metadata: authors: 0: author: Bibas, Ruben
1: author: Hallegatte,Stephane
2: author: Lemoyne de Forges, Sabine
count: World
docna: 0: docna: A dynamic model of extreme risk coverage :
            resilience and efficiency in the global reinsurance market
docty: Policy Research Working Paper
subtopic: Science of Climate Change,Climate Change and Health,Economic Theory & Research,Social Policy,Legal Products,International Law,Judicial System Reform,Industrial Economics,Climate Change and Environment,International Trade and Trade Rules,Economic Growth,Legal Reform,Treaties,Legislation,Regulatory Regimes,Natural Disasters
teratopic: Private Sector Development,Macroeconomics and Economic Growth,Finance and Financial Sector Development
abstracts: cdata!: This paper presents a dynamic model of
            the reinsurance market for catastrophe risks. The model is
            based on the classical capacity-constraint assumption.
            Reinsurers choose every year the quantity of risk they cover
            and the level of external capital they raise to cover these
            risks. The model exhibits time dependency and reproduces a
            market dynamics that shares many features with the real
            market. In particular, market price increases and
            reinsurance coverage decreases after large shocks, and a
            series of smaller losses may have a deeper impact than one
            larger loss. There is a significant oligopoly effect
            reducing reinsurance supply, and the market is segregated
            into strategic large actors that influence market prices and
            price-taker smaller firms. A regulation trade-off between
            market efficiency and resilience is identified and
            quantified: improving the ability of the market to cope with
            exceptional events increases the cost of reinsurance. This
            model provides an interesting basis to analyze further
            capacity needs for the insurance industry in view of growing
            worldwide exposure to catastrophic risks and climate change.
repnme: repnme: A dynamic model of extreme risk coverage :
            resilience and efficiency in the global reinsurance market
display_title: A dynamic model of extreme risk
            coverage : resilience and efficiency in the global
            reinsurance market
listing_relative_url: /research/2011/09/15112575/dynamic-model-extreme-risk-coverage-resilience-efficiency-global-reinsurance-market
topicv3: Economic Growth m1326241 471,Climate Change m1326829 567,Trade m1326924 698,Treaties m1327063 1026,Earthquake m1503308 3154
publishtoextweb_dt: 2011-09-01T00:00:00Z
historic_topic: Private Sector Development,Macroeconomics and Economic Growth,Finance and Financial Sector Development
no_of_pages: 30
author_keys: 0: author_key: 1103816
model_extractions: WPS5807 Policy Research Working Paper 5807 A Dynamic Model of Extreme Risk Coverage Resilience and Efficiency in the Global Reinsurance Market _Sabine Lemoyne de Forges Ruben Bibas Stéphane Hallegatte_ The World Bank Sustainable Development Network Office of the Chief Economist September 2011
Policy Research Working Paper 5807

# **Abstract**

This paper presents a dynamic model of the reinsurance market for catastrophe risks. The model is based on the classical capacity-constraint assumption. Reinsurers choose every year the quantity of risk they cover and the level of external capital they raise to cover these risks. The model exhibits time dependency and reproduces a market dynamics that shares many features with the real market. In particular, market price increases and reinsurance coverage decreases after large shocks, and a series of smaller losses may have a deeper impact than one larger loss. There is a significant oligopoly effect reducing reinsurance supply, and the market is segregated into strategic large actors that influence market prices and price-taker smaller firms. A regulation trade-off between market efficiency and resilience is identified and quantified: improving the ability of the market to cope with exceptional events increases the cost of reinsurance. This model provides an interesting basis to analyze further capacity needs for the insurance industry in view of growing worldwide exposure to catastrophic risks and climate change.

This paper is a product of the Office of the Chief Economist, Sustainable Development Network. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at hallegatte@centre-cired.fr.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
A dynamic model of extreme risk coverage: Resilience and eciency in the global reinsurance market Sabine Lemoyne de Forges<sup>_a,b_</sup> , Ruben Bibas<sup>_c_</sup> , Stéphane Hallegatte<sup>_d,e,∗_</sup>

> _a_ Ecole Polytechnique, Département d'économie, Palaiseau, France.

> _b_ AgroParisTech ENGREF, Paris, France.

> _c_ Centre International de Recherche sur l'Environnement et le Développement, Nogent-sur-Marne, France.

> _d_ The World Bank, Sustainable Development Network, Washington, D.C., USA.

> _e_ Météo-France, Toulouse, France.

> _∗_ Corresponding author: hallegatte@centre-cired.fr.

Keywords: regulation, reinsurance, dynamic model, natural disasters JEL classication: G28, G22, C63, Q54 1
# 1 Introduction

Over the last 50 years, insured losses caused by natural disasters have followed an increasing trend.<sup>1</sup> Population increase and economic growth on the one hand, the development of insurance markets on the other hand, have expanded the global insurance industry. In the future, continued growth and development in at-risk zones and possible changes in climate are likely to maintain or amplify this trend. This evolution would represent a signicant challenge for the insurance industry (Herweijer et al., 2009). Reinsurance rms are particularly exposed to these issues as they supply disaster coverage to insurance companies for hurricanes, windstorms or earthquakes, among others.

The reinsurance market resilience is a concern for the insurance sector. In particular, insurers need to assess the ability of the reinsurance sector to absorb large shocks. More specically, they need to know how important losses impact the reinsurance market capacity, reinsurance prices and the time needed for the sector to recover.<sup>2</sup> From a public policy perspective, one may ask if a modied solvency regulation increases the system resilience, and how it impacts the market. Resilience is a key issue in the nance industry in general. It has been raised again following the nancial crisis, as the ability of the nancial markets to withstand shocks has been questioned. The economic literature has examined the question of regulations for nancial industries and especially insurance industry (see Plantin and Rochet (2007) for a review). Our study focuses on the specic case of the robustness of the reinsurance market to large shocks but its conclusions are of interest for the nancial sector in general.

In this paper, we analyze the impact of important catastrophic losses on the reinsurance market, using a dynamic model of the reinsurance sector based on a solvency constraint for reinsurance rms. We thus enrich Winter (1994) rst dynamic model of the competitive insurance market. This solvency constraint is represented by a bounded probability of default, and the model is related to the classical capacity constraint hypothesis. Furthermore, we take into account the strategic eects of rms individual behavior on the market equilibrium, completing the traditional competitive view of the insurance industry (Hardelin and Lemoyne de Forges, 2009).

Model results show a market dynamics that reproduces many real-world observations, suggesting that the capacity-constraint hypothesis is able to explain the observed variability in reinsurance prices and quantities, provided that strategic behavior is taken into account. We nd path-dependency, i.e. the fact that the consequence of a one shock depends on previous years losses and reinsurers response. We characterize and quantify the resilience of the sector through the time necessary to restore reinsurance capacity after large disasters. We provide a sensitivity analysis of the key parameters including higher loss levels and series of losses. We show that improving the ability of the reinsurance market to cope with large losses and reducing reinsurer default probability that is consolidating reinsurance rms with regard to risk has a cost, due to its negative impact on market prices and capacity. Thus we exhibit a regulation trade-o between market resilience and eciency.

The paper is organized as follows: Section 2 briey describes the main characteristics of the reinsurance market for natural catastrophe risks. Section 3 presents the underlying economic

> 1See Munich Re's long-term statistics: http://www.munichre.com/en/ts/geo\_risks/ natcatservice/long-term\_statistics\_since\_1950/default.aspx

> 2Capacity refers to the term used by Cummins et al. (2002) and dened as follows: For any losses for which the reinsurance companies are liable, the capacity of the reinsurance market is the proportion of the liabilities that is deliverable to their customers, given the nancial resources of the companies, and all their risk management arrangements (retrocession, catastrophe bonds).

2
model and its main assumptions and limits. Section 4 describes the modeling approach and details the data used for the calibration. Section 5 proposes a reference scenario and compares it with the stylized facts described in the literature. Section 6 provides a sensitivity analysis on model parameters. Section 7 discusses the regulation trade-o beween eciency and resilience. Section 8 concludes.

# 2 The reinsurance market for natural catastrophes - An overview

The reinsurance market is a relatively small market compared to the insurance industry. Its capitalization, all lines included, was estimated at the end of 2008 at $309 bn (Aon-Beneld, 2009). As underlined by Plantin (2006), the demand for reinsurance can be explained from two dierent points of view. Reinsurance can be used by insurance companies as a risk management tool (Borch, 1962; Blazenko, 1986; Lewis and Murdock, 1996; Froot, 2001; Froot and O'Connell, 2008) or from a capital structure perspective (Doherty and Tinic, 1981; Mayers and Smith, 1990; Garven and Tennant, 2003). In the specic case of natural disasters insurance, the role of reinsurance is actually threefold: (i) reinsurers provide insurance companies with additional capacity, thereby making it possible for them to increase insurance supply; (ii) reinsurance allows for a worldwide mutualization of losses, thus reducing the cost of risks; and (iii) reinsurers provide an expertise on the risks supported by insurers. This third aspect is particularly important for insurance companies of medium and small sizes, which do not always have access to sucient knowledge on disasters risks.

The reinsurance market for natural catastrophe risks is a peculiar market as described in Froot (2001) and Cummins and Trainar (2009). Contracts, referred to as treaties, are passed between a ceding insurer, called the cedant, and the reinsurance company. They determine the covered portfolio, the underlying risk and the level of coverage. For natural catastrophes coverage, the most common contracts are excess-of-loss treaties based on non-proportional reinsurance.<sup>3</sup> The cedants transfer a layer of the risk from a dened portfolio to the reinsurer at a contractually dened price. A layer is dened by a deductible (the risk remaining at the charge of the insurance companies) and a limit (that is the maximum indemnity that can be paid by the reinsurance companies). For example, an insurer gets coverage for a property portfolio exposed to hurricanes with a treaty of $20 million in excess of $5 million. In this case, if an event (dened in the contract) occurs causing a loss of $12 million, $7 million will be paid by the reinsurer. If an event causes losses amounting to more than $25 million, the reinsurer only pays $20 million to the cedant. Typically, a cedant builds a reinsurance program, dividing coverage among dierent reinsurers even for the same layer of the risk, thus diversifying its exposure to each reinsurer, and taking into account its quality (for instance its default risk, its claims management system, etc.). Note that there is no standardized market for such risks, and data on these contracts are dicult to obtain.

To model the ability of the reinsurance industry to withstand large disasters, a good understanding of its specicities is necessary. Numerous contributions see a limited systemic risk from disasters in the sector, as mentioned by the reports of The Group of Thirty (2005) and The Geneva Association Systemic Risk Working Group (2010). But even if capital depletion in the reinsurance industry does not increase the risk of default signicantly, it has however an important impact

> 3Other types of reinsurance contracts are described in Cummins and Trainar (2009).

3
on reinsurance prices (Guy Carpenter, 2009) and thus on the capacity of insurance companies to maintain the same level of coverage. As a consequence, it impacts signicantly insurance prices and the wider economy.

Several market characteristics have to be taken into account to understand better this mechanism which is complicated by the entangled aspect of the reinsurance market and its opacity. First, the market has a small number of participants who share a lot of business risks (exposition to large disasters, asset risks). Second, the reinsurers may themselves use retrocession, i.e. they transfer part of their risk to other companies.<sup>4</sup> Third, the reinsurance market is characterized by several market imperfections that explain the limited use of reinsurance. Indeed, the market volume of the global reinsurance market is quite low compared to the insurance one, and the price of reinsurance has been found quite high, up to several times the actuarial price (the expected loss of the underlying risk's distribution). Market imperfections, as listed by Froot (2001), include important frictional costs linked to the illiquidity of reinsurance treaties; moral hazard (Bohn and Hall, 1999; Doherty and Smetters, 2005); adverse selection (Cutler and Zeckhauser, 1999; Jean-Baptiste and Santomero, 2000); interventions on the reinsurance market by third-parties, from disaster reliefs of all forms to state guaranty funds (Bohn and Hall, 1999); and agency issues. However, K. Froot underlines two aspects of the market that are particularly important: (i) capital market imperfections that lead to capacity shortage; and (ii) reinsurance market power. Our modeling exercise is built on these two mechanisms.

Another characteristic of the industry is the time-variability of prices and capacities, which is often referred to as a cycle (Meier, 2006). A large literature exists on the reinsurance cycle, on which a review has been conducted by Weiss (2007). It provides a wider theoretical analysis of the dynamics of insurance and reinsurance markets. Thus, reinsurance prices are high during hard market and low during soft markets. Even if this denomination of cycle is debated (Kessler, 2005), the special role of disasters is noted as they may induce an important depletion of capacity (Weiss, 2007). This hypothesis is reinforced by the observation of emerging transitory means for reinsurance companies to gain access to additional capital after large shocks, like catastrophe bonds or newer vehicles (Lane, 2007a; Cummins, 2008), or the emergence of new reinsurance companies as the Bermudians (Lane, 2007b).<sup>5</sup> Figure 1 represents the evolution of natural catastrophe reinsurance price index (given by the rate-on-line) as well as new capital ows into the natural catastrophe reinsurance industry.<sup>6</sup> A distinction is made between equity and IPO (Initial Public Oering) in dark grey on the graph and Insurance Linked Securities (ILS) as catastrophe bonds and sidecars in light grey on the graph. The impact of Andrew (1992), the World-Trade Center's attack (2001) and Katrina (2005) can be observed. Following these losses, the price of natural catastrophe reinsurance increases. There was also an increase of the ow in external capital to the market: $10 bn in 1992/93, $16 bn in 2001/02, and more than $35 bn in 2005/06 for the whole industry (Guy Carpenter, 2009). The development of the ILS market can be seen on the graph

> 4Retrocession qualies the cession of risk by reinsurance companies to retrocession companies (i.e. other companies devoted to retrocession or other reinsurance companies). The Group of Thirty (2005) estimate its volume at 10 to 15% of the reinsurance risk.

> 5Risk-linked securities enable insurance or reinsurance companies to transfer a share of their risks to the capital market. Catastrophe bonds are dened by Cummins (2008) as a fully collateralized instrument that pays o on the occurrence of a dened catastrophic event , side cars are.

> 6The rate-on-line corresponds to the reinsurance limit (that is the maximum indemnity that can be paid by the reinsurance company) of a contract divided by the premium paid. It is an imperfect proxy for reinsurance prices, but it is widely used in the industry.

4
<!-- Start of picture text --> 450 20<br>400 18<br>ee<br>=<br>8 300<br>a / 12<br>3ww 250 a —/ i q<br>4wo 200 il aia<br>a 8<br>Wl<br>a iso<br>o&io) 100 | :<br>50 : 3<br>u<br>: i<br>1990 1991 1992 1995 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008<br>GEM New capital flows (Equity and IPO) New Capital flows (Cat Bonds and Sidecars) ===Nat cat reinsurance rates<br><!-- End of picture text -->
# 3 The model

Our approach is similar to that of Dynamic Stochastic General Equilibrium (DSGE) models. It is based on a succession of Nash equilibriums on the reinsurance market in which market conditions depend on past losses and reinsurers' actions. Our approach shares common features with Winter (1994) model of the dynamics of competitive insurance markets, especially the model recursive form. But we extend this approach considering a solvency constraint (and no limited liability), an oligopolistic market, and by providing long-term simulation of the market variables.

## 3.1 Reinsurance demand

Modeling the process of risk sharing between several insurers and reinsurers is dicult. We propose here a simple approach: we only consider one aggregate demand for reinsurance. This demand consists of one catastrophe layer to be covered by the whole market. Each reinsurance company provides supply for the coverage of a fraction of this layer. These assumptions, even if they are restrictive, provide a tractable way to understand the reinsurance market. This is a proxy since it does not distinguish between dierent disasters and aected regions, and between risk layers.

We suppose that market demand is characterized by _d_ ( _p_ ) , the share of the insured underlying risk covered by the reinsurance market, in other words the fraction of the insured losses that are transferred to reinsurers. This coinsurance factor only depends on the market reinsurance price _p_ : For notational simplicity, the subscript _t_ , which refers to time, is omitted in all equations. The parameter _δ_ characterizes the elasticity of demand to market price. The parameter _d_ 0 is such that for an actuary fair premium _p_<sup>_a_</sup> , _d_ ( _p_<sup>_a_</sup> ) = 1 and all insured losses are reinsured. The function _d_ ( _p_ ) is decreasing and convex in _p_ .

## 3.2 Reinsurance supply

We consider a market with _NR_ reinsurance rms, indexed by _r_ = 1 _, .., NR_ , that produce equivalent goods _Or_ . The variable _Or_ is the amount of reinsurance coverage (in terms of coinsurance factor) that is supplied to a unique global insurance market, i.e. to a continuum of insurance companies. For example, in presence of insured losses _L_<sup> ̃</sup> , reinsurer _r_ will have to pay _OrL_<sup> ̃</sup> to its customer insurers when losses are distributed homogeneously among reinsurers (see Section 3.4 for details).

At the beginning of the period, each reinsurer has an initial wealth _Wr_ that arises from the capital issued in the past and accumulated surplus. This capital is costly to hold, even if it corresponds to the accumulation of retained earnings.

At each period, the reinsurance company chooses _Or_ . The reinsurer can also acquire some external capital _Er ≥_ 0 , if this additional wealth allows it to supply more coverage and to increase the premiums it receives in such a way that it increases its prots in spite of higher capital costs. This new external capital is costly to raise on short notice (Froot and Stein, 1998). Also, the reinsurer can reduce its wealth by _Er <_ 0 , through purchases of its own shares to shareholders. This is the case if the cost of capital is larger than benets that can be derived from it in the 6
reinsurance business.

Our model relies on a strong assumption of perfect information. Each reinsurer knows the loss distribution perfectly. Then, when it decides the risk amount it covers, the external capital acquisition and given an initial wealth, it is able to compute a nal wealth for all possible losses and quantify its bankruptcy risk.

The probability of default _πD_ of a given reinsurer is a key variable in our dynamic model. In a nutshell, it represents the quantied risk that this reinsurer goes bankrupt, i.e. the risk that the realized loss at this year added to all costs outweighs the sum of the capital and earned premiums. We then make the following assumption: (A1) Reinsurers limit their default probability under an exogenous below a limit _πlim_ that corresponds to an exogenous solvency constraint, linked to regulatory requirements.

# Reinsurer's program

We model the reinsurance companies from a managerial perspective (Shleifer and Vishny, 1997). We assume, therefore, that all reinsurers aim at maximizing the rm total prots rather than maximizing the return on equity. Prots depend on the quantity of risk subscribed by the rm and its price, the level of capital raised or bought back, and the initial wealth of the rm. Raising additional capital allows for a decrease of the reinsurer default risk, and thus larger reinsurance supply. The following equation denes expected prots for one period of time (subscript _t_ is omitted).

where _pOr_ is the reinsurer revenue from premium, and E _L_<sup> ̃</sup> _r_ is the expected loss reinsured by reinsurer _r_ (see Section 3.4 for details). Earned premium and capital (new external and initial wealth) are invested at the risk-free rate _α_ . Furthermore, there is a dead weight cost of raising additional capital ( _Er >_ 0 ): We assume that there is no cost for reducing the amount of capital through share purchase programs and special dividend. The term _Sr_ ( _Er_ + _Wr_ ) corresponds to the cost of carrying capital, where _Sr_ is the remuneration of capital asked by reinsurer shareholders. The dierence ( _Sr − α_ ) is the corresponding risk premium. The model does not distinguish between the long-term cost of internal and external capital.

A complete treatment of the reinsurers program at time _t_ would involve the denition of an expected actualized value function integrating the expected actualized prots in following periods.<sup>7</sup> Here, we propose a simple approach where at each period _t_ , the reinsurance companies maximize their expected prots over one year, taking into account their solvency constraints. Moreover, a limit is introduced on share repurchase and special dividend distribution.

> 7Indeed, the program could be written in a form recalling Bellman's equations. The problem could then be broken apart following Bellman's principle of optimality, as the choice for ( _Or_<sup>_t, E_</sup> _r_<sup>_t_)atperiod</sup><sup>_t_canbeseparated</sup> from all choices for further periods. Such a dynamic model is very dicult to solve due to the recursive term of the equation. Most of the time, it is resolved through linearization or at the stationary mode. However, our situation can not be satised with such practices as (1) the equations of the valuation of the rm are quite complicated and cannot be easily linearized, and (2) our interest resides in the analysis of the non stationary (non linear) dynamics after great shocks that can deplete the rms from a large share of their wealth.

7
Indeed, two situations can be encountered at time _t_ :

- The reinsurance company is under-capitalized and thus is in a situation where the solvency constraint is saturated. In such case, we consider that short term considerations overcome the inter-temporal issues, and the optimization is done over one year only.

- The reinsurance company is overcapitalized at the beginning of the period and thus its quantity choice is not restrained by the solvency constraint. If the reinsurance company only considered short-term eects at the beginning of period _t_ , it would redistribute to shareholders as much capital as it could, until the solvency constraint is saturated. However, the intertemporal part of the program gives an incentive to limit the redistribution to shareholders, taking into account the value of keeping more wealth in case of losses in the years to come or of better business opportunities in case prices increase. To represent these inter-temporal eects in a simple way, we assume that the quantity of capital that the rm can redistribute over a year is limited.

This simplication leads to the following assumption: (A2) Reducing the amount of capital through share purchase and special dividend is restricted to a share _κ_ of the reinsurer's wealth at the beginning of the period.

From Assumptions (A1) and (A2), we consider that managers choose the quantity _Or_ they supply to the market and their level of external capital _Er_ , in order to maximize their expected prot under their solvency constraint and the limit on the amount of capital they can redistribute: EΠ<sup> ̃</sup> _r_ ( _Or, Er_ ) is the expected prot of the rm detailed in Eq.(2).

# 3.3 Market equilibrium

At each period, we consider that reinsurance companies, under perfect information, maximize their expected actualized prot by choosing their reinsurance supply and their capital level, either by raising new capital or buying capital back. We look for a Nash equilibrium where no reinsurer has an interest to deviate from its choice of quantity and capital. We suppose here that (A3): reinsurers compete in quantity. The equilibrium price is such that market demand equals reinsurance supply.

The demand is given by the inverse demand function _d_ ( _p_ ) , where, at equilibrium, the aggregate output of the industry is _d_ ( _p_ ) =<sup>�</sup><sup>_R_</sup> _r_ =1<sup>_Or_(</sup><sup>_p_).Bydenition,as</sup><sup>_Or_correspondtotheshareofthe</sup> market risk layer covered by the reinsurance companies, we have _d_ ( _p_ ) _≤_ 1 . The equilibrium price is such that market demand equals reinsurance supply.

The equilibrium is dened by the following system at each period (subscript _t_ omitted): 8
We solve this system by computing the rst order conditions. Details of the methods can be found in the Appendix.

## 3.4 Model dynamics

At each period _t_ , the model follows a series of steps. At the beginning of the period, each reinsurance company is endowed with an initial level of internal capital _Wr_<sup>_t_,derivedfromthe</sup> previous time step. Our model is dynamic and can be written in the following recursive form, dening the level of internal capital for all reinsurers at time _t_ , _W_<sup>_t_</sup> = [ _W_ 1<sup>_t, .., W_</sup> _R_<sup>_t_]:</sup> At time step _t_ + 1 , reinsurers' initial wealth depends on the wealth they inherited from the preceding period _t_ , and the loss _L_<sup>_t_</sup> they incurred at the end of time step _t_ .

Stage 1 Each reinsurer chooses simultaneously its market share and its level of capital to maximize its prot, under the constraint of market equilibrium. The market prices _p_<sup>_t_</sup> , reinsurer supplies _Or_<sup>_t_,capitalchoices</sup><sup>_E_</sup> _r_<sup>_t_,anddefaultprobabilities</sup><sup>_π_</sup> _r_<sup>_d,t_</sup> are determined.

Stage 2 Losses are realized following the loss distribution function. The nal wealth of all reinsurers is computed as: where _L_<sup>_t_</sup> _r_<sup>istheamountoflossesreinsuredbythereinsurer</sup><sup>_r_.Weproceedinthefollowing</sup> manner: when losses are drawn at the end of each period, they are distributed randomly among reinsurers. Each of them supports a loss: where ̃ _ε_ is drawn in a normal distribution _N_ (0 _,_ 0 _._ 5) . When computing their default probability and choosing their capital and quantities, reinsurers anticipate this random distribution of losses. We dene a bankruptcy event when a reinsurer wealth goes below a lower limit threshold wealth at the end of the period. This is anticipated in the estimation of reinsurers' probability of default.

Figure 2 summarizes the main steps of model computation.

Table 1 summarizes the variables and parameters of the model.

# 4 Materials & methods

## 4.1 Algorithm

For the resolution of the market equilibrium each year, we decided to use the trust-regionreective algorithm. This algorithm is a subspace trust-region method and is based on the interiorreective Newton method described in Coleman and Li (1994) and Coleman and Li (1996). Each 9
|Going to _t_ +1|Reinsurers<br>Capital at _t_|
|---|---|
||Reinsurers<br>|
|Expected|MarketEquilibrium<br>choose|
|Losses|<br>_Or_, _Er_<br>_Market Price_<br>_π_<sup>_d_</sup><br>_r_<br>Realized Loss<br>Final wealth|
|Figur<br>Table <br>|Possible bankruptcy<br>e 2: Model timing schematics<br> 1: Variables and Parameters|
|Market variables||
| ̃_L_<br>_pa_<br>_p_<sup>_t_</sup>|Distribution of market loss to be reinsured<br>Actuarial price of losses<br>Market price of reinsurance at time _t_|
|Market parameters||
|_d_0<br>_δ_<br>_α_<br>_πlim_<br>_κ_<br>_cr_<br>_Sr_|Demand parameter<br>Price demand elasticity<br>Rate of return of risk-free investments<br>Limit probability of default<br>Limit percentage of capital that reinsurers can buy back<br>Cost of external capital<br>Shareholder cost|
|Reinsurers' specic variables||
|_r_<br>_NR_<br>_W_ <sup>0</sup><br>_r_<br>_W _<sup>_t_</sup><br>_r_<br>_E_<sup>_t_</sup><br>_r_|Reinsurer index<br>Number of reinsurance rms<br>Initial wealth of the rm<br>Wealth of the rm at the beginning of time _t_<br>External capital acquired by the rm at time _t_|
|_O_<sup>_t_</sup><br>_r_|Share of the market risk covered by reinsurer _r_ at time _t_|
|_π_<sup>_d,t_</sup><br>_r_|Probability of default of the rm at time _t_|

iteration involves the approximate solution of a large linear system using the method of Preconditioned Conjugate Gradients (PCG).

# 4.2 Scenario

## Reinsurance loss coverage

To build the insurance market demand for reinsurance, we proceed in the following manner. We use model-based aggregate insured loss data from Risk Management Solutions Inc. for the four main reinsurance disaster markets: Japan Earthquake, European Windstorm, US Hurricane and US Earthquake, including four types of risk covered (agricultural, commercial, residential and 10
business). We suppose that there are no correlations between losses in these markets.

For each line _l_ = 1 _.._ 4 , we consider the industry-wide demand for coverage. We note _L_<sup> ̃</sup><sup>_l_</sup> _I_<sup>theloss</sup> to which the insurance industry is exposed. Let us consider _L_<sup> ̃</sup><sup>_l_</sup> the demand for reinsured loss in the _l_<sup>_th_</sup> line of business as a function of _L_<sup> ̃</sup><sup>_l_</sup> _I_<sup>.Wesuppose,atrst,thattheinsuranceindustrychooses</sup> to cover losses for each line of business between a xed limit and a xed exhaustion point dened by their return periods, respectively _TL_<sup>_l_and</sup><sup>_T l_</sup> _E_<sup>thatdenetwoleveloflosses</sup><sup>_Ll_</sup> _min_<sup>and</sup><sup>_Ll_</sup> _max_<sup>.</sup> This could be interpreted as if there was one layer of the risk that could be covered for each line of risk. In practice, each reinsurer would dene its own layers (limits and retentions) for each risks. This allows the generation of a cumulative distribution function of potential reinsured losses for the whole disaster market. In this version of the model, we assume that _TL_<sup>_l_=10yearsand</sup> _TE_<sup>_l_=300years.Weobtainthepotentialmarketforreinsuredlossesbylookingattheaggregate</sup> potentially reinsured loss distribution. We t this distribution with a log-normal distribution. Figure 3 represents the obtained loss distribution, in billion of dollars. The stairs arise from the limits and exhaustion points calculation. Expected loss is of $22 bn, standard deviation is $26 bn, and maximum market loss is $568 bn.

<!-- Start of picture text --> 140<br>120<br>100<br>80<br>60<br>40<br>20<br>0<br>0 50 100 150 200 250 300 350 400<br>Return period (years)<br>Reinsurable losses (bn US$)<br><!-- End of picture text --> Figure 3: Distribution of market loss considered for reinsurance coverage.

This model of reinsurance demand is quite limited as reinsurance companies do not all have an equal exposition to catastrophic risks (e.g. earthquake, wind etc...). However, we account here for the main problem reinsurers have to address that is the covariance of the risks and the systematic component of catastrophe risk. One limitation of our model is that we do not take into account diversication aspects as only one line of risk is considered.

# 5 Reference simulation: The oligopoly eect

The reference simulation is a starting point to understand the main eects reproduced by the model as well as its limitations. It is based on an ctive market in which all reinsurers are identical. We consider a 50-year simulation with a scenario of annual market losses, assuming 11
that the form of reinsurance demand does not evolve over time (it only depends on the price). Each year, losses are randomly drawn from the distribution described above. For the purpose of illustration, we introduced a 100-year loss at year 30 and use the same loss scenario for the whole paper. No stochastic distribution of losses among reinsurers is carried out in this rst simulation, and reinsurers incur losses proportional to their market shares (i.e. _ε_ = 0 in Equation 8).

The aim is here to understand the dynamics of the present market and check that the model is able to reproduce a realistic dynamics. For these simulations, a bankruptcy event is dened when reinsurer wealth falls below 0.

Table 2 presents the values of the parameters used in this reference case. There are little data available on most on these values in the literature. We base our reference parameters on the available studies.<sup>8</sup> The value of _Sr_ is consistent with the magnitude obtained in Zanjani (2002).<sup>9</sup> The value of _πlim_ corresponds to a return period of 200 years, which is used to calculate the level of required capital in Solvency II.<sup>10</sup> Concerning _cr_ , we calibrated this parameter such that the recourse to external capital following a high loss would be of a magnitude comparable with the one shown on Figure 1, but without considering new insurers. Finally, available information concerning share repurchase program are from Aon Beneld (2010). Share repurchase are estimated at 2%, peaking at 11% of shareholders fund for three of their reinsurance companies in the rst semester of 2010. This cannot be used directly to calibrate _κ_ , since most of these reinsurers are multiline, but it provides orders of magnitude for this process. To assess the robustness of our results, a sensitivity analysis is conducted on each of these parameters.

Table 2: Parameters

|Parameters|Reference simulation|
|---|---|
|elasticity<sup>a</sup>|0_._2|
|_α_|4%|
|_Sr_|12%|
|_cr_|4_._5_·_10<sup>4</sup>|
|_πlim_<br>_κ_|0_._5%<br>1%|

> aThe elasticity parameter is dened as _δ_ =<sup>_elasticit_</sup> E _L_<sup>_<u>y</u>_</sup> .

We consider 5 reinsurers that each have an initial wealth _W_ 0 equal to $ 20 bn. Our reference market can appear as highly capitalized but as reinsurers are monolines, there is no diversication eects that can play here.

# 5.1 Simulation results

Results are presented on Figure 5 and commented below. All the graphs in this paper will follow the same pattern. The dierent output graphs are disposed following the order presented on Figure 4. The black curves represent market variables, and the gray curves individual reinsurance companies variables.

> 8Moreover most empirical studies are based on data that do not include years following Katrina.

> 9A more detailed analysis by Cummins and Phillips (2005) is focused on the insurance market.

> 10See http://www.gccapitalideas.com/2010/02/17/higher-pressure-on-cat-risk-under-solvency-ii-part-istandard-formula-approach/ for a CEIOPS discussion on Solvency II cat risk approach.

12
The model reproduces a market showing many characteristics similar to those observed empirically. In particular, prices are higher than actuarial prices, market capacity decreases after high loss shocks, as the market price increases, and additional capital is raised by reinsurers after large shocks. Since losses are distributed proportionally to market shares, the reinsurers remain identical during the entire simulation (this assumption is relaxed in Section 6.2).

|Graph 1|Graph 2|Graph 3|
|---|---|---|
|Graph 4|Graph 5|Graph 6|

<!-- Start of picture text --> Graph 4<br><!-- End of picture text --> <!-- Start of picture text --> Graph 5<br><!-- End of picture text --> <!-- Start of picture text --> Graph 6<br><!-- End of picture text --> Figure 4: Output graphs layout

- Graph 1: random reinsurance market losses scenario. The $90 bn loss at year 30 corresponds to a return period of about 100 years. From year 38 to 45, there is a cluster of several medium losses.

- Graph 2: market price of reinsurance as the ratio of price on expected loss. A price equal to 1 corresponds to the expected loss. Market price always exceeds twice the actuarial price. The price spikes after high loss events. Depending on the size of the shock, the time needed to return to the lower prices is dierent.

- Graph 3: reinsurers' probability of default. The probability of default is capped by _πLim_ as our solvency constraint requires. After loss events, the solvency constraint is saturated until market capacity has replenished itself.

- Graph 4: market wealth at the end of the year. The total market capitalization (black curve) is the sum of reinsurer capitalizations (gray curve). Each important loss is followed by capital depletion. After the loss, the market rebuilds itself as capitalization rises. Time needed to recapitalize depends on the intensity of the shock.

- Graph 5: reinsurance supply. The gray curve corresponds to the risk covered by one reinsurer, and the black curve to the risk covered by all the market. The total market supply always remains below a ceiling value. This corresponds to an oligopoly eect: serving a higher part of the market would decrease the market price, and not be as beneciary to the reinsurers. As a consequence, reinsurers voluntarily ration the market to drive prices upward.

- Graph 6: market external capital acquisition. The sum of all external capital movements (black curve) corresponds to the sum of the identical individual reinsurers' external capital movements (gray curve). At the beginning of the simulation, it is negative corresponding to a purchase of reinsurer's own shares, as it is the case each time the solvency constraint is not saturated. In such a situation, the market is overcapitalized. Most of the time, external capital is positive and small, but after high losses it becomes much higher (year 15 and 31), as well as in the middle of the series of medium losses (year 41).

13
will introduce a more sophisticated treatment of reinsurance demand.

- Impact of a series of medium losses: From years 38 to 43, several losses occur, which have a return period between 25 and 30 years. The market has no time to recover between them. Therefore, market capacity remains low and reinsurance companies keep acquiring external capital although in a limited amount. Market supply decreases to 50%, showing the importance of considering series of shocks and introducing path dependency and the dynamics of the market. This clearly indicates a path dependency in the market. This result emphasizes the need to take into account possible clustering in extreme events, as suggested by Bunde et al. (2005).

This reference simulation already exhibits some of the characteristics of the reinsurance market dynamics. The main ones are (i) the rise of the price following an extreme event, (ii) the correlation between market capacity and the price of risk, and (iii) the fact that reinsurers raise signicant amounts of capital after large shocks. These eects arise from the capacity constraint due to the limit of 0.5% of probability of default. The high rise of the price after year 30 does not take into account the impact of a change in reinsurance demand, nor of a risk reevaluation by reinsurance companies, that Froot (2001) and Guy Carpenter (2009), among others, also mention to explain market reaction.

# 5.3 Sensitivity analysis

Little data are available on the reinsurance market and there is a large uncertainty in all model parameters. In such a situation, it is useful to carry out a sensitivity analysis to assess the robustness of our results and to better understand important mechanisms. It is based on the reference reinsurance-market scenario detailed above. We carry out sensitivity analysis on the main parameters presented on Table 1. For each of those, we take varying values around the one used in the reference scenario. Exact values and gures are presented in Appendix (Figures 11 to 15). Results are the following:

- Demand elasticity ( _e_ ): The higher the demand elasticity, the lower the share of the market covered after the large loss of year 33. This result is consistent with intuition: the increase in prices after a large shock reduces reinsurance demand for insurers. For the year following a high loss, higher elasticity leads to lower ex-post prices. It is in line with the economic theory literature: Cagle and Harrington (1995) show that in the case of capacity constraint and endogenous solvency risk (with limited liability) the increase in price after a shock with inelastic demand price increases is lower than the amount necessary to oset the shock. The higher the elasticity, the lower the impact of shocks on prices. However, when considering market recovery over several years, a lower elasticity leads to an earlier recovery of the market.

- Cost of carrying capital ( _Sr_ ): As can be expected, the higher the cost of holding capital, the less capitalized the reinsurance companies are, and the less they are able to supply the market. Hence, the equilibrium capital level (i.e. the stabilized level if losses are set to the expected loss) depends on the cost of carrying capital. Following a large shock, external capital acquisition is less important when the share cost is high.

- Cost of external capital ( _cr_ ): This parameters has a main impact after important losses when the reinsurers acquire external capital. The lower the cost of external capital, the more the

15
rms relay on it, and the less the market correction on prices. This leads to a higher capacity of the reinsurance market to supply coverage after important losses when cost of external capital is low. This impact is only visible during tight markets.

- Impact of initial capital ( _Wr_<sup>0):Inthepresentcasewithaperfectlysymmetricmarketwithno</sup> entry nor exit, all reinsurers have an optimal capital target that they all tend to reach. A dierence in initial capital disappears after periods of soft market during which the reinsurers can replenish their capital.

- Impact of the number of reinsurers ( _Nr_ ): When we distribute the same total capital among dierent numbers of reinsurers, the oligopoly eect is quite clear: the fewer the reinsurers, the lower market capacity and the higher the prices during soft markets. There is therefore a clear gain from increased competition in the market. However, impact of important losses lead to the same depletion of capacity.

- Impacts of the solvency constraint ( _πlim_ ): Results show that the lower the solvency constraint, i.e. the higher _πlim_ , the higher the capacity of the reinsurance market in soft market and the higher the time needed for recovery after an important loss. However, as path dependency is important, a rened analysis of the impact of this constraint, particularly on the resistance to high shocks, is conducted in Section 7.

- Impacts of the limit share of buy backs ( _κ_ ): The sensitivity analysis shows that market capitalization stays higher over the scenario when _κ_ is small, as rms can not buy back as many shares as they would for higher level of _κ_ . Consequently, market resilience to large losses is higher.

# 6 Asymmetric market and reinsurer bankruptcy

## 6.1 Market segmentation between two sizes of reinsurers

To investigate a more realistic market, we build a ctive reinsurance market from available market data. The reinsurance market is characterized by a certain concentration that has increased in the last decade. The current market involves big reinsurance rms such as Swiss Re and Munich Re, smaller reinsurers, and new reinsurers as the Bermudian. However, as reinsurance companies oer coverage on several markets (life, disasters...), we do not have specic data on the capital linked to disaster risks and on the market shares on the natural-disaster market. We use as a proxy the market shares as obtained from Standard and Poor's (2008) on the reinsurance market for all lines of risk: in 2007, the four biggest reinsurance companies accounted for 48 % of the market. We consider a market with 4 large reinsurers that each have an initial wealth _W_ 0 of $ 22 _._ 5 bn, and 10 small reinsurers that have an initial wealth of $ 3 bn. This market is still ctive, but it is consistent with the type of concentration observed in the industry. The Geneva Association Systemic Risk Working Group (2010) presents a concentration curve for the reinsurance market where the 10 biggest reinsurers share 80% of the market and the ve biggest more than 50% in 2008. The market parameters used are the same than in the reference reinsurance-market simulation. Figure 6 presents the results of this simulation. The smaller reinsurance companies are represented in solid gray and the larger ones in dashed gray. The interpretation of the output graphs is essentially the same as in the reference simulation. However, several interesting features appear due to the 16
# ~~<u><mark>ce pe</mark></u>~~
- ~~u~~ ~~<mark>l)</mark>~~ ~~<u><mark>Pre [ee4 aEn</mark></u>~~ ~~<mark>=,</mark>~~
simple representation of the real market, and a deeper study of the impacts of the market structure would be of great interest.

The model allows to get a rst insight on the importance of market structure on the price and capacity dynamics. Interesting results are obtained, as the model reproduces a segmented market with both strategic and follower reinsurers that evolve through time depending on losses impact on them. The former are large enough to have a real impact on prices and to reduce the amount of reinsurance they supply to drive the price upward, especially in hard markets. The latter do not have the same inuence on the market, and behave in some ways as price-taker agents and benets from the price driven by the other ones, as shown on the asymmetric idealized market.

One major insight from this article is the modeling of new capital ows into the market. Two dierent kinds of ows currently exist. First, there is a ow of external capital in surviving reinsurance companies, corresponding to equity emission or development of sidecars and catbonds. Second, one can observe the development of new reinsurance companies when the reinsurance market price is high (Lane, 2007b). The total of these two ows is important. Following hurricanes Katrina, Rita and Wilma, for instance, Guy Carpenter (2009) estimates that more than $35 _bn_ were injected in the industry through these various alternative means of capital. This eect has taken more importance during the last decade, and impacts on prices have been less pronounced since hurricanes Andrew (The Geneva Association Systemic Risk Working Group, 2010). Some calibration is still needed to better assess ows of capital in the market. The model presented in this paper does only model the rst of these two ows, namely the ow of external capital into existing reinsurers. A forthcoming version of the model shall include new reinsurers entry on the market.

# 7 A regulation trade-o: Eciency vs. resilience

Our modeling of reinsurer's risk aversion can be interpreted as a regulatory constraint: reinsurers are not allowed to exceed an exogenous probability of default. This is of course a very simple way to model insurance solvency regulations. Such a constraint has an impact on the market behavior of the rms, as holding more capital to cover the same quantity of risk is costly for the rms. On Figure 8, we represent the sensitivity analysis conducted on the symmetric market for variations of the threshold default probability _πLim_ between 0 _._ 1% and 0 _._ 9% . The impact of this parameter on market variables is important. When the solvency constraint is strong ( _πLim_ low), the share of total loss covered by the reinsurance industry is low, prices are high, and the probability of defaults of reinsurers is saturated even in soft markets. As the limit probability of default is lower, reinsurers need a higher capitalization to cover the supply they provide. Market prices evolve correspondingly.

This sensitivity analysis suggests the existence of a classical trade-o between a higher market resistance to large and rare shocks with more stringent regulation and an eciency criterion in the most frequent situations. Intuitively, a strong solvency constraint increases market resilience. But simultaneously, it decreases market eciency as holding the same amount of risk requires more capital - and thus is more expensive. To understand better the impact of this solvency constraint, 19
<!-- Start of picture text --> Losses Market pri ce<br>100 10<br>50 5<br>0 0 0.001<br>0 10 20 30 40 50 0 10 20 30 40 50 0.003<br>Year Year<br>0.005<br>Total external capital Market supply<br>0.007<br>20 1<br>0.009<br>0 0.5<br>−20 0<br>0 10 20 30 40 50 0 10 20 30 40 50<br><!-- End of picture text --> Figure 8: Sensitivity analysis for the solvency constraint we rst studied the impact of a loss on the market price. To do so, we impose a large loss on a stabilized market of 5 symmetric reinsurers, assuming they were exposed to the expected loss level during the 10 previous years (to remove path dependency from the analysis). Figure 9 represents the price in the year following the shock for the limit default probabilities 0.1%, 0.5%, and 1%. This range is centered on a 200- ~~year risk.~~ <!-- Start of picture text --> 20<br>18<br>16<br>14<br>12<br>0.1%<br>0.25<br>10 0.5%<br>0.75%<br>1%<br>8<br>6<br>4<br>2<br>0<br>0 20 40 60 80 100 120 140 160 180 200<br>Losses amplitude (billion dollars)<br>Price just after shock<br><!-- End of picture text --> Figure 9: Non linearity between price reaction and loss amplitude We rst observe a non linearity between price and the magnitude of the loss. When the regulation is stricter (lower limit probability of default), the price after small losses is higher, because it requires more capital to supply the same amount of reinsurance capacity. So, the market is most of the time less ecient with a strict regulatory constraint. But after a large loss, the market collapse with a weak constraint (e.g., for losses larger than $80 billion if the limit is at 1%), while the loss can be absorbed with a stricter regulation (however at the expense of a large increase in price). Indeed, the highest loss the market is able to absorb is a market loss of $190 bn for a limit probability of default of 0.1% (1000-year loss), $110 bn for a probability of 0.5% (200-year loss) and $80 bn for a probability of 1% (100-year loss).

A closer look at the impact of the default probability on the market is provided on Figure 10.

20
On the typical scenario we used in Section 5, we compute the mean price and mean covered risks on the market during the 50<sup>_th_</sup> years.<sup>14</sup> We see again that average market price increases with the regulatory constraint whereas correspondingly the capacity of the reinsurance market decreases. Thus tighter regulation leads during a regular time period with limited losses to higher prices and less available capacity for the same amount of capital. This is a typical example of an eciency vs. resilience trade-o as can be found in ecological systems.

<!-- Start of picture text --> 80 5<br>Mean Coverage<br>Mean Market Prices<br>70 4<br>60 3<br>50 2<br>0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br>Limit default probability<br>Mean total market share covered (%)<br>Mean market prices (multiples of actuarial price)<br><!-- End of picture text --> Figure 10: Reinsurance price and market capacity in function of the limit probability of default This analysis is of course quite limited for several reasons: (i) we only consider one line of business for the reinsurers with only one capital charge; (ii) we only take into account shocks that may arise on the liability side of the reinsurers balance-sheet while asset-side shock play a crucial role; (iii) we disregard any potential correlation between nancial market risks and natural disasters. Furthermore, the reinsurance industry reaction to shocks on their asset sides can induce impacts on the nancial markets.

The justication and the choice of a proper regulation is a dicult question that Plantin and Rochet (2007) analyze. Agency considerations and moral hazard issues are crucial: policyholders are mostly very diluted and do not exert sucient monitoring.<sup>15</sup> Furthermore, the complexity of the reinsurance business makes it very dicult to assess the solvency risks linked to each company for technical as well as moral hazard reasons. In this paper, we do not ask the question of the optimal level nor the justication of regulation as we do not consider any welfare perspective and only look at the impact on the reinsurance market. But the eciency/resilience trade-o in this simple way provides a clear illustration of one of the key regulator issues: what is the acceptable risk for such an industry and at what cost can it be hedged? This question is at the center of regulation questions following the nancial crisis, especially following AIG's buy-out (Harrington, 2009). Certainly, the reinsurance industry does not have as much a macroeconomic impact as the insurance industry, but its proper operation is an important condition for the insurance industry to be able to still cover large risks.

> 14A proper analysis would require running the full model with loss randomization. This will be done in a further version of the model.

> 15In the case of reinsurance companies, it could be argued that insurance companies could have the ability to monitor them eciently.

21
# 8 Conclusion

The model outputs shed a better light on the way classical economic assumptions allow for understanding and reproducing the reinsurance market dynamics. Compared with previous analyses, our approach investigates the issues of reinsurance capacity to withstand important losses in a dynamic framework. Our model relies on a capacity constraint analysis, with a threshold probability of default for reinsurance rms. This factor may enhance the role of regulatory measures on the dynamics and the resilience of the reinsurance market either through the resistance to severe losses or through the time needed for capacity replenishment. Furthermore, we take into account the impacts of the rm choice on the market equilibrium to model potential strategic behaviors of large reinsurers. Our approach is innovative as we compute a dynamics of the market, based on these clear and tractable assumptions.

This analysis is only the rst step of a broader research agenda. Incorporating several lines of risk will rene the reinsurance demand model. A better calibration would allow a better understanding of the market features that inuence the market resilience. Recent data on reinsurance contracts would be needed to do so, and they are dicult to access. The long-term objectives of this model are to develop insights on the needed market capacity and to understand the insurance industry limits in providing disaster coverage. This question is critical when considering the current trends in insured and reinsured exposure. In addition to the classical economic trend issues (Hallegatte, 2011), the IPCC (2007) suggests a probable rise in the frequency and intensity of some natural disasters (e.g., storm surge and coastal oods). Several studies have proposed projections of future exposure due to these trends (see for instance Hanson et al. (2011) on the exposure in coastal cities) and the consideration of dierent scenarios will be necessary. How could the reinsurance industry be able to cover increasing risks, and how could specic policies and regulations enhance its ability to do so?

# 9 Acknowledgements

The authors wish to thank Patrice Dumas, Antoine Leblois and Valentin Przyluski for insightful discussions, and Pierre Picard, Dominique Henriet, Jean-Marc Tallon and Vincent Martinet for useful comments on a previous version of this paper. The authors also thank Risk Management Solutions, Inc., and especially Auguste Boissonnade and Patricia Grossi, for providing invaluable data to the analysis. The views expressed in this paper are the sole responsibility of the authors. They do not necessarily reect the views of the World Bank, its executive directors, or the countries they represent.

# References

- Aon-Benfield 2009. Reinsurance market outlook. resilient without assistance. Technical report, AonBeneld.

Aon Benfield 2010. Reinsurance market outlook. Technical report, Aon Beneld.

Blazenko, G. 1986. The economics of reinsurance. The <u>Journal</u> of <u>Risk</u> and Insurance 53:258277.

- Bohn, J. G. and Hall, B. J. 1999. The moral hazard of insuring the insurers, pp. 363390. In K. Froot (ed.), The Financing of Catastrophe Risk, chapter 1. The University of Chicago Press.

22
Borch, K. 1962. Equilibrium in a reinsurance market. Econometrica 30:424444.

- Bunde, A., Eichner, J., Kanterlhardt, J., and Havlin, S. 2005. Long-term memory: a natural mechanism for the clustering of extreme events and anomalous residual times in climate records. Physical <u>Review</u> Letters 94:48701.

- Cagle, J. A. B. and Harrington, S. E. 1995. Insurance supply with capacity constraints and endogenous insolvency risk. <u>Journal</u> of <u>Risk</u> and <u>Uncertainty</u> 11:219232.

- Coleman, T. F. and Li, Y. 1994. On the convergence of interior-reective newton methods for nonlinear minimization subject to bounds. Mathematical Programming 67:189224.

- Coleman, T. F. and Li, Y. 1996. A reective newton method for minimizing a quadratic function subject to bounds on some of the variables. <u>SIAM Journal</u> on <u>Optimization</u> 6:1040.

- Cummins, J. D. 2008. Cat bonds and other risk-linked securities: State of the market and recent developments. <u>Risk</u> Management and Insurance <u>Review</u> 11:2347.

- Cummins, J. D. and Danzon, P. M. 1997. Price, nancial quality, and capital ows in insurance markets. <u>Journal</u> of Financial Intermediation 6:338.

- Cummins, J. D., Doherty, N., and Lo, A. 2002. Can insurers pay for the "big one"? measuring the capacity of the insurance market to respond to catastrophic losses. <u>Journal</u> of Banking <u>&</u> Finance 26:557583.

- Cummins, J. D. and Phillips, R. D. 2005. Estimating the cost of equity capital for property-liability insurers. <u>Journal</u> of <u>Risk &</u> Insurance 72:441478.

- Cummins, J. D. and Trainar, P. 2009. Securitization, insurance, and reinsurance. The <u>Journal</u> of <u>Risk</u> and Insurance 76:463492.

- Cummins, J. D. and Weiss, M. A. 2009. Convergence of insurance and nancial markets: Hybrid and securitized risk-transfer solutions. <u>Journal</u> of <u>Risk &</u> Insurance 76:493545.

- Cutler, D. M. and Zeckhauser, R. 1999. Reinsurance for catastrophes and cataclysms, pp. 233275. In K. Froot (ed.), The Financing of Catastrophe Risk, chapter 1. The University of Chicago Press.

- Doherty, N. and Smetters, K. 2005. Moral hazard in reinsurance markets. <u>Journal</u> of <u>Risk &</u> Insurance 72:375391.

- Doherty, N. A. and Tinic, S. M. 1981. Reinsurance under conditions of capital market equilibrium: A note. The <u>Journal</u> of Finance 36:949953.

- Froot, K. A. 2001. The market for catastrophe risk: a clinical examination. <u>Journal</u> of Financial Economics 60:529571.

- Froot, K. A. and O'Connell, P. G. J. 2008. On the pricing of intermediated risks: Theory and application to catastrophe reinsurance. <u>Journal</u> of Banking <u>&</u> Finance 32:6985.

- Froot, K. A. and Stein, J. C. 1998. Risk management, capital budgeting, and capital structure policy for nancial institutions: An integrated approach. <u>Journal</u> of Financial Economics 47:5582.

- Garven, J. R. and Tennant, J. L. 2003. The demand for reinsurance: Theory and empirical tests. Insurance and <u>Risk</u> Management 7:217237.

- Gron, A. 1994. Capacity constraints and cycles in property-casualty insurance markets. The <u>RAND Journal</u> of Economics 25:110127.

Group of Thirty 2005. Reinsurance.

Guy Carpenter 2009. World catastrophe reinsurance market. Technical report, Guy Carpenter.

- Hallegatte, S. 2011. How economic growth and rational decisions can make disaster losses grow faster than wealth. Technical report, World Bank Policy Research Working Paper 5617.

- Hanson, S., Nicholls, R., Ranger, N., Hallegatte, S., Corfee-Morlot, J., Herweijer, C., and Chateau, J. 2011. A global ranking of port cities with high exposure to climate extremes. <u>Climatic Change</u> 104:89111. 10.1007/s10584-010-9977-4.

23
- Hardelin, J. and Lemoyne de Forges, S. 2009. Raising capital in an insurance oligopoly market. Working Paper - Ecole Polytechnique.

- Harrington, S. E. 2009. The nancial crisis, systemic risk, and the future of insurance regulation. Issue Analysis .

- Herweijer, S., Ranger, N., and Ward, R. 2009. Adaptation to climate change: Threats and opportunities for the insurance industry. The <u>Geneva</u> Papers on <u>Risk</u> and Insurance - Issues and Practices 34:360380.

- Jean-Baptiste, E. L. and Santomero, A. M. 2000. The design of private reinsurance contracts. <u>Journal</u> of Financial Intermediation 9:274297.

- Kessler, D. 2005. La n du cycle traditionnel en assurance et réassurance de dommages ? <u>Revue</u> d'économie nancière .

- Lai, G. C., Witt, R. E., Fung, H.-G., MacMinn, R. D., and Brockett, P. L. 2000. Great (and not so great) expectations: An endogenous economic explanation of insurance cycles and liability crises. <u>Journal</u> of <u>Risk &</u> Insurance 67:617652.

- Lane, M. 2007a. Of sidecars and such. Trade Notes, Lane Financial.

Lane, M. 2007b. Recapitalizing reinsurers, a never ending story? Trade Notes, Lane Financial.

- Lewis, C. M. and Murdock, K. C. 1996. The role of government contracts in discretionary reinsurance markets for natural disasters. The <u>Journal</u> of <u>Risk</u> and Insurance 63:567597.

- Mayers, D. and Smith, Clifford W., J. 1990. On the corporate demand for insurance: Evidence from the reinsurance market. The <u>Journal</u> of Business 63:1940.

Meier, U. B. 2006. Existence and Causes of Insurance Cycles in Dierent Countries. Haupt.

Plantin, G. 2006. Does reinsurance need reinsurers? The <u>Journal</u> of <u>Risk</u> and Insurance 73:153168.

- Plantin, G. and Rochet, J.-C. 2007. When Insurers Go Burst. Pinteton University Press.

- Shleifer, A. and Vishny, R. W. 1997. A survey of corporate governance. The <u>Journal</u> of Finance 52:737783.

- Standard and Poor's 2008. Global reinsurance highlights - 2008 edition. Technical report, Standard and Poor's.

- The Geneva Association Systemic Risk Working Group 2010. Systemic risk in insurance: An analysis of insurance and nancial stability. Special report, The Geneva Association.

- Weiss, M. A. 2007. Underwriting cycles: a synthesis and further direction. <u>Journal</u> of Insurance Issues 30:3145.

- Weiss, M. A. and Chung, J.-H. 2004. U.s. reinsurance prices, nancial quality, and global capacity. <u>Journal</u> of <u>Risk &</u> Insurance 71:437467.

- Winter, R. A. W. 1994. The dynamics of competitive insurance markets. <u>Journal</u> of Financial Intermediation 3:379415.

- Zanjani, G. 2002. Pricing and capital allocation in catastrophe insurance. <u>Journal</u> of Financial Economics 65:283305.

# A.1 Market Equilibrium Resolution

Intuitively, two situations shall be distinguished for modeling for each reinsurer whether its solvency constraint is saturated or not. First, in the saturated case, the choice of external capital _Er_ is an implicit function of the choice of supply _Or_ through the solvency constraint.<sup>16</sup> The equilibrium constraint is then equivalent for the concerned reinsurers to the maximization of their expected prot as a function of reinsurance supply only.

> 16In practice, due to the form of the capital costs, there may be two solutions _Er_ to this constraint that lead to the same expected prots. In this case, we take the lower level of capital _Er_ as it gives the highest yield.

24
In the case where the constraint is not saturated, the choices of external capital and reinsurance supply are not linked any more. The rst order conditions for the concerned reinsurers are composed of two dierent elements: ( _FOCO_ ) determining the reinsurance supply choice and ( _FOCE_ ) determining the external capital level that corresponds to the exogenous limit for reducing the amount of capital through share purchase programs and special dividend.

# A.2 Sensitivity analysis

Sensitivity analyses are conducted on key parameters. Parameters are changed individually from the reference reinsurance-market scenario. The order of the graphs is the same as for the reference simulations. To be able to better read the results, a shorter time period is shown for each analysis, depending on the parameter considered.

25
Figure 11 presents the sensitivity analysis for the elasticity ~~of demand. The elasti~~ city p ~~arameter~~ _e_ = E _L ∗ δ_ (0.2 corresponds to a low elasticity, and 0.6 to a high <u>demand elasticity with respect to price.</u> Note that for a low elasticity, pale gray, reinsurers buy share back much more frequently since price and quantity can be large together.

<!-- Start of picture text --> Losses Market pri ce<br>100 10<br>50 5<br>0 0 0.100<br>0 10 20 30 40 50 0 10 20 30 40 50 0.150<br>Year Year<br>0.200<br>Total external capital Market supply<br>0.250<br>20 1<br>0.300<br>0 0.5<br>−20 0<br>0 10 20 30 40 50 0 10 20 30 40 50<br>Figure 11: Sensitivity analysis for elasticity<br><!-- End of picture text --> Figure 12 presents the sensitivit ~~y ana~~ lysis for the cost of carrying <u>capital. The parameter</u> _S_ _~~r~~_ <u>, which</u> represents this cost, varies from 10 ~~% to~~ 14%.The model is sensitive to this parameter, but qualitative results remain unchanged.

<!-- Start of picture text --> Losses Market pri ce<br>100 10<br>50 5<br>0 0 0.100<br>0 10 20 30 40 50 0 10 20 30 40 50 0.110<br>Year Year<br>0.120<br>Total external capital Market supply<br>0.130<br>20 1<br>0.140<br>0 0.5<br>−20 0<br>0 10 20 30 40 50 0 10 20 30 40 50<br>Figure 12: Sensitivity analysis for sharecost<br><!-- End of picture text --> 26
Figure 13 presents the sensitivity analysis on the initial capital of the rms. After 10 years, all simulations converge to the same paths. It illustrates the fact that reinsurers <u>target an optimal</u> level of <u>capital</u> that depends on the number of rms of the market. This converge ~~nce is linked to the comp~~ lete sy ~~mmetry~~ of the market, and the fact that there is no entry nor exit of reinsurers. The legend indicates the sum of all initial wealth.

<!-- Start of picture text --> Losses Market pric e<br>100 10<br>50 5<br>0 0 60.000<br>0 10 20 30 40 50 0 10 20 30 40 50 80.000<br>Year Year<br>100.000<br>Total external capital Market supply<br>120.000<br>20 1<br>140.000<br>0 0.5<br>−20 0<br>0 10 20 30 40 50 0 10 20 30 40 50<br>Figure 13: Sensitivity analysis for W 0<br><!-- End of picture text --> Figure 14 presents the sensitivi ~~ty an~~ alysis for the cost of raising <u>new capital. It mainly</u> impacts <u>the</u> quantity of capital raised after larg ~~e shoc~~ k and the time necessary for the market to recover. It has only a limited impact on the market othe ~~r varia~~ bles.

<!-- Start of picture text --> Losses Market pri ce<br>100 10<br>50 5<br>0 0 0.001<br>0 10 20 30 40 50 0 10 20 30 40 50 0.003<br>Year Year<br>0.005<br>Total external capital Market supply<br>0.007<br>50 1<br>0.009<br>0 0.5<br>−50 0<br>0 10 20 30 40 50 0 10 20 30 40 50<br><!-- End of picture text --> Figure 14: Sensitivity analysis for cost of external capital _c_ 27
Figure 15 presents the sensitivity analysis for the number of <u>reinsurers sharing the same</u> amount <u>of</u> capital. It illustrates the oligopoly eect: when reinsurers are fewer, they deliver less capacity to the market and prices are higher during soft market, the impact of important <u>losses is however slightly</u> equivalent.

<!-- Start of picture text --> Losses Market pri ce<br>100 10<br>50 5<br>0 0 3.000<br>0 10 20 30 40 50 0 10 20 30 40 50 4.000<br>Year Year<br>5.000<br>Total external capital Market supply<br>6.000<br>20 1<br>7.000<br>0 0.5<br>−20 0<br>0 10 20 30 40 50 0 10 20 30 40 50<br>Figure 15: Sensitivity analysis for the number of reinsurers<br><!-- End of picture text --> Figure 16 corresponds to the sen ~~sitivit~~ y analysis for _κ_ . The lower _κ_ , the more capitalized the industry, and the lower the market prices.

<!-- Start of picture text --> Losses Market pri ce<br>100 10<br>50 5<br>0 0 0.010<br>0 10 20 30 40 50 0 10 20 30 40 50 0.033<br>Year Year<br>0.055<br>Total external capital Market supply<br>0.077<br>10 1<br>0.100<br>0 0.5<br>−10 0<br>0 10 20 30 40 50 0 10 20 30 40 50<br>Figure 16: Sensitivity analysis for κ<br><!-- End of picture text --> The Figure corresponding to the ~~sensi~~ tivity analysis for _πlim_ can be found in Section 7 (Figure 8) .

28