Fh

j~1(z6 )N,

uhj(z6 )"argmax

@r@ yKrz6G!D~Fh

j~1(z6 )A

1

2p2u2#h(k

1u)~BH

implying uhj(z6 )"0 as D~Fh

j~1(z6 )50.


7Note that Qh(z)"supMQh(z, f,u): !14f4Kfz, Du D4KrzN. Fitzpatrick and Fleming(1991) also impose bounds on the controls to avoid control dependent denominators. Theirscheme corresponds to replacing Qh(z, f,u) with a constant Qh given by Qh"supMQh(z, f,u) :z3Rh,!14f4K,!K4u4KN for some constant K. Of course, this will increase the prob-ability of staying at a state which tends to slow down convergence. Since the advantages of havinga constant Qh and, hence, a constant *th, are very limited, the Qh(z) probability scheme is adopted here.

Convergence of the policy space algorithm will, other things equal, be faster,the faster the probability mass &spreads'. A locally consistent Markov chain withph(z,z D f,u)"0 is obtained by replacing the denominator Qh(z) of ph(z, z$h D f,u)with the control-dependent denominator

Qh(z, f,u)"p2u2#d2(1!o2)z2#h( D k D z# D k1u D# D f D ).

This procedure does not require bounding of the controls, but the policyimprovement step is now signi"cantly more complicated, since the controls thenenter both the numerator and the denominator of the probabilities. Therefore,we prefer the scheme with bounded controls.7

4. Numerical results

In this section we present and discuss results from an implementation of thenumerical method outlined in Section 3. The basic economic parameter valuesare taken to be

c"0.5, r"0.1, b"0.15, k"0.05,

b"0.2, p"0.3, d"0.1, o"0.0(24)

unless otherwise indicated. The auxiliary parameters are then

bK "0.17625, k1"0.05, k

2"0.055, k"0.055, A+0.27222.

Kf and Kr must be determined experimentally and are, of course, dependent onthe other parameters. They are chosen as small as possible under the conditionthat their values do not a!ect the optimal controls. Unnecessarily high valuesslow down the method.

4.1. Properties of the numerical method

The Markov chain approximation approach does not provide any measuresof the precision of the numerical results. The reduced problem has a structuresimilar to Merton's no-income consumption/investment problem for which the


8A standard way to assess the speed and smoothness of the convergence of a numerical method isto look at the so-called experimental order of convergence, which can be estimated by examining thevariations in the numerically computed quantity when the grid re"nement h (or, equivalently, I) isvaried. With the default parameter values in Eq. (24) the experimental order of convergence istypically very unstable as the grid re"nement h is varied. For higher values of the contractionparameter bK the convergence is much smoother and the convergence order is roughly one for boththe value function and the controls. (For high levels of z relative to z6 the value function convergesnicely, but for both controls, most pronouncedly for u, the computed experimental order indicatesa lack of convergence.) With such a smooth convergence the Richardson extrapolation techniquecan be successfully superimposed, but since our focus is on lower and economically more appealingb-values we cannot bene"t from extrapolation methods.

9The main determinant of the running time is the number of iterations the policy iterationalgorithm requires for convergence. With the stated value of e the policy space iteration algorithmtypically converges in 5}10 iterations. A smaller e will result in additional time-consuming andessentially futile iterations.

solution is as shown in Section 2.2. An indication of the numerical properties ofour proposed scheme can, therefore, be obtained by studying the performance ofthe Markov chain approximation approach on Merton's problem. Such ananalysis is contained in Munk (1998). The numerically computed value functionwas found to be very precise over the entire range of the wealth level, which is theonly state variable of that problem. The numerically computed optimal controlswere rather imprecise near the arti"cially imposed upper bound on the state vari-able, but otherwise also very precise. For our problem there is also no reason tobelieve that the numerically computed optimal controls at the upper bound areanywhere near the true optimal controls. The error introduced this way willpropagate to lower values of the state variable. The range of states in which thenumerically computed controls can be considered reliable is highly dependent onthe value of the contraction parameter of the DPE, which in our case is theparameter bK . The higher the contraction parameter, the wider the reliable range.

For practical applications the particular values of the controls might not bethat important, as long as they induce a utility level very close to the true, butunknown value function. But since we want to study the economic properties ofthe optimal control policies, we cannot ignore this feature of the numericalscheme. Therefore, the numerical results discussed below do not rely on valuesnear the arti"cial upper bound on z.8

The numerical values presented below are computed with a grid re"nement ofI"20 000 (the corresponding h-values depend, of course, on z6 ). The policyiteration algorithm is implemented with tolerance e"10~5. With these speci-"cations running the computer program on an HP9000/E35 128MB computertakes about 5}20 s depending on z6 and the parameter values.9

Our numerical results con"rm the analytical "ndings of DFSZ concerning thebehavior of F( ) ) near zero and that the value function and optimal policies of theunreduced problem asymptotically are equal to the value function and optimal


Fig. 1. The value function, v(x, y).

policies of Merton's no-income problem as the ratio of wealth to income goes toin"nity.

4.2. The computed value function and optimal controls

Next, we present the computed value function and optimal controls fornon-extreme values of the wealth/income ratio. Results discussed in this subsec-tion are obtained from an implementation with z6 "100. Fig. 1 shows the valuefunction and Figs. 2 and 3 depict the optimal controls. All three surfaces aresmooth and increasing in both coordinate directions as one would expect themto be.

Recall that in Merton's no-income problem, both the optimal consumptionrate and the optimal risky investment are constant fractions of wealth. Withundiversi"able income risk and liquidity constraints these two fractionsdepend on wealth and income. For relatively high levels of wealth the fractionsof consumption and risky investment to wealth are nearly constant acrossincome rates, but they are rapidly increasing with the initial income rate forsmall levels of wealth. When measured relative to the initial income rate bothcontrols are rather insensitive to the level of wealth as long as income is largerelative to wealth, but for small levels of the income rate both the consump-tion/income ratio and the investment/income ratio increase rapidly as wealthincreases.


Fig. 2. The optimal consumption rate, C(x, y).

Fig. 3. The amount optimally invested in the risky asset, P(x, y).


Fig. 4. Convergence of the implicit value of income multipliers to the complete markets incomemultiplier. The results are from an implementation with z6 "200.

4.3. The implicit value of an uncertain income stream

In this subsection we consider the implicit value that the investor associateswith her stochastic income stream given the initial income rate. As discussed inSection 2.3, we can measure this value by B(z)y, where B(z) is given by Eq. (11), oralternatively by BH(z)y, where BH(z) is given by Eq. (15). Fig. 4 shows the twomultipliers B(z) and BH(z) of the implicit value of the income stream as a functionof z. The multipliers are strictly increasing in z, so the value associated witha given uncertain stream of income increases with wealth. Intuitively this isbecause the importance of the liquidity constraint, i.e. the non-negative wealthconstraint, decreases as wealth increases. Recall that for the complete marketscase the multiplier is 1/j. With the parameters in Eq. (24), 1/j"15, so theimplicit value attached to the income stream is much smaller in the presence ofliquidity constraints and non-spanning. Since the magnitude of B(z) and BH(z) isrelatively insensitive to the value of the correlation parameter o, as will bediscussed in the next section, the large di!erence between the complete and theincomplete market income valuation must be imputed mainly to the liquidityconstraints.


Fig. 5. The dependence of the relative optimal risky investment P(x, y)/x on the correlationparameter o. The results are from an implementation with z6 "50.

10Since u(z)"P(x, y)/y!doz/p we have P(x, y)/x"u(z)/z#do/p where z"x/y.

5. Parameter sensitivity

In this subsection we examine the sensitivity of results to the income processparameters o, d, and k, and to the time preference rate b. We shall not presenta similar analysis for the remaining parameters, since the comparative staticswith respect to those parameters are qualitatively the same as for the well-known complete market problem.

The correlation parameter, o. Fig. 5 shows the ratio of optimal risky invest-ment to wealth, P(x,y)/x, as a function of o for four di!erent values of z"x/y.10This ratio, the portfolio weight of the risky asset, is obviously a decreasingfunction of o for all four values of z. Intuitively this is because a negativecorrelation between changes in the income rate and changes in the risky assetprice provides insurance against negative wealth: If the risky asset price andtherefore the value of the investment decreases, the income rate will tend toincrease and vice versa. For high values of z the portfolio weight is nearlyconstant, whereas for low z it is steeply decreasing in o. The intuition for thisproperty is straightforward: When the "nancial wealth is small relative to theincome rate it is very important to hedge against future changes in the income


Fig. 6. The dependence of the relative optimal consumption rate C(x, y)/x on the correlationparameter o. The results are from an implementation with z6 "50.

11Since f(z)"C(x,y)/y!1, we have C(x, y)/x"(1#f(z))/z.

12Any linearly increasing transformation of the utility function of consumption will give the sameoptimal policies, but may change the resulting value function drastically. Therefore, one shouldgenerally avoid assessing the magnitude of changes in the value function. Instead, one can translatethe value function into a measure which is independent of the speci"c representation of preferences.One such measure is the constant consumption equivalent CM (x, y) de"ned as

CM (x, y)"infGc50: P=

0

e~btu(c) dt"v(x, y)H.Whenever we address changes in the value function in the text, it is to be understood as changes inthe constant consumption equivalent.

rate and therefore the portfolio weight is more sensitive to the correlationparameter.

Fig. 6 shows the ratio of the optimal consumption rate to wealth, C(x,y)/x, asa function of o.11 The ratio is increasing in o, but nearly constant for highwealth/income ratios. The value function is decreasing in o, although onlyslightly, due to the fact that a lower o implies that the risky investment providesa better implicit hedge of the income risk.12

Fig. 7 displays the dependence of the income multiplier BH(z) on o. For lowwealth/income ratios the income multiplier is only slightly decreasing in o,intuitively because with low wealth the liquidity constraint is more important


Fig. 7. The dependence of the implicit value of income multiplier BH(z) on the correlation parametero. The results are from an implementation with z6 "50.

than the ability to hedge income risk in the future. Note that the implicit value ofthe uncertain income stream is substantially lower than in the complete marketscase. For higher wealth/income ratios the liquidity constraint is less of a prob-lem and the hedging possibilities are more important. Therefore, the implicitvalue of the income stream decreases signi"cantly with o. The lower thecorrelation is, the better is the implicit income risk hedging properties ofa positive investment in the risky asset and, hence, the more valuable is anygiven income process to the investor.

In Figs. 5}7 we have included the computed optimal policies and incomemultiplier for the cases o"$1, although the analytical derivations leading tothe numerical approach presume that D o D(1. The case of o"1 corresponds toa model where the income rate process is perfectly positively correlated with therisky asset price, so there is no undiversi"able income risk. For this particularcase El Karoui and Jeanblanc-PicqueH (1996, Theorem 4.3) have deriveda closed-form expression of the current wealth/income ratio z"x/y as a func-tion of the ratio of optimal consumption to current income, C(x,y)/y. Table 1shows the ratio of optimal consumption to wealth, C(x,y)/x, for o"1 computedboth with our numerical method and the El Karoui and Jeanblanc-PicqueHrelation for representative values of z. The numerically computed consump-tion/wealth ratios are within a few percent of the ratios computed with theclosed-form relation. This indicates that our numerical method provides reliable


Table 1The optimal consumption rate relative to current wealth, C(x,y)/x, for various values of thewealth/income ratio, z"x/y, in the case of perfectly positive correlation, o"1

z"x/y 0.2 1 5 20

Our numerical method 5.962 1.675 0.641 0.389El Karoui and Jeanblanc 5.670 1.609 0.625 0.384

Fig. 8. The dependence of the relative optimal consumption rate C(x, y)/x on the income volatilityparameter d. The results are from an implementation with z6 "50.

results also for the case where o"1, which again allows us to separate thee!ects of liquidity constraints from the e!ects of undiversi"able income risk.

The income rate volatility, d. Higher income risk induces a risk-averse agent toincrease total savings and decrease consumption in order to protect the wealthagainst possible declines in future income. Consistent with that argumentationFig. 8 shows that the ratio of optimal consumption to initial wealth is a decreas-ing function of d. The decrease is most pronounced for small values of thewealth/income fraction where the precautionary motive is strongest.

It is less clear how the allocation of total savings on the riskless and the riskyasset varies with d. Obviously, a certain income stream acts as a substitute forthe riskless asset, but, as d increases, the income stream becomes more likea risky asset. Therefore, the fraction of wealth invested in the risky asset shoulddecrease with d, ceteris paribus. For the complete market case, Eq. (13) implies


Fig. 9. The dependence of the relative optimal risky investment P(x, y)/x on the income volatilityparameter d. The results are from an implementation with z6 "50.

13Koo (1995) "nds a similar relation for his discrete-time model which we brie#y described in theIntroduction.

that the fraction optimally invested in the risky asset is a linearly decreasingfunction of d. The higher the wealth/income ratio, the #atter the line. WhileFig. 9 con"rms the complete market intuition for moderate and high levels ofwealth relative to income, it also shows that for a very low wealth/income ratiothe fraction invested in the risky asset actually increases with d.13

Fig. 10 shows that the income multiplier BH(z) is a concavely decreasingfunction of d, so that the implicit value of an uncertain income stream decreaseswith the income rate volatility. In the complete market case, the incomemultiplier B(z)"1/j is a convex, decreasing function of d for the set of valueschosen here for the other parameters. The value function is also a decreasing,concave function of d. However, the changes of both the controls and the valuefunction are relatively small when d is varied from 0 up to about 0.2. Hence, forinvestors with an income which is not extremely volatile, the precise value of d isnot that important.

The income rate drift, k. The optimal consumption rate is an increasingfunction of k for all levels of the wealth/income ratio, as can be seen from Fig. 11.The higher the wealth/income ratio, the #atter the curve. For low wealth/incomeratios, a higher k implies that a signi"cantly higher consumption rate is possible


Fig. 10. The dependence of the implicit value of income multiplier B*(z) on the income volatilityparameter d. The results are from an implementation with z6 "50.

Fig. 11. The dependence of the relative optimal consumption rate C(x, y)/x on the income driftparameter k. The results are from an implementation with z6 "50.

without leading the investor into bankruptcy. The dependence of the proportionof wealth invested in the risky asset on the income rate drift is less intuitive.Fig. 12 unveils that the risky asset weight is a decreasing function of k for smallvalues of the wealth/income ratios, but a slightly increasing function of k for


Fig. 12. The dependence of the relative optimal risky investment P(x, y)/x on the income driftparameter k. The results are from an implementation with z6 "50.

14 In his related discrete-time model, Koo (1995) presents some results for two di!erent expectedgrowth rates of the income stream. His results are consistent with our "ndings.

large wealth/income ratios.14 As expected, the value function increases con-vexely with k. The income multipliers B(z) and BH(z) increase signi"cantly,also in a convex manner, with k.

The time preference rate, b. The value of the time preference rate, b, has agreat impact on the optimal policies and the implicit value of the income stream.With a high time preference rate, the investor is very eager to transform someof the future income to current consumption. Due to the liquidity constraints,this is not possible, and therefore the implicit value of the income stream ismuch smaller than in the unconstrained case. Liquidity constraints arethus more restrictive for investors with a high time preference rate. Fig. 13depicts the value of the income multiplier BH(z) for di!erent values of b.For relatively high values of b, the income multiplier is much smaller than inthe complete markets case, even for very high values of the wealth/incomeratio z.


Fig. 13. The implicit value of income multipliers BH(z) for di!erent values of the time preference rate,b. The results are from an implementation with z6 "50.

6. Concluding remarks

In this paper we have studied the properties of the optimal consumption andinvestment policies of a liquidity constrained investor with an uncertain, non-spanned income process in a continuous-time set-up. The underlying optimiza-tion problem was solved numerically with the Markov chain approximationapproach and the numerical solution was shown to converge to the true solutionof the problem. We have found that the implicit value the investor associateswith the entire income process is much smaller in the presence of liquidityconstraints and undiversi"able income risk than without such imperfections.Our results suggest that this is mainly due to the presence of liquidity con-straints.

Acknowledgements

I appreciate comments and suggestions received at presentations at the ThirdNordic Research Symposium on Contingent Claims Analysis in Finance (Reyk-javik, May 1996), at the Third International Conference on Computing inEconomics and Finance (Stanford, July 1997), at the 24th European FinanceAssociation Meeting (Vienna, August 1997), and at Odense University and the


Copenhagen Business School. In particular, I thank Bent Jesper Christensen,Anders Damgaard, Darrell Du$e, Bjarne Astrup Jensen, Kristian R. Miltersen,J+ril Mvland, Peter Schotman, Carsten S+rensen, T+rres Trovik, and anony-mous referees and the editor Berc7 Rustem. Financial support from theDanish Research Councils for Natural and Social Sciences is gratefully acknow-ledged.

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