Research Article Portfolio Selection with Subsistence Consumption Constraints and CARA Utility

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1 Mathematical Problems in Engineering Volume 14, Article ID , 6 pages Research Article Portfolio Selection with Subsistence Consumption Constraints and CARA Utility Gyoocheol Shim 1 and Yong Hyun Shin 1 Department of Financial Engineering, Ajou University, Suwon , Republic of Korea Department of Mathematics, Sookmyung Women s University, Seoul 14-74, Republic of Korea Correspondence should be addressed to Yong Hyun Shin; yhshin@sookmyung.ac.kr Received January 14; Revised 5 March 14; Accepted 6 April 14; Published 16 April 14 Academic Editor: Pankaj Gupta Copyright 14 G. Shim and Y. H. Shin. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the optimal consumption and portfolio choice problem with constant absolute risk aversion (CARA) utility and a subsistence consumption constraint. A subsistence consumption constraint means there exists a positive constant minimum level for the agent s optimal consumption. We use the dynamic programming approach to solve the optimization problem and also give the verification theorem. We illustrate the effects of the subsistence consumption constraint on the optimal consumption and portfolio choice rules by the numerical results. 1. Introduction Following the seminal research works of Merton [1, ], various problems of continuous-time optimal consumption and portfolio selection have been considered under various financial/economic constraints. One of the interesting research topics in a continuous-time portfolio selection problem is the optimization problem subject to a subsistence consumption constraint (or a downside consumption constraint). A subsistence constraint means that there exists a positive lower bound level for the agent s optimal consumption rate. hus this constraint affects the agent s financial decision including her optimal portfolio. LaknerandNygren[3] havestudiedtheportfoliooptimization problem subject to a downside constraint for consumption and an insurance constraint for terminal wealth with a martingale approach. Gong and Li [4] haveinvestigated the role of index bonds in the optimal consumption and portfolio selection problem with constant relative risk aversion (CRRA) utility and a real subsistence consumption constraint using the dynamic programming approach. Shin et al. [5] have also considered a similar problem to that of Gong and Li [4]. hey have studied the portfolio selection problem with a general utility function and a downside consumption constraint using the martingale approach. Yuan and Hu [6] have investigated the optimal consumption and portfolio selection problem with a consumption habit constraint and a terminal wealth downside constraint using the martingale approach. In this paper we use the dynamic programming method based on Karatzas et al. [7] to derive the value function and the optimal policies in closed-form with a constant absolute risk aversion (CARA) utility function and a subsistence consumption constraint. Lim et al. [8]haveconsidered a similar portfolio optimization problem combined with the voluntary retirement choice problem. Shin and Lim [9] have analyzed the effects of the subsistence consumption constraint for behavior of investment in the risky asset. he rest of this paper proceeds as follows. Section introduces the financial market. In Section 3 we consider the main optimization problem. We use the dynamic programming principle to derive the solutions in closed form with CARA utility and a subsistence consumption constraint. We also give some numerical results and the solutions derived by the martingale method. Section 4 concludes.. he Financial Market Setup We assume that there are two assets in the financial market: one is a riskless asset with constant interest rate r>,andthe

2 Mathematical Problems in Engineering other is a stock whose price process S t } t evolves according to the stochastic differential equation (SDE) ds t S t = μdt + σdb t, for t, (1) where μ is the constant expected rate of return of the stock, σ > is the constant volatility of the stock, and B t isastandardbrownianmotiononaprobabilityspace (Ω, F, P) endowed with the filtration F t } t which is the augmentation under P of the natural filtration generated by the standard Brownian motion B t } t. We assume that μ =r so that the market price of risk θ is not zero: θ μ r σ =. () Let X t be an economic agent s wealth at time t, π t the amount of money invested in the stock at time t, andc t the consumption rate at time t. heportfolioprocessπ t } t is adapted to F t } t and satisfies, for all t,almostsurely (a.s.), t π s ds <, (3) andtheconsumptionrateprocessc t } t is a nonnegative process adapted to F t } t such that, for all t, a.s., t c s ds <. (4) We assume that there is a subsistence consumption constraint which restricts the minimum consumption level. hat is, the consumption process should satisfy c t R, t, (5) where R>is a constant lower bound for the consumption rates. hus the agent s wealth process X t } t follows the SDE dx t =[rx t +π t (μ r) c t ]dt+σπ t db t, (6) with an initial endowment X = x > R/r. (We need this restriction on the initial endowment for the positive consumption rate. See Lemma 3.1 of Gong and Li [4]). A consumption-portfolio plan (c, π) := (c t } t,π t } t ) satisfying the above conditions is called admissible at x>r/r if X t R/r,forallt.WeletA(x) denote the class of admissible controls at x>r/r. 3. he Optimization Problem Now the agent s optimization problem with initial wealth X =x>r/ris to choose (c, π) A(x) to maximize the following expected life-time utility: E [ e βt u(c t )dt]. (7) Here, β > is the subjective discount factor and u( ) is aconstantabsoluteriskaversion(cara)utilityfunction defined by u (c) e c, (8) where >is the agent s coefficient of absolute risk aversion. hustheagent svaluefunctionisgivenby V (x) sup (c,π) A(x) E [ dt]. (9) Bellman equation corresponding to the optimization problem for x > R/r is max c R,π [rx+π(μ r) c}v (x) + 1 σ π V (x) βv (x) e c ] =. (1) We assume that the wealth process X t must satisfy a transversality condition lim t e βt V(X t )=. (11) We will find the solution V(x), as the candidate value function,to Bellman equation (1) under the conditions that V (x) > and V (x) < for x > R/r and V (x) = u (R) = e R for a real number x>r/r. After obtaining the solution, we can check these conditions. Under these conditions, in particular, the first-order condition (FOC), V (x)+u (c) = with respect to c R, is binding if R/r<x<x so that the maximizing c Rin Bellman equation (1)is R in this case. hus, from the first-order conditions (FOCs) of Bellman equation (1), we derive the candidate optimal consumption and portfolio R, if R c r <x<x = log V (x)}, if x x, π = θv (x) σv (x). (1) Remark 1. For later use, we consider two quadratic algebraic equations: rm (r+β+ θ )m+β=, (13) with two roots m 1 ( < m 1 <1)and m >1and θ n +(r β θ )n r=, (14) with two roots n 1 <and n >.

3 Mathematical Problems in Engineering 3 heorem. Let V(x) be given by C 1 (x R m r ) 1 e R β, 1 θ V (x) = (r + β n 1)D 1 e (n 1 )ξ 1 r e ξ, where if R r <x<x if x x, (15) rm D 1 = 1 β θ / >, (16) r e n1r (1 ((1 m 1 )/)n 1 ) x= 1 m 1 (n 1 D 1 e n 1R + 1 r )+R r > R r, (17) C 1 = e R (x R 1 m1 m 1 r ) >, (18) and ξ is determined from the algebraic equation x=d 1 e n 1ξ + 1 r ξ+ 1 r hen it satisfies Bellman equation (1). θ (r β ). (19) Proof. By using Remark 1, we can check the inequalities in (16) and(17). he inequality in (18) holdsby(17). Define the function X(c) of c on [R, ) by X (c) =D 1 e n 1c + 1 r c+ 1 r By using (16)and(17), onecan check θ (r β ). () X (R) = x. (1) Since the function X(c) is increasing in c, ithastheinverse function. Let C(x) for x x be the inverse function of X(c). In particular, we have By (19), we have C (x) =R. () ξ=c(x) for x x. (3) By using Remark 1, (16), (18), (), and (3), we can show thatthefunctionv(x) defined by (15)iscontinuous.Byusing Remark 1,(), (3),andtheinverserelationshipbetweenX and C,wecanobtain V (x) =e C(x), V (x) = e C(x) X (C (x)), for x>x. (4) By (15), (18), (), (), and (4), we get the smooth-pasting (C 1 ) condition V (x ) =m 1 C 1 (x R m r ) 1 1 =e R =V (x+), (5) and the high-contact (C ) condition V (x ) =m 1 (m 1 1)C 1 (x R r ) m 1 = e R X (R) =V (x+). (6) hus, the function V(x) is twice continuously differentiable. Furthermore, V (x) > and V (x) < for x > R/r and V (x) = u (R) = e R. For R/r < x < x,ifwesubstitutefocs(1)intobellman equation (1), we obtain the changed Bellman equation (rx R) V (x) 1 θ (V (x)) V (x) βv(x) e R =. (7) We can easily check that V(x) is the solution to (7)forR/r < x<x. For x x, we also obtain the changed Bellman equation from (1): rxv (x) 1 θ (V (x)) V (x) + V (x) βv(x) (log V (x) 1)=. (8) If we substitute (4) intobellmanequation(8), then we obtain the equation rx (c) e c + θ X (c) e c βv(x (c)) e c (c + 1) =. By using (15)and(3), we can check that (9)holds. (9) Now we can derive the candidate optimal policies with the functionv( ) in heorem. heorem 3. he candidate optimal policies are given by (c,π ) such that c t = R, ξ t, θ π σ(1 m t = 1 ) (X t R r ), θ σ (n 1D 1 e n 1ξ t + 1 r ), if R r <X t < x, if X t x, if R r <X t < x, if X t x, where ξ t is determined from the optimal wealth process X t =D 1 e n 1ξ t + 1 r ξ t + 1 r (3) (31) θ (r β ). (3)

4 4 Mathematical Problems in Engineering heorem 4 (verification theorem). he value function of the optimization problem (9) is equal to V(x) in heorem. hat is, V (x) = V(x). Consequently the candidate consumption and portfolio in heorem 3 are the optimal policies of problem (9). Proof. For arbitrary given consumption and portfolio plan (c, π) A(x) and (, ),wehave E [ dt] Consumption Wealth level E [ e βt (rx t +π t (μ r) c t )V (X t ) + 1 σ π t V (X t ) βv(x t )} dt] Figure 1: his figure is the optimal consumption rate c t when β=.5, r =.1, μ =.7, σ =., R =., and=3. Solid line gives the optimal consumption rate with a subsistence consumption constraint and dotted line gives the optimal consumption rate without the constraint. = E [ d(e βt V(X t ))] + E [ e βt σπ t V (X t )db t ] =V(x) E [e β V(X )], (33) = E [ d(e βt V(X t ))] + E [ e βt σπ t V (X t )db t ] =V(x) E [e β V(X )], (36) where the inequality is obtained from Bellman equation (1), the first equality from applying Itô s formula to e βt V(X t ), and the second equality from E[ e βt σπ t V (X t )db t ]=. aking andusing transversality condition (11), we have V (x) E [ dt], (34) for arbitrary given consumption and portfolio plan (c, π) A(x);thatis, V (x) sup (c,π) A(x) E [ dt]. (35) Now we consider the candidate optimal consumption and portfolio plan (c, π ) A(x) in heorem 3.For (, ), we have e βt c t E [ dt] = E [ e βt (rx t +π t (μ r) c t )V (X t ) + 1 σ (π t ) V (X t ) βv(x t )} dt] where the first equality is obtained from Bellman equation (1),thesecondfromapplyingItô s formula to e βt V(X t ),and the third from E[ e βt σπ t V (X t )db t ]=.aking and using transversality condition (11), we have e βt c t V (x) = E [ dt]. (37) hus, from (35) and(37), we show that V(x) which is the solution to Bellman equation (1) isarealvaluefunctionof the optimization problem (9). Now we compare our solution in heorem 3 with the Merton s solution with CARA utility. he optimal consumption and portfolio policies without the subsistence constraint are given by c M t =rx t + 1 r (β r+1 θ ), π M t = θ σr, (38) respectively. Figures 1 and give the numerical results for the optimal consumption and portfolio. Remark 5. Dynamic programming principle can be also applied to the CRRA utility function with a subsistence consumption constraint following our approach. See Gong and Li [4] and Lee and Shin [1]. Remark 6. Following Shin et al. [5] we can use the martingale method to derive a similar solution with CARA utility. We will give (rough) sketch of the derivation.

5 Mathematical Problems in Engineering 5 Portfolio Wealth level Figure : his figure is the optimal portfolio π t when β=.5, r =.1, μ =.7, σ =., R =., and = 3. Solid line gives the optimal portfolio with a subsistence consumption constraint and dotted line gives the optimal portfolio without the constraint. From the agent s wealth dynamics X t in (6), we obtain the budget constraint E [ c t H t dt] x, (39) where the state price density H t is defined by H t e (r+(1/)θ )t θb t. (4) A dual utility function u(y) of the CARA utility function u(c) = e c / is derived by u(y)=sup [ e c c R cy]= [1 y log y 1 y] 1 <y y} he undetermined coefficients of V(y) are given by c 1 = ((p 1)/r )(β r+(1/) θ )+p /β 1/r (p + p ) e R(1 p +), d = ((p + 1)/r )(β r+(1/) θ )+p + /β 1/r (p + p ) e R(1 p ). (44) hen we use the Legendre transform inverse formula to obtain the value function V m (x) as follows: V m (x) where = inf [V (y) + xy] y> = p 1 ( 1 1/(p 1) ) p p d e R β, (1 p + )c 1 (y ) p + 1 r (y ), if x x m, (x R r ) p /(p 1) if R r <x<x m, (45) x m = p + c 1 e R(p + 1) + R r + 1 r (r β 1 θ ), (46) and y is determined from the following algebraic equation: +[ e R Ry]1 y>y}, (41) x= p + c 1 (y ) p r log (y )+ 1 r (r β 1 θ ). (47) where y=e R.hedualvaluefunctionV(y) is given by V (y) = E y=y [ e βt [ 1 y t log (y t ) 1 y t} 1 <yt y} + e R c 1 y p y log y r = + 1 r (β r + 1 θ )y, Ry t} 1 yt >y}] dt] if <y y, d y p R r y e R β, if y>y, (4) where y t ye βt H t and p + >1and p <are two real roots of the quadratic algebraic equation 1 θ p +(β r 1 θ )p β=. (43) Actually we can show that V m (x) in (45)andx m in (46)agree with V(x) in (15)andx in (17), respectively, if we set y =e ξ and show that m 1 C 1 = ( p d ) 1/(1 p ), D1 = p + c 1. (48) Refer to Lee and Shin [1]. Remark 7. We simplify the calculation by using the dynamic programming approach, where, instead of Legendre transformation in the martingale approach, we introduce the function X(c) in () whoseinverseisc(x) in (3). he link between the two methods is the relation y = e ξ = e C(x) = u (C(x)) = V (x) for x x,wherethelastequalityisin(4). 4. Concluding Remarks We have considered the optimal consumption and portfolio choice problem with constant absolute risk aversion (CARA) utility and a subsistence consumption constraint. he existence of a subsistence consumption constraint is

6 6 Mathematical Problems in Engineering realistic since one needs a minimum level of consumption to live. For example, we cannot live without necessities. We have obtained the closed form solution to optimization problem by using the dynamic programming approach. We have illustrated the effects of the subsistence consumption constraint on the optimal consumption and portfolio by the numerical results. Furthermore one can consider the optimization problem under regime switching as future research. Conflict of Interests he authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments he authors are indebted to two anonymous referees for their helpful comments and insightful suggestions. he research of the corresponding author (Yong Hyun Shin) was supported by Sookmyung Women s University Research Grants 1. References [1] R. C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time case, Review of Economics and Statistics, vol. 51, no. 3, pp , [] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Economic heory, vol. 3, no. 4, pp , [3] P. Lakner and L. M. Nygren, Portfolio optimization with downside constraints, Mathematical Finance,vol.16,no.,pp , 6. [4] N. Gong and. Li, Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption, Applied Mathematics and Computation, vol.174,no.1,pp , 6. [5] Y. H. Shin, B. H. Lim, and U. J. Choi, Optimal consumption and portfolio selection problem with downside consumption constraints, Applied Mathematics and Computation, vol. 188, no., pp , 7. [6] H. Yuan and Y. Hu, Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints, Insurance: Mathematics & Economics,vol.45,no.3,pp.45 49,9. [7] I. Karatzas, J. P. Lehoczky, S. P. Sethi, and S. E. Shreve, Explicit solution of a general consumption/investment problem, Mathematics of Operations Research, vol. 11, no., pp , [8] B. H. Lim, Y. H. Shin, and U. J. Choi, Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints, Mathematical Analysis and Applications,vol.345,no.1,pp.19 1,8. [9] Y. H. Shin and B. H. Lim, Comparison of optimal portfolios with and without subsistence consumption constraints, NonlinearAnalysis:heory,Methods&Applications,vol.74,no.1, pp.5 58,11. [1] H. S. Lee and Y. H. Shin, An optimal consumption and investment problem with subsistence consumption constraints: a dynamic programming approach, Working Paper, 14.

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