論文題目 : Catastrophe Risk Management and Credit Enhancement by Using Contingent Capital

Transcription

1 論文題目 : Catastrophe Risk Management and Credit Enhancement by Using Contingent Capital 報名編號 :B0039

2 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 1

3 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 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 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. 2

4 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 is written mathematically as: where. If the option is exercised, then the payoff of CatEPut at maturity time T and it 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 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 3,

5 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 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; 4

6 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: where denotes the instantaneous interest rate at time t; κ is mean-reverting force measurement; is the long-run mean of the interest rate; 5 (1) is the volatility parameter for the interest rate; and

7 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: where the term are respectively defined as,,, 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 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 ( where subscript x= I for insurer, and x=r for reinsurer; ) are governed by the following process (at time t): (2) (3) 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 6,, is a Wiener process under the risk-neutralized pricing measure and is independent of, is the market price of asset risk 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.

8 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 ( reinsurance company ( described as follows (at time t): ) and ) while incorporating the effects of the above considerations can be, (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. 7

9 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 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) 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, as:, can be written, (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 8

10 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: here, measure, and denotes the expectations taken on the issuing date under a risk-neutral pricing (11), 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 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 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: to another case, insurer takes counterparty risk into consideration, the insurer's default probability is determined as follows: 9

11 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 ( price and assumed out of money (K = ). ) is assumed to be 1, and the strike price is dependent on the mean of the stock 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) 10

12 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 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 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 11,

13 mean and variance are equal to λ. The results are in Table 4, the result indicates the occurrence intensities of catastrophe losses are , 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 (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 = and = , respectively, and the value of the log-likelihood function is The next goodness-of-fit is Gamma distribution, the estimates are = and = , respectively, and the value of the log-likelihood 12

14 function is Table 5 Estimation of parameters for catastrophe losses Distribution Parameter Estimate Log-likelihood Exponential λ (0.0003) Gamma (0.0213) (0.0067) Weibull (0.0012) (0.0276) Pareto κ (3.3904) (0.0000) Lognormal (1.1368) (0.8220) Note: The values in parentheses denote standard error 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, , ), the counterparty risk premium is about basis points, while the counterparty risk premium can go as high as basis points for the case of (, ) = (0.0026, ), and λ = 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 and the occurrence intensity (λ) is , the CatEPuts premium increases from to when the scale parameter ( ) increases from to , while the CatEPuts premium will rises to when the scale parameter increases to 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, ) when λ decreases from to , respectively. 13

15 Table 6 CatEPuts Premium and Counterparty Risk Premium with Weibull distribution no counterparty risk( ) counterparty risk( ) counterparty risk premium( ) = 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( ) = 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. 14

16 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, ), the probability of default will drop 307 basis points when increase from to and fall another 206 basis points as goes to Table 8 Probability of Default without CatEPuts λ 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 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, ), the probability of default goes from to percent when λ from up to , an increment of 57 basis points. The increment of probability of default raise up to about 64 basis 15

17 points when is at 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= λ= 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, ), the improvement ratio will increase about 46 percent when the occurrence intensities increase from 16

18 to , especially when the shape parameter ( ) is , scale parameter ( ) is , and occurrence intensities (λ) are , 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= λ= 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, ), the probability of default only 1 basis points when λ rises from to In addition, the improvement ratio can go as high as 9.62 and percent, respectively. Table 12 Default Probability and Improvement Ratio with CatEPuts and Counterparty Risk (Weibull distribution) Probability of default Improvement ratio (%) λ= λ= 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 17

19 close and the range only from zero to 0.3 percent. We note that the improvement ratio become insignificant when = , 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= λ= 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 = and catastrophe losses stand for Weibull distribution Figure 2 Improvement ratio with and without counterparty risk when λ = 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 18