Catastrophe (CAT) bonds are one of the more recent financial derivatives to be traded on the world markets. In the mid-1990s a market in catastrophe insurance risk emerged in order to facilitate the direct transfer of reinsurance risk associated with natural catastrophes from corporations, insurers and re-insurers to capital market investors. The primary instrument developed for this purpose was the CAT bond.
CAT bonds are more specifically referred to as insurance-linked securities (ILS) The distinguishing feature of these bonds is that the ultimate repayment of principal depends on the outcome of an insured event. The basic CAT bond structure can be summarized as follows (Lane; 2004):
There are three types of ILS triggers: indemnity, index and parametric. An indemnity trigger involves the actual losses of the bond-issuing insurer. For example the event may be the insurer's losses from an earthquake in a certain area of a given country over the period of the bond. An industry index trigger involves, in the US for example, an index created from property claim service (PCS) loss estimates. A parametric trigger is based on, for example, the Richter scale readings of the magnitude of an earthquake at specified data stations. In this chapter we address the issue of pricing CAT bonds with indemnity and index triggers.
Until fairly recently, property reinsurance was a relatively well understood market with efficient pricing. However, naturally occurring catastrophes, such as earthquakes and hurricanes, are beginning to have a dominating impact on the industry. In part, this is due to the rapidly changing, heterogeneous distribution of high-value property in vulnerable areas. A consequence of this has been an increased need for a primary and secondary market in catastrophe related insurance derivatives. The creation of CAT bonds, along with allied financial products such as catastrophe insurance options, was motivated in part by the need to cover the massive property insurance industry payouts of the early- to mid-1990s. They also represent a ``new asset class'' in that they provide a mechanism for hedging against natural disasters, a risk which is essentially uncorrelated with the capital market indices (Doherty; 1997). Subsequent to the development of the CAT bond, the class of disaster referenced has grown considerably. As yet, there is almost no secondary market for CAT bonds which hampers using arbitrage-free pricing models for the derivative.
Property insurance claims of approximately USD 60 billion between 1990 and 1996 (Canter, Cole, and Sandor; 1996) caused great concern to the insurance industry and resulted in the insolvency of a number of firms. These bankruptcies were brought on in the wake of hurricanes Andrew (Florida and Louisiana affected, 1992), Opal (Florida and Alabama, 1995) and Fran (North Carolina, 1996), which caused combined damage totalling USD 19.7 billion (Canter, Cole, and Sandor; 1996). These, along with the Northridge earthquake (1994) and similar disasters (for the illustration of the US natural catastrophe data see Figure 4.1), led to an interest in alternative means for underwriting insurance. In 1995, when the CAT bond market was born, the primary and secondary (or reinsurance) industries had access to approximately USD 240 billion in capital (Canter, Cole, and Sandor; 1996; Cummins and Danzon; 1997). Given the capital level constraints necessary for the reinsuring of property losses and the potential for single-event losses in excess of USD 100 billion, this was clearly insufficient. The international capital markets provided a potential source of security for the (re-)insurance market. An estimated capitalisation of the international financial markets, at that time, of about USD 19 trillion underwent an average daily fluctuation of approximately 70 basis points or USD 133 billion (Sigma; 1996). The undercapitalisation of the reinsurance industry (and their consequential default risk) meant that there was a tendency for CAT reinsurance prices to be highly volatile. This was reflected in the traditional insurance market, with rates on line being significantly higher in the years following catastrophes and dropping off in the intervening years (Sigma; 1997; Froot and O'Connell; 1997). This heterogeneity in pricing has a very strong damping effect, forcing many re-insurers to leave the market, which in turn has adverse consequences for the primary insurers. A number of reasons for this volatility have been advanced (Cummins and Danzon; 1997; Winter; 1994).
CAT bonds and allied catastrophe related derivatives are an attempt to address these problems by providing effective hedging instruments which reflect long-term views and can be priced according to the statistical characteristics of the dominant underlying process(es). Their impact, since a period of standardisation between 1997 and 2003, has been substantial. As a consequence the rise in prices associated with the uppermost segments of the CAT reinsurance programs has been dampened. The primary market has developed and both issuers and investors are now well-educated and technically adept. In the years 2000 to 2003, the average total issue exceeded USD 1 billion per annum (McGhee; 2004). The catastrophe bond market witnessed yet another record year in 2003, with total issuance of USD 1.73 billion, an impressive 42 percent year-on-year increase from 2002’s record of USD 1.22 billion. During the year, a total of eight transactions were completed, with three originating from first-time issuers. The year also featured the first European corporate-sponsored transaction (and only the third by any non-insurance company). Electricité de France, the largest electric utility in Europe, sponsored a transaction to address a portion of the risks facing its properties from French windstorms. Since 1997, when the market began in earnest, 54 catastrophe bond issues have been completed with total risk limits of almost USD 8 billion. It is interesting to note that very few of the issued bonds receive better than ``non-investment grade" BB ratings and that almost no CAT bonds have been triggered, despite an increased reliance on parametric or index based payout triggers.
Capitalisation of insurance and consequential risk spreading through share issue, is well established and the majority of primary and secondary insurers are public companies. Investors in these companies are thus de facto bearers of risk for the industry. This however relies on the idea of risk pooling through the law of large numbers, where the loss borne by each investor becomes highly predictable. In the case of catastrophic natural disasters, this may not be possible as the losses incurred by different insurers tend to be correlated. In this situation a different approach for hedging the risk is necessary. A number of such products which realize innovative methods of risk spreading already exist and are traded (Sigma; 2003; Cummins and Danzon; 1997; Aase; 1999; Litzenberger, Beaglehole, and Reynolds; 1996). They are roughly divided into reinsurance share related derivatives, including Post-loss Equity Issues and Catastrophe Equity Puts, and asset-liability hedges such as Catastrophe Futures, Options and CAT Bonds.
In 1992, the Chicago Board of Trade (CBOT) introduced the CAT futures. In 1995, the CAT future was replaced by the PCS option. This option was based on a loss index provided by PCS. The underlying index represented the development of specified catastrophe damages, was published daily and eliminated the problems of the earlier ISO index. The options traded better, especially the call option spreads where insurers would appear on both side of the transaction, i.e. as buyer and seller. However, they also ceased trading in 2000. Much work in the reinsurance industry concentrated on pricing these futures and options and on modelling the process driving their underlying indices (Canter, Cole, and Sandor; 1996; Aase; 1999; Embrechts and Meister; 1997). CAT bonds are allied but separate instruments which seek to ensure capital requirements are met in the specific instance of a catastrophic event.
In this chapter we investigate the pricing of CAT Bonds. The methodology developed here can be extended to most other catastrophe related instruments. However, we are concerned here only with CAT specific instruments, e.g. California Earthquake Bonds (Sigma; 1996; McGhee; 2004; Sigma; 1997,2003), and not reinsurance shares or their related derivatives.
In the early market for CAT bonds, the pricing of the bonds was in the hands of the issuer and was affected by the equilibrium between supply and demand only. Consequently there was a tendency for the market to resemble the traditional reinsurance market. However, as CAT bonds become more popular, it is reasonable to expect that their price will begin to reflect the fair or arbitrage-free price of the bond, although recent discussions of alternative pricing methodologies have contradicted this expectation (Lane; 2003). Our pricing approach assumes that this market already exists.
Some of the traditional assumptions of derivative security pricing are not correct when applied to these instruments due to the properties of the underlying contingent stochastic processes. There is evidence that certain catastrophic natural events have (partial) power-law distributions associated with their loss statistics (Barton and Nishenko; 1994). This overturns the traditional log-normal assumption of derivative pricing models. There are also well-known statistical difficulties associated with the moments of power-law distributions, thus rendering it impossible to employ traditional pooling methods and consequently the central limit theorem. Given that heavy-tailed or large deviation results assume, in general, that at least the first moment of the distribution exists, there will be difficulties with applying extreme value theory to this problem (Embrechts, Resnick, and Samorodnitsky; 1999). It would seem that these characteristics may render traditional actuarial or derivatives pricing approaches ineffective.
There are additional features to modelling the CAT bond price which are not to be found in models of ordinary corporate or government issue (although there is some similarity with pricing defaultable bonds). In particular, the trigger event underlying CAT bond pricing is dependent on both the frequency and severity of natural disasters. In the model described here, we attempt to reduce to a minimum any assumptions about the underlying distribution functions. This is in the interests of generality of application. The numerical examples will have to make some distributional assumptions and will reference some real data. We will also assume that loss levels are instantaneously measurable and updatable. It is straightforward to adjust the underlying process to accommodate a development period.
There is a natural similarity between the pricing of catastrophe bonds and the pricing of defaultable bonds. Defaultable bonds, by definition, must contain within their pricing model a mechanism that accounts for the potential (partial or complete) loss of their principal value. Defaultable bonds yield higher returns, in part, because of this potential defaultability. Similarly, CAT bonds are offered at high yields because of the unpredictable nature of the catastrophe process. With this characteristic in mind, a number of pricing models for defaultable bonds have been advanced (Jarrow and Turnbull; 1995; Duffie and Singleton; 1999; Zhou; 1997). The trigger event for the default process has similar statistical characteristics to that of the equivalent catastrophic event pertaining to CAT bonds. In an allied application to mortgage insurance, the similarity between catastrophe and default in the log-normal context has been commented on (Kau and Keenan; 1996).
With this in mind, we have modelled the catastrophe process as a compound doubly stochastic Poisson process. The underlying assumption is that there is a Poisson point process (of some intensity, in general varying over time) of potentially catastrophic events. However, these events may or may not result in economic losses. We assume the economic losses associated with each of the potentially catastrophic events to be independent and to have a certain common probability distribution. This is justifiable for the Property Claim Loss indices used as the triggers for the CAT bonds. Within this model, the threshold time can be seen as a point of a Poisson point process with a stochastic intensity depending on the instantaneous index position. We make this model precise later in the chapter.
In the article of Baryshnikov, Mayo, and Taylor (1998) the authors presented an arbitrage-free solution to the pricing of CAT bonds under conditions of continuous trading. They modelled the stochastic process underlying the CAT bond as a compound doubly stochastic Poisson process. Burnecki and Kukla (2003) applied their results in order to determine no-arbitrage prices of a zero-coupon and coupon bearing CAT bond. In Section 4.2 we present the doubly stochastic Poisson pricing model. In Section 4.3 we study -year catastrophe loss data provided by Property Claim Services. We find a distribution function which fits the observed claims in a satisfactory manner and estimate the intensity of the non-homogeneous Poisson process governing the flow of the
natural events. In Section 4.4 we illustrate the values of different CAT bonds associated with this loss data with respect to the threshold
level and maturity time. To this end we apply Monte Carlo simulations.