chapter6 pricing reinsurance contracts · reinsurance agreements are usually tailored to meet...

Chapter 6Pricing Reinsurance Contracts

Andrea Consiglio and Domenico De Giovanni

Abstract Pricing and hedging insurance contracts is hard to perform if we subscribeto the hypotheses of the celebrated Black and Scholes model. Incomplete marketmodels allow for the relaxation of hypotheses that are unrealistic for insurance andreinsurance contracts. One such assumption is the tradeability of the underlyingasset. To overcome this drawback, we propose in this chapter a stochastic program-ming model leading to a superhedging portfolio whose final value is at least equal tothe insurance final liability. A simple model extension, furthermore, is shown to besufficient to determine an optimal reinsurance protection for the insurer: we proposea conditional value at risk (VaR) model particularly suitable for large-scale probleminstances 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 dueto market swings. The issue of pricing and hedging embedded options in insurancecontracts is also ruled by the International Financial Reporting Standards (IFRS 4)and by the new regulatory capital framework for insurers Solvency II. The newreporting standards prescribe that the cost of options and guarantees embedded ininsurance 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 applyin the insurance context. As a primary obstacle, the underlying of an insurancecontract is usually a non-tradeable asset, thus making impossible the trading activ-ity needed to hedge the protection seller risk exposure. Moreover, the liabilitiesgenerated by an insurance contract are long-term ones, and unexpected shortfallsin the asset returns are not envisaged by the geometric Brownian motion under-neath the celebrated Black and Scholes model. These unpredictable events can

A. Consiglio (B)Department of Statistics and Mathematics “Silvio Vianelli”, University of Palermo, Palermo, Italye-mail: [email protected]

M. Bertocchi et al. (eds.), Stochastic Optimization Methods in Finance and Energy,International Series in Operations Research & Management Science 163,DOI 10.1007/978-1-4419-9586-5_6, C© Springer Science+Business Media, LLC 2011

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126 A. Consiglio and D. De Giovanni

generate serious losses and rolling the hedge forward, as witnessed by the Met-allgesellschaft default (Mello and Parsons 1995), may lead to unexpected severelosses.

In options theory such limitations are known as sources of market incomplete-ness, and recently, scholars have started coping with pricing and hedging options inincomplete markets. For example, Consiglio and De Giovanni (2008) adopt a super-replication model to determine the hedging portfolio of insurance policies whosefinal liabilities depend on a minimum guarantee option and a bonus distributionscheme. 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 toleave the contract before maturity. An alternative approach is proposed by Colemanet al. (2007). They address and solve a similar incomplete-market hedging problemextending the traditional Black and Scholes price process to include Merton’s jumpdiffusion process. The insurance claim is then hedged, by using the underlying assetand a set of standard options expiring before the claim’s maturity. The hedgingstrategy is here determined by applying the minimum local hedging risk principleby 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 witha short insurance position but also the exposure induced by a reinsurance agree-ment. To this aim we propose a novel approach to the asset–liability manage-ment of such contracts. A general definition of reinsurance contract is that ofa coverage purchased by an insurance company (insurer) from typically anotherinsurance company (reinsurer) to transfer an original risk exposure. The reinsur-ance agreement specifies the basis for which the reinsurer will pay the insurer’slosses (excess of loss or proportional) and the reinsurance premium paid by thereinsured.

The chapter is organized as follows. Section 6.2 introduces the reinsurance prob-lem. Section 6.3 deals with the mathematical formulation of the stochastic pro-gramming models, including: (i) the notation, (ii) the pricing of European contin-gent claims, and (iii) an efficient method to build scenario trees. The asset–liabilitymanagement 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 Propertyand Casualty Market

Broadly speaking, reinsurance is the insurance of insurance liabilities. Insurancefirms 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 rein-surance protection.

6 Pricing Reinsurance Contracts 127

Popular agreements in the reinsurance market are proportional reinsurance andaggregate excess reinsurance. With proportional contracts, the transferring partycedes to the reinsurance company a fixed proportion of the risk associated withthe portfolio, while with aggregate excess treaty, the reinsurance company promisesto pay the claims exceeding a given retention (see Straub (1997) for more details onreinsurance contracts).

We denote by LT the stochastic cash flow representing the future liabilities ofthe company, and by L0 the premium income collected. LT is the sum of all thepayments occurring during the time interval [0, T ]. A typical aggregate excess rein-surance contract is represented by the stop-loss agreements, where the reinsurerpays the part of LT which exceeds a certain amount, R. The reinsurer obligationis usually limited to a given amount, M . More precisely, the payoff of a stop-losscontract can be summarized as follows:

YT =

⎧⎪⎪⎨⎪⎪⎩

0, LT ≤ R

(LT − R) , R < LT < R + M.

M, LT ≥ 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 thetransferring parts.

Reinsurance agreements are usually evaluated either by standard actuarial meth-ods (see Straub 1997) or by equilibrium techniques, using the so-called financialreinsurance method. In the former case, the price is computed as the sum of theexpected value of the future payments and a risk premium which is usually pro-portional to the variance of the distribution of LT . In the latter case, the insuranceliability is assumed to be highly correlated to a traded asset, and the price followsby application of the capital asset pricing model of Sharpe (1964). Application offinancial reinsurance and the design of reinsurance contract have been studied inde Lange et al. (2004).

In the sequel of the chapter we describe a stochastic programming model toevaluate reinsurance contract by super-replication. This is done by recognizing thatthe contract YT is a European contingent claim (ECC) written on the non-tradedunderlying liability LT .

We also consider an asset and liability management (ALM) problem, where aninsurance firm with a future liability LT needs to determine the optimal investmentstrategy in order to maximize its expected profit. The asset universe in which thecompany 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 expectedprofit, given a set of regulatory constraints to be fulfilled. In fact, internationalaccounting standards require that insurance companies meet specific obligations,expressed in terms of risk margins which are usually proportional to the value atrisk (VaR) of the loss distribution.


6.3 A Stochastic Programming Model for Super-Replication

The problem of option pricing in incomplete markets is currently an active area ofresearch. The different methodologies proposed over the years differ on the way theoption price is defined. In what follows, we describe a superhedging method basedon a stochastic programming representation of the hedging problem. An alternativeapproach is the quadratic hedging of Föllmer and Schweizer (1989) (see also Dahland Møller 2006, Dahl et al. 2008, for recent applications in the life insurance con-text). Global risk minimization (see Cerny and Kallsen 2009, and reference therein)based on quadratic hedging is a promisingly new area of research. Utility-basedpricing 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 appli-cation to the insurance sector.

6.3.1 Notation and Probabilistic Structure

We introduce a financial market where security prices and other payments arediscrete random variables supported by a finite-dimensional probability space(Ω,F , P). The atoms ω ∈ Ω are sequences of real-valued vectors (asset valuesand payments) over the discrete time periods t = 0, 1, . . . , T . The path historiesof the security prices up to time t correspond one-to-one with nodes n ∈ Nt . Theset N0 consists of the root node n = 0 and the leaf nodes n ∈ NT correspondone-to-one with the probability atoms ω ∈ Ω . This probabilistic structure can bemodeled by a discrete, non-recombining scenario tree. In the scenario tree, everynode n ∈ Nt , t = 1, . . . , T , has a unique ancestor a(n) ∈ Nt−1, and every noden ∈ Nt , t = 0, . . . , T − 1, has a non-empty set of child nodes C(n) ⊂ Nt . Thecollection of all the nodes is denoted by N ≡ ⋃T

t=0 Nt . The information arrivalcan be modeled by associating a set of σ -algebras {Ft }t=0,...,T with F0 = {φ,�},Ft−1 ⊆ Ft , and FT = F .

We model the probability measure P by attaching weights pn > 0 to each leafnode n ∈ NT so that

∑n∈NT

pn = 1. The probability of each non-final node n ∈Nt , t �= T is recursively determined by

pn =∑

m∈ C(n)pm . (6.2)

The market consists of J + 1 tradeable securities indexed by j = 0, 1, . . . , Jone of which, say security 0, is the risk-free security. We model the risk-free assetby assuming the existence of a market for zero coupon bonds for each trading datet = 0, . . . , T − 1. At time t + 1 the risk-free value in node n, S0

n , corresponds to theprevious value S0

a(n) capitalized with the return given by the bond Bnt,t+1 in node n.

That is, S0n is a one-dimensional stochastic process defined on the finite probability


space (Ω,F , P) and Ft+1-measurable.1 This allows us to introduce in the marketmodel the stochastic dynamics of the yield curve. Other securities prices (stocks,future, etc.) are described by a nonnegative-valued vector S0 =

(S1

0 , . . . , S J0

)of

initial (known) market prices and nonnegative-valued random vectors Sn : Ω →R

J , Ft -measurable, ∀n ∈ Nt 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 account-ing standards Solvency II. In the context of this chapter, the term structure is used toinclude in the universe of investment opportunities an asset in which investors cantrade with no risk.

In presence of risk factors other than the traded securities, the process St isaugmented by J + 1 real-valued variables ξt =

(ξ1

0 , . . . , ξJt

)whose path histo-

ries match the nodes n ∈ Nt , for each t = 0, 1, 2, . . . , T . This is certainly thecase of the property and casualty market, where liabilities are generated by factorsexogenous to the financial markets (car accidents, earthquakes, etc.). In the frame-work described above, an ECC is a security whose owner is entitled to receive theFt -adapted stochastic cash flow F = {Ft }t=0,...,T . The above definition of ECCis general enough to encompass a large variety of derivatives, including futures andexotic options. Options written on non-traded underlying variables—which is one ofthe major source of market incompleteness—are also embraced. The stop-loss rein-surance contract described in (6.1) clearly falls in the class of ECCs, with Fn = Yn

for n ∈ NT and Fn = 0 for n ∈ Nt , t = 0, . . . , T − 1. Here Yn is the value ofthe reinsurance contract given as a function of the liability Ln occurred at node nand corresponds to the underlying asset of the ECC. In the property and casualtymarkets, contrary to the financial markets, one cannot trade with the underlyingliabilities. This makes the reinsurance contract difficult to hedge by trading in theunderlying, and thus we must switch to the theory of option pricing in incompletemarkets.

In the general framework of incomplete markets, contingent claims cannot beuniquely priced by no-arbitrage arguments. Rather, there exists the so-called arbi-trage interval which describes the range of all arbitrage-free evaluations. Lower andupper bounds of the arbitrage interval are determined by the buyer price and sellerprice, respectively. Informally, the buyer price is the maximum amount of moneyan investor is willing to pay in order to buy the contingent claim. Likewise, theseller price is the minimum amount of money that is needed to the writer in order tobuild 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 concentratingour attention on the seller problem only. For details about the construction of thebuyer price and the relation between the arbitrage interval and the set of equivalentmartingale measure we refer to King (2002).

1 This means that, at node n, we already know the ending-period value S0m , ∀m ∈ C(n) but, ∀n,m ∈

Nt with n �= m, S1n and S0

m could be different.


6.3.2 Super-Replication of ECCs

The reinsurer (protection seller) receives an amount, V , corresponding to the pre-mium of the reinsurance contract and she agrees to pay the protection buyer thepayoff of the reinsurance contract. The seller’s objective is to select a portfolio oftradeable assets that enables her to meet her obligations without risk (i.e., on allnodes n ∈ N ). The portfolio process must be self-financing, i.e., the amount ofassets bought has to offset the amount of assets sold. Put it differently, there areno inflows or outflows of money, since the amount of money available at node n isfunded only by price movements at the ancestor node, a(n), and at the node itself.

Using the tree notation, we denote by Z jn 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∑j=0

S j0 Z j

0 = V . (6.3)

At each node n ∈ Nt , t = 1, 2, . . . , T , the self-financing constraints are defined asfollows:

J∑j=0

S jn Z j

n =J∑

j=0

S jn Z j

a(n). (6.4)

To sum up, the stochastic programming model for the insurance evaluation problemcan be written as follows:

Problem 3.1 (Writer problem for ECCs)

MinimizeZ j

n∈V (6.5)

J∑j=0

S j0 Z j

0 = V, (6.6)

J∑j=0

S jn Z j

n =J∑

j=0

S jn Z j

a(n) for all n ∈ Nt , (6.7)t = 1, 2, . . . , T − 1,

J∑j=0

S jn Z j

n + Ln =J∑

j=0

S jn Z j

a(n) for all n ∈ NT , (6.8)

J∑j=0

S jn Z j

n ≥ 0 for all n ∈ NT , (6.9)

Z jn ∈ R for all n ∈ N , (6.10)

j = 0, 1, . . . , J.


We highlight here some important points:

1. Problem 3.1 is a linear programming model where the objective function mini-mizes the value of the hedging portfolio or rather the price of the option.

2. The payout process {Fn}n∈N is a parameter of Problem 3.1. This impliesthat any complicated structure for Fn does not change the complexity of themodel.

3. Constraints (6.9) ensure that at each final node the total position of the hedgingportfolio is not short. In other words, if short positions are allowed, the portfolioprocess must end up with enough long positions so that a positive portfolio valueis delivered.

6.3.3 Tree Generation

We generate the tree for the underlying price process S by matching the firstM moments of its unknown distribution. Our approach is based on the momentmatching method of Høyland and Wallace (2001), where the user provides a setof moments, M, of the underlying distribution (mean, variance, skewness, covari-ance, or quantiles), and then, prices and probabilities are jointly determined bysolving a non-linear system of equations or a non-linear optimization problem.The method also allows for intertemporal dependencies, such as mean reverting orvolatility clumping effect. For a review on alternative scenario generation methods,see Dupacová et al. (2000) and references therein.

The following system of non-linear equations formalizes the moment matchingproblem:

Problem 3.2 (Moment matching model)

fk (S, p) = τk ∀k ∈M, (6.11)∑m∈C(n)

pm = 1, (6.12)

pm ≥ 0, m ∈ C(n), (6.13)

where fk (S, p) is the algebraic expression for the statistical property k ∈ M andτk is its target value. For example, let us consider the problem of matching theexpected values μS0 , . . . , μS J , the variances, σ 2

S0 , . . . , σ2S J , and the correlations,

ρ j,k , j = 0, 1, . . . , J, and k �= j , of the log-returns of each asset. The momentmatching problem becomes

Problem 3.3 (Moment matching model: an example)

∑m∈C(n)

r jm pm = μS j , j = 0, . . . , J, (6.14)


∑m∈C(n)

(r j

m − μS j

)2pm = σ 2

S j , j = 0, 1, . . . , J, (6.15)

∑m∈C(n)

(r j

m − μS j

) (rk

m − μSk

)pm

σS jσSk= ρ j,k, j = 0, 1, . . . , J and k �= j,

(6.16)

ln

(S j

m

S jn

)= r j

m, j = 1, 2, . . . , n, (6.17)

∑m∈C(n)

pm = 1, (6.18)

pm ≥ 0, m ∈ C(n). (6.19)

The non-linearity of Problem 3.2 could lead to infeasibility. An alternative strat-egy is to formulate Problem 3.2 as a goal programming model. We can minimizethe weighted distance between the statistical properties of the tree, and their targetvalues, that is

Problem 3.4 (Moment matching model: goal programming)

MinimizeS,p

∑k∈M

ωk ( fk (S, p)− τk)2 , (6.20)

∑m∈C(n)

pm = 1, (6.21)

pm ≥ 0, m ∈ C(n), (6.22)

where ωk is the weight which measures the importance associated with the statisticalproperty k ∈ M. Problem 3.4 is easier to solve with respect to the previous non-linear system, but a perfect match is generally difficult to meet. A hybrid model, inwhich only some statistical properties are required to be perfectly matched, may beadopted to avoid bad match of the specifications. In detail, we split the set M ofparameters to fit in two subsets, M1 and M2, and implement the following non-linear programming problem:

Problem 3.5 (Moment matching model: mixed goal programming)

MinimizeS,p

∑k∈M1

ωk ( fk (S, p)− τk)2 , (6.23)

fi (S, p) = τi , i ∈M2, (6.24)∑m∈C(n)

pm = 1, (6.25)

pm ≥ 0, m ∈ C(n). (6.26)


Although these three different strategies can be implemented, there is no generalrule that suggests the choice of a particular one. Each of these strategies has its ownpros and cons, and the choice depends on the specific problem to be faced with.

To be consistent with financial asset pricing theory, arbitrage opportunitiesshould be avoided. In the setting described in Section 6.3, the no-arbitrage conditionis fundamental. In fact, if the market allows for arbitrages, the stochastic program-ming problem described in this chapter will end up with an unbounded solution.

Following Gulpinar et al. (2004), we can preclude arbitrages by imposing theexistence of a strictly positive martingale measure such that

S jn = e−rn

∑m∈C(n)

S jmπm, (6.27)

where rn 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 whoseobjective is to determine the optimal hedging strategy for the contract and to price itaccordingly. We now turn to the position of the buyer of the protection. The goal isto manage the liabilities she has to face by making use of both financial investmentsand the reinsurance contract. We assume that the price of the reinsurance contractis determined by the protection seller and that the company has to satisfy regulatoryconstraints expressed in terms of value at risk of the loss distribution. To be morespecific, we define the loss function

�(n) = −J∑

j=0

S jn Z j

n − xYn + Ln, n ∈ NT , (6.28)

which corresponds to the value of the portfolio after the payments for the liabil-ity Ln have been made. According to (6.28): �(n) = −Ψ (n), where Ψ (n) is theprofit realized at node n after the liability has been paid. Following Rockafellar andUryasev (2000), the conditional value at risk (CVar) of the loss distribution at theprobability level α is defined by the following set of equations:

ζ + 1

(1− α)1

|NT |∑

n∈NT

φ(n), (6.29)

φ(n) = �(n)− ζ, n ∈ NT , (6.30)

φ(n) ≥ 0, n ∈ NT . (6.31)


The endogenous variable ζ can be shown to represent the value at risk of the lossdistribution. Notice that the shortfalls in (6.30) are defined to be non-negative. Asa consequence, in order to bound ζ , an upper limit on (6.29) is sufficient. The fol-lowing linear program determines the optimal investment strategy for an insurancecompany seeking to maximize its expected profits, given that the α-CVar of the lossdistribution cannot exceed the limit ω:

Problem 4.1 (Conditional VaR model)

MaximizeZ j

n ,x,ζ

1

|NT |∑

n∈NT

Ψ (n), (6.32)

J∑j=0

S j0 Z j

0 + xY0 = L0, (6.33)

J∑j=0

S jn Z j

n xYn − Ln = Ψ (n), n ∈ NT , (6.34)

J∑j=0

S jn

(Z j

n − Z ja(n)

)= 0 for all n ∈ Nt , t = 1, 2, . . . , T, (6.35)

−∑

n∈NT

�(n)− ζ ≤ φ(n), n ∈ NT , (6.36)

ζ + 1

(1− α)1

|NT |∑

n∈NT

φ(n) ≤ ω, (6.37)

φ(n) ≥ 0, (6.38)

x, Z jn , ≥ 0. (6.39)

6.5 Implementation Notes and Results

We consider a tree structure with six stages and perform all the experiments byassuming a time horizon T = 10 years. The time step between two consecutivestages is thus fixed to 1.67 years. In the tree, each non-final node branches into fivesuccessor nodes. The resulting tree has exactly 19,531 total nodes and 56 = 15,625final nodes.

The financial market consists of three risky securities (stocks) plus a risk-freeasset. The risk-free asset has initial value 100 and is assumed to grow at the con-tinuously compounded annual rate of 3%. We generate the risky assets by usingthe tree generation model described in Problem 3.4 and by restricting the momentmatching problem to fit the expected values, standard deviations, and correlationsof the continuously compounded asset returns as displayed in Table 6.1. For all theassets, the skewness parameter and kurtosis are set, respectively, to β1 = 0 and


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 1 0.04 0.13 1 – –Asset 2 0.045 0.15 0.6 1 –Asset 3 0.054 0.2 0.45 0.24 1

We built scenario trees by matching means, standard deviations, and correlations shown in thetable. Skewness and kurtosis parameters are set, respectively, to β1 = 0 and β2 = 3

β2 = 3. This is equivalent to assume a market where asset returns are Gaussianwith parameters specified as in Table 6.1. This is a simplifying assumption thatdoes not affect the relevance of our framework. In a more general setup, we candetermine probability distributions that match higher moments (see Consiglio andDe Giovanni, forthcoming). Finally, the initial value of all the assets is set equal to100.

6.5.1 Pricing Reinsurance Contracts

Our experiments are based on four case studies (CS). More specifically, we considerfour different distributions for the random variable LT that represents the actuarialclaim the insurance company will face with. Accordingly, we generate four differentscenario trees for LT , one for each CS, using Problem 3.4. Table 6.2 displays theexpected values and the standard deviation of each CS. The skewness and kurtosisparameter are set, respectively, to β1 = 0 and β2 = 3, while the financial assets andthe actuarial claims are assumed to be uncorrelated.

Table 6.3 displays the prices of the reinsurance contract with fixed values of Rand M . The pattern shown in Table 6.3 is not surprising. The higher the expectedvalue and the variance of the liability distribution, the higher the price of the rein-surance contract. It is, however, interesting to look at the initial composition of thestrategies produced to super-replicate the reinsurance contract. Such results are dis-played in Fig. 6.1. We observe that to super-replicate the contract the reinsurer islong 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-1 180 22CS-2 190 40CS-3 195 45CS-4 200 50

We built scenario trees by matching means and standard deviations as displayed. The skewnessand kurtosis parameter are set, respectively, to β1 = 0 and β2 = 3. No correlation between thefinancial market and the actuarial risk is assumed


Table 6.3 Prices of thereinsurance contract for twodifferent levels of theparameters R and M

R = 160, M = 200 R = 140, M = 220

CS_1 21.264 36.473CS_2 49.494 64.921CS_3 59.051 74.542CS_4 64.356 79.835

–60%

–40%

–20%

0%

20%

40%

60%

Ass

et a

lloca

tio

n

CS_1 CS_2 CS_3 CS_4

RF Asset_1 Asset_2 Asset_3

–60%

–40%

–20%

0%

20%

40%

60%

80%

CS_1 CS_2 CS_3 CS_4

Ass

et a

lloca

tio

n

RF Asset_1 Asset_2 Asset_3

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 LT ) and the risky assets areuncorrelated. This prevents the reinsurer to exploit the co-movements between thetraded 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 toshort-sell in the super-replicating strategy is much higher than that of asset 3 which


corresponds to the most risky asset in the market. The latter comes into play whenthe riskiness of the liability exceeds a certain level that in this experiment is givenby CS_3 and CS_4.

6.5.2 Risk Management with Reinsurance Contracts

We use the liability structure based on the simulation CS_1 in Table 6.1, and setthe initial premium collected by the insurance company to L0 = 145. Two differentparametrizations of the reinsurance contract are considered with (1) R = 160 andM = 200 and (2) R = 140 and M = 220. The prices of the two contracts are setto their super-replication values: Y0 = 21.264 and Y0 = 36.473, respectively. Wethen run Problem 4.1 with increasing levels of the risk tolerance, from ω = 0 toω = 3. The resulting efficient frontiers are displayed in Fig. 6.2, while the optimalportfolios for each level of tolerance are displayed in Fig. 6.3. Note that the problemallows for a feasible optimal solution when ω = 0. This is because the reinsurancecontract completely offset any possible loss, thus eliminating any risk.

From the experiment we learn some important facts. First, the optimal portfo-lios 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-replicationprices. 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 theinvestors are willing to trade in the derivative. Second, the amount of reinsurancedeclines with the increase of the level of riskiness allowed ω. This is a confirmationof the rational choice of the model which reduces the level of the reinsurance if therisk exposure is increased by the decision maker. We strongly believe this could be

70

80

90

100

110

120

130

140

150

160

170

0 0.5 1 1.5 2 2.5 3 3.5Conditional VaR

Exp

ecte

d p

rofi

t

R = 160 M = 200

R = 140 M = 220

Fig. 6.2 Efficient frontiers. Conditional VaR of the losses vs expected profit of the portfolio forconfidence level α = 99%. The efficient frontiers start at ω = 0 as the losses can be totally offsetby purchasing a reinsurance contract


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Conditional VaR

Ass

et a

lloca

tio

n

x RF AA_1 AA_2 AA_3

0 0.6 1.2 1.8 2.4 3

x RF AA_1 AA_2 AA_3

0%10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0.6 1.2 1.8 2.4 3Conditional VaR

Ass

et a

lloca

tio

n

Fig. 6.3 Optimal portfolios for R = 160, M = 200 (top panel) and R = 140, M = 220 (bottompanel)

a useful tool for regulators which can ascertain whether a given level of reinsuranceis safe for the liability structure of a company, and in case of negative result forcesthe company toward a more consistent level of protection.

6.6 Conclusions

We propose in this chapter a stochastic programming model to cope with the priceand management of reinsurance contracts. We show that the pricing in incom-plete 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 non-tradeability of the underlying asset. The model is flexible enough to be embedded


in a wider asset–liability model to determine the optimal level of trade-off betweenexpected profit and conditional VaR.

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chapter6 pricing reinsurance contracts · reinsurance agreements are usually tailored to meet...

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