Company value with ruin constraint in a discrete model

Hipp, Christian Article Company value with ruin constraint in a discrete model Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Hipp, Christian (2018) : Company value with ruin constraint in a discrete model, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 6, Iss. 1, pp. 1-14, https://doi.org/10.3390/risks6010001 This Version is available at: https://hdl.handle.net/10419/195798 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/ Article Company Value with Ruin Constraint in a Discrete Model Christian Hipp Karlsruhe Institute of Technology, Karlsruhe 76131, Germany; [email protected] Received: 6 December 2017; Accepted: 4 January 2018; Published: 7 January 2018 Abstract: Optimal dividend payment under a ruin constraint is a two objective control problem which—in simple models—can be solved numerically by three essentially different methods. One is based on a modified Bellman equation and the policy improvement method (see Hipp (2003)). In this paper we use explicit formulas for running allowed ruin probabilities which avoid a complete search and speed up and simplify the computation. The second is also a policy improvement method, but without the use of a dynamic equation (see Hipp (2016)). It is based on closed formulas for first entry probabilities and discount factors for the time until first entry. Third a new, faster and more intuitive method which uses appropriately chosen barrier levels and a closed formula for the corresponding dividend value. Using the running allowed ruin probabilities, a simple test for admissibility—concerning the ruin constraint—is given. All these methods work for the discrete De Finetti model and are applied in a numerical example. The non stationary Lagrange multiplier method suggested in Hipp (2016), Section 2.2.2, also yields optimal dividend strategies which differ from those in all other methods, and Lagrange gaps are present here. Keywords: stochastic control; optimal dividend payment; ruin probability constraint 1. Introduction Let S(t) , t= 0, 1, ... be the time t surplus of a company and D(t) , t= 0, 1, ... the adapted non-decreasing sequence of accumulated dividends. For a fixed discount factor 0 <r< 1 the dividend value under D(t) = d(1) + ... +d(t)is given by VD(s) = E"∞ ∑ 0 rtd(t)|S(0) = s#, where s≥0 is the initial surplus. The with dividend ruin time of the company is τD=inf{t≥0 : S(t)−D(t)<0}, and ψD(s)is the corresponding with dividend ruin probability ψD(s) = P{τD<∞|S(0) = s}. (1) We assume in the following that dividends are never paid at or after ruin. The object to be investigated is V(s,α) = sup D {VD(s):ψD(s)≤α},s≥0, (2) for a given value α . The quantity V(s , 1 ) is sometimes called value of the company. A lot of research has been done on this quantity, starting with the seminal work of De Finetti (1957) and Gerber (1969), Choulli et al. (2003) and Albrecher and Thonhauser (2008) as well as Schmidli (2007), Section 2.4, and Risks 2018,6, 1; doi:10.3390/risks6010001 www.mdpi.com/journal/risks Risks 2018,6, 1 2 of 14 Loeffen (2008), Avanzi (2009) and Feng et al. (2015) for related work. The concept leading to V(s , α) is a possible answer to the problem posed in Borch (1963) who wrote: If the general manager of our insurance company wants to run the company strictly as a business enterprise, he will probably always seek out the decisions which maximize V(s , 1 ) .If, however, he is concerned with the social responsibility of the company, and the security which it offers to policy holders, he may also consider ψ0(s)[the ruin probability without dividend payment] when making his decisions. He will probably try to balance the two elements, but it is not easy to specify how this should be done. One approach for the computation of the value function V(s , α) is based on a modified Hamilton-Jacobi-Bellman equation for the corresponding stationary Markov process with a bivariate state variable (see Hipp (2003)). This approach needs a fine discretization of the values for the ruin probability, and a large number of iteration steps. In the ruin probability grid, a complete search was necessary in the old version of the policy improvement method. A second approach is the iteration method presented in Hipp (2016). Here, we have shorter but still long computation times. Again we have a complete search, but in the much smaller set of possible surplus values. The purpose of this paper is to study the form of optimal dividend strategies and use running allowed ruin probabilities to speed up the computation of the first method. This enables a big number of iterations for this first method even for fine discretizations. Compared with the iteration method, the second approach, we obtained slightly higher company values caused by the larger number of iterations. Finally, we show that optimal dividend strategies are of barrier type, and we present analytic formulas for the dividend value of these barrier type strategies. In a numerical example we show how appropriate barrier levels can be found. The quantity company value under a ruin constraint should later serve as an objective function for finding optimal reinsurance or investment strategies. For this we need simple algorithms for the computation of V(s , α) with a possible chance to use them also in the corresponding control problem. We restrict ourselves to the following very simple space and time discrete model in which such algorithms can be more easily found. We consider a simple random walk S(t) , t= 0, 1, ..., on the integers starting at s and going up or down by 1 with probability p or q= 1 −p , respectively. This is the classical De Finetti model which is skip free (upwards and downwards). In the insurance framework, t labels periods in which premia of size 1 come in and claims of size 2 go out. In this discrete model, each dividend payment can be assumed to be integral (see Schmidli (2007), Lemma 1.9). In Hipp (2016) it is shown that for rp >1/2 (3) and for fixed s≥ 0 the function α→V(s , α) is continuous (notice that the continuity statement in Hipp (2003), Lemma 2e, is not correct, and its proof has a gap; a correct proof can be found in Hipp (2016), Lemma 2). This shows that a purely discrete model can lead to a situation with a continuous parameter α . To avoid technical problems we will assume in the following that (3) holds. The function α→V(s , α) is strictly increasing on ψ0(s)≤α≤ 1, and V(s , α) = 0 for α≤ψ0(s) . In the De Finetti model the survival probability 1 −ψ0(s) satisfies the following difference equation for functions f(s),s≥ −1, with f(−1) = 0 : f(s) = p f (s+1) + q f (s−1),s≥0. (4) This equation is homogeneous, and the set of solutions is one-dimensional. If 0 ≤s<B then the probability p(s,B)that S(t)reaches Bfrom sbefore ruin satisfies (4), and p(B,B) = 1 leads to p(s,B) = (1−ψ0(s))/(1−ψ0(B)). Risks 2018,6, 1 3 of 14 The company value V(s , 1 ) satisfies a similar difference equation for functions f(s) , s≥ − 1, satisfying f(−1) = 0 : f(s) = r(p f (s+1) + q f (s−1)),s≥0, (5) which holds in the range without dividend payment: let W(s) be the unique solution of (5) with W(1) = 1. Find M≥0 for which W(M+1)−W(M)≤W(s+1)−W(s)for all s≥0. Then V(s, 1) = W(s)/(W(M+1)−W(M)),s≤M, where M is the barrier for dividend payment: V(s , 1 ) = V(M , 1 ) + s−M for s≥M . Also equation (5) is homogeneous, and the set of solutions has dimension 1. So, for the waiting time τ(s , B) to reach B from s before ruin, the expected discount factor W(s , B) = E[rτ(s,B)] is a solution of (5), and W(s , B) is proportional to the function W(s): E[rτ(s,B)] = W(s)/W(B). 2. Methods 2.1. A Modified Bellman Equation Our first numerical method for the company value with ruin constraint is based on a modified Bellman equation. We use the following dynamic equations for V(s , α) (see Hipp (2003), formula (4)): V(s,α) = max{V(s−1, α) + 1, G(s,α)}, (6) G(s,α) = sup A(s,α) {rpV(s+1, β1) + rqV(s−1, β2)}(7) A(s,α) = {(β1,β2)∈B(s,α):pβ1+qβ2=α}(8) B(s,α) = {(β1,β2):ψ0(s+1)≤β1≤1, ψ0(s−1)≤β2≤1}. (9) These equations hold in the range s= 0, 1, 2, ... and ψ0(s)≤α≤ 1, and we use the values V(− 1, α) = 0 and ψ0(− 1 ) = 1. The dynamic equations define the optimal dividend strategy in feedback form: Equation (6) tells us when a dividend of size 1 is paid. Equation (7) gives the value function when no dividend is paid, depending on the next period in which the surplus can go up with probability p or down with probability q . The number α is the running allowed ruin probability, which changes to β1 or β2 in the next period depending on an upor down-move of the surplus. Equation (8) implies that the process of running allowed ruin probabilities is a martingale with mean α . Computation is based on an iteration which is the well known policy improvement procedure (see Hipp (2003)): we start from V0(s , α) = 0, and when Vn(s , α) is given for all s and α , we compute Vn+1(s , α) from Equations (6)–(9) where we use the functions Vn on the right hand side of (7) and obtain Vn+1on the left hand side of (6): Gn(s,α) = sup A(s,α) {rpVn(s+1, β1) + rqVn(s−1, β2)}(10) Vn+1(s,α) = max{Vn+1(s−1, α) + 1, Gn(s,α)}. (11) One can show that the sequence of functions Vn(s , α) is non-decreasing and bounded, and its limit is a solution of the dynamic equations above (see Hipp (2003), Lemma 2a). The classical verification argument yields that the limit is the value function of our control problem, and a solution to the dynamic Equations (6)–(9), see also Hipp (2003), Lemma 2b–d. By continuity of α→V(s , α) , the supremum in (7) is attained at some (β1 , β2)∈A(s , α) . Let α(t) be the process of allowed ruin probabilities defined by α(t+ 1 ) = α(t) when a dividend of size 1 is paid at t , and otherwise α(t+ 1 ) = β1 or α(t+ 1 ) = β2 when S(t)goes up or down, respectively. Notice that for all t≥0 we have Risks 2018,6, 1 4 of 14 ψ(S(t)) ≤α(t)≤1. (12) Using the bivariate process (S(t) , α(t)) we can define the optimal dividend strategy in feedback form: in state (s , α) we pay a dividend of size 1 whenever the maximum in (6) is at V(s− 1, α) + 1. The second component α(t) makes the optimal dividend strategy path dependent. Each payment of size 1 does not change the state, so during dividend payment we stay in the same state until the next claim (downward jump). This implies that there exists a function M(α) such that dividends are paid above M(α) when the allowed ruin probability equals α . The function M(α) is a non-increasing step function. Below, we study the running allowed ruin probabilities α(t) in more detail. In the above computation based on the modified Bellman equation we first used a complete search for the maximizer β1 , β2 . Here we replaced each complete search by an easy computation of running allowed ruin probabilities which speeds up a lot. 2.2. Iteration Method The iteration method is based on the observation that, starting at initial surplus s , we either pay dividends immediately, or we wait until we arrive at some larger surplus B . If at B the ruin probability a(B) is allowed, then we continue with a dividend strategy producing a dividend value (close to) V(B , a(B)) . If we start with an initial function V0(s , α) (e.g., V0(s , α) = 0 ) , and if Vn−1(s , α) is given, then our iteration reads Vn(s,α) = max B≥s{W(s,B)V(s,a(B))}(13) Vn(s,α)≥Vn(s−1, α) + 1 if ψ(s−1)≤α, (14) α=p(s,B) + (1−p(s,B))a(B). (15) 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[rτ(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 might converge to the value function V(s , α) . The first Equation (13) covers the case in which no dividends are paid before reaching B , while Equation (14) allows for immediate dividend payment at surplus s . The numerical results verify that the optimal dividend strategies are of barrier type. 2.3. Running Allowed Ruin Probabilities The running allowed ruin probabilities are ruin probabilities for optimal dividend strategies: if D is the optimal dividend strategy with initial surplus s and allowed ruin probability α , then the ruin probability of the with dividend process S(u)−D(u) , u≥ 0, equals α . At time t the dividend strategy Dt(u) = D(t+u) is the optimal strategy for (S(t) , a(t)) , and so a(t) is the ruin probability for the with dividend process S(t+u)−Dt(u) , u≥ 0. Let B0≥s0 be the surplus above which dividends are paid first, i.e., dividends of size 1 are paid at state B0+ 1 which produces a constant value B0 for the with dividend process until the next claim (downward jump). Since no dividends are paid when s0≤S(t)≤B0 , we can write a(t) = a0(S(t)) , where a0(s) satisfies (4) with a0(− 1 ) = 1. This implies that for 0 ≤s≤B0 we have a0(s) = 1 −γ0+γ0ψ0(s) for some 0 <γ0≤ 1, and γ0 can be computed from a0(s0) = α0 . During dividend payment, a(t) stays on the level α0=a0(B0) , it leaves this level at the first claim. Let B1≥B0 be the level above which we first pay dividends after leaving B0 . Repeating the above reasoning with B1 instead of B0 and B0− 1 instead of s0 , we obtain a function a1(s) , s≤B1 , which is the ruin probability of the with dividend process for the initial pair (B0 , α0) . Since the transition from B0 to B0− 1 is certain, we get a1(B0− 1 ) = α0 . This value determines γ1 in the representation a1(s) = 1 −γ1+γ1ψ(s) . Proceeding in this way, for a non-decreasing sequence of barriers Bi,i≥0, we obtain a non-decreasing sequence of numbers γi,i≥0 satisfying the recursion Risks 2018,6, 1 5 of 14 γi+1=γi 1−ψ0(Bi) 1−ψ0(Bi−1). (16) The dividend strategy D which pays dividends at the levels Bi satisfies the ruin constraint ψD(s0)≤a0provided sup i {γi} ≤ 1. (17) If we stop the sequence Bi at some finite number n , this means that after visiting n barrier levels we stop paying dividends for ever, i.e., γi=1 for i>n. 2.4. The Barrier Method The barrier method does not use iterations or discretizations, it is more interactive and simpler. We start with a (finite) sequence of barrier levels B(i) , i= 1, ..., n and compute the dividend value with an analytic formula in which all dividends which are paid on one of these levels are appropriately discounted and added. The value of dividend payments on the level Bi , discounted to the time when we reach Bi+1 after leaving Bi−1−1, does not depend on iand equals A= ∞ ∑ k=0 pnrn=1/(1−rp). So the dividend value consists of the sum of all these payments, discounted over the times elapsed between s and B0+ 1 (for the payments at level B0 ), then over this time plus the time elapsed between B0− 1 and B1+ 1 plus the time spent on level B0 (for the payments at level B1 ), and so on. The discount factor for the time spent on level Biis again independent of i, it equals C= ∞ ∑ k=1 qpn−1rn=qr/(1−rp). The present value for payments on level B0is AW(s) W(B0+1), for level B1we obtain the present value AW(s) W(B0+1)CW(B0−1) W(B1+1) and so on. A closed formula for the total dividend value of the dividend strategy Dis VD(s) = AW(s) W(B0+1) ∞ ∑ k=0 Ckk ∏ i=1 W(Bi−1−1) W(Bi+1). (18) One method to find barrier levels uses the function M(α) , which might come from the computation with one of the above numerical methods: M(α) = min{s:V(s+1, α) = V(s,α) + 1}. Notice that for all s≥M(α) we have V(s+ 1, α) = V(s , α) + 1, since the running allowed ruin probability equal α for s≥M(α) (use α=pβ1+qβ2 which holds for β2=α only if β1=α ). The function M(α) (see Figure 1left below) is combined with the running ruin probabilities ai(s) defined sequentially as follows: a0(s) is computed from the initial data (s0 , α0) . The intersection of a0(s) with M(α) , plotted in the same diagram, is barrier B0 . From the data (B0 , a0(B0)) we compute a1(s) , and so on: the intersection points of ai(s) with M(α) are the barriers Bi . Figure 1right above shows the functions ai(s),i=0, ..., 15 intersecting M(α)at level s=4 or s=5. Risks 2018,6, 1 6 of 14 0 0.1 0.2 0.3 4 6 8 10 12 0 2 4 6 8 10 0.05 0.1 0.15 0.2 0.25 0.3 Figure 1. The function M(α)and the running ruin probabilities ai(s). Another, more precise method is an (almost) complete search in the vectors of non-decreasing n− tuples of numbers k , k+ 1, ..., K , where k is the barrier in the unconstrained problem and K a suitable limit of the state space for s . Search for a smallest – in pointwise order – vector for which the maximal γi is smaller than 1. Finally we apply formula (18) to this smallest vector. The computation of the γ0s is very simple, and the test checks for an appropriate with dividend ruin probability. A numerical example is given below. Following our intuition we searched for a barrier sequence only in the set of all non decreasing sequences. That intuition does not fail in this situation can be seen with the following argument. The functions ai(s) are defined by ai(− 1 ) = 1, Equation (4) for 0 ≤s≤Bi− 1, and some value for ai(s0) with 0 ≤s0≤Bi . The functions are concatenated by the value in which the with dividend surplus jumps after leaving the barrier level Bi . For Bi+1≥Bi− 1 this produces the recursion (16), but for Bi+1<Bi− 1 after a jump to Bi− 1 we pay out dividends immediately which leads us to Bi+1. In this case the recursion reads γi+1=γi 1−ψ0(Bi) 1−ψ0(Bi+1). With a next barrier Bi+2≥Bi+1−1 we obtain γi+2=γi 1−ψ0(Bi) 1−ψ0(Bi+1−1)(19) If we replace Biby ˆ Bi=Bi+1+1<Biwe obtain for the barriers ˆ Bi,Bi+1,Bi+2a parameter ˆ γi+2=γi 1−ψ0(ˆ Bi) 1−ψ0(Bi+1−1)≤γi+2, and the same value appears for the non decreasing threetuple Bi+1 , ˆ Bi , Bi+2 . The dividend value for these barriers is larger than before, since we pay dividends earlier. Repeating this argument step by step, we can replace an arbitrary admissible sequence of barriers by an admissible non decreasing one which leads to a higher dividend value. 2.5. The Lagrange Multiplier Approach For the Lagrange multiplier method we choose a constant L> 0 and maximize the company value minus the weighted corresponding ruin probability: V(s,L) = sup D {VD(s)−LψD(s)},s≥0. (20) Risks 2018,6, 1 7 of 14 We used a non-stationary approach and computed the quantities for time t V(s,L,t) = sup D {VD(s,t)−LψD(s,t)},s≥0, VD(s,t) = E"∞ ∑ t rud(u)|S(t) = s#, ψD(s,t) = P{S(u)−D(u)<0 for some u≥t|S(t) = s} via the recursion V(s,L,t−1) = max{V(s−1, L,t−1) + rt−1,pV(s+1, L,t) + qV(s−1, L,t)}(21) with V(− 1, L , t) = −L . The resulting optimal dividend strategy is a time dependent barrier strategy M(t) with which dividends are paid at t when the with dividend surplus is above M(t) . Using the barrier function M(t) one can compute the ruin probability for the optimal dividend strategy via the recursion ψ(s,t−1) = max(pψ(s+1, t) + qψ(s−1, t),ψ(M(t−1),t). The value V(s,L) = V(s,L, 0)can efficiently be approximated via a backward recursion starting at V(s , L , T) = −Lψ(s) and ψ(s , T) = ψ0(s) for some large T , a computation which turned out to be easy. Numerical experiments indicate that the approach produces dividend strategies which differ from the ones computed with the other methods: The resulting optimal dividend strategies for V(s , L) are state and time dependent, but not path dependent. The proposed policy improvement method without dynamic equation works also for more general models which are skip-free upwards and have independent stationary increments, e.g., classical Lundberg models with arbitrary claim size distribution or Brownian motions with drift. For these models the fist entrance probabilities and the discount factors for first entry waiting times are available. For Lundberg models the policy improvement method based on a modified Bellman equation can probably be applied, in particular with the explicit form of running allowed ruin probabilities. For the barrier method a continuous state space might cause problems: after discretization the resulting grid will be too large for an easy selection of optimal barriers. 3. Numerical Example All computations in this section are done with MatLab (modified Bellman, policy improvement, and Lagrange) or with Maple (Barrier method). We consider the case with parameters p= 0.7, r=1/1.03, s0=4 and a0=0.2. We have ψ0(s) = (q/p)s+1,s≥0, W(s) = Kzs 1+ (1−K)zs 2,s≥0, z1=1.07142857142857142, z2=0.4, K=1.5957446808510638298, A=103/33, C=10/11, γ0=0.804988026. We used the iteration method with 150 repetitions and a step size 1/100, 000 for αand obtained V(4, 0.2) = 12.8162. Risks 2018,6, 1 8 of 14 The unconstrained company value is V(4, 1) = 13.1004. This shows that a ruin constraint is rather cheap. The method using the modified Bellman equation described in Hipp (2003) is done—slightly modified—with the same step size 1 / 100, 000 for α, which results with 800 iterations and interpolation in a somewhat larger value: V(4, 0.2) = 12.817618. The modification, which speeds up a lot and allows for a small step size and a large number of iterations, is the specification of the maximizers β1 and β2 when s and α are given. We use again the running ruin probabilities for states without dividend payment a(x) = 1 −γ+γψ0(x) with γ derived from a(s) = αand set β1=a(s+1),β2=a(s−1). (22) The larger value obtained with the old method indicates that the iteration method was used with an insufficient number of repetitions. Furthermore, interpolation reduces the effect of a discretization. Since the iteration method uses a complete search over the possible surplus values (reducing the search to one over a small region leads to wrong results), larger numbers of iterations are not acceptable even for a patient user. Finally, for the iteration method we do not have a proof for convergence to the value function. Of course the best results can be obtained using the barrier method which is based on exact formulas. We computed V( 4, 0.2 ) from given barrier levels B0 , ..., B100 . Stopping dividend payment after visiting 100 not necessarily different barriers produces a numerical result below the true value, but the small size of this error can be seen in the (worst) case α= 1 :V( 4, 1 ) = 13.1003845, while with 100 steps we obtain 13.1003469. We used the barriers Bi=4, 0 ≤i≤6, Bi=5, 7 ≤i≤14, Bi=8, 15 ≤i≤19, Bi=12, 20 ≤i≤30, Bi=15, 31 ≤i≤40, Bi=18, 41 ≤i≤50, Bi=24, 51 ≤i≤100. All corresponding γiare smaller than 1. With these we obtained the value V(4, 0.2) = 12.9099. The barriers are found in an interactive procedure: we started with three regions [ 0, ..., 6 ] , [ 7, ..., 13 ] , [ 14, ..., 19 ] in which all barriers have the same value a , b , c , respectively. We took a= 4 which is the barrier in the unconstrained problem, b=6 and b=7. All other barriers are K. To avoid γi> 1 we increased step by step to c= 8. Then we reduced the size of barriers in the remaining groups. We are close to the optimal value when γK< 1 is very close to one. The difference between the dividend values 12.817618 and 12.9099 for V( 4, 0.2 ) is caused by the discretization of α ; even a step size of 1/100, 000 results in a rather big error due to the large number of calculations. For the Lagrange multiplier method we wanted to use the above numerical methods with a factor L for which the ruin probability equals 0.2. This L , however, does not exist, there is a Lagrange gap at this point. These gaps always exist in De Finetti models, see Hipp (2017). We computed with L= 2.94 the values α= 0.1998175 and V( 4, α) = V( 4, L) + αL= 12.84498, and for L= 2.93 the values are