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## Enhancing insurer value using reinsurance and Value-at-Risk criterion, The Geneva Risk and Insurance Review, (2012)

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### BibTeX

@MISC{Tan12enhancinginsurer,

author = {Ken Seng Tan and Chengguo Weng},

title = {Enhancing insurer value using reinsurance and Value-at-Risk criterion, The Geneva Risk and Insurance Review,},

year = {2012}

}

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### Abstract

The quest for optimal reinsurance design has remained an interesting problem among insurers, reinsurers, and academicians. An appropriate use of reinsurance could reduce the underwriting risk of an insurer and thereby enhance its value. This paper complements the existing research on optimal reinsurance by proposing another model for the determination of the optimal reinsurance design. The problem is formulated as a constrained optimization problem with the objective of minimizing the value-at-risk of the net risk of the insurer while subjecting to a profitability constraint. The proposed optimal reinsurance model, therefore, has the advantage of exploiting the classical tradeoff between risk and reward. Under the additional assumptions that the reinsurance premium is determined by the expectation premium principle and the ceded loss function is confined to a class of increasing and convex functions, explicit solutions are derived. Depending on the risk measure's level of confidence, the safety loading for the reinsurance premium, and the expected profit guaranteed for the insurer, we establish conditions for the existence of reinsurance. When it is optimal to cede the insurer's risk, the optimal reinsurance design could be in the form of pure stop-loss reinsurance, quota-share reinsurance, or a combination of stop-loss and quota-share reinsurance. The Geneva Risk and Insurance Review (2012) 37, 109-140. doi:10.1057/grir.2011; published online 23 August 2011 Keywords: value-at-risk (VaR); optimal reinsurance; expectation premium principle; linear programming in infinite dimensional spaces Introduction The importance of sound risk management for financial institutions and insurance enterprises has been dramatically highlighted by the subprime crisis. Risk professionals are constantly seeking better risk measures to quantify risks associated with market, credit, operational, catastrophic, and many others. Risk measures such as value-at-risk (VaR) and conditional VaR or conditional tail expectation (CTE) have been proposed. Among these risk The Geneva Risk and Insurance Review, 2012, 37, (109-140) 3 The popularity of VaR in part is due to its simplicity and in part is driven by the regulatory requirement. 4 Initial utilization of VaR is predominantly confined to quantifying market risk exposure for traders, corporate treasurers, and dealers. 5 Subsequently the use of VaR (as well as other risk measures such as CTE) as a criterion for optimal portfolio construction has sparked considerable interest among practitioners and researchers. 6 While these results shed some insights on the optimal construction of portfolios, researchers have also acknowledged the computational difficulties arising from adopting VaR in an optimization model. For example, as pointed out in Gaivoronski and Pflug 7 [t]he VaR optimization problem is nonconvex, may exhibit many local minima and is of combinatorial character, i.e. exhibits exponential growth in computational complexity. In recent years, these risk measures have also been exploited for determining the optimal policy in the context of insurance and reinsurance applications. For example, Wang et al., 8 Huang, 9 and Zhou and Wu 10 demonstrated that employing VaR as a constraint could assist an insured in determining his or her optimal insurance policy. By minimizing VaR or CTE of the total risk exposure of an insurer, Cai and Tan 11 and Cai et al. 12 derived explicitly the optimal reinsurance treaties for the insurer. 13 While analytic solutions have been derived in the above reinsurance models, these results can be criticized on the ground that the optimality is based exclusively on minimizing an insurer's risk exposure. In practice, an insurer is concerned not only with its exposure to risk but also its profitability of insuring the underlying risks. To elaborate this point, let us first note that when an insurer uses reinsurance to cede (or transfer) part of its loss to a reinsurer, the insurer is liable to pay reinsurance 1 The Geneva Risk and Insurance Review 110 premium to the reinsurer upfront. Furthermore the greater the expected loss that is ceded to a reinsurer, the higher the reinsurance premium. The additional cost associated with the reinsurance has a direct impact on the overall profitability of the insurer. Hence, it is unsatisfactory to determine the optimal reinsurance treaty by just focusing on minimizing an insurer's risk exposure. A viable optimal reinsurance treaty should reflect both risk exposure and cost (or equivalently profitability); this is the classical risk and reward tradeoff. The above argument implies that an insurer is faced with the conflicting objectives of risk transfer and profitability. Consequently, a more desirable optimal reinsurance model is the one that minimizes an insurer's risk exposure while at the same time taking into consideration its profitability. This is precisely the motivation of the proposed study in this paper. We formulate the optimal reinsurance model as a linear programme using VaR as the pertinent measure of the insurer's risk exposure. Furthermore, by confining to a class of increasing convex ceded loss function, we explicitly derive the optimal reinsurance policy. There are at least four reasons for imposing the ceded loss function being increasing and convex. The first reason is driven by the market practice. Commonly available reinsurance treaties such as quota-share reinsurance, stoploss reinsurance, and change-loss reinsurance (a combination of quota-share and stop-loss reinsurance) are increasing and convex. The second is motivated by the theoretical findings. Many well-known results in the literature have supported that the stop-loss type reinsurance treaty is optimal under various optimal reinsurance models. The third is due to the tractability and finally, without imposing the increasing and convex assumptions, the resulting optimal ceded loss function could be counter-intuitive and could even lead to moral hazard. We will further elaborate these reasons in the next section. The remainder of the paper is organized as follows: the next section specifies the reinsurance model that is of interest to the paper. The section after that reformulates the proposed reinsurance model as an equivalent linear programming problem. The subsequent section presents the optimal solutions together with some remarks. We also provide two numerical examples to highlight and contrast the importance of incorporating the profitability constraint in our proposed reinsurance model. The last section concludes the paper. The technical details and the proofs of the main results are relegated to Appendix A and Appendix B. Optimal reinsurance model Let X denote the (aggregate) loss initially assumed by an insurer. The random variable X is assumed to be nonnegative with cumulative distribution function F X (x)¼Pr{Xpx}, survival function S X (x)¼1ÀF X (x)¼Pr{X>x}, and mean [X]oN. To simplify our discussions, we assume that X has a continuous strictly increasing distribution function on (0, N) with a possible jump at 0, which allows X to be a random sum P i¼1 N X i . This is an important special case in the actuarial loss model. 11 Given the initial exposure to X, we assume that an insurer is interested in using reinsurance to manage its risk. Under the reinsurance arrangement, the insurer cedes part of its loss, denoted by f(X), to a reinsurer while retaining the remaining loss I f (X). The function f(x), satisfying the indemnity constraint 0pf(X)pX, is known as the ceded loss function and I f (x)¼xÀf(x) is the retained loss function. In exchange of ceding part of its loss to a reinsurer, the insurer incurs an upfront cost in the form of a reinsurance premium. By P f (X) we denote as the resulting reinsurance premium for a given reinsurance policy f written on the aggregate loss X, and by T f (X) we define as the sum of the retained loss and the reinsurance premium. In other words, we have the following relationship: Reducing T f (X) by p, the aggregate insurance premium received by the insurer from the insureds (or policyholders) for insuring X, we obtain the net cost or the net risk of insuring risk X in the presence of reinsurance. Using NC f (X) to denote the resulting net risk random variable, we have We now make three remarks with respect to NC f (X). First, it is more instructive to consider the random variable NC f (X) as opposed to the random variable T f (X) as in Cai and Tan 11 and Cai et al. 12 The random variable T f (X) can be interpreted as the total risk (or the total cost) of the insurer in the presence of reinsurance. NC f (X), on the other hand, takes into consideration both insurance premium inflow and reinsurance premium outflow. From a risk management point of view and the viability of the underlying business, it is therefore more prudent to focus on NC f (X). Second, (2) clearly demonstrates the intricate roles of the loss random variable X, the reinsurance policy f, the insurance premium p, and the reinsurance premium P f (X) on NC f (X). In particular, the choice of the ceded loss function f could have a tremendous impact on NC f (X) since for an initially insured loss X, the aggregate insurance premium p is fixed but the reinsurance premium P f is critically dependent on f. Third, the classical risk and reward is highlighted in (2). An insurer could reduce its risk exposure by transferring most of its expected risk to a reinsurer but at the expense of higher upfront reinsurance premium. On the other hand, if the insurer were to reduce its cost of reinsurance, this could be achieved by exposing to a higher risk exposure. The interplay between the ceded loss The Geneva Risk and Insurance Review 112 function f and the reinsurance premium P f and their overall effect on the net cost random variable NC f (X) are the key to the determination of the optimal reinsurance policy. An appropriate choice of ceded loss function could provide an effective way of reducing the risk exposure of an insurer. Let us now briefly review VaR and some of its properties. Formally, the VaR of a random variable X at a confidence level 1Àa, 0oao1, is defined as where the parameter a is typically a small value such as 1 or 5 per cent. Note that if X has a continuous strictly increasing distribution function on (0, N), then we have VaR a (X)¼S X À1 (a), where S X À1 is the inverse of the survival function S X . In addition, the following two properties on the VaR will be useful in our subsequent discussions. If function is increasing and continuous, then See Dhaene et al. 14 for the proof of the above property. For any constant c, the VaR satisfies the translation invariance property; that is Exploiting VaR as a measure of risk exposure for an insurer, an optimal reinsurance design could be defined as the solution to the following optimization problem: min f VaR a ðNC f ðXÞÞ subject to E½ÀNC f ðXÞXP and 0pfðxÞpx: In the above formulation, the objective function VaR a (NC f (X)) corresponds to the VaR of the net cost random variable NC f (X). Ideally the risk exposure of the insurer, as measured by VaR a (NC f (X)), should be as low as possible for a chosen ceded loss function. The expectation E [ÀNC f (X)] can be interpreted as the expected profit and hence the inequality ensures that the expected profit of the insurer under the ceded loss function f is at least P. The optimal reinsurance treaty that solves the above optimization problem (5) therefore minimizes the risk exposure of the insurer while guaranteeing a certain level of expected profit P. The second constraint 0p f (x)px ensures that the loss 14 The optimization problem (5) can be simplified further by first noting that the objective of minimizing VaR a (NC f (X)) over function f is equivalent to minimizing VaR a (T f (X)). This is due to the property (4). Second, under the additional assumption that the reinsurance premium P f (X) is determined by the expectation premium principle; that is, where y>0 is the safety loading factor, it follows from (2) and (6) that the profitability condition Third, the minimization in (5) is assumed to be taken over the class of ceded loss functions F consisting of all increasing and convex functions f(x) defined on (0, N). As alluded in the introduction, there are four reasons for imposing the increasing and convex conditions on the ceded loss function. First, these assumptions are consistent with practice in that the reinsurance treaties such as the quota-share reinsurance with f (x)¼ax and I f (x)¼(1Àa)x, 0oap1, the stop-loss reinsurance with f (x)¼(xÀd) þ ¼max{0, xÀd} and I f (x)¼min{x, d}, dX0, and the change-loss reinsurance (combination of quota-share and stoploss reinsurance) with f (x)¼a(xÀd) þ and I f (x)¼(1Àa)x þ a min{x, d} are common in the marketplace. All of these treaties are increasing and convex. Second, the reinsurance treaties mentioned above are by far the most widely analysed contracts in the literature. Just to name a few, it is well known that the stop-loss reinsurance is optimal in that it yields the lowest variance of retained loss among the class of ceded loss functions with the same expectations; see, for example, Bowers et al. explicitly adopts VaR in a reinsurance model but the optimal solutions are derived through a numerical method. Finally, not imposing the increasing condition on the ceded loss function could be counter-intuitive and could even lead to moral hazard. To elaborate, the results derived in both Bernard and Tian 21 and Weng 22 indicate that the truncated stop-loss function could be optimal. The truncated stop-loss function has the peculiar property that for any loss less than a certain threshold level, the ceded loss function mimics a stop-loss in the sense that the amount ceded to the insurer is zero for loss less than the deductible and increases linearly for loss above the deductible until the threshold level. Beyond the threshold, the loss ceded to the reinsurer drops to zero. Such a phenomenon is counter-intuitive in that the insurer should be more concerned with extreme losses. Yet with the truncated stop-loss function, the insurer is only protected for moderate losses and no coverage for large extreme losses. In addition, there is a potential issue with the moral hazard induced by the truncated stop-loss function. See also Kaluszka, 23 Kaluszka and Okolewski, 24 and Bernard and Tian. 21 For these reasons, we investigate the optimal ceded loss functions by confining to a class of increasing and convex functions F . In summary, the above arguments imply that the optimal reinsurance model (5) becomes min f2F VaR a ðT f ðXÞÞ subject to E½ fðXÞpB and 0pfðxÞpx for all xX0: The indemnity constraint leads to 0pE [ f(X)]pE [X]. This suggests that when BXE[X], the constrained optimization problem (8) reduces to an unconstrained problem since the profitability condition in (8) will have no effect. The optimal solution to this special case is addressed in Cai et al. 12 The objective of this paper is to solve the optimal reinsurance model (8) and obtain explicitly the optimal reinsurance treaties, as discussed in the following sections. 21 Optimal reinsurance model reformulation As pointed out in Gaivoronski and Pflug, 7 the optimization problem associated with VaR, in general, is a non-trivial exercise. In order to obtain the solution of our proposed optimal reinsurance model, it is convenient to reformulate (8) as a linear programming with respect to s-finite positive measures on the Borel measurable space Lemma 1 An increasing convex function defined on (0, N) can be represented as the following form: for some positive s-additive measure m on B Lemma 2 For any f (x)A F, I f (x)¼xÀf (x) is increasing and concave in x. The proofs of these two lemmas can be found in Cardin and Pacelli 25 and Cai et al. (Lemma A.1), 12 respectively. We now make the following two remarks: Remark 1 Note that for any ceded loss function fAF, f (0)¼0 and hence by Lemma 1 the ceded loss function f has the following representation: with a positive s-finite measure m on B. Furthermore, by Fubini theory, Remark 2 It follows from Lemma 2 that for any fAF , I f (x) is continuous and hence together with property (3), we have Note that when aXS X (0), then VaR a (X)¼0, the goal function VaR a (T f (X)) in model (8) depends on E [ f(X)] and y so that it is optimal for the insurer 25 Cardin and Pacelli (2007 Consequently, our proposed optimal reinsurance model where M þ denotes the set of all s-finite positive measure on the measurable space Optimal solutions and examples The optimal solutions to the above linear programming (13) are formally derived in Appendix A. It should, however, be emphasized that solving (13) is non-trivial. We use an approximating approach, which is a routine approach regarding the optimization over a measured space. More specifically, this entails us first approximating (13) by a series of linear programming with optimization over a certain set of discrete measures with a particular structure (see (A.2) in Appendix A), and then reformulating the programming as some equivalent models (A.6), which are optimization problems over Euclidean space. By first obtaining the optimal solutions to (A.6) (and equivalently (A.2)), we then justify that these solutions are also the required solutions to (13) and (8) (or equivalently (5) with f restricted in F ). Furthermore, the VaR a * in column five is the corresponding minimal VaR a (T f (X)) for f over the class of increasing and convex functions such that 0pf (x)px, 8xX0. Note that the optimal ceded loss function f *(x) is related to m* via (10). As pointed out earlier that Cai et al. 12 proposed an unconstrained optimal reinsurance model that determines the optimal ceded loss function by minimizing certain risk measures of the insurer's total risk exposure. The approach described in this paper, on the other hand, is a generalization of their model in the sense that we impose an additional profitability constraint. The model proposed in this paper is intuitively more appealing since it takes into account both risk and Column 2 states the conditions for each case. For each case, column 3 gives the optimal measure to problems (13) and (A.2), while column 4 presents the corresponding optimal ceded loss function to the model (8). Column 5 tabulates the minimal value of VaR a (T f (X)). The Geneva Risk and Insurance Review 118 reward, consistent with practice. We now make the following remarks to compare and contrast the results obtained in this paper to that in Cai et al. 12 Remark 3 Except for the first two cases, the optimal ceded loss functions presented in Remark 4 Recall that when B>E [X], the expected profit constraint in our proposed constrained optimal reinsurance model will have no impact on the solution. Consequently, this special case reduces to the results of Cai et al. 12 We can also examine the range of insurers' expected profit P corresponding to this special case. Note that when B>E[X], the inequality B4 Hence if the expected profit P of the insurer is less than the quantity on the right hand side of the above inequality, the profitability constraint becomes redundant. In fact, in this situation, Case (iv) in 12 Remark 5 To understand the impact of imposing the profitability constraint on the optimal reinsurance model, let us compare Case (iv) to Case (v) and assume l(d r )X0. When an insurer becomes more aggressive so that it requires an expected profit greater than the quantity on the right-hand-side of the inequality (19) (i.e. b(d r )o1), the optimal ceded loss function is f *( . By contrasting these results to the unconstrained model as in To conclude this section, we provide two examples to illustrate our results. Example 1 Assume X is exponentially distributed with mean E [X]¼1,000. Then S X (x)¼e À0.001x , xX0, and S X (0)¼1. Assume further that the loading factors for the reinsurer and insurer are 20 and 15 per cent, respectively. This implies y¼0.2 and d r ¼S X À1 (1/ for a given ceded loss function. In practice it is to be expected that the loading factor for the reinsurer is higher than the insurer's. Consequently, the achievable expected profits P are in the range [0, 150] so that BA [0, 750]. The impact of the expected profit P (or equivalently B) on optimal reinsurance is also clearly demonstrated. First, if we were to decrease the minimum level of the expected profits, the optimal retention d does not change as long as b(d o )p1. The optimal c, however, increases accordingly as asserted in part (i), Proposition A.4(b). This is also consistent with our numerical results. Second, when the condition b(d o )>1 is satisfied as we further decrease P, the optimal reinsurance design becomes a pure stop-loss contract with the optimal retention d that also declines with P (see Eq. .33 and part (b) of Proposition A.3 can be used to determine the optimal ceded loss function. In this case, the upper constraint B has no impact on the optimization problem and it reduces to the unconstrained problem, as studied in Cai et al. 12 In our example, VaR a * is US$1182.3 with optimal retention d r ¼182.3. See also Remark 4. The unconstrained optimal reinsurance design also serves as a benchmark to our proposed constrained optimization problem. For instance at a¼1 per cent, if the insurer were to seek an expected profit of US$145, the insurer would need to sustain more than three times the risk exposure relative to the unconstrained case (compare US$3715.5 to 1182.3). Example 2 In this example, we assume X has a Pareto distribution with S X (x)¼((2,000)/(x þ 2,000)) 3 , xX0 so that its E [X]¼1,000 is the same as the previous example. We also assume y¼0.2 and p¼1150. Conclusion In this paper, we extended and improved the optimal reinsurance model of Cai et al. 12 by formulating the model as a constrained optimization problem. By minimizing the VaR of the total risk of an insurer while subjecting to a profitability constraint, explicit solutions for the optimal reinsurance were derived. We formally established that the optimal reinsurance could be in the form of a pure stop-loss reinsurance, a quota-share reinsurance, a combination of them, or not to cede at all, depending on the risk measure's level of confidence, the safety loading for the reinsurance premium, and the expected profit guaranteed for the insurer. The numerical examples highlighted the importance of incorporating the profitability condition. The key assumptions in our proposed optimal reinsurance model are (i) reinsurance premium is determined by the expectation premium principle, (ii) ceded loss function is confined to a class of increasing and convex functions, and (iii) risk exposure of the insurer is captured by its VaR measure. It will be of interest to investigate the impact on the optimal reinsurance design if any of these assumptions are modified. The expectation premium principle is a linear function. Incorporating other premium principles in a reinsurance model usually incurs additional mathematical complexity. In particular, when the objective is to minimize the VaR risk measure, a nonlinear reinsurance premium principle usually results in a notoriously nontractable model. At this point, it might be interesting to mention the thesis by Weng, 22 who developed a numerical method in analysing the optimal solution to the VaR minimization model. 26 His proposed numerical solution, however, is only applicable to the expectation reinsurance premium principle. Consequently, exploring the optimal solution to the VaR-based reinsurance models under general premium principles is an ongoing and challenging problem. Besides VaR, there are other risk measures such as the CTE that can be incorporated in the proposed reinsurance model. See for example Gajek and Zagrodny 18 and Balba´s et al. 27 However, it should be emphasized that the CTE minimization reinsurance model is usually much more tractable due to the convexity of CTE. In fact under the expectation reinsurance premium principle, Tan et al. 28 explicitly derived the optimal solutions over the class of all general reinsurance treaties. Their results again confirm that a stop-loss reinsurance treaty is optimal.