Finance Research Letters. Maximizing utility of consumption subject to a constraint on the probability of lifetime ruin

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1 Finance Research Leers 5 ( Conens liss available a ScienceDirec Finance Research Leers Maximizing uiliy of consumpion subjec o a consrain on he probabiliy of lifeime ruin Erhan Bayrakar, Virginia R. Young Deparmen of Mahemaics, Universiy of Michigan, Ann Arbor, MI 4819, USA aricle info absrac Aricle hisory: Received 29 July 28 Acceped 29 Augus 28 Available online 5 Sepember 28 JEL classificaion: G11 G1 C61 G19 In his paper, we explicily solve he problem of maximizing uiliy of consumpion (unil he minimum of bankrupcy and he ime of deah wih a consrain on he probabiliy of lifeime ruin, which can be inerpreed as a risk measure on he whole pah of he wealh process. 28 Elsevier Inc. All righs reserved. Keywords: Uiliy maximizaion from consumpion Probabiliy of lifeime ruin consrain Nonconvex risk consrain on he enire pah of he wealh process 1. Inroducion In our pas work, we deermined he opimal invesmen sraegy of an individual who arges a given rae of consumpion and who seeks o minimize he probabiliy of going bankrup before she dies, also known as lifeime ruin. For an economic jusificaion of minimizing lifeime probabiliy of ruin as an invesmen crierion see, for example, Bayrakar and Young (27b. Young (24 considered his problem when he individual coninuously consumes eiher a consan (real dollar amoun or a consan proporion of wealh. Bayrakar and Young (27b inroduced borrowing consrains. Bayrakar and Young (27a esablished when he wo problems of minimizing a funcion of lifeime minimum wealh and of maximizing uiliy of lifeime consumpion resul in he same opimal invesmen sraegy on a given open inerval in wealh space. On he oher hand, Bayrakar and Young * Corresponding auhor a: Universiy of Michigan, Deparmen of Mahemaics, Eas Hall, 53 Church Sree, Ann Arbor, MI, USA. addresses: erhan@umich.edu (E. Bayrakar, vryoung@umich.edu (V.R. Young /$ see fron maer 28 Elsevier Inc. All righs reserved. doi:1.116/j.frl

2 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( (28 considered he case for which he consumpion is sochasic and is correlaed o he wealh. By using a convex dualiy relaionship beween he sopper and conroller problems, we showed ha he minimal probabiliy of lifeime ruin in his se up, whose sochasic represenaion does no have a classical form as he uiliy maximizaion problem does, is he unique classical soluion of is Hamilon Jacobi Bellman (HJB equaion, which is a nonlinear boundary-value problem. When we presened our work, several of our colleagues suggesed ha we consider he probabiliy of lifeime ruin as a risk consrain. Porfolio opimizaion problems wih risk consrains on erminal wealh were considered by Basak and Shapiro (21 and Boyle and Tian (27, for example. Their soluion can be obained by pahwise maximizaion since he problem can be formulaed in erms of choosing he erminal opimal wealh. However, he probabiliy of lifeime ruin is a risk consrain on he enire pah of he wealh process (he consrain depends on wheher he lifeime minimum reaches he bankrupcy level before he ime of deah, and such an approach is no possible. Risk consrains on he enire pah of he process recenly were also considered by Cheridio e al. (25. By using he resuls of Karazas e al. (1986, we will show ha opimal sraegies invesmen and consumpion sraegies exis and can be numerically obained using a bisecion search. The res of he paper is organized as follows: In Secion 2.1, we provide a precise saemen of he problem. In Secion 2.2, we give sufficien condiions for opimaliy and relae he problem o an unconsrained opimizaion problem (Lemma 2.1, which was analyzed by Karazas e al. (1986. In Secion 3.1, wesummarizeheresulsofkarazas e al. (1986 ha we need o prove our main resul. In Secion 3.2, we prove our main resul, Theorem 3.1, wih he help of he auxiliary resuls Lemmas These resuls poin ou ha a bisecion search can be carried ou o deermine an opimal policy (see Remark Maximizing uiliy of consumpion subjec o a probabiliy of ruin consrain In Secion 2.1, we presen he financial marke and define he problem of maximizing expeced uiliy of lifeime consumpion subjec o a consrain on he probabiliy of lifeime ruin. In Secion 2.2, we show ha he mehod of Lagrange mulipliers allows us o apply he work of Karazas e al. (1986 o solve his problem Saemen of he problem In his secion, we firs presen he financial ingrediens ha make up he agen s wealh, namely, consumpion, a riskless asse, and a risky asse. We, hen, define he problem of maximizing uiliy of consumpion subjec o a consrain on he ruin probabiliy. We assume ha he individual invess in a riskless asse whose price S a ime follows he process ds = rs d, for some fixed rae of ineres r >. Also, he individual invess in a risky asse whose price a ime, S, follows geomeric Brownian moion given by ds = μs d + σ S dz, in which μ > r, σ >, and Z is a sandard Brownian moion wih respec o a filered probabiliy space (Ω, F, P, {F.LeX denoe he wealh a ime of he invesor. We allow him o specify a consumpion process and an invesmen sraegy {π, in which c denoes he rae of consumpion and π denoes he dollar amoun invesed in he risky asse a ime. (We will consider admissible consumpion and invesmen pairs, π such ha is a nonnegaive process adaped o {F and saisfies c s ds < almos surely for all, and such ha {π is adaped o {F and saisfies π 2 s ds < almos surely for all. The remaining wealh, namely X π, is invesed in he riskless asse. Thus, wealh follows he process dx = [ rx + (μ rπ c d + σπ dz, X = x. (2.2 We assume ha he invesor chooses admissible consumpion and invesmen sraegies o maximize his expeced uiliy of consumpion before he dies or before he bankrups, whichever occurs firs. Le U denoe a sricly increasing, sricly concave uiliy funcion on (, whose firs hree (2.1

3 26 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( derivaives are coninuous; we exend U o [, by defining U( = lim c U(c; similarly, for U (. Le τ d denoe he random ime of deah of he invesor. We assume ha τ d is exponenially disribued wih parameer β> (ha is, wih expeced ime of deah equal o 1/β; his parameer is also known as he hazard rae of he invesor. We assume ha τ d is independen of he Brownian moion Z driving he price process of he risky asse. The invesor is subjec o a consrain on he probabiliy of bankrupcy, or ruin, before he dies. Le τ denoe he firs ime ha wealh equals ; ha is, τ = inf{ : X =. τ is he ime of ruin or bankrupcy. For noaional compleeness, we should wrie τ,π o indicae he dependence of τ on he consumpion and invesmen sraegies. However, for simpliciy, we will wrie τ,ifhe consumpion and invesmen sraegies are undersood by he conex. Thus, he invesor chooses admissible and {π o maximize [ τ τ d W,π (x := E x U(c d, (2.3 subjec o he consrain ψ(x := P x (τ < τ d ϕ(x. (2.4 Here, ϕ(x is a given hreshold of olerance for he probabiliy of lifeime ruin. We assume ha ϕ(x is less han or equal o he probabiliy of ruin when he individual maximizes (2.3 wihou any consrain on he probabiliy of ruin. E x and P x denoe he expecaion and probabiliy, respecively, condiional on X = x. Remark 2.1. Noe ha because τ d Exp(β is independen of he Brownian moion driving he price process of he risky asse, we can express W,π as follows: [ τ d W,π (x = E x U(c 1 { τ d [ = E x U(c 1 { τ [ = E x e β U(c 1 { τ d [ = E x βe βs dsd s βe βs U(c 1 { τ d ds [ τ = E x e β U(c d. (2.5 Similarly, we can rewrie he ruin probabiliy in (2.4 as follows: [ ψ(x = E x 1 {τ βe β d [ = E x τ 1 {τ < βe β d = E x[ e βτ 1 {τ < = E x [ e βτ. (2.6 Remark 2.2. To ensure ha maximizing (2.3 subjec o (2.4 resuls in a finie value funcion, we make he following assumpion concerning he uiliy funcion U. Define he posiive consan γ by γ = 1 ( μ 2 r, 2 σ and le λ < 1 and λ + > denoehesoluionsof γ λ 2 (r β γ λ r =. (2.7 (2.8

4 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( We assume ha c for all c >. (U (θ λ <, ( Sufficien condiions for opimaliy For any real number P and admissible sraegy, π,define [ τ V,π (x; P = E x e β U(c d + Pe βτ. (2.1 Lemma 2.1. For a fixed x >, assume ha here exiss an admissible sraegy, π such ha P x( τ,π τ d = ϕ(x, and assume ha here exiss a consan P such ha, π solves (2.11 V (x; P := sup V,π (x; P = V,π,π (x; P. (2.12 Then, for any admissible sraegy, π saisfying P x (τ,π τ d ϕ(x, wehaveha [ τ,π E x e β U(c d [ τ,π E x e β U(c d. (2.13 Proof. By assumpion [ τ,π E x e β U(c d + Pe βτ,π [ τ,π E x e β U(c d + Pe βτ,π. (2.14 On he oher hand, again from our assumpion, E x [e βτ,π =ϕ(x E x [e βτ,π for any admissible, π. Using his las inequaliy in (2.14 yields he resul. Noe ha he expressions for V,π in (2.1 and V in (2.12 are idenical o he corresponding expressions in (1.6 and (2.7 in Karazas e al. (1986. Thanks o he resuls of ha paper, we will be able o idenify P,, and {π saisfying he assumpions of Lemma Main resul In his secion, we will firs describe how o obain V for a given real number P using he resuls presened in Karazas e al. (1986. Then, we will give our main resul and show he exisence of P,, and {π. We will also describe how hese can be numerically compued efficienly Soluion of he unconsrained opimizaion problem in (2.12 Karazas e al. (1986, Theorem 4.1 and Remark 4.2 prove he following verificaion heorem concerning V :

5 28 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( Lemma 3.1. For P finie, suppose V : (, (P, is a C 2 funcion saisfying he Hamilon Jacobi Bellman (HJB equaion β V (x = rxv ( (x + max U(c cv (x ( + max (μ rπ V (x + 1 σ 2 π 2 V (x. (3.1 c π 2 (a If U( is finie, hen V (x V (x for x >. (b If U( = and if E x [ τ eβ max(, U(c d < for all admissible sraegies, hen V (x V (x for x >. When he nonnegaiviy consrain on consumpion is no acive, hen Karazas e al. (1986 (Secions 6 12 use consumpion as an inermediae variable o solve he HJB equaion (3.1. When he nonnegaiviy consrain on consumpion is acive, hen in Secion 13, hey use y = (V (x as an inermediae, or dual, variable. We shall follow he laer approach in boh cases and hereby unify our presenaion of he resuls of Karazas e al. (1986. Define he funcion I : (, U ( [, o be he inverse of U.ExendI o (, by seing I on[u (,. IfV is C 2 and sricly concave, hen he HJB equaion (3.1 becomes β V (x = [ rx ( I V (x V (x + ( ( U I V (x γ (V (x 2 V, x >. (3.2 (x Le ρ < and ρ + > 1denoehesoluionsof γρ 2 (r β + γ ρ r =. Noe ha ρ ± = 1 + λ ±.Fora, B, and A, define he following funcions for < y < U (a: { X (y; a, B = By λ + + I(y 1 y λ I(y + r γ (λ + λ λ + (U (θ λ + yλ + λ (U (θ λ (3.4 a I(y (3.3 and J (y; a, A = Ay ρ + + U(I(y β { 1 y ρ + γ (λ + λ ρ + I(y a (U (θ λ + + yρ ρ I(y (U (θ λ. (3.5 Now, X is sricly decreasing, and lim y X (y; a, B =.DefineX (U (a; a, B = lim y U (a X (y; a, B. Thus, X ( ; a, B maps (, U (a ono [X (U (a; a, B,, and is inverse funcion Y( ; a, B is C 2, sricly decreasing, and maps [X (U (a; a, B, ono (, U (a. When U( and U ( are finie, and when 1 β U( P < P := 1 β U( (U ( ρ βλ (U (θ λ, (3.6 we se a = and exend he definiion of X and J o all y >. In his case, X is sricly decreasing for all y >, and lim y X (y;, B = lim y By λ +.IfP = U(/β, henseb =, so X ( ;, B maps (, ono (, and has surjecive inverse Y( ;, B : (, (,. IfU(/β < P < P, hen se β B(P = (P γ (λ + λ [ȳ(p ρ 1β U(, (3.7 + in which ȳ(p is defined by ( [ȳ(p ρ =βλ P 1 [ 1 β U( (U (θ λ. (3.8

6 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( I follows ha I(ȳ = and X (ȳ;, B =. We consider X ( ;, B resriced o (, ȳ, which has range [, and surjecive inverse Y( ;, B :[, (, ȳ. Define V by ( V (x; a, B = J Y(x; a, B; a, λ + B, (3.9 ρ + wih he domain of V given by he domain of Y. Define candidae opimal consumpion and invesmen sraegies in feedback form by c = I ( V (X ; a, B, π = μ r σ 2 V (X ; a, B V (X ; a, B. (3.1 Afer hese preliminaries, we are ready o give V (x; P in erms of V (x; a, B he opimal consumpion and invesmen sraegies by specifying he values of a and B in (3.9 and (3.13 case by case; see Table 1 on p. 292 of Karazas e al. (1986. Firs, we dispense wih he case for which P lim c U(c/β. In his case, i is opimal o consume all of one s wealh immediaely o bankrupcy; no coninuous opimal consumpion sraegy exiss. Lemma 3.2. Depending on he uiliy funcion and is derivaive a zero, he value funcion V (x; P can be compued in erms of V (x; a, B as follows: (i When P U(/β, V (x; P = V (x;,. NoehaV(x;, is independen of P. (ii When U ( = and U(/β < P < lim c U(c/β, V (x; P = V (x, a(p, B(P, inwhichgivenp, a(p is he unique posiive roo of he sricly decreasing funcion F (c = ρ + P (U (c ρ γ λ ρ and B(P is defined by B(P (U (a λ + + a r (U (θ λ ρ + β U(c + λ + r cu (c (3.11 c (U (a λ γ λ (λ + λ (U (θ λ a =. (3.12 (iii When U ( is finie and U(/β < P < P,V (x; P = V (x;, B(P, inwhichb(p is given by (3.7. (iv U ( is finie and P = P,V (x; P = V (x;, B(P, inwhichb(p is given by (3.12. (v U ( is finie and P < P < lim c U(c/β, V (x; P = V (x, a(p, B(P, inwhicha(p and B(P are as in Case (ii. The sraegy c P = I ( (V (X ; P, π P saisfies V P,π P (x; P = V (x; P. = μ r σ 2 (V (X ; P (V (X ; P, (3.13 (3.14 Excep for Case (iii, he nonnegaiviy consrain on he rae of consumpion is no acive. For Case (iii, he opimal rae of consumpion is for wealh beween and x and posiive for wealh greaer han x, in which x = X (U (;, B(P wih B(P given by ( Probabiliy of lifeime ruin Define a process {Y by Y := (V (X, in which V is given in Lemma 3.2. Karazas e al. (1986 show ha bankrupcy occurs when he process {Y his ȳ(p, in which ȳ(p is given by

7 21 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( (3.8 for Case (iii and equals U (a(p for he remaining cases; see Remarks 7.1 and 13.3 in ha aricle. Noe ha U (a migh be infinie, in which case he probabiliy of bankrupcy is. Karazas e al. (1986 show ha he process {Y follows geomeric Brownian moion. Specifically, dy =(r βy d μ r σ Y dz. (3.15 Recall from (2.6 ha he probabiliy of lifeime ruin equals ψ(x = E x (e βτ. Thus,fory(P saisfying X (y; a(p, B(P = x (or y(p = (V (x; P, he probabiliy of lifeime ruin is ψ(x = ˆψ(y(P in which ˆψ(y solves β ˆψ(y =(r βy ˆψ (y + γ y 2 ˆψ (y, y ȳ(p; ˆψ ( ȳ(p = 1. (3.16 From (3.16, i is sraighforward o prove he following resul. Lemma 3.3. The probabiliy of lifeime ruin ψ is given by ψ(x = ˆψ(y, wih y saisfying X (y; a, B = x, in which ˆψ equals ˆψ(y; P = ( y(p/ȳ(p ρ +, y [, ȳ(p. (3.17 Recall ha ρ + > 1 is he posiive roo of (3.3. Remark 3.1. ȳ = in Case (i in Lemma 3.2. WhenP U(/β, hen he probabiliy of bankrupcy, P x (τ <, is zero (see Remarks 7.1 and 13.3 in Karazas e al., 1986, and herefore he probabiliy of lifeime ruin, P x (τ < τ d,isalsoequalozero. Now, given a hreshold ϕ for he probabiliy of lifeime ruin, we wish o deermine a corresponding Lagrange muliplier P in (2.1. Then, by he soluion of (2.12 given in Lemma 3.2, we have he corresponding opimal invesmen and consumpion sraegies, as saed here. Firs, we show ha he probabiliy of lifeime ruin in (3.17 is increasing wih respec o P. Lemma 3.4. As a funcion of he penaly P [U(/β, lim c U(c/β, he probabiliy of lifeime ruin ˆψ is increasing. Moreover, when P = U(/β, he probabiliy of lifeime ruin equals, and as P approaches lim c U(c/β, he probabiliy of lifeime ruin approaches 1. Proof. The limiing values of he probabiliy wih respec o P follow from Remark 3.1 and he discussion preceding Lemma 3.2. Nex,leP 1 < P 2 ; we wish o show ha E x (e βτ under he opimal consumpion and invesmen sraegy (1, π (1 corresponding o P 1 is less han or equal o E x (e βτ under he opimal consumpion and invesmen sraegy (2, π (2 corresponding o P 2. From he opimaliy of (2, π (2 for P 2,wehave E x [ τ (1,π (1 e β U ( c (1 and we can rewrie his inequaliy as [ τ d + e βτ (1,π (1 P 2 E x (2,π (2 e β U ( c (2 d + e βτ (2,π (2 P 2, (3.18

8 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( E x [ τ (1,π (1 E x [ τ e β U ( c (1 (2,π (2 e β U ( c (2 d + e βτ (1,π (1 P 1 + e d + e βτ (2,π (2 P 1 + e (1,π (1 βτ (P 2 P 1 (2,π (2 βτ On he oher hand, from he opimaliy of (1, π (1 for P 1,wehave E x [ τ (2,π (2 e β U ( c (2 From (3.19 and (3.2, i follows ha E x[ e (1,π (1 βτ E x [ e which is wha we wished o show. [ τ d + e βτ (2,π (2 P 1 E x (2,π (2 βτ (1 (P 2 P 1,π (1 e β U ( c (1. (3.19 d + e βτ (1,π (1 P 1. (3.2, (3.21 Remark 3.2. Noe ha if P =, hen he probabiliy of lifeime ruin corresponding o V in (2.12 equals he probabiliy of ruin when we maximize W in (2.3 wihou he consrain in (2.4. Therefore, P = aains he maximum allowable value for he hreshold ϕ(x, namely, he probabiliy of ruin when he individual maximizes W wihou any consrain on he probabiliy of ruin. Lemma 3.5. The probabiliy of lifeime ruin defined in (3.17 is a coninuously differeniable funcion of P. Proof. To show ha ˆψ is a coninuously differeniable implici funcion of P, we will need o deermine ha (given x y and ȳ are coninuously differeniable funcions of P.Noehay is a soluion X (y; a(p, B(P = x (for proper values of a and B, whose choice depends on he value funcion U as described by Lemma 3.2. Since X ( ; a, B is sricly decreasing and coninuously differeniable, he implici funcion heorem implies ha y is a coninuously differeniable funcion of P. On he oher hand, ȳ is eiher given by (3.8, byu (a, in which a is he unique roo of he sricly decreasing funcion F in (3.11, orbyu (. One only needs o prove ha U (a is a coninuously differeniable funcion of P. Bu, his resul again follows from he implici funcion heorem hanks o he fac ha F is sricly decreasing and differeniable. Theorem 3.1. For a given wealh x >, here exiss a consan P such ha he sraegy P, π P, defined by (3.13, maximizes expeced uiliy of consumpion (2.3 subjec o he probabiliy of ruin consrain (2.4. Proof. Le U(/β. Recall ha if P U(/β in (2.12, ruin is impossible under he opimal sraegy. Therefore, from Remark 3.2, we can choose P =. Now, le U(/β <. Thanks o Lemmas 3.4 and 3.5 and o Remark 3.2, for a fixed x >, we can deermine a P saisfying P x (τ P,π P τ d = ϕ(x [, 1. The resul follows now from Lemmas 2.1 and 3.2. Remark 3.3. I follows from Lemmas 3.4 and 3.5 ha for a given x >, he consan P in Theorem 3.1 can be compued by bisecion search using (3.17 along wih X (y; a, B = x and he expression for ȳ described in Lemma 3.2.

9 212 E. Bayrakar, V.R. Young / Finance Research Leers 5 ( Acknowledgmens E. Bayrakar hanks he Naional Science Foundaion for financial suppor. V.R. Young hanks he Cecil J. and Ehel M. Nesbi Professorship for financial suppor. References Basak, S., Shapiro, A., 21. Value a risk based risk managemen: Opimal policies and asse prices. Journal of Business 78, Bayrakar, E., Young, V.R., 27a. Correspondence beween lifeime minimum wealh and uiliy of consumpion. Finance and Sochasics (ISSN , Bayrakar, E., Young, V.R., 27b. Minimizing he probabiliy of lifeime ruin under borrowing consrains. Insurance Mahemaics and Economics (ISSN , Bayrakar, E., Young, V.R., 28. Proving he regulariy of he minimal probabiliy of ruin via a game of sopping and conrol, Technical repor, Universiy of Michigan, URL: hp://arxiv.org/abs/ v2. Boyle, P., Tian, W., 27. Porfolio managemen wih consrains. Mahemaical Finance (ISSN , Cheridio, P., Delbaen, F., Kupper, M., 25. Coheren and convex moneary risk measures for unbounded càdlàg processes. Finance and Sochasics (ISSN , Karazas, I., Lehoczky, J.P., Sehi, S.P., Shreve, S.E., Explici soluion of a general consumpion/invesmen problem. Mahemaics of Operaions Research (ISSN X 11, Young, V.R., 24. Opimal invesmen sraegy o minimize he probabiliy of lifeime ruin. Norh American Acuarial Journal (ISSN ,