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Optimal consumption and portfolio selection problem with downside consumption constraints
Applied Mathematics and Computation 188 (2007) 1801–1811 www.elsevier.com/locate/amc
Optimal consumption and portfolio selection problem with downside consumption constraints Yong Hyun Shin *, Byung Hwa Lim, U Jin Choi Department of Mathematics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305701, Republic of Korea
Abstract We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some positive constant. We obtain the general optimal policies in an explicit form using martingale method and Feynman–Kac formula. We derive some numerical results of optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Consumption; Portfolio selection; Utility maximization; Downside consumption constraint; Martingale method
1. Introduction We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some positive constant. We obtain the general optimal policies in explicit forms using martingale method and Feynman–Kac formula. We derive properties of the optimal policies and some numerical results of the optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function. Historically, Merton [7,8] introduced the dynamic programming method in order to study the optimal consumption and portfolio selection problem in continuous-time. Karatzas et al. [3] extended this work to the general utility function. They presented an explicit solution of a general consumption-portfolio problem using the dynamic programming method. Cox and Huang [1] and Karatzas et al. [4] introduced the martingale method independently. Lakner and Nygren [6] solved the portfolio optimization problem with both consumption and terminal wealth downside constraints using the gradient operator and Clark–Ocone formula in Malliavin calculus on a finite horizon. Since we only consider an infinite horizon case in this paper, we need not consider the *
Corresponding author. E-mail addresses: [email protected] (Y.H. Shin), [email protected] (B.H. Lim), [email protected] (U.J. Choi).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.053
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terminal wealth downside constraint. Sethi et al. [9] solved a single investor’s consumption and portfolio selection problem with positive subsistence consumption and bankruptcy. This work is an extension of Karatzas et al. [3]. Gong and Li [2] studied the role of index bonds in a consumption and asset allocation model with real subsistence consumption using dynamic programming method. They obtained the optimal policies in the case of the CRRA utility function, but we do it for the general utility function. The rest of this paper proceeds as follows. Section 2 describes the financial market setup. In Section 3, we solve an optimization problem. We gain the optimal policies in explicit forms. Section 4 gives an example in the case of the CRRA utility function. We give the optimal policies and some numerical results of the optimal consumption and portfolio. Section 5 concludes. 2. The financial market setup We consider the continuous time financial market with an infinite horizon time interval [0, 1). Assume that there are one riskless asset with a constant interest rate r and one stock, which is governed by the stochastic differential equation (SDE) dS t =S t ¼ ldt þ rdBt ; where l is the constant return of a stock St, r is the constant volatility of a stock St and Bt is the standard 1 Brownian motion on a probability space ðX; F; PÞ endowed with an augmented filtration fFgt¼0 generated by the Brownian motion Bt. We define the market price of risk, the discount process, the exponential martingale, and the state–price– density, respectively, by lr 1 2 h, ; ft , expfrtg; Z t , exp hBt h t ; H t , ft Z t : r 2 Let Xt be an investor’s wealth process at time t, pt be the amount invested in the stock St at time t, and ct be the consumption rate process at time t. Assume that the portfolio process pt is Ft -measurable, adapted such that for all t P 0 Z 1 p2s ds < 1; almost surelyða:s:Þ 0
and the consumption rate process ct is progressively measurable with respect to Ft , positive such that for all tP0 Z 1 cs ds < 1; a:s: 0
So the investor’s wealth dynamics are given by dX t ¼ ½rX t þ pt ðl rÞ ct dt þ pt r dBt ;
ð1Þ
with initial wealth X0 = x > 0. A consumption-portfolio plan pair (c, p) is called admissible if Xt > 0 for all t P 0. For a given T > 0, we define the auxiliary probability measure (or equivalent martingale measure) PQ ðAÞ , E½Z 0 ðT Þ1A ;
for A 2 FT :
By Girsanov’s Theorem, we obtain the process BQ t ¼ Bt þ ht;
ð2Þ
0 6 t 6 T; Q
which is a standard Brownian motion under the new measure P . Then, by Eq. (2), the wealth process (1) is rewritten as dX t ¼ ðrX t ct Þdt þ pt rdBQ t :
ð3Þ
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
Thus, we obtain the budget constraints from the wealth process (3) Z 1 E ct H t dt 6 x:
1803
ð4Þ
0
(See Remark 9.3 of Chapter 3 of Karatzas and Shreve [5].) Now we define a general utility function. Definition 1. A function u : ½R; 1Þ ! R is called a utility function if it is strictly increasing, strictly concave, continuously differentiable and satisfies u0 ðRþÞ , lim u0 ðcÞ; c#R
lim u0 ðcÞ ¼ 0: c"1
3. The optimization problem The investor’s problem is to maximize her expected utility Z 1 J ðx; c; pÞ , E ebt uðct Þdt ; 0
subject to the downside consumption constraint which restricts the consumption such that for all t P 0
ct P R;
for fixed R > 0 and the budget constraint (4). Here b > 0 is the subjective discount factor. Assumption 1 x>
R : r
This assumption is needed for the positive consumption rate. (See Lemma 3.1 of [2].) Thus, we obtain the value function of our problem V ðxÞ ¼ sup J ðx; c; pÞ; ðc;pÞ2A
where A is the set of all admissible pairs (c, p) such that Z 1 ebt u ðct Þdt < 1; E 0
where u , max(u,0). For a Lagrange multiplier k > 0, we define a dual value function Z 1 Z 1 bt bt e V ðkÞ þ kx , sup J ðx; c; pÞ kE ct H t dt þ kx ¼ E e ~uðke H t Þdt þ kx; ðc;pÞ2A
0
0
where ~ uðyÞ ¼ supðuðcÞ cyÞ: cPR
Here u˜(y) can be obtained by ~uðyÞ ¼ ½uðIðyÞÞ yIðyÞ1f0
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^ct ¼
Iðy t Þ; R;
if 0 < y 6 ~y ; if y P ~y :
ð5Þ
So Ve ðkÞ is given by Z 1 Ve ðkÞ ¼ E ebt ðfuðIðy kt ÞÞ y kt Iðy kt Þg1f0
y kt
bt
where ¼ ke H t . Let y t ¼ k expfðb r 12 h2 Þt hBt g, then, by Itoˆ’s formula, we obtain the SDE dy t ¼ y t fðb rÞdt h dBt g:
ð6Þ
Here we can easily see that y kt is a unique strong solution to (6) with initial value y k0 ¼ k. We consider the following problem: Z 1 y t ¼y bs /ðt; yÞ ¼ E e ðfuðIðy s ÞÞ y s Iðy s Þg1f0
ð7Þ
t
Then, by Feynman–Kac formula, we obtain the following equivalent partial differential equations (PDEs) from the Eq. (7) L/ðt; yÞ þ ebt fuðIðyÞÞ yIðyÞg ¼ 0; if 0 < y 6 ~y ð8Þ L/ðt; yÞ þ ebt fuðRÞ Ryg ¼ 0; if y P ~y ; where the partial differential operator is L,
o o 1 o2 þ ðb rÞy þ h2 y 2 2 : ot oy 2 oy
The next proposition provides a solution to the PDEs (8). Proposition 1. We consider the function 8 R y zIðzÞuðIðzÞÞ Ry n 2y n < C 1 y nþ þ 2 2y þ dz nþ þ1 2 ~ y z h ðnþ n Þ h ðnþ n Þ ~y vðyÞ ¼ uðRÞ n R : D2 y y þ ; if y P ~y ; r b
zIðzÞuðIðzÞÞ zn þ1
dz;
if 0 < y 6 ~y ; ð9Þ
where C1 ¼
R ðn r
uðRÞ 1Þ nb~ y
ðnþ n Þ~y nþ 1
and D2 ¼
R ðnþ r
uðRÞ 1Þ nþb~ y
ðnþ n Þ~y n 1
:
~ yÞ ¼ ebt vðyÞ is a solution to the PDEs (8). Then /ðt; Proof. First we consider the following PDE: L/ðt; yÞ þ ebt fuðIðyÞÞ yIðyÞg ¼ 0;
0 < y 6 ~y :
ð10Þ
If we set a trial solution of the form /(t,y) = ebtv(y), then from PDE (10), we obtain the ordinary differential equation (ODE) 1 2 2 00 h y v ðyÞ þ ðb rÞyv0 ðyÞ bvðyÞ þ fuðIðyÞÞ yIðyÞg ¼ 0: 2
ð11Þ
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1805
In the homogeneous case, a general solution to (11) can be derived by the quadratic equation 1 2 2 1 2 h n þ b r h n b ¼ 0; 2 2
ð12Þ
with two roots n+ > 1 and n < 0. So the homogenous solution is given by vh ðyÞ ¼ C 1 y nþ þ C 2 y n for some constants C1 and C2. But by the growth condition of v(y), C2 must be zero. We derive the particular solution by using the variation of parameters. That is, Z y Z y 2y nþ zIðzÞ uðIðzÞÞ 2y n zIðzÞ uðIðzÞÞ nþ vðyÞ ¼ C 1 y þ 2 dz 2 dz: nþ þ1 z zn þ1 h ðnþ n Þ ~y h ðnþ n Þ ~y Similarly, for y P ~y , we obtain vðyÞ ¼ D2 y n
R uðRÞ yþ : r b
Now we determine the constants C1 and D2. We use the smooth condition and C1-condition at y ¼ ~y such that R uðRÞ vð~y Þ ¼ C 1 ~y nþ ¼ D2 ~y n ~y þ r b and v0 ð~y Þ ¼ C 1 nþ ~y nþ 1 ¼ D2 n ~y n 1
R ; r
respectively. So we can obtain C1 ¼
R ðn r
uðRÞ 1Þ nb~ y
ðnþ n Þ~y nþ 1
and D2 ¼
R ðnþ r
uðRÞ 1Þ nþb~ y
ðnþ n Þ~y n 1
:
Using Eq. (7) and Proposition 1, we derive Ve ðkÞ from /(t, y) at t = 0 and y = k, and consequently Ve ðkÞ ¼ vðkÞ. So we can derive the value function V(x) using the following proposition. Proposition 2. If Ve ðkÞ exists and is differentiable for k > 0, then V ðxÞ ¼ inf ð Ve ðkÞ þ kxÞ;
ð13Þ
k>0
for any x 2 (0, 1). Proof. The value function V(x) is obtained from Ve ðkÞ by the Legendre transform inverse formula. See Section 3.8 of Karatzas and Shreve [5]. h Using Eqs. (9) and (13), we define the wealth boundary ~x , C 1 nþ ~y nþ 1 ¼ D2 n ~y n 1 þ
R : r
ð14Þ
Thus from Proposition 2, we can obtain the value function V(x). Theorem 1. The value function is given by 8 n 1
> < D2 Rrx n 1 þ x R Rrx n 1 þ uðRÞ ; if R=r < x 6 ~x; r D2 n D2 n b V ðxÞ ¼ R R k zIðzÞuðIðzÞÞ nþ > k n 2ðk Þ zIðzÞuðIðzÞÞ 2ðk Þn : C 1 ðk Þ þ þ 2 dz dz þ ðk Þx; nþ þ1 2 ~ y zn þ1 z h ðn n Þ h ðn n Þ ~y þ
þ
ð15Þ if x P ~x;
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where k* is determined from the following algebraic equation n 1 Z k n 1 Z k 2nþ ðk Þ þ zIðzÞ uðIðzÞÞ 2n ðk Þ zIðzÞ uðIðzÞÞ n 1 dz þ dz ¼ x: C 1 nþ ðk Þ þ 2 znþ þ1 zn þ1 h ðnþ n Þ ~y h2 ðnþ n Þ ~y
ð16Þ
Remark 2. It is easily seen the one-to-one correspondence between k 2 ð0; ~y Þ and x 2 ð~x; 1Þ in (16). Since Ve ðÞ is decreasing, we get this fact using decreasing property of (16) with respect to k*. Remark 3. For R=r < x 6 ~x, we also define an algebraic equation with respect to k like Eq. (16). Using Eq. (9) in Proposition 1 and (13) in Proposition 2 we derive R ¼ x: ð17Þ r Now we determine optimal policies. Let y kt and y kt be solutions of SDE (6) with initial values y0 = k* and ** y0 = k , respectively. In order to find the optimal policies, we consider the optimal wealth processes which are obtained by substituting y kt for k* into (16) and y kt for k** into (17). Then we see that D2 n ðk Þ
n 1
þ
X t
¼
n 1 C 1 nþ ðy kt Þ þ
2nþ ðy kt Þnþ 1 2 h ðnþ n Þ
Z ~y
y kt
zIðzÞ uðIðzÞÞ 2n ðy kt Þn 1 dz þ znþ þ1 h2 ðnþ n Þ
Z
y kt
~y
zIðzÞ uðIðzÞÞ dz zn þ1
ð18Þ
and
k X t ¼ D2 n ðy t Þ
n 1
þ
R ; r
ð19Þ
respectively. Theorem 2. The optimal policies are provided by (c*, p*) such that R; if R=r < X t 6 ~x ct ¼ Iðy kt Þ; if X t P ~x; and
8h ðn 1Þ Rr X t ; > r > > > n uðRÞ > > k Þnþ 1 R y k k Þn 1 R y k > 2n ðn 1Þðy 2n ðn 1Þðy zIðzÞuðIðzÞÞ : þ þ þ t t t t dz 2 n þ1 2 þ ~ y ~y z h ðn n Þ h ðn n Þ þ
þ
if R=r < X t 6 ~x
zIðzÞuðIðzÞÞ dz zn þ1
o
;
if X t P ~x:
Proof. We have already determined an optimal consumption (5) in Remark 1. So we need to show that the optimal consumption and portfolio processes generate the optimal wealth processes X t of (18) and X t of (19) and determine an optimal portfolio. For R=r < X t 6 ~x, applying Itoˆ’s formula to Eq. (19), we obtain o 1n n 3 2 k n 2 D2 n ðn 1Þðn 2Þðy kt Þ dX ðdy kt Þ þ ðdy kt Þ t ¼ D2 n ðn 1Þðy t Þ 2 o 1n D2 n ðn 1Þðn 2Þðy kt Þn 1 h2 dt ¼ D2 n ðn 1Þðy kt Þn 1 fðb rÞdt h dBt g þ 2 1 2 2 1 2 n 1 ¼ D2 n h n b r h n þ b ðy kt Þ dt 2 2 n o
n 1 n 1 þ D2 h2 n2 þ D2 ðb rÞn D2 h2 n D2 n b ðy kt Þ dt þ h D2 n ðn 1Þðy kt Þ dBt n 1 R n 1 ¼ r D2 n ðy kt Þ þ dt R dt þ h2 ðD2 n ðn 1Þðy kt Þ Þdt r n o n 1 þ h D2 n ðn 1Þðy kt Þ dBt :
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
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Here we can see that the first term of right hand side of the third equality is equal to zero, since n is a solution of the quadratic Eq. (12). For the last term, if we choose o hn n 1 D2 n ðn 1Þðy kt Þ ; p t ¼ r then we have
dX t ¼ rX t þ pt ðl rÞ ct dt þ rpt dBt : That is, the optimal portfolio is h R X t ; for R=r < X t 6 ~x: pt ¼ ðn 1Þ r r Similarly for X t P ~x, applying Itoˆ’s formula to Eq. (18), we obtain " Z k 2nþ ðnþ 1Þðy kt Þnþ 2 y t zIðzÞ uðIðzÞÞ k nþ 2 dz dX t ¼ nþ ðnþ 1ÞC 1 ðy t Þ znþ þ1 h2 ðnþ n Þ ~y # n 2 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ þ dz 2 ðdy kt Þ n þ1 k Þ2 z h h2 ðnþ n Þ ðy ~y t " k nþ 3 Z y kt 1 n 3 2n ðn 1Þðn 2Þðy zIðzÞ uðIðzÞÞ þ þ þ k þ t Þ dz þ nþ ðnþ 1Þðnþ 2ÞC 1 ðy t Þ 2 2 znþ þ1 h ðnþ n Þ ~y n 3 Z y k t 2n ðn 1Þðn 2Þðy kt Þ zIðzÞ uðIðzÞÞ þ dz 2 zn þ1 # h ðnþ n Þ ~y 2ðnþ n 3Þ y kt Iðy kt Þ uðIðy kt ÞÞ 2 Iðy kt Þ ðdy kt Þ2 3 2 2 2 k k h h ðy t Þ ðy t Þ n 1 ¼ nþ ðnþ 1ÞC 1 ðy kt Þ þ n 1 Z y k t 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ dz 2 znþ þ1 h ðnþ n Þ ~y Z k 2n ðn 1Þðy kt Þn 1 y t zIðzÞ uðIðzÞÞ þ dz zn þ1 h2 ðnþ n Þ ~y 2 y kt Iðy kt Þ uðIðy kt ÞÞ 1 n 1 fðb rÞdt hdBt g þ nþ ðnþ 1Þðnþ 2ÞC 1 ðy kt Þ þ 2 k y 2 h t Z k 2nþ ðnþ 1Þðnþ 2Þðy kt Þnþ 1 y t zIðzÞ uðIðzÞÞ dz znþ þ1 h2 ðnþ n Þ ~y n 1 Z y k t 2n ðn 1Þðn 2Þðy kt Þ zIðzÞ uðIðzÞÞ þ dz 2 zn þ1 h ðnþ n Þ ~y 2ðnþ n 3Þ y kt Iðy kt Þ uðIðy kt ÞÞ 2 2 Iðy kt Þ h2 dt 2 k y h h t " # k nþ 1 Z y kt k n 1 Z y kt n 1 2n ðy Þ zIðzÞ uðIðzÞÞ 2n ðy Þ zIðzÞ uðIðzÞÞ þ t t ¼ r nþ C 1 ðy kt Þ þ 2 dz þ 2 dz dt znþ þ1 zn þ1 h ðnþ n Þ ~y h ðnþ n Þ ~y " n 1 Z y k t n 1 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ dz þ h2 nþ ðnþ 1ÞC 1 ðy kt Þ þ þ 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 dt Iðy kt Þdt 2 n þ1 k z yt h h ðnþ n Þ ~y 1 2 2 1 2 n 1 nþ C 1 h nþ þ b r h nþ b ðy kt Þ þ dt 2 2
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Z y k n 1 t 2nþ ðy kt Þ þ 1 2 2 1 2 zIðzÞ uðIðzÞÞ h h n þ b r b dz dt n þ þ 2 2 znþ þ1 h ðnþ n Þ 2 ~y Z y k n 1 t 2n ðy kt Þ 1 2 2 1 2 zIðzÞ uðIðzÞÞ h n þ b r h n b þ 2 dz dt 2 zn þ1 h ðnþ n Þ 2 ~y 2ðb rÞ y kt Iðy kt Þ uðIðy kt ÞÞ nþ þ n 1 þ dt y kt h2 " n 1 Z y k t n 1 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ dz þ h nþ ðnþ 1ÞC 1 ðy kt Þ þ þ 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dBt dz þ 2 zn þ1 y kt h h2 ðnþ n Þ ~y " # k nþ 1 Z y kt k n 1 Z y kt n 1 2n ðy Þ zIðzÞ uðIðzÞÞ 2n ðy Þ zIðzÞ uðIðzÞÞ þ t t ¼ r nþ C 1 ðy kt Þ þ 2 dz þ 2 dz dt znþ þ1 zn þ1 h ðnþ n Þ ~y h ðnþ n Þ ~y " n 1 Z y k t 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ 2 k nþ 1 þ h nþ ðnþ 1ÞC 1 ðy t Þ þ dz 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 dt Iðy kt Þdt 2 n þ1 k z yt h h ðnþ n Þ ~y " n 1 Z y k t n 1 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ þ h nþ ðnþ 1ÞC 1 ðy kt Þ þ þ dz 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 dBt : zn þ1 y kt h h2 ðnþ n Þ ~y
Here we can also see that some terms of the third equality are zeros because of the same reason. For the last term, if we choose " n 1 Z y k t h 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ pt ¼ nþ ðnþ 1ÞC 1 ðy kt Þnþ 1 þ dz 2 r znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 ; zn þ1 y kt h h2 ðnþ n Þ ~y then we have dX t ¼ ½rX t þ pt ðl rÞ ct dt þ rpt dBt : That is, the optimal portfolio is " uðRÞ Rr ð1 n Þ nb~ h 2 y kt Iðy kt Þ uðIðy kt ÞÞ y k nþ 1 nþ ðnþ 1Þ ðy Þ þ pt ¼ t r y kt ðnþ n Þ~y nþ 1 h2 Z k 2nþ ðnþ 1Þðy kt Þnþ 1 y t zIðzÞ uðIðzÞÞ 2n ðn 1Þðy kt Þn 1 þ dz znþ þ1 h2 ðnþ n Þ h2 ðnþ n Þ ~y # Z y kt zIðzÞ uðIðzÞÞ dz ; for X t P ~x: zn þ1 ~y
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
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4. Example: a solution to a CRRA utility function Now we consider a CRRA utility function which is defined by uðcÞ ,
c1c ; 1c
where c > 0 (c 5 1) is an investor’s coefficient of relative risk aversion. Now we define the Merton constant K. Assumption 2 br c1 2 h > 0: þ c 2c2
K ,rþ
Then, from the previous results, a dual function u˜( Æ ) of u( Æ ) is given by 1c c 1c R c ~ y 1f0
1 cn ð K 1c
n þ 1Þ þ nr1 bð1cÞ
nþ n
R1cþcnþ
and d2 ¼
1 K
cnþ 1c
nþ þ 1 þ nþr1 bð1cÞ nþ n
R1cþcn :
The wealth boundary ~x of (14) is given by ~x , c1 nþ ~y nþ 1 þ
1 1c R ~y ¼ d 2 n ~y n 1 þ : K r
And the value function (15) in Theorem 1 is given by 8 n
1 > < d 2 Rrx n 1 þ x R Rrx n 1 þ R1c ; if R=r < x 6 ~x; r bð1cÞ d 2 n d 2 n V ðxÞ ¼ 1c > : c ðk Þnþ þ c ðk Þ c þ ðk Þx; if x P ~x; 1 Kð1cÞ where k* is determined from the following algebraic equation: c1 nþ ðk Þ
nþ 1
þ
1 1c ðk Þ ¼ x: K
The optimal wealth process in (18) becomes 1 k 1c ðy Þ : K t The optimal consumption and portfolio in Theorem 2 are given by ( R; if R=r < X t 6 ~x; ct ¼ 1 ðy kt Þc if X t P ~x
X t ¼ c1 nþ ðy kt Þ
nþ 1
þ
1810
and
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
8
< rh ðn 1Þ Rr X t ; if R=r < X t 6 ~x; o pt ¼ h n 1 1 : r c1 nþ ðnþ 1Þðy kt Þnþ 1 þ Kc ðy kt Þc ;
if X t P ~x:
Also we obtain the some numerical results for the optimal consumption and portfolio. See Figs. 1 and 2.
Consumption
0.6
5
4
0.58 0.56 0.54 0.52 0.5 0.48
Consumption
0.46 50
50.1
50.2
50.3
50.4
50.5
Wealth Level
3
2
1
50
55
60
65
70
Wealth Level
Fig. 1. This figure is the optimal consumption rate when u(c) is a CRRA utility function (b = 0.07, r = 0.01, l = 0.05, r = 0.2, c = 2 and R = 0.5). Dotted line gives the optimal consumption rate for the case where the investor does not have a downside consumption constraint, i.e. the classical Merton case with an infinite horizon. The solid line gives the optimal consumption rate for the case in our model. The figure on the upper right hand side shows the specific region of wealth level from 50 to 50.5 at the solid line.
40
Portfolio
30
20
10
50
55
60
65
70
Wealth Level
Fig. 2. This figure is the optimal investment in the stock when u(c) is a CRRA utility function (b = 0.07, r = 0.01, l = 0.05, r = 0.2, c = 2 and R = 0.5). Dotted line gives the optimal investment in the risky asset for the case where the investor does not have a downside consumption constraint, i.e. the classical Merton case with an infinite horizon. The solid line gives the optimal investment in the stock for the case in our model.
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5. Concluding remarks In this paper, we solve a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. We derive the explicit solutions, the numerical results for the optimal consumption and portfolio in the case of a CRRA utility function, specifically. Acknowledgements We are grateful to Hyeng Keun Koo, Kyoung Jin Choi and an anonymous referee for helpful comments. This work was supported by BK 21 project. References [1] J.C. Cox, C.F. Huang, Optimum consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory 49 (1989) 33–83. [2] N. Gong, T. Li, Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption, Appl. Math. Comput. 174 (2006) 710–731. [3] I. Karatzas, J.P. Lehoczky, S.P. Sethi, S.E. Shreve, Explicit solution of a general consumption/investment problem, Math. Oper. Res. 11 (1986) 261–294. [4] I. Karatzas, J.P. Lehoczky, S.E. Shreve, Optimal portfolio and consumption decisions for a small investor on a finite horizon, SIAM J. Control Optim. 25 (1987) 1557–1586. [5] I. Karatzas, S.E. Shreve, Methods of Mathematical Finance, Springer, 1998 (Chapter 3). [6] P. Lakner, L.M. Nygren, Portfolio optimzation with downside constraints, Math. Financ. 16 (2006) 283–299. [7] R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat. 51 (1969) 247–257. [8] R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econ. Theory 3 (1971) 373–413. [9] S.P. Sethi, M.I. Taksar, E.L. Presman, Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy, J. Econ. Dyn. Control 16 (1992) 747–768.
Optimal consumption and portfolio selection problem with downside consumption constraints Yong Hyun Shin *, Byung Hwa Lim, U Jin Choi Department of Mathematics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305701, Republic of Korea
Abstract We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some positive constant. We obtain the general optimal policies in an explicit form using martingale method and Feynman–Kac formula. We derive some numerical results of optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Consumption; Portfolio selection; Utility maximization; Downside consumption constraint; Martingale method
1. Introduction We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some positive constant. We obtain the general optimal policies in explicit forms using martingale method and Feynman–Kac formula. We derive properties of the optimal policies and some numerical results of the optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function. Historically, Merton [7,8] introduced the dynamic programming method in order to study the optimal consumption and portfolio selection problem in continuous-time. Karatzas et al. [3] extended this work to the general utility function. They presented an explicit solution of a general consumption-portfolio problem using the dynamic programming method. Cox and Huang [1] and Karatzas et al. [4] introduced the martingale method independently. Lakner and Nygren [6] solved the portfolio optimization problem with both consumption and terminal wealth downside constraints using the gradient operator and Clark–Ocone formula in Malliavin calculus on a finite horizon. Since we only consider an infinite horizon case in this paper, we need not consider the *
Corresponding author. E-mail addresses: [email protected] (Y.H. Shin), [email protected] (B.H. Lim), [email protected] (U.J. Choi).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.053
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terminal wealth downside constraint. Sethi et al. [9] solved a single investor’s consumption and portfolio selection problem with positive subsistence consumption and bankruptcy. This work is an extension of Karatzas et al. [3]. Gong and Li [2] studied the role of index bonds in a consumption and asset allocation model with real subsistence consumption using dynamic programming method. They obtained the optimal policies in the case of the CRRA utility function, but we do it for the general utility function. The rest of this paper proceeds as follows. Section 2 describes the financial market setup. In Section 3, we solve an optimization problem. We gain the optimal policies in explicit forms. Section 4 gives an example in the case of the CRRA utility function. We give the optimal policies and some numerical results of the optimal consumption and portfolio. Section 5 concludes. 2. The financial market setup We consider the continuous time financial market with an infinite horizon time interval [0, 1). Assume that there are one riskless asset with a constant interest rate r and one stock, which is governed by the stochastic differential equation (SDE) dS t =S t ¼ ldt þ rdBt ; where l is the constant return of a stock St, r is the constant volatility of a stock St and Bt is the standard 1 Brownian motion on a probability space ðX; F; PÞ endowed with an augmented filtration fFgt¼0 generated by the Brownian motion Bt. We define the market price of risk, the discount process, the exponential martingale, and the state–price– density, respectively, by lr 1 2 h, ; ft , expfrtg; Z t , exp hBt h t ; H t , ft Z t : r 2 Let Xt be an investor’s wealth process at time t, pt be the amount invested in the stock St at time t, and ct be the consumption rate process at time t. Assume that the portfolio process pt is Ft -measurable, adapted such that for all t P 0 Z 1 p2s ds < 1; almost surelyða:s:Þ 0
and the consumption rate process ct is progressively measurable with respect to Ft , positive such that for all tP0 Z 1 cs ds < 1; a:s: 0
So the investor’s wealth dynamics are given by dX t ¼ ½rX t þ pt ðl rÞ ct dt þ pt r dBt ;
ð1Þ
with initial wealth X0 = x > 0. A consumption-portfolio plan pair (c, p) is called admissible if Xt > 0 for all t P 0. For a given T > 0, we define the auxiliary probability measure (or equivalent martingale measure) PQ ðAÞ , E½Z 0 ðT Þ1A ;
for A 2 FT :
By Girsanov’s Theorem, we obtain the process BQ t ¼ Bt þ ht;
ð2Þ
0 6 t 6 T; Q
which is a standard Brownian motion under the new measure P . Then, by Eq. (2), the wealth process (1) is rewritten as dX t ¼ ðrX t ct Þdt þ pt rdBQ t :
ð3Þ
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
Thus, we obtain the budget constraints from the wealth process (3) Z 1 E ct H t dt 6 x:
1803
ð4Þ
0
(See Remark 9.3 of Chapter 3 of Karatzas and Shreve [5].) Now we define a general utility function. Definition 1. A function u : ½R; 1Þ ! R is called a utility function if it is strictly increasing, strictly concave, continuously differentiable and satisfies u0 ðRþÞ , lim u0 ðcÞ; c#R
lim u0 ðcÞ ¼ 0: c"1
3. The optimization problem The investor’s problem is to maximize her expected utility Z 1 J ðx; c; pÞ , E ebt uðct Þdt ; 0
subject to the downside consumption constraint which restricts the consumption such that for all t P 0
ct P R;
for fixed R > 0 and the budget constraint (4). Here b > 0 is the subjective discount factor. Assumption 1 x>
R : r
This assumption is needed for the positive consumption rate. (See Lemma 3.1 of [2].) Thus, we obtain the value function of our problem V ðxÞ ¼ sup J ðx; c; pÞ; ðc;pÞ2A
where A is the set of all admissible pairs (c, p) such that Z 1 ebt u ðct Þdt < 1; E 0
where u , max(u,0). For a Lagrange multiplier k > 0, we define a dual value function Z 1 Z 1 bt bt e V ðkÞ þ kx , sup J ðx; c; pÞ kE ct H t dt þ kx ¼ E e ~uðke H t Þdt þ kx; ðc;pÞ2A
0
0
where ~ uðyÞ ¼ supðuðcÞ cyÞ: cPR
Here u˜(y) can be obtained by ~uðyÞ ¼ ½uðIðyÞÞ yIðyÞ1f0
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^ct ¼
Iðy t Þ; R;
if 0 < y 6 ~y ; if y P ~y :
ð5Þ
So Ve ðkÞ is given by Z 1 Ve ðkÞ ¼ E ebt ðfuðIðy kt ÞÞ y kt Iðy kt Þg1f0
y kt
bt
where ¼ ke H t . Let y t ¼ k expfðb r 12 h2 Þt hBt g, then, by Itoˆ’s formula, we obtain the SDE dy t ¼ y t fðb rÞdt h dBt g:
ð6Þ
Here we can easily see that y kt is a unique strong solution to (6) with initial value y k0 ¼ k. We consider the following problem: Z 1 y t ¼y bs /ðt; yÞ ¼ E e ðfuðIðy s ÞÞ y s Iðy s Þg1f0
ð7Þ
t
Then, by Feynman–Kac formula, we obtain the following equivalent partial differential equations (PDEs) from the Eq. (7) L/ðt; yÞ þ ebt fuðIðyÞÞ yIðyÞg ¼ 0; if 0 < y 6 ~y ð8Þ L/ðt; yÞ þ ebt fuðRÞ Ryg ¼ 0; if y P ~y ; where the partial differential operator is L,
o o 1 o2 þ ðb rÞy þ h2 y 2 2 : ot oy 2 oy
The next proposition provides a solution to the PDEs (8). Proposition 1. We consider the function 8 R y zIðzÞuðIðzÞÞ Ry n 2y n < C 1 y nþ þ 2 2y þ dz nþ þ1 2 ~ y z h ðnþ n Þ h ðnþ n Þ ~y vðyÞ ¼ uðRÞ n R : D2 y y þ ; if y P ~y ; r b
zIðzÞuðIðzÞÞ zn þ1
dz;
if 0 < y 6 ~y ; ð9Þ
where C1 ¼
R ðn r
uðRÞ 1Þ nb~ y
ðnþ n Þ~y nþ 1
and D2 ¼
R ðnþ r
uðRÞ 1Þ nþb~ y
ðnþ n Þ~y n 1
:
~ yÞ ¼ ebt vðyÞ is a solution to the PDEs (8). Then /ðt; Proof. First we consider the following PDE: L/ðt; yÞ þ ebt fuðIðyÞÞ yIðyÞg ¼ 0;
0 < y 6 ~y :
ð10Þ
If we set a trial solution of the form /(t,y) = ebtv(y), then from PDE (10), we obtain the ordinary differential equation (ODE) 1 2 2 00 h y v ðyÞ þ ðb rÞyv0 ðyÞ bvðyÞ þ fuðIðyÞÞ yIðyÞg ¼ 0: 2
ð11Þ
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
1805
In the homogeneous case, a general solution to (11) can be derived by the quadratic equation 1 2 2 1 2 h n þ b r h n b ¼ 0; 2 2
ð12Þ
with two roots n+ > 1 and n < 0. So the homogenous solution is given by vh ðyÞ ¼ C 1 y nþ þ C 2 y n for some constants C1 and C2. But by the growth condition of v(y), C2 must be zero. We derive the particular solution by using the variation of parameters. That is, Z y Z y 2y nþ zIðzÞ uðIðzÞÞ 2y n zIðzÞ uðIðzÞÞ nþ vðyÞ ¼ C 1 y þ 2 dz 2 dz: nþ þ1 z zn þ1 h ðnþ n Þ ~y h ðnþ n Þ ~y Similarly, for y P ~y , we obtain vðyÞ ¼ D2 y n
R uðRÞ yþ : r b
Now we determine the constants C1 and D2. We use the smooth condition and C1-condition at y ¼ ~y such that R uðRÞ vð~y Þ ¼ C 1 ~y nþ ¼ D2 ~y n ~y þ r b and v0 ð~y Þ ¼ C 1 nþ ~y nþ 1 ¼ D2 n ~y n 1
R ; r
respectively. So we can obtain C1 ¼
R ðn r
uðRÞ 1Þ nb~ y
ðnþ n Þ~y nþ 1
and D2 ¼
R ðnþ r
uðRÞ 1Þ nþb~ y
ðnþ n Þ~y n 1
:
Using Eq. (7) and Proposition 1, we derive Ve ðkÞ from /(t, y) at t = 0 and y = k, and consequently Ve ðkÞ ¼ vðkÞ. So we can derive the value function V(x) using the following proposition. Proposition 2. If Ve ðkÞ exists and is differentiable for k > 0, then V ðxÞ ¼ inf ð Ve ðkÞ þ kxÞ;
ð13Þ
k>0
for any x 2 (0, 1). Proof. The value function V(x) is obtained from Ve ðkÞ by the Legendre transform inverse formula. See Section 3.8 of Karatzas and Shreve [5]. h Using Eqs. (9) and (13), we define the wealth boundary ~x , C 1 nþ ~y nþ 1 ¼ D2 n ~y n 1 þ
R : r
ð14Þ
Thus from Proposition 2, we can obtain the value function V(x). Theorem 1. The value function is given by 8 n 1
> < D2 Rrx n 1 þ x R Rrx n 1 þ uðRÞ ; if R=r < x 6 ~x; r D2 n D2 n b V ðxÞ ¼ R R k zIðzÞuðIðzÞÞ nþ > k n 2ðk Þ zIðzÞuðIðzÞÞ 2ðk Þn : C 1 ðk Þ þ þ 2 dz dz þ ðk Þx; nþ þ1 2 ~ y zn þ1 z h ðn n Þ h ðn n Þ ~y þ
þ
ð15Þ if x P ~x;
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where k* is determined from the following algebraic equation n 1 Z k n 1 Z k 2nþ ðk Þ þ zIðzÞ uðIðzÞÞ 2n ðk Þ zIðzÞ uðIðzÞÞ n 1 dz þ dz ¼ x: C 1 nþ ðk Þ þ 2 znþ þ1 zn þ1 h ðnþ n Þ ~y h2 ðnþ n Þ ~y
ð16Þ
Remark 2. It is easily seen the one-to-one correspondence between k 2 ð0; ~y Þ and x 2 ð~x; 1Þ in (16). Since Ve ðÞ is decreasing, we get this fact using decreasing property of (16) with respect to k*. Remark 3. For R=r < x 6 ~x, we also define an algebraic equation with respect to k like Eq. (16). Using Eq. (9) in Proposition 1 and (13) in Proposition 2 we derive R ¼ x: ð17Þ r Now we determine optimal policies. Let y kt and y kt be solutions of SDE (6) with initial values y0 = k* and ** y0 = k , respectively. In order to find the optimal policies, we consider the optimal wealth processes which are obtained by substituting y kt for k* into (16) and y kt for k** into (17). Then we see that D2 n ðk Þ
n 1
þ
X t
¼
n 1 C 1 nþ ðy kt Þ þ
2nþ ðy kt Þnþ 1 2 h ðnþ n Þ
Z ~y
y kt
zIðzÞ uðIðzÞÞ 2n ðy kt Þn 1 dz þ znþ þ1 h2 ðnþ n Þ
Z
y kt
~y
zIðzÞ uðIðzÞÞ dz zn þ1
ð18Þ
and
k X t ¼ D2 n ðy t Þ
n 1
þ
R ; r
ð19Þ
respectively. Theorem 2. The optimal policies are provided by (c*, p*) such that R; if R=r < X t 6 ~x ct ¼ Iðy kt Þ; if X t P ~x; and
8h ðn 1Þ Rr X t ; > r > > > n uðRÞ
þ
if R=r < X t 6 ~x
zIðzÞuðIðzÞÞ dz zn þ1
o
;
if X t P ~x:
Proof. We have already determined an optimal consumption (5) in Remark 1. So we need to show that the optimal consumption and portfolio processes generate the optimal wealth processes X t of (18) and X t of (19) and determine an optimal portfolio. For R=r < X t 6 ~x, applying Itoˆ’s formula to Eq. (19), we obtain o 1n n 3 2 k n 2 D2 n ðn 1Þðn 2Þðy kt Þ dX ðdy kt Þ þ ðdy kt Þ t ¼ D2 n ðn 1Þðy t Þ 2 o 1n D2 n ðn 1Þðn 2Þðy kt Þn 1 h2 dt ¼ D2 n ðn 1Þðy kt Þn 1 fðb rÞdt h dBt g þ 2 1 2 2 1 2 n 1 ¼ D2 n h n b r h n þ b ðy kt Þ dt 2 2 n o
n 1 n 1 þ D2 h2 n2 þ D2 ðb rÞn D2 h2 n D2 n b ðy kt Þ dt þ h D2 n ðn 1Þðy kt Þ dBt n 1 R n 1 ¼ r D2 n ðy kt Þ þ dt R dt þ h2 ðD2 n ðn 1Þðy kt Þ Þdt r n o n 1 þ h D2 n ðn 1Þðy kt Þ dBt :
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
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Here we can see that the first term of right hand side of the third equality is equal to zero, since n is a solution of the quadratic Eq. (12). For the last term, if we choose o hn n 1 D2 n ðn 1Þðy kt Þ ; p t ¼ r then we have
dX t ¼ rX t þ pt ðl rÞ ct dt þ rpt dBt : That is, the optimal portfolio is h R X t ; for R=r < X t 6 ~x: pt ¼ ðn 1Þ r r Similarly for X t P ~x, applying Itoˆ’s formula to Eq. (18), we obtain " Z k 2nþ ðnþ 1Þðy kt Þnþ 2 y t zIðzÞ uðIðzÞÞ k nþ 2 dz dX t ¼ nþ ðnþ 1ÞC 1 ðy t Þ znþ þ1 h2 ðnþ n Þ ~y # n 2 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ þ dz 2 ðdy kt Þ n þ1 k Þ2 z h h2 ðnþ n Þ ðy ~y t " k nþ 3 Z y kt 1 n 3 2n ðn 1Þðn 2Þðy zIðzÞ uðIðzÞÞ þ þ þ k þ t Þ dz þ nþ ðnþ 1Þðnþ 2ÞC 1 ðy t Þ 2 2 znþ þ1 h ðnþ n Þ ~y n 3 Z y k t 2n ðn 1Þðn 2Þðy kt Þ zIðzÞ uðIðzÞÞ þ dz 2 zn þ1 # h ðnþ n Þ ~y 2ðnþ n 3Þ y kt Iðy kt Þ uðIðy kt ÞÞ 2 Iðy kt Þ ðdy kt Þ2 3 2 2 2 k k h h ðy t Þ ðy t Þ n 1 ¼ nþ ðnþ 1ÞC 1 ðy kt Þ þ n 1 Z y k t 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ dz 2 znþ þ1 h ðnþ n Þ ~y Z k 2n ðn 1Þðy kt Þn 1 y t zIðzÞ uðIðzÞÞ þ dz zn þ1 h2 ðnþ n Þ ~y 2 y kt Iðy kt Þ uðIðy kt ÞÞ 1 n 1 fðb rÞdt hdBt g þ nþ ðnþ 1Þðnþ 2ÞC 1 ðy kt Þ þ 2 k y 2 h t Z k 2nþ ðnþ 1Þðnþ 2Þðy kt Þnþ 1 y t zIðzÞ uðIðzÞÞ dz znþ þ1 h2 ðnþ n Þ ~y n 1 Z y k t 2n ðn 1Þðn 2Þðy kt Þ zIðzÞ uðIðzÞÞ þ dz 2 zn þ1 h ðnþ n Þ ~y 2ðnþ n 3Þ y kt Iðy kt Þ uðIðy kt ÞÞ 2 2 Iðy kt Þ h2 dt 2 k y h h t " # k nþ 1 Z y kt k n 1 Z y kt n 1 2n ðy Þ zIðzÞ uðIðzÞÞ 2n ðy Þ zIðzÞ uðIðzÞÞ þ t t ¼ r nþ C 1 ðy kt Þ þ 2 dz þ 2 dz dt znþ þ1 zn þ1 h ðnþ n Þ ~y h ðnþ n Þ ~y " n 1 Z y k t n 1 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ dz þ h2 nþ ðnþ 1ÞC 1 ðy kt Þ þ þ 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 dt Iðy kt Þdt 2 n þ1 k z yt h h ðnþ n Þ ~y 1 2 2 1 2 n 1 nþ C 1 h nþ þ b r h nþ b ðy kt Þ þ dt 2 2
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Z y k n 1 t 2nþ ðy kt Þ þ 1 2 2 1 2 zIðzÞ uðIðzÞÞ h h n þ b r b dz dt n þ þ 2 2 znþ þ1 h ðnþ n Þ 2 ~y Z y k n 1 t 2n ðy kt Þ 1 2 2 1 2 zIðzÞ uðIðzÞÞ h n þ b r h n b þ 2 dz dt 2 zn þ1 h ðnþ n Þ 2 ~y 2ðb rÞ y kt Iðy kt Þ uðIðy kt ÞÞ nþ þ n 1 þ dt y kt h2 " n 1 Z y k t n 1 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ dz þ h nþ ðnþ 1ÞC 1 ðy kt Þ þ þ 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dBt dz þ 2 zn þ1 y kt h h2 ðnþ n Þ ~y " # k nþ 1 Z y kt k n 1 Z y kt n 1 2n ðy Þ zIðzÞ uðIðzÞÞ 2n ðy Þ zIðzÞ uðIðzÞÞ þ t t ¼ r nþ C 1 ðy kt Þ þ 2 dz þ 2 dz dt znþ þ1 zn þ1 h ðnþ n Þ ~y h ðnþ n Þ ~y " n 1 Z y k t 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ 2 k nþ 1 þ h nþ ðnþ 1ÞC 1 ðy t Þ þ dz 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 dt Iðy kt Þdt 2 n þ1 k z yt h h ðnþ n Þ ~y " n 1 Z y k t n 1 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ þ h nþ ðnþ 1ÞC 1 ðy kt Þ þ þ dz 2 znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 dBt : zn þ1 y kt h h2 ðnþ n Þ ~y
Here we can also see that some terms of the third equality are zeros because of the same reason. For the last term, if we choose " n 1 Z y k t h 2nþ ðnþ 1Þðy kt Þ þ zIðzÞ uðIðzÞÞ pt ¼ nþ ðnþ 1ÞC 1 ðy kt Þnþ 1 þ dz 2 r znþ þ1 h ðnþ n Þ ~y # n 1 Z y k t 2n ðn 1Þðy kt Þ zIðzÞ uðIðzÞÞ 2 y kt Iðy kt Þ uðIðy kt ÞÞ dz þ 2 ; zn þ1 y kt h h2 ðnþ n Þ ~y then we have dX t ¼ ½rX t þ pt ðl rÞ ct dt þ rpt dBt : That is, the optimal portfolio is " uðRÞ Rr ð1 n Þ nb~ h 2 y kt Iðy kt Þ uðIðy kt ÞÞ y k nþ 1 nþ ðnþ 1Þ ðy Þ þ pt ¼ t r y kt ðnþ n Þ~y nþ 1 h2 Z k 2nþ ðnþ 1Þðy kt Þnþ 1 y t zIðzÞ uðIðzÞÞ 2n ðn 1Þðy kt Þn 1 þ dz znþ þ1 h2 ðnþ n Þ h2 ðnþ n Þ ~y # Z y kt zIðzÞ uðIðzÞÞ dz ; for X t P ~x: zn þ1 ~y
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
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4. Example: a solution to a CRRA utility function Now we consider a CRRA utility function which is defined by uðcÞ ,
c1c ; 1c
where c > 0 (c 5 1) is an investor’s coefficient of relative risk aversion. Now we define the Merton constant K. Assumption 2 br c1 2 h > 0: þ c 2c2
K ,rþ
Then, from the previous results, a dual function u˜( Æ ) of u( Æ ) is given by 1c c 1c R c ~ y 1f0
1 cn ð K 1c
n þ 1Þ þ nr1 bð1cÞ
nþ n
R1cþcnþ
and d2 ¼
1 K
cnþ 1c
nþ þ 1 þ nþr1 bð1cÞ nþ n
R1cþcn :
The wealth boundary ~x of (14) is given by ~x , c1 nþ ~y nþ 1 þ
1 1c R ~y ¼ d 2 n ~y n 1 þ : K r
And the value function (15) in Theorem 1 is given by 8 n
1 > < d 2 Rrx n 1 þ x R Rrx n 1 þ R1c ; if R=r < x 6 ~x; r bð1cÞ d 2 n d 2 n V ðxÞ ¼ 1c > : c ðk Þnþ þ c ðk Þ c þ ðk Þx; if x P ~x; 1 Kð1cÞ where k* is determined from the following algebraic equation: c1 nþ ðk Þ
nþ 1
þ
1 1c ðk Þ ¼ x: K
The optimal wealth process in (18) becomes 1 k 1c ðy Þ : K t The optimal consumption and portfolio in Theorem 2 are given by ( R; if R=r < X t 6 ~x; ct ¼ 1 ðy kt Þc if X t P ~x
X t ¼ c1 nþ ðy kt Þ
nþ 1
þ
1810
and
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
8
< rh ðn 1Þ Rr X t ; if R=r < X t 6 ~x; o pt ¼ h n 1 1 : r c1 nþ ðnþ 1Þðy kt Þnþ 1 þ Kc ðy kt Þc ;
if X t P ~x:
Also we obtain the some numerical results for the optimal consumption and portfolio. See Figs. 1 and 2.
Consumption
0.6
5
4
0.58 0.56 0.54 0.52 0.5 0.48
Consumption
0.46 50
50.1
50.2
50.3
50.4
50.5
Wealth Level
3
2
1
50
55
60
65
70
Wealth Level
Fig. 1. This figure is the optimal consumption rate when u(c) is a CRRA utility function (b = 0.07, r = 0.01, l = 0.05, r = 0.2, c = 2 and R = 0.5). Dotted line gives the optimal consumption rate for the case where the investor does not have a downside consumption constraint, i.e. the classical Merton case with an infinite horizon. The solid line gives the optimal consumption rate for the case in our model. The figure on the upper right hand side shows the specific region of wealth level from 50 to 50.5 at the solid line.
40
Portfolio
30
20
10
50
55
60
65
70
Wealth Level
Fig. 2. This figure is the optimal investment in the stock when u(c) is a CRRA utility function (b = 0.07, r = 0.01, l = 0.05, r = 0.2, c = 2 and R = 0.5). Dotted line gives the optimal investment in the risky asset for the case where the investor does not have a downside consumption constraint, i.e. the classical Merton case with an infinite horizon. The solid line gives the optimal investment in the stock for the case in our model.
Y.H. Shin et al. / Applied Mathematics and Computation 188 (2007) 1801–1811
1811
5. Concluding remarks In this paper, we solve a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. We derive the explicit solutions, the numerical results for the optimal consumption and portfolio in the case of a CRRA utility function, specifically. Acknowledgements We are grateful to Hyeng Keun Koo, Kyoung Jin Choi and an anonymous referee for helpful comments. This work was supported by BK 21 project. References [1] J.C. Cox, C.F. Huang, Optimum consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory 49 (1989) 33–83. [2] N. Gong, T. Li, Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption, Appl. Math. Comput. 174 (2006) 710–731. [3] I. Karatzas, J.P. Lehoczky, S.P. Sethi, S.E. Shreve, Explicit solution of a general consumption/investment problem, Math. Oper. Res. 11 (1986) 261–294. [4] I. Karatzas, J.P. Lehoczky, S.E. Shreve, Optimal portfolio and consumption decisions for a small investor on a finite horizon, SIAM J. Control Optim. 25 (1987) 1557–1586. [5] I. Karatzas, S.E. Shreve, Methods of Mathematical Finance, Springer, 1998 (Chapter 3). [6] P. Lakner, L.M. Nygren, Portfolio optimzation with downside constraints, Math. Financ. 16 (2006) 283–299. [7] R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat. 51 (1969) 247–257. [8] R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econ. Theory 3 (1971) 373–413. [9] S.P. Sethi, M.I. Taksar, E.L. Presman, Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy, J. Econ. Dyn. Control 16 (1992) 747–768.