Catastrophe swap valuation with counterparty default...

Catastrophe swap valuation with counterpartydefault risk

March 09, 2019

Abstract

In this study we develop a CAT swap pricing model and incorporate thee¤ect of counterparty default risk on the CAT swap valuation. We estimatethe spreads of CAT swaps by using the Monte Carlo simulation method. Thenumerical results allow us to measure the e¤ects of counterparty default risk, thechanges in interest rates, catastrophe intensities, trigger settings, and maturitieson the values of CAT swap spreads.

Key Words: Swap; Catastrophe Risk; Counterparty default risk; Spread.

1

1 Introduction

Due to the climate change and the population concentration the losses from natural

or man-made disasters have both grown steadily during the past several decades.

The Sigma Report reveals that natural catastrophes and man-made disasters in 2012

caused economic losses of USD 186 billion with the lost of approximately 14 000

lives. The worldwide average insured catastrophic loss was US$5.1 billion per year

before 1990, but increases to US$27.1 billion per year during the period of 1990-2009.

Insurance companies typically transfer their catastrophe risks from the coverage of

reinsurance contracts. Huge catastrophic losses stress the capacity of the insurance

industry signicantly and threaten the credit risk of many reinsurers (Cummins et

al., 2002). Froot (1999, 2001) and Harrington and Niehaus (2003) pointed out that

the catastrophe reinsurance market have found that the series of catastrophe events

limited availability of catastrophic reinsurance coverage in the market. Therefore,

the insurance industry has to seek solutions other than traditional reinsurance to

increase the capacity for providing catastrophe coverage and nd ways to diversify

their catastrophe risk. Meanwhile, the capital markets have developed alternative risk

transfer instruments to provide (re)insurers with vehicles for hedging such catastrophe

risk. These instruments include catastrophe (CAT) swap , contingent surplus notes,

CAT futures/options, CAT equity put options, CAT bonds, and so on. They provide

the insurance industry alternative channels to raise capital from capital market and

increase overall underwriting capacity.

Catastrophe (CAT) swaps are insurance-linked securities and are a bilateral con-

tract through which catastrophe losses can be transferred between two counterpar-

ties.1 In a typical CAT swaps, the protection buyers (xed payers) agrees to pay

1The rst exchange-traded insurance-linked securities were launched by the Chicago Board ofTrade (CBOT), which introduced CAT futures and CAT futures call spreads in 1992 and 1993,

2

periodic payments (premiums) to protection sellers (oating payers) in exchange for

a predetermined compensation contingent on the occurrence of triggered catastrophic

losses. An example o¤ered in the current market is the Deutsche Bank Event Loss

Swaps (ELS). The Deutsche Bank Event Loss Swaps for US wind storms were launched

in the late of 2006. The swaps are available with alternative thresholds of US $ 20

billion, US $ 30 billion or US $ 50 billion, while the attachment levels for earthquakes-

based contracts to be set at US $ 01 billion and US $ 15 billion. A similar contract for

US wind and earthquake events is the Swiss Re Natural Catastrophe Swaps (SNaC-

STM) launched by the Swiss Re. An alternative type of CAT swaps is a termed as pure

risk swap. In the pure risk swaps, two counterparties (typically two (re)insurers) ex-

change uncorrelated or low correlated catastrophic risk exposures in their businesses

in order to improve diversication in their underwriting portfolio. The (re)insurer

whose business is highly concentrated in a particular area/ particular line can replace

a portion of its core risk with another type of perils they may not be able to access.

This can enable (re)insurers to operate with less equity capital. An example is the

swap executed by Mitsui Sumitomo Insurance and Swiss Re in 2003. The Mitsui

Sumitomo Insurance swapped US $ 12 billion of Japanese typhoon risk against US $

50 million each of North Atlantic and European windstorm risk.

respectively. The CBOT launched CAT options in 1995 and so did the Bermuda CommoditiesExchange (BCE) in 1997. Both contracts were withdrawn from markets due to lack of trading. Thehuge losses of the 2005 Hurricane Katrina erodes the underwriting capacity of the U.S. insuranceindustry and motivates the exchanges, the Chicago Mercantile Exchange (CME), the InsuranceFutures Exchange (IFEX), and the New York Mercantile Exchange (NYMEX), to develop contractson U.S. hurricane risk in 2007. The lack of trading results in the withdrawn of the NYMEX contracts,but the CME and IFEX contracts are still listed. Another attempt to securitize catastrophe riskis insurance-linked debt contracts, e.g. CAT bonds. The CAT bond is a liability hedge instrumentfor (re)insurers. The rst CAT bond is issued by the Hannover Re in 1994. The design of debt-forgiveness provision of CAT bonds allows the payment of interest or the return of principal tobe forgiven when the specied event is triggered. Catastrophic equity put (CatEPut) is anothercapital market solution to help (re)insurers to hedge their catastrophic losses. The rst CatEPutwas launched on behalf of RLI Corporation in 1996. CatEPuts give the buyer the right to sella certain amount of its shares at a predetermined price if catastrophe losses surpass a speciedtrigger. CatEPuts provide (re)insurers with additional equity capital when they need funds to facehuge catastrophe claims.

3

There is a series of studies discussed the innovation of insurance-linked securi-

ties from practical and regulatory points of views. For example, Cox and Schwebach

(1992) discussed the advantages and disadvantages from the (re)insurers point of view

to hedge catastrophic risk by the insurance derivatives. Niehaus and Mann (1992)

developed a theoretical model to price the insurance futures. Harrington, Mann,

and Niehaus (1995) point out that the insurance-linked security marked provide way

for (re)insurers to lower the cost to manage their catastrophe risk. Doherty (1997,

2000) noted that the trade-o¤ e¤ect between the moral hazard and basis risk when

(re)insurers hedge their catastrophe risk by using alternative insurance-linked secu-

rities. Doherty and Richter (2002) argued that the optimal strategy by combination

of insurance-linked security instruments and indemnity-based reinsurance to cover

the gap of basis risk. At the same time there is some studies focus on the pricing

of catastrophe-linked securities. For example, Cox and Schwebach (1992), Cummins

and Geman (1995), and Chang, Chang, and Yu (1996) valued the CAT futures and

CAT call spreads on the property claims services (PCS) index under deterministic

interest rate. Geman and Yor (1997) valued the CAT options on a jump-di¤usion loss

process. Bakshi and Madan (2002) provided a solution for pricing CAT options on a

mean-reverting Markov process. The CAT bond is the most successful instrument to

date. There is a series studies focus on developing pricing models of CAT bond. For

example, Litzenberger, Beaglehole, and Reynolds (1996) priced a zero-coupon CAT

bond and compared the CAT bond price calculated by the bootstrap approach with

that estimated values. Louberg? Kellezi, and Gilli (1999) numerically estimated the

CAT bond price under the binomial interest rate process. Lane (2000) and Lane and

Mahul (2008) applied an actuarial methodology to examine the CAT bonds prices.

Embrechts and Meister (1997) valued CAT futures in a utility maximization context.

4

Cox and Pederson (2000) provided an approach to value CAT bonds under incom-

plete market. Lee and Yu (2002) provided a theoretical model to value CAT bonds

under the stochastic interest rate of Cox, Ingersoll, and Ross (1985) while considering

the moral hazard and basis risk. Vaugirard (2003, 2004) applied the methodology for

pricing barrier options to value CAT bonds under the interest rate based on Vasicek

(1977). For the valuation of instruments for raising contingent capital, there is also

a few studies have been proposed. For example, Cox, Fairchild and Pedersen (2004)

priced CatEPuts while assuming the process of (re)insurers share price to be driven

by a geometric Brownian motion with downward jumps initiated by the occurrence of

catastrophic events. Jaimungal and Wang (2006) priced CatEPuts under stochastic

interest rates. Lo, Lee, and Yu (2013) applied the structure model of Merton (1977)

to value CatEPuts while considering the endogenous problem.

In contrast to the fruitful studies focusing on CAT bonds and other insurance-

linked securities, there is little studies pay attention on the CAT swaps. Braun (2011)

applied the contingent claim approach to value CAT swaps without considering the

e¤ect of counterparty risk. Cummins and Barrieu (2012) and Cummns (2012) noted

that nonperformance risk and basis risk would increase the transaction costs when

the (re)insurer hedge their catastrophe risk by CAT swaps. Therefore, in this study,

we intend to develop a pricing model that includes the e¤ects of counterparty risk

and basis risk on the CAT swaps valuation.

2 Model

We consider a insurance company make a multi-period CAT swap with a reinsurance

company to hedge its catastrophe risk. The insurance company take the xed leg of

the CAT swap and pay a xed percentage (spread) of notional amount (N) to the

5

reinsurance company at the beginning of each period, t1; t2; :::; tn = T . The reinsur-

ance company takes the oating leg of the CAT swap and agrees to pay the insurance

company the notional amount, N, if the underlying catastrophic losses/numbers of

catastrophes exceeds a predetermined level. Due to the CAT swap is a bilateral con-

tract in which the xed payer of CAT swap pays regular payments in exchange for a

contingent payment on a specic catastrophe events. A CAT swap contract is usually

designed to ensure the both sides of the risk to achieve parity. That is the separate

value of each leg of the CAT swap is set to be equal through the fair spread on the

xed leg. In this study, we extend the Mertons structural approach to incorporate

the e¤ect of counterparty nonperformance risk on the CAT swap valuation. Under

the structural method, the capital positions of both sides are explicitly specied as

follows.

2.1 Dynamics of xed/oating payers asset value

We consider the case in which an insurance company take a CAT swap with a rein-

surance company to transfer its catastrophe risk. We follow Merton (1977) and Cum-

mins (1988) to assume the asset values of the xed payer (Vfixed;t) and oating payer

(Vfloating;t) are governed by geometric Brownian motions. The dynamics of asset

values of both legs can be described as follows:

dVX;tVX;t

= �VXdt+ �VXdrt + �VXdWVX;t ; X = fixed ; oat ing (1)

where the subscript X = fixed for CAT swaps xed payer (insurance company), and

X = floating for CAT swaps oating payer (reinsurance company). rt is the instan-

taneous interest rate at time t. �VX is the volatility of the credit risk of X-companys

asset returns. �VX is the instantaneous interest rate elasticity of the assets of X-

company. WVX;t is the Wiener process referring to the credit risk on the X-companys

6

asset return. The term credit risk refers to all risks that are orthogonal to the inter-

est rate risk, rt. The correlation coe¢ cient between the asset returns on the swaps

xed payer and oating payer (i.e. WVfixed;t and WVoat ing;t ) is �V .The instantaneous

interest rate is set to be governed by the squared-root process of Cox, Ingersoll, and

Ross (1985). The dynamic of instantaneous interest rate can be described as follows:

drt = �(m� rt)dt+ �rprtdZt; (2)

where � is the mean-reverting force measurement; m is the long-run mean of the

interest rate; �r is the volatility parameter for the interest rate. Zt is a Wiener

process independent of WVX;t . Combining (1) and (2), the dynamics of X-compnay

can be rewritten as follows:

dVX;tVX;t

= (rt + �VX + �VX�m� �VX�rt)dt+ �VXvprtdZt + �VXdWVX ;t: (3)

The dynamic of instantaneous interest rate under the risk-neutralized pricing mea-

sure, denoted by Q, can be written as follows:

drt = ��(m� � rt)dt+ �r

prtdZ

�t ; (4)

where k�, m�, and Z� are respectively dened as

�� = �+ �r

m� =�m

�+ �r

dZ�t = dZt +�rprt

�rdt:

The term �r denotes the market price of interest rate risk and is set to be constant

under the framework of Cox, Ingersoll, and Ross (1985). Z�t is a Wiener process under

Q.

7

The dynamics of X-companys asset return under the risk-neutral probability

measure Q can be described as follows:

dVX;tVX;t

= rtdt+ �VX�rprtdZ

�t + �VIndW

�VIn;t

; (5)

where W �VX;t is a Wiener process under Q and is independent of the Z�t .

2.2 Dynamics of xed/oating payers liabilities

In this study, we consider the case in which both the xed payer and oating payer are

(re)insurance companies. In contrast to the liability dynamic of a typical company,

there usually exists jumps in the value of (re)insurers liability when catastrophic

events occur. We follow Duan and Yu (2005), Lee and Yu (2008) and Lo, Lee, and

Yu (2013) to value the insurers total contractual liabilities at time t, LX;t, as the

time-t value of the (re)insurers future claims related to the outstanding policies. The

change in value of contractual liabilities of the (re)insurer is assumed to consist of two

components. The rst one reects the fact that the (re)insurer faces large jumps in

liabilities raised from catastrophe claims. As the frameworks of Cummins (1988) and

Shimko (1992), the component of jump risk is assumed to be governed by a compound

Poisson process. The second component reects the normal variation in liabilities and

is modeled as a di¤usion process. Since the (re)insurers total contractual liabilities

is the present value of all future estimated claims, the continuous component should

reect the e¤ects of interest rate changes and day-to-day small shocks. The dynamic

of liabilities of the xed/oating payers while incorporating the e¤ects of the above

considerations can be described as follows:

dLX;t =�rt + �LX � �e

�yX+ 12�2yIn

�LX;t�dt+�LXLX;t�drt+�LXLIn;t�dWLX ;t+YPLX ;tLX;t�dPL;t;

(6)

8

where �LX denotes the risk premium for small shocks in the insurers liabilities,

which is denoted by a pure di¤usion process with the volatility of �LX . �LX is the

instantaneous interest rate elasticity of the X-companys liabilities. WLX ;t is a Weiner

process summarizing all continuous shocks that are not related to the asset risk of the

X-company. YPLX ;t is a sequence of independent and identically-distributed positive

random variables describing the percentage change in the X-companys liabilities due

to catastrophes. We assume that ln(YPLX ;t) has a normal distribution with mean

�yX and standard deviation �yX . This assumption ensures that a catastrophe always

rises the (re)insurers liabilities even though the magnitude is random. PL;t is a

Poisson process with intensity parameter �P and is independent of other variables.

The term �e�yX+12�2yX o¤sets the drift arising from the compound Poisson component

YPLX ;tLX;t�dPL;t.

2.3 Payo¤s of CAT swap

We extend Braun (2011) to consider alternative trigger events set in the CAT swaps.

Specically, triggers set by the accumulated catastrophic losses and the accumulated

number of catastrophe events are taken into account. The underlying accumulated

catastrophic losses can be determined by the xed payers own catastrophic loss pay-

ment or a specied composite index of catastrophic losses. The CAT swap is a

bilateral contract in which the xed payer pays a xed amount (a xed percentage

(') of the notional amount (N)), also known as the spread of the CAT swap) to

the oating payer to exchange for the contingent payment of notional amount by the

oating payer if the trigger is pulled. Under the case without considering the default

of oating payer, the payo¤s of oating leg of the CAT swap can be described as

follows:

9

POoat ing = N1� tigger�T ; (7)

where POfloating denotes the contingent payment paid by the oating payer. 1� trigger

2.3.1 Counterparty default risk

We extend the passage model developed by Black and Cox (1976) to determine the

time to default. The default event can be initiated by both parties of the CAT swap.

Therefore, the CAT swap will be terminated if one of the parties of CAT swap can

not fulll its obligation. Therefore, the default occurs at the rst time, � default, when

the xed payers or oating payers assets fall below their liabilities. The default time

can be described as follow:

� default = minn� fixeddefault; �

oat ingdefault

o; (9)

� fixeddefault = inf ftijVfixed;ti < Lfixed;tig ;

�oat ingdefault = inf ftijVoat ing;ti < Loat ing;tig ;

where � fixeddefault and �

oat ingdefault Vfixed;t denote the times when the xed payer and oat-

ing payer become insolvent and default on the CAT swap, respectively. Lfixed;t; Vfixed;t;

Loat ing;t and Voat ing;t are described in the previous section. Therefore, while consid-

ering the e¤ect of counterparty default risk, the rst stopping time can be described

as follows:

� � = minf� trigger; � defaultg: (10)

2.3.2 Payo¤s of CAT swap with default-risky counterparty

The payo¤s of the both legs of CAT swap under the cases with considering the e¤ect

of counterparty default depend on the abilities of both parties to fulll their total

liabilities, which includes the liabilities for ordinary operation and the obligation on

11

the CAT swap payment. While taking into account the counterpartys default risk,

POdoat ing, under the alternative designs of underlying catastrophic losses set by can

be described as follows:

POd;Lioat ing;T =

8

2.3.3 Accumulated Catastrophic Losses

The catastrophic losses faced by the xed payer, Lfixed;ti, during the period of ti�1 to

ti can be described as total liabilities minus non-catastrophic liabilities:

Lfixed;ti = LCfixed;ti�1�LCfixed;ti�1 = Lfixed;ti�Lfixed;ti�1�exp

��

NtiXj=1+Nti�1

ln(1+YPLfixed;j )

�;

(13)

since

Lfixed;ti = Lfixed;ti�1 � exp�Z ti

ti�1

rsds��12�2Lfixed + �e

�yPfixed+ 12�2yPfixed

�(ti � ti�1)

+�Lfixed

�W �Lfixed;ti �W

�Lfixed;ti�1

�+

N(T )Xj=1

ln(1 + YPLfixed ;j)

�:

2.4 CAT swap spread

The CAT swap market-to-market value will uctuate after the CAT swap has been

entered. However, the CAT swap is designed to ensure two sides of the risk achieve

parity at the initial day. That is the expected present value of the two sides are

equivalent. In the case of without considering the counterpartys default risk, the

expected present value of aggregated payments of xed leg at time 0, PV POfixed;0,

can be described as follow:

PV POfixed;0 = EQ0

"� triggerXi=1

e�R ti�10 rsdsPOfixed;ti

#=

� triggerXi=1

he�

R ti�10 rsds'N1��>ti

i:

(14)

The expected present value of the aggregated payo¤s of oating leg can be described

as follows:

PV POoat ing;0 = EQ0

"� triggerXi=1

e�R ti�10 rsdsPOoat ing;ti

#= EQ0

"� triggerXi=1

e�R ti�10 rsdsN1� tigger�T

#:

(15)

13

Therefore, the fair value of the spread, ', can be estimated bas follows:

� triggerXi=1

he�

R ti�10 rsds'N1ti�1>� trigger>ti

i= EQ0

"� triggerXi=1

e�R ti�10 rsdsN1� tigger�T

#(16)

In the case of considering the default possibilities of both parties, the fair value of

the spread, ', can be estimated bas follows:

��Xi=1

he�

R ti�10 rsds'N1ti�1>� trigger>ti

i= EQ0

"��Xi=1

e�R ti�10 rsdsN1��T

#: (17)

3 Numerical analysis

3.1 Parameter values

The pricing model has been well developed. We do not expect to have analytical so-

lution for the comprehensive model. In stead, we estimater the CAT swap spreads by

using the Monte Carlo simulation method. The simulations are run on weekly basis

with 20,000 paths. For a reference point of numerical analysis, a set of parameters is

established and summarized in Table 1. Deviations from the base values are set to

assess the comparative e¤ects of the parameters on the spreads of CAT swaps.

We consider an insurer intends to hedge his catastrophic losses by engaging in a

CAT swap contract with a reinsurer. The insurer pays a xed amount, the spread

multiplied by the notional amount of CAT swap, to exchange the contingent payment,

the notional amount of CAT swap, paid by the reinsurer as the trigger of CAT swap

is pulled. As the typical swap contract, the insurer plays the role of xed leg and the

reinsurer plays the alternative oating leg of the CAT swap. The notional amount of

CAT swap is set at 10%, 30%, and 50% of the insurers initial liabilities, respectively.

The initial insurers asset/liability ratios is set to be 1.2. The maturities of CAT swap

are set to be one year, two years, and three years, respectively. The asset volatilities

caused by the credit risk are both set to be 5% for insurer and reinsurer. The interest

14

rate parameters for the Cox, et al. (1985) model are set to be the estimates reported

in Duan and Simonato (1999). The mean-reverting force is equal to 0.2249. The

volatility of interest rate is 0.07. The market price of interest rate risk is equal to

-0.11111. The initial spot interest rate and the long-run interest rate are both equal

to 0.0613. The arrive of catastrophes assumed to be governed by a Poisson Process

with an intensity �. The values of � are set to be 0.1, 0.33, 0.5, and 1 to reect

the frequencies of catastrophe events per year, respectively. The occurrence of a

catastrophe event increases the liabilities of insurer and reinsurer in the magnitudes

of YPL;fixed and YPL;oat ing percents, respectively. Where YPL;fixed and YPL;oat ing are

both assumed to be lognormal distributed. The values of correlation coe¢ cient of

YPL;fixed and YPL;oat ing , �Y , are set to be 0, 0.2, 0.5, 0.8, and 1, respectively. The

combinations of logarithmic means and standard deviations set at the values of -

2.3075851 (-1.6294389) and 0.1 (0.2) to imply an average 10% (20%) jump in the

(re)insurers liabilities when a catastrophe occurs. The volatilities of pure liability

di¤usion process, �Lfixed and �Loat ing , are both set to be 3%.

3.2 Default-Free CAT Swap

Table 2 reports the speads of CAT swaps without considering the default possibilities

of counterparties. Panel A shows the spreads of CAT swaps while the trigger of

contingent payment of oating leg (i.e. the reinsurer) is set by a specied number

of catastrophes occurs during the life time of contract. We observe that the spread

decreases with the trigger number of CAT swap. It indicates that the higher trigger

number of catastrophes, the lower possibility of reinsurer to pay contingent payment

is. The spreads of CAT swap in which the contingent payment is triggered by 10%,

30%, and 50% of xed payers accumulated catastrophic loesses are presented in Panel

B. The left and right three columns of Panel B shows the spreads estimated while

15

the average jump of (re)insurers liability being set at 10% and 20%, respectively. We

observe that the spread increases with the maturity of CAT swap. For instance, in

the case of the catastrophe intensity (�) and jump size of catastrophe being set at

0.33 and 10%, respectively, the spread of CAT swap with trigger being set at 10%

of the insurers accumulated catastrophic losses increases from 0.04887 to 0.06916

and 0.08518 while the maturity of CAT swap increasing from one year to two years

and three years. It indicates that the longer maturity of CAT swap is, the higher

possibility of the trigger being pulled is. Comparing the left 3 columns of panel B

and the corresponding right 3 columns of Panel B, we observe that the average jump

size in catastrophic losses rises with the spread of CAT swap. For instance, in the

case of the catastrophe intensity being 0.33, the spread of 2-year CAT swap increases

from 0.00175 to 0.03132 while the average jump size in catastrophic losses increases

from 10% to 20%. We also observe that the spread decreases with the trigger level,

especially in the case of the insurer facing catastrophic losses with lower intensity.

3.3 Default-Risky CAT Swap

The spreads estimated with considering the default risk are presented in the Tables 3-

6. Tables 3 shows the spreads estimated in the case of the contingent payment trigger

being set by the numbers of catastrophes faced by the insurer and the notional amount

being set at the 10% of the insurers initial liabilities. Panel A and Panel B show

the spreads estimated in the cases of the average jump magnitudes of catastrophe are

set at 10% and 20%. We observe that the spread with considering the counterparty

default risk increases with the maturity of CAT swap and catastrophe intensity, but

decreases with the correlation between the catastrophic losses faced by the insurer and

reinsurer, �Y . Comparing the values in Table 3 with the values in the corresponding

cells of Table 2, we observe that the spreads of default-risky CAT swaps are smaller

16

than the spreads of default-free CAT swaps. In addition, in contrast to the spread

of default-free CAT swap increasing with the average jump size of catastrophe, the

spread decreases with the average jump size of catastrophe while taking into account

of the counterparty default risk. The result indicates that the counterparty default

risk is an important factor to determine the spread of CAT swap. Tables 4-5 show

the spreads estimated with trigger of contingent payment being set by the numbers

of catastrophes faced by the insurer and the notional amount being set at 30% and

50% of the insurers initial liabilities, respectively. It is similar to the result shown

in Table 3, the spread increases with the maturity of CAT swap and catastrophe

intensity, but decreases with the average jump size of catastrophe and the correlation

between the catastrophic losses faced by the insurer and reinsurer. In contrast to the

spread of default free CAT swap be irrelevant to the level of notional amount, we

observe that the spread decreases with the level of notional amount. For instance,

in the case of 2-year CAT swap with trigger set at 2 catastrophe events faced by the

insurer and the catastrophe intensity (�), the correlation between catastrophic losses

of insurer and reinsurer (�Y ), and average jump size of catastrophe being 0.33, 0.5,

and 10%, respectively, the spread decreases from .02715 to 0.02088 and 0.01593 when

the notional amount of CAT swap increases from 10% of the insurers initial liabilities

to 30% and 50%. It reects that the sudden payment of large notional amount rises

the possibility of reinsurer to fulll his obligation and therefore the value of CAT

swap.

Table 6 shows the spreads of default-risky CAT swap with the contingent pay-

ment of reinsurer being triggered by the accumulated catastrophic losses faced by the

insurer. The spreads are estimated when the case of the trigger level is equal to the

notional amount. Panels A and B show the spreads estimated while the average jump

17

size of catastrophe being 10% and 20%. It is similar to the result of CAT swap with

trigger set by the numbers of catastrophes faced by the insurer. The spread increases

with the maturity of CAT swap and the catastrophe intensity, but decreases with

the correlation between the catastrophic losses faced by the insurer and reinsurer. In

contrast to the spread of CAT swap with trigger set by the numbers of catastrophes

faced by the insurer decreases with the average jump size of catastrophe, we observe

that the spread increases with the average jump size of catastrophe. It is because

that the higher average jump size of catastrophe increases the possibility of trigger

being pulled early and its e¤ect on the value of CAT swap dominates that results

from the default of the reinsurer.

3.4 Default Risk Premium

Because the CAT swaps are bilateral OTC contracts without the settlement guaranty

provided by the typical exchange. An insurance company faces thecounterparty de-

fault risk when he hedges catastrophic risk by engaging in the CAT swap. The default

risk premiums of CAT swap can be measured by the di¤erence between the values of

default free CAT swap and default-risky CAT swap. Tables 7-9 show the default risk

premiums in the cases of the trigger being set by the numbers of catastrophe occurs

and the notional amount being set at 10%, 30%, and 50% of the insurers initial

liabilities, respectively. Panels A and B of Table 7 show the default risk premiums

calculated in the situation where the average jump size of catastrophe are set at 10%

and 20%. Comparing the values in Panel A with the values in corresponding cells

in Panel B, we observe that the default risk premiums calculated while the average

jump size of catastrophe being set at 10% are much smaller than those calculated

while the average jump size of catastrophe being set at 20%. For instance, in the case

of 2-year CAT swap with trigger set by 2 catastrophe events faced by the insurer and

18

the catastophe intensity (�) and correlation between the catastrophic losses faced by

the insurer and reinsurer (�Y ) being set at 0.33 and 0.5, the default risk premium in-

creases from 0.02662 to 0.04674 when the average jump size of catastrophe rises from

10% to 20%. The default risk premiums amount about 22% and 39% of the spreads

of CAT swap. Comparing the values in Table 7 with the values in the corresponding

cells in Tables 8 and 9, we observe that the default risk premium increases with the

notional amount of CAT swap. For instance, in the case of 2-year CAT swap with

trigger set by 2 catastrophe events faced by the insurer and the catastrophe intensity

(�) and correlation between the catastrophic losses faced by the insurer and reinsurer

(�Y ) being set at 0.33 and 0.5, the default risk premium increases from 0.02662 to

0.03289 and 0.3784 when the notional amount increases from 10% of the insurers

initial liabilities to 30% and 50%, respectively. It indicates that the high notional

amount of CAT swap decreases the possibility of reinsurer to fulll his obligation and

the insurer will pay less to engage in the contract. Table 10 shows the default risk

premiums of CAT swaps with trigger being set by the accumulated catastrophic losses

faced by the insurer. We observe that the default risk premium increases with the

maturity of CAT swap, catastrophe intensity, correlation between the catastrophic

losses faced by the insurer and reinsurer, and the average jump size of catastrophe.

4 Summary Remark

This study develops a model to value CAT swaps under an environment of stochastic

interest rates for two types of trigger mechanisms. The model allows us to measure

how counterparty default risk a¤ects the value of CAT swap. We estimate the spreads

of default-free CAT swaps with trigger being set by the number of catastrophe events

and insurers accumulated catastrophic losses, respectively. The results show that the

19

spread increases with the catastrophe occurrence intensity, maturity of CAT swap,

notional amount and average jump size of catastrophe, but decreases with the trigger

numbers of catastrophes. We then estimate the spreads of default-risky CAT swaps.

We observe that the spreads of default-risky CAT swaps are much lower than the

spreads of default free CAT swaps. It indicates that the counterparty default risk

is substantial on the CAT swap valuation. The default risk premiums are measured

by the di¤erence between the spreads of default free CAT swap and default-risky

CAT swap. The results indiacte that default risk premium increases with the corre-

lation between catastrophic losses faced by the insurer and reinsurer and the average,

the catastrophe occurrence intensity, the average jump size of catastrophe, and the

notional amount of CAT swap.

References

[1] Bakshi, G. and D. Madan, 2002, Average Rate Claims with Emphasis onCatastrophe Loss Options, Journal of Financial and Quantitative Analysis37(1), 93115.

[2] Bantwal, V. J., and H. C. Kunreuther, 2000, A CAT Bond Premium Puzzle?,Journal of Psychology and Financial Markets 1(1), 76-91.

[3] Barone-Adesi, G. and R. E. Whaley, 1987, E¢ cient Analytic Approximation ofAmerican Option Values, Journal of Finance 42(2), 301-320.

[4] Black, F., 1976, The Pricing of Commodity Contracts, Journal of FinancialEconomics 3, 167-179.

[5] Braun, A., 2011, Pricing Catastrophe Swaps: A Contingent Claims Approach,Insurance: Mathematics and Economics 49, 520-536.

[6] Chang, C. W., J. S. Chang and W. Lu, 2010, Pricing Catastrophe Optionswith Stochastic Claim Arrival Intensity in Claim Time, Journal of Banking andFinance 34(1), 24-32.

20

[7] Chang, C. W., J. S. Chang and M.-T. Yu, 1996, Pricing Catastrophe InsuranceFutures Call Spreads: A Randomized Operational Time Approach, Journal ofRisk and Insurance 63, 599-617.

[8] Cox, S. H., J. R. Fairchild and H. W. Pederson, 2004, Valuation of StructuredRisk Management Products, Insurance: Mathematics and Economics 34, 259-272.

[9] Cox, J., J. Ingersoll and S. Ross, 1985, The Term Structure of Interest Rates,Econometrica 53, 385-407.

[10] Cox, S.H., Pedersen, H.W., 2000, Catastrophe Risk Bonds, North AmericanActuarial Journal 4(4), 5682.

[11] Cox, S. H. and R. G. Schwebach, 1992, Insurance Futures and Hedging InsurancePrice Risk, Journal of Risk and Insurance 59, 628-644.

[12] Cummins, J. D., 1988, Risk-Based Premiums for Insurance Guaranty Funds,Journal of Finance 43, 593-607.

[13] Cummins, J. D., 2008 , Cat Bonds and Other Risk-Linked Securities: State ofthe Market and Recent Developments, 2008, Risk Management and InsuranceReview 11, 23-47.

[14] Cummins, J. D. and P. Barrieu, 2012, Innovations in Insurance Markets: Hy-brid and Securitized Risk-transfer Solutions, in George Dionne, ed., Handbook ofInsurance, 2ed. (Boston: Kluwer Academic Publishers).

[15] Cummins, J. D. and N. A. Doherty, 2002, Capitalization of the property-liabilityInsurance Industry: Overview, Journal of Financial Services Research 21, 5-14.

[16] Cummins, J. D. and H. Geman, 1995, Pricing Catastrophe Futures and CallSpreads: An Arbitrage Approach, Journal of Fixed Income, March, 46-57.

[17] Cummins, J. D., D. Lalonde and R. D. Phillips, 2004, The Basis Risk ofCatastrophic-Loss Index Securities, Journal of Financial Economics 71, 77-111.

[18] Cummins, J. D. and M. A. Weiss, 2009, Convergence of Insurance and FinancialMarkets: Hybrid and Securitized Risk-transfer Solutions, Journal of Risk andInsurance 76, 493-545.

[19] Doherty, N. A., 1997, Innovations in Managing Catastrophe Risk, Journal ofRisk and Insurance 64, 713-178.

21

[20] Doherty, N. A., 2000, Innovation in Corporate Risk Management: The Caseof Catastrophe Risk, in G.Dionne, ed., Handbook of Insurance (Boston, MA:Kluwer Academic Publishers).

[21] Doherty, N. A., and A. Richter, 2002, Moral Hazard, Basis Risk, and Gap Insur-ance, Journal of Risk and Insurance, 69, 9-24.

[22] Duan, J.-C. and M.-T. Yu, 2005, Fair Insurance Guaranty Premia in the Presenceof Risk-Based Capital Regulations, Stochastic Interest Rate and CatastropheRisk, Journal of Banking and Finance 29(10), 2435-2454.

[23] Finken, S. and C. Laux, 2009, Catastrophe Bonds and Reinsurance: The Com-petitive E¤ect of Information-Intensity Triggers, Journal of Risk and Insurance76(3), 579-605.

[24] Froot, K. A. (Ed.), 1999, The Financing of Catastrophe Risk, University ofChicago Press, Chicago.

[25] Froot, K. A., 2001, The Market for Catastrophe Risk: A Clinical Examination,Journal of Financial Economics 60(2), 529-571.

[26] Geman, H. and M. Yor, 1997, Stochastic Time Changes in Catastrophe OptionPricing, Insurance: Mathematics and Economics 21, 183-218.

[27] Harrington, S. E., S. V. Mann, and G. Niehaus, 1995, Insurer Capital Struc-ture Decisions and the Viability of Insurance Derivatives, Journal of Risk andInsurance 62, 483-508.

[28] Harrington, S. E. and G. Niehaus, 2003, Capital, Corporate Income Taxes, andCatastrophe Insurance, Journal of Financial Intermediation 12(4), 365-389.

[29] Jaimungal, S. and T. Wang, 2006, Catastrophe Options with Stochastic InterestRates and Compound Poisson Losses, Insurance: Mathematics and Economics38, 469-483.

[30] Jarrow, R. A., 2010, A Simple Robust Model for CAT Bond Valuation, FinanceResearch Letters 7, 72-79.

[31] Jarrow, R. and A. Rudd, 1982, Approximate Option Valuation for ArbitraryStochastic Processes, Journal of Financial Economics 10, 347-369.

[32] Lane, M.N., 2000, Pricing risk transfer transactions, Astin Bulletin 30 (2), 259293.

22

[33] Lane, M., and O. Mahul, 2008, Catastrophe Risk Pricing: An Empirical Analysis,World Bank Policy Research Working Paper 4765.

[34] Lee, J.-P. and M.-T. Yu, 2002, Pricing Default-Risky CAT Bonds With MoralHazard and Basis Risk, Journal of Risk and Insurance 69(1), 25-44.

[35] Lee, J.-P. and M.-T. Yu, 2007, Valuation of Catastrophe Reinsurance with CATBonds, Insurance: Mathematics and Economics 41, 264-278.

[36] Litzenberger, R. H., D. R. Beaglehole and C. E. Reynolds, 1996, AssessingCatastrophe Reinsurance-Linked Securities as a New Asset Class, Journal ofPortfolio Management, Special Issue, 76-86.

[37] Lo, C.-L., J.-P. Lee, and M.-T. Yu, 2013, Valuation of insurerscontingent capitalwith counterparty risk and price endogeneity, Journal of Banking and Finance37, 5025-5035.

[38] Louberge H., E. Kellezi and M. Gilli, 1999, Using Catastrophe-Linked Securitiesto Diversify Insurance Risk: A Financial Analysis of CAT Bonds, Journal ofInsurance Issues 22, 125-146.

[39] Merton, R., 1977, An Analytic Derivation of the Cost of Deposit Insurance andLoan Guarantees, Journal of Banking and Finance 1, 3-11.

[40] Niehaus, G., and S. V. Mann, 1992, The Trading of Underwriting Risk: AnAnalysis of Insurance Futures Contracts and Reinsurance, Journal of Risk andInsurance 59, 601-627.

[41] Vasicek, O., 1977, An Equilibrium Characterization of the Term Structure, Jour-nal of Financial Economics 5(2), 177188.

[42] Vaugirard, V.E., 2003, Pricing Catastrophe Bonds by an Arbitrage Approach,Quarterly Review of Economics and Finance 43(1), 119132.

[43] Vaugirard, V.E., 2004, A Canonical First Passage Time Model to Pricing Nature-linked Bonds, Economics Bulletin 7(2), 17.

23

Table 1Parameters Denitions and Base Values

Interest Rate Parameters Valuesr Initial instantaneous interest rate 6:13%� Magnitude of mean reverting force 0:2249m Long-run mean of interest rate 6:13%v Volatility of interest rate 0:07�r Market price of interest rate risk �0:1111

Asset ParametersVfix Fixed payers initial asset value Vfixed;0=Loat ing;0 = 1:2Voat ing Floating payers initial asset value Voat ing;0=Vfixed;0 = 5�Vfixed ; �Vfloat ing Volatility of credit risk 5%�Vfixed ; �float ing Interest rate elasticity of xed/oating payers asset 0;�7WVfixed ;WVfloat ing Wiener process for credit shock

Liability ParametersLfixed Initial value of xed payers liabilities 1Loat ing Initial value of oating payers liabilities Vfixed;0=Lfixed;0 = 1:2�Lfixed ; �Lfloat ing Volatility of pure liability risk 3%�Lfixed ; �Lfloat ing Interest rate elasticity of xed/oating payers liabilities 0;�3WLfixed ;WLfloat ing Wiener process for pure liability risk

Catastrophe Loss Parameters� Catastrophe intensity 0:1; 0:33; 0:5; and 1N(t) The number process of catastrophe losses�yIn ; �yRe Mean of the logarithm of CAT losses for the xed/oating payer �2:3075851; �1:6294389�yIn ; �yRe Standard deviation of the logarithm of CAT losses for the xed/oating payer 10%; 20%�Y Correlation coe¢ cient of the logarithms of CAT losses of the xed/oating payer 0:2; 0:5; 0:8; and 1

Other ParametersT Time to maturity 1; 2; and 3n Trigger numbers 1; 2; and 3L Trigger levels 10%; 20%; 30% of the Vfixed;0

Table2: Spread of Default-Free Cat Swaps(VIn=DIn = 1:2; VRe=DRe = 1:2; VRe=Vi = 5)

Panel A Panel BCatastrophe Number as Trigger Catastrophe Loss as Trigger1 2 3 10% 30% 50% 10% 30% 50%

e�yIni+12�

2yIn = 10% e�yIn+

12�

2yIn = 20%

T=1� =0.1 0.09020 0.00404 0.00000 0.00978 0.00000 0.00000 0.08996 0.00154 0.000050.33 0.26962 0.03518 0.00253 0.04887 0.00024 0.00000 0.26689 0.01634 0.000190.5 0.37647 0.07502 0.00715 0.08874 0.00043 0.00000 0.37710 0.03834 0.001101 0.60895 0.23812 0.04313 0.24576 0.00680 0.00005 0.60951 0.13173 0.00728T=2� =0.1 0.08923 0.00639 0.00010 0.01478 0.00005 0.00000 0.08805 0.00288 0.000120.33 0.26765 0.05377 0.00606 0.06916 0.00175 0.00000 0.26445 0.03132 0.001560.5 0.37412 0.10873 0.01998 0.12097 0.00412 0.00010 0.37640 0.00603 0.005661 0.30780 0.29937 0.10674 0.30836 0.03206 0.00164 0.60175 0.19249 0.03437T=3� =0.1 0.08910 0.00822 0.00031 0.02103 0.00022 0.00000 0.08818 0.00450 0.000240.33 0.26590 0.06668 0.01177 0.08518 0.00459 0.00014 0.26368 0.04322 0.004070.5 0.37610 0.13155 0.03483 0.14451 0.01060 0.00053 0.37598 0.08655 0.012930.1 0.60746 0.32885 0.15212 0.33587 0.06550 0.00826 0.59695 0.21927 0.06541All estimates are computed using 20,000 simulation runs.

Table 3: Prices of Default-Risky Cat SwapsNotional amount is set at the 10% of the xed payers initial liabilities

(Catastrophe Number as Trigger; VIn=DIn = 1:2; VRe=DRe = 1:2; VRe=VIn = 5)

Panel A: e�yIn+12�

2yIn = 10%

No. 1 2 3�Y

0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1T=1� = 0:1 0.07365 0.07351 0.07356 0.07346 0.00115 0.00116 0.00125 0.00125 0.00000 0.00000 0.00000 0.000000.33 0.23451 0.23394 0.23370 0.23351 0.01527 0.01560 0.01546 0.01546 0.00033 0.00028 0.00033 0.000330.5 0.33439 0.33492 0.33535 0.33558 0.03643 0.03677 0.03659 0.03644 0.000176 0.00171 0.00157 0.001571 0.57921 0.57949 0.58012 0.57957 0.14021 0.13988 0.13907 0.13883 0.01271 0.01257 0.01276 0.01300T=2� = 0:1 0.06942 0.06944 0.06931 0.06925 0.00212 0.00212 0.00220 0.00212 0.00003 0.00003 0.00003 0.000050.33 0.22804 0.22764 0.22743 0.22720 0.02707 0.02715 0.02723 0.02728 0.00174 0.00169 0.00169 0.001660.5 0.32974 0.32841 0.32886 0.32893 0.06580 0.06598 0.06579 0.06566 0.00616 0.00616 0.00608 0.006081 0.57851 0.57855 0.57859 0.57863 0.20997 0.21004 0.20971 0.20953 0.05281 0.05289 0.05322 0.05316T=3� = 0:1 0.06958 0.06958 0.06944 0.06944 0.00325 0.00326 0.00329 0.00321 0.00009 0.00009 0.00009 0.000130.33 0.22857 0.22822 0.22805 0.22774 0.03719 0.03729 0.03742 0.03744 0.00371 0.00366 0.00366 0.003640.5 0.32896 0.32947 0.32974 0.32967 0.08408 0.08426 0.08445 0.08412 0.01416 0.01424 0.01408 0.014121 0.57879 0.57895 0.57929 0.57899 0.24588 0.24593 0.24570 0.24560 0.09197 0.09197 0.09230 0.09237

Panel B: e�yIn+12�

2yIn = 20%

T=1� = 0:1 0.03967 0.03947 0.03922 0.03983 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000000.33 0.14261 0.14255 0.14363 0.14392 0.00133 0.00147 0.00129 0.00124 0.00000 0.00000 0.00000 0.000000.5 0.22576 0.22517 0.00539 0.00591 0.00443 0.00462 0.00490 0.00495 0.00000 0.00000 0.00000 0.000001 0.44134 0.44116 0.44079 0.44259 0.04039 0.04126 0.04336 0.04504 0.00024 0.00043 0.00062 0.00066T=2� = 0:1 0.04272 0.04254 0.04247 0.04260 0.00008 0.00010 0.00010 0.00008 0.00000 0.00000 0.00000 0.000000.33 0.16186 0.16183 0.16192 0.16204 0.00674 0.00703 0.00703 0.00695 0.00005 0.00005 0.00013 0.000130.5 0.25398 0.25406 0.25436 0.25458 0.02199 0.02254 0.02301 0.02317 0.00046 0.00049 0.00057 0.000681 0.48092 0.48088 0.48047 0.48173 0.13078 0.13184 0.13418 0.13466 0.01929 0.01969 0.02006 0.02043T=3� = 0:1 0.04684 0.04671 0.04680 0.04672 0.00040 0.00044 0.00044 0.00044 0.00000 0.00000 0.00000 0.000000.33 0.17352 0.17334 0.17341 0.17354 0.01431 0.01452 0.01468 0.01468 0.00046 0.00048 0.00052 0.000540.5 0.26909 0.26914 0.26942 0.26998 0.04266 0.04329 0.04379 0.04402 0.00306 0.00324 0.00347 0.003631 0.49136 0.49133 0.49098 0.49211 0.18200 0.18294 0.18508 0.18537 0.05932 0.05979 0.06034 0.06107All estimates are computed using 20,000 simulation runs.

Table 4: Prices of Default-Risky CatEPutsNotional amount is set at the 30% of the xed payers initial liabilities



2yIn = 10%

No. 1 2 3�Y

0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1T=1� = 0:1 0.05970 0.05943 0.05919 0.05933 0.00083 0.00083 0.00078 0.00078 0.00000 0.00000 0.00000 0.000000.33 0.19336 0.19295 0.19344 0.19303 0.01017 0.01008 0.01008 0.00994 0.00010 0.00010 0.00010 0.000100.5 0.28275 0.28293 0.28321 0.28281 0.02423 0.02423 0.02414 0.02397 0.00097 0.00102 0.00107 0.001111 0.50802 0.50811 0.50802 0.50744 0.10041 0.10073 0.10001 0.10049 0.00822 0.00822 0.00854 0.00882T=2� = 0:1 0.05773 0.05767 0.05766 0.05764 0.00166 0.00169 0.00163 0.00159 0.00003 0.00003 0.00003 0.000030.33 0.19349 0.19328 0.19363 0.19318 0.02093 0.02088 0.02081 0.02073 0.00129 0.00123 0.00123 0.001180.5 0.28289 0.28303 0.28338 0.28326 0.05207 0.05212 0.05205 0.05195 0.00478 0.00476 0.00489 0.005021 0.51752 0.51733 0.51736 0.51715 0.17277 0.17322 0.17297 0.17297 0.04035 0.04311 0.04344 0.04360T=3� = 0:1 0.05919 0.05906 0.05905 0.05901 0.00325 0.00326 0.00329 0.00321 0.00009 0.00009 0.00009 0.000090.33 0.19699 0.19678 0.19006 0.19657 0.03088 0.03087 0.03084 0.03079 0.00304 0.00298 0.00300 0.002960.5 0.28740 0.28751 0.28778 0.28787 0.07044 0.07045 0.07038 0.07050 0.01217 0.01215 0.01225 0.012301 0.52125 0.52103 0.52106 0.52082 0.21108 0.21142 0.21129 0021120 0.08104 0.08095 0.08117 0.08114


2yIn = 20%


Table 5: Prices of Default-Risky Cat SwapsNotional amount is set at the 50% of the xed payers initial liabilities



2yIn = 10%

No. 1 2 3�Y

0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1T=1� = 0:1 0.03976 0.03923 0.03967 0.03984 0.00060 0.00052 0.00051 0.00051 0.00000 0.00000 0.00000 0.000000.33 0.13110 0.13126 0.13042 0.13046 0.00645 0.00650 0.00659 0.00641 0.00010 0.00010 0.00049 0.000100.5 0.19661 0.19727 0.19688 0.19662 0.01578 0.01566 0.01566 0.01567 0.00044 0.00040 0.00049 0.000531 0.36446 0.36520 0.36603 0.36679 0.06874 0.06896 0.06944 0.07033 0.00519 0.00515 0.00516 0.00512T=2� = 0:1 0.04247 0.04207 0.04251 0.04252 0.00127 0.00124 0.00124 0.00122 0.00003 0.00003 0.00003 0.000030.33 0.14364 0.14376 0.14334 0.14359 0.01596 0.01593 0.01608 0.01598 0.00096 0.000947 0.00096 0.000940.5 0.21431 0.21447 0.21413 0.21420 0.04050 0.04051 0.04036 0.04029 0.00354 0.00354 0.00372 0.003751 0.39960 0.40012 0.40050 0.40094 0.13903 0.13888 0.13936 0.13941 0.03440 0.03423 0.03434 0.03428T=3� = 0:1 0.04548 0.04548 0.04560 0.04559 0.00272 0.00276 0.00274 0.00270 0.00008 0.00008 0.00008 0.000080.33 0.15217 0.15217 0.15178 0.15205 0.02528 0.02535 0.02528 0.02530 0.00255 0.00250 0.00252 0.002480.5 0.22538 0.22559 0.22525 0.22543 0.05850 0.05847 0.05841 0.05845 0.01006 0.01008 0.01032 0.010371 0.41026 0.41076 0.41104 0.41153 0.17808 0.17806 0.17844 0.17869 0.06987 0.06976 0.06994 0.06977


2yIn = 20%


Table 6: Spreads of Default-Risky Cat Swaps ,V_Re=V_In = 5(Catastrophe Loss as Trigger ; VIn=DIn = 1:2; VRe=DRe = 1:2; VRe=VIn = 5)


2yIn = 10%

notional amount set by a portion of the xed payers initial liabilities10% 30% 50%

�Y0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1

T=1� = 0:1 0.00609 0.00590 0.00585 0.00575 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000000.33 0.02718 0.02704 0.02651 0.02632 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000000.5 0.05014 0.05028 0.04975 0.04927 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000001 0.15804 0.15885 0.15842 0.15737 0.00058 0.00058 0.00063 0.00058 0.00000 0.00000 0.00000 0.00000T=2� = 0:1 0.00922 0.00914 0.00911 0.00901 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000000.33 0.04070 0.04074 0.04062 0.04048 0.00035 0.00033 0.00030 0.00030 0.00000 0.00000 0.00000 0.000000.5 0.07475 0.07460 0.07415 0.07376 0.00072 0.00073 0.00068 0.00068 0.00000 0.00000 0.00000 0.000001 0.22236 0.22310 0.22284 0.22202 0.00847 0.00849 0.00847 0.00852 0.00008 0.00010 0.00011 0.00011T=3� = 0:1 0.01292 0.01284 0.01273 0.01264 0.00003 0.00003 0.00003 0.00003 0.00000 0.00000 0.00000 0.000000.33 0.05263 0.05277 0.05259 0.05252 0.00104 0.00103 0.00099 0.00107 0.00000 0.00000 0.00000 0.000000.5 0.09518 0.09505 0.09470 0.09437 0.00288 0.00291 0.00284 0.00284 0.00000 0.00000 0.00000 0.000001 0.25417 0.25489 0.25179 0.25421 0.02629 0.02635 0.02644 0.02655 0.00126 0.00121 0.00121 0.00125


2yIn = 20%


Table 7: Credit Premiums for Default-Risky Cat Swaps ,V_Re=V_In = 5notional amount set at 10% of xed payers initial liabilities

(Catastrophe No. as Trigger ; VIn=DIn = 1:2; VRe=DRe = 1:2; VRe=VIn = 5)


2yIn = 10%

1 2 3�Y

0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1T=1� = 0:1 0.01655 0.01669 0.01664 0.01674 0.00289 0.00288 0.00279 0.00279 0.00000 0.00000 0.00000 0.000000.33 0.03511 0.03568 0.03592 0.03611 0.01991 0.01958 0.01972 0.01972 0.00220 0.00225 0.00220 0.002200.5 0.04208 0.04155 0.04112 0.04089 0.03859 0.03825 0.03843 0.03858 0.00078 0.00083 0.00097 0.000971 0.02974 0.02946 0.02883 0.02938 0.09791 0.09824 0.09905 0.09929 0.03042 0.03056 0.03037 0.03013T=2� = 0:1 0.01981 0.01979 0.01992 0.01998 0.00427 0.00427 0.00419 0.00427 0.00007 0.00007 0.00007 0.000050.33 0.03961 0.04001 0.04022 0.04045 0.02670 0.02662 0.02654 0.02649 0.00432 0.00437 0.00437 0.004400.5 0.04438 0.04571 0.04526 0.04519 0.04293 0.04275 0.04294 0.04307 0.01382 0.01382 0.01390 0.013901 0.02929 0.02925 0.02921 0.02917 0.08940 0.08933 0.08966 0.08984 0.05393 0.05385 0.05352 0.05358T=3� = 0:1 0.01952 0.01952 0.01966 0.01966 0.00497 0.00496 0.00493 0.00501 0.00022 0.00022 0.00022 0.000180.33 0.03733 0.03768 0.03785 0.03816 0.02949 0.02939 0.02926 0.02924 0.00806 0.00811 0.00811 0.008130.5 0.04714 0.04663 0.04636 0.04643 0.04747 0.04729 0.04710 0.04743 0.02067 0.02059 0.02075 0.020711 0.02867 0.02851 0.02817 0.02847 0.08297 0.08292 0.08315 0.08325 0.06015 0.06015 0.05982 0.05975


2yIn = 20%





2yIn = 10%

1 2 3�Y

0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1T=1� = 0:1 0.03050 0.03077 0.03101 0.03087 0.00321 0.00321 0.00326 0.00326 0.00000 0.00000 0.00000 0.000000.33 .07626 0.07667 0.07618 0.07659 0.02501 0.02510 0.02510 0.02524 0.00243 0.00243 0.00243 0.002430.5 0.09372 0.09354 0.09326 0.09366 0.05079 0.05079 0.05088 0.05105 0.00157 0.00152 0.00147 0.001431 0.10093 0.10084 0.10093 0.10151 0.13771 0.13739 0.13811 0.13763 0.03491 0.03491 0.03459 0.03431T=2� = 0:1 0.03150 0.03156 0.03157 0.03159 0.00473 0.00470 0.00476 0.00480 0.00007 0.00007 0.00007 0.000070.33 0.07416 0.07437 0.07402 0.07447 0.03284 0.03289 0.03296 0.03304 0.00477 0.00483 0.00483 0.004880.5 0.09123 0.09109 0.09074 0.09086 0.05666 0.05661 0.05668 0.05678 0.01520 0.01522 0.01509 0.014961 0.09028 0.09047 0.09044 0.09065 0.12660 0.12615 0.12640 0.12640 0.06639 0.06363 0.06330 0.06314T=3� = 0:1 0.02991 0.03004 0.03005 0.03009 0.00497 0.00496 0.00493 0.00501 0.00022 0.00022 0.00022 0.000220.33 0.06891 0.06912 0.07584 0.06933 0.03580 0.03581 0.03584 0.03589 0.00873 0.00879 0.00877 0.008810.5 0.08870 0.08859 0.08832 0.08823 0.06111 0.06110 0.06117 0.06105 0.02266 0.02268 0.02258 0.022531 0.08621 0.08643 0.08640 0.08664 0.11777 0.11743 0.11756 0.11765 0.07108 0.07117 0.07095 0.07098


2yIn = 20%





2yIn = 10%

1 2 3�Y

0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1T=1� = 0:1 0.05044 0.05097 0.05053 0.05036 0.00344 0.00352 0.00353 0.00353 0.00000 0.00000 0.00000 0.000000.33 0.13852 0.13836 0.13920 0.13916 0.02873 0.02868 0.02859 0.02877 0.00243 0.00243 0.00204 0.002430.5 0.17986 0.17920 0.17959 0.17985 0.05924 0.05936 0.05936 0.05935 0.00210 0.00214 0.00205 0.002011 0.24449 0.24375 0.24292 0.24216 0.16938 0.16916 0.16868 0.16779 0.03794 0.03798 0.03797 0.03801T=2� = 0:1 0.04676 0.04716 0.04672 0.04671 0.00512 0.00515 0.00515 0.00517 0.00007 0.00007 0.00007 0.000070.33 0.12401 0.12389 0.12431 0.12406 0.03781 0.03784 0.03769 0.03779 0.00510 0.00512 0.00510 0.005120.5 0.15981 0.15965 0.15999 0.15992 0.06823 0.06822 0.06837 0.06844 0.01644 0.01644 0.01626 0.016231 0.20820 0.20768 0.20730 0.20686 0.16034 0.16049 0.16001 0.15996 0.07234 0.07251 0.07240 0.0724T=3� = 0:1 0.04362 0.04362 0.04350 0.04351 0.00550 0.00546 0.00548 0.00552 0.00023 0.00023 0.00023 0.000230.33 0.11373 0.11373 0.11412 0.11385 0.04140 0.04133 0.04140 0.04138 0.00922 0.00927 0.00925 0.009290.5 0.15072 0.15051 0.15085 0.15067 0.07305 0.07308 0.07314 0.07310 0.02477 0.02475 0.02451 0.024461 0.19720 0.19670 0.19642 0.19593 0.15077 0.15079 0.15041 0.15016 0.08225 0.08236 0.08218 0.08235


2yIn = 20%


Table 10: Credit Premiums for Default-Risky Cat Swaps ,V_Re=V_In = 5(Catastrophe Loss as Trigger ; VIn=DIn = 1:2; VRe=DRe = 1:2; VRe=VIn = 5)


2yIn = 10%

notional amount set at the portion of the xed payers initial liabilities10% 30% 50%

�Y0.2 0.5 0.8 1 0.2 0.5 0.8 1 0.2 0.5 0.8 1

T=1� = 0:1 0.00369 0.00388 0.00393 0.00403 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000000.33 0.02169 0.02183 0.02236 0.02255 0.00024 0.00024 0.00024 0.00024 0.00000 0.00000 0.00000 0.000000.5 0.03860 0.03846 0.03899 0.03947 0.00043 0.00043 0.00043 0.00043 0.00000 0.00000 0.00000 0.000001 0.08772 0.08691 0.08734 0.08839 0.00622 0.00622 0.00617 0.00622 0.00005 0.00005 0.00005 0.00005T=2� = 0:1 0.00556 0.00564 0.00567 0.00577 0.00005 0.00005 0.00005 0.00005 0.00000 0.00000 0.00000 0.000000.33 0.02846 0.02842 0.02854 0.02868 0.00140 0.00175 0.00145 0.00145 0.00000 0.00000 0.00000 0.000000.5 0.04622 0.04637 0.04682 0.04721 0.00340 0.00412 0.00344 0.00344 0.00010 0.00010 0.00010 0.000101 0.08600 0.08526 0.08552 0.08634 0.02359 0.03148 0.02359 0.02354 0.00156 0.00154 0.00153 0.00153T=3� = 0:1 0.00811 0.00819 0.00830 0.00839 0.00019 0.00019 0.00019 0.00019 0.00000 0.00000 0.00000 0.000000.33 0.03255 0.03241 0.03259 0.03266 0.00355 0.00356 0.00360 0.00352 0.00014 0.00014 0.00014 0.000140.5 0.04933 0.04946 0.04981 0.05014 0.00772 0.00769 0.00776 0.00776 0.00053 0.00053 0.00053 0.000531 0.08170 0.08098 0.08408 0.08166 0.03921 0.03915 0.03906 0.03895 0.00700 0.00705 0.00705 0.00701


2yIn = 20%


Catastrophe swap valuation with counterparty default...

Documents

Transcript of Catastrophe swap valuation with counterparty default...