Insurance: Mathematics and Economics 49 (2011) 520–536

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Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Pricing catastrophe swaps: A contingent claims approachAlexander Braun ∗

Institute of Insurance Economics, University of St. Gallen, Kirchlistrasse 2, CH-9010 St. Gallen, Switzerland

a r t i c l e i n f o

Article history:Received June 2010Received in revised formJuly 2011Accepted 18 August 2011

JEL classification:G1G2G13G22

Keywords:Catastrophe swapsContingent claims pricing approachDoubly stochastic Poisson processMean-reverting Ornstein–Uhlenbeckintensity

Counterparty default riskImplied intensitiesExploratory factor analysisFirst order autoregressive process

a b s t r a c t

In this paper, we comprehensively analyze the catastrophe (cat) swap, a financial instrument which hasattracted little scholarly attention to date. We begin with a discussion of the typical contract design, thecurrent state of the market, as well as major areas of application. Subsequently, a two-stage contingentclaims pricing approach is proposed, which distinguishes between the main risk drivers ex-ante aswell as during the loss reestimation phase and additionally incorporates counterparty default risk.Catastrophe occurrence is modeled as a doubly stochastic Poisson process (Cox process) with mean-revertingOrnstein–Uhlenbeck intensity. In addition,we fit various parametric distributions to normalizedhistorical loss data for hurricanes and earthquakes in the US and find the heavy-tailed Burr distributionto be the most adequate representation for loss severities. Applying our pricing model to market quotesfor hurricane and earthquake contracts, we derive implied Poisson intensities which are subsequentlycondensed into a common factor for each peril bymeans of exploratory factor analysis. Further examiningthe resulting factor scores, we show that a first order autoregressive process provides a good fit. Hence, itscontinuous-time limit, the Ornstein–Uhlenbeck process should be well suited to represent the dynamicsof the Poisson intensity in a cat swap pricing model.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Since the early 1990s, insurance and reinsurance companieshave been using innovative financial instruments to lay off nat-ural disaster risk in the capital markets. To date the most popu-lar of these alternative risk transfer tools is the catastrophe (cat)bond, a security which pays regular coupons to the investor un-less a catastrophic event occurs, leading to full or partial lossof principal. The principal is held by a special purpose vehicle(SPV) in the form of highly-rated securities and paid out to thehedging (re)insurer to cover its losses if the trigger condition,which has been defined in the bond indenture, is fulfilled. In ad-dition to cat bonds, catastrophe derivatives can be employed toaccess the capital markets. In 1992 the Chicago Board of Trade(CBOT) initiated exchange-traded catastrophe futures and optionsbased on its own loss index (see Swiss Re, 2009). Due to hum-ble trading activity, these contracts were soon replaced by op-tions based on Property Claim Services (PCS) indices, a measurefor catastrophe losses in nine geographical regions of the United

∗ Tel.: +41 71 243 3653; fax: +41 71 243 4040.E-mail address: alexander.braun@unisg.ch.

0167-6687/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2011.08.003

States. However, PCS-options, which paid off for the buyer in casethe underlying index exceeded the strike price, were eventuallyalso discontinued in 2000 (see Cummins and Weiss, 2009). De-spite their earlier demise, catastrophe derivatives with an exclu-sive focus on US hurricane risk have recently been re-launchedby several exchanges. Catastrophe event-linked futures (ELFs),which are co-offered by Deutsche Bank and the Insurance Fu-tures Exchange (IFEX), feature a binary payoff contingent on re-gional PCS losses.1 Similar contracts are listed on the EuropeanExchange (EUREX). Apart from insurance futures, the ChicagoMer-cantile Exchange (CME) and the New York Mercantile Exchange(NYMEX) provide catastrophe options as well. While the formerreference the so-called CMEHurricane Index,2 NYMEX options set-tle based on the Re-Ex index by Gallagher Re (see Cummins, 2008).

Although there is a growing body of literature on catastrophebonds, futures and options, it seems that another innovative risktransfer instrument of increasing importance for risk managers

1 IFEX contacts are traded on the Chicago Climate Futures Exchange. Refer to theIFEX website for more information.2 This index, which had been developed by the reinsurance intermediary Carvill,

was formerly known as the Carvill Hurricane Index (CHI). In 2009, CME grouppurchased the rights and renamed it to CMEHurricane Index. Seewww.artemis.bm.

A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536 521

and investors has been neglected so far: the catastrophe swap. Catswaps are over-the-counter (OTC) contracts, allowing (re)insurersto tap additional risk capacity by synthetically passing a portionof their insurance risk on to a counterparty. The latter could bean investor, who thereby gains unfunded exposure to naturaldisaster risk. As indicated by industry experts, catastrophe swapshave been steadily gaining ground over the past few years anddue to recent progress with regard to contract documentationand standardization, the market outlook is fairly promising. Earlyreferences to catastrophe swaps can be found in Borden and Sarkar(1996) and Canter et al. (1997), who mentioned the instrumentin their articles on insurance-linked securities. Furthermore,Cummins (2008) and Cummins and Weiss (2009) briefly describethe general mechanism behind a catastrophe swap contract. Apartfrom these publications, however, the instrument has, to the bestof our knowledge, not attracted scholarly attention. This paper isintended to fill this gap.

The remainder of the article is structured as follows. InSection 2, we briefly review the extant literature on the pricing ofcatastrophe-linked instruments. Furthermore, Section 3 contains adiscussion of the main characteristics of cat swap contracts and anoverview of the current state of the market as well as major areasof application. A two-stage approach for the pricing of catastropheswaps ex-ante and in the loss reestimation phase that additionallyincorporates counterparty default risk is presented in Section 4. InSection 5, we fit different parametric distributions to normalizedhistorical hurricane and earthquake loss data for the US to selectan adequate loss severity distribution. Subsequently, the pricingmodel is applied to back out implied Poisson intensities from catswap market data which we condense into a common factor foreach peril by means of exploratory factor analysis. The resultingfactor score times series are then used to estimate a first orderautoregressive process and evaluate its fit, thereby shedding somelight on the adequacy of a mean-reverting process for the Poissonintensity in our cat swap model. Finally, in Section 6, we state ourconclusion.

2. Literature review

While no article has been devoted to potential pricingapproaches for catastrophe swaps yet, a few authors havediscussed the pricing of cat bonds, futures, and options. In thisregard, models based on three different theoretical foundationshave been brought forward.3 First of all, within their empiricalexamination of cat bond prices, Lane (2000) as well as Lane andMahul (2008) apply an actuarial pricing methodology, therebyacknowledging the resemblance of catastrophe-linked capitalmarket instruments to traditional reinsurance.

In contrast to that, utility-based approaches are centeredaround the notion that insurance markets are generally incom-plete, implying that it is not possible to find a unique equivalentmartingale measure by merely ruling out arbitrage opportuni-ties. Embrechts and Meister (1997) provide a generic discussion ofcatastrophe futures pricing in a utility maximization context. Fur-thermore, Aase (1999) treats catastrophe risk as systematic andresorts to a partial equilibrium framework with constant abso-lute risk aversion to derive pricing formulae for cat futures, caps,call options, and spreads. Similarly, Cox and Pedersen (2000) de-rive a pricing approach for cat bonds in an incomplete marketssetting based on equilibrium pricing theory and time separableutility. Christensen and Schmidli (2000) introduce an exponentialutility model for cat futures which includes loss reporting lags.

3 Galeotti et al. (2009) provide an empirical comparison of some of theseapproaches.

Amending his earlier work on cat derivatives pricing by employ-ing a Markov model for the dynamics of underlying, Aase (2001)proposes a competitive equilibrium approach which assumes con-stant relative risk aversion of the representative agent. In addition,Young (2004) computes the indifference price of cat bonds basedon exponential utility investor preferences. Utility-based pricing ofcat bonds is also considered by Egami and Young (2008).Moreover,Dieckmann (2009) proposes a dynamic equilibrium model for catbonds with an external habit process as in Campbell and Cochrane(1999).

However, the majority of papers on the pricing of cat bondsandderivatives proposes preference free no-arbitrage frameworks.Cummins and Geman (1994, 1995) value cat futures and callspreadswith an Asian option approach, assuming a jump-diffusionprocess with constant jump amplitude for the claim dynamics.Besides, Chang et al. (1996) develop a cat option model based ona stochastic time change linked to insurance futures transactions,allowing them to convert a compound Poisson into a pure diffusionprocess for which risk-neutral valuation is readily applicable and aparsimonious closed formula can be derived. Similarly, by meansof stochastic time change and Laplace transform, Geman andYor (1997) present a semi-analytical solution for the price of catoptions on a loss index which follows a jump-diffusion process.Having priced simple cat bonds under the Black and Scholes(1973) assumptions in the first section of their paper, Loubergéet al. (1999) subsequently consider a compound Poisson processin combination with a simple binomial model for the interestrate. Another no-arbitrage pricing model for cat bonds built upona compound Poisson process is presented by Baryshnikov et al.(2001) and Lee and Yu (2002) additionally contemplate default riskof the cat bond issuer as well as issues of moral hazard and basisrisk, adopting a structural credit model with stochastic interestrates as in Cox et al. (1985). Furthermore, Bakshi andMadan (2002)provide a closed-form solution for (PCS) cat option prices basedon the assumption that losses follow a mean-reverting Markovprocess with one-sided jumps. A compound doubly stochasticPoisson process (Cox process) is used by Burnecki and Kukla(2003) to value zero-coupon and coupon cat bonds and by Dassiosand Jang (2003) to model stop-loss reinsurance contracts andcat derivatives. Muermann (2003) assumes a compound Poissonloss process and values cat derivatives relative to observedpremiums of insurance contracts on the same underlying risks.Moreover, in his model for options on a PCS index, Schmidli(2003) distinguishes between catastrophe occurrence and lossdevelopment period, which are governed by a compound Poissonprocess and a Geometric Brownian Motion, respectively. A barrieroption framework for the price of a cat bond is proposed byVaugirard (2003a,b, 2004), who assumes a jump-diffusion processfor the underlying physical index and stochastic interest ratesbased on a Vasicek (1977) model. In addition, Cox et al. (2004)consider the valuation of double trigger catastrophe put optionswhen losses are generated by a compound Poisson process andJaimungal andWang (2006) generalize theirwork by incorporatingstochastic interest rates. While Lee and Yu (2007) apply insightsfrom their earlier work on cat bonds to reinsurance contracts,Biagini et al. (2008) use a Fourier transform to derive an analyticalsolution for the price of an option with catastrophe occurrenceand loss development period. Muermann (2008) applies a catcall option model based on a compound Poisson process for theunderlying loss index to extract the market price of insurance riskfrom quotes of traded cat derivatives. Chang et al. (2008, 2010)generalize their concept from the mid-1990s from a completemarket continuous-time to an incomplete market discrete-timeframework to price Asian-style cat optionswith a doubly-binomialmodel. Additionally, they consider stochastic Poisson intensitiesdescribed by a mean-reverting Ornstein–Uhlenbeck process andreduce the computational effort through a stochastic time changefrom calendar to claim time. Härdle and Cabrera (2010) price a

522 A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536

hybrid cat bond for earthquakes, assuming a doubly stochasticPoisson process for the flow of catastrophic events. Finally, Wuand Chung (2010) employ a doubly stochastic Poisson processwithOrnstein–Uhlenbeck intensity in combination with a Cox et al.(1985)model for the interest rate and the framework of Jarrow andYu (2001) for counterparty default risk to price catastrophe bonds,futures, and options.

3. Catastrophe swaps

3.1. Contract design

Catastrophe swaps are financial instruments through whichnatural disaster risk can be transferred between counterparties.In a typical contract, the protection buyer (fixed payer) agrees tomake periodic premiumpayments to the protection seller (floatingpayer) in exchange for a predetermined binary compensationpayment4 contingent on the occurrence of a trigger event (coveredevent) in the covered territory (see, e.g., Swiss Re, 2006). Whilethe covered territory defines the country or geographic region inwhich a catastrophe has to strike in order to be relevant under theswap transaction, a trigger event is determined by the so-calledreference peril (reference catastrophe), the associated referencelosses (reference amount), and the contract’s thresholds. The termreference peril means the type of disaster which is covered underthe swap, e.g., wind storms including namedhurricanes.Wheneversuch a catastrophe occurs, the appointed loss report providerassigns a serial number to it and publishes an initial estimateof the resulting insurance industry losses. This loss estimate issubsequently refreshed on a regular basis, with the final lossreport usually being released no later than six months after theevent. In this regard, an important characteristic of the cat swapis that the reference losses from different natural disasters are notaggregated but tracked separately. Hence, the trigger mechanismrelates to the losses of individual catastrophes, not a sum oflosses. Cat swaps typically exhibit two thresholds: an event anda slightly higher acceleration threshold.5 If, during the term ofthe contract, a final loss estimate for a reference peril reachesthe event threshold (attachment level), it results in an immediatepayoff to the protection buyer and the subsequent terminationof the contract. Similarly, the payoff under the swap is triggeredinstantaneously by an interim loss estimate in excess of theacceleration threshold. Finally, the protection buyer receives apayoff at maturity if an interim loss estimate is equal to or higherthan the event threshold.

A concrete example for cat swap contracts offered in currentmarket practice are ‘‘Deutsche Bank Event Loss Swaps’’ (ELS).6ELS for US wind storms, i.e., hurricanes and tornadoes, have beenlaunched in late 2006 and are available with thresholds of USD20 billion, USD 30 billion or USD 50 billion, while the attachmentlevels for earthquake-based contracts can be set at USD 10 billionand USD 15 billion. The standard maturity is one calendar yearand notional amounts are staggered in lots of USD 5 million.Similar standardized contracts for US wind and earthquake eventscalled ‘‘Swiss Re Natural Catastrophe Swaps’’ (SNaCSTM) have beenlaunched by Swiss Re (see Swiss Re, 2009). For US transactions,the Property Claim Services (PCS) division of Insurance ServicesOffice, Inc. (ISO) acts as loss report provider. PCS has accessto a nationwide network of industry representatives, claim

4 This is usually the full notional value. Alternatively, the payoff profile can belinearly increasing in the underlying losseswithin the layer between an attachmentlevel and a cap.5 The acceleration threshold is commonly set ten percent above the event

threshold. See ISDA documentation template.6 This information is based on a press release by Deutsche Bank.

departments and adjusters, insurance agents, meteorologists, andpublic authorities through which it gathers loss information. InJanuary 2009, several leading firms in the insurance industry havefounded the European index provider PERILS AG, which collectsinsurance data and provides a benchmark measure for lossescaused by natural catastrophes in Europe.7

An alternative to the above mentioned contract structure is atransaction format termed pure risk swap (or portfolio swap). Ina pure risk swap, two (re)insurance companies exchange uncorre-lated catastrophe risk exposures from their existing books in orderto improve portfolio diversification and potentially reduce regu-latory capital requirements (see Bruggeman, 2007). Thereby, in-surers whose business is locally concentrated in an area which isparticularly susceptible to natural disasters can replace a portionof their core risk with another type of peril that they may not beable to access directly. Risk swaps can be executed through inter-mediaries, via the web-based Catastrophic Risk Exchange (CATEX)or directly in the OTC market (see Mutenga and Staikouras, 2007;Cummins, 2008). Like standard catastrophe swaps, these contractsare usually set up such that the present values of the two swapsides exactly balance and there are no up-front payments betweenthe counterparties. Instead, money is only exchanged in case of aqualifying event. This requires an alignment of the triggers as wellas precise riskmodeling in order tomatch expected losses throughthe configuration of the terms and conditions of the contract. Aprominent risk swap example is a 2003 transaction in which Mit-sui Sumitomo Insurance swapped USD 100 million of Japanese ty-phoon risk against USD 50 million of North Atlantic hurricane riskand USD 50 million of European windstorm risk with Swiss Re(see Cummins, 2008). Since risk swaps reference the counterpar-ties’ insurance portfolios, they are indemnity-based and thus donot constitute financial instruments. Such contracts are not the fo-cus of this paper.

3.2. Market development

Although cat bonds and insurance derivatives have been aroundfor almost two decades, the market for catastrophe swaps is veryyoung and, due to its OTC character, information on transactionsis currently largely anecdotal (see Cummins and Weiss, 2009).Nevertheless, industry experts claim that the market size is in-creasing rapidly (see, e.g., Cummins, 2008) and the World Eco-nomic Forum (2008) estimates that cat swaps, together with in-dustry loss warranties (ILWs), currently account for about USD10 billion in outstanding notional. In May 2009, the Interna-tional Swaps and Derivatives Association (ISDA) released a doc-umentation template for catastrophe swap transactions refer-encing US windstorm events.8 The goal of these standardizeddefinitions for key terms is to reduce uncertainty, improve liq-uidity and transparency, and encourage growth in the market.ISDA documentation for other reference catastrophes, such asCalifornia earthquakes, is already planned. The introduction ofISDA standards is an important step with regard to the devel-opment of the catastrophe swap market as well as the accep-tance of the instrument among investors and is expected to re-sult in increasing trading volumes (see Swiss Re, 2009). To date,swap counterparties are mainly insurance and reinsurance com-panies. Yet, as for insurance futures, new participants, such as in-vestment banks, hedge funds, and other institutional investors,could soon be encouraged to establish themselves as market

7 Founding shareholders of PERILS AG include Allianz SE, AXA, AssicurazioniGenerali, Groupama, Guy Carpenter, Munich Re, Partner Re, Swiss Re, and ZurichFinancial Services. For more information refer to the company website.8 This template can be accessed on the ISDA website.

A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536 523

makers.9 Illiquidity of swaps has been cited as a significant short-coming relative to tradable catastrophe securities and insuranceoptions (see, e.g., Cummins andWeiss, 2009). However, increasingstandardization and the introduction of ISDA standards now alsoenables swap counterparties to assign or unwind a contract in or-der to close out their position, thus enhancing liquidity.

3.3. Areas of application

By no-arbitrage reasoning, catastrophe swaps should behavesimilar to cat bonds, i.e., they should share their properties ofcomparatively high yields and immaterial return correlation withother asset classes.10 Consequently, from an investor’s perspectivecatastrophe swaps are an attractive means to gain syntheticexposure to natural disaster risk, i.e., without requiring to fundthe purchase of a cat bond. In addition, it is not necessary for theprotection buyer to actually hold a book of insurance contracts, nordoes the protection seller need to have the status of a regulatedinsurance entity to be eligible as swap counterparty. Therefore,apart from hedging insurance risks, catastrophe swaps could alsobe applied for investment purposes. An example are negative basistrades between cat swaps and bonds. This risk-arbitrage strategy,which is common in the credit markets, aims at exploiting pricediscrepancies between the cash and derivative instrument. If a catbond spread is sufficiently larger than the spread on an adequatelymatching catastrophe swap, i.e., if the basis is negative, a positivecarry can be locked-in by buying the bond and simultaneouslybuying protection under the swap agreement.11 The idea is thatthe occurrence of an event should trigger both instruments suchthat the loss on the bond is (at least partially) compensated bythe payoff from the swap. Another potential field of applicationfor cat swaps are synthetic Collateralized Debt Obligations (CDOs)of catastrophic risks. In general, a CDO is a securitization of apool of assets. These assets are purchased and held by an SPV,which funds the transaction through the issuance of rated securitytranches of differing seniority. In contrast to a so-called true sale orcash CDO structure, a synthetic CDO does not involve the physicaltransfer of assets. Instead, the SPV gains risk exposure by sellingprotection under swap contracts. While offering substantial risktransfer capacity to (re)insurers, cat CDOs enable investors to takea position in a diversified portfolio of natural disaster risks byselecting a tranche which matches their specific risk appetite. Aconcrete example is the USD 200 million transaction ‘‘Fremantle2007-I’’ arranged by ABN AMRO, which featured three classes ofnotes rated AAA, BBB+ and BB− by Fitch Ratings and utilized catswaps to transfer the risk to the SPV. For a further discussion ofinsurance risk CDOs refer to Forrester (2008).

3.4. Accounting and regulation

In contrast to ILWs, catastrophe swaps are financial instru-ments, not insurance contracts. Their regulatory and accountingtreatment is unambiguous: Under International Financial Report-ing Standards (IFRS) as well as US GAAP, pure index contracts,such as catastrophe swaps, are not eligible for reinsurance account-ing and consequently do not influence the underwriting result. In-stead, they have to be accounted for at fair value, which can leadto elevated volatility in the income statement of the (re)insurer,

9 See IFEX website for additional information on the insurance futures marketstructure.10 These characteristics of cat bond returns have been documented by severalauthors, see, e.g., Litzenberger et al. (1996), Bantwal and Kunreuther (2000), GuyCarpenter (2008) or Cummins and Weiss (2009).11 In analogy to the credit markets, the basis can be defined as catastrophe swapminus catastrophe bond spread.

since technical liabilities are currently not marked to market (see,e.g., World Economic Forum, 2008). In addition, the current Sol-vency framework in Europe as well as the National Association ofInsurance Commissioners (NAIC) regulation in the US only acceptinstruments with an indemnity trigger (i.e., instruments withoutbasis risk) as (re)insurance contracts. However, it appears that un-der the upcoming Solvency II, which is currently still being refined,all instrumentswhich accomplish an effective economic risk trans-fer could result in regulatory capital relief (see, e.g., Swiss Re, 2009).In the US, on the contrary, new reserving rules, employing a moreeconomic stance with regard to risk mitigation instruments, arecurrently not envisioned. Klein andWang (2009) believe this to bea major impediment for US insurers to use swaps on a larger scale.

3.5. Comparison to other risk transfer instruments

Although they do not include an indemnity trigger, catastropheswaps are economically largely equivalent to ILWs. In solelyreferencing indices, they are not subject to issues of asymmetricinformation and moral hazard and, as a result, do not requirean intensive underwriting process. Thus, they can serve as cost-effective substitutes for traditional reinsurance contracts. Verymuch like ILWs and most cat bonds, catastrophe swaps doexpose the protection buyer to basis risk if the transaction isaimed at hedging a specific portfolio of underwritten insurancecontracts. Basis risk arises as the industry index is typically notperfectly correlated with the losses which the (re)insurer sufferson his book of business. Cat swaps are highly standardized andavoid the structural complexities and costs associated with theissuance of full-fledged insurance securitizations, such as settingup an SPV and entering a total return swap. Consequently, fromthe perspective of a (re)insurance company, they are simple toinitiate and can be executed much more rapidly than a cat bondtransaction. Compared to ILWs, cat swaps bear the additionalpotential of becoming more liquidly tradable once the marketfully takes off. Swaps in general are unfunded transactions andthus by design not fully collateralized. However, since fluctuationsin a contract’s mark-to-market value will effect regular margincalls between the swap partners, counterparty default risk for theprotection buyer is limited. Table 1 summarizes the main points ofthis comparison.

4. Pricing model

Below we introduce a two-stage pricing framework forcatastrophe swaps. Ex-ante themain risk drivers of the instrumentare the random number of natural disasters, their timing, and theassociated final loss estimates.12 Apart from determining the faircat swap spread before the occurrence of a catastrophe, however,market participants will also need to value their contracts inthe special situation when a catastrophe has already struck andan initial loss estimate has been published. In this case, whichwill be termed the loss reestimation phase in the following,the timing of the catastrophe and an approximate range for thecorresponding losses are known. Yet, there is still uncertainty asto the further development of the loss estimates until the finalloss report or maturity. Consequently, we distinguish between thepricing of cat swaps before a catastrophe (ex-ante) and during thereestimation phase.13 Both model components will be developedin a continuous-time contingent claims framework without bid-ask spreads, transaction costs, short-selling constraints, taxes, orother market frictions.

12 Refer to Section 3.1.13 Practitioners call exchange traded catastrophe instruments ‘‘live cat’’ beforea catastrophic event and ‘‘dead cat’’ during the loss reestimation phase. Seewww.theifex.com.

524 A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536

Table 1Catastrophe swaps, cat bonds, ILWs, and reinsurance contracts in comparison.

Catastrophe Swaps Cat Bonds ILWs Reinsurance contracts

Triggers Industry index Indemnity-basedIndustry indexPure parametricParametric indexModeled loss

Double trigger: industry index andindemnity trigger

Indemnity-based

Moral hazard None None if pure index trigger Low HighBasis risk High None if pure indemnity trigger High NoneStandardization Very high Low High LowTransaction cost Low High Low HighCounterparty default risk Partially collateralized Low (collateralized) Collateralization possible Collateralization possibleAccounting treatment Financial instrument Depends on trigger Reinsurance Reinsurance

4.1. Risk-neutral valuation of catastrophe derivatives

Since catastrophe swaps are not insurance contracts, it seemsadequate to price them with financial rather than actuarial ap-proaches.14 Based on the no-arbitrage principle modern optionpricing theory as constituted by Black and Scholes (1973) as wellas Merton (1973) has established the preference free pricing ofderivative instruments under the risk-neutral (equivalent mar-tingale) measure. The absence of arbitrage opportunities in thecapital markets implies the existence of such an equivalentmartingale measure. For a single unique equivalent martingalemeasure to arise, however, markets also need to be complete,meaning that all contingent claims are replicable through availablesecurities (see Harrison and Kreps, 1979). Hence, the general in-completeness of insurance markets prevents the uniqueness ofthe equivalent martingale measure. Nevertheless, the literatureon catastrophe derivatives and bonds is dominated by contingentclaims valuation frameworks, irrespective of the non-traded un-derlying and the fact that natural disaster risk cannot be hedgedwith traditional securities.15 Different solutions have been pro-posed to tackle the ambiguity with regard to the change ofmeasure.

Some authors assume that market completeness is preserveddue to the existence of other tradable and sufficiently liquidinstrumentswhich are driven by the same source of randomness.16Consequently, replicating portfolios can be formed and a uniquerisk-neutral measure is obtainable by recovering a market price ofcatastrophe risk from observed quotes.17 While early attempts byCBOT to establish a liquid market for exchange-traded catastrophederivatives have failed, several exchanges have recently re-introduced various types of contracts. Among these are optionsand futures on the CME hurricane index and futures on the PCSindices (see Swiss Re, 2009). Although the market is still juvenile,its long-term outlook is promising.18 Alternatively, investorscould turn to OTC catastrophe derivatives or cat bonds in orderto mimic movements of cat swap positions. In addition, asargued by Muermann (2003), insurance and reinsurance contractspermit indirect trading in the underlying catastrophe risk. Lastly,

14 Although a discussion with industry experts indicated that actuarial pricingcurrently seems to prevail in practice.15 Refer to the literature review in Section 1.16 Examples are Cummins and Geman (1994, 1995), Chang et al. (1996), Gemanand Yor (1997), Baryshnikov et al. (2001), Muermann (2003), Vaugirard (2003a,b,2004), Muermann (2008) and Chang et al. (2008, 2010).17 A brief illustration of the reasoning behind this proceeding is given in theAppendix. Concrete examples for arbitrage portfolios in the context of PCS optionscan be found in the empirical study of Balbás et al. (1999), who demonstrate thegeneral applicability of financial theory to the sphere of catastrophe derivatives.18 However, due to currently still restricted trading volumes and liquidity, thesemay not be ideal instruments to approximate an instantaneous riskless portfolioyet.

Cummins and Geman (1995) and Vaugirard (2003a,b, 2004)mention that certain energy, commodity, and weather derivativescould be suitable to track continuous changes in catastrophe losses,since geological and meteorological determinants of insuranceclaims impact the value of these instruments, too.

Another common approach to specify a unique pricingmeasuredespite an incompletemarkets set-up dates back toMerton (1976).Analogous to his reasoning, natural disasters can be treated asunsystematic shocks to the overall economy which are fullydiversifiable, implying risk-neutrality of themarket participants.19Thus, there is no cat risk premium and model parameters underthe physical and equivalent martingale measure are identical.This stance is supported by the empirical studies of Hoyt andMcCullough (1999) as well as Cummins and Weiss (2009), whoprovide evidence for the zero-beta characteristics of catastrophe-linked instruments.

Finally, the issue of incomplete markets can be overcome byselecting a particular change of measure such as the well-knownEsscher transform (see Gerber and Shiu, 1996).20 Another suchdistortion operator for the risk-neutral valuation of insurancecontracts has been introduced by Wang (2000).

We follow Biagini et al. (2008) as well as Wu and Chung(2010) in not further discussing the choice of martingales and thechange from the physical measure P to the risk-neutral measureQ.Instead, we will assume that Q has been predetermined accordingto one of the above-mentioned alternatives and directly proceed toa risk-neutral formulation of ourmodel framework. The equivalentmartingale measure Q is restricted to the class which only correctsparameters, while stochastic processes and distributions retain thesame characteristics as under P.

4.2. Pricing catastrophe swaps ex-ante

The first, more general component of our pricing approach ismeant to be applied before a catastrophe has occurred in thecovered territory. As mentioned above, the major uncertainty inthis phase of a cat swap transaction centers around the stochasticnumber and timing of natural disasters during the term of thecontract as well as the ultimate losses they cause. Hence, wewill focus on these underlying sources of randomness, whileabstracting from any uncertainty surrounding the reestimationprocess of claims. Let (Ω, F , Q) denote a probability space withthe set of all possible outcomes Ω , a filtration F for the relevantsubsets of Ω , and the equivalent probability measure Q. In linewith the prevailing practice in the literature on catastrophederivatives, we assume that the number of natural disasters nt,t+1t

19 This stance is adopted in Bakshi andMadan (2002), Lee and Yu (2002), Cox et al.(2004), Jaimungal and Wang (2006), as well as Lee and Yu (2007).20 See, e.g., Embrechts and Meister (1997), Schmidli (2003) and Dassios and Jang(2003).

A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536 525

in any time interval (t, t+1t] is Poisson distributedwith intensityλt,t+1t :

nt,t+1t ∼ Pλt,t+1t

, ∀t ∈ (0, T − 1t], (1)

where λt,t+1t = t+1tt λ(u)du. Furthermore, we follow Chang

et al. (2008, 2010) as well as Wu and Chung (2010) and allowfor cyclicality in the occurrence of catastrophes by incorporatinga stochastic Poisson intensity, which is assumed to adhere to themean-reverting Ornstein–Uhlenbeck process

dλ(t) = κ (µλ − λ(t)) dt + σλdWQλ (t), (2)

with mean reversion rate κ , long-term mean µλ, volatility of theintensity σλ, and a standard Brownian motion dWQ

λ (t). Thus, thearrival of natural disasters is governed by a doubly stochasticPoisson process (Cox process), i.e., two-stage randomizationprocedure: the Ornstein–Uhlenbeck process in Eq. (2) generatesthe intensity for the Poisson distribution of nt,t+1t . Intuitivelythis makes sense particularly for periodic climate patterns suchas the El Niño phenomenon which recurs on average in five yearintervals or the annual Atlantic hurricane season in the US fromJune to November. However, it also seems suitable to capturethe typical clustering of earthquakes. We aim to provide someempirical support for this assumption in Section 5.3.

Each catastrophe i is associated with a stochastic final lossestimate, represented by positive independent and identicallydistributed (i.i.d.) random variables Yi with distribution functionFY (x). We further assume that nt,t+1t and Yi are stochasticallyindependent and that there is no time delay between theoccurrence of the catastrophe and the issuance of the final lossreport.21 Consequently, the aggregate final loss estimates due tonatural disasters in any time interval (t, t + 1t] can be expressedas a compound Poisson process with expected value λt,t+1tEQ(Yi):

Lt,t+1t =

nt,t+1t−i=1

Yi, ∀t ∈ (0, T − 1t]. (3)

Recall from Section 3.1 that the payoff of a cat swap transactionis triggered immediately when the final loss estimate for a singlenatural disaster equals or exceeds the event threshold. Hence,in contrast to the usual procedure for other cat instruments, wemust not aggregate losses from different events over the wholeterm of the contract. Instead, each final loss estimate is separatelycompared to the event threshold at the time of its occurrence.Since the instrument terminates prematurely if a trigger event hasbeen identified, its timing is crucial for valuation purposes anda pricing model needs to capture path dependency. We achievethis by sequentially reevaluating the loss process in Eq. (3) forinfinitesimally small time steps dt from t = 0, i.e., the outset ofthe contract until its maturity t = T :

lim1t→0

Lt,t+1t = Lt,t+dt ≡ dLt , ∀t ∈ (0, T − 1t]. (4)

Consequently, instead of one process for the whole term, wegenerate a series of compound Poisson processes. Under this set-up, arrivals in non-overlapping intervals are independent, theprobability of exactly one catastrophe per marginal interval oflength dt is approximately λt,t+dt , and the probability ofmore thanone catastrophe is negligible.22

21 It is straightforward to extend the model with a deterministic or stochasticwaiting time between occurrence of the catastrophe and the issuance of the finalloss report. We abstain from this additional layer of complexity as it is neitheremphasized in the literature nor central to the cat swap pricing problem.22 Alternatively the occurrence of a catastrophe at each time step could bemodeled as a Bernoulli trial, implying a binomially distributed sum. With regard tothe total number of events until maturity this differentiation is less relevant, since,for a large number of trials, the Binomial converges to the Poisson distribution.

As described in Section 2, a cat swap consists of a fixed leg,which comprises the streamof regular premiums by the protectionbuyer, and a floating leg, which is the compensation payment bythe protection seller contingent on a trigger event. Swap pricinggenerally entails the separate valuation of each leg in a transaction,with the goal of balancing their present values the through the fairspread scat:PVfloating = PVfixed(scat). (5)

We define the first passage time (or stopping time) τ for theseries of compound Poisson processes as the earliest instant inwhich a final loss estimate is equal to or higher than the eventthreshold ET :τ ≡ inft | Lt,t+dt ≥ ET . (6)

Consider a catastrophe swap contract withmaturity T , notionalN , and a payoff which is determined as a preset percentage α of thenotional. Assume that a trigger event can occur at any given pointin time and causes an immediate payoff under the swap contract.23Then the following is an expression for the present value of thefloating leg:

PVfloating = EQ0 [e−rταN1τ≤T ]. (7)

Here, 1τ≤T is the indicator function which equals 1 if τ ≤ Tand 0 otherwise. In addition, r is the instantaneous risk-free rate,i.e., the term structure is assumed to be flat and deterministic.Although relatively recent publications in the context of cat bondvaluation have introduced stochastic interest rates based on thewell-known term structure models of Vasicek (1977) or Coxet al. (1985) (see, e.g., Lee and Yu, 2002; Vaugirard, 2003a,b; Wuand Chung, 2010), we decide to dismiss this possibility in favorof computational efficiency. Since catastrophe swap contractsare exclusively available with one year maturities, the effect ofrandom changes in the risk-free term structure should have anotably lesser impact than onmediumor longer-term instruments.Consequently, this assumption does not severely influence ourresults.24

Furthermore, the fixed leg of a cat swap consists of regularpremiums as well as an accrual payment in case the first passagetime does not coincide with a premium payment date:PVfixed(scat) = PVpremiums(scat) + PVaccrual(scat). (8)

Given a final loss estimate has not exceeded the eventthreshold, the buyer of cat swap protection makes payments ofscatN1ti on premium dates ti, where i = 1, . . . , n, 1ti is the lengthof a premium period (ti − ti−1), and scat is the fair spread we arelooking for.

Consequently, we can value the premium part of the fixed legas follows:

PVpremiums(scat) =

n−i=1

e−rtiEQ0 [scatN1ti1τ>ti ]. (9)

Finally, the present value of the expected accrual payment forthe time period since the last premium date (τ − ti−1) can beexpressed as:

PVaccrual(scat) = EQ0 [e−rτ scatN(τ − ti−1)1ti−1≤τ≤ti ]. (10)

Evidently, pricing the cat swap involves solving a first passagetime problem. Due to the compound Poisson process in Eq. (3),however, a closed-form solution for the first passage time cannotbe derived (see Kou and Wang, 2003). Therefore, one needs toresort to Monte Carlo techniques for an estimation of the fairspread scat.

23 Since we model the final loss estimate for each catastrophe, we do not needto allow for a development period of the losses after τ . However, through thisassumption we do abstract from delays in payment collection.24 Young (2004) argues that term structure models are appropriate for instru-ments with a maturity of more than one year.

526 A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536

4.3. Pricing catastrophe swaps in the loss reestimation phase

After a cat swaphas been entered, itsmark-to-market valuewillfluctuate in accordance with the occurrence of natural disastersas well as the development of their associated loss estimates.More specifically, a particularly sharp increase in value shouldbe observed when interim loss estimates approach the contract’sthresholds. However, the ex-ante pricing approach introducedabove does not reflect this sensitivity of the instrumentwith regardto interim loss reports. Therefore, it needs to be complementedto price cat swaps during the reestimation phase, i.e., after acatastrophe has occurred and an initial loss estimate has beenissued. At that time it is still unclear what the final loss estimatewill be and whether the preset acceleration threshold will beexceeded by an interim loss estimate before maturity. In theremainder of this section, we aim to capture this uncertainty withregard to the reestimation of losses by means of a parsimoniousbarrier option framework under which closed-form expressionsfor the cat swap spread can be derived. Assume that, under the risk-neutralmeasureQ, the dynamics of the interim loss estimates Li(t)referenced by the catastrophe swap are adequately described by aGeometric Brownian Motion:

dLi(t)Li(t)

= rdt + σdWQLi (t), (11)

with volatility σ and a standard Q-Wiener process dWQLi (t). The

choice of a diffusion process for the development of catastrophicloss estimates is common in the literature on cat bond andderivative pricing (see, e.g., Bakshi and Madan, 2002; Schmidli,2003; Biagini et al., 2008). The Geometric Brownian motion inparticular has been applied by quite a few authors to model theaccrual of losses and unpredictability in reporting over time.25Apart from that, it ensures analytical tractability and the resultinglognormally distributed estimates are in line with the empiricalfindings of Levi and Partrat (1991) and Burnecki et al. (2000) forPCS loss data.

We define the distance to the acceleration threshold AT (≥ET )

at time t as follows:

DAT (t) =ATLi(t)

. (12)

Similarly, we will refer to the ratio of ET to the loss estimate Li(t)as the distance to the event threshold:

DET (t) =ETLi(t)

. (13)

Again, we need tomatch both legs of the swap transaction throughscat:

PVfloating = PVfixed(scat). (14)

We begin with the floating leg. Assume that at time s (0 ≤

s ≤ T ) an initial or interim loss estimate Li(s) for a specificcatastrophe is available. Although Li(s) is unknown at the outset ofa contract, i.e., in t = 0, it is deterministic during the reestimationphase. The interim loss process determined by Eq. (11) startsat Li(s) and the protection buyer receives a payment wheneveran interim loss estimate Li(t) during the remaining term of thecontract breaches the acceleration threshold for the first time, i.e.,

25 Examples are Cummins and Geman (1994, 1995), Geman and Yor (1997),Loubergé et al. (1999), Gatzert et al. (2007) and Wu and Chung (2010).

when DAT (t) hits unity. This payoff profile equals a binary up-and-in one-touch barrier option on the interim loss estimates. Definethe first passage time τd for the diffusion process as26:

τd ≡ inft | Li(t) ≥ AT . (15)

From Rubinstein and Reiner (1991a,b) we know that in thepresent setting, the following analytic expression for the firstpassage time density applies:

h(τd) =ln(DAT (s))√2πστ

3/2d

exp

−12

− ln(DAT (s)) +

r −

σ 2

2

τd

σ√

τd

2 . (16)

Now, recall from Section 3.1, that a compensation payment αNby the protection seller can also be due at maturity T in case aninterim loss estimate is equal to or higher than the event thresh-old, i.e., Li(T ) ≥ ET which implies DET (T ) ≤ 1. Thus, ET can beinterpreted as the strike price of a binary European-type call op-tion. However, in order not to price certain states twice, this optionalso needs to include a knock-out feature so that it lapses when-ever Li(t) ≥ AT , i.e., when the payoff from the previously discussedone-touch option is triggered. To see this consider the case whereLi(T ) = AT ≥ ET . Here both the up-and-in one-touch and a sim-ple binary European call option would pay off. Hence, in order tovalue the floating leg,weneed to combine the up-and-in one-touch(UAIone-touch) with a binary up-and-out call (UAOcall) option whichpays off if and only if Li(T ) ≥ ET and AT has not been hit duringthe term of the contract:

PVfloating = UAIone-touch + UAOcall. (17)

Applying results from Rubinstein and Reiner (1991a,b), theprice of the up-and-in one-touch (binary) in t = s can be expressedas:

UAIone-touch =

∫ T

sαNe−rτdh(τd)dτd

= αN∫ T

se−rτdh(τd)dτd

= αNQ (T ), (18)

with

Q (u) =

∫ u

se−rτdh(τd)dτd

= DAT (s)a+bΦ(d1) + DAT (s)a−bΦ(d2),

where s < u, Φ(x) is the standard normal cumulative distributionfunction (cdf), and

a =rσ 2

−12, b =

r −

σ 2

2

2+ 2rσ 2

σ 2,

d1 =− ln(DAT (s)) − bσ 2u

σ√u

,

d2 =− ln(DAT (s)) + bσ 2u

σ√u

.

26 Due to the design of typical catastrophe swap contracts with a binary payoffas soon as the acceleration threshold is reached, we do not need to allow for adevelopment period of the losses after τd has occurred. Again, we do abstract fromdelays in payment collection. Also note that the model ignores so-called extensionthresholds which, if exceeded by the interim loss estimates, cause a prespecifiedextension of the contract’s maturity to allow for a further accumulation of losses.

A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536 527

Furthermore, the up-and-out cash-or-nothing call (binary) int = s is worth (see Rubinstein and Reiner, 1991a,b):

UAOcall = αN (B1(T ) − B2(T ) + B3(T ) − B4(T )) (19)

with

B1(T ) = e−rTΦ(x1 − σ√T ),

B2(T ) = e−rTΦ(x2 − σ√T ),

B3(T ) = e−rTDAT (s)2aΦ(−y1 + σ√T ),

B4(T ) = e−rTDAT (s)2aΦ(−y2 + σ√T ),

and

x1 =− ln(DET (s)) + (a + 1)σ 2T

σ√T

,

x2 =− ln(DAT (s)) + (a + 1)σ 2T

σ√T

,

y1 =ln(AT 2/(Li(s)ET )) + (a + 1)σ 2T

σ√T

,

y2 =ln(DAT (s)) + (a + 1)σ 2T

σ√T

.

The fixed leg again comprises the stream of spread paymentsand an accrual which accounts for the fact that the contract can betriggered in between two scheduled premium dates:

PVfixed(scat) = PVpremiums(scat) + PVaccrual(scat). (20)

The protection buyer pays scatN1ti on the remaining premiumdates ti (s < ti ≤ T )with1ti being the length of a premium period(ti − ti−1). Defining the survival probability of the contract fromtime s to time u (i.e., the probability that a trigger event does notoccur before u) as (1 − H(u))withH(u) =

us h(τd)dτd, the stream

of premium payments has the following value in t = s:

PVpremiums(scat) =

n−i=1

scatN1tie−rti (1 − H(ti))

= scatNn−

i=1

1tie−rti (1 − H(ti))

= scatNΣpremiums, (21)

where

Σpremiums =

n−i=1

1tie−rti (1 − H(ti)) ,

and (see, e.g., Vaugirard, 2003b)

H(u) =

∫ u

sh(τd)dτd = DAT (s)2aΦ(z1) + Φ(z2),

with

z1 =

− ln(DAT (s)) −

r −

σ 2

2

u

σ√u

,

z2 =

− ln(DAT (s)) +

r −

σ 2

2

u

σ√u

.

Moreover, the present value in t = s of the expected accrualpayment can be expressed as:

PVaccrual(scat) =

n−i=1

∫ ti

ti−1

scatN(τd − ti−1)e−rτdh(τd)dτd

= scatN

n−

i=1

∫ ti

ti−1

τde−rτdh(τd)dτd

−

n−i=1

ti−1

∫ ti

ti−1

e−rτdh(τd)dτd

= scatN

n−

i=1

(J(ti) − J(ti−1))

−

n−i=1

ti−1(Q (ti) − Q (ti−1))

= scatNΣaccrual, (22)

where

Σaccrual =

n−

i=1

(J(ti) − J(ti−1)) −

n−i=1

ti−1 (Q (ti) − Q (ti−1))

,

and (see, e.g., Gil-Bazo, 2006)

J(u) =

∫ u

sτde−rτdh(τd)dτd

=− ln(DAT (s))

bσ 2

DAT (s)a+bΦ(d1) − DAT (s)a−bΦ(d2)

.

Inserting (17)–(22) into Eq. (14), the fair cat swap spread can becalculated as follows:

PVfloating = PVfixed(scat)

(UAIone-touch + UAOcall) = scatNΣpremiums + Σaccrual

scat =

(UAIone-touch + UAOcall)

NΣpremiums + Σaccrual

. (23)

During the loss reestimation phase, this spread should be addedto the result of the ex-ante pricing model.

Finally, we complete our discussion of cat swap valuation byillustrating the instrument’s sensitivity with regard to interim lossestimates. If Li(s) is too low, the subsequent reestimation processdoes not constitute a major risk driver and the instrument will bevirtually worthless under the barrier option approach. This is dueto the fact that small Li(s) are associated with large distances tothe thresholds, which implies low probabilities of the up-and-inone-touch being triggered and the up-and-out call ending up inthe money. This effect is shown in Fig. 1(a) by means of a simplenumerical example. It is based on a cat swap contract with ET =

20 bn, AT = 22 bn, a remaining time to maturity of T = 0.5,and up-front premium instead of regular spread payments, i.e., wesimply price the floating leg as shown in Eq. (17). The followingparameter values have been assumed: σ = 0.2, r = 0.02, α =

1,N = 1, 1t = 1/252 (discretization based on trading days peryear), and λt,t+1t = λ1t = 21t . Evidently, an increase in Li(t),i.e., a shrinking distance to ET and AT inflates the cat swap spreadexponentially since it results in a higher probability of a payoff tothe protection buyer. In Fig. 1(b) we have plotted the sensitivitywith regard to the remaining time to maturity T . The dotted linein the center shows the prices from the ex-ante approach (MonteCarlo simulation with 10,000 paths) while the solid line at the topand the dashed line at the bottom represent the additional valuesimplied by the barrier option approach for current a loss estimateof Li(s) = 18 bn and Li(s) = 17 bn, respectively. We observe thatfor Li(s) = 17 bn, the additional value suggested by the barrieroption approach is almost as high as the price we get from the ex-ante model across all T . For Li(s) = 18 bn it is already considerablyhigher.

528 A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536

(a) Sensitivity with regard to interim loss estimates. (b) Values from the barrier option and ex-ante pricing model.

Fig. 1. Illustration of the barrier option pricing approach.

4.4. Incorporating counterparty default risk

Apart from the underlying natural disaster risk, cat swapcontracts are susceptible to counterparty default risk. Twoscenarios are important in this regard. Firstly, if the protectionseller in a cat swap defaults during the term of the contract, theprotection buyer loses its hedge and needs to enter a new contractat the current market conditions, which could have changed to hisdisadvantage. Secondly, the protection seller might default whena trigger event under the cat swap contract has occurred, thusbecoming unable to provide the due compensation payment, infull or at least in part. Since the cat swap pricing model introducedabove does not incorporate counterparty default risk, the resultingspread should be adjusted accordingly. The most straightforwardway to accomplish this task is to subtract the credit default swap(CDS) spread, i.e., the most widely used measure of default risk inthe capital markets, from the default-free cat swap spread:

sdfcat = scat − scds, (24)

where sdfcat denotes the cat swap spread incorporating default riskand scds is the spread on a CDS which references the cat protectionseller and whose characteristics match those of the cat swapcontract under consideration. By additionally entering such a CDS,the protection buyer can virtually eliminate the default risk fromthe cat swap contract.27 Hence, Eq. (24) implies that the higherthe default risk of the protection seller in a cat swap transaction,the lower the spread the protection buyer is prepared to pay.28Due to the fact that the CDS market is highly liquid, all availableinformation with regard to the financial health of the respectivereference entity, including its dependence on the occurrence ofsevere natural disasters, should be included in the CDS spread.

Although CDS contracts are traded for many common cat swapcounterparties, a cat swap investor might face situations where nomarket quotes are available. In addition, existing contracts haveto be marked-to-market on a regular basis. Both of these casesrequire the application of a CDS pricingmodel,many ofwhich havebeen proposed in the finance literature. In the following, we brieflyrecapitulate a model proposed by Houweling and Vorst (2005),which seemswell suited to complement our cat swap set-up, since

27 Note that there is still some chance for a so-called double default, meaning thatthe protection seller under the CDS and the underlying reference entity default atthe same time. In general, however, the probability for such an event is negligible.28 On 29/07/2011 the average one year CDS spread for the reference entitiesquoted on the Thomson Reuters Insurance Linked Securities Community websitewas 94.45 basis points. This provides an indication for the size of the default riskinherent in a typical cat swap transaction.

it is based on discrete premiumdates, includes an accrual payment,and allows defaults to occur at any time (not only on premiumpayment dates). As for the cat swap contract, the starting pointto determine the fair CDS spread scds is the general swap pricingidentity:

PVcdsfloating = PVcds

fixed(scds). (25)

Let τ cds be the stochastic time of default, represented by the firstjump of a Poisson process with default intensity (hazard rate) λcds

under Q. To facilitate things, we assume that λcds is constant andconsequently independent of the catastrophe Poisson intensityλt,t+1t faced by the insurer.29 This approach allows for corporatedefaults to take place before, after or at the same time as a naturaldisasters. As indicated above, the CDS contract needs to match thecharacteristics of the cat swap. Hence, we have the same maturityT and notional N . Furthermore, in order for the payoffs in case ofa trigger and a default event to match, we choose α = (1 − δ).Here, δ denotes the recovery rate of the CDS, which we assume tobe fixed at inception of the contract. Thus, the present value of thefloating leg of the CDS can be expressed as follows:

PVcdsfloating = EQ

0 [e−rτ cds(1 − δ)N1τ cds≤T ] (26)

In addition, the fixed leg is based on regular premiums scdsN1ti,which are made on the same dates ti as those of the cat swap, aswell as an accrual payment scdsN(τ cds

− ti−1) in case the defaultdoes not coincide with a premium payment date:

PVcdsfixed =

n−i=1

e−rtiEQ0 [scdsN1ti1τ cds>ti ]

+ EQ0 [e−rτ cds

scdsN(τ cds− ti−1)1ti−1≤τ cds≤ti ]. (27)

5. Empirical analysis

5.1. Severity distributions for natural disasters in the US

In this section we want to select a distribution for the severityof final losses Yi, which is a crucial component of our ex-ante pricing model (see Section 4.2). The class of heavy-taileddistributions is particularly relevant in the context of catastrophe-related claims, since it allows to properly account for the low

29 Since, to the best our knowledge, there is no publicly available empiricalevidence for a statistically significant correlation between default and catastropherisks, we deem it acceptable to abstract from a correlation parameter in our model.

A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536 529

(a) Normalized annual hurricane losses. (b) Normalized annual earthquake losses.

Fig. 2. Natural disaster losses in the US (1900–2005).

frequency high severity character of natural disasters by assigningcomparatively large probabilities to extreme losses.30 Althoughsome authors have applied distributions with lighter tails, suchas the gamma and exponential distribution,31 extant empiricalevidence indicates that those are outperformed by heavy-taileddistributions. Levi and Partrat (1991), e.g., estimate the lognormal,Pareto, and exponential distribution based on US hurricane lossesbetween 1954 and 1986 and conclude that the former provides thebest fit. Their result is confirmed by the work of Burnecki et al.(2000), who examine the time series of the quarterly national PCSloss index from 1950 to 2000, fitting lognormal, Pareto, Burr, andgamma distributions. Consistent with this empirical evidence, thelognormal distribution remains by far themost common choice forthe modeling of catastrophe loss amounts in the literature.32 Incontrast to that, Milidonis and Grace (2008) analyze catastropheloss data for the state of Florida between 1949 and 2004 andfind that the lognormal is outperformed by the Pareto as wellas the Burr distribution. Based on a likelihood ratio test, theysubsequently decide to employ the latter.

Since their samples differ in terms of time period, geographicfocus, and loss normalization method, there is some ambiguityas to which of these previous empirical studies best suits ourpurpose. The article of Milidonis and Grace (2008) is the mostrecent and at least partially covers the last decade which has beenparticularly prominent with regard to Atlantic hurricane activity.However, it exclusively focuses on the state of Florida while Leviand Partrat (1991) and Burnecki et al. (2000) use national data. Inaddition, despite their larger dataset, Milidonis and Grace (2008)only report estimation results for the shorter time series from1990to 2004, thus ignoring once in a century events such as the 1906 SanFrancisco Earthquake or the Great Miami Hurricane of 1926. Yet,on a normalized scale, damages caused by these extreme incidentshave been shown to considerably exceed those of more recentnatural disasters.33

30 Heavy-tailed distributions have a density function which converges to zeromore slowly than an exponential function, i.e., compared to the exponentialdistribution they exhibit more probability mass in the tails. Typical examples arethe lognormal,Weibull (with a shape parameter<1), Pareto, and Burr distributions.For a more detailed discussion of this class of distributions see Bryson (1974).31 Bakshi and Madan (2002) as well as Dassios and Jang (2003), e.g., use theexponential distribution and Aase (1999) as well as Jaimungal and Wang (2006)employ the gamma distribution.32 Examples are Chang et al. (1996), Loubergé et al. (1999), Schmidli (2003), Leeand Yu (2002), Burnecki and Kukla (2003), Vaugirard (2003a,b, 2004), Lee and Yu(2007), and Wu and Chung (2010).33 In 2005 Dollars, the 1906 San Francisco Earthquake and the 1926 Great MiamiHurricane would have caused losses of USD 284 billion and USD 161 billion,respectively (see Pielke et al., 2008; Vranes and Pielke, 2009). This compares to USD116 billion for Katrina, which was the second most severe hurricane in US history.

Hence, we aim to conduct an empirical investigation ourselvesin order to select a distribution which is capable of adequatelycapturing the tail characteristics of catastrophe losses. We use thedatasets of Pielke et al. (2008) and Vranes and Pielke (2009), who,in their recent empirical work, normalize estimates of economiclosses fromallmajor US hurricanes and earthquakes between 1900and 2005 to 2005 Dollars. They have published their results inextensive appendices to the articles, thereby providing a reliablebasis for the estimation of catastrophe loss distributions.34 Theirnormalizationmethodology is based on inflation, wealth, and pop-ulation growth in the affected areas and has proven its capabilityto effectively adjust historical loss data for societal factors. The in-flation and wealth adjustment are based on the implicit gross do-mestic product price deflater and the fixed assets and consumerdurable goods statistic, respectively. Both magnitudes are avail-able from the US Bureau of Economic Analysis (BEA). Moreover,the population adjustment is conducted with county-level statis-tics from the US Department of Census. Fig. 2 shows the time seriesof normalized annual US hurricane and earthquake losses. The cor-responding histograms can be found in Fig. 3.

Although the histogram for earthquake losses is a little morepointy than for hurricanes, the shape of both indicates an asym-metrical distribution with a particularly heavy right tail. Hence,we will fit the lognormal, Burr, Pareto, and Weibull distribution tothe data by obtaining estimates for the parameters of their respec-tive cumulative distribution function (cdf):• lognormal cdf for µ ∈ R and σ > 0

F(x) =

∫ x

0

1√2πσu

e−(lnx−µ)2

2σ2 = Φ

lnx − µ

σ

, (28)

• Burr cdf with a (shape 1), b (shape 2), and c (scale) >0 (alsocalled Burr XII distribution)

F(x) = 1 −

1 +

xc

b−a

, (29)

• Pareto cdf with a (shape) and c (scale) >0 (also called Paretotype II or Lomax distribution)35

F(x) = 1 −

1 +

xc

−a, (30)

34 Note, however, that economic losses regularly exceed insured losses. As aconsequence, our fitted loss distributions must be viewed as quite conservative forthe purpose of pricing of cat instruments.35 Note that this is a special case of the Burr XII distribution with shape parameterb = 1. The Pareto distribution is often used in a generalized form with a third, so-called location parameter which allows for a lower limit above zero. We waive thatgenerality here.

530 A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536

(a) Histogram of normalized hurricane losses. (b) Histogram of normalized earthquake losses.

Fig. 3. Histograms of normalized natural disaster losses.

Table 2Descriptive statistics for (non-zero) disaster losses (1900–2005).

Hurricane losses (USD bn)

max. 161.3311 Quantiles (%)min. 0.0190 0 0.0190Mean 12.1362 25 0.4003Median 3.0555 50 3.0555s.d. 24.7117 75 13.2370Skewness 3.7786 100 161.3311Excess kurtosis 16.7313

Earthquake losses (USD bn)

max. 283.7353 Quantiles (%)min. 0.0019 0 0.0019Mean 9.0238 25 0.0526Median 0.1707 50 0.1707s.d. 41.2054 75 1.3028Skewness 6.1166 100 283.7350Excess kurtosis 37.5699

• Weibull cdf with a (shape) and c (scale) >0

F(x) = 1 − e−( xc )

a. (31)

In addition, the gamma and exponential distribution will beconsidered for comparison purposes:

• gamma cdf with a (shape), c (scale), and β = 1/c (rate) >0

F(x) =

∫ x

0

ua−1e−u/c

Γ (a)cadu, (32)

• exponential cdf for β (rate) >0

F(x) = 1 − e−βx. (33)

While some of these distributions are only defined for strictlypositive values, our dataset also includes years without damages.To tackle this issue we follow Burnecki et al. (2000), remove thecharacteristic spike at zero (see histograms), and estimate thecdfs on the subset of positive observations. Table 2 provides somedescriptive statistics with regard to the empirical distribution ofnon-zero losses.

In order to assess the fit of the above parametric distributions,we apply theKolmogorov–Smirnov and theAnderson–Darling test,which are based on a comparison of the empirical distributionfunction F(x) =

1n

∑ni=1 1xi≤x and the theoretical distribution func-

tion F(x, θ ), where θ represents a vector of parameter estimates.The null hypothesis for both tests is that the sample at hand comesfrom the specified distribution (H0: F(x) = F(x, θ )). Althoughthe chi-square goodness of fit test is also very common, it willnot be considered due to its sensitivity to the binning of the dataand its low power for small sample sizes. Since empirical samples

such as our dataset of cat losses generally contain a rather smallamount of extreme observations, it is questionable whether thechi-square test would reveal a severe misfit of the tail. The Kol-mogorov–Smirnov test, in contrast, is more suitable for small sam-ples. In addition, the Anderson–Darling test, a modification of theKolmogorov–Smirnov test, is one of the globally most powerfulgoodness of fit tests (see Levi and Partrat, 1991). It puts a higherweight on the tail of the distribution and is therefore the most ad-equate statistic for our purpose.

Tables 3 and 4 contain maximum likelihood estimation (MLE)results for the cdf parameters as well as the Kolmogorov–Smirnov(KSn) and Anderson–Darling (ADn) goodness of fit test statistics andtheir corresponding p-values. The lower the respective test statistic(the higher the p-value), the better the fit of the distribution. Asexpected, the exponential distribution is rejected on all reasonablesignificance levels. While the gamma distribution does very poorlywith respect to the earthquake dataset (p-values <0.01), it fits thehurricane losses surprisingly well. In fact, its ADn value for theearthquake dataset is even smaller than that of the more heavy-tailed Pareto distribution. Furthermore, the lognormal distributionseems to be a reasonable choice for both samples, although it isoutperformed in the tail by the Burr and the Weibull distributionfor hurricane losses and by the Burr and the Pareto distributionfor earthquake losses (see respective ADn values). Overall, the Burrdistribution exhibits the lowest Anderson–Darling statistics forboth datasets, leading us to conclude that it is the most suitablecandidate for the severity of natural disaster damages. Note thatthis confirms the empirical results of Milidonis and Grace (2008).Thus, in the context of the following analysis, we adopt a Burrloss severity distribution with the parametrizations as shown inTable 3.

5.2. Derivation of implied Poisson intensities

The vast majority of authors opts for a simple homogeneousPoisson process to represent the arrival of claims due to catastro-phes.36 The adequacy of this choice has been underlined by the em-pirical studies of Levi and Partrat (1991) and Milidonis and Grace(2008), who test the goodness of fit of the Poisson distributionwith ISO/PCS data and find it to be superior to the alternative bi-nomial distribution. Hence, the choice of a general Poisson distri-bution for the frequency of natural disasters seems hardly ques-tionable. However, more advanced modeling frameworks have

36 See Cummins and Geman (1994, 1995), Chang et al. (1996), Embrechts andMeister (1997), Geman and Yor (1997), Aase (1999), Loubergé et al. (1999),Christensen and Schmidli (2000), Bakshi and Madan (2002), Lee and Yu (2002),Vaugirard (2003a,b, 2004), Cox et al. (2004), Jaimungal and Wang (2006), Lee andYu (2007), and Muermann (2008).

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Table 3Parameter estimates and test statistics for the lognormal, Burr, and Pareto distribution.

Lognormal Burr ParetoEarthquake Hurricane Earthquake Hurricane Earthquake Hurricane

µ −1.3778 0.8064 a 0.4027 4.8358 a 0.4602 0.6620σ 2.5835 2.1346 b 1.1018 0.5886 c 0.0503 1.1300

c 0.0426 70.1560KSn 0.1218 0.0683 KSn 0.0712 0.0784 KSn 0.0703 0.0926p-value 0.4038 0.7699 p-value 0.9420 0.6101 p-value 0.9472 0.3982ADn 0.8275 0.6403 ADn 0.3264 0.4958 ADn 0.3451 1.5995p-value 0.4610 0.6102 p-value 0.9167 0.7506 p-value 0.9004 0.1545

Table 4Parameter estimates and test statistics for the Weibull, gamma, and exponential distribution.

Weibull Gamma ExponentialEarthquake Hurricane Earthquake Hurricane Earthquake Hurricane

a 0.3557 0.5252 a 0.2050 0.3909 β 0.1108 0.0824c 0.9774 6.3610 β 0.0227 0.0322KSn 0.1673 0.0787 KSn 0.2440 0.0993 KSn 0.6546 0.3084p-value 0.1025 0.6054 p-value 0.0037 0.3159 p-value 0.0000 0.0000ADn 1.9962 0.5264 ADn 4.3858 1.4651 ADn 81.275 27.4791p-value 0.0925 0.7196 p-value 0.0057 0.1851 p-value 0.0000 0.0000

been based on the time-inhomogeneous Poisson process37 or thedoubly stochastic Poisson process (Cox process).38 Just recently,e.g., Chang et al. (2010) as well as Wu and Chung (2010) sug-gested to employ a mean-reverting intensity process, which wealso adopted within our ex-ante model framework in Section 4.2.Since, to the best of our knowledge, there is little empirical ev-idence to support these dynamics for the Poisson intensity as ofyet, in this section we employ our model to derive intensities im-plied by cat swap market quotes as a basis for a time series anal-ysis. In analogy to implied volatilities in option markets we de-fine the implied intensity as the fixed value λt,t+1t = λ1t ∀t ∈

(0, T − 1t] which, if used in our cat swap pricing framework,generates a theoretical spread equal to the observed marketspread.

Due to their OTC character, market quotes for catastrophe-linked instruments are generally scarce and hardly publicly avail-able. Yet, we obtained ILW quotes from the BMS pricing gridwhich is published by the Thomson Reuters Insurance Linked Se-curities community on a regular basis. Where possible, the fig-ures have been cross-checked with expert judgment based onvarious sources as well as direct market intelligence. Interviewswith industry practitioners revealed that, from a pricing perspec-tive, ILWs and cat swaps are currently not differentiated, whichconfirms the suitability of ILW premiums for our research pur-pose. The dataset comprises monthly time series of up-front pricesfrom August 2005 to September 2010 for US hurricane and earth-quake contracts with one year maturities and event thresholds ofUSD 20 bn, USD 25 bn, USD 30 bn, USD 40 bn, as well as USD 50bn.39 In case of a trigger event each contract pays off its full no-tional. Table 5 contains some descriptive statistics for the time se-ries. We notice that, for each attachment level, the prices of hurri-cane contracts exhibit a higher mean and standard deviation thanthose of earthquake contracts. To acquire protection against hurri-cane losses in excess of USD 20 bn, e.g., a protection buyer needed

37 See Embrechts and Meister (1997), Schmidli (2003) and Biagini et al. (2008).38 Christensen and Schmidli (2000), Basu and Dassios (2002) and Dassios and Jang(2003), e.g., use a Cox process with gamma, lognormal, and shot noise intensity,respectively.39 Since these prices reflect market averages instead of quotes by a specificcounterparty and the average CDS spread of common cat swap issuers is relativelysmall (see Section 4.4), we decide to simplify the empirical analysis by abstractingfrom an explicit treatment of default risk.

Table 5Descriptive statistics: time series of cat swap prices.

Attachment Hurricane contracts20 bn 25 bn 30 bn 40 bn 50 bn

Mean 0.2680 0.2203 0.1870 0.1437 0.1170s.d. 0.0509 0.0467 0.0452 0.0386 0.0324max. 0.3750 0.3167 0.2667 0.2083 0.1750min. 0.1200 0.0800 0.0575 0.0450 0.0350

Attachment Earthquake contracts20 bn 25 bn 30 bn 40 bn 50 bn

Mean 0.1433 0.1159 0.0924 0.0786 0.0676s.d. 0.0239 0.0231 0.0197 0.0162 0.0146max. 0.2000 0.1750 0.1500 0.1200 0.1000min. 0.1000 0.0800 0.0600 0.0500 0.0425

to pay an average up-front premium of 26.80% of the notional be-tween 08/2005 and 09/2010. The USD 20 bn earthquake contract,in contrast, was available for an average price of 14.33%. Further-more, while the minimum premiums for corresponding hurricaneand earthquake contracts differ only slightly, the maximums forthe former are considerably larger. Finally, of course both hurri-cane and earthquake contracts are on average more expensive forlower thresholds.

In the following, we employ the ex-ante pricing frameworkfrom Section 4.2 to back out implied intensity time series fromthe market quotes.40 Recall that the model has been formulatedunder the risk-neutral measure Q. Thus, for it to be applicable toreal data, we need to determine a change of measure as discussedin Section 4.1. Due to extant empirical support for the zero betacharacteristics of cat instruments, we assume that catastrophe riskis unsystematic. As a result, the model parameters under Q remainthe same as under the physical measure P. Since a closed-formsolution for the cat swap price is unavailable, our calculations arebased on Monte Carlo simulations. For this purpose, we discretizethe model and evaluate the underlying loss process of Eq. (3)for a total of 10,000 sample paths, each one consisting of 252trading days per year (i.e.,1t = 1/252). The intuition is that, whilea catastrophe can occur on any day, the official declaration of

40 A similar proceeding is applied by Härdle and Cabrera (2010), who back out theimplied intensity for a Mexican cat bond and compare it to figures derived from thereinsurance market as well as historical data.

532 A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536

(a) Hurricane contracts. (b) Earthquake contracts.

Fig. 4. Monthly time series of annualized implied intensities from 08/2005 to 09/2010.

Table 6Descriptive statistics and p-values of normality tests for the implied intensity time series.

Attachment Hurricane contracts Earthquake contracts20 bn 25 bn 30 bn 40 bn 50 bn 20 bn 25 bn 30 bn 40 bn 50 bn

Mean 2.1221 2.0840 2.0951 2.1742 2.3011 1.5346 1.4059 1.3077 1.1472 1.0675s.d. 0.4718 0.5009 0.5658 0.6355 0.7113 0.3627 0.3126 0.3084 0.2472 0.2370max. 3.2618 3.2778 3.2561 3.4430 3.8274 2.7309 2.3719 2.1101 1.6771 1.5221min. 0.8609 0.7055 0.5829 0.5870 0.6346 0.9860 0.8704 0.7806 0.7361 0.6423JB p-value 0.7438 0.4999 0.3289 0.8600 0.9867 0.0070 0.0528 0.1715 0.0709 0.0859KSn p-value 0.3339 0.5438 0.4606 0.2829 0.6632 0.2900 0.2316 0.3923 0.0432 0.0635ADn p-value 0.4789 0.6039 0.4283 0.3647 0.7672 0.3956 0.4169 0.5682 0.1104 0.0790

a trigger event, i.e., a final loss estimate in excess the of eventthreshold, and the resulting transfer of cash flows will only takeplace on trading days. Furthermore, the risk free interest rate isthe monthly yield on 1-year US T-Bills41 and, as suggested byour results in the previous section, we employ a Burr distributionwith the parametrizations from Table 3 for the respective lossseverities Yi of hurricanes and earthquakes. The proceeding tocapture the implied intensities works as follows: instead of thestochastic process of Eq. (2), we assume a deterministic annualλ which corresponds to an intensity of λ1t per trading day.Then we embed the valuation framework into a one-dimensionaloptimization algorithm which searches for λ by recalculating themodel price until it matches the monthly market quote for eachcontract. In doing so, we extract ten time series of annualizedimplied intensities, i.e., one for each hurricane and earthquakecontract. The results are displayed in Fig. 4. Table 6 contains somedescriptive statistics.

From a theoretical perspective, the market implied intensitiesfor each type of peril at any point in time should not varyacross event thresholds (attachment levels), since all contractsare driven by the same catastrophes. Yet, there seem to be slightdifferences in the cross sections: we observe a tendency for themeans and standard deviations of hurricane implied intensities toincrease and those of earthquake implied intensities to decreasewith the attachment level (see Table 6). Unreported results ofWelch’s t-test indicate that the differences in all pairs of means ofthe implied intensities for hurricane contracts are insignificant.42For earthquake contracts, in contrast, the means of the impliedintensity time series seem to differ significantly, suggesting thatthe model might be somewhat less suitable to capture thecharacteristics of cat swaps on earthquake risk. From the sampleof Pielke et al. (2008) we derive a historical number of 1.95 UShurricanes per year between 1900 and 2005. Similarly, records ofthe Insurance Information Institute show that an average of 1.80

41 The rates can be accessed on www.ustreas.gov.42 Welch’s test allows us to compare the means of two samples without assumingthat their variances are equal. The null hypothesis is equality of the means.

hurricanes per yearmade landfall between 1990 and 2009.43 Thesefigures lie well within the overall range of implied intensities foreach hurricane cat swap contract in Table 6. In contrast to that,historical experience from the Vranes and Pielke (2009) datasetindicates that, between 1900 and 2005, 0.76 major earthquakesoccurred in the US per year. This figure is at the lower bound of theranges of implied intensities we derived from the market prices ofthe earthquake contracts, thus again alluding to some refinementpotential.

5.3. The stochastic process of implied Poisson intensities

Having considered the cross-sectional characteristics of theimplied intensities, we now want to focus on our actual researchgoal: the time series analysis. More specifically, we aim todetermine whether a mean-reverting Ornstein–Uhlenbeck typeprocess is an adequate assumption for the dynamics of cat swapimplied intensities. A first indication is provided through Fig. 4.We observe a clear sign of cyclicality: implied intensities seemto adhere to some sort of wave pattern over time. Since anOrnstein–Uhlenbeck process is the continuous-time limit of adiscrete-time first order autoregressive process, i.e., an AR(1),we could now simply apply the Box–Jenkins methodology toevery single implied intensity time series and assess the fit ofdifferent models. Instead, however, we choose a slightly moreefficient way to tackle the issue. Table 7 shows the correlationmatrices for the implied intensity time series of hurricane andearthquake cat swaps from 08/2005 through 09/2010. Evidently,the implied intensities for contracts on the same type of perilare highly correlated,44 suggesting that there is at least onecommon underlying driver which can be revealed by means ofexploratory factor analysis (EFA). EFA is a statistical techniquethat describes the covariance (correlation) structure of observedrandom variables in terms of a smaller number of latent variablescalled factors. Once we have identified these factors for the two

43 Refer to www.iii.org.44 All correlations are significant on the one percent level.

A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536 533

Table 7Correlation matrices and factor loadings for the implied intensity time series.

Attachment Hurricane contracts Earthquake contracts20 bn 25 bn 30 bn 40 bn 50 bn 20 bn 25 bn 30 bn 40 bn 50 bn

20 bn 1.0000 1.000025 bn 0.9734 1.0000 0.9525 1.000030 bn 0.9546 0.9606 1.0000 0.9126 0.9446 1.000040 bn 0.9672 0.9679 0.9667 1.0000 0.8975 0.8996 0.9122 1.000050 bn 0.9125 0.9237 0.9370 0.9645 1.0000 0.7639 0.7712 0.8188 0.8789 1.0000Loading 0.9775 0.9802 0.9767 0.9914 0.9578 0.9609 0.9765 0.9645 0.9377 0.8260

(a) Time series of factor scores (standardized). (b) One month forecasts for the hurricane factor.

Fig. 5. Intensity factor scores and out-of-sample forecast example.

perils and derived their respective factor scores, we can focus ouranalysis on their time series instead of those of the individualcontracts. The following is an analytical representation of thegeneral EFA model:

X = Λξ + δ (34)

where X is the vector of observed variables (indicators), Λ

represents the matrix of factor loadings, ξ is the vector of latentvariables (factors), and δ stands for the vector of unique factors(residuals). Applyingmatrix algebra, one can derive the covariancematrix Σ implied by the model:

Σ = ΛΦΛ′+ Ψδ (35)

with Φ being the covariance matrix of the factors and Ψδ

being the covariance matrix of the residuals. The parameters(factor loadings and residual variances) for the EFA model aredetermined by means of maximum likelihood estimation (MLE)such that themodel implied covariance (correlation) matrix fits itsempirically observed counterpart as closely as possible. StandardEFA assumes multivariate normality of the indicator variables. Inorder to check this prerequisite, the well-known Jarque–Bera testas well as the previously introduced Kolmogorov–Smirnov andAnderson–Darling tests have been applied to the implied intensitytime series (see Table 6 for the respective p-values). Since all testresults but two are insignificant on the five percent level, i.e., wecannot reject the null hypothesis that the sample has been drawnfrom a normal distribution, we reason that multivariate normalityis given.45 The adequacy of our sample for an EFA is underlined by aKaiser–Mayer–Olkin (KMO) Measure of 0.88 for the hurricane and0.86 for the earthquake implied intensities. While 0 < KMO <1,KMO > 0.5 indicates that an EFA can be performed and if KMO> 0.8, the sample is particularly well suited for the analysis (seeKaiser, 1974). Furthermore, conducting Bartlett’s test of sphericity,we reject the null hypothesis of all pairwise correlations being

45 A vector of random variables is multivariate normally distributed if all of itselements follow a univariate normal distribution.

zero on all reasonable significance levels with a χ2 test statistic of679.05 for hurricanes and 474.57 for earthquakes.46 Consequently,we can proceed and apply EFA to the sample.

Initial factor extraction is conducted by means of principalcomponents analysis, which provides as many factors as thereare indicator variables, i.e., five for each peril. We find that,for the hurricane contracts, the first factor explains 96.23% ofthe variance of the implied intensity series. Similarly, for theearthquake contracts, it explains 90.10% of the variance. Thus, aspreviously suspected, a one-factor solution is an adequate choicewith regard to the dimensionality of the EFAmodel. The last row ofTable 7 contains the factor loadings we obtained for the one-factorEFA based on the previously generated correlation matrices.47Apart from one exception all factor loadings are higher than 0.90,underlining a strong influence of the common factor for each perilon the implied intensity time series for the individual contracts.48

In addition, factor score estimates ξ have been computed bymeansof the so-called regression method, which employs the samplecovariance matrix Σ and the estimated factor loadings matrix Λ

as follows:

ξ = Λ′Σ−1X . (36)

The time series of the standardized factor scores for hurricaneand earthquake contracts are plotted in Fig. 5(a). Again we observea clear cyclical pattern. Having the time series of the factorscores at hand, it is now straightforward to test which stochasticprocess provides a satisfactory fit to the data. Before estimatingthe parameters, one commonly preselects reasonable modelspecifications through the patterns observed in the autocorrelationfunction (ACF) and partial autocorrelation function (PACF) of thetime series, which have been plotted in Figs. 6 and 7. Both the

46 Note that Bartlett’s test also requires multivariate normality.47 EFA is commonly performed with standardized variables. Hence, all indicatorvariables are demeaned and divided by their standard deviation before the analysis.48 Note that the factor loadings can be interpreted as correlation coefficientsbetween indicator variables and factors.

534 A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536

(a) Autocorrelation function (ACF). (b) Partial autocorrelation function (PACF).

Fig. 6. ACF and PACF for the hurricane intensity factor.

(a) Autocorrelation function (ACF). (b) Partial autocorrelation function (PACF).

Fig. 7. ACF and PACF for the earthquake intensity factor.

factor scores for hurricane and for earthquake contracts exhibit thetheoretical characteristics of an AR(1): an ACF which successivelydecays towards zero and a PACF with a significant spike at thefirst lag while all other lags are insignificant. Since the spikein the PACF is positive, we can expect a positive coefficientin the autoregressive process.49 The high value of the partialautocorrelation at lag one suggests that the process is nearintegrated or might have a unit root. Hence, we conduct theDickey–Fuller unit root test with asymptotic and small sample(MacKinnon) critical values.50 In addition, the KPSS test forstationarity is considered. The p-values for these tests with a laglength of one can be found in the left part of Tables 8 and 9. Sincethe unit root tests are statistically significant at least on the fivepercent level (i.e., we reject the null hypothesis of a unit-root) andthe stationarity tests are insignificant, we conclude that there doesnot seem to be a unit root problem.

We now estimate an AR(1) on the hurricane and earthquakefactor score series. Tables 8 and 9 contain the results. Forcomparison purposes we have also included an AR(2). Generally, ifa model is suitable to capture the pattern inherent in a time series,its residuals should be white noise. To test the null hypothesisof independent residuals, we calculate the Ljung–Box (LB) Q -statistic with a lag of 3 and 10.51 Since these tests turn outto be insignificant (p-values >0.1000), the residuals of both theAR(1) and the AR(2) should not be autocorrelated. Due to theirdifferent parameter specifications, the in-sample performance of

49 In the absence of a negative coefficient, the oscillating decline in the ACF couldbe a sign for seasonality.50 Since the factor score series are standardized, we do not include an intercept inthe equation of the test regression.51 It is common to conduct the Ljung–Box test for a short and a long lag length.

these models is compared by means of two common goodness offit criteria: the Akaike information criterion (AIC) and the SchwarzBayesian criterion (SBC) which incorporates a relatively largerpenalty term for the number of parameters. For both criteria themodel with the smaller value is considered superior. Althoughthe AR(1) appears slightly worse under the AIC, it is associatedwith a better SBC for the hurricane and earthquake factor. Incombination with the fact that the second coefficient of the AR(2)is insignificant on the five per cent level, this leads us to decidein favor of the AR(1).52 Consequently, the Ornstein–Uhlenbeckprocess, which is the continuous-time equivalent of the AR(1)seems to be an adequate choice for the intensity dynamics in a catswap pricing model. In addition, it could be considered for out-of-sample forecasts of implied intensities and, in turn, cat swapspreads (see Fig. 5(b) for an example). Before such an application,however, one should conduct further analyses of the short- andlong-term forecasting performance.

6. Summary and conclusion

In this paper, we contribute to the literature through acomprehensive analysis of the catastrophe swap, a relativelynew financial instrument which has attracted little scholarlyattention to date. We begin with a brief discussion of the typicalcontract design, the current state of the market, as well asmajor areas of application. Subsequently, a two-stage contingentclaims pricing approach for catastrophe swaps is proposed, which

52 Unreported results for pure moving average (MA), higher order AR, andcombined ARMA models indicate their inferiority due to insignificant coefficients,non-white noise residuals, or worse AIC/SBC values.

A. Braun / Insurance: Mathematics and Economics 49 (2011) 520–536 535

Table 8Hurricane intensity factor: unit root tests and model estimation results.

p-values for unit root tests AR(1) AR(2)

Dickey–Fuller 0.0075 Coefficients 0.9463 1.1249 −0.2008MacKinnon 0.0031 p-value 0.0000 0.0000 0.0606

LB Q(3)/Q(10) p-value 0.1124 0.2835 0.2293 0.4341KPSS (stationarity test) >0.1000 AIC/SBC 67.6732 71.9275 67.3087 73.6901

Abbreviations: AR(1): first order autogregressive process; AR(2): second order autoregressive process; LB Q (t): Ljung–Box Q -statistic forlag t; AIC: Akaike information criterion; SBC: Schwarz Bayesian criterion.

Table 9Earthquake intensity factor: unit root tests and model estimation results.

p-values for unit root tests AR(1) AR(2)

Dickey–Fuller 0.0241 Coefficients 0.8751 1.0470 −0.2027MacKinnon 0.0089 p-value 0.0000 0.0000 0.0548

LB Q(3)/Q(10) p-value 0.5643 0.5577 0.5715 0.3637KPSS (stationarity test) >0.1000 AIC/SBC 94.4151 98.6694 93.9122 100.2936

Abbreviations: AR(1): first order autogregressive process; AR(2): second order autoregressive process; LB Q (t): Ljung–Box Q -statistic forlag t; AIC: Akaike information criterion; SBC: Schwarz Bayesian criterion.

distinguishes between the main risk drivers ex-ante and duringthe loss reestimation phase. The occurrence of catastrophes ismodeled as a doubly stochastic Poisson process with mean-reverting Ornstein–Uhlenbeck intensity. In addition, we fit variousparametric distributions to normalized historical loss data forhurricanes and earthquakes in the US and find the heavy-tailedBurr distribution to be the most adequate representation for lossseverities. Applying our ex-ante pricing model to market quotesfor hurricane and earthquake contracts, we then derive impliedintensities which are subsequently condensed into a commonfactor for each peril by means of exploratory factor analysis.Further examining the resulting factor scores, we show that anAR(1) provides a good fit. Hence, its continuous-time limit, i.e., theOrnstein–Uhlenbeck process should bewell suited to represent thedynamics of the Poisson intensity in a cat swap pricing model.

Future research could be centered around refinements of thepricing model such as the inclusion of stochastic interest rateswhich might be reasonable in case longer term cat swap contractsbegin to be traded.Moreover, itwould be interesting to use impliedintensities from cat swap transactions in valuation models forother, less standardized and liquid catastrophe-linked instrumentssuch as cat bonds or even reinsurance. In doing so, one could ensureconsistent pricing across different markets for catastrophe riskand eliminate potential arbitrage opportunities. Finally, the AR(1)could be applied to produce implied intensity and cat swap spreadforecasts, the accuracy of whichwould need to be assessed relativeto natural competitors such as the random walk or an AR(1) withseasonality.

Appendix. The market price of cat risk

In the case of derivatives on non-traded underlyings marketcompleteness can be preserved if other liquidly tradable instru-ments exist, which are driven by the same source of risk. Hence,replicating portfolios canbe formed and aunique risk-neutralmea-sure is obtainable by recovering a market price of risk from ob-served quotes. Below, we briefly recapitulate the reasoning of Hull(2008) for the simple case where the underlying process is a Geo-metric BrownianMotion. Consider twoderivativesG1(ξ) andG2(ξ)on the variable ξ . Suppose the dynamics of ξ , which by itself is nota traded asset, are adequately described through the following dif-fusion process, with drift a and volatility b:

dξt = aξtdt + bξtdWt (37)

where dWt is a standard Wiener process. Further, assume thatG1(ξ) and G2(ξ) adhere to the processes:

dG1(ξt) = µ1G1(ξt)dt + σ1G1(ξt)dWt , (38)dG2(ξt) = µ2G2(ξt)dt + σ2G2(ξt)dWt , (39)

such that the Wiener process dWt , which originates from thedynamics of the underlying, is the only source of uncertaintyaffecting the derivative prices. By buying σ2G2(ξt) of the firstderivative and sellingσ1G1(ξt) of the second derivative, an investorcould now form a portfolio Ω , which would be instantaneouslyrisk-free (i.e., dWt is eliminated):

Ω(ξt) = σ2G2(ξt)G1(ξt) − σ1G1(ξt)G2(ξt). (40)

Using (38) and (39), the marginal change in value of this portfoliocan be expressed as follows:

dΩ = σ2G2(ξt)dG1 − σ1G1(ξt)dG2

= [σ2G2(ξt)µ1G1(ξt) − σ1G1(ξt)µ2G2(ξt)] dt. (41)

Consequently, this portfolio must yield the risk-free interest rateover the next marginal time period:

dΩ = rΩdt. (42)

Inserting (40) and (41) and eliminating G(ξ)1G(ξ)2dt , this isequivalent to σ2µ1 − σ1µ2 = rσ2 − rσ1, or

µ1 − rσ1

=µ2 − r

σ2= λξ . (43)

λξ is the market price of risk for the underlying ξ . If the no-arbitrage principle holds, λξ at any given point in time has tobe the same for any derivative dependent on ξ , regardless ofits specification. The market price of risk gages the risk-returntradeoff for financial instruments based on ξ . Multiplied by thevolatility (i.e., the quantity of risk) of the asset under consideration,it represents the risk premium over and above the risk-free rate,which investors require to hold this asset: µ − r = λξσ .

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