Ewa Marciniak, Jakub Trybuła OPTIMAL DIVIDEND POLICY IN DISCRETE TIME

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1 DEMONSTRATIO MATHEMATICA Vol XLVII No Ewa Marciniak, Jakub Trybuła OPTIMAL DIVIDEND POLICY IN DISCRETE TIME Abstract A problem of optimal dividend policy for a firm with a bank loan is considered A regularity of a value function is established A numerical example of calculating value function is given 1 Introduction A firm which net income is uncertain must choose between paying dividends and creating cash reserves in order to maximize the expected present value of all payoffs until bankruptcy time In influential papers [DV] and [JS] some continuous time problems were formulated and studied In [DV], Décamps and Villeneuve considered a firm with a growth option, which gives the opportunity to invest in a new technology in order to increase its profit rate They showed when it is optimal to postpone dividend payoffs, to accumulate cash and to invest at a certain date This work deals with the optimal dividend and growth option problem, where expenses are covered by a loan We are concerned with a discrete time model 2 Formulation of the problem Consider a firm whose activities generate at moment n an income Y n and a cash process X n Moreover, the firm took out a loan that have to be paid in constant rates C at each moment n That may cause bankruptcy if the cash process is too low The manager of the firm acts in the best interest of shareholders and at any moment n may decide to pay out dividends The aim of this paper is to maximize the expected discounted satisfaction of payoffs paid out till bankruptcy time The mathematical formulation of this problem is as follows Assume we have a probability space pω, F, Pq and a sequence ξ 1, ξ 2, of independent 2010 Mathematics Subject Classification: Primary 93E20; Secondary 60J20 Key words and phrases: optimal dividend policy, stochastic control in discrete time DOI: /dema c Copyright by Faculty of Mathematics and Information Science, Download Warsaw Date University 11/20/17 of 12:13 Technology PM

2 Optimal dividend policy in discrete time 259 identically distributed random variables such that P pξ 1 ą 1q 1 and E ξ 1 p ă 8, for some p ą 1 Let r 0 be a risk free rate of return Define F px, y, b, ξq : y p1 ` ξq h pb, xq, Gpx, y, a, bq : px Cqp1 ` rqp1 a bq ` y, where function h is positive, increasing and bounded from above by a constant M ą 0 Starting with initial values px 0, Y 0 q px, yq, we have the following recursive formula Y n`1 F px n, Y n, b n, ξ n`1 q, X n`1 G px n, Y n`1, a n, b n q, where a n and b n represent respectively dividend paid out and investment at time n A control policy π pa π n, b π nq npn is said to be admissible if and only if pa π n, b π nq is adapted to the filtration generated by the process px n, Y n q and pa π n, b π nq P U, for all n P N, where Define the bankruptcy time τ 0 by and performance functional U tpa, bq : a 0, b 0 and a ` b 1u τ 0 : inf tn 0: X n C ă 0u J π px, yq E x,y ÿ 8 ı γ n g pa π npx n Cqq I tτ0 ąnu, n 0 where γ P p0, 1q is a given discount factor and g is a utility function from the class " K : f : R Ñ r0, 8q : f pxq 0 for all x 0, * f pxq f is continuous and }f} K : xpr 1 ` x ă 8 The objective is to calculate the value function v 8 and an optimal dividend policy π 8 pa 8 n, b 8 n q npn, that is v 8 and π 8 such that v 8 px, yq : J π px, yq J π8 px, yq, where remum is taken over all admissible strategies Notice that since the time of bankruptcy is included in the function J π, we need to extend a

3 260 E Marciniak, J Trybuła state space putting E : tδu Y E 1 Y E 2, where E 1 : tpx, yq P r0, 8q 2 : x C ă 0u, E 2 : tpx, yq P r0, 8q 2 : x C 0u, δ P R 2 zr0, 8q 2 We equipped E with a natural topology and assume that if px n, Y n q P tδu Y E 1, then px n`1, Y n`1 q δ Further, for the admissible strategy π pa π n, b π nq npn, we have that pa π n, b π nq P A px n, Y n q, for each n P N, where A px, yq tpa, bq : a 0, b 0 and a ` b 1u, for px, yq P E 2, A px, yq tp0, 0qu, for px, yq P E 1 Y tδu Finally, we will denote by px π n, Y π n q npn the process corresponding to the strategy π pa π n, b π nq npn Note that J π px, yq E x,y 8 ÿ n 0 ı γ n g pa π n px n Cqq 3 Solution by the Bellman Dynamic Programming For the convenience of the reader, we recall the definitions of upper semicontinuous functions and set-valued mappings Below ps, τ q and pr, ρq are topological spaces and 2 R denotes the set of all subsets of R Definition 31 A function φ: S Ñ R is called upper semi-continuous (usc) if for each open set B Ă R, the set ts P S : φ psq P Bu is open Definition 32 A set-valued mapping ψ : S Ñ 2 R is called upper semicontinuous (usc) if for each open set F Ă R, the set ts P S : ψ psq Ă F u is open Let A be a positive constant We will consider the following set of functions " W : f : E Ñ r0, 8q : f is usc on E, }f} W : * f px, yq 0 for each px, yq P E 1 Y tδu, with the weight function w px, yq # 1 ` x ` Ay, if px, yq P Ez tδu, 1, if px, yq δ f px, yq px,yqpe w px, yq ă 8, Obviously pw, } } W q is a complete metric space Define an operator T γ on

4 W by the formula where T γ f px, yq : Optimal dividend policy in discrete time 261! ) g pa px Cqq ` γp a,b f px, yq, P a,b fpx, yq : EpfpX n`1, Y n`1 q X n x, Y n y, a n a, b n bq Theorem 33 Let α : max t1 ` r, M p1 ` Eξ 1 qu Suppose that g P K and γα ă 1 Then there exists A ą 0 such that the value function v belongs to W and is the unique solution to the equation T γ v px, yq v px, yq, v P W Moreover, there are measurable functions a : E Ñ r0, 1s and b : E Ñ r0, 1s, such that for all px, yq P E, pa px, yq, b px, yqq P A px, yq and T γ v px, yq g pa px, yq px Cqq ` γp a px,yq,b px,yq v px, yq, and the optimal strategy π pa n, b nq npn is of the form a pa n, b nq X π n, Y π n, b X π n, Y π n, n P N To prove Theorem 33, we need the following general result on measurable selectors (for proofs see [S1] and [S2]) Theorem 34 Let X be a metric space, U be a separable metric space, and let U : X Ñ 2 U be a usc set-valued measurable mapping, such that for each x P X, U pxq is nonempty and compact in U Assume that Ψ : tpx, uq : x P X and u P U pxqu is measurable with respect to the product topology on X ˆ U and that ψ : Ψ Ñ R is measurable, usc and bounded from above Then there exists a measurable function f : X Ñ U such that for all x P X, f pxq P U pxq and ψ pxq : ψ px, uq ψ px, f pxqq upupxq Furthermore, ψ is measurable, usc and bounded from above on X Proof of Theorem 33 Let us first prove that the operator T γ transforms W into W To this end, notice that for any function f P W, we have (1) f px, yq }f} W w px, yq, for all px, yq P E Consider two cases:

5 262 E Marciniak, J Trybuła (2) (3) 1 If px, yq P tδu Y E 1 then A px, yq tp0, 0qu and 2 If px, yq P E 2 then T γ f px, yq Since g P K, we have T γ f px, yq g p0q ` γef pδq 0, T γ f px, yq 0 px,yqptδuye 1 w px, yq! ) g pa px Cqq ` γp a,b f px, yq tg pa px Cqq ` γ }f} W p1 ` x p1 ` rq ` y p1 ` Aq M p1 ` Eξ 1 qqu (4) g pa px Cqq }g} K p1 ` xq, for all 0 a 1 and x 0 By (3) and (4), T γ f px, yq w px, yq Therefore, in particular we have In conclusion (5) }T γ f} W px,yqpe ˆ ˆ }g} K ` γ α ` M p1 ` Eξ 1q }f} A W T γ f px, yq ă 8 px,yqpe 2 w px, yq T γ f px, yq w px, yq ă 8, for all f P W Furthermore, from (2) we have T γ f px, yq 0, for all px, yq P tδu Y E 1 Our next claim is that T γ f is usc for any f P W Fix an arbitrary f P W and define where φ px, yq : T γ f px, yq }T γ f} W w px, yq ψ px, y, a, bq, ψ px, y, a, bq : g pa px Cqq ` γp a,b f px, yq }T γ f} W w px, yq Notice that by (5), the function ψ is non positive To use Theorem 34, we need ψ to be usc on Ψ : tpx, y, a, bq : px, yq P E, pa, bq P A px, yqu As g and w are continuous, it is sufficient to prove that P a,b f px, yq is usc Consider an arbitrary sequence px n, y n, a n, b n q n 1 Ă Ψ such that px n, y n, a n, b n q Ñ px, y, a, bq Applying (1), we obtain q n p : f px n`1, y n`1 q p q p1 ` ξ 1 p q, P as,

6 Optimal dividend policy in discrete time 263 for some q ą 0 Hence, as E ξ 1 p ă 8, the sequence tq n u n 1 is uniformly integrable and by the Fatou lemma, for all px n, y n, a n, b n q Ñ px, y, a, bq, we have lim P an,bn f px n, y n q P a,b f px, yq nñ8 Therefore, P a,b f px, yq is usc on Ψ Applying Theorem 34, we conclude that φ is usc and there exists a measurable selector pâ, ˆbq such that φ px, yq ψ x, y, â px, yq, ˆb px, yq, for all px, yq P E Hence, since w is continuous on E, the function T γ f is usc on E and T γ f px, yq g pâ px, yq px Cqq ` γp âpx,yq,ˆbpx,yq f px, yq Finally, T γ f P W for all f P W To show that T γ is a contraction on W, consider two arbitrary functions f 1, f 2 P W Then ˇ T γ f 1 px, yq T γ f 2 px, yq γ ˇP a,b ˇ rpf 1 f 2 q px, yqsˇ Using (1) we obtain T γ f 1 px, yq T γ f 2 px, yq γ }f 1 f 2 } W P a,b w px, yq We need to consider 3 cases: 1 If px, yq δ then since A px, yq tp0, 0qu and P 0,0 w pδq 1, we have T γ f 1 pδq T γ f 2 pδq w pδq γ }f 1 f 2 } W 2 If px, yq E 1 then A px, yq tp0, 0qu and P 0,0 w px, yq w pδq 1 Moreover, since w 1, for all px, yq E 1, we have T γ f 1 px, yq T γ f 2 px, yq w px, yq 3 If px, yq P E 2 then A px, yq γ }f 1 f 2 } W γ }f 1 f 2 } w px, yq W! ) pa, bq P r0, 1s 2 : a ` b 1 and P a,b w px, yq 1 ` p1 ` rq x ` y p1 ` Aq M p1 ` Eξ 1 q

7 264 E Marciniak, J Trybuła Therefore, T γ f 1 px, yq T γ f 2 px, yq 1 ` x ` Ay 1 ` p1 ` rq x ` y p1 ` Aq M p1 ` Eξ 1 q γ }f 1 f 2 } W px,yqpe 2 1 ` x ` Ay ym p1 ` Eξ 1 q γ }f 1 f 2 } W α ` y 0 1 ` Ay ˆ γ }f 1 f 2 } W α ` M p1 ` Eξ 1q A ˆ γ α ` M p1 ` Eξ 1q }f 1 f 2 } A W Summarizing, we have where }T γ f 1 T γ f 2 } W γη }f 1 f 2 } W, η : ˆ α ` M p1 ` Eξ 1q A Since we assumed that γα ă 1, therefore γη ă 1 for A large enough This proves that T γ is a contraction on W Thus by the Banach fixed point theorem, T γ has a unique fixed point v in W It remains to prove that lim NÑ8 γn E x,y v px π N, Y π Nq 0, for all px, yq P E and all admissible strategies π Notice that for any px, yq P E 2 and pa, bq P A px, yq, (6) P a,b w px, yq 1 ` p1 ` rqx ` yp1 ` AqMp1 ` Eξq ηw px, yq For px, yq P E 1 Y tδu, A px, yq tp0, 0qu and (7) P 0,0 w px, yq w pδq 1 ηw px, yq Combining (6) and (7), we obtain P a,b w px, yq ηw px, yq, for all px, yq P E, pa, bq P A px, yq Going recursively backwards, we get that for any initial state px, yq P E and admissible strategy π pa π n, b π nq npn, (8) E x,y w px π n, Y π n q η n w px, yq We proved that v P W Therefore, combining (1) and (8) gives E x,y v px π N, Y π Nq }v} W E x,y w px π N, Y π Nq }v} W η N w px, yq

8 Optimal dividend policy in discrete time 265 Hence, we have lim NÑ8 γn E x,y v pxn, π YNq π 0, and we can use the arguments from the proof of Theorem 836 in [HL] 4 Numerical example By Theorem 33, the value function can be obtained by iteration of T γ In this section, we give a numerical example Suppose that the state space is of the form E tδu Y E 2, where δ : p0, 0q, E 2 : rc, Ls ˆ r0, Ss and 0 ă C ă L, 0 ă S In order to make numerical approximations, we need to discretize the state space We shall consider a sequence p m q mpn of partitions of E 2 defined as follows with the convention C x 1 ă x 2 ă ă x lpmq L, 0 ă y 1 ă y 2 ă ă y spmq S, } m } : px k`1 x k q ` k 1,,lpmq 1 k 1,,spmq 1 py k`1 y k q, lim } m} 0 mñ`8 Therefore, we replace the original state space by the discrete one E pmq tp0, 0qu Y px, yq: x P 0, x 1,, x lpmq (, y P 0, y1,, y spmq (( Moreover, we assume that A px, yq tpa, bq : a, b P t0, 01,, 1u and a ` b 1u, for px, yq P E 2, Then for all n P N, we have Y S n`1 : min tf px n, Y n, b n, ξq, Su, A px, yq tp0, 0qu, for px, yq δ X L n`1 : min tg px n, Y n`1, a n, b n q, Lu, and the value function is of the form V pl,sq px, yq : E x,y ÿ 8 γ n g `a π n n 0 `XL n C ı First, we estimate the value function V pl,sq px, yq from below Define $ & 0, if x ă x 1, Π X mpxq : x i, if x i x ă x i`1, where i P t1,, lpmq 1u, % L, if x L,

9 266 E Marciniak, J Trybuła $ & 0, if y ă y 1, Π Y mpyq : y j, if y j y ă y j`1, where j P t1,, spmq 1u, % S, if y S, and consider a new control problem Namely, for all n P N, we have Y n`1 : F m px n, Y n, b n, ξq : F `Π X m px n q, Π Y m py n q, b n, ξ, X n`1 : G m `Xn, Y n`1, a n, b n : G `ΠX m px n q, Y n`1, a n, b n, with an initial condition `X0, Y 0 : `ΠX m px 0 q, Π Y m py 0 q and the value function V pl,sq m px, yq : E x,y ÿ 8 ı γ n g pa π n px n Cqq n 0 To estimate the value function V pl,sq px, yq from above, we need to consider one more problem Define $ 0, if x ă x 1, & Π X x 1, if x x 1, mpxq : x i`1, if x i ă x x i`1, where i P t1,, lpmq 1u, % L, if x L, $ 0, if y ă y 1, & Π Y y 1, if y y 1, mpyq : y j`1, if y j ă y y j`1, where j P t1,, spmq 1u, % S, if y S For all n P N, we have Y n`1 : F m `Xn, Y n, b n, ξ : F Π X `Xn Y m, Π m `Y n, bn, ξ, X n`1 : G m `Xn, Y n`1, a n, b n : G Π X `Xn m, Y n`1, a n, b n, with an initial condition `X0, Y 0 : Π X m px 0 q, Π Y m py 0 q The value function of this problem is V pl,sq m px, yq : E x,y ÿ 8 γ n g `a π n `Xn C ı n 0

10 Notice that, for all px, y, a, b, ξq Optimal dividend policy in discrete time 267 F m F m`1 F F m`1 F m, and G m G m`1 G G m`1 G m Hence (9) V ps,lq m V ps,lq m`1 V ps,lq V ps,lq m`1 V ps,lq m From p9q, the limits lim mñ8 V ps,lq m and lim mñ8 V ps,lq m raises some natural question Namely exist This result Question 1 If lim V ps,lq mñ8 m px, yq V ps,lq px, yq lim mñ8 V ps,lq Now we present numerical solutions for parameters m px, yq, for all px, yq P E? r 003, C 1, γ 035 and M 2 Suppose S 45, L 143 and the partition of E 2 is of the form Let C 1 ă 11 ă 12 ă 13 ă 18 ă 23 ă 28 ă 33 ă 38 ă 48 ă 58 ă 68 ă 93 ă 118 ă 143 L, 0 ă 075 ă 15 ă 3 ă 45 S hpb, xq 2 1 bpx Cq ` 1, gpzq? z, and the random variable # 01, where P pξ 01q 06, ξ 01, where P pξ 01q 04 Note that all the assumptions of Theorem 33 are satisfied In the Appendix we show the optimal strategies, the value functions and the difference between the value functions Acknowledgements The authors are very grateful to Professor S Peszat for encouragement, helpful comments and suggestions

11 268 E Marciniak, J Trybuła 5 Appendix The optimal consumption a State State x y The optimal investment b State State x y The optimal consumption a State State x y The optimal investment b State State x y

12 Optimal dividend policy in discrete time The value function V ps,lq px, yq State State x y The value function V ps,lq px, yq State State x y The difference V ps,lq px, yq V ps,lq px, yq State State x y

13 270 E Marciniak, J Trybuła References [DV] J P Décamps, S Villeneuve, Optimal dividend policy and growth option, Finance Stoch 11 (2007), 3 27 [HL] O Hernández-Lerma, J B Lasserre, Further Topics on Discrete-Time Markov Control Processes, Springer, New York, 1999 [JS] M Jeanblanc-Picqué, A N Shiryaev, Optimalization of the flow of dividends, Russian Math Surveys 50 (1995), [S1] M Schäl, A selection theorem for optimization problems, Arch Math 25 (1974), [S2] M Schäl, Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal, Z Wahrs Verw Geb 32 (1975), [Z] J Zabczyk, Chance and Decision Stochastic Control in Discrete Time, Quaderni, Scuola Normale Superiore, Pisa, 1996 Ewa Marciniak AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY FACULTY OF APPLIED MATHEMATICS al Mickiewicza KRAKÓW, POLAND dalczyns@wmsmataghedupl Jakub Trybuła JAGIELLONIAN UNIVERSITY FACULTY OF MATHEMATICS AND COMPUTER SCIENCE ul Łojasiewicza KRAKÓW, POLAND JakubTrybula@imujedupl Received February 28, 2012; revised version September 24, 2012