CATASTROPHE RISK BONDS potx

Equilibrium pricing theory is used to develop a pricing method based on a model of the term structure of interest rates and a probability structure for the catastrophe risk.. Unlike corp

Samuel H Cox* and Hal W Pedersen

ABSTRACT

This article examines the pricing of catastrophe risk bonds Catastrophe risk cannot be hedged by

traditional securities Therefore, the pricing of catastrophe risk bonds requires an incomplete

markets setting, and this creates special difficulties in the pricing methodology The authors briefly

discuss the theory of equilibrium pricing and its relationship to the standard arbitrage-free valuation

framework Equilibrium pricing theory is used to develop a pricing method based on a model of the

term structure of interest rates and a probability structure for the catastrophe risk This pricing

methodology can be used to assess the default spread on catastrophe risk bonds relative to

traditional defaultable securities

“It is indeed most wonderful to witness such

desolation produced in three minutes of time.”

—Charles Darwin commenting on the February

20, 1835, earthquake in Chile

1 INTRODUCTION

Catastrophe risk bonds provide a mechanism for

direct transfer of catastrophe risk to capital

mar-kets, in contrast to transfer through a traditional

reinsurance company The bondholder’s cash

flows (coupon or principal) from these bonds are

linked to particular catastrophic events such as

earthquakes, hurricanes, or floods Although

sev-eral deals involving catastrophe risk bonds have

been announced recently, the concept has been

around awhile Goshay and Sandor (1973)

pro-posed trading reinsurance futures in 1973 In

1984, Svensk Exportkredit launched a private

placement of earthquake bonds that are

immedi-ately redeemable if a major earthquake hits Japan

(Ollard 1985) Insurers in Japan bought the

bonds, agreeing to accept lower-than-normal

cou-pons in exchange for the right to put the bonds

back to the issuer at face value if an earthquake

hits Japan This is the earliest catastrophe riskbond deal we know about

In the early 1990s, the Chicago Board of Tradeintroduced exchange-traded futures (which werelater dropped) and options based on industry-wide loss indices More recently, catastrophe riskhas been embedded in privately placed bonds,which allows the borrower to transfer risk to thelender In the event of a catastrophe, a catastro-phe risk bond behaves much like a defaultablecorporate bond The “default” of a catastropherisk bond is triggered by a catastrophe as defined

by the bond indenture

Unlike corporate bonds, the default risk of acatastrophe risk bond is uncorrelated with theunderlying financial market variables such as in-terest rate levels or aggregate consumption(Froot, Murphy, Stern, and Usher 1995) Conse-quently, the payments from a catastrophe riskbond cannot be hedged1 by a portfolio of tradi-tional bonds or common stocks The pricing ofcatastrophe risk bonds requires an incompletemarkets framework because no portfolio of prim-itive securities replicates the catastrophe riskbond Fortunately, the fact that catastrophe risk

is uncorrelated with movements in underlyingeconomic variables renders the incomplete mar-kets theory somewhat simpler than the case of

*Samuel H Cox, F.S.A., Ph.D., is a Professor of Actuarial Science in the

Department of Risk Management and Insurance at Georgia State

University, P.O Box 4036, Atlanta, GA 30302-4036, e-mail,

samcox@gsu.edu.

†

Hal W Pedersen, A.S.A., Ph.D., is Assistant Professor in the Actuarial

Science Program, Department of Risk Management and Insurance at

Georgia State University, P.O Box 4036, Atlanta, GA 30302-4036,

e-mail, inshwp@panther.gsu.edu.

1 Financial economists would say that the payments from catastrophe risk bonds cannot be spanned by primitive assets (ordinary stocks and bonds).

56

significant correlation We use this to develop a

simple approach to pricing catastrophe risk

bonds

The model we present for pricing catastrophe

risk bonds is based on equilibrium pricing The

model is practical in that the valuation can be

done in a two-stage procedure First, we select or

estimate the interest rate dynamics2in the states

of the world that do not involve the catastrophe

Constructing a term structure model is a

rela-tively well-understood and practiced procedure

Second, we estimate the probability3of the

catas-trophe occurring Valuation for the full model is

then accomplished by combining the probability

of the catastrophe occurring and the interest rate

dynamics from the term structure model We

show how one can implement valuation under the

full model using the standard tool of a risk-neutral

valuation measure The full model is

arbitrage-free but incomplete

Section 2 describes how catastrophe risk bonds

arise from the securitization of liabilities We also

describe some recent catastrophe risk bond deals

Section 3 provides a quick overview and

motiva-tion of how pricing can be carried out for

catas-trophe risk bonds We work out a numerical

ex-ample that illustrates the principles underlying

catastrophe risk bond deals Section 4 details the

inherent pricing problems one faces with

catas-trophe risk bonds because of the incomplete

mar-kets setting Section 5 describes our formal model

and provides a numerical example, and Section 6

concludes the paper

2 CATASTROPHE REINSURANCE AS A

HIGH-YIELD BOND

Most investment banks, some insurance brokers,

and most large reinsurers developed

over-the-counter insurance derivatives by 1995 This is a

form of liability securitization, but instead of

be-ing treated like exchange-traded contracts, these

securities are handled like private placements or

customized forwards or options Tilley (1995,

1997) describes securitized catastrophe

reinsur-ance in terms of a high-yield bond Froot et al.(1995) describe a similar one-period product.These products illustrate how catastrophe riskcan be distributed through capital markets in anew way The following description is an abstrac-tion and simplification but is useful for illustratingthe concepts

Consider a one-period reinsurance contractunder which the reinsurer agrees to pay a fixed

amount L at the end of the period if a defined

catastrophic event occurs It pays nothing if no

catastrophe occurs L is known when the policy

is issued If q cat denotes the probability of a

catastrophic event and P is the price of the

reinsurance, then the fair value of the ance is

reinsur-P⫽1⫹ r1 q cat L, where r is the one-period effective, default-free

interest rate This defines a one-to-one spondence between bond prices and probabili-ties of a catastrophe Since the reinsurance

corre-market will determine the price P, it is natural

to denote the corresponding probability with a

subscript “cat.” That is, q cat is the reinsurancemarket’s assessment of the probability of a ca-tastrophe

From where does the capital to support thereinsurer come? Astute buyers and regulatoryauthorities will want to be sure that the reinsurerhas the capital to meet its obligations if a catas-trophe occurs Usual risk-based capital require-ments based on diversification over a portfolio donot apply since the reinsurer has a single largerisk The appropriate risk-based capital require-ment is full funding That is, the reinsurer willhave no customers unless it can convince them

that it has secure capital at least equal to L To

obtain capital before it sells the reinsurance, thereinsurer borrows capital by issuing a defaultablebond, i.e., a junk bond Investors know when theybuy a junk bond that it might default, but theybuy the instrument anyway because these bonds

do not often default and they have higher returnsthan more reliable bonds (Indeed, we will seethat the recent deals were popular with inves-tors.) The reinsurer issues enough bonds to raise

an amount of cash C determined so that

共P ⫹ C兲共1 ⫹ r兲 ⫽ L.

2 Those familiar with state prices will find that this step is equivalent to

estimating local state prices for states of the world that are

indepen-dent of the catastrophe.

3

More generally, one estimates the probability distribution for the

varying degrees of severity of the catastrophe risk.

This satisfies the reinsurer’s customers They see

that the reinsurer has enough capital to pay for a

catastrophe The bondholders know that the

bonds will be worthless if there is a

catastro-phe—in which case, they get nothing If there is

no catastrophe, they get their cash back plus a

coupon R ⫽ L ⫺ C The bond market will

deter-mine the price per unit of face value In terms of

discounted expected cash flow, the price per unit

can be written in the form

1

1⫹ r 共1⫹ c兲共1 ⫺ q b兲,

where c ⫽ R/C is the coupon rate, and q bdenotes

the bondholders assessment of the probability of

default on the bonds We can assume that the

investment bank designing the bond contract sets

c so that the bonds sell at face value Thus, c is

determined so that investors pay 1 in order to

receive 1⫹ c one year later, if there is no

catas-trophe This is expressed as

1⫽1⫹ r 共1 1⫹ c兲共1 ⫺ q b兲

Of course, default on the bonds and a catastrophe

are equivalent events The probabilities might

dif-fer because bond investors and reinsurance

cus-tomers might have different information about

catastrophes The reinsurance company sells

bonds once c is determined to raise the required

capital C The corresponding bond market

prob-ability is found by solving for q b:

q b⫽1c ⫺ r ⫹ c.The implied price for reinsurance is

P b⫽1⫹ r1 1c ⫺ r ⫹ c L⫽1⫹ r1 q b L.

Provided the reinsurance market premium P

(the fair price determined by the reinsurance

market) is at least as large as P b, the

reinsur-ance company will function smoothly It will

collect C from the bond market and P from the

reinsurance market at the beginning of the

pol-icy period The sum invested for one period at

the default-free rate will be at least L This is

easy to see mathematically using the relation

exceed P, or equivalently, as long as

q catⱖ1c ⫺ r ⫹ c,there will be an economically viable market forreinsurance capitalized by borrowing in the bondmarket Borrowing (issuing bonds) to financelosses is not new In the late 1980s, when U.S.liability insurance prices were high and interestrates were moderate, some traditional insurancecustomers replaced insurance with self-insuranceprograms financed by bonds Of course, this is not

a securitization of insurance risk but it is anexample of insurance customers turning to thecapital markets to finance losses More recently,several state-run hurricane and windstorm poolsextended their claims-paying ability with bank-arranged contingent borrowing agreements inlieu of reinsurance (Neidzielski 1996) The catas-trophe property market in the 1990s has seenlower prices than the 1980s Providing prices arehigh enough to permit the structuring of dealsthat are attractive to investors and to entice cap-ital market advocates such as Froot et al (1995),Lane (1997), and Tilley (1995, 1997) to offer catrisk products, it is natural that these deals willcontinue to proliferate Thus far, a catastropherisk bond market is developing

In our model, the fund always has adequatecash to pay the loss if a catastrophic event occurs

If no catastrophe occurs, the fund goes to thebond owners From the bond owners’ perspective,the bond contract is like lending money subject tocredit risk, except the risk of “default” is reallythe risk of a catastrophic event Tilley describesthis as a fully collateralized reinsurance contractsince the reinsurer has adequate cash at the be-ginning of the period to make the loss payment

with probability one This scheme is a simple

version of how a traditional reinsurer works, with

the following differences:

● The traditional reinsurance company owners

buy shares of stock instead of bonds

● Traditional reinsurer losses affect investors

(stockholders) on a portfolio basis rather than a

single-exposure basis

● Simplifying and specializing makes it possible

to sell single exposures through the capital

mar-kets, in contrast to shares of stock of a

rein-surer, which are claims on the aggregate of

out-comes

Tilley (1995, 1997) demonstrates this

tech-nique in a more general setting in which the

reinsurance and bond are N period contracts.

This one-period model illustrates the key ideas

Now we describe three catastrophe bonds that

have recently appeared on the market In Section

5, we describe a hypothetical example that

illus-trates how catastrophe bonds increase insurer

capacity to write catastrophe coverages

2.1 USAA Hurricane Bonds

USAA is a personal lines insurer based in San

Antonio It provides personal financial

manage-ment products to current or former U.S military

officers and their dependents Zolkos (1997a), in

reporting on the USAA deal, described USAA as

“overexposed” to hurricane risk in its personal

automobile and homeowners business along the

U.S Gulf and Atlantic coasts In June 1997, USAA

arranged for its captive Cayman Islands

rein-surer, Residential Re, to issue $477 million face

amount of one-year bonds with coupon and/or

principal exposed to property damage risk to

USAA policyholders due to Gulf or East Coast

hurricanes Residential Re issued reinsurance to

USAA based on the capital provided by the bond

sale

The bonds were issued in two series (also called

tranches), according to an article in The Wall

Street Journal (Scism 1997) In the first series,

only coupons are exposed to hurricane risk—the

principal is guaranteed For the second series,

both coupons and principal are at risk The risk is

defined as damage to USAA customers on the Gulf

or East Coast during the year beginning in June

The coupons and/or principal will not be paid to

investors if these losses exceed $1 billion That is,the risk begins to reduce coupons at $1 billion,and at $1.5 billion the coupons in the first seriesare completely gone, and in the second series thecoupons and principal are lost The coupon-onlytranche has a coupon rate of the London Inter-bank Offered Rate (LIBOR) plus 2.73%, The prin-cipal and coupon tranche has a coupon rate ofLIBOR ⫹ 5.76%

The press reported that the issue was scribed,” meaning there were more buyers thananticipated The press reports indicated that thebuyers were life insurance companies, pensionfunds, mutual funds, money managers, and, to avery small extent, reinsurers As a point of referencefor the risk involved, we note that industry lossesdue to hurricane Andrew in 1992 amounted to

“oversub-$16.5 billion and USAA’s Andrew losses amounted

to $555 million Niedzielski reported in the

National Underwriter that the cost of the coverage

was about 6% rate on line plus expenses.4According

to Niedzielski’s (unspecified) sources, the ble reinsurance coverage is available for about 7%rate on line The difference is probably more thanmade up by the fees related to establishing Residen-tial Re and the fees to the investment bank forissuing the bonds The rate on line refers only to thecost of the reinsurance The reports did not give thesale price of the bonds, but the investment bankprobably set the coupon so that they sold at facevalue

compara-As successful as this issue turned out (the tastrophe provision was not triggered and thebonds matured as scheduled), it was a long timecoming Despite advice of highly regarded advo-cates such as Morton Lane and Aaron Stern (seeFroot et al 1995, Lane 1995, Niedzielski 1995),catastrophe bonds have developed more slowlythan many experts expected According to pressreports, USAA has obtained 80% of the coverage

ca-of its losses in the $1.0 to $1.5 billion layer withthis deal On the other hand, we have to wonderwhy it is a one-year deal Perhaps it is a matter ofgetting the technology in place The off-shore re-

4 Rate on line is the ratio of premium to coverage layer The ance agreement provides USAA with 80% of $500 million in excess of

reinsur-$1 billion The denominator of the rate on line is (0.80)($500) ⫽

$400 million, so this implies USAA paid Residential Re a premium of about (0.06)($400) ⫽ $24 million.

insurer is reusable And the next time USAA goes

to the capital market, investors will be familiar

with these exposures If the traditional

catastro-phe reinsurance market gets tight, USAA will

have a capital market alternative The cost of this

issue is offset somewhat by the gain in access to

alternative sources of reinsurance

2.2 Winterthur Windstorm Bonds

Winterthur is a large insurance company based in

Winterthur, Switzerland In February 1997,

Win-terthur issued three-year annual coupon bonds

with a face amount of 4,700 Swiss francs The

coupon rate is 2.25%, subject to risk of windstorm

(most likely hail) damage during a specified

ex-posure period each year to Winterthur

automo-bile insurance customers The deal was described

in the trade press and Schmock (1999) has

writ-ten an article in which he values the coupon cash

flow The deal has been mentioned in U.S

publi-cations (for example, Investment Dealers Digest

[Monroe 1997]), but we had to go to Euroweek

(1997) for a published report on the contract

details If the number of automobile windstorm

claims during the annual observation period

ex-ceeds 6,000, the coupon for the corresponding

year is not paid The bond has an additional

fi-nancial wrinkle It is convertible at maturity;

each face amount of 4,700 Swiss francs is

con-vertible to five shares of Winterthur common

stock at maturity

2.3 Swiss Re California Earthquake Bonds

The Swiss Re deal is similar to the USAA deal in

that the bonds were issued by a Cayman Islands

reinsurer, evidently created for issuing

catastro-phe risk bonds, according to Zolkos (1997b)

However, unlike USAA’s deal, the underlying

Cal-ifornia earthquake risk is measured by an

indus-try-wide index rather than Swiss Re’s own

port-folio of risks The index is developed by Property

Claims Services Evidently, the bond contract is

written on the same (or similar) California index

underlying the Chicago Board of Trade (CBOT)

Catastrophe Options The CBOT options have

been the subject of numerous scholarly and trade

press articles (Cox and Schwebach 1992; D’Arcy

and France 1992; D’Arcy and France 1993;

Em-brechts and Meister 1995)

Zolkos (1997b) reported details on the Swiss Re

bonds in Business Insurance There were earlier

reports that Swiss Re was looking for a 10-yeardeal This is not it, so perhaps they are still look-ing According to Zolkos, SR Earthquake Fund (acompany Swiss Re apparently set up for this pur-pose) issued Swiss Re $122.2 million in Californiareinsurance coverage based on funds provided bythe bond sale In the next section, we will provide

a numerical example that illustrates the ples underlying these three deals

princi-3 MODELING CATASTROPHE RISK BONDS

In the previous section, we discussed the tization underlying catastrophe risk bonds Inthis section, we adopt a standardized definition of

securi-a csecuri-atsecuri-astrophe risk bond for the purposes of securi-ansecuri-a-lyzing this security using financial economics Weare informal in this section, leaving the definition

ana-of some technical terms until Section 5

A catastrophe risk bond with a face amount of

$1 is an instrument that is scheduled to make a

coupon payment of c at the end of each period

and a final principal repayment of $1 at the end of

the last period (labeled time T) as long as a

spec-ified catastrophic event (or events) does not cur.5 The investment banker designing the bondknows the market well enough to know what cou-pon is required for the bond to sell at face value.However, we will take the view that the coupon isset in the contract, and we will determine themarket price This is an equivalent approach

oc-We will focus most of our attention on bondsthat have coupons and principal exposed to ca-tastrophe risk These are defined as follows Thebond coupons are made with only one possiblecause of default—a specified catastrophe The

bond begins paying at the rate c per period and continues paying to T with a final payment of

1 ⫹ c, if no catastrophe occurs If a catastrophe

should occur during a coupon period, the bondmakes a fractional coupon payment and a frac-tional principal repayment that period and is thenwound up The fractional payment is assumed to

be of the fraction f so that if a catastrophe occurs,

the payment made at the end of the period in

5

In practice, catastrophe risk bonds will vary by the contractual manner in which catastrophes affect the payment of coupons and repayment of principal Therefore, this is too narrow a definition to capture the variety of features one finds in these bonds.

which the catastrophe occurs is equal to f(1 ⫹ c).

At present, we are not allowing for varying

sever-ity in the claims associated with the catastrophe

Varying severity would occur in practice We

mention this modeling issue later

Financial economics theory tells us that when

an investment market is arbitrage-free, there

ex-ists a probability measure, which we denote by⺡,

referred to as the risk-neutral measure, such that

the price at time 0 of each uncertain cash-flow

stream {c(k) 兩 k ⫽ 1, 2, , T} is given by the

following expectation under the probability

The process {r(k) : k ⫽ 1, , T ⫺ 1} is the

stochastic process of one-period interest rates

We denote the price at time 0 of a nondefaultable

zero-coupon bond with a face amount of $1

ma-turing at time n by P(n) Therefore we have, for

n ⫽ 1, 2, , T,

关1⫹r共0兲兴关1⫹r共1兲兴· · ·关1⫹r共n⫺1兲兴册

(3.2)

We shall let ␶ denote the time of the first

oc-currence of a catastrophe.6 A catastrophe might

or might not occur prior to the scheduled

matu-rity of the catastrophe risk bond at time T If a

catastrophe occurs, then␶ 僆 {1, 2, , T} For a

catastrophe bond with coupons and principal at

risk (like the second tranche of the USAA bond

issue or the Swiss Re bonds), the cash-flow

stream to the bondholder can be described (using

indicator functions7) as follows:

factor f(1 ⫹ c) in Equation (3.3) by fc and adjust

the payment in the event ␶ ⫽ T to reflect the

return of principal guarantee:

Let us assume that we are trading catastropherisk bonds in an investment market that is arbi-trage-free with risk-neutral valuation measure⺡.The time of the catastrophe is independent of theterm structure under the probability measure⺡

We shall formalize these notions8 in Section 5

We can relate Equation (3.1) to the cash-flowstream in Equation (3.3) and find that the price attime 0 of the cash-flow stream provided by thecatastrophe risk bond is given by the expression

The term ⺡(␶ ⬎ k) is the probability under the

risk-neutral valuation measure that the

catastro-phe does not occur within the first k periods The

other probabilistic terms can be verbalized larly No assumption has been made about thedistribution of␶ but the assumption that only one

simi-6

Since we are working in discrete-time, to say that a catastrophe

occurs at time k means that in real time the catastrophe occurred

after time k ⫺ 1 and before or at time k (i.e., the catastrophe

occurred in the interval (k ⫺ 1, k]).

7

For an event A, the indicator function is the random variable, which

is one if A occurs and zero otherwise It is denoted 1.

8

These are the assumptions made by Tilley (1995, 1997) although they are not stated in quite this terminology.

degree of severity can occur is clearly being used

here Of course, the distribution of␶ will depend

on the structure of the catastrophe risk exposure

Equation (3.5) expresses the price of the

catas-trophe risk bond in terms of known parameters,

including the coupon rate c As we described at

the beginning of this section, the principal

amount of the catastrophe risk bond is fixed at

the time of issue and the coupon rate is varied to

ensure that the price of the cash flows provided

by the bond are equal to the principal amount

One can apply the valuation Equation (3.5) to

obtain a formula for the coupon rate as

The bondholder’s cash flow, X, given a

catastro-phe occurs, could be random, requiring an

adjust-ment to the model Let G( x) denote the

condi-tional severity distribution of the bondholders’

cash flow X, given a catastrophe occurs Under

Tilley’s assumptions, Equation (3.5) becomes

When comparing Equations (3.5) and (3.7), we

see that there is little difference between the two

formulas Generally, the conditional severity

dis-tribution is embedded as part of the risk-neutral

measure⺡

Let us suppose that the catastrophe risk

struc-ture is such that the conditional probability

un-der the risk-neutral measure of no catastrophe for

a period is equal to a constant ␪0 Furthermore,

suppose that should a catastrophe occur, there is

a single severity level resulting in a payment

equal to f(1 ⫹ c) at the end of the period in which

the catastrophe occurs Let ␪1 ⫽ 1 ⫺ ␪0 In this

case, Equation (3.5) simplifies to the expression

given by Tilley (1995, 1997) for the price at time

0 of the catastrophe risk bond, namely

␪1 has not been related to the empirical tional probability of a catastrophe occurring.Therefore, Equation (3.8) is not quite “closed.” Inorder to close the model, we need to link thevaluation formula in Equation (3.8) with observ-able quantities that can be used to estimate theparameters needed to apply the valuation model.Although we began the discussion of the pricingmodel with an assumption about the existence of

condi-a vcondi-alucondi-ation mecondi-asure⺡, it is possible to justify aninterpretation of ␪1 as the empirical conditionalprobability of a catastrophe occurring We shalladdress and clarify this point in Section 5

4 INCOMPLETENESS IN THE PRESENCE OF

CATASTROPHE RISK

The introduction of catastrophe risk into a rities market model implies that the resultingmodel is incomplete The pricing of uncertaincash-flow streams in an incomplete model is sub-stantially weaker in the interpretation of the pric-ing results that can be obtained than is pricing incomplete securities markets In this section, wediscuss market completeness and explain the na-ture of the incompleteness problem for modelswith catastrophe risk exposures For simplicity,

secu-we work with a one-period model, although ilar notions can be developed for multiperiodmodels Let us consider a single-period model inwhich two bonds are available for trading, one ofwhich is a one-period bond and the other a two-period bond For convenience we shall assumethat both bonds are zero-coupon bonds We fur-ther assume that the financial markets will evolve

sim-to one of two states at the end of the period—

“interest rates go up” or “interest rates godown”—and that the price of each bond will be-

have according to the binomial model depicted in

Figures 1 and 2

The bond prices for this model could be derived

from the equivalent information in the tree

dia-gram in Figure 2 for which the one-period model

is embedded We specified the bond prices

di-rectly to avoid bringing a two-period model into

our discussion of the one-period case The prices

that are reported in Figure 1 have been rounded

from what one would compute directly from

Fig-ure 2 For example, we rounded

1 1.06 共1

2兲共 1 1.07⫹ 1 1.05兲

to 0.8901

Suppose that we select a portfolio of the

one-period and two-one-period bonds Let us denote the

number of one-period bonds held in this portfolio

by n1and the number of two-period bonds held in

this portfolio by n2 This portfolio will have a

value in each of the two states at time 1 Let us

represent the state dependent price of each bond

at time 1 using a column vector Then, we can

represent the value of our portfolio at time 1 by

the following matrix equation:

period bonds held multiplied by today’s price of

one-period bonds plus the number of two-period

bonds held multiplied by today’s price of

two-period bonds

The 2 ⫻ 2 matrix of bond prices at time 1

appearing in Equation (4.1) is nonsingular.Therefore, any vector of cash flows at time 1 can

be generated by forming the appropriate portfolio

of these two bonds For instance, if we want thevector of cash flows at time 1 given by the columnvector,

Upon substituting for n1and n2as determined byEquation (4.4) into the expression for the cost ofthe portfolio given by Equation (4.5), one findsthat the price of each cash flow of the form ofEquation (4.3) is given by the expression

共1

2兲 1 1.06 c u⫹ 共1

2兲 1 1.06 c d ⫽ 0.4717c u ⫹ 0.4717c d

(4.6)Since every such set of cash flows at time 1 can beobtained and priced in the model we say that the

one-period model is complete The notion of

pric-ing in this complete model is justified by the factthat the price we assign to each uncertain cash-flow stream is exactly equal to the price of theportfolio of one-period and two-period bonds thatgenerates the value of the cash-flow stream attime 1

Let us see how the model changes when

catas-Figure 1

One-Period Bond Versus Two-Period Bond*

*Prices have been rounded: 1/1.06 ⬇ 0.9434, 1/1.07 ⬇ 0.9346, and

1/1.05 ⬇ 0.9524.

Figure 2

The Two-Period Term Structure Model

trophe risk exposure is incorporated as part of the

information structure Suppose that we have the

framework of the previous model with the

addi-tion of catastrophe risk Furthermore, let us

sup-pose that the catastrophic event occurs

indepen-dently of the underlying financial market

variables Therefore, there will be four states in

the model that we can identify as follows:

兵interest rate goes up, catastrophe occurs其 ⬅ 兵u, ⫹ 其

兵interest rate goes up, no catastrophe occurs其 ⬅ 兵u, ⫺ 其

兵interest rate goes down, catastrophe occurs其 ⬅ 兵d, ⫹ 其

兵interest rate goes down, no catastrophe occurs其 ⬅ 兵d, ⫺ 其

(4.7)

The reader will note that the symbol {u, ⫹} is

shorthand for “interest rates go up” and

“catas-trophe occurs,” and so forth This information

structure is represented on a single-period tree

with four branches as shown in Figure 3

The values at time 1 of the one-period bond and

the two-period bond are not linked to the

occur-rence or nonoccuroccur-rence of the catastrophic

event, and therefore, do not depend on the

cata-strophic risk variable We can represent the

prices of the one-period and two-period bond in

the extended model as shown in Figure 4 In

contrast to Equation (4.1), the value at time 1 of

a portfolio of the one-period and two-period

bonds is now given by the following matrix

The most general vector of cash flows at time 1

in this model is of the following form:

冤 c u,⫹

c u,⫺

c d,⫹

On reviewing Equation (4.8), we see that the span

of the assets available for trading in the model(that is, the one-period and two-period bonds) isnot sufficient to span all cash flows of the form inEquation (4.9) Since there are cash flows at time

1 that cannot be obtained by any portfolio of thetwo bonds (one-period and two-period) we haveavailable for trade, this one-period model is said

to be incomplete Consequently, we cannot

de-rive a pricing relation such as Equation (4.6) that

is valid for all cash-flow vectors of the form ofEquation (4.9) The best we can do is obtainbounds on the price of a general cash-flow vector

so that its price is consistent with the absence ofarbitrage A discussion follows

Our one-period securities market model is bitrage-free, if and only if, there exists a vector(see Panjer et al 1998, chapter 5, or Pliska 1997,chapter 1):

ar-⌿ ⬅ 关ar-⌿u,⫹, ⌿u,⫺, ⌿d,⫹, ⌿d,⫹兴; (4.10)each component of which is positive, such that,

Figure 3

Information Structure

Figure 4

Prices of One-Period and Two-Period Bonds*

*Prices have been rounded: 1/1.06 ⬇ 0.9434, 1/1.07 ⬇ 0.9346, and 1/1.05 ⬇ 0.9524.

can solve Equation (4.11) for all such vectors to

find that the class of all state price vectors for this

model is of the form

⌿ ⫽ 关0.4717 ⫺ s, s, 0.4717 ⫺ t, t兴, (4.12)

for 0⬍ s ⬍ 0.4717 and 0 ⬍ t ⬍ 0.4717 For each

cash flow of the form in Equation (4.9), there is a

range of prices that are consistent with the

ab-sence of arbitrage This is given by the expression

0.4717c u,⫹⫹ 0.4717c d,⫹⫹ s共c u,⫺⫺ c u,⫹兲

⫹ t共c d,⫺⫺ c d,⫹兲, (4.13)

where s and t range through all feasible values

0⬍ s ⬍ 0.4717 and 0 ⬍ t ⬍ 0.4717 Note that a

security with cash flows that do not depend on

the catastrophe10are uniquely priced This is not

true of catastrophe risk bonds For instance, the

price of the cash-flow stream that pays 1 if no

catastrophe occurs and 0.5 if a catastrophe

oc-curs has the price range given by the expression

0.4717共0.5兲 ⫹ 0.4717共0.5兲 ⫹ s共1 ⫺ 0.5兲

⫹ t共1 ⫺ 0.5兲 ⫽ 0.4717 ⫹ 共s ⫹ t兲共0.5兲.

The range of prices for this cash-flow stream is

the open interval (0.4717, 0.9434) These price

bounds are not very tight However, this is all that

can be said, if working solely from the absence of

arbitrage

Let us consider the case of a one-period

catas-trophe risk bond with f ⫽ 0.3 In return for a

principal deposit of $1 at time 0, the investor will

receive an uncertain cash-flow stream at time 1 of

the form:

共1 ⫹ c兲冤 0.3

1.00.3

We may apply the relation in Equation (4.13) tofind that the range of values on the coupon thatmust be paid to the investor lie in the open inter-val (0.06, 2.5333) The coupon rate of the catas-trophe risk bond is not uniquely defined Indeed,there is but a range of values for the coupon thatare consistent with the absence of arbitrage Al-though this is a very wide range of coupon rates,this is the strongest statement about how thecoupon values can be set subject only to thecriterion that the resulting securities market isarbitrage-free Evidently, we need to bring insome additional theory in order to obtain useful,benchmark pricing formulas for catastrophe riskbonds In fact, we shall see that we can tightenthese bounds, even to the point of generating anexplicit price, by embedding in the model theprobabilities of the catastrophe occurring Forthis example, let us assume that investors agree

on the probability q of a catastrophe and they

agree that the catastrophe bond price should beits discounted expected value The expected cashflow11to the bondholder at time 1 is

共1 ⫹ c兲共0.3q ⫹ 1.0共1 ⫺ q兲兲, (4.15)and it thus remains to discount appropriately.This bond has the same (expected) value in each

interest rate state, so its price V is that value

times the price of the one-year default-free bond.Thus,

V ⫽ 共1 ⫹ c兲共0.3q ⫹ 1.0共1 ⫺ q兲兲 1

1.06

Now, we could determine the coupon c so that the bond sells at par (that is, V ⫽ 1) initially, or wecould determine the price for a specified coupon.Given the probability distribution of the catastro-phe and the assumption that prices are dis-counted expected values (over both risks), we canthen obtain unique prices

This section has illustrated the difficulties ent in applying modern financial theory to analyzecatastrophe risk Generally, prices can no longer be

inher-9 The reader may check that the components of the state price vector

are precisely the risk-neutral probabilities of each state discounted by

the short rate.

10

Mathematically, if the cash flows do not depend on the catastrophe

then, c u,⫺ ⫽ c u,⫹ and c d,⫺ ⫽ c d,⫹

11

The expression in Equation (4.15) is the average cash-flow at time

1 over all states— catastrophic and noncatastrophic.

justified by arbitrage considerations alone because

this notion is based on the principle that the price of

a set of cash flows is equal to the cost of a portfolio

of existing assets that has the same payoffs as the

cash flows we are interested in pricing; generally,

there is no such portfolio In short, the presence of

catastrophe risk results in nonuniqueness of prices

and unique prices can only be recovered at the

expense of introducing the probability distribution

of the catastrophe risk Such is the nature of

incom-plete markets In the following section we shall

de-scribe a method of obtaining explicit prices for

ca-tastrophe risk bonds and describe some examples

5 A FORMAL MODEL

In Section 3, we gave a preliminary presentation

of the basic formulation of a valuation model for

catastrophe risk bonds and discussed the type of

valuation formulas described in Tilley (1995,

1997) The discussion offered in Section 3 should

be considered as motivation for the formal model

that we now develop The formal model we

de-scribe is designed to combine primary financial

market variables with catastrophe risk variables

to yield a theoretical valuation model for

catas-trophe risk bonds Of course, the mathematics of

the model can be used in other contexts

regard-less of the interpretation we give to the

compo-nents of the model The formalization of the

model requires some technical measure theoretic

definitions, but the intuition of the model

re-mains concrete

5.1 Information and Probabilistic

Structure for the Model

The financial market variables are assumed to

be modeled on the filtered probability space

(⍀(1), ᏼ(1), ⺠1) We briefly review the relevant

concepts and notation on the run.12 The sample

space ⍀(1) is taken to be finite,13 and it

repre-sents all the paths the financial variables can take

over the times k ⫽ 0, 1, , T The time T ⬍ ⬁ is

interpreted as the end of the trading interval Thefiltration ᏼ(1)represents the way in which infor-mation evolves in the financial market and can bethought of as an information tree More precisely,the filtration is an increasing sequence

ᏼ共1兲⫽ 兵ᏼ0 共1兲債 ᏼ1 共1兲債 · · · 債 ᏼT共1兲其 (5.1)

of sets of events indexed by time k ⫽ 0, 1, , T.

The events in ᏼk

(1) represent the investment

in-formation available to the market at time k In

practice, this investment information consists ofpast security prices.14The increasing feature for-malizes the idea that no information is lost fromone time to the next The probability measure⺠1

is defined on the sigma-algebraᏼT

(1), and thus, bythe increasing property in Equation (5.1),⺠1( A)

is defined for all events A 僆 ᏼk

(1)for k ⱕ T.

The catastrophic risk variables are assumed to

be modeled on the filtered probability space(⍀(2), ᏼ(2), ⺠2) ⺠2 is the probability measuregoverning the catastrophe structure.15 The filtra-tion ᏼ(2) is indexed over the same times k ⫽ 0,

1, , T as the filtration for the financial market

variables The probability measure⺠2is the ical probability measure governing catastrophicevents In other words,⺠2is the probability mea-sure used to compute the probability of a cata-strophic event

phys-The sample space for our full model is taken to

be the product space:

⍀ ⫽def ⍀共1兲⫻ ⍀共2兲.Therefore, a typical element of the sample spacefor the full model is of the form

␻ ⫽ 共␻共1兲, ␻共2兲兲 with ␻共1兲僆 ⍀共1兲, ␻共2兲僆 ⍀共2兲.Such an element can be interpreted as jointlydescribing the state of the financial market vari-ables and the catastrophic risk variables Itshould again be emphasized that under this con-struction, the embedded sample space ⍀(1) rep-resents the primary financial market variables16

12 The reader may consult Panjer (1998, chapter 5) or Pliska (1997,

chapter 3) for the all of the background details A very concrete

discussion of filtration in discrete models can be found in Panjer

(1998, chapter 11).

13 The major results of the paper also hold for infinite sample spaces.

We assume the sample space is finite to make the mathematics more

ca-while the embedded sample space⍀(2)represents

the catastrophic exposure risk variables

The probability measure on the sample space⍀

is given by the natural product measure

struc-ture Therefore, the probability of a generic state

of the world,␻ ⫽ (␻(1),␻(2)), is

⺠共␻兲 ⫽ ⺠1共␻共1兲兲⺠2共␻共2兲兲

This assumption implies the independence of

events that depend only on economic risk variables

and those that depend only on catastrophe risk

variables This is formalized in Lemma 5.1 The

filtration for the product measure space, denoted by

ᏼ, is taken to be the natural product filtration

gen-erated by the rectangles inᏼ(1)⫻ ᏼ(2) Formally,

ᏼk ⫽def ᏼk共1兲⫻ ᏼk共2兲 for k ⫽ 0, 1, , T (5.2)

Thus, the probability space for the full model is

the triple (⍀, ᏼ, ⺠)

In order to discuss random variables in our full

model that depend only on financial variables or

catastrophic risk variables, we require some

tech-nical definitions We begin by defining two new

filtrationsᏭ(1)and Ꮽ(2)

Ꮽk共1兲 ⫽def ᏼk共1兲⫻ 兵À, ⍀共2兲其 for k ⫽ 0, 1, , T

and

Ꮽk共2兲 ⫽def 兵À, ⍀共1兲其 ⫻ ᏼk共2兲 for k ⫽ 0, 1, , T.

(DRV-FIN) A random variable X on (⍀, ᏼ, ⺠) is

said to depend only on financial risk variables if X is measurable

with respect toᏭT

(1).Intuitively, this corresponds to the functional no-

tion that

X共␻共1兲, ␻共2兲兲 ⬅ X共␻共1兲兲

A similar concept applies for catastrophic risk

(DRV-CAT) A random variable X on (⍀, ᏼ, ⺠) is

said to depend only on strophic risk variables if X is mea-

cata-surable with respect to ᏭT

(2).Analogous concepts apply to the evolution of sto-

chastic processes

(DSP-FIN) A stochastic process Y is said to

evolve through dependence only

on financial risk variables if Y is

adapted to Ꮽ(1)

(DSP-CAT) A stochastic process Y is said to

evolve through dependence only

on catastrophic risk variables if Y

(1)does not containevents that are defined in (⍀, ᏼ, ⺠) becauseevents in our full probability space have the prod-uct structure of Equation (5.2) This is why weneeded to formalize the independence notion us-ingᏭ(1)andᏭ(2) The idea thatᏼT

(1)andᏼT

(2)areindependent under ⺠ is captured through sets ofthe form in Equation (5.3) We also point out thatLemma 5.1 does not depend on any of the as-sumptions about aggregate consumption that aremade in Section 5.2

5.2 The Valuation Set-up for the Model

The benchmark financial economics techniqueused to price uncertain cash-flow streams in anincomplete markets setting is the representativeagent We now describe this technique in the con-text of the probability structure we have just de-fined The representative agent technique consists

of an assumed representative utility function and anaggregate consumption process The agent uses theutility function to make choices about consumptionstreams These consumption streams are permitted

to depend only on observable information, and are

thus constrained to be adapted17 stochastic

pro-cesses We will denote a generic consumption

stream by

兵c共k兲兩k ⫽ 0, 1, 2, , T其.

The aggregate consumption process is the

to-tal consumption available in the economy at each

point in time and in each state We shall denote

the aggregate consumption stochastic process by

兵C*共k兲兩k ⫽ 0, 1, 2, , T其.

C*( ␻, k) is the amount of the consumption good18

endowed to the entire economy in state␻ at time

k Only the aggregate consumption amount C*(0)

is known with certainty at time k ⫽ 0 Each of

C*(k), k ⱖ 1 are random and could be formally

expressed as C*( ␻, k), which reflects the

depen-dence on the random state ␻ Generally, we will

follow the usual convention of suppressing the

explicit indication of randomness in stochastic

processes by omitting the␻

We shall assume that the representative agent’s

utility is additively separable as well as

differen-tiable Additively separable means that there are

utility functions u0, u1, , u T, such that the

agent’s expected utility for any generic

consump-tion process {c(k) 兩k ⫽ 0, 1, , T} is given by

E⺠ 冋 冘

k⫽0

T

u k 共c共k兲兲册 (5.4)

It follows from the theory of the representative

agent19 that the price V(d) of a generic future

cash flow process d ⫽ {d(k) 兩 k ⫽ 1, 2, , T} at

time 0 is given by the expectation

More generally, viewed from time n, the price of the

remaining portion of the generic future cash-flow

process, which would be {d(k) 兩k ⫽ n ⫹ 1, n ⫹

2, , T}, is given by the conditional expectation

by relating the pricing relation to the valuationmeasure approach of arbitrage-free pricing

In order to relate the representative agent uation formula to the usual valuation measureapproach from arbitrage-free pricing, we need todefine the one-period interest rates implicit in therepresentative agent pricing model We define21

val-the one-period interest rates

We will now define a new probability measure⺡

in terms of⺠ and a positive random variable, calledthe Radon-Nikodym derivative of⺡ with respect to

⺠ This change is very convenient because, underthe new measure, all prices are discounted expected

17Adapted means that the consumption taken at time k can depend

only on the information available at time k The information available

at time k is represented by the sigma-algebraᏼk.

18

For our purposes, the consumption good can be thought of as

money.

19

See Huang and Litzenberger (1988), Karatzas (1997), Magill and

Quinzii (1996), or Panjer et al (1998) for details on the theory of the

representative agent Embrechts and Meister (1995) apply a related

method from an alternative viewpoint.

20

For an exchange economy in equilibrium, the aggregate tion is equal to the aggregate endowment in all states at all times Thus, some presentations of the representative agent model will refer

consump-to C* as the aggregate endowment process.

21

In fact, this is the standard correspondence One can motivate this

as follows Suppose that we are in state ␻ at time k If there is

available a cash-flow stream that pays a unit one period from now with certainty and nothing else, by Equation (5.6), the price of this cash-flow stream in state␻ at time k is equal to