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Indifference Prices of Structured Catastrophe (CAT) Bonds

Masahiko Egami∗

Virginia R. Young†

17 August 2007

Abstract

We present a method for pricing structured CAT bonds based on utility indifference pricing. The CAT

bond considered here is issued in two distinct notes called tranches, specifically senior and junior

tranches each with its own payment schedule. Our contributions to the literature of CAT bond pric-

ing are two-fold. First, we apply indifference pricing to structured CAT bonds. We find a price for the

senior tranche as a relative indifference price, that is, relative to the price of the junior tranche. Alterna-

tively, one could take the approach that the senior tranche is priced first and the price of the junior tranche

is relative to that. Second, instead of simply supposing that the “not-issue-a-CAT-bond” strategy of the

reinsurer is to do nothing, we suppose that the reinsurer reduces its risk by reinsuring proportionally less

of the claims. We assume the reinsurance claims follow a (Poisson) jump-diffusion process.

Key Words: Catastrophe (CAT) bond, structured derivative security, indifference price, exponential

utility, jump diffusion, reinsurance strategy

JEL Classification: G22, G13

1Introduction

Catastrophe bonds (CAT bond, thereafter) are one of the most important insurance-linked financial secu-

rities. Investors purchase CAT bonds from the issuer, a special purpose vehicle that simultaneously enters

into a reinsurance contract with a reinsurance company. The coupon and principal payments depend on the

performance of a pool or index of natural catastrophe risks. The default of a CAT bond occurs when catas-

trophic events (such as earthquakes, hurricanes and floods) of some degree occurs. In a simple transaction,

if the default occurs, no cash is paid to the investor thereafter. Otherwise, the coupons and principal are paid

out to the buyer of the CAT bond. Due to the magnitude of losses caused by a large catastrophe, a CAT bond

is an innovative financial instrument by which a reinsurance company transfers the risk of a possible large

payment caused by catastrophic events to the capital market.

∗Corresponding author. Tel:+81-75-753-3430; Fax:+81-75-753-3492. Department of Mathematics, University of Michi-

gan, Ann Arbor, MI 48109, USA / Graduate School of Economics, Kyoto University, Kyoto, 606-8501, Japan.

egami@econ.kyoto-u.ac.jp.

†Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, Email: vryoung@umich.edu.

Email:

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The recent paper by Lee and Yu [5] cites McGhee [6] regarding the size of the CAT bond market,

namely a total issuance of $1.22 billion in 2002 and $1.73 billion in 2003. According to Swiss Re,1“the

market is growing: the volume of bonds outstanding has steadily increased since 1997 to a current level of

approximately USD 3 billion.” Due to the recent event of Hurricane Katrina in 2005, it can be reasonably

assessed that the market size of a new issuance is around $3-4 billion in recent years. To address diverse

investors with different risk-return preferences, a structured CAT bond is usually issued in the market. In

these transactions, CAT bonds are issued in separate notes (called “tranches”) with different payment sched-

ules and seniority. Senior tranches usually have investment grades from the rating agencies since the holders

of senior tranches have higher priority of being paid. On the other hand, junior tranches have speculative

grades, but the higher risk is compensated by larger coupon payments. See, for example, Shimpi [9] (page

179) for a transaction completed by Swiss Re.

In a broad sense, CAT bonds are categorized as catastrophe derivative securities. In the literature, there

are several papers with respect to catastrophe derivatives. Geman and Yor [3] analyze catastrophe options

with payoff (L(T) − K)+where L is the aggregate claim process modeled by a jump-diffusion process.

Dassios and Jang [2] use a doubly-stochastic Poisson process for the claim process to price catastrophe

reinsurance contract and derivatives. Jaimungal and Wang [4] study the pricing and hedging of catastrophe

put options (CatEPut) under stochastic interest rates with a compound Poisson process. Among them, only

few papers directly attempt to price CAT bonds: Cox and Pedersen [1] price a CAT bond under a term

structure model together with an estimation of the probability of catastrophic events. Lee and Yu [5] adopt

a structural approach borrowed from the literature of credit risk modeling in corporate finance (see Merton

[7] for the first model) to value the reinsurance contract. They allow the reinsurer to transfer the risk to

the capital market via CAT bonds and, in effect, to reduce the risk of the reinsurer’s default risk. Since the

payments from CAT bonds cannot be replicated by the ordinary types of securities available in financial

markets, the pricing has to be done in the incomplete market model.

One common technique used for pricing in incomplete markets is indifference pricing via expected util-

ity. Young [10] finds indifference prices under a stochastic interest rate for the case where investors receive

unity if a catastrophe does not occur and 0 otherwise; see Young [10] for further references on indifference

pricing. Our paper extends the result of [10] by incorporating a more complex payment structure. Namely,

the CAT bond is issued in two tranches, specifically senior and junior tranches each with its own payment

schedule. We price each tranche in the framework of “relative” indifference pricing. To our knowledge, this

is the first paper that deals with a structured catastrophe derivative.

The remainder of the paper is organized as follows: In Section 2, we assume that reinsurance claims

follow a (Poisson) jump-diffusion process, so the corresponding surplus process of the reinsurer also follows

a jump-diffusion because we ignore investment earnings. We also describe the structured CAT bond that the

reinsurer will issue in order to reduce its risk. In Section 3, we present our method for pricing the CAT bond;

we rely on utility indifference pricing. In Section 4, we demonstrate our method with a number of examples

while assuming that utility is exponential. First, we assume that the claims only follow a diffusion process;

that is, we assume that there is no jump risk. Second, we assume that there is only a jump risk in the claim

www.swissre.com/INTERNET/pwswpspr.nsf/

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process, with no diffusion in the claim process nor the index process. We end Section 4 with a numerical

example in which there is both jump and diffusion risk in the claim process. Section 5 ends the paper.

2 Problem Description

Assume that a reinsurer faces a reinsurance contract with claim process

Ct= at − σWt+

Yi,C0= 0,

(2.1)

in which a and σ are positive constants, and Y1,Y2. are identically distributed positive random variables

with common distribution F(dy). W is a Brownian motion, N is a Poisson process with a constant arrival

rate λ, and N is independent of the Y ’s and of the Brownian motion W. The Y ’s represent infrequent

claims, while we let the Brownian motion approximate the frequent claims.

The reinsurer receives insurance premiums continuously at the constant rate (1 + θ)(a + λEY ) with

a relative risk loading of θ > 0. We ignore investment earnings of the reinsurer. Therefore, the reinsurer’s

surplus process X has state space I = R with dynamics

Xt= x0+ (1 + θ)(a + λEY )t − at + σWt−

Yi=: x0+ µt + σWt−

Yi,

(2.2)

in which

µ. (1 + θ)λEY + θa,

(2.3)

and x0is the initial surplus assigned to this contract. Note that there is positive probability that the process

X hits the ruin state 0. However, we assume that the reinsurer has a large capital (besides x0) and this

particular reinsurance contract takes up only a small percentage of the total wealth of the reinsurer. In other

words, even if the process X hits the zero state, the reinsurer has enough capital so that it does not fall into

insolvency.

There are two ways for the reinsurer to reduce its risk: First, the reinsurer could accept proportionally

less reinsurance with correspondingly proportionally less premium, so that the claim and surplus processes