Optimal proportional and excess-of-loss reinsurance for multiple classes of insurance business

In this paper we consider a reinsurance strategy which combines a proportional and an excess-of-loss reinsurance in a risk model with multiple dependent classes of insurance business. Under the assumption that the claim number of the classes has a multivariate Poisson distribution, the aim is to maximize the expected utility of terminal wealth. In a general setting, after deriving the corresponding Hamilton–Jacobi–Bellman equation, we prove a Verification Theorem and identify sufficient conditions for the optimality. Then, in a special case with exponential utility, an explicit solution is found by solving an intricate associated static constrained optimization problem.


Introduction
Over the past two decades, the studies devoted to the search of an optimal reinsurance strategy have registered considerable advancements and relevance in the actuarial literature.A great attention has been given to the classical proportional and excessof-loss reinsurance strategies, or both as a mixed contract (see Brachetta and Ceci 2019a, b;Brachetta and Schmidli 2020;Irgens and Paulsen 2004;Liu and Ma 2009 and references therein), which have been addressed under different optimization criteria.In many cases, the aim has been to minimize the probability of ruin (e.g.see Bai and Guo 2008;Browne 1995;Liang and Guo 2008;Promislow and Young 2005); in many others the goal has been to maximize the adjustment coefficient or the expected utility of terminal wealth (e.g.see Brachetta and Ceci 2019a, b;Centeno 2005;Liang and Guo 2011;Liang and Yuen 2016;Xu et al. 2017;Zhang et al. 2009).In all the cases the insurer's surplus process has been modelled either as a jump process or as a diffusion approximation.(We refer to Schmidli 2018 for a valuable introduction to risk models.) In this extremely vast literature, the recent relevant contributions Brachetta and Ceci (2019a, b) and Brachetta and Schmidli (2020) deserve close attention: in these papers the problem of optimal reinsurance and investment is addressed assuming that the insurer's surplus is affected by an exogenous stochastic factor, introduced to model the change in the number of policyholders and the risk modifications.The goal is to maximize either the wide-used expected exponential utility of terminal wealth or a member of the SAHARA class utility functions, which includes CRRA and CARA utility functions as limiting cases.Both the cases of proportional and excess-of-loss reinsurance are treated.On the other hand, restricting to the past two decades, the paper (Centeno 2005) represents one of the first contributions to consider the expected utility of terminal wealth maximization problem in the case of two dependent classes of insurance business.Afterwards, various authors considered two or more classes of dependent risks and, under different assumptions, studied the optimal dynamic reinsurance with the analogous objective to maximize the expected utility of terminal wealth.For instance, Liang and Yuen (2016) considers two classes of insurance business and assumes that the insurer's company is allowed to invest all its surplus in a risk-free asset with interest rate r employing the variance premium principle, while in Yuen et al. (2015) a model with more than two classes of insurance business is introduced and used.Both the aforementioned papers share the goal to find the optimal proportional reinsurance.The same goal appears in Bai and Guo (2008) where the insurer is allowed to invest in a risk-free asset and n risky assets, and the risk model is of diffusion type; further, under the constraint of no-shorting, the two problems of maximizing the expected exponential utility of terminal wealth and of minimizing the probability of ruin are considered.Two other contributions, especially relevant to our work, are Gosio et al. (2016) and Liang and Guo (2011) that, with the common aim to maximize the expected utility of terminal wealth, combine the proportional and the excess-of-loss reinsurance: Liang and Guo (2011) considers independent classes of insurance business, while (Gosio et al. 2016) works with two dependent classes.
Considering a risk model with multiple dependent classes of insurance, our paper characterizes the reinsurance strategy which combines a proportional and an excessof-loss reinsurance, and that maximizes the expected utility of wealth at a terminal finite time.We set up the model in continuous time and we assume that the claim number of the insurance classes follow a multivariate Poisson distribution.Moreover, we allow for n ≥ 2 dependent classes of insurance business, thus generalizing the risk model proposed in Gosio et al. (2016).Our analysis starts by deriving the rigorous control-theoretic setup where to formulate the optimal reinsurance problem.Then, invoking the dynamic programming principle, we derive the Hamilton-Jacobi-Bellman (HJB) equation for the general multiple dependent risk model problem and formulate a Verification Theorem.This provides a set of sufficient conditions for the optimality of a (candidate) admissible reinsurance strategy.Under the hypothesis that the claim size random variables are exponentially distributed and that the utility function is exponential, an educated guess on the form of the value function of the problem reduces the HJB equation to an intricate static optimization problem which we handle via the classical Karush-Kuhn-Tucker (KKT) conditions.By performing an accurate analysis of all the possible cases arising from the system of KKT conditions, we manage to explicitly determine an optimal reinsurance strategy which we prove to satisfy the conditions of the Verification Theorem.As a special case we analyse and solve the completely symmetric situation, in which all the parameters related to each class of insurance risk are the same.This study allows us to evaluate the behaviour of both the value function and the optimal strategies when the number of insurance classes increases.Further, we consider the case of two dependent classes of insurance business and relate our work to some existing literature, highlighting that our results agree with the findings of Gosio et al. (2016), when the proportional reinsurance is excluded, and of Centeno (2005), in which only the excess-of-loss reinsurance is considered.Finally, in the case n = 2, we provide a numerical comparative statics and discuss the effect of the parameters on the problem's solution.
The paper is organized as follows: Sect. 2 is devoted to the model presentation; Sect. 3 contains the HJB equation and the Verification Theorem.Section 4 focuses on the case of n classes of insurance business under the assumption that the utility is exponential and the claim size random variables corresponding to each class of insurance is exponentially distributed.In particular, in Sect. 4 we present an explicit solution of the HJB equation, show its optimality by means of the Verification Theorem and analyse the special symmetric case.Section 5 presents some numerical examples to show the computed optimal solution's behaviour under parameters' variations when dealing with 2 classes of insurance business.Section 6 contains some final remarks and the "Appendix" collects the proof of some technical results.

Model formulation
In this section, under the well-known assumptions of collective risk theory, we model the total claim amount charged to the insurance company by means of the classical compound Poisson process (see Shreve 2004, Sects. 11.2-11.3).
We consider the finite time horizon [0, T ], with 0 < T < +∞.Let ( , F, P) be a complete probability space on which is given a filtration F := {F s } 0≤s≤T satisfying the usual conditions.All stochastic processes introduced below are supposed to be adapted with respect to F.
For each class i of insurance risk, we let X i1 , X i2 , . . .be a sequence of identically distributed random variables with density f i , where f i is of class L 1 , f i (x) = 0 for x ≤ 0 and μ i := ∞ 0 x f i (x)dx < +∞.The random variables X i1 , X i2 , . . .represent the claim size corresponding to the ith class of insurance business, they are assumed to be independent of one another and also independent of the Poisson process N i (s), s ∈ [0, T ]; is the compound Poisson process with jumps of size X i j .
The insurance company has the possibility to reinsure each class i of insurance risk either by a pure quota-share or by an excess-of-loss strategy as follows.We let a i (s) ∈ [0, 1] and b i (s) ∈ [0, +∞] be the decision variables representing the retention limits of proportional and excess-of-loss reinsurance at time s.The quantity min{a i (s)Q i (s), b i (s)} will be retained by the insurer from the ith claim Q i (s).We highlight two limit cases: a i (s) = 0, for any value b i (s) ∈ [0, +∞], refers to full reinsurance, whereas a i (s) = 1 and b i (s) = +∞ mean that no reinsurance occurs.We assume that the coefficients (a, b) := (a 1 (s), . . ., a n (s), b 1 (s), . . ., b n (s)) are the control parameters belonging to the set of admissible strategies F-adapted, and such that a i (s) and assuming, from now on, the convention β i 0 = +∞, for each β i ∈ [0, +∞].The total claim amount charged to the insurance company at time s and referred to the i-type claim is (2.2) In order to define the surplus process, we assume that all the premiums are computed according to the expected value principle and are subject to a safety loading.For each i ∈ {1, . . ., n}, let η i > 0 be the ith safety loading coefficient and recall that μ i is the expectation of X i j .Thus, the insurance premium rate is and the reinsurance premium rate is where, by convention, +∞ • 0 = 0 and we assumed γ i > η i , i.e. reinsurance is more expensive than first insurance, a reasonable assumption in the actuarial practice, see also Liang and Guo (2011).Therefore, given the reinsurance strategy (a, b), the inflow of premium rate over time for the insurance company is )), so using (2.3) and (2.4): (2.5) It follows that the surplus process R (a,b) satisfies Starting from the state x at time t, t ∈ [0, T ], the insurer company aims to maximize the expected utility of terminal wealth using a utility function u : R → R, representing the preferences.Namely, we consider the performance criterion The value function associated to the stochastic control problem is We make the following assumptions on the utility function.
Assumption 2.1 The utility function u : R → R satisfies the following assumptions: Remark 2.2 In our model the wealth may become negative.This may be interpreted by implicitly assuming the presence of a riskless asset with r = 0 in the market which allows the company to borrow money.Therefore, our model considers a null interest rate, simplifying a situation occurred during recent years (for instance, until recently the European Central Bank (ECB) fixed a negative Deposit facility rate).If r = 0 the equation of the surplus process would satisfy and the subsequent formulas would change accordingly.For simplicity, we consider the case r = 0.Moreover, as pointed out in the literature, see Brachetta and Ceci (2019a, b); Liang and Young (2018), the absence of a risky asset represents no loss of generality as, under mild assumptions, the optimal reinsurance strategy turns out to depend only on the risk-free asset.

Preliminary estimates on the value function
In the following result we prove that J (t, x, (a, b)) is well-defined for each Proposition 2.3 The functional J (t, x, (a, b)) is well-defined for each (t, x) ∈ [0, T ] × R and (a, b) ∈ A t and satisfies where Consequently, (2.9) Further, using (2.5) the premium rate P (a,b) (s) can be bounded from below as follows and, from (2.1) and (2.2), the ith total claim amount has the following upper bound (2.11) Therefore, using (2.10) and (2.11) in (2.8) yields Using again Assumption 2.1 and recalling the hypothesis Further, by applying Shreve (Shreve 2004, formula (11.3.2)) to each compound Poisson process Q i (T ) and using the hypothesis ϕ i (K 1 ) < ∞, it follows (2.12) Recalling (2.7), the result on the value function follows from (2.9) and (2.12).

The HJB equation and the Verification Theorem
In this section we use the dynamic programming approach to write the Hamilton-Jacobi-Bellman (HJB) equation for the stochastic control problem defined in (2.7), we show that V (t, x) is its viscosity solution and formulate a Verification Theorem (see Theorem 3.2), and the latter provides a set of sufficient conditions for the optimality of a given (candidate) admissible reinsurance strategy.
Let C := [0, 1] n × [0, +∞] n ; recalling (2.6), the dynamic of the surplus process R (a,b)  Øksendal and Sulem (2005) we write the following HJB equation: with terminal boundary condition where, by standard arguments, the infinitesimal generator is (3.4) Then: (a) For any admissible control (a, b) Proof Let M > 0; we define the stopping time − − → T as M → +∞.Ito's formula, see also Øksendal and Sulem (2005),Theorem 1.23, applied in the time interval [t, τ M ] yields (3.7) We now use the HJB equation, and then take limits as M → +∞ to obtain, thanks to the hypothesis of uniform integrability (3.5) and the dominated convergence theorem, that: Then, employing the terminal boundary condition (3.3) and the definition of J (t, x, (a, b)), it follows that J (t, x, (a, b)) ≤ W (t, x), so that claim (a) is proved.
In order to prove claim (b), we evaluate formula (3.7) at the admissible control exploiting the hypothesis of uniform integrability (3.5), we obtain again by the dominated convergence theorem that 123 The proof is thus completed.

A case study with explicit solution
In this section we consider a special case of the above problem.Assumption 4.1 (i) The claim size random variables X i j , i ∈ {1, . . ., n}, j = 1, 2, . .., corresponding to the ith class of insurance business is exponentially distributed, with density f i (s Given the structure of the problem we look for a solution of the HJB equation (3.2) of the form (see for instance Gosio et al. 2016;Liang and Guo 2011;Liang and Yuen 2016): with terminal boundary condition W (T , x) = u(x), and so with Z (T ) = 0. We recall the notion of zth elementary symmetric polynomial in n variables x 1 , . . ., x n denoted by σ z (x 1 , . . ., x n ):

Remark 4.3
We list some observations on Theorem 4.2.We notice that all the retention levels a * i , i = 1, . . ., n are useless: in fact, the reinsurer pays the quantity , so that the combination of a proportional and an excess-of-loss reinsurance ends with a purely excess-of-loss reinsurance.

Remark 4.4
In the special case of 2 classes of insurance business, we observe that the results of Theorem 4.2, specialized for n = 2, agree with the findings of Gosio et al. (2016), see formula (30),( 31),(32)and Centeno (2005), see Result 2.2.Indeed, assume that then the system of equations are optimal strategies starting at any (t, x) ∈ [0, T ] × R. For a proof of this facts we refer to Proposition A.2.In Sect. 5 we treat this simple case from a numerical point of view, introducing explicit examples aimed to highlight the effect of the different parameters variation on the optimal strategies.
We end the section with the analysis of the symmetric case, in which we assume that all the involved parameters are equal: . . . , n}.This study allows us to evaluate the behaviour of both the value function and the optimal strategies when n goes to infinity.To this aim, in the involved variables we make explicit the dependence on n, so that we denote V = V n and b i = b n,i , for each i ∈ {1, . . ., n}.
Theorem 4.5 In the symmetric case, that is when be the unique solution of the equation , where Z n (t) is given by Proof By Theorem 4.2 and using symmetry, we obtain that sup 3) becomes equivalent to (4.17).The unique solution of (4.17) is b * n ∈ 0, 1 β ln(1 + γ ) such that b * n → 0 when n → +∞ (see "Appendix", Proposition A.3).The value function is V n (t, x) = − 1 β e −β(x−Z n (t)) where Z n (t), deriving from Theorem 4.2, is given by (4.18).Finally V n (t, x) → 0 when n → +∞.

Numerical examples
In this section we consider models with two classes of insurance business and assume that the claim sizes X 1 j and X 2 j are exponentially distributed with parameters k 1 and k 2 .Using the results of Theorem 4.2 and Remark 4.4, we present some numerical examples to highlight the effect of the different parameters variation on the optimal strategies.Yuen et al. (2015).In fact, when the claim sizes are exponentially distributed, Yuen et al. (2015) show that X 1 j (X 2 j respectively) has no effect on the optimal reinsurance strategy b * 2 (b * 1 respectively), while in our case X 1 j (X 2 j respectively) has almost no effect on the optimal reinsurance strategy b * 1 (b * 2 , respectively).In the second case k 1 = 2 and k 2 ∈ [1, 10]; condition (4.14) holds again.The results are shown in Fig. 1b which present a situation symmetrical to Fig. 1a, so all above observations hold.

Conclusions
In this paper we consider a risk model with multiple dependent classes of insurance business.Under the assumption that the insurance company combines the proportional and the excess-of-loss reinsurance, our aim is to maximize the expected utility of its wealth at the end of a finite time horizon.In a general setting we derive the associated HJB equation and formulate a Verification Theorem to detect the strategies optimality.
In a special case with exponential utility we specialize the HJB equation and transform it into an optimization problem and, using the classical KKT conditions and a detailed discussion of its feasible solutions, we explicitly determine a reinsurance strategy and prove its optimality by the stated Verification Theorem.When only two classes of insurance business are considered, our explicit solutions agree with the results of Gosio et al. (2016), in the case the proportional reinsurance is excluded, and of Centeno (2005), in which only the excess-of-loss reinsurance is considered.This observation suggests us that the excess-of-loss reinsurance could be predominant with respect to the proportional one.The paper ends with some numerical examples to effectively show the optimal results and the impact of the different parameters' variations on the optimal strategies.The results obtained in the paper suggest possible future developments in the field, such as the use of different claim size random variables distributions or of other principles for the premium computation, e.g. the variance principle.

Example 5 . 1
We let β = 0.05, γ 1 = γ 2 = γ = 0.3, θ 1 = 3, θ 2 = 4, θ 0 = 2 and we consider variations of the parameters k 1 and k 2 .In the first case k 1 ∈ [1, 10] and k 2 = 3; condition (4.14) holds.The results are shown in Fig. 1a.We note that the variations of k 1 affect the values of both b * 1 and b * 2 , but this effect is much more evident on b * 2 than on b * 1 .In particular, a greater value of k 1 yields greater values of b * 2 and almost constant values of b * 1 .Our interpretation of the phenomenon agrees with the observations made in Yuen et al. (2015, Example 5.1): since a smaller value of k 1 means to have a more risky claim size but also a greater expected value, the insurer's optimal strategy consists to retain less.Nevertheless, our results are symmetrical with respect to the results of 1 The values of b * 1 (solid line), b * 2 (dashed line) and y = 1 β ln(1 + γ ) (dotted line) of Example 5.1