Irreversible reinsurance: Minimization of capital injections in presence of a fixed cost

Federico, Salvatore; Ferrari, Giorgio; Torrente, Maria Laura Working Paper Irreversible reinsurance: Minimization of capital injections in presence of a fixed cost Center for Mathematical Economics Working Papers, No. 682 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Federico, Salvatore; Ferrari, Giorgio; Torrente, Maria Laura (2023) : Irreversible reinsurance: Minimization of capital injections in presence of a fixed cost, Center for Mathematical Economics Working Papers, No. 682, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29834173 This Version is available at: https://hdl.handle.net/10419/283395 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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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/ 682 September 2023 Irreversible Reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost Salvatore Federico, Giorgio Ferrari, and Maria Laura Torrente Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en IRREVERSIBLE REINSURANCE: MINIMIZATION OF CAPITAL INJECTIONS IN PRESENCE OF A FIXED COST SALVATORE FEDERICO, GIORGIO FERRARI, AND MARIA LAURA TORRENTE Abstract. We propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cram´er-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any specific functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model’s parameters. Keywords: reinsurance; fixed cost; capital injections; diffusive risk model; optimal stopping. MSC2010 subject classification: 97M30, 91B30, 60G40, 49L20. JEL classification: C61, G22, C41. 1. Introduction Reinsurance contracts usually run for long time periods (at least for longer than the typical maturity of financial contracts) and are exposed to high frictional costs. As a result, reinsurance negotiations are costly, lengthy, and can be thought of as irreversible, cf. [5]. As noticed by [6], it is indeed the case that “although reinsurance, in principle, is reversible, in practice reversing a reinsurance transaction exposes the insurer to relatively high transaction costs as well as additional charges to protect the reinsurer against adverse selection.” Furthermore, many external factors can interfere with changes in the reinsurance contracts. It is recent news that “2023’s renegotiation of reinsurance policies has been the most challenging in years as reinsurers respond to pressure from spiralling inflation and large losses from natural catastrophes, as well as the fallout from Russia’s invasion of Ukraine” (cf. Ian Smith, Insurance Correspondent of the “Financial Times”, January 3 2023∗). Optimal reinsurance decisions are typically formulated in terms of regular control problems, thus neglecting the aforementioned irreversibility feature. Given the vastity of the related literature, we refrain here form providing a list of references (that would necessarily result in being not exhaustive) and we simply refer to the discussion in Chapter 2 of [21] or in Chapter 11 of [1] for models and solutions. However, in the last decade the actuarial literature has started experiencing models of optimal irreversible reinsurance. In [2] it is investigated an optimal reinsurance problem under fixed cost, for an insurance company aiming at maximizing exponential expected utility at terminal time. The problem of optimal reinsurance negotiations with implementation delay and fixed cost is considered in [8], the optimal timing for the activation of an excess-of-loss reinsurance with fixed costs is studied in [18], while the presence of additional proportional transaction costs for a company Date: September 28, 2023. ∗https://www.ft.com/content/f5f9d450-c539-47a7-bc5c-44a8db57e74e 1 2 FEDERICO, FERRARI, AND TORRENTE minimizing the ruin probability is treated in [17]. Finally, a singular stochastic control model for the optimal sequential adjustment of reinsurance contracts has been recently formulated in [26]. Our paper contributes to that bunch of literature by proposing a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a (perpetual) reinsurance contract. Assuming that investors inject capital to avoid bankruptcy of the company, the insurance company aims at minimizing the sum of the expected value of the discounted fixed activation cost and of the cumulative discounted flow of capital injections. Capital injection models have been introduced by Dickson and Waters in [7]. Therein, starting from the observation that ruin occurs almost surely when the company pays dividends by following the optimal strategy of the de Finetti’s problem, a model has been suggested in which the shareholders are obliged to inject capital in order to avoid bankruptcy. We also refer to [13], [14], [19], [20], [22], [27] and reference therein for works related to the optimal dividends’ distribution in presence of capital injections. In the context of optimal reinsurance problems, the employment of the cumulative discounted flow of capital injections as a risk measure alternative to the ruin probability has firstly been proposed in [10], and later also used in [9] and [12]. As a matter of fact, the use of the ruin probability as measure of risk presents drawbacks: first of all, it is not a coherent risk measure, this potentially leading to decisions that are not economically sounded; second of all, it does not provide information about neither the time of ruin nor the severity of ruin. In order to define a unified framework for the evaluation of a variety of risk quantities, and in particular to give indications about the deficit at ruin and the time of ruin, Gerber and Shiu proposed in [15] the so-called expected discounted penalty function – also known as Gerber-Shiu function – of which the capital injection criterion represents an example (see, e.g., Section 2.4.3 in [16] or Chapter 4 in [25]). In this paper, we assume that the surplus process of an insurance company evolves according to the diffusive approximation of the Cram´er-Lundberg model. Claims arrive at a fixed constant rate and the distribution of their sizes is general. Furthermore, we do not specify any form of the retention level, which is simply assumed to be a continuous function, strictly increasing with respect to the reinsurance parameter. The company can choose the time τat which buying reinsurance and the desired retention level, which will then be kept from time τon. Those once-for-all actions involve a fixed cost, which is immediately withdrawn from the company’s surplus at time τ. The aim of the company is to take actions in such a way that the total discounted costs of capital injection and of the reinsurance contract are minimized. We provide an explicit solution to this problem which we show can be solved via a two-step procedure (see also [2] and [17], among others). We first solve for the optimal retention level, which is uniquely identified through the solution to a nonlinear algebraic equation. Then, given the optimal retention level, we look for the optimal time at which it is worth activating the irreversible reinsurance contract. This turns out to be given as the solution to a one-dimensional optimal stopping problem for a reflected drifted Brownian motion. We use the classical guess-and-verify approach by determining a smooth solution to the corresponding variational inequality with Neumann boundary condition and then by verifying the actual optimality of the candidate policy. It is worth noticing that, given the reflecting condition of the surplus process at zero, the verification argument requires quite some technical work in order to check that the variational inequality is indeed satisfied by the candidate value function (see the proofs of Proposition 3.10 and of Theorem 3.11 below). We find that a barrier-strategy is optimal and that reinsurance should be bought when the insurance company’s surplus process is sufficiently large, in particular larger than an endogenously determined trigger level (free boundary) that depends on the model’s parameters. Interestingly, we observe that the solution to our problem is consistent with that of [10], where, given the absence of a fixed transaction cost, reinsurance is bought immediately. Namely, the optimal retention level in our model is the same as that in [10], and, when the fixed cost K↓0, the free boundary converge to zero as well, implying that immediate reinsurance is in fact optimal. IRREVERSIBLE REINSURANCE WITH FIXED COST 3 We finally illustrate our results in the two relevant cases of proportional and excess-of-loss reinsurance, when the distribution of the claims’ sizes are Exponential or Pareto with parameters (ζ, α), for some α > 2 and ζ > 0. We solve numerically the equations that uniquely determine the optimal retention level and the free boundary and we study the dependency of those quantities with respect to relevant model’s parameters. We observe that both the optimal retention level and the free boundary exhibit a monotonic behavior with respect to the considered parameters and we provide explanations of these findings. Furthermore, we show (for fixed values of the model’s parameters) that, when the claim’s size is exponentially distributed, the value function one has in the case of proportional reinsurance is smaller than the one related to an excess-of-loss reinsurance, while no uniform comparison can be made in the case of Pareto-distributed claim’s size. The rest of the paper is organized as follows. Section 2presents the problem, which is then solved in Section 3. Section 4illustrates numerically the theoretical findings in the case of proportional and excess-of-loss reinsurance, while a final appendix collects most of the technical proofs of the paper. 2. Problem Formulation Let (Ω,F,F:= (Ft)t,P) be a complete probability space, rich enough to accommodate a onedimensional F-Brownian motion (Wt)t≥0and an independent square-integrable random variable Z, taking values in Z ⊂ R+, and with law νZunder P. Within this probabilistic setting, we consider the unaffected surplus process ( b Xt)t≥0of an insurance company, with initial value b X0=x > 0, evolving through the diffusion approximation of the classical Cram´er-Lundberg model (see, e.g., Appendix D in [21] or Section 8 in Chapter IV of [1]) b Xx t=x+ληµt +σ√λWt, t ≥0.(2.1) Here, µ:= RZzνZ(dz)>0 and σ2:= RZz2νZ(dz)>0 are, respectively, the mean and the standard deviation of the generic claim size Z,λis the arrival time parameter of the claims, ηis the safety loading. In order to avoid bankruptcy, investors are asked by the insurance company to inject capital whenever the surplus level attempts to become negative. Assuming that investors are impatient agents, it is clear that those injections of capital are made only when strictly necessary. The cumulative amount of capital injections (It)twill then reflect (`a la Skorokhod) the surplus process at x= 0, so that the resulting dynamics are Xx t=x+ληµt +σ√λWt+It=b Xx t+It, t ≥0,(2.2) with It= sup 0≤s≤th−b Xx si+, t ≥0. Within this model, we consider the possibility for the insurer of adopting a reinsurance strategy. More precisely, we consider a continuous function r:Z ×[0,1] →R+, which represents the retention level of the insurer — that is, the part of risk remaining in her charge — whose value depends on the chosen level b∈[0,1]. It is assumed r(z, ·) is strictly increasing and that b= 1 corresponds to no reinsurance and b= 0 corresponds to full reinsurance; that is, r(z, 1) = z, r(z, 0) = 0.(2.3) Typical examples are the case of proportional reinsurance, for which r(z, b) = bz,(2.4) and that of the excess-of-loss reinsurance, for which r(z, b) = z∧b 1−b.(2.5) 4 FEDERICO, FERRARI, AND TORRENTE Remark 2.1. It is worth noticing that choosing the reinsurance parameter b∈[0,1] allows us to cover the relevant reinsurance models using a unique parametrization within a unified setting. In the case of excess-of-loss reinsurance (cf. (2.5)above), this leads to a deviation from the classical formula of the reinsurance retention level which assumes b≥0(cf. [21]). In our model, we assume that the reinsurance policy is irreversible. This means that at a properly picked F-stopping time τthe insurer chooses the level bτ, which will then be kept from time τon. Formally, the reinsurance policy is thus a couple a:= (τ, bτ)∈ A := T ×MFτ, where T:= {τ: Ω →R+F-stopping time},MFτ:= {B: (Ω,Fτ)→[0,1] measurable}. To implement the reinsurance strategy a, the insurer faces a fixed transaction cost K≥0 at the time τ∈ T at which the reinsurance contract is signed. From time τon, according to the expected value principle, the insurer pays to the reinsurance company a perpetual premium rate with value θλ(µ−M1(bτ)),where M1(b) := ZZ r(z, b)νZ(dz),(2.6) for θ > η. On the other hand, the risk exposure of the insurance company is reduced, leading to the diffusion coefficient M2(bτ),where M2(b) := ZZ r(z, b)2νZ(dz).(2.7) In particular, from (2.3) it follows that M1(1) = µand M2(1) = σ2. Consequently, for t≥τ, the insurer only faces the outflows relative to her part of risk, represented by the retention level r. All in all, the dynamics with capital injection of the surplus process after time τare Xx t=Xx τ−−K+λ(θM1(bτ)−(θ−η)µ)(t−τ) + pλM2(bτ)(Wt−Wτ) + It, t ≥τ,(2.8) where (It)tis now such that It= sup τ≤s≤th−Xx τ−−K+λ(θM1(bτ)−(θ−η)µ)(t−τ) + pλM2(bτ)(Wt−Wτ)i+, t ≥τ. In the sequel, in order to stress the dependency of Ion the reinsurance policy and x, we shall write Ix,a, when needed. Following [9], [10], and [11], we assume that the insurance company employs the expected total amount of discounted capital injections as a measure of risk and thus aims at determining an admissible irreversible reinsurance policy a∗∈ A such that a∗∈argmin a∈A EZ∞ 0 e−ρtdIx,a t, where ρ > 0 is a subjective intertemporal discount rate. For future frequent use we also define U(x) := inf a∈A EZ∞ 0 e−ρtdIx,a t, x ≥0.(2.9) 3. Solution to the problem In this section, we determine the explicit solution to (2.9). To accomplish that, we shall first reformulate the problem in an handier way (cf. Section 3.1), then we shall obtain the optimal level (cf. Section 3.2) and, finally, the optimal time for reinsurance (cf. Section 3.3). IRREVERSIBLE REINSURANCE WITH FIXED COST 5 3.1. Reformulation of the problem. In order to obtain an handy representation of U, we compute the injection costs associated to a fixed retention parameter b∈[0,1] taken at t= 0; that is, given y≥0 and b∈[0,1], we calculate Gb(y) := EZ∞ 0 e−ρtdHy,b t,(3.1) where Hy,b t:= sup 0≤s≤th−Yy,b si+, t ≥0,(3.2) with Yy,b t:= y+λ(θM1(b)−(θ−η)µ)t+pλM2(b)f Wt, t ≥0, for another F-Brownian motion (f Wt)t≥0. Following [24], we know that, when y≥0, the function Gbis the solution to the differential problem        1 2λM2(b)G00 b(y) + λ(θM1(b)−(θ−η)µ)G0 b(y)−ρGb(y)=0, G0 b(0) = −1,lim y→+∞Gb(y)=0. (3.3) It then follows from (3.3) that Gb(y) = −1 γ−(b)eγ−(b)y,∀y≥0,(3.4) where γ−(b)<0 is the negative solution to the equation Φ(b, γ) = 0, with Φ(b, γ) := 1 2λM2(b)γ2+λ(θM1(b)−(θ−η)µ)γ−ρ, γ ∈R.(3.5) On the other hand, when y < 0, we have Gb(y) = −y+Gb(0) = −y−1 γ−(b).(3.6) With the help of the previously defined quantities (cf. (3.1) and (3.2)), an application of the strong Markov property allows us to rewrite Uas follows: U(x) = inf a∈A E"Zτ− 0 e−ρtdIx,a t+Z∞ τ− e−ρtdIx,a t# = inf a∈A E"Zτ− 0 e−ρtdIx,a t+EZ∞ τ− e−ρtdIx,a tFτ−# = inf a∈A EZ∞ 0 e−ρtdHx,1 t−Z∞ τ− e−ρtdHx,1 t+EZ∞ τ− e−ρtdIx,a tFτ− =G1(x) + inf a∈A Ee−ρτ (Gbτ(Xx τ−−K)−G1(Xx τ−)) =: G1(x) + inf a∈A J(x, a). Letting V(x) := inf a∈A J(x, a),(3.7) where J(x, a) := Ee−ρτ fbτ(Xx τ−), fb(y) := Gb(y−K)−G1(y),(3.8) 6 FEDERICO, FERRARI, AND TORRENTE we have U(x) = G1(x) + V(x).(3.9) We now continue our analysis by determining the optimal a∗= (τ∗, b∗) s.t. V(x) = J(x, a∗). Clearly, such an a∗will also be optimal for (2.9). 3.2. Optimal reinsurance. Recall that the function γ−(b) has been defined as the negative solution to the equation Φ(b, ·) = 0, with Φ as in (3.5). Its explicit expression is γ−(b) = −(θM1(b)−(θ−η)µ) + q(θM1(b)−(θ−η)µ)2+ 2ρM2(b) λ M2(b), b ∈[0,1].(3.10) The next assumption will be standing throughout the rest of the paper. Assumption 3.1. Recall (2.6)and (2.7). We have M1, M2∈C1([0,1]; R). Under Assumption 3.1, we clearly have γ−∈C1([0,1]; R) and (γ−)0(b) = −γ−(b)φ(b) D(b),∀b∈[0,1],(3.11) where φ(b) := 1 2γ−(b)M0 2(b) + θM0 1(b)(3.12) D(b) := M2(b)γ−(b) + θM1(b)−(θ−η)µ.(3.13) A relevant fact is that the minimum points of γ−in [0,1] coincide with the minimum points of Gb(x) for any x≥0, as it is shown in the following result. Proposition 3.2. For any x≥0, we have argminb∈[0,1]γ−(b) = argminb∈[0,1]Gb(x). Proof. Explicit calculations give ∂Gb ∂b (x) = (γ−)0(b) (γ−(b))2eγ−(b)x(x−γ−(b)),∀x≥0. Since x−γ−(b)>0 for each b∈[0,1] and x≥0, then the functions b7→ Gb(x) and b7→ γ−(b) have the same monotonicity and the claim follows.  Remark 3.3. It is worth noticing that the function γ−(b),b∈[0,1], as defined in (3.10)coincides with the opposite of the function β(b),b∈[0,eb], defined in [10](when eb= 1). In particular, up to a parametrization, any optimizer b∗of γ−on [0,1] does also optimize βin [10], and viceversa. We shall see in the next Theorem that, as in [10], optimizers of γ−actually give the optimal level to be adopted. The optimal timing for reinsurance is then determined given the optimal level b∗(see Section 3.3 below). Taking into account Proposition 3.2, for any x≥0, we denote B∗:= argminb∈[0,1] γ−(b) = argminb∈[0,1] Gb(x). The next result shows how to reduce the solution to (3.7) to a pure optimal timing problem. Theorem 3.4. Recall (3.7)and (3.8). Let b∗∈ B∗and let τ∗(b∗)∈ T such that τ∗(b∗)∈argminτ∈T Ee−ρτ fb∗(Xx τ−)=argminτ∈T J(x, (τ, b∗)), with the convention e−ρτ fb∗(Xx τ)=0on {τ=∞}. Then, the couple a∗:= (τ∗(b∗), b∗)∈ A is an optimal reinsurance strategy (with b∗thought of as a constant random variable). IRREVERSIBLE REINSURANCE WITH FIXED COST 7 Proof. Since Gb∗(x) = minb∈[0,1] Gb(x) (see Proposition 3.2) then U(x) = G1(x) + V(x) ≥G1(x) + inf τ∈T Ee−ρτ (Gb∗(Xx τ−−K)−G1(Xx τ−)) =G1(x) + inf τ∈T J(x, (τ, b∗)) =G1(x) + J(x, (τ∗(b∗), b∗)). On the other hand U(x) = G1(x) + V(x) ≤G1(x) + Ehe−ρτ∗(b∗)Gb∗(Xx τ∗(b∗)−−K)−G1(Xx τ∗(b∗)−)i =G1(x) + J(x, (τ∗(b∗), b∗)). Consequently U(x) = G1(x) + J(x, (τ∗(b∗), b∗)) and (τ∗(b∗), b∗) is optimal.  Theorem 3.4 provides sufficient conditions needed to identify an optimal reinsurance parameter b∗. If b∗∈ B∗, then the level corresponding to a random variable with constant value belonging to the set B∗is the second component of an optimal reinsurance strategy. It is then worthful to give at least sufficient conditions under which such optimal level is nontrivial, i.e. to rule out the case 1∈ B∗. This is provided in Proposition 3.6 below, under the following assumption. Assumption 3.5. Recall (3.12). There exists δ > 0such that (3.14) φ(b)<0,∀b∈(1 −δ, 1). Proposition 3.6. If Assumption 3.5 holds true, then 1/∈ B∗. Proof. See Appendix.  Remark 3.7. Notice that, instead of (3.14), a stronger sufficient condition that would rule out the case 1∈ B∗is φ(1) <0.(3.15) We will see that the proportional reinsurance case treated in Section 4.1 satisfies (3.15), whereas the excess-of-loss reinsurance presented in Section 4.2 satisfies the weaker Assumption 3.5. 3.3. Optimal reinsurance timing. Given Theorem 3.4, in order to solve the optimization problem (3.7) we need to solve, for a fixed b∗∈ B∗, the optimal stopping problem (cf. (3.8)) Fb∗(x) := inf τ∈T J(x, (τ, b∗)) = inf τ∈T Ee−ρτ fb∗(Xx τ−), x ∈[0,∞).(3.16) Before addressing problem (3.16) we collect properties of the obstacle function fb∗. Proposition 3.8. The following hold true: (a) fb∗is strictly decreasing in [0,ˆxb∗]and strictly increasing in [ˆxb∗,∞), where ˆxb∗:= γ−(b∗) γ−(b∗)−γ−(1)K∈(K, ∞).(3.17) (b) limx→∞ fb∗(x)=0. (c) fb∗is bounded. Precisely, we have the following two cases: (i) If −Kγ−(1)γ−(b∗)≤γ−(b∗)−γ−(1) <0, then fb∗(0) ≥0(†), and −γ−(1) −γ−(b∗) γ−(1)γ−(b∗)e−γ−(1)γ−(b∗) γ−(1)−γ−(b∗)≤fb∗(x)≤K+γ−(b∗)−γ−(1) γ−(1)γ−(b∗),∀x∈[0,∞). †With fb∗(0) = 0 if and only if −K=γ−(b∗)−γ−(1) γ−(1)γ−(b∗). 14 FEDERICO, FERRARI, AND TORRENTE By some computations, we find M1(b) =        b 1−bif b < ζ 1 + ζ ζ α−1 α−ζ1−b bα−1!if b≥ζ 1 + ζ M2(b) =          b 1−b2 if b < ζ 1 + ζ ζ2 α−2 α−2ζ1−b bα−2!if b≥ζ 1 + ζ. Equation (3.5) now reads Φ(b, γ) :=    1 2λb 1−b2γ2+λhθb 1−b−ζ+ηζiγ−ρif b < ζ 1+ζ 1 2λζ2 α−2α−2ζ1−b bα−2γ2+λhθζ α−11−ζ1−b bα−1+ηζiγ−ρif b≥ζ 1+ζ, with γ∈R. We illustrate numerically the sensitivity of the optimal level b∗and of the optimal reinsurance boundary x∗ b∗with respect some relevant model’s parameters. We choose benchmark values of the parameters as it follows: θ= 0.5, η= 0.3, λ= 0.05, ρ= 0.04, ζ= 10, K= 10. With such a choice, by (4.8) and Proposition 3.10, we obtain b∗= 0.5195 and x∗ b∗= 11.7572. Figure 6shows how b∗depends on the parameters ρand ζ, while Figure 7plots x∗ b∗as a function of the parameters ρ,ζand K. We observe behaviors of b∗and x∗ b∗which are similar to those observed with respect to ρ,µand Kin the case of an Exponential distribution of the claim size, and which can be then explained through the same rationale. As a matter of fact, in the Pareto distribution, the average and the variance of the sizes of the claims are increasing functions of the only parameter ζ, just as they are functions of µin the case of an Exponential distribution. b∗(ρ)b∗(ζ) Figure 6. Dependency of b∗with respect to ρand ζ. 4.3. Comparison of the value function in the case of proportional and excess-of-loss reinsurance. In Figure 8we collect drawings of the value function in the case of proportional reinsurance (solid line) and excess-of-loss reinsurance (dashed line), when the claim size Z∼Exp(1/µ) (left panel) and Z∼Pareto(ζ, α) (right panel). We observe that, when the claim’s size is exponentially distributed, the value function one obtains in the case of proportional reinsurance is smaller than the one related to an excess-of-loss reinsurance, while no uniform comparison can be made in the case of Pareto-distributed claim’s size. We thus conclude that (at least for the benchmark values of the parameters that we have used) excess-of-loss reinsurance is not necessarily favourable to proportional reinsurance, a finding that is in contrast to that of Figure 3 in [10] (see also the subsequent discussion at page 13 therein). IRREVERSIBLE REINSURANCE WITH FIXED COST 15 x∗ b∗(ρ)x∗ b∗(ζ)x∗ b∗(K) Figure 7. Dependency of x∗ b∗with respect to ρ,ζand K. Z∼Exp(1/µ)Z∼Pareto(ζ, α) Figure 8. Value function (zoomed image in the bottom panels) in the case of proportional reinsurance (solid line) and excess-of-loss reinsurance (dashed line), when the claim sizes Z∼Exp(1/µ) (left panel) and Z∼Pareto(ζ, α) (right panel). Appendix Proof of Proposition 3.6 Proof. The symmetric axis of the parabola Φ(b, γ) = 0 is bγ(b) = −θM1(b)−(θ−η)µ M2(b); as a consequence, since γ−(b) is the negative solution to the equation Φ(b, γ) = 0, D(b) = M2(b)γ−(b) + θM1(b)−(θ−η)µ < 0,∀b∈[0,1].(4.9) 16 FEDERICO, FERRARI, AND TORRENTE Then, by (3.11), (3.14) and (4.9), we obtain (γ−)0(b)>0 for each b∈(1 −δ, 1), which implies 1/∈ B∗. Proof of Proposition 3.8 Proof. From (3.4), (3.6) and (3.8), the explicit expression of fb∗is: fb∗(x) =      −(x−K)−1 γ−(b∗)+1 γ−(1)eγ−(1)x0≤x≤K, −1 γ−(b∗)eγ−(b∗)(x−K)+1 γ−(1)eγ−(1)xx > K, (4.10) from which fb∗(0) = K+γ−(b∗)−γ−(1) γ−(1)γ−(b∗)and limx→∞ fb∗(x) = 0. We compute f0 b∗(x) = (−1 + eγ−(1)x0< x < K, −eγ−(b∗)(x−K)+eγ−(1)xx > K. Then, fb∗is strictly decreasing in [0,ˆxb∗] and strictly increasing in [ˆxb∗,∞), where ˆxb∗is defined in (3.17). The point ˆxb∗is the unique global minimum point of fb∗, whose minimum value is: fb∗(ˆxb∗) = −γ−(1) −γ−(b∗) γ−(1)γ−(b∗)e−γ−(1)γ−(b∗) γ−(1)−γ−(b∗). If −Kγ−(1)γ−(b∗)≤γ−(b∗)−γ−(1) <0, then fb∗(0) ≥0 and item (i) follows. Otherwise, if γ−(b∗)−γ−(1) <−Kγ−(1)γ−(b∗), then fb∗(0) <0 and item (ii) follows.  Proof of Theorem 3.9 Proof. Let x≥0, T > 0, τn:= inf{t≥0 : Xx t≥n},n≥0, and τ∈ T. Applying a change of variable formula for semimartingales (see e.g. [4], Theorem 2.1) to {e−ρtw(Xx t), t ∈[0, τn∧τ∧T]} and then taking expectations we find: Ehe−ρ(τn∧τ∧T)w(Xx τn∧τ∧T)i=w(x)+EZτn∧τ∧T 0 e−ρs ((L−ρ)w) (Xx s)ds +Zτn∧τ∧T 0 e−ρsw(Xx s)dIs, where the Brownian-local martingale term has vanished in expectation since w∈C1([0,∞); R) (by Sobolev embedding) and because of the definition of τn. With regards to the fact that wsolves (3.19) and t7→ Itincreases on {t≥0 : Xx t= 0}, we have (after rearranging terms) w(x)≤Ehe−ρ(τn∧τ∧T)fb∗(Xx τn∧τ∧T)i. As fb∗is bounded (cf. Proposition 3.8), by sending n↑ ∞ and T↑ ∞, by the dominated convergence theorem we obtain w(x)≤Ee−ρτ fb∗(Xx τ). Given the arbitrariness of τ∈ T and x≥0, we have w≤Fb∗on [0,∞). Repeating now the same arguments above, but with τreplaced by τ∗(b∗), we find (by definition of τ∗(b∗)) w(x) = Ehe−ρτ∗(b∗)fb∗Xx τ∗(b∗)i. Hence, w(x)≥inf τ∈T Ee−ρτ fb∗(Xx τ)=Fb∗(x). Given the arbitrariness of x≥0, we have w≥Fb∗, which, together with the previously proved w≤Fb∗, implies that w=Fb∗and that τ∗(b∗) is optimal.  IRREVERSIBLE REINSURANCE WITH FIXED COST 17 Proof of Proposition 3.10 Proof. Step 1. We here prove existence and uniqueness of x∗ b∗. Using (3.21), we rewrite problem (3.20) as follows    C1γ−(1) + C2γ+(1) = 0 C1eγ−(1)x+C2eγ+(1)x=fb∗(x) C1γ−(1)eγ−(1)x+C2γ+(1)eγ+(1)x=f0 b∗(x), (4.11) where the explicit expression of fb∗is given in (4.10). The first equation yields: C2=−C1 γ−(1) γ+(1),(4.12) whereas the second and third equation depend on the expression of fb∗. We consider two distinct cases. (i) If 0 ≤x≤K, then the second and the third equations of (4.11) become    C1γ−(1) eγ−(1)x−γ−(1) γ+(1) eγ+(1)x=−γ−(1)(x−K)−γ−(1) γ−(b∗)+eγ−(1)x C1γ−(1) eγ−(1)x−eγ+(1)x=−1 + eγ−(1)x. (4.13) Since γ−(1) 6=γ+(1), the previous system yields: −γ−(1)(x−K)−γ−(1) γ−(b∗)+eγ−(1)x eγ−(1)x−γ−(1) γ+(1) eγ+(1)x=−1 + eγ−(1)x eγ−(1)x−eγ+(1)x, which can be rewritten as F1(x) = F2(x),(4.14) where F1(x) := x−K+1 γ−(b∗)e−γ−(1)x−e−γ+(1)x F2(x) := 1 γ−(1) 1−e−γ+(1)x−1 γ+(1) 1−e−γ−(1)x. We have F1(0) = F2(0) = 0 and F0 1(x) = γ+(1) x−K+1 γ−(b∗)−1 γ+(1)e−γ+(1)x−γ−(1) x−K+1 γ−(b∗)−1 γ−(1)e−γ−(1)x F0 2(x) = γ+(1) γ−(1)e−γ+(1)x−γ−(1) γ+(1)e−γ−(1)x. From (3.22) we note that 1 γ−(1) +1 γ+(1) =γ−(1) + γ+(1) γ−(1)γ+(1) =ληµ ρ>0. Consequently 1 γ−(b∗)<1 γ+(1) +1 γ−(1) and, for each x∈[0, K], it holds: F0 1(x) = γ+(1) x−K+1 γ−(b∗)−1 γ+(1)e−γ+(1)x−γ−(1) x−K+1 γ−(b∗)−1 γ−(1)e−γ−(1)x ≤γ+(1) 1 γ−(b∗)−1 γ+(1)e−γ+(1)x−γ−(1) 1 γ−(b∗)−1 γ−(1)e−γ−(1)x <γ+(1) γ−(1)e−γ+(1)x−γ−(1) γ+(1)e−γ−(1)x=F0 2(x). 18 FEDERICO, FERRARI, AND TORRENTE Hence, the unique solution of (4.14) in [0, K] is x= 0, and (4.13) and (4.12) yield C1=C2= 0. (ii) If x>K, then the second and the third equation of (4.11) become    C1γ−(1) eγ−(1)x−γ−(1) γ+(1) eγ+(1)x=−γ−(1) γ−(b)eγ−(b∗)(x−K)+eγ−(1)x C1γ−(1) eγ−(1)x−eγ+(1)x=−eγ−(b∗)(x−K)+eγ−(1)x. (4.15) Since γ−(1) 6=γ+(1), the previous system yields: −γ−(1) γ−(b∗)eγ−(b∗)(x−K)+eγ−(1)x eγ−(1)x−γ−(1) γ+(1) eγ+(1)x=−eγ−(b∗)(x−K)+eγ−(1)x eγ−(1)x−eγ+(1)x, which can be rewritten as (3.23); that is, Θ(x) = D(K),(4.16) where Θ(x) := γ+(1)(γ−(b∗)−γ−(1))e(γ−(b∗)−γ+(1))x−γ−(1)(γ−(b∗)−γ+(1))e(γ−(b∗)−γ−(1))x and D(K) := γ−(b∗)(γ+(1) −γ−(1))eγ−(b∗)K<0. We have Θ(0) = γ−(b∗)(γ+(1) −γ−(1)) ≤γ−(b∗)(γ+(1) −γ−(1))eγ−(b∗)K=D(K). and Θ0(x)=(γ−(b∗)−γ−(1))(γ−(b∗)−γ+(1)) γ+(1)e−γ+(1)x−γ−(1)e−γ−(1)xeγ−(b∗)x>0. Since Θ is strictly increasing and limx→+∞Θ(x) = 0, (4.16) (and (3.23)) admits in [0,+∞) the unique positive solution x∗ b∗=x∗ b∗(K)=Θ−1(D(K)) >0. Further, since D0(K)=(γ−(b∗))2(γ+(1)− γ−(1))eγ−(b∗)K>0, then x∗ b∗is strictly increasing in K. Step 2. We show that x∗ b∗≤ˆxb∗. Because Θ(ˆxb∗) = γ+(1)(γ−(b∗)−γ−(1))e γ−(b∗)−γ+(1) γ−(b∗)−γ−(1) γ−(b∗)K−γ−(1)(γ−(b∗)−γ+(1))eγ−(b∗)K =γ+(1)(γ−(b∗)−γ−(1))e γ−(1)−γ+(1) γ−(b∗)−γ−(1) γ−(b∗)K−γ−(1)(γ−(b∗)−γ+(1))eγ−(b∗)K ≥γ+(1)(γ−(b∗)−γ−(1)) −γ−(1)(γ−(b∗)−γ+(1))eγ−(b∗)K =γ−(b∗)(γ+(1) −γ−(1))eγ−(b∗)K=D(K), then the monotonicity of Θ yields x∗ b∗≤ˆxb∗=γ−(b∗) γ−(b∗)−γ−(1)K. Step 3. We now aim at proving that x∗ b∗≥K. Notice that Θ(K) = eγ−(b∗)Kγ+(1)(γ−(b∗)−γ−(1))e−γ+(1)K−γ−(1)(γ−(b∗)−γ+(1))e−γ−(1)K. Let S(K) := γ+(1)(γ−(b∗)−γ−(1))e−γ+(1)K−γ−(1)(γ−(b∗)−γ+(1))e−γ−(1)K, K ≥0. IRREVERSIBLE REINSURANCE WITH FIXED COST 19 It holds S(0) = γ−(b∗)(γ+(1) −γ−(1)) <0 and S0(K) = −γ+(1)2(γ−(b∗)−γ−(1))e−γ+(1)K+γ−(1)2(γ−(b∗)−γ+(1))e−γ−(1)K =−γ+(1)2(γ−(b∗)−γ−(1))e−γ−(1)Ke(γ−(1)−γ+(1))K−γ−(1)2(γ−(b∗)−γ+(1)) γ+(1)2(γ−(b∗)−γ−(1)). Since γ−(1) + γ+(1) <0, then γ−(1)2(γ−(b∗)−γ+(1)) γ+(1)2(γ−(b∗)−γ−(1)) >1. Therefore, S0(K)<0 for each K≥0, so that Θ(K) = eγ−(b∗)KS(K)≤eγ−(b∗)KS(0) = γ−(b∗)(γ+(1) −γ−(1))eγ−(b∗)K=D(K) and K≤x∗ b∗by monotonicity of Θ. Step 4. Finally, using (4.15) and (4.12) we get C1=1 γ−(1)Band C2=1 γ+(1)B, where B=eγ−(b∗)(x∗ b∗−K)−eγ−(1)x∗ b∗ eγ+(1)x∗ b∗−eγ−(1)x∗ b∗ .(4.17) We rewrite (3.23) (or equivalently Θ(x) = D(K)) as follows: eγ−(b∗)(x∗ b∗−K)H(x∗ b∗) = γ−(b∗)γ+(1) −γ−(1)e(γ−(1)+γ+(1))x∗ b∗,(4.18) where H(x∗ b∗) is given in (3.27). Using (4.18) in (4.17) we get B=eγ−(b∗)(x∗ b∗−K)−eγ−(1)x∗ b∗ eγ+(1)x∗ b∗−eγ−(1)x∗ b∗ = γ−(b∗) (γ+(1) −γ−(1)) eγ+(1)x∗ b∗ H(x∗ b∗)−1!eγ−(1)x∗ b∗ eγ+(1)x∗ b∗−eγ−(1)x∗ b∗ =γ+(1)(γ−(b)−γ−(1))(eγ+(1)x∗ b∗−eγ−(1)x∗ b∗) H(x∗ b∗) eγ−(1)x∗ b∗ eγ+(1)x∗ b∗−eγ−(1)x∗ b∗ =γ+(1)(γ−(b)−γ−(1))eγ−(1)x∗ b∗ H(x∗ b∗)=: B(x∗ b∗) as given by (3.26). Since γ−(b)< γ−(1) and H < 0, then B(x∗ b∗)>0; further, H < γ+(1)(γ−(b)− γ−(1))eγ−(1)x∗ b∗and therefore B(x∗ b∗)<1.  Proof of Theorem 3.11 Proof. In order to prove that w≡V, we need to show that: w(x)≤fb∗(x),∀x > 0 wis s.t. 0 ≤(L−ρ)w(x),∀x > 0, (4.19) which is implied by the following two conditions: (i) w(x)≤fb∗(x),∀0< x ≤x∗ b∗ (ii) 0 ≤(L−ρ)w(x),∀x>x∗ b∗. We start proving (i). By Proposition 3.10 w(x) = B(x∗ b∗)1 γ−(1)eγ−(1)x−1 γ+(1)eγ+(1)x, 20 FEDERICO, FERRARI, AND TORRENTE where B(x∗ b∗) is given by (3.26). We compute w(0) = B(x∗ b∗)1 γ−(1) −1 γ+(1)and fb∗(0) = K− 1 γ−(b∗)+1 γ−(1), so that w(0) ≤fb∗(0) if and only if K≥B(x∗ b∗)1 γ−(1) −1 γ+(1)+1 γ−(b∗)−1 γ−(1) =(γ−(b∗)−γ+(1))(γ−(b∗)−γ−(1)) γ−(b∗) eγ+(1)x∗ b∗−eγ−(1)x∗ b∗ H(x∗ b∗)=Q(x∗ b∗), where, for x > 0, Q(x) := (γ−(b∗)−γ+(1))(γ−(b∗)−γ−(1)) γ−(b∗) eγ+(1)x−eγ−(1) H(x), and (cf. (3.27)) H(x) := γ+(1) γ−(b∗)−γ−(1)eγ−(1)x−γ−(1) γ−(b∗)−γ+(1)eγ+(1)x<0. By some computations we have Q0(x) = (γ−(b∗)−γ+(1))(γ−(b∗)−γ−(1))(γ+(1) −γ−(1))2e(γ+(1)+γ−(1))x (H(x))2>0. Recalling that x∗ b∗=x∗ b∗(K) = Θ−1(D(K)) (see Step 1 in the proof of Proposition (3.10)), it holds Q(x∗ b∗(0)) = Q(0) = 0. Further, ∂Q(x∗ b∗(K)) ∂K =Q0(x∗ b∗(K)) ·(x∗ b∗)0(K). Since x∗ b∗is strictly increasing in Kit follows that Q(x∗ b∗(K)) is strictly increasing in Kand ∂Q(x∗ b∗(0)) ∂K =Q0(0) ·(x∗ b∗)0(0) =(γ−(b∗)−γ+(1))(γ−(b∗)−γ−(1)) γ−(b∗)2·γ−(b∗)2 (γ−(b∗)−γ−(1))(γ−(b∗)−γ+(1)) = 1. In order to study the concavity of K7→ Q(x∗ b∗(K)), we compute ∂2Q(x∗ b∗(K)) ∂K2=Q00(x∗ b∗(K)) ·(x∗ b∗)0(K)2+Q0(x∗ b∗(K)) ·(x∗ b∗)00(K).(4.20) We find Q00(x)=(γ−(b∗)−γ+(1))(γ−(b∗)−γ−(1))(γ+(1) −γ−(1))3e(γ+(1)+γ−(1))x (H(x))3A(x) where A(x) := γ+(1)(γ−(b∗)−γ−(1))eγ−(1)x+γ−(1)(γ−(b∗)−γ+(1))eγ+(1)x. Since γ+(1)+γ−(1) = −2ηµ σ2<0, then A(x∗ b∗(0)) = A(0) = γ−(b∗)(γ+(1)+γ−(1))−2γ−(1)γ+(1) >0; further, A0(x) = γ+(1)γ−(1) (γ−(b∗)−γ−(1))eγ−(1)x+ (γ−(b∗)−γ+(1))eγ+(1)x>0, therefore A(x)≥0 for each x≥0, and consequently, since H(x)<0 for each x≥0, it holds Q00(x)≤0. In particular, Q00(x∗ b∗(K)) ≤0,for each K≥0.(4.21) IRREVERSIBLE REINSURANCE WITH FIXED COST 21 Using that x∗ b∗=x∗ b∗(K) is the unique solution to the equation Ψ(x, K) = Θ(x)−D(K) = 0 (see again step 1 of the proof of Proposition 3.10), we have (see [3]) (x∗ b∗)00(K) = ∂Ψ ∂K ∂2Ψ ∂K∂x −∂Ψ ∂x ∂2Ψ ∂K2/∂Ψ ∂x 2 , so that the sign of (x∗ b∗)00(K) coincides with the sign of the following quantity: ∂Ψ ∂K ∂2Ψ ∂K∂x −∂Ψ ∂x ∂2Ψ ∂K2= Θ0(x)·D00(K)<0.(4.22) Using (4.20), (4.21) and (4.22), we get ∂2Q(x∗ b∗(K) ∂K2<0; consequently, K≥Q(x∗ b∗(K)), and w(0) ≤ fb∗(0). Further, wis negative and, because w0(x) = Beγ−(1)x−eγ+(1)x<0, wis strictly decreasing. We recall by Proposition 3.8 that fb∗is strictly decreasing in [0,ˆxb∗]. If x∈[0, K] then w0(x) = Beγ−(1)x−eγ+(1)x< eγ−(1)x−eγ+(1)x< eγ−(1)x−1 = f0 b∗(x). Consequently, it holds w(x)≤fb∗(x) for each x∈[0, K]. On the other hand, since x∗ b∗> K > γ−(b)K γ−(b)−γ+(1), then for each x∈[K, x∗ b∗] w0(x) = Beγ−(1)x−eγ+(1)x< eγ−(1)x−eγ+(1)x< eγ−(1)x−eγ−(b)(x−K)=f0 b∗(x), which, coupled with w(K)< fb∗(K), yields w(x)≤fb∗(x) for each x∈[K, x∗ b∗]. We now prove (ii). For each x>x∗ b∗> K we have (L−ρ)fb∗(x)=(L−ρ)−1 γ−(b∗)eγ−(b∗)(x−K)+1 γ−(1)eγ−(1)x =−1 γ−(b∗)eγ−(b∗)(x−K)1 2λσ2(γ−(b∗))2+λ(θµ −(θ−η)µ)γ−(b∗)−ρ =−1 γ−(b∗)eγ−(b∗)(x−K)Φ(1, γ−(b∗)).(4.23) Since γ−(1) is the unique negative solution to Φ(1, γ) = 0 and because γ−(b∗)< γ−(1), then Φ(1, γ−(b)) >0 and, by (4.23), (L−ρ)fb∗(x)>0 for each x>x∗ b∗. Proof of Proposition 4.1 Proof. Since condition (4.3) is equivalent to (3.15), by Proposition 3.10 and Remark 3.7 it is enough to prove that 1 6∈ B∗implies (4.3). From (3.10) and (4.4) we explicitly compute the expressions of γ−(1) = −µ σ2 η+sη2+2ρσ2 λµ2!<0(4.24) and (γ−)0(1) = −γ−(1)σ2γ−(1) + µθ σ2γ−(1) + µη.(4.25) From the hypothesis 1 6∈ B∗it follows that (γ−)0(1) <0. 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Donato 5, 40126, Bologna, Italy E-mail address:[email protected] G. Ferrari: Center for Mathematical Economics (IMW), Bielefeld University, Universit¨ atsstrasse 25, 33615, Bielefeld, Germany E-mail address:[email protected] M.L. Torrente: Dipartimento di Economia, Universit` a di Genova, Via Vivaldi 5, 16126 Genova, Italy E-mail address:[email protected]