Chapter 6 Pricing Reinsurance Contracts

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1 Chapter 6 Pricing Reinsurance Contracts Andrea Consiglio and Domenico De Giovanni Abstract Pricing and hedging insurance contracts is hard to perform if we subscribe to the hypotheses of the celebrated Black and Scholes model. Incomplete market models allow for the relaxation of hypotheses that are unrealistic for insurance and reinsurance contracts. One such assumption is the tradeability of the underlying asset. To overcome this drawback, we propose in this chapter a stochastic programming model leading to a superhedging portfolio whose final value is at least equal to the insurance final liability. A simple model extension, furthermore, is shown to be sufficient to determine an optimal reinsurance protection for the insurer: we propose a conditional value at risk (VaR) model particularly suitable for large-scale problem instances and rationale from a risk theoretic point of view. Keywords Reinsurance Option pricing Incomplete markets 6.1 Introduction Hedging a liability is the best practice to mitigate the potential negative impact due to market swings. The issue of pricing and hedging embedded options in insurance contracts is also ruled by the International Financial Reporting Standards (IFRS 4) and by the new regulatory capital framework for insurers Solvency II. The new reporting standards prescribe that the cost of options and guarantees embedded in insurance contracts are measured consistently with the market and, for this purpose, it is suggested to split the risk into hedgeable and non-hedgeable component. The hedging process, as described in finance textbooks, however is hard to apply in the insurance context. As a primary obstacle, the underlying of an insurance contract is usually a non-tradeable asset, thus making impossible the trading activity needed to hedge the protection seller risk exposure. Moreover, the liabilities generated by an insurance contract are long-term ones, and unexpected shortfalls in the asset returns are not envisaged by the geometric Brownian motion underneath the celebrated Black and Scholes model. These unpredictable events can A. Consiglio (B) Department of Statistics and Mathematics Silvio Vianelli, University of Palermo, Palermo, Italy consiglio@unipa.it M. Bertocchi et al. (eds.), Stochastic Optimization Methods in Finance and Energy, International Series in Operations Research & Management Science 163, DOI / _6, C Springer Science+Business Media, LLC

2 126 A. Consiglio and D. De Giovanni generate serious losses and rolling the hedge forward, as witnessed by the Metallgesellschaft default (Mello and Parsons 1995), may lead to unexpected severe losses. In options theory such limitations are known as sources of market incompleteness, and recently, scholars have started coping with pricing and hedging options in incomplete markets. For example, Consiglio and De Giovanni (2008) adopt a superreplication model to determine the hedging portfolio of insurance policies whose final liabilities depend on a minimum guarantee option and a bonus distribution scheme. An extension of such a model (Consiglio and De Giovanni forthcoming) allows the pricing of insurance contract with a surrender option, that is, the option to leave the contract before maturity. An alternative approach is proposed by Coleman et al. (2007). They address and solve a similar incomplete-market hedging problem extending the traditional Black and Scholes price process to include Merton s jump diffusion process. The insurance claim is then hedged, by using the underlying asset and a set of standard options expiring before the claim s maturity. The hedging strategy is here determined by applying the minimum local hedging risk principle by Föllmer and Schweizer (1989). The aim of this chapter is to extend the model in Consiglio and De Giovanni (2008, forthcoming) to handle not only the primary risk exposure associated with a short insurance position but also the exposure induced by a reinsurance agreement. To this aim we propose a novel approach to the asset liability management of such contracts. A general definition of reinsurance contract is that of a coverage purchased by an insurance company (insurer) from typically another insurance company (reinsurer) to transfer an original risk exposure. The reinsurance agreement specifies the basis for which the reinsurer will pay the insurer s losses (excess of loss or proportional) and the reinsurance premium paid by the reinsured. The chapter is organized as follows. Section 6.2 introduces the reinsurance problem. Section 6.3 deals with the mathematical formulation of the stochastic programming models, including: (i) the notation, (ii) the pricing of European contingent claims, and (iii) an efficient method to build scenario trees. The asset liability management problem to manage reinsurance contracts is described in Section 6.4, while implementation notes and discussion of the results are reported in Section 6.5. Section 6.6 concludes the chapter and highlights the major findings. 6.2 Stop-Loss Reinsurance Contracts in the Property and Casualty Market Broadly speaking, reinsurance is the insurance of insurance liabilities. Insurance firms with a specified risk exposure might decide or are forced to by regulators to transfer part of their own risk exposure to third-party companies by buying reinsurance protection.

3 6 Pricing Reinsurance Contracts 127 Popular agreements in the reinsurance market are proportional reinsurance and aggregate excess reinsurance. With proportional contracts, the transferring party cedes to the reinsurance company a fixed proportion of the risk associated with the portfolio, while with aggregate excess treaty, the reinsurance company promises to pay the claims exceeding a given retention (see Straub (1997) for more details on reinsurance contracts). We denote by L T the stochastic cash flow representing the future liabilities of the company, and by L 0 the premium income collected. L T is the sum of all the payments occurring during the time interval [0, T ]. A typical aggregate excess reinsurance contract is represented by the stop-loss agreements, where the reinsurer pays the part of L T which exceeds a certain amount, R. The reinsurer obligation is usually limited to a given amount, M. More precisely, the payoff of a stop-loss contract can be summarized as follows: 0, L T R Y T = (L T R), R < L T < R + M. M, L T R + M (6.1) It is worth mentioning that no standardized contracts can be found on the market. Reinsurance agreements are usually tailored to meet specific requirements of the transferring parts. Reinsurance agreements are usually evaluated either by standard actuarial methods (see Straub 1997) or by equilibrium techniques, using the so-called financial reinsurance method. In the former case, the price is computed as the sum of the expected value of the future payments and a risk premium which is usually proportional to the variance of the distribution of L T. In the latter case, the insurance liability is assumed to be highly correlated to a traded asset, and the price follows by application of the capital asset pricing model of Sharpe (1964). Application of financial reinsurance and the design of reinsurance contract have been studied in de Lange et al. (2004). In the sequel of the chapter we describe a stochastic programming model to evaluate reinsurance contract by super-replication. This is done by recognizing that the contract Y T is a European contingent claim (ECC) written on the non-traded underlying liability L T. We also consider an asset and liability management (ALM) problem, where an insurance firm with a future liability L T needs to determine the optimal investment strategy in order to maximize its expected profit. The asset universe in which the company might invest consists of a set of risky assets (stocks), a risk-free security, and the reinsurance contract. The goal of the company is to maximize its expected profit, given a set of regulatory constraints to be fulfilled. In fact, international accounting standards require that insurance companies meet specific obligations, expressed in terms of risk margins which are usually proportional to the value at risk (VaR) of the loss distribution.

4 128 A. Consiglio and D. De Giovanni 6.3 A Stochastic Programming Model for Super-Replication The problem of option pricing in incomplete markets is currently an active area of research. The different methodologies proposed over the years differ on the way the option price is defined. In what follows, we describe a superhedging method based on a stochastic programming representation of the hedging problem. An alternative approach is the quadratic hedging of Föllmer and Schweizer (1989) (see also Dahl and Møller 2006, Dahl et al. 2008, for recent applications in the life insurance context). Global risk minimization (see Cerny and Kallsen 2009, and reference therein) based on quadratic hedging is a promisingly new area of research. Utility-based pricing algorithms have been proved to be effective in different area of applications. We refer to Carmona (2008) for a book-length treatment of the topic including application to the insurance sector Notation and Probabilistic Structure We introduce a financial market where security prices and other payments are discrete random variables supported by a finite-dimensional probability space (Ω, F, P). The atoms ω Ω are sequences of real-valued vectors (asset values and payments) over the discrete time periods t = 0, 1,...,T. The path histories of the security prices up to time t correspond one-to-one with nodes n N t.the set N 0 consists of the root node n = 0 and the leaf nodes n N T correspond one-to-one with the probability atoms ω Ω. This probabilistic structure can be modeled by a discrete, non-recombining scenario tree. In the scenario tree, every node n N t, t = 1,...,T, has a unique ancestor a(n) N t 1, and every node n N t, t = 0,...,T 1, has a non-empty set of child nodes C(n) N t.the collection of all the nodes is denoted by N T t=0 N t. The information arrival can be modeled by associating a set of σ -algebras {F t } t=0,...,t with F 0 ={φ, }, F t 1 F t, and F T = F. We model the probability measure P by attaching weights p n > 0 to each leaf node n N T so that n N T p n = 1. The probability of each non-final node n N t, t = T is recursively determined by p n = m C(n) p m. (6.2) The market consists of J + 1 tradeable securities indexed by j = 0, 1,...,J one of which, say security 0, is the risk-free security. We model the risk-free asset by assuming the existence of a market for zero coupon bonds for each trading date t = 0,...,T 1. At time t + 1 the risk-free value in node n, Sn 0, corresponds to the previous value Sa(n) 0 capitalized with the return given by the bond Bn t,t+1 in node n. That is, Sn 0 is a one-dimensional stochastic process defined on the finite probability

5 6 Pricing Reinsurance Contracts 129 space (Ω,F, P) and F t+1 -measurable. 1 This allows us to introduce in the market model the stochastic dynamics of the yield curve. Other securities prices (stocks, future, etc.) are described by a nonnegative-valued vector S 0 = ( S0 1,...,S 0 J ) of initial (known) market prices and nonnegative-valued random vectors S n : Ω R J, F t -measurable, n N t and t = 0,...,T. The yield curve plays an important role in the insurance industry. It is used to determine the best estimate of future liabilities as provided by the European accounting standards Solvency II. In the context of this chapter, the term structure is used to include in the universe of investment opportunities an asset in which investors can trade with no risk. In presence of risk factors other than the traded securities, the process S t is augmented by J + 1 real-valued variables ξ t = ( ξ0 1 ),...,ξj t whose path histories match the nodes n N t, for each t = 0, 1, 2,...,T. This is certainly the case of the property and casualty market, where liabilities are generated by factors exogenous to the financial markets (car accidents, earthquakes, etc.). In the framework described above, an ECC is a security whose owner is entitled to receive the F t -adapted stochastic cash flow F ={F t } t=0,...,t. The above definition of ECC is general enough to encompass a large variety of derivatives, including futures and exotic options. Options written on non-traded underlying variables which is one of the major source of market incompleteness are also embraced. The stop-loss reinsurance contract described in (6.1) clearly falls in the class of ECCs, with F n = Y n for n N T and F n = 0forn N t, t = 0,...,T 1. Here Y n is the value of the reinsurance contract given as a function of the liability L n occurred at node n and corresponds to the underlying asset of the ECC. In the property and casualty markets, contrary to the financial markets, one cannot trade with the underlying liabilities. This makes the reinsurance contract difficult to hedge by trading in the underlying, and thus we must switch to the theory of option pricing in incomplete markets. In the general framework of incomplete markets, contingent claims cannot be uniquely priced by no-arbitrage arguments. Rather, there exists the so-called arbitrage interval which describes the range of all arbitrage-free evaluations. Lower and upper bounds of the arbitrage interval are determined by the buyer price and seller price, respectively. Informally, the buyer price is the maximum amount of money an investor is willing to pay in order to buy the contingent claim. Likewise, the seller price is the minimum amount of money that is needed to the writer in order to build a portfolio whose payout is at least as equal as to that of the contingent claim. In this chapter we assume that the policyholder are price-taker, thus concentrating our attention on the seller problem only. For details about the construction of the buyer price and the relation between the arbitrage interval and the set of equivalent martingale measure we refer to King (2002). 1 This means that, at node n, we already know the ending-period value Sm 0, m C(n) but, n, m N t with n = m, Sn 1 and S0 m could be different.

6 130 A. Consiglio and D. De Giovanni Super-Replication of ECCs The reinsurer (protection seller) receives an amount, V, corresponding to the premium of the reinsurance contract and she agrees to pay the protection buyer the payoff of the reinsurance contract. The seller s objective is to select a portfolio of tradeable assets that enables her to meet her obligations without risk (i.e., on all nodes n N ). The portfolio process must be self-financing, i.e., the amount of assets bought has to offset the amount of assets sold. Put it differently, there are no inflows or outflows of money, since the amount of money available at node n is funded only by price movements at the ancestor node, a(n), and at the node itself. Using the tree notation, we denote by Z j n the number of shares held at each node n N and for each security j = 0, 1,...,J. At the root node, the value (price) of the hedging portfolio plus any payout to be covered is the price of the option, that is J S j 0 Z j 0 = V. (6.3) At each node n N t, t = 1, 2,...,T, the self-financing constraints are defined as follows: J Sn j Zn j = J Sn j Z j a(n). (6.4) To sum up, the stochastic programming model for the insurance evaluation problem can be written as follows: Problem 3.1 (Writer problem for ECCs) Minimize V (6.5) Zn j J S j 0 Z j 0 = V, (6.6) J Sn j Zn j = J Sn j Zn j + L n = J Sn j Z j a(n) for all n N t, (6.7) t = 1, 2,...,T 1, J Sn j Z j a(n) for all n N T, (6.8) J Sn j Zn j 0 for all n N T, (6.9) Z j n R for all n N, (6.10) j = 0, 1,...,J.

7 6 Pricing Reinsurance Contracts 131 We highlight here some important points: 1. Problem 3.1 is a linear programming model where the objective function minimizes the value of the hedging portfolio or rather the price of the option. 2. The payout process {F n } n N is a parameter of Problem 3.1. This implies that any complicated structure for F n does not change the complexity of the model. 3. Constraints (6.9) ensure that at each final node the total position of the hedging portfolio is not short. In other words, if short positions are allowed, the portfolio process must end up with enough long positions so that a positive portfolio value is delivered Tree Generation We generate the tree for the underlying price process S by matching the first M moments of its unknown distribution. Our approach is based on the moment matching method of Høyland and Wallace (2001), where the user provides a set of moments, M, of the underlying distribution (mean, variance, skewness, covariance, or quantiles), and then, prices and probabilities are jointly determined by solving a non-linear system of equations or a non-linear optimization problem. The method also allows for intertemporal dependencies, such as mean reverting or volatility clumping effect. For a review on alternative scenario generation methods, see Dupačová et al. (2000) and references therein. The following system of non-linear equations formalizes the moment matching problem: Problem 3.2 (Moment matching model) f k (S, p) = τ k k M, (6.11) p m = 1, (6.12) p m 0, m C(n), (6.13) where f k (S, p) is the algebraic expression for the statistical property k M and τ k is its target value. For example, let us consider the problem of matching the expected values μ S 0,...,μ S J, the variances, σ 2 S 0,...,σ 2 S J, and the correlations, ρ j,k, j = 0, 1,...,J, and k = j, of the log-returns of each asset. The moment matching problem becomes Problem 3.3 (Moment matching model: an example) rm j p m = μ S j, j = 0,...,J, (6.14)

8 132 A. Consiglio and D. De Giovanni ( r j m μ S j ) 2 pm = σ 2 S j, j = 0, 1,...,J, (6.15) ( r j m μ S j ) (r k m μ S k ) pm σ S j σ S k ( ln S j m S j n = ρ j,k, j = 0, 1,...,J and k = j, (6.16) ) = rm, j j = 1, 2,...,n, (6.17) p m = 1, (6.18) p m 0, m C(n). (6.19) The non-linearity of Problem 3.2 could lead to infeasibility. An alternative strategy is to formulate Problem 3.2 as a goal programming model. We can minimize the weighted distance between the statistical properties of the tree, and their target values, that is Problem 3.4 (Moment matching model: goal programming) Minimize ω k ( f k (S, p) τ k ) 2, (6.20) S,p k M p m = 1, (6.21) p m 0, m C(n), (6.22) where ω k is the weight which measures the importance associated with the statistical property k M. Problem 3.4 is easier to solve with respect to the previous nonlinear system, but a perfect match is generally difficult to meet. A hybrid model, in which only some statistical properties are required to be perfectly matched, may be adopted to avoid bad match of the specifications. In detail, we split the set M of parameters to fit in two subsets, M 1 and M 2, and implement the following nonlinear programming problem: Problem 3.5 (Moment matching model: mixed goal programming) Minimize ω k ( f k (S, p) τ k ) 2, (6.23) S,p k M 1 f i (S, p) = τ i, i M 2, (6.24) p m = 1, (6.25) p m 0, m C(n). (6.26)

9 6 Pricing Reinsurance Contracts 133 Although these three different strategies can be implemented, there is no general rule that suggests the choice of a particular one. Each of these strategies has its own pros and cons, and the choice depends on the specific problem to be faced with. To be consistent with financial asset pricing theory, arbitrage opportunities should be avoided. In the setting described in Section 6.3, the no-arbitrage condition is fundamental. In fact, if the market allows for arbitrages, the stochastic programming problem described in this chapter will end up with an unbounded solution. Following Gulpinar et al. (2004), we can preclude arbitrages by imposing the existence of a strictly positive martingale measure such that S j n = e r n S j mπ m, (6.27) where r n is the risk-free interest rate observed in state n. We refer to Klaassen (2002) for alternative, more stringent, no-arbitrage conditions. 6.4 Asset and Liability Management with Reinsurance Contracts In Section 6.3 we have considered the position of the reinsurance company whose objective is to determine the optimal hedging strategy for the contract and to price it accordingly. We now turn to the position of the buyer of the protection. The goal is to manage the liabilities she has to face by making use of both financial investments and the reinsurance contract. We assume that the price of the reinsurance contract is determined by the protection seller and that the company has to satisfy regulatory constraints expressed in terms of value at risk of the loss distribution. To be more specific, we define the loss function (n) = J Sn j Zn j xy n + L n, n N T, (6.28) which corresponds to the value of the portfolio after the payments for the liability L n have been made. According to (6.28): (n) = Ψ(n), where Ψ(n) is the profit realized at node n after the liability has been paid. Following Rockafellar and Uryasev (2000), the conditional value at risk (CVar) of the loss distribution at the probability level α is defined by the following set of equations: 1 1 ζ + φ(n), (1 α) N T n N T (6.29) φ(n) = (n) ζ, n N T, (6.30) φ(n) 0, n N T. (6.31)

10 134 A. Consiglio and D. De Giovanni The endogenous variable ζ can be shown to represent the value at risk of the loss distribution. Notice that the shortfalls in (6.30) are defined to be non-negative. As a consequence, in order to bound ζ, an upper limit on (6.29) is sufficient. The following linear program determines the optimal investment strategy for an insurance company seeking to maximize its expected profits, given that the α-cvar of the loss distribution cannot exceed the limit ω: Problem 4.1 (Conditional VaR model) ζ + Maximize Z j n,x,ζ 1 N T Ψ(n), (6.32) n N T J S j 0 Z j 0 + xy 0 = L 0, (6.33) J Sn j Zn j xy n L n = Ψ(n), n N T, (6.34) J ( ) Sn j Zn j Z j a(n) = 0 for all n N t, t = 1, 2,...,T, (6.35) (n) ζ φ(n), n N T n N T, (6.36) 1 1 φ(n) ω, (1 α) N T n N T (6.37) φ(n) 0, (6.38) x, Zn, j 0. (6.39) 6.5 Implementation Notes and Results We consider a tree structure with six stages and perform all the experiments by assuming a time horizon T = 10 years. The time step between two consecutive stages is thus fixed to 1.67 years. In the tree, each non-final node branches into five successor nodes. The resulting tree has exactly 19,531 total nodes and 5 6 = 15,625 final nodes. The financial market consists of three risky securities (stocks) plus a risk-free asset. The risk-free asset has initial value 100 and is assumed to grow at the continuously compounded annual rate of 3%. We generate the risky assets by using the tree generation model described in Problem 3.4 and by restricting the moment matching problem to fit the expected values, standard deviations, and correlations of the continuously compounded asset returns as displayed in Table 6.1. For all the assets, the skewness parameter and kurtosis are set, respectively, to β 1 = 0 and

11 6 Pricing Reinsurance Contracts 135 Table 6.1 Statistical properties for the returns of securities used in the experiments Correlations Mean Std. Dev Asset 1 Asset 2 Asset 3 Asset Asset Asset We built scenario trees by matching means, standard deviations, and correlations shown in the table. Skewness and kurtosis parameters are set, respectively, to β 1 = 0andβ 2 = 3 β 2 = 3. This is equivalent to assume a market where asset returns are Gaussian with parameters specified as in Table 6.1. This is a simplifying assumption that does not affect the relevance of our framework. In a more general setup, we can determine probability distributions that match higher moments (see Consiglio and De Giovanni, forthcoming). Finally, the initial value of all the assets is set equal to Pricing Reinsurance Contracts Our experiments are based on four case studies (CS). More specifically, we consider four different distributions for the random variable L T that represents the actuarial claim the insurance company will face with. Accordingly, we generate four different scenario trees for L T, one for each CS, using Problem 3.4. Table 6.2 displays the expected values and the standard deviation of each CS. The skewness and kurtosis parameter are set, respectively, to β 1 = 0 and β 2 = 3, while the financial assets and the actuarial claims are assumed to be uncorrelated. Table 6.3 displays the prices of the reinsurance contract with fixed values of R and M. The pattern shown in Table 6.3 is not surprising. The higher the expected value and the variance of the liability distribution, the higher the price of the reinsurance contract. It is, however, interesting to look at the initial composition of the strategies produced to super-replicate the reinsurance contract. Such results are displayed in Fig We observe that to super-replicate the contract the reinsurer is long in the risk-free security and short in the risky assets. The cause of this strategy Table 6.2 Statistical properties for the actuarial claims used in the experiments Mean Std. Dev CS CS CS CS We built scenario trees by matching means and standard deviations as displayed. The skewness and kurtosis parameter are set, respectively, to β 1 = 0andβ 2 = 3. No correlation between the financial market and the actuarial risk is assumed

12 136 A. Consiglio and D. De Giovanni Table 6.3 Prices of the reinsurance contract for two different levels of the parameters R and M R = 160, M = 200 R = 140, M = 220 CS_ CS_ CS_ CS_ % RF Asset_1 Asset_2 Asset_3 40% Asset allocation 20% 0% 20% 40% 60% CS_1 CS_2 CS_3 CS_4 80% RF Asset_1 Asset_2 Asset_3 60% Asset allocation 40% 20% 0% 20% 40% 60% CS_1 CS_2 CS_3 CS_4 Fig. 6.1 Initial values, in percentage, of the super-replicating portfolio for R = 160, M = 200 (top panel) and R = 140, M = 220 (bottom panel) is that the underlying asset (the insurance liability L T ) and the risky assets are uncorrelated. This prevents the reinsurer to exploit the co-movements between the traded assets and the underlying (see the discussion in Consiglio and De Giovanni, forthcoming, section 5). We also observe that the amount of asset 1 and asset 2 to short-sell in the super-replicating strategy is much higher than that of asset 3 which

13 6 Pricing Reinsurance Contracts 137 corresponds to the most risky asset in the market. The latter comes into play when the riskiness of the liability exceeds a certain level that in this experiment is given by CS_3 and CS_ Risk Management with Reinsurance Contracts We use the liability structure based on the simulation CS_1 in Table 6.1, and set the initial premium collected by the insurance company to L 0 = 145. Two different parametrizations of the reinsurance contract are considered with (1) R = 160 and M = 200 and (2) R = 140 and M = 220. The prices of the two contracts are set to their super-replication values: Y 0 = and Y 0 = , respectively. We then run Problem 4.1 with increasing levels of the risk tolerance, from ω = 0to ω = 3. The resulting efficient frontiers are displayed in Fig. 6.2, while the optimal portfolios for each level of tolerance are displayed in Fig Note that the problem allows for a feasible optimal solution when ω = 0. This is because the reinsurance contract completely offset any possible loss, thus eliminating any risk. From the experiment we learn some important facts. First, the optimal portfolios include a high percentage of reinsurance in both cases and for all levels of ω (Fig. 6.3) even if the contracts themselves are evaluated at their super-replication prices. This empirical finding is supported by theoretical arguments in King (2002, section 8), where the author proves that in the presence of a liability structure the investors are willing to trade in the derivative. Second, the amount of reinsurance declines with the increase of the level of riskiness allowed ω. This is a confirmation of the rational choice of the model which reduces the level of the reinsurance if the risk exposure is increased by the decision maker. We strongly believe this could be Expected profit R = 160 M = 200 R = 140 M = Conditional VaR Fig. 6.2 Efficient frontiers. Conditional VaR of the losses vs expected profit of the portfolio for confidence level α = 99%. The efficient frontiers start at ω = 0 as the losses can be totally offset by purchasing a reinsurance contract

14 138 A. Consiglio and D. De Giovanni Asset allocation 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% x RF AA_1 AA_2 AA_ Conditional VaR Asset allocation 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% x RF AA_1 AA_2 AA_ Conditional VaR Fig. 6.3 Optimal portfolios for R = 160, M = 200 (top panel) andr = 140, M = 220 (bottom panel) a useful tool for regulators which can ascertain whether a given level of reinsurance is safe for the liability structure of a company, and in case of negative result forces the company toward a more consistent level of protection. 6.6 Conclusions We propose in this chapter a stochastic programming model to cope with the price and management of reinsurance contracts. We show that the pricing in incomplete markets is a feasible alternative to the stronger hypotheses of completeness, where, in this case, the main obstacle in hedging the insurance contract is the nontradeability of the underlying asset. The model is flexible enough to be embedded

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