# «Abstract Optimal contracts have widely been studied in the literature, yet the bargaining for optimal prices has remained relatively unexplored. ...»

Price of Reinsurance Bargaining with

Monetary Utility Functions

Tim J. Boonen∗, Ken Seng Tan†, Sheng Chao Zhuang†

July 29, 2015

Abstract

Optimal contracts have widely been studied in the literature, yet the

bargaining for optimal prices has remained relatively unexplored. Therefore

the key objective of this paper is to analyze the price of reinsurance con-

tracts. We use a novel way to model the bargaining powers of the insurer

and reinsurer, which allows us to generalize the contracts according to the Nash bargaining solution, full competition, and the equilibrium contracts.

We illustrate these pricing functions by means of inverse-S shaped distortion functions of the insurer and the Value-at-Risk for the reinsurer.

1 Introduction This paper analyzes optimal reinsurance design and its pricing when ﬁrms are endowed with monetary utility functions. Broadly speaking there are two recent streams of literature that consider risk sharing with monetary utility functions.

Both streams study roughly the same objective function in mathematical terms, but with diﬀerent motivations. First, several authors study optimal risk sharing and Pareto equilibria (see, e.g., Filipovi´ and Kupper, 2008; Jouini et al., 2008;

c Ludkovski and Young, 2009; Boonen, 2015). Second there is a stream in the literature that studies optimal (re)insurance contract design with a given premium principle analogue to a monetary utility function (see, e.g., Asimit et al., 2013; Cui et al., 2013; Chi and Meng, 2014; Assa, 2015; Boonen et al., 2015; Cheung and Lo, 2015). This paper combines both settings in the sense that we use a bargaining ∗ Department of Quantitative Economics/Actuarial Science, University of Amsterdam, Valck- enierstraat 65-67, 1098 XE Amsterdam, The Netherlands † Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 1 approach for optimal risk sharing contracts in the context of optimal reinsurance contract design. To the best of our knowledge, we are the ﬁrst to explicitly combine both streams of literature.

Pricing of insurance and reinsurance contracts is typically done by assuming full competition. Then, the price is set such that the reinsurer or insurer is indiﬀerent to selling the contract or not. In this way, one determines the zero-utility premium.

Moreover, there is a stream in the literature that focuses on empirical data on insurance prices, and try to derive the implied pricing functions. The problem with such approach is that the number of transactions in reinsurance is typically limited. Our approach is diﬀerent from both approaches. We determine the prices via a cooperative bargaining process.

Kihlstrom and Roth (1982), Schlesinger (1984), and Quiggin and Chambers (2009) all use the Nash bargaining solution for an insurance contract between a client and an insurer. Moreover, Aase (2009) uses the Nash bargaining problem to price reinsurance risk as well. Speciﬁcally for longevity risk, Boonen et al. (2012) and Zhou et al. (2012) use Nash bargaining solutions to price longevity-linked Over-The-Counter contracts. All these authors focus on ﬁrms that maximize Von Neumann-Morgenstern expected utility. In contrast, we use a cooperative bargaining approach to derive optimal reinsurance contracts and their corresponding prices via the Nash bargaining solutions. Moreover, we let ﬁrms maximize a monetary utility function. We provide a unique mechanism that allows us to generalize optimal contracts even if there is asymmetric bargaining power such as for the asymmetric Nash bargaining solution (Kalai, 1977). This mechanism allows us to include full competition as well, which leads to the extreme case that one ﬁrm is indiﬀerent from trading, and the other ﬁrm gains maximally. This assumption is popular in the economic and actuarial literature, dating back from the concept of Bertrand equilibria (Bertrand, 1883).

This paper contributes to the literature in the following ways. We characterize the optimal hedge beneﬁts (alternative interpreted as welfare gains) from bilateral bargaining for reinsurance. In the special case in which the preferences are given by a distortion risk measure, we derive a simple expression of the hedge beneﬁts.

Moreover, we derive bounds on the individual rational prices of a speciﬁc Pareto optimal contract, and provide to any price a corresponding bargaining power for the asymmetric Nash bargaining solution. To highlight our results, we illustrate the construction of the premium principle under the special case that the insurer is endowed with preferences given by an inverse-S shaped distortion risk measure, and the reinsurer optimizes a trade-oﬀ between the expected value and the Value-at-Risk (VaR). This leads to a discontinuous pricing function. Inverse-S shaped distortion risk measures are getting more popular to use as preferences since Quiggin (1982, 1991, 1992) and Tversky and Kahneman (1992).

2 This paper is set out as follows. Section 2 provides all general results on bargaining with monetary utility functions. Section 3 shows how there results translate to preferences given by a distortion risk measure. Section 4 provides important insight on the premium principle for the class of inverse-S shaped distorted preferences and the well-known Value-at-Risk.

2 Model formulation We consider a one-period model involving two ﬁrms, with one ﬁrm representing an insurer (I) and the other ﬁrm representing the reinsurer (R). Let (Ω, F, P) be a probability space, and L∞ (Ω, F, P) be the class of bounded random variables on it. When there is no confusion, we simply write L∞ = L∞ (Ω, F, P). The total insurance liabilities that the insurer faces is given by the non-negative, bounded risk X ∈ L∞. Here we assume that the insurer is interested in transferring a part of this risk to a reinsurer. Let us denote M = esssup X = inf{a ∈ R : P(X a) = 0}.

The reinsurance contract is given by the tuple (f, π), where f (X) is the indemnity paid by the reinsurer to the insurer and π ≥ 0 is the price (or premium) paid by the insurer to the reinsurer. It is natural to assume that f ∈ F, where F = {f : IR+ → IR+ |0 ≤ f (x) − f (y) ≤ x − y, ∀x ≥ y ≥ 0, f (0) = 0 }, i.e., we assume that the reinsurance contract f ∈ F is 1-Lipschitz. 1-Lipschitz contracts account for moral hazard (Bernard and Tian, 2009), and is often used in the literature on reinsurance contract design (see, e.g., Young, 1999; Asimit et al., 2013; Chi and Meng, 2014; Assa, 2015; Xu et al., 2015).

By denoting Wk as the deterministic initial wealth for ﬁrm k, where k ∈ {I, R}, and πI as the premium received by the insurer for accepting risk X, then without the reinsurance the wealth at a pre-determined future time for the insurer and reinsurer are WI + πI − X and WR, respectively. If the insurer were to transfer part of its risk to a reinsurer using f (X) with corresponding price π, then the wealth at a pre-determined future time for the insurer becomes

To assess if there should be a risk transfer between both ﬁrms, we need to make additional assumption on how ﬁrms evaluate such preference. In particular, we assume that ﬁrm k uses a monetary utility function Vk. A preference relation Vk is monetary if it is monotone with respect to the order of L∞, satisﬁes the 3 normalization condition Vk (0) = 0 and has the cash-invariance property Vk (X + a) = Vk (X) + a for every X ∈ L∞ and a ∈ IR. Moreover, we assume that Vk is comonotonic additive, i.e., if X and Y are comonotonic, then Vk (X + Y ) = Vk (X)+Vk (Y ). This implies that the Choquet representation of Schmeidler (1986) can be used to formalize Vk. It is well-known that the initial wealth and πI are irrelevant for preferences given by monetary utility functions.

Pareto optimal reinsurance contracts are such that there does not exist another contract that is weakly better for both ﬁrms, and strictly better for at least one ﬁrm. Since we consider 1-Lipschitz reinsurance contracts, we have that −X + f (X) and −f (X) are comonotonic for all f ∈ F. Furthermore, the comonotonic additivity of Vk implies that the utility function Vk is additive (and hence concave) on the domain F.

In risk sharing, the problem of ﬁnding an f ∈ F and a price π ∈ IR is analogous to the problem of ﬁnding comonotonic risk sharing contracts. If there is no constraints on π, it is shown by Jouini et al. (2008) that all Pareto optimal reinsurance contracts are obtained by max VI (−X + f (X)) + VR (−f (X)). (3) f ∈F Note that Wk, k ∈ {I, R}, πI and π do not appear in the above objective function due to the cash invariance property of Vk, k ∈ {I, R}; the π’s cancel out each other.

The non-negativity constraint inﬂuences the Pareto optimal set only for contracts with VR (−f (X) + π) VR (0), which holds true for all π 0 and f ∈ F. The following asserts the existence of the Pareto optimal reinsurance contract.

Proposition 2.1 Under the assumption that f ∈ F and Vk ∞, k ∈ {I, R} are monetary utility function, then there exists a Pareto optimal reinsurance contract f ∗ ; i.e. f ∗ is the optimal solution to (3).

Proof By deﬁning the norm d(f 1, f 2 ) = maxt∈[0,M ] | f 1 (t) − f 2 (t) |, for any f 1, f 2 ∈ F, then the set F is compact under this norm d. Moreover, we have Vk (X) ∞ for k ∈ {I, R}. Hence, an optimal solution to (3) exists.

We now consider the beneﬁts of risk sharing to both ﬁrms. Recall that without risk sharing, the utility of the insurer for insuring risk X is VI (WI + πI − X) and the utility of the reinsurer is simply VR (WR ). If both ﬁrms agree to a risk sharing indemnity function f (X) with corresponding price π, then the resulting utility of the insurer changes to VI (WI + πI − X + f (X) − π) so that the diﬀerence

4 can be interpreted as the hedge beneﬁt to the insurer using the risk sharing strategy f ∈ F. The right hand side of the above equation follows from comonotonic additivity and cash invariance of VI. Similarly, from the perspective of the reinsurer its hedge beneﬁt can easily be shown to be

Positive diﬀerence implies that there is an incentive with the risk sharing due to the gain in monetary utility. By denoting HB(f ) as the aggregate hedge beneﬁts or the aggregate utility gains in the market for exercising the risk sharing strategy f ∈ F, then we have

Note that HB(f ) is simply the sum of the hedge beneﬁt of both insurer and reinsurer and hence for brevity we refer to HB(f ) as the (aggregate) hedge beneﬁt for a given risk sharing f ∈ F. Note also that HB(f ) can be positive, negative, or zero, depending on f (X) and the heterogeneous preferences of insurer and reinsurer. Since the utility functions are monetary, the hedge beneﬁt HB(f ) is expressed in monetary terms as well.

If the risk sharing strategy corresponds to a Pareto optimal f ∗, then the maximum achievable hedge beneﬁt of the market is given by HB ∗ ≡ HB(f ∗); i.e.

HB ∗ = HB(f ∗ ) = VR (−f ∗ (X)) − VI (−f ∗ (X)) ≥ 0. (6) The inequality follows immediately from the fact that f (X) = 0 is a feasible strategy in F. Also if VI = VR, then the comonotonic additivity of Vk leads to HB ∗ = 0; i.e. there is no gain in welfare in the market regardless of the risk sharing f ∈ F.

Depending on the market conditions, the hedge beneﬁt HB ∗ will be shared among both ﬁrms. Particularly, we require that the following two conditions are

**satisﬁed:**

• Pareto optimality,

• individual rationality, or both ﬁrms are better oﬀ than when they do not trade: VI (−X + f (X) − π) ≥ VI (−X) and VR (−f (X) + π) ≥ 0.

Recall that the Pareto optimal f ∗ does not depend on the price π, the following proposition establishes the lower and upper bounds of the individual rational price corresponding to f ∗. The key to deriving these bounds is based on the minimum acceptable price that a reinsurer is willing to accept the risk from an insurer and the maximum price that an insurer is willing pay to transfer its risk to a reinsurer.

5 Proposition 2.2 The set of Pareto optimal and individual rational prices is given by the interval [−VR (−f ∗ (X)), −VI (−f ∗ (X))], where f ∗ is a solution of (3).

Proof For any π ≥ 0, the solution f ∗ is optimal (see Proposition 2.1). Note that due to cash-invariance of V, we get that VI (−X + f ∗ (X) − π) is strictly decreasing and continuous in π, and VR (−f ∗ (X) + π) is strictly increasing and continuous in π. Hence, the set of individual rational pricing is given by an interval, where the lower bound is such that VR (−f ∗ (X) + π) = VR (0), and the upper bound such that VI (−X + f ∗ (X) − π) = VI (−X). The lower bound follows directly from cash-invariance and VR (0) = 0, and the upper bound follows directly from cashinvariance, comonotonic additivity, and the fact that −X + f ∗ (X) and −f ∗ (X) are comonotonic. Finally, −VR (−f ∗ (X)) ≤ −VI (−f ∗ (X) follows from (6). This concludes the proof.

Suppose now for a given indemnity function f, we deﬁne α ∈ [0, 1] as the proportion of the hedge beneﬁt that is allocated to the insurer; i.e. αHB(f ) hedge beneﬁt is assigned to the insurer and the remaining (1 − α)HB(f ) hedge beneﬁt to the reinsurer. Corresponding to the indemnity function f and the hedge beneﬁt allocation α, it is of interest to determine the resulting price of the reinsurance contract. To do this, it is useful to interpret the price π as a function of both f and α so that π ≡ π(α, f ) represents the price of a reinsurance contract f (X) with the insurer receives αHB(f ) hedge beneﬁt and reinsurer receives the remaining (1 − α)HB(f ) hedge beneﬁt. Note that for a given allocation α the posterior utility of the insurer and the reinsurer are given by VI (WI + πI − X) + αHB(f ) and VR (WR ) + (1 − α)HB(f ), respectively.

For a given f ∈ F, the function VI (WI + πI − X + f (X) − π) is continuous and strictly decreasing in π, and the function VR (WR − f (X) + π) is continuous and strictly increasing in π. Therefore, we get that the pricing function for a given

**f ∈ F and α can be deﬁned as the solution to the following optimization problem:**