論文題目:

Catastrophe Risk Management and Credit

Enhancement by Using Contingent Capital

報名編號:B0039

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Abstract

Catastrophe risk comprises exposure to losses from man-made and natural disasters, and recently

such disasters occur more frequently and with greater severity, which was faced with insufficient

capital and close down by insurer. Catastrophe equity puts (“CatEPut”) provide an additional pipeline

to obtain funds, and insurer can issue equity securities at the agreed price to obtain sufficient funds to

cover its catastrophe losses when catastrophe events occurred. This paper develops a structural

framework to value CatEPuts with counterparty risk, and utilize structure model to estimate default

probability and credit improvement ratio of insurer with CatEPuts. In empirical analysis, we use

hurricane losses and number available from the U.S. database on spatial hazard events and losses to

estimate the parameters of severity to find the goodness-of-fit distribution. In numerical analysis, we

first price the CatEPuts premium with/without counterparty risk, and investigate the impacts of the

counterparty risk, loss parameters on the CatEPuts premiums. Second, we investigate the level of

credit enhancement of insurer when using CatEPuts. Based on numerical analysis, we find that scale

parameter has the greatest influence on the CatEPuts premium, and we find that CatEPuts premiums

and counterparty risk have the higher impact on default probability and improvement ratio of insurer.

Keywords: Contingent capital, catastrophe equity put, counterparty risk, default probability

1. Introduction

Since the worldwide financial crisis occurred in 2008, capital markets have suffered unprecedented

volatility. Some highly-rated financial institutions in the United States and Europe, deemed “too big

to fail,” either completely failed or was rescued by their governments. The consensus among

regulators was that banks were too highly leveraged and that they should be required to hold more

capital in the future. The financial crisis also prompted regulators to reexamine how much capital is

enough for the insurance industry.

Catastrophe risk comprises exposure to losses from man-made and natural disasters, such as

earthquakes, hurricanes, and floods, which have a low probability of occurrence. In recent years,

such disasters occur more frequently and with greater severity, cause insurance and reinsurance

companies subject to substantial financial losses. A.M. Best Company’s Best Wire News reported

recently that from 2007 through 2010 four secondary peril flood events in Australia caused insured

losses exceeding A$1 billion each. A report released at the World Economic Forum in February

2011 by Swiss Reinsurance Co. Ltd. revealed that the financial consequences of the 2010

earthquake in Haiti produced losses that were 114% of the island nation’s gross domestic product.

In the last two years, some of the world’s largest banks started issuing capital instruments of a more

hybrid nature. These newer vehicles are designed to provide financial institutions with ready access

to capital upon the occurrence of certain triggering events. This movement toward such instruments,

commonly known as “contingent capital,” signifies a noticeable shift in market preference to a form

of capital readily available before a crisis or disaster strikes. In December 2010, SCOR launched a

three-year € 150m equity line facility, which is triggered when SCOR has experienced total

aggregated losses from natural catastrophes above certain thresholds occurring over a three-year

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period; To satisfy an estimated 50% of Suisse high-trigger capital requirements, Credit Suisse

issued approximately CHF 6 billion in capital notes in February 2011 to Middle Eastern

shareholders to be paid up no earlier than October 2013 for cash or in exchange for capital notes

issued in 2008.

Contingent capital instruments are generally issued as debt that is convertible into common or

preferred equity. Other forms of contingent capital instruments can include contingent debt facilities,

contingent surplus notes, CatEPut or standby loans. Contingent capital loans are typically backed by

assets and may be issued as fixed or floating rate loans. In this paper, we focus on CatEPut, the first

CatEPut was issued on behalf of RLI Corporation in October 1996. Other records of the issuing

CatEPut, please refers to following table.

Table 1 The issue of CatEPuts

Date Insured or Cedent Issue Size Date Insured or Cedent Issue Size

Oct 1996 RLI $ 50 million Jan 1999 Horace Mann $ 100 million

Mar 1997 Horace Mann $ 100 million May 1999 Intrepid Re $ 100 million

Jul 1997 LaSalle Re $ 100 million Jan 2001 Trenwick Group $ 55 million

This paper contributes to use CatEPut to discuss the catastrophe risk management and credit

enhancement of the insurer. We first find that the Weibull (Gamma) distribution is the best (second)

goodness-of-fit for the U.S. database on spatial hazard events and losses. Second, we examine the

effect of loss parameters, occurrence intensities on CatEPut value when with/without considering

counterparty risk. Furthermore, because that CatEPut has provided the insurer a useful channel to

raise additional capital to hedge against CAT losses and thus maintain even enhance the credit

quality of the insurer. Based on Weibull and Gamma distribution, we further investigate the level of

credit enhancement of the insurers when they use CatEPut. From the numerical results, we find that

for most possible scenarios the counterparty risk premium can be substantial in the presence of CAT

risk and should not be ignored in the valuation of the reinsurance policy. The results also show that

counterparty risk reduce the CatEPut premium of the reinsurance policy and raise its counterparty

risk premium. And, we find that scale parameter have higher impact on CatEPut premium. Finally,

taking counterparty risk into account when computing probability of default of the insurer, we find

that default probability of the insurer decrease and improvement ratio increase significantly under

Weibull distribution, whereas no significant changes in Gamma distribution.

The remainder of this paper is organized as follows. Section 2 shows the literatures reviews and

introduces the CatEPut and credit default models. Section 3 provides a model to value CatEPut

under stochastic interest rates with/without considering the credit risk and the probability of default

of insurer are also modeled. Section 4 provides the numerical analysis and discusses the results.

Section 5 summarizes the principal conclusions of the paper.

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2. Literature Reviews

2.1. Catastrophe equity put option (CatEPut)

CatEPut is an agreement entered into before any natural catastrophe losses occur, enabling the

insurance company to raise cash by selling stock at prearranged terms following natural catastrophe

losses that exceeds a certain threshold. The owner (e.g. an insurance company) and the investor or

stakeholder (a reinsurance company, in usual), whereby the owner pays a commitment fee (i.e. a

premium) and purchases CatEPut to the investor, much like a regular put option; however, the right

can be exercised by two conditions, one is the stock price is at or below a certain level, the other is

the catastrophe losses exceed the trigger. The payoff of CatEPut depends on the stock price when

the losses exceed . If the option is exercised, then the payoff of CatEPut at maturity time T and it

is written mathematically as:

,

where is the specified level of losses, X denotes the aggregate losses of the insured over the time

period [0,T) and denotes the indicator function for the event . S denotes the share

value, while K represents the strike price at which the issuer is obligated to purchase unit shares in

the event that losses exceed x.

If an insurer suffers a loss of capital due to a catastrophe, then its stock price is likely to fall,

lowering the amount it would receive for newly issued stock. CatEPut gives the insurer the right to

sell a certain amount of its stock to investors at a predetermined price if catastrophe losses surpass a

specified trigger. Thus, the CatEPut can provide insurers with additional equity capital precisely

when they need funds to cover catastrophe losses. A major advantage of CatEPut is that they make

equity funds available at a predetermined price when the insurer needs them the most. However, the

insurer that uses CatEPut faces a credit risk - the risk that the seller of the CatEPut will not have

enough cash available to purchase the insurer’s stock at the predetermined price. For the investors

of CatEPut they also face the risk of owning shares of an insurer that is no longer viable.

2.2. Pricing CatEPut

Cox et al. (2004) are the first to introduce such a model for pricing catastrophe linked financial

options. They use a pure Poisson process to model the aggregate catastrophe losses of an insurance

company and the underlying asset price process is driven by a geometric Brownian motion with

additional downward jumps of a pre-specified size in the event of a catastrophe occurring. They

assume that only a catastrophe event influences the underlying asset price whereas the size of the

catastrophe itself is unrelated. In their model, the pricing formula of CatEPuts is assumed as a

standard European put option and with two important assumptions. One is the constant arrival rates

of catastrophe events; the other is the constant impact on the market price of the insurance

company's stock in catastrophe events. Since CatEPuts can be a long-term contract due to their

maturity date, interest rate risk plays a significant role in pricing CatEPuts. Jaimungal and Wang

(2006) provide a generalized pricing model for evaluating CatEPuts in a stochastic interest rate

environment, assuming that losses follow a compound Poisson process and that the share value

price is affected by the total loss level, instead of the total number of losses, but maintain the

assumption of the constant arrival rate of a catastrophe. Chang and Hung (2009) analyze the

valuation of CatEPuts in the case of constant and stochastic interest rates when the underlying asset

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price is modeled as a Lévy process with finite activity, and using the jump-diffusion model with

negative exponentially distributed jumps. Numerical examples show that diffusion volatility and

jump frequency have a positive impact on CatEPut prices, and stochastic interest rates play a

significant role in determining prices of CatEPuts.

Cox et al. (2004), and Jaimungal and Wang (2006) assume that the catastrophe option is of

European style. In practice, however, the CatEPut is of American style and can be exercised

anytime once the aggregate losses exceed the trigger. Lin and Wang (2009) further explore the use

of the expected discounted penalty function of Gerber and Shiu (1998) and mathematical tools

developed for the function to evaluate perpetual American catastrophe equity put options.

Cox et al. (2004), and Jaimungal and Wang (2006) denote a constant percentage to represent the

drop in the stock price per unit of the loss, and this implies that each specific loss has the same

effect on the stock price. However, in practice, different specific losses should have different

impacts on the drop in the underlying asset price. Lin, Chang, and Powers (2009) propose a doubly

stochastic Poisson process to model the arrival process for catastrophe events and the pricing

formulas of contingent capital are derived by the Merton measure. Based on the data from PCS loss

index and the annual number of natural catastrophe events during 1950 to 2004, numerical example

shows that the CatEPut price under the doubly stochastic Poisson process is larger than that under

the pure Poisson process as the instantaneous growth rate of catastrophe intensity rises. Furthermore,

with a higher arrival rate, mean of the loss, and standard deviation of the loss, there is a higher

CatEPut price. Wu and Chung (2010) incorporate these catastrophe characteristics and counterparty

risk into the valuation of catastrophe products to investigate the mean-reverting arrival rate with

those of the constant arrival rate and the two-state regime-switching arrival rate with two drifting

Brownian motions. Empirical results show that a mean-reverting arrival rate captures the

occurrence of catastrophe events better than the constant arrival rate and the regime-switching

arrival rate. In addition, under a consistent pricing framework, the American-type CatEPut is more

appropriate than the European-type one, the prices of the two options can be computed by the FST

method and the non-negligible price difference means the premium of timely funds. Lin, Chang,

and Yu (2011) propose Markov Modulated Poisson process (MMPP) where the underlying state is

governed by a homogenous Markov chain to model the arrival process for catastrophe events.

Numerical results show that when the transition rate increases or arrival rate of catastrophe events

decreases, then the decreasing of jump rate makes the CatEPut price decreasing, and higher the

percentage drop in share value due to catastrophe events results in higher the CatEPut price.

2.3. Credit default model

Structural models use the evolution of firms' structural variables, such as asset and debt values, to

determine the time of default. Merton's model (1974) was the first modern model of default and is

considered the first structural model. In Merton's model, a firm defaults if, at the time of servicing

the debt, its assets are below its outstanding debt, and the firm's capital structure is assumed to be

composed by equity and a zero-coupon bond with maturity T and face value of D, whose values at

time t are denoted by and z (t, T) respectively, for 0 ≤ t ≤ T. The firm’s asset value is

simply the sum of equity and debt values. Under these assumptions, equity represents a European

call option with maturity T and strike price D on the asset value. If at maturity T the firm’s asset

value is enough to pay back the face value of the debt D, the firm does not default and

shareholders receive . Otherwise ( ) the firm defaults, bondholders take control of the

firm, and shareholders receive nothing. The rest of assumptions Merton (1974) adopts are the

inexistence of transaction costs, bankruptcy costs, taxes or problems with indivisivilities of assets;

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continuous time trading; unrestricted borrowing and lending at a constant interest rate r; no

restrictions on the short selling of the assets; the value of the firm is invariant under changes in its

capital structure (Modigliani-Miller Theorem) and that the firm’s asset value follows a diffusion

process. The firm’s asset value is assumed to follow a diffusion process given by

,

where is the (relative) asset volatility and is a Brownian motion.

The premium to equityholders and bondholders at time T under the assumptions of this model are

respectively, max and , i.e.

,

applying the Black-Scholes pricing formula, the value of equity at time t (0 ≤ t ≤ T) is given by

where (.) is the distribution function of a standard normal random variable and and are

given by

,

the probability of default at time T is given by

This approach assumes a very simple and unrealistic capital structure and implies that default can

only happen at the maturity of the zero-coupon bond. The shortcomings of the model is that default

would only occur at maturity T, without regard to other factors that make the company default, such

as lack of liquidity and other circumstances. Black and Cox (1976) modified Merton's assumption

that default only occur at maturity, they allow default to take place at any time. The model through

company's bankruptcy boundary condition before maturity to limit option pricing formula of the

solution by Black and Scholes (1973), default occur first time the firm's asset value goes below a

certain lower threshold, i.e. the firm is liquidated immediately after the default event, thus Black

and Cox (1976) is the first of the so-called First Passage Models (FPM). When the default barrier is

exogenously fixed, as in Black and Cox (1976) and Longstaff and Schwartz (1995), it acts as a

safety covenant to protect bondholders. Alternatively it can be endogenously fixed as a result of the

stockholders' attempt to choose the default threshold which maximizes the value of the firm.

3. Catastrophe equity put pricing models

3.1. The interest rate dynamics

In this paper, we consider insurance and reinsurance companies operate in an environment where

interest rates are stochastic. We use the square-root process of Cox, Ingersoll, and Ross (CIR, 1985)

to describe the stochastic interest rate. This setting avoids the negative interest rate that may appear

in Vasicek's model (Vasicek, 1977). The instantaneous interest rate process can be written as:

(1)

where denotes the instantaneous interest rate at time t; κ is mean-reverting force measurement;

is the long-run mean of the interest rate; is the volatility parameter for the interest rate; and

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is a Wiener process. For derivative pricing, it is standard to use the device of a risk-neutralized

pricing measure. The dynamics for the interest process under the risk-neutralized pricing measure,

denoted by Q, can be written as:

(2)

where

are respectively defined as

,

,

,

the term is interpreted as the market price of interest rate risk and is constant under the Cox et

al. (1985) assumption, and is a Wiener process under Q.

3.2. The asset value dynamics

In the previous literature, the asset value dynamic is typical modeled by a lognormal diffusion

process; for example, as in Merton (1977) and Cummins (1988). The asset value of the insurance

company ( ) and reinsurance company ( ) are governed by the following process (at time t):

(3)

where subscript x= I for insurer, and x=R for reinsurer; is the value of the (re)insurer’s assets

at time t;

is the instantaneous expected return on x-company’s assets; is the total

volatility of x-company’s asset returns, and is a Wiener process that denotes the credit risk,

this model assumes that is independent of . Applying Ito’s lemma to the logarithm of the

asset value, Equation (3) becomes:

,

and according to Girsanov’s theorem, we can solve the process for the (re)insurer’s asset value

under the risk-neutralized world, the solution is:

, (4)

where is denoted as

,

,

the term is a Wiener process under the risk-neutralized pricing measure and is independent

of , is the market price of asset risk.

3.3. The liability dynamics

For the liability of the (re)insurance company, we assume that all of its contractual obligations are

honored. With the possibility of default, the market value at time t of its liabilities must be less than

. The cost to the guaranty fund precisely reflects the event in which the insurer’s asset value is

less than . The change in the total contractual liabilities is assumed to be stochastic and consists

of two components. The first component reflects the fact that the (re)insurer is subject to large

jumps in liabilities, i.e., catastrophes. In property insurance, for example, a major hurricane or

terrorist attack could cause widespread damages to an area such that an insurer could face a

staggering amount of claims. We use a compound Poisson process to model this change in liabilities.

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Cummins (1988) and Shimko (1992) make the same assumption regarding the jump risk in their

valuation of insurance liabilities. The second component of the change represents normal variation

in liabilities and is modeled as a continuous diffusion process.

In addition to the liability of providing catastrophe reinsurance coverage, the reinsurer also faces a

liability that comes from providing reinsurance coverage for other lines. Since the total contractual

liabilities represent the present value of future claims, the continuous component reflects the effects

of other day-to-day small shocks. The liability dynamics of an insurance company ( ) and

reinsurance company ( ) while incorporating the effects of the above considerations can be

described as follows (at time t):

, (5)

where is the instantaneous expected return on x-company’s liabilities, is the total

volatility of x-company’s liabilities returns, is a sequence of independent and

identically-distributed positive random variables describing the percentage change in liabilities due

to catastrophes; and is a Poisson process with intensity parameter λ independent of all other

random variables, we assume the insurance and reinsurance companied share the same catastrophe

intensity, and is a Wiener process of x-company’s liability.

For valuation purpose, we follow Merton (1976) and Cummins (1988) to know the liability

dynamic under the risk-neutral pricing measure. In Merton Jump Diffusion Model, the jump-risk is

diversifiable and uncorrelated with aggregate market returns, thus the distribution of the

jump-arrivals and the jump-sizes do not change under the change of measure. This paper also

assume that the overall economy is only marginally influenced by localized catastrophes and. This

assumption implies that and pertain to idiosyncratic shocks to the insurer's liabilities, we

will assume a zero risk premium for these risk. Similarly, by Ito’s lemma and Girsanov's theorem,

we can have the following the process of the x-company's liabilities under the risk-neutral measure:

where

, are respectively defined as:

,

, ,

,

where is the market price of liability risk and is a Wiener process summarizing all

continuous shocks that are not related to the interest rate or asset risk of the x-company; that is

assumed to be independent of

. Note that the term

offsets the

drift arising from the compound Poisson component .

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3.4. Premium of CatEPuts

Once the risk-neutral processes of asset, liability, and interest rate dynamics are obtained, we can

price the catastrophe reinsurance contract by the discounted expectation of its various premium in

the risk-neutral world. This subsection specifies the premium of CatEPuts for alternative

considerations.

3.4.1. Share price

We consider the case where an insurer with shares outstanding intends to hedge its catastrophe

risk by purchasing units of CatEPuts from a reinsurer (i.e. the writer of CatEPut). denotes

the insurer's share price at time t, and the share price can be obtained as follows:

. (7)

3.4.2. Accumulated Catastrophe Losses as Trigger

In the case of Jaimungal and Wang (2006) where CatEPuts become exercisable when the

accumulated catastrophe losses faced by the insurer is larger than a specified amount (denoted as

). Each CatEPut allows the insurer to sell one share of its stock to the reinsurer at a specified price

of K, the exercise price of the CatEPut. The premium of CatEPuts at maturity, , can be written

as:

, (8)

where denotes the trigger level of the insurer's initial asset value, and denotes the process

for the accumulated catastrophe losses faced by the insurer at time t. It can be described as follows:

, (9)

that is if the catastrophe losses faced by the insurer is larger than a specified percentage of its initial

asset value the insurer has the right to exercise CatEPuts.

In this study, counterparty risk is also taken into consideration one of factors to value the premium.

In a typical insurance transaction, the primary insurance company and the reinsurance company

enter into a contract, whereby the insurer pays the premium to the reinsurer, once the catastrophe

events occur, and the insurer's share price ( ) fall to a certain threshold, when the insurer fulfill

CatEPuts, but the reinsurer becomes unable to pay its obligations to the insurer, counterparty risk

exists. When counterparty risk exists, the premium of the reinsurance contract at maturity, ,

can be described as follows:

, (10)

where and denote the values of the reinsurer's assets and liabilities, respectively. In this

case the premium of the reinsurance contract depend on the values of the reinsurance company's

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financial position. We assume that the reinsurer's catastrophe reinsurance liabilities and other

liabilities have the same priority of clam, and they share the assets on the pro rata basis when the

reinsurer becomes insolvent.

According to the premium structures, the catastrophe losses and the dynamics for the specified

above, the CatEPut can be valued as follows:

(11)

here, denotes the expectations taken on the issuing date under a risk-neutral pricing

measure, and , denotes the premium of CatEPut under the trigger threshold

of catastrophe losses. We note that CatEPut prices are determined by percentage of the insurer's

initial share price and estimated by the Monte Carlo simulation.

3.5. Probability Default

This paper unitizes structural model to calculate the probability of default for the insurer. Under

structural model, a default event is deemed to occur for a (re)insurer when its assets reach a

sufficiently low level compared to its liabilities. We first consider the case where the primary

insurance did not purchase the CatEPuts from a reinsurer, the probability of default for the insurer

without CatEPuts at time T is given by

The second case, the insurance company purchased CatEPuts contract and paid premium to

coverage catastrophe losses. The contingent savings from the debt forgiven of the CatEPuts can be

considered as an infusion of capital of the insurer and can be used to pay off liability claims. Hence,

the probability of default relates not only to the capital position of the insurance company, but also

CatEPuts premium and catastrophe losses. At maturity, the default probability for the insurer with

the CatEPuts when without considering the reinsurer's default risk, can be specified as follows:

another case, insurer takes counterparty risk into consideration, the insurer's default probability is

determined as follows:

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4. Empirical Analysis

4.1. Parameter values

This section estimates the premium of CatEPuts and its default probability for alternative scenarios

by using the Monte Carlo method. The simulations are run on a yearly basis with 10,000 paths. As a

reference point for the numerical analysis, a set of parameters and base values that are established and

summarized in Table 2, and deviations from the base values provide insight into how changes in the

characteristics of the catastrophe losses parameters and the intensities of catastrophe losses affect the

CatEPuts options prices. The initial spot interest rate ( ) and the long-run interest rate ( ) are both

set at 5 percent. The mean-reverting force ( ) is set to be 0.2, and the volatility of the interest rate ( )

is set at 7 percent. The (re)insurer's initial asset/liability position (A/L) ratios is 1.2, the volatility of

(re)insurer's asset return that is caused by the credit risk, and the volatility of (re)insurer's total

liabilities are both set at 0.2. These term structure parameters are all within the ranges typically used

in the previous literature (Duan and Simonato, 1999). In addition, the option term is assumed to be

one year, the trigger level ( ) is determined by percentage of the insurer's initial assets, the insurer

shares outstanding ( ) is assumed to be 1, and the strike price is dependent on the mean of the stock

price and assumed out of money (K = ).

Table 2 Parameters Definitions and Base Values

Interest Rate Parameters Values

Instantaneous interest rate =5%

Magnitude of mean-reverting force 20%

Long-run mean of interest rate 5%

Volatility of interest rate 7%

Wiener process for interest rate shock

Asset Parameters

Insurer’s assets =1.2

Reinsurer’s assets =1

Total volatility of (re)insurer’s asset return 20%

Wiener process for credit shock, x=I or R

Liability Parameters

Insurer’s liabilities 1

Reinsurer’s liabilities =1.2

Total volatility of (re)insurer’s liabilities 20%

Wiener process for pure liabilities risk , x=I or R

Catastrophe Loss Parameters

λ Catastrophe intensity

Mean of the logarithm of CAT losses for the (re)insurer

Standard deviation of the logarithm of CAT Losses for the (re)insurer

Other Parameters

T Time to maturity 3

Trigger levels 50%

Shares outstanding 1

Units of CatEPuts 50%

Insurer's initial share price 0.2

K Strike price 0.5 (out of the money)

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4.2. Estimation

This section evaluates CatEPuts whose total losses are linked to hurricane losses available from the

U.S. database on spatial hazard events and losses. The frequency data and insured loss, shown in

Table 3, are collected from 1960 to 2008.

Table 3 U.S. Frequency of and insured losses from 1960 to 2008

Year Frequency Insured loss US$ Trillion Year Frequency Insured loss US$ Trillion

1960 5 1.05786 1985 11 7.58283

1961 3 5.60439 1986 3 2.75

1962 1 5 1987 1 5

1963 4 5.215 1988 3 5.55

1964 11 1.60211 1989 3 2.191105

1965 3 5.53488 1990 0 0

1966 2 5.64434 1991 1 1.56

1967 6 1.50011 1992 3 1.5

1968 3 5.41278 1993 2 5.625

1969 2 5.515 1994 1 5

1970 2 4.91842 1995 6 3.531821

1971 4 1.08 1996 7 3.53428

1972 4 5.1 1997 1 6.55

1973 0 0 1998 7 1.468902

1974 1 4.79166 1999 12 4.06268

1975 2 5 2000 1 5

1976 2 1.055 2001 7 5.12939

1977 0 0 2002 10 7.06529

1978 1 5 2003 11 8.04786

1979 3 6.54717 2004 12 1.8221605

1980 2 5 2005 16 4.9576865

1981 0 0 2006 5 5.3902

1982 1 5 2007 2 2.57

1983 1 5 2008 18 3.782

1984 2 1.6055

According to previous studies (e.g., Louberge et al., 1999; Lee and Yu 2002; Cox et al., 2004;

Jaimungal and Wang, 2006), catastrophe intensities stand for the Pure Poisson distribution with a

constant intensity λ so that the probability of a catastrophe in a small interval of time is λ .

The probability density function of Poisson distribution is

,

where λ is a intensity parameter, equal to the average incidence of random events per unit time

whose rate is constant and must be positive; for the Poisson distribution with parameters λ, both the

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mean and variance are equal to λ. The results are in Table 4, the result indicates the occurrence

intensities of catastrophe losses are 4.2449, to reflect the frequencies of catastrophe events per year;

in other word, on average, four catastrophe events occur every year. We use a simple t-test for

difference in means assuming unequal catastrophe occurrence intensities to establish statistical

significance. From Table 4, we can find that the t-statistic exhibits statistically significant

differences in means at the 5% or greater level of significance.

Table 4 Estimation of parameters for catastrophes number

Distribution λ

Poisson 4.2449(0.2943)

Note: The value in parentheses denotes standard error.

We use maximum likelihood estimation to estimate the parameter values of catastrophe losses by

using several popular distributions, and then take the comparison of these models. The distribution

includes Exponential, Gamma, Weibull, Pareto, and Lognormal, and their probability density

function are respectively defined as

Exponential distribution:

where is rate parameter; the mean and standard deviation are both

λ.

Gamma distribution:

,

where is shape parameter, is scale parameter and is the Gamma function; the mean of

Gamma distribution with parameters and is and the variance is .

Weibull distribution:

where is scale parameter and is shape parameter; the mean of Weibull distribution with

parameters and is and the variance is .

Pareto distribution:

,

where κ is shape parameter and κ ≠ 0, is scale parameter and is threshold parameter; the

mean of Pareto distribution with parameters is and the variance is

.

Lognormal distribution:

,

where and are the mean and standard deviation, respectively, of the associated normal

distribution; the mean and variance of lognormal random variable are functions of and

are

and

.

The empirical result is shown in Table 5, from Table 5, we find that the Weibull distribution is the

best goodness-of-fit, the estimates are = 0.0014 and = 0.2274, respectively, and the value of

the log-likelihood function is 1006.3643. The next goodness-of-fit is Gamma distribution, the

estimates are = 0.1387 and = 0.0145, respectively, and the value of the log-likelihood

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function is 426.8655.

Table 5 Estimation of parameters for catastrophe losses

Distribution Parameter Estimate Log-likelihood

Exponential λ 0.0020(0.0003) 255.1831

Gamma 0.1387(0.0213) 426.8655

0.0145(0.0067)

Weibull 0.0014(0.0012) 1006.3643

0.2274(0.0276)

Pareto κ 22.7206(3.3904) 373.8278

0.0000(0.0000)

Lognormal -11.8117(1.1368) 409.3646

7.7571(0.8220)

Note: The values in parentheses denote standard error.

4.2.1. CatEPuts Price

We consider the case where the insurer purchases CatEPuts and report the premium under with

(without) considering counterparty risk and alternative sets of occurrence intensities and parameter

estimates. The difference between the premium without counterparty risk and premium with

counterparty risk is the counterparty risk premium. Table 6 presents the premium of the reinsurance

contract when catastrophe losses stand for Weibull distribution. Comparing of with and without

considering counterparty risk, we observe that counterparty risk reduces the value of the premium

irrespective of parameter estimates of size and occurrence intensities. This is because when the

reinsurance company's capital structure is more robust, the lower the probability of default, the

insurer facing counterparty default risk is lower, which heighten the ability for the reinsurance

company to pay catastrophe claims. In addition, the impact of counterparty risk is more obvious in

the cases of high occurrence intensity, high scale parameter, and low shape parameter. For example,

in the case of ( , , λ) = (0.0014, 0.1998, 3.9506), the counterparty risk premium is about 681.1

basis points, while the counterparty risk premium can go as high as 871.7 basis points for the case

of ( , ) = (0.0026, 0.1998), and λ =4.5392. We also note that the counterparty risk premium

decreases with shape parameter and increases with scale parameter and occurrence intensities, this

is because that the greater the value of shape parameter, the smaller the mean of catastrophe losses,

and thus the lower the CatEPuts premium.

When catastrophe losses stand for Gamma distribution, the results are shown in Table 7, we find

that counterparty risk still substantially lower the CatEPuts premium, thus counterparty risk should

not be ignored in the valuation of reinsurance policy; this point is in line with Table 6. Observe that

the CatEPuts premium increases with scale parameter, shape parameter and occurrence intensities

for most cases. For instance, in the case where the shape parameter ( ) is 0.1174 and the occurrence

intensity (λ) is 3.9506, the CatEPuts premium increases from 0.000056 to 0.000831 when the scale

parameter ( ) increases from 0.0786 to 0.1452, while the CatEPuts premium will rises to 0.002452

when the scale parameter increases to 0.2118. Further note that the increment of without

counterparty risk is higher than that with counterparty risk. For example, the increment of CatEPuts

premium about 4.5 and 0.2 basis points for the case of ( , ) = (0.16, 0.1452) when λ decreases

from 4.2449 to 3.9506, respectively.

14

Table 6 CatEPuts Premium and Counterparty Risk Premium with Weibull distribution

no counterparty risk( ) counterparty risk( ) counterparty risk premium( )

=3.9506 4.2449 4.5392 3.9506 4.2449 4.5392 3.9506 4.2449 4.5392

0.0002 0.1998 38.605 40.895 42.540 0.381 0.397 0.391 38.224 40.498 42.149

0.2274 23.068 23.445 22.784 0.207 0.223 0.243 22.861 23.222 22.541

0.2550 10.018 11.022 12.380 0.071 0.112 0.101 09.947 10.91 12.279

0.0014 0.1998 68.723 71.045 74.794 0.613 0.638 0.643 68.110 70.407 74.151

0.2274 47.279 47.301 49.844 0.487 0.470 0.454 46.792 46.831 49.39

0.2550 29.987 30.215 29.749 0.276 0.288 0.271 29.711 29.927 29.478

0.0026 0.1998 83.217 80.191 87.879 0.720 0.656 0.711 82.497 79.535 87.168

0.2274 55.788 58.308 61.870 0.545 0.546 0.598 55.243 57.762 61.272

0.2550 39.262 41.811 40.471 0.374 0.455 0.372 38.888 41.356 40.099

Table 6 presents CatEPuts premium and counterparty risk premium under Weibull distribution. Counterparty risk

premium is the difference between premium without counterparty risk and premium with counterparty risk. Premium

are calculated and reported for alternative sets of catastrophe intensities (λ). , represent Weibull distribution with

scale and shape parameters, respectively. All estimates are computed using 10,000 simulation runs.

Table 7 CatEPuts Premium and Counterparty Risk Premium with Gamma distribution

no counterparty risk( ) counterparty risk( ) counterparty risk premium( )

=3.9506 4.2449 4.5392 3.9506 4.2449 4.5392 3.9506 4.2449 4.5392

0.1174 0.0786 0.056 0.274 0.169 0.002 0.006 0.005 0.054 0.268 0.164

0.1452 0.831 1.169 1.030 0.023 0.028 0.060 0.808 1.141 0.97

0.2118 2.452 2.736 2.912 0.064 0.078 0.076 2.388 2.658 2.836

0.1387 0.0786 0.000 0.110 0.058 0.000 0.002 0.001 0.000 0.108 0.057

0.1452 0.748 1.147 0.936 0.021 0.028 0.024 0.727 1.119 0.912

0.2118 3.179 2.906 2.708 0.082 0.089 0.071 3.097 2.817 2.637

0.1600 0.0786 0.054 0.000 0.114 0.001 0.000 0.003 0.053 0.000 0.111

0.1452 0.463 0.913 1.278 0.012 0.032 0.095 0.451 0.881 1.183

0.2118 2.237 2.849 2.890 0.128 0.110 0.076 2.109 2.739 2.814

Table 7 presents CatEPuts premium and counterparty risk premium under Gamma distribution. Counterparty risk

premium is the difference between premium without counterparty risk and premium with counterparty risk. Premium

are calculated and reported for alternative sets of catastrophe intensities (λ). , represent Gamma distribution with

shape and scale parameters, respectively. All estimates are computed using 10,000 simulation runs.

Further comparing Table 6 with Table 7, we find that the CatEPuts premium under Weibull

distribution are more valuable to the insurance company and have higher values than their

corresponding Gamma distribution. Most previous researches (e.g. Hsieh and P-H, 2004; Jaimungal

and Wang, 2006; Wang et. al, 2011) point that catastrophe losses stand for Gamma distribution, but

we use actual data to find the Weibull distribution is the best goodness-of-fit, hence the CatEPuts

premium and counterparty risk premium will be underestimated when we assume that catastrophe

losses stand for Gamma distribution. Moreover, we also find that whether a Weibull or Gamma

distribution, scale parameter has the greatest influence on the CatEPuts premium and counterparty

risk premium.

15

4.2.2. Default Probability

In this subsection, we investigate the effect of CatEPuts and counterparty risk on the default

probability of the insurer. We consider the first case where the insurer does not purchase CatEPuts

and catastrophe losses stand for Weibull distribution, the results are shown in Table 8. Table 8

shows that the values almost increase with and decrease with , this is because that it would

increase the mean of catastrophe losses one year when rise or decline. Turning to the

parameter of catastrophe losses, if the occurrence intensities (λ) increase, the default possibility of

the insurer will increase. For example, in the case where ( , λ) = (0.0002, 3.9506), the probability

of default will drop 307 basis points when increase from 0.1998 to 0.2274 and fall another 206

basis points as 0.2274 goes to 0.2550.

Table 8 Probability of Default without CatEPuts

λ

3.9506 4.2449 4.5392

0.0002 0.1998 0.6553 0.6554 0.6573

0.2274 0.6246 0.6274 0.6279

0.2550 0.6040 0.6051 0.6071

0.0014 0.1998 0.6938 0.6989 0.7035

0.2274 0.6634 0.6657 0.6673

0.2550 0.6404 0.6393 0.6413

0.0026 0.1998 0.7183 0.7124 0.7239

0.2274 0.6781 0.6823 0.6827

0.2550 0.6535 0.6563 0.6561

Table 8 presents the default probability of insurer for alternative sets of catastrophe intensities ( ). , represent

Weibull distribution with scale and shape parameters, respectively. All estimates are computed using 10,000 simulation

runs.

Table 9 Probability of Default without CatEPuts (Gamma distribution)

λ

3.9506 4.2449 4.5392

0.1174 0.0786 0.4135 0.4127 0.4123

0.1452 0.4245 0.4230 0.4240

0.2118 0.4290 0.4342 0.4347

0.1387 0.0786 0.4125 0.4140 0.4142

0.1452 0.4244 0.4236 0.4255

0.2118 0.4325 0.4338 0.4358

0.1600 0.0786 0.4128 0.4127 0.4144

0.1452 0.4210 0.4264 0.4261

0.2118 0.4315 0.4338 0.4379

Table 9 presents the default probability of insurer for alternative sets of catastrophe intensities (λ). , represent

Gamma distribution with shape and scale parameters, respectively. All estimates are computed using 10,000 simulation

runs.

Table 9 exhibits the default possibility of the insurer when catastrophe losses stand for Gamma

distribution. Observe that most of the value increases with the catastrophe occurrence intensities,

shape parameter and scale parameter. This is because the higher of , and λ, the higher of the

average of catastrophe intensity and losses one year, and thus this makes the default probability of

the insurance company increase. For instance, in the case where ( , ) = (0.1174, 0.2118), the

probability of default goes from 42.90 to 43.47 percent when λ from 3.9506 up to 4.5392, an

increment of 57 basis points. The increment of probability of default raise up to about 64 basis

16

points when is at 0.1600.

We further consider the case that when the reinsurer sells CatEPuts to the insurer, they will

simultaneously hedge their position to avoid taking on very large losses, whereas fail to facing the

counterparty risk, the units of CatEPuts ( ) are set at 0.2. We investigate the level of credit

enhancement of the insurers when they use CatEPuts, the improvement ratio of the default

probability for the insurer is calculated as follows:

Improvement ratio =

,

if improvement ratio is greater than 0, means that the insurer purchased CatEPut, indeed enhance

the insurer's credit, reduce the insurer's default risk. Table 10 represents the probability of default

for the insurer and improvement ratio of the default probability, and we assume that catastrophe

losses obey Weibull distribution. From Table 10, we find that the default probability of the insurer

increases with scale parameter and decreases with shape parameter, scale parameter ( ) and shape

parameter ( ) are significantly moderately positive and negative related to improvement ratio,

respectively. The improvement ratio are all greater than 0 and increase with occurrence intensities

(λ) for most cases. In addition, comparing the values of Table 10 with Table 8, we observe that

CatEPuts reduce the default probability of the insurer. This is because CatEPuts can provide

insurers with additional equity capital when they need funds to cover catastrophe losses. Further

comparison in Table 10 and Table 11, we observe that the impact of a CatEPuts issuance is more

obvious in the cases of the improvement ratio is better than the results of Table 11.

Table 10 Default Probability and Improvement Ratio with CatEPuts and without Counterparty Risk (Weibull distribution)

Probability of default Improvement ratio (%)

λ=3.9506 4.2449 4.5392 λ=3.9506 4.2449 4.5392

0.0002 0.1998 0.5847 0.5858 0.5854

10.7737 10.6195 10.9387

0.2274 0.5749 0.5760 0.5752 7.9571 8.1925 8.3931

0.2550 0.5704 0.5716 0.5699 5.5629 5.5363 6.1275

0.0014 0.1998 0.6006 0.6026 0.6026 13.4333 13.7788 14.3426

0.2274 0.5880 0.5891 0.5909 11.3657 11.5067 11.4491

0.2550 0.5793 0.5799 0.5793 9.5409 9.2914 9.6679

0.0026 0.1998 0.6095 0.6089 0.6134 15.1469 14.5284 15.2645

0.2274 0.5922 0.5961 0.5968 12.6677 12.6337 12.5824

0.2550 0.5850 0.5845 0.5853 10.4820 10.9401 10.7910

Table 10 presents the default probability of insurer with CatEPuts issuance, but no counterparty risk. Probability of

default is calculated and reported for alternative sets of catastrophe intensities (λ). Improvement ratio is calculated

based on Table 8. , represent Weibull distribution with scale and shape parameters, respectively. All estimates are

computed using 10,000 simulation runs.

Table 11 reports the default probability of the insurer if catastrophe losses obey Gamma distribution

and the insurer fail to shouldering counterparty risk. As we expected, the CatEPuts drive down the

probability of default of the insurer. Find that the default probability of the insurer increases with

occurrence intensities and shape parameter and scale parameter, although the change in magnitude

is not large. Note also that the improvement ratio increases with occurrence intensities and shape

parameter and scale parameter, for example, in the situation of ( , ) = (0.1387, 0.1452), the

improvement ratio will increase about 46 percent when the occurrence intensities increase from

17

4.2449 to 4.5392, especially when the shape parameter ( ) is 0.1600, scale parameter ( ) is 0.2118,

and occurrence intensities (λ) are 4.5392, the improvement ratio is the highest.

Table 11 Default Probability and Improvement Ratio with CatEPuts and without Counterparty Risk (Gamma distribution)

Probability of default Improvement ratio (%)

λ=3.9506 4.2449 4.5392 λ=3.9506 4.2449 4.5392

0.1174 0.0786 0.3957 0.3958 0.3958

4.3047 4.0950 4.0019

0.1452 0.3961 0.3959 0.3963 6.6902 6.4066 6.5330

0.2118 0.3967 0.3971 0.3969 7.5291 8.5444 8.6957

0.1387 0.0786 0.3957 0.3957 0.3957 4.0727 4.4203 4.4664

0.1452 0.3962 0.3962 0.3960 6.6447 6.4684 6.9330

0.2118 0.3971 0.3969 0.3965 8.1850 8.5062 9.0179

0.1600 0.0786 0.3957 0.3957 0.3958 4.1424 4.1192 4.4884

0.1452 0.3960 0.3962 0.3962 5.9382 7.0826 7.0171

0.2118 0.3969 0.3974 0.3973 8.0185 8.3910 9.2715

Table 11 presents the default probability of insurer with CatEPuts issuance, but no counterparty risk. Probability of

default is calculated and reported for alternative sets of catastrophe intensities (λ). Improvement ratio is calculated

based on Table 9. , represent Gamma distribution with shape and scale parameters, respectively. All estimates are

computed using 10,000 simulation runs.

In the last case of the insurer purchases CatEPuts from the reinsurer, and considers the risk of the

reinsurer may not be able to fulfill the contract. Table 12 and Table 13 are the results when

catastrophe losses stand for Weibull and Gamma distribution, respectively. From Table 12 and Table

13, we find that the probability of default under alternative occurrence intensities and parameter

estimates are very close and within the range of 7 basis points for most cases, implies that no

significant correlation between three parameters and the probability of default. For example, in the

case of ( , ) = (0.1174, 0.1452), the probability of default only 1 basis points when λ rises from

4.2449 to 4.5329. In addition, the improvement ratio can go as high as 9.62 and 21.48 percent,

respectively.

Table 12 Default Probability and Improvement Ratio with CatEPuts and Counterparty Risk (Weibull distribution)

Probability of default Improvement ratio (%)

λ=3.9506 4.2449 4.5392 λ=3.9506 4.2449 4.5392

0.0002 0.1998 0.5685 0.5686 0.5686

13.2458 13.2438 13.4946

0.2274 0.5681 0.5681 0.5685 9.0458 9.4517 9.4601

0.2550 0.5682 0.5681 0.5682 5.9272 6.1147 6.4075

0.0014 0.1998 0.5684 0.5685 0.5687 18.0744 18.6579 19.1613

0.2274 0.5681 0.5686 0.5683 14.3654 14.5861 14.8359

0.2550 0.5683 0.5684 0.5682 11.2586 11.0903 11.3987

0.0026 0.1998 0.5685 0.5688 0.5684 20.8548 20.1572 21.4809

0.2274 0.5684 0.5684 0.5687 16.1776 16.6935 16.6984

0.2550 0.5681 0.5683 0.5681 13.0681 13.4085 13.4126

Table 12 presents the default probability of insurer with CatEPuts issuance and considers counterparty risk. Probability

of default is calculated and reported for alternative sets of catastrophe intensities (λ). Improvement ratio is calculated

based on Table 8. , represent Weibull distribution with scale and shape parameters, respectively. All estimates are

computed using 10,000 simulation runs.

In view of counterparty risk, we first compare Table 12 with Table 10, we observe that counterparty

risk reduce the probability of default and raise the improvement ratio, the improvement ratio with

counterparty risk can go as high as 21 percent than one without counterparty risk increased by 6

percent. Further comparison in Table 13 and Table 11, observe that the improvement ratio are very

18

close and the range only from zero to 0.3 percent. We note that the improvement ratio become

insignificant when = 0.1174, the range only from zero to 0.02 percent.

Table 13 Default Probability and Improvement Ratio with CatEPuts and Counterparty Risk (Gamma distribution)

Probability of default Improvement ratio (%)

λ=3.9506 4.2449 4.5392 λ=3.9506 4.2449 4.5392

0.1174 0.0786 0.3957 0.3957 0.3957

4.3047 4.1192 4.0262

0.1452 0.3957 0.3957 0.3958 6.7845 6.4539 6.6509

0.2118 0.3957 0.3958 0.3957 7.7622 8.8439 8.9717

0.1387 0.0786 0.3957 0.3957 0.3957 4.0727 4.4203 4.4664

0.1452 0.3957 0.3958 0.3957 6.7625 6.5628 7.0035

0.2118 0.3959 0.3958 0.3957 8.4624 8.7598 9.2015

0.1600 0.0786 0.3957 0.3957 0.3957 4.1424 4.1192 4.5125

0.1452 0.3957 0.3957 0.3957 6.0095 7.1998 7.1345

0.2118 0.3957 0.3959 0.3958 8.2966 8.7367 9.6141

Table 13 presents the default probability of insurer with CatEPuts issuance and considers counterparty risk. Probability

of default is calculated and reported for alternative sets of catastrophe intensities (λ).Improvement ratio is calculated

based on Table 9. , represent Gamma distribution with shape and scale parameters, respectively. All estimates are

computed using 10,000 simulation runs.

The improvement ratio of default probability with and without counterparty risk is illustrated, under

alternative parameter values and different distributions in Figures 2 and 3. From Figure 2, the

significant improvement ratio different illustrates that counterparty risk is an important factor and

should be taken into account when pricing the probability of default and improvement magnitude

along with scale parameter ( ) rises or shape parameter ( ) decline, the different is more significant.

Figures 3 shows that improvement ratio for alternative sets of shape parameter ( ) and scale

parameter ( ) are very similar. This implies that counterparty risk and improvement ratio are shown

not to be significantly related under Gamma distribution.

Figure 1 Improvement ratio with and without

counterparty risk when λ = 4.2449 and catastrophe losses

stand for Weibull distribution

Figure 2 Improvement ratio with and without counterparty

risk when λ = 4.2449 and catastrophe losses stand for

Gamma distribution.

5. Conclusions

This paper develops a model for measuring catastrophe risk, counterparty risk, and catastrophe

losses distributions that is associated with the valuation of CatEPut, and examines the impact of

19

these factors on default probability and improvement ratio of the insurer. This paper utilizes the

hurricane losses data from the U.S. database during 1960 to 2008 to engage numerical analysis.

From the empirical results, we can summarize as the following: First, catastrophe occurrences are

drawn from a Poisson distribution, catastrophe intensity parameters of 4.2449, implying four

catastrophes every year, seem to be on the high side. Second, we utilize maximum likelihood

estimation to estimate the parameter values of catastrophe losses by using difference distributions,

we find the best goodness-of-fit is Weibull distribution, next is Gamma distribution.

From the numerical results, we price the CatEPut premium with/without counterparty risk and find

that for most possible scenarios the counterparty risk premium can be substantial in the presence of

catastrophe risk and should not be ignored in the valuation of the reinsurance policy. The results

also show that counterparty risk reduce the CatEPut premium of the reinsurance policy and raise its

counterparty risk premium. And, we can know that scale parameter have higher impact on premium.

Forth, CatEPuts drive down the probability of default of insurance company, when catastrophe

losses stand for Gamma distribution, the impact of CatEPuts on default probability of the insurer is

more substantial for the cases of higher catastrophe intensity, higher shape parameter and higher

scale parameter, but in the case of Weibull distribution, the greatest impact in the case of high

catastrophe intensity, high scale parameter and low shape parameter. Finally, taking counterparty

risk into account when computing probability of default of the insurer, we find that default

probability of the insurer decrease and improvement ratio increase significantly under Weibull

distribution, whereas no significant changes in Gamma distribution.

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