Company value with ruin constraint in Lundberg models

Hipp, Christian Article Company value with ruin constraint in Lundberg models Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Hipp, Christian (2018) : Company value with ruin constraint in Lundberg models, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 6, Iss. 3, pp. 1-15, https://doi.org/10.3390/risks6030073 This Version is available at: https://hdl.handle.net/10419/195865 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Company Value with Ruin Constraint in Lundberg Models Christian Hipp Institute of Finance, Banking and Insurance, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany; [email protected]; Tel.: +49-2202-188-2031 Received: 17 May 2018; Accepted: 10 July 2018; Published: 20 July 2018   Abstract: In this note we study the problem of company values with a ruin constraint in classical continuous time Lundberg models. For this, we adapt the methods and results for discrete de Finetti models to time and state continuous Lundberg models. The policy improvement method works also in continuous models, but it is slow and needs discretization. Better results can be obtained faster using the barrier method for discrete models which can be adjusted for Lundberg models. In this method, dividend strategies are considered which are based on barrier sequences. In our continuous state model, optimal barriers can be computed with the Lagrange method leading to a backward recursion scheme. The resulting dividend strategies will not always be optimal: in the case without ruin constraint, there are examples in which band strategies are superior. We also develop equations for optimal control of dynamic reinsurance to maximize the company value under a ruin constraint. These identify the optimal reinsurance strategy in no action regions and allow for an interactive computation of the value function. We apply the methods in a numerical example with exponential claims. Keywords: stochastic control; optimal dividend payment; ruin probability constraint; Lundberg models MSC: 93E20; 93E25; 49L20 1. Introduction We consider a classical Lundberg model for the surplus S(t)of an insurer at time t: S(t) = s+ct −X1−... −XN(t),t≥0, (1) with initial surplus s≥ 0, premium rate c and claim sizes X , X1 , X2 , ... which are iid. We will assume throughout that Xis non-atomic, i.e., P{X=x}=0 for all x≥0. The claims arrival process N(t) is a homogeneous Poisson process with constant intensity λ . We always assume the net profit condition c>λE[X] which guarantees that τ=inf{t:S(t)< 0 } is infinite with positive probability, i.e., the ruin probability ψ(s) = P{τ<∞|S( 0 ) = s}< 1, and on {τ=∞}we have S(t)→∞. For a discount rate δ> 0 we compute the value of the insurance company as the expected discounted future sum of dividends: V(s) = sup DEZ∞ 0e−δtdD(t)|S(0) = s, (2) Risks 2018,6, 73; doi:10.3390/risks6030073 www.mdpi.com/journal/risks Risks 2018,6, 73 2 of 15 where the supremum is taken over all predictable dividend strategies with total payment D(t) up to time t. We always assume that no dividends are paid at or after ruin. For a given dividend strategy D , the ruin probability of the with dividend risk process is defined by ψD(s) = P{S(t)−D(t)<0 for some t≥0|S(0) = s}. In this note we consider the company value with ruin constraint V(s,α) = sup DEZ∞ 0e−δtdD(t)|S(0) = s:ψD(s)≤α(3) which is defined for s≥0 with ψ(s)≤α≤1. Dividend optimization with a ruin related constraint has been considered in several earlier papers: Albrecher and Thonhauser (2007), Hernandez et al. (2017) as well as Junca et al. (2018) deal with the time value of ruin. In these problems both objective functions are discounted which allows for explicit solutions. Without a ruin constraint, the optimal dividend problem is often solved by a barrier strategy, i.e., there exists a constant B∗≥0 such that D(t) = sup 0≤u≤t (S(u)−B∗)+, (4) V(s) = v(s)/v0(B∗),s≤B∗, (5) V(s) = V(B∗) + s−B∗,s≥B∗, (6) B∗=arg min{v0(s),s≥0}, (7) where v(s) is the scale function given below. The corresponding with dividend process S(t)−D(t) has certain ruin for each initial surplus s. The optimal dividend strategy defining the company value can also be a band strategy in which two or more bands exist: these are disjoint closed intervals [ai , bi] , i= 1, ..., m , with 0 =a1<a2< ... <am in which no dividends are paid. The with dividends surplus process lives in the union of these intervals: dividends are paid at bi , and if the process drops below ai , then no dividends are paid when we reach a point in one of the intervals [aj , bj] , or dividends are paid such that the resulting surplus lies in the largest value bj below the surplus before dividend payment. For situations in which barrier strategies are optimal for spectrally negative Lévy processes see Loeffen (2008) and Loeffen and Renaud (2018). For the characterization and computation of the value function and the corresponding optimal strategy see (Schmidli 2007, sct. 2.4). When the solution v(x) of Equation (12) with v( 0 ) = 1 has a continuous derivative v0(x) with v0(x) decreasing for x<B and v0(x) increasing for x>B , then we have optimality of a barrier strategy, and B=B∗is the value of this barrier. For initial surplus below b1 , the upper bound of the first band, a barrier strategy is optimal, since the process stays in the interval [a1 , b1] . The problem of optimality for barrier strategies is unsolved in the case with ruin constraints. There, it might be possible to have optimal dividend strategies which are band strategies with bands depending on the allowed ruin probability α. In our numerical example we checked for optimality of barrier strategies. Using the iteration method, we could not find a second band in the range s≤50, for exponentially distributed claims. For dividend strategies D(t) representing (or approximating) the company value with ruin constraint, the with dividend process S(t)−D(t)may not be bounded, i.e., P{sup t [S(t)−D(t)] <∞}=0, for otherwise we would have certain ruin which is excluded when α< 1. With dividend processes which are unbounded can be obtained, e.g., with linear barriers (see Albrecher et al. (2005) and Gerber (1981)). Risks 2018,6, 73 3 of 15 Remark 1. With barriers B(t) = at +b,t≥0, 0 <a<c,b≥0, at time t the amount exceeding B(t) is paid out as dividend. We write U(s , a , b) for the dividend value of these strategies and ψ(s , a , b) for the corresponding ruin probability. In our numerical example below (exponential claims with mean 1, premium rate 2 and discount rate 0.03) the dividend value U(s , a , b) is considerably smaller than our numerical values for V(s,α)whenever ψ(s,a,b)≤α. For this we consider the surplus s= 2 and the allowed ruin probability α= 0.2. For 0 <a<c we choose b=b(a)> 0 such that ψ(s , a , b) equals α .The company value computed with the barrier method equals V=V(s , α) = 20.15151719. Since for all a , b> 0 we have U(s , a , b)<v(s)/v0(b) which is the unconstrained dividend value for a barrier b, we obtain from v(2)/v0(6.35) = 20.0832891 and v(2)/v0(14.2) = 20.1146463 that U(2,a,b(a)) < V for b(a) < 6.35 and b(a) > 14.2. For a = 0.188 we obtain b(a) = 14.2955096, so U(2,a,b(a)) < V for a≤ 0.188. For a = 0.385 we obtain b(a) = 6.322805359, so U(2,a,b(a)) < V for a≥ 0.385. The interval 0.188 ≤a≤ 0.385 was checked with step size 0.001. We found a maximum of the corresponding U(s , a , b(a)) values of 17.304735 at a = 0.267. The curves of b(a) and U(2,a,b(a)) are shown in Figure 1. For the computation we used the formulas in Gerber (1981) (18), (20), (23), (24), (27) for ψ(s , a , b) and (20), (48)–(51) for U(s , a , b) . We checked our source code with the numerical results in Albrecher et al. (2005). 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 6 7 8 9 10 11 12 13 14 15 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 15.5 16 16.5 17 17.5 Figure 1. Plots of a→b(a)(Left) and a→U(2, a,b(a)) (Right). 2. Methods 2.1. Policy Improvement without Bellman Equation This method is conceptually and formally equivalent to the iteration method described in Hipp (2017) . We repeat the iteration procedure for the sake of completeness: we start with an appropriate initial value function V0(s,α)such as V0(s,α) = 0 Risks 2018,6, 73 4 of 15 or V0(s,α) = V(s−s(α)),s≥s(α), V0(s,α) = 0, s<s(α), α=E[ψ(s(α)−Y)]. Here, Y is the claim causing ruin when dividends are paid using a dividend strategy with barrier B∗ . The second initial function is the dividend value for the strategy which pays out the total surplus above the barrier B∗+s(α) and stops paying dividends forever when the surplus is below the value s(α). For the iteration we define Vn+1(s,α) = max B≥s{W(s,B)Vn(s,a(B))}(8) Vn+1(s,α)≥Vn+1(s−1, α) + 1 if ψ(s−1)≤α, (9) α=p(s,B) + (1−p(s,B))a(B). (10) Here, p(s , B) is the probability that the without dividend process S(t) falls below zero before reaching B , and W(s , B) is the discounting factor E[exp(−δτ(s , B))] for τ(s , B) the waiting time to reach B from s before ruin. This device produces a monotone sequence of functions Vn which converge to a function which is a possible candidate for the value function V(s , α) . The first Equation (8) covers the case in which no dividends are paid before reaching B , while Equation (9) allows for immediate dividend payment at surplus s . The functions p(s , B) and W(s , B) can be written with the survival function and the scale function given below. 2.2. Barrier Method For the construction of optimal dividend strategies the barrier method has been used for time and state discrete models in Hipp (2018). There, an increasing sequence of barriers had been selected at which dividends are paid, and the dividend value as well as th corresponding ruin probability were computed. We adjust this for the continuous Lundberg model and use an optimal selection of barriers which is based on the Lagrange multiplier method. The two fundamental ingredients for the Lundberg model are the survival probability f(s) = 1−ψ(s)which satisfies f(s) = 0 for s<0 and the equation λE[f(s−X)−f(s)] + c f 0(s) = 0, (11) as well as the scale function v(s)which is the unique solution of the equation −δv(s) + λE[v(s−X)−v(s)] + cv0(s) = 0, (12) satisfying v(s) = 0, s<0, and v(0) = 1. We also consider the functions g(s) = E[f(s−X)] and w(s) = E[v(s−X)] . With formulas (11) and (12) these can be computed as g(s) = λf(s)−c f 0(s),s≥0, (13) w(s) = (λ+δ)v(s)−cv0(s),s≥0. (14) Many ruin related quantities can be expressed via f(s) , and most dividend values are connected with v(s) . Examples are first entry probabilities (before ruin) and discount factors corresponding to the time of first entry (see Hipp (2018)). So, e.g., the functions p(s,B)and W(s,B)are given by Risks 2018,6, 73 5 of 15 p(s,B) = f(s)/f(B)and W(s,B) = v(s)/v(B). (15) The Equations (11) and (12) have solutions with continuous first derivative when the claim size distribution is non-atomic. All solutions of (11) vanishing for s< 0 are proportional. The same is true for Equation (12). Fix an initial surplus s> 0 and an allowed ruin probability α with ψ(s)<α≤ 1. We first define the running ruin probabilities introduced in Hipp (2017) for a dividend strategy D which is defined via a finite non-decreasing sequence s<B0<B1< ... <Bn : whenever we reach Bi , i≤n , all incoming premia are paid out as dividends until the next claim happens. When the surplus reaches Bn , we pay out all premia as dividends until the next claim happens, and then we stop dividend payment forever. For 0 ≤x<B0 the ruin probability ψD(x) of the (with dividend) risk process S(t)−D(t) is proportional to ψ(x), and ψD(s) = αimplies that it can be written as ψD(x) = 1−γ0+γ0ψ(x),γ0=1−α 1−ψ(s). (16) These running allowed ruin probabilities had been introduced in Hipp (2018) for a discrete model. Let Bi be one of the barrier levels at which all premia are paid out as dividends. Then the corresponding running ruin probability ri(x),x≤Biafter hitting Bihas the form 1−ri(x) = γi(1−ψ(x)) = γif(x). We leave level Bi at the first claim during dividend payment. The ruin probability ri+1(x) after leaving the level Bi and before hitting Bi+1 is also of the above form with γi+1 derived from ri(Bi) = E[ri+1(Bi−X)] = γif(Bi) = γi+1g(Bi)or γi+1=γi f(Bi) g(Bi). The ruin probability ψD(x) for the with dividend process S(t)−D(t) satisfies ψD(s)≤α whenever γn=max iγi≤1. Lemma 1. For fixed surplus s≥ 0 and allowed ruin probability 0 <α≤ 1 let s<B0<B1< ... be an infinite sequence with Bi→∞ .Assume that the corresponding factors γi converge to 1. Then ψD(s) = α ,where D is the dividend strategy which pays dividends on the barrier levels Bi. Proof. The survival probability 1 −ψD(s) is the probability that from surplus s we reach B0 , after leaving B0 at a claim we reach B1 and so on. For B≥x≥ 0 let p(x , B) be the probability that the without dividend process S(t)starting at xwill reach Bbefore ruin. Then p(x,B) = f(x) f(B). The events Ai , i= 1, 2, ... that after dividend payment at Bi−1 the surplus will reach Bi are independent and have probability E[p(Bi−1−X,Bi)] = g(Bi−1) f(Bi). Risks 2018,6, 73 6 of 15 The event A0that S(t)starting at swill reach B0has probability f(s) f(B0). For i≥0 we have g(Bi) f(Bi+1)=γi γi+1 f(Bi) f(Bi+1) and so we obtain the survival probability f(s) f(B0) ∞ ∏ i=0 g(Bi) f(Bi+1)=f(s)γ0=1−α. (17) With a finite sequence s<B0<B1<B2< ... <Bn we represent a dividend strategy in which dividend payment is stopped forever when the surplus leaves Bn , i.e., γn+1= 1 and Bn+1=∞ . The survival probability after reaching Bn equals g(Bn) . For this strategy we obtain the survival probability 1−ψD(s) = f(s) f(B0) n−1 ∏ i=0 g(Bi) f(Bi+1)g(Bn) = (1−α)/γn≥1−α. The present value of dividends paid on a barrier Bidoes not depend on i, it is A=Z∞ 0Zt 0cexp(−δu)λexp(−λt)dt =c λ+δ. This dividend value is discounted to the time at which Bi is visited first. The discount factor for the time spent at barrier level Biequals C=Z∞ 0λexp(−λt)exp(−δt)dt =λ λ+δ. (18) The discount factor for the time between leaving Biand hitting Bi+1≥Biequals Gi:=E[w(Bi)/v(Bi+1)] = λ+δ λ v(Bi) v(Bi+1)−c λ v0(Bi) v(Bi+1). (19) The dividend value for a strategy Dwith barriers B0,B1, ..., Bnsatisfying s≤B0≤B1≤... ≤Bn is given by VD(s) = Aw(s)/w(B0) n ∑ i=0 Ui, (20) where U0= 1 and Ui+1=CGiUi . The present value of dividends paid at level B0 equals Av(s)/v(B0) . The present value for dividends paid at level B1 equals Av(s)/v(B0)CG0 , and the next barrier level contributes Av(s)/v(B0)C2G0G1, and so on. For equal barriers Bi=Mwe easily obtain that for s≤Mand with G=Gi V(s) = Av(s) v(M) 1 1−CG =v(s) v0(M) Risks 2018,6, 73 7 of 15 which is the well known correct value (compare with Renaud and Zhou (2007)). We first consider dividend strategies which have only one barrier B at which dividends are paid afinite number of times. This means that barriers are paid at B0=B , ..., BK=B , and after paying dividends on barrier level B K + 1 times we stop paying dividends forever. The barrier B can be chosen such that the allowed ruin probability is exact, and the corresponding dividend value can be given explicitly. Lemma 2. For any given pair s , α with ψ(s)<α and K≥ 1we can find B such that the above dividend strategy D(t) yields the with dividend ruin probability P{S(t)−D(t)<0 for some t> 0 |S( 0 ) = s}=α . The corresponding dividend value is V=v(s) v0(B) 1−1−cv0(B) (λ+δ)v(B)K+1!. (21) Proof. The with dividend survival probability is f(s) f(B) K ∏ i=1 g(B) f(B)g(B)(22) which equals 1 −αwhenever 1−α=f(s)g(B) f(B)K+1 , so the appropriate choice for Bis the solution of g(B) f(B)=γ1/(K+1) 0. (23) Equation (21) follows from (18)–(20). These one barrier strategies are clearly suboptimal: the dividend value converges to zero when K→∞ , since the corresponding barriers converge to infinity. However, for moderate values of K the corresponding dividend values are surprisingly good. These can be used for barrier strategies for the tail of a finite barrier sequence s<B0<B1< ... <Bn: we choose K large enough such that the barrier B corresponding to γn and K is larger than Bn , and then the sequence of barriers in which Bn is followed by K values of B has the exact allowed ruin probability. The same method applies to the choice of B0: if γN< 1 then we can decrease B0 to get an exact allowed ruin probability (and a slightly larger dividend value). A manual search for good sequences of barrier sequences is tedious: we used 1. linear sequences of the form Bi=B0+i∆,i=1, ..., n, 2. sequences satisfying the recursion g(Bi+1)/f(Bi+1) = g(Bi)/f(Bi)ρ,i=0, ..., n−1, (24) 3. barriers s<B0<... <Bnwith small nwhich are selected manually. Linear sequences produce dividend values which are almost optimal, and sequences from recursion (24) can do even better, but are still suboptimal. We come close to optimal results when, for large N , we maximize the dividend value V=G(B0 , ..., BN) using the Lagrange multiplier method. The problem has a high dimension, but using the structure of the function G we could find some simplifications. As a first step, we look at a related but simpler problem. Risks 2018,6, 73 8 of 15 Problem: For N ≥1and a smooth increasing function F(u)maximize the function G(u0, ..., uN) = N ∑ n=0 n ∏ i=0 F(ui) under the constraint u0+u1+... +uN=R. We start with the Lagrange system of equations ∂G ∂uk = N ∑ n=k n ∏ i=0 F(ui)F0(uk) F(uk)=L,k=0, ..., N. For n =N we get L= N−1 ∏ i=0 F(ui)F0(uN) which leads to the recursive relations F0(ui−1) F(ui−1)=F(ui)...F(uN−1)F0(uN) 1+F(ui) + F(ui)F(ui+1) + ... +F(ui)...F(uN). For the calculation of ui−1we would need that F0(u)/F(u)is monotone. In our control problem we want to maximize the following term by the choice of barriers B0,B1, ..., BN: with F(x) = w(x)/v(x)and Z(x) = 1/v(x) G(B0, ..., BN) = V Av(s)=Z(B0) + DF(B0)Z(B1) + D2F(B0)F(B1)Z(B2) + ... +DNF(B0)...F(BN−1)Z(BN). As in the above problem, we fix BN and use the Lagrange equations with a multiplier L and H(x) = g(x)/f(x) ∂ ∂Bi G(B0, ..., BN) = LH0(Bi),i=0, ..., N. (25) With i=Nwe get a formula for L: F(B0)...F(BN−1Z0(BN) = H0(BN). (26) Our function G(B) allows for the following simplification: for i= 0, ..., N− 1 the barrier Bi solves A1H0(x)F(x) = Z0(x) + A2F0(x), (27) A1=CN−iF(Bi)...F(BN−1)Z0(BN)H0(Bi)/H0(BN), (28) A2=CZ(Bi+1) + C2F(Bi+1)Z(Bi+2) + ... +CN−iF(Bi+1)...F(BN−1)Z(BN). (29) Here, we first omit the factor Av(s) and all terms in which Bi does not occur, then we divide both sides by F(B0) ... F(Bi−1)Ci . Equation (27) can be solved easily and quickly using Newton or the false position (regula falsi) method. Some care is needed since the method has some uncommon features: We start with the last barrier BN which has practically no impact on the value of the company or the ruin probability, but in the recursion it determines all barriers. If we increase a large N by 1, then the company value as well as the corresponding ruin probability will change a lot, since we add a new first barrier which has a major impact on both numbers. Finally, we want to maximize G(x0 , ..., xN) under the constraint s≤x0≤ ... ≤xN . Nevertheless, the method is stable and accurate, even with the numerical precision of MatLab, and it produces admissible solutions. For N= 200 we obtain B0= 12.061058869 and V(2, 0.2) = 20.15151719. Risks 2018,6, 73 15 of 15 Junca, Mauricio, Harold Moreno-Franco, José Pérez, and Kazutoshi Yamazaki. 2018. Optimality of refraction strategies for a constrained dividend problem. arXiv, arXiv:1803.08492. Loeffen, Ronnie L. 2008. On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Annals of Applied Probability 18: 1669–80. [CrossRef] Loeffen, Ronnie L., and Jean-François Renaud. 2009. De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insurance: Mathematics and Economics 46: 98–108. [CrossRef] Renaud, Jean-François, and Xiaowen Zhou. 2007. Distribution of the present value of dividend payment in a Lévi risk model. Journal of Applied Probability 44: 420–27. [CrossRef] Schmidli, Hanspeter. 2007. Stochastic Control in Insurance. Heidelberg: Springer. ISBN 978-1-84800-003-2. c 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).