Engineering Seismic Risk Assessment Based Pricing Model For Catastrophe Bond

With the rapid economic development, population, prosperous business, density of buildings, transportation systems, indispensable infrastructures and perfect facilities in a metropolitian area are significantly increased. Many destructive earthquakes in recent years show that the world we live is more vulnerable to an earthquake disaster. If such disaster is intensive enough, it will become a catastrophe. What we can do is to reduce the vulnerability of our society, which is what managing and reducing catastrophic risk are all about.Earthquake risk management is a very complex process involved with engineering mitigation measures and non-engineering countermeasures. The former, prescribed in seismic design codes and prevention counter plans, played a great role in earthquake disaster reduction of China in decades. The latter must be emphasized as the rapid economic development and more opening of China towards the world, especially after joining in the WTO, while the management system is moving from the planned model to the market model. In order to diversify the salvation for earthquake disasters, some instruments, such as insurance, reinsurance and catastrophe insurance derivatives experienced in developed countries and regions may be used as references.Hurricane Andrew and Northridge Earthquake caused the insurance industry to realize the impact that a catastrophe could have on the industry's solvency, and have caused the industry to reevaluate its exposure to loss from a single event or multiple major events in a short period of time, and to look for additional sources of capital to finance or spread the risk. Traditionally, insurers transfer insurance risk through the purchase of reinsurance. However, the reinsurance industry has a limited capacity available to deal with catastrophes. Major carriers, if they were to choose to purchase, could not purchase enough reinsurance to cover a catastrophe, and the surplus of these carriers is exposed in the event of a catastrophe or a series of moderate-sized disasters within a short period. Even if reinsurance could be purchased to cover the catastrophe, the aggregation of exposure to reinsurers could be so large that several reinsurers could become insolvent.Catastrophes that might seriously impair the capital base of the worldwide (re)insurance industry may cause only a ripple if spread through the global capital markets. The capital market can provide (re)insurers with far more financing capacity than what has been available previously, and it can provide insurers with a vehicle for dispersing risk widely. The evolution of catastrophe insurance derivatives is reviewed in this dissertation, and some popular catastrophe insurance derivatives, including CAT Future, CAT Options, CAT Swaps, Insurance Loss Warranties, Credit Line, Contingent Surplus Notes, Catastrophe Equity Puts and CAT bonds, are introduced. It is important to determine a trigger event in pricing for a CAT bond. There are three general types of trigger events, including issuer's loss, industry index and pure parametric trigger, which will be introduced in detail. And there are also other trigger events, like modeled losses and parametric index.The result from a research group in Beijing University in 2005 showed that the accumulative potential loss in China was about $1.3 trillion among effective domestic policies of flood and storm according to a preliminary statistic analysis, in which about 80%-90% of the loss was contributed from the home damage. At present, the claim payments for natural disasters in China are mainly from the government and public endowment, and insurance and reinsurance are only the supplements. So only a fraction of above-mentioned potential losses is insured. Since the insurance industry in China is still at the developing stage, the premium increases each year but the claim rate is low, the profit of insurers is getting higher, which causes insurers, to some extent, weakened the sense of reinsurance. Actually, risk will be accumulated quickly as the expansion of business, and the stabilization of insurers will be threatened. Consequently, the demand of reinsurance must be large. On the other hand, according to the WTO commitments, the proportion of mandatory reinsurance should be reduced by 5% each year, so there was no mandatory reinsurance in 2006. Meanwhile some transnational (re)insurers, like Swiss Re and Munich Re, have entered into the market. In order to retain the market competitive, Chinese insurers have to expand the coverage. Catastrophe insurance already issued by these transnational (re)insurers is in insistent demand. It is also a necessary instrument on the view of the evolution of catastrophic risk management.As we can see, in the evolution, the catastrophic risk management is turning from passive defense to active management and the effective financial instruments became more and more important in some developed countries. However in China, it is still at the groping stage. A key factor in dealing with risk from catastrophic losses is the critical role of the government. The Chinese government emphasizes the natural disaster mitigation all the time, and some financial instruments have been studied especially in the reformation of the management system. And the research of earthquake insurance is currently in its preliminary stage. However, the insurance industry cannot bear the catastrophic risk by itself. It is necessary to diversify the financing channels through the catastrophe insurance derivatives and make use of the capacity of the capital market to disperse risk. From the evolution of earthquake insurance and the utilization of insurance funds in China, it can be concluded: (1) the catastrophe insurance is an indispensable instrument not only for catastrophic losses mitigation but also for the social stabilization; (2) the insurance funds must be reinvested to improve the capacity of insurers; (3) the catastrophe insurance derivatives, especially CAT bonds, are expected to be powerful instruments to supple earthquake insurance on both time and spatial domains. In this process, the key point is to develop a pricing model combined with the Chinese market practice and to build a bridge between insurance and capital markets. It is also important to provide an institutional guarantee from the aspects of legal and policy for the effective operation of these derivatives. The purpose of this dissertation is to build a pricing model for a CAT bond combined with seismic risk assessment.In some pricing models, the distribution of losses, claims or loss indices etc are simulated based on the loss experience and the trading market from the view of financial economics. However, for earthquake disasters, losses including property and life are mainly derived from the damage of engineering structures, so the amount of losses cannot represent the component of losses. The proposed pricing model will be combined with the engineering seismic risk assessment closely. An exceeding probability curve of loss rate, the rate of earthquake losses to Industry Value Added (IVA) in last year, is the joint of two parts in this dissertation: the engineering seismic risk assessment, and the financial instruments for earthquake disaster reduction. The former involves with problems in the current seismic risk assessment methodology combining with financial instruments, such as CAT bonds; the latter, based on the former, involves with pricing a CAT bond from the view of engineering in the development of earthquake risk management instruments. In other words, the occurrence probability of the defined catastrophe from the former is a parameter in the pricing model for the CAT bond.Engineering seismic risk assessment, consisting of seismic hazard analysis and vulnerability evaluation, is really a very comprehensive assessment as the first and fundamental step in earthquake disaster prevention process. In China, seismic hazards and vulnerabilities of many cities were assessed in the past few decades. Seismic hazard analysis is a fundamental aspect of seismic risk assessment. The widely adopted probabilistic approach of seismic hazard analysis involves with local seismicity, potential earthquake sources and attenuations of ground motion. The vulnerability evaluation is to infer a conditional probability of the potential damage of engineering structure given various levels of earthquake intensity. It describes a relationship between damages and ground motion for a specific type of structure. Typically, some vulnerability matrixes for structures in many cities and/or regions are established based on the earthquake damage experience in China and founded on macro intensity. In general, the assessments are completed separately and then they are combined together for the loss estimation. These two parts are completed by seismological team and engineering team separately. Seismic hazard expression must be improved to combine these together, especially for the application in financial instruments such as insurance and CAT bonds. Some suggestions are presented in the first part.(1) In the seismic risk assessment procedure adopted in many cities of China, the expected total loss of buildings in a given period of time can be calculated as the product of the losses, including indoor property loss, of Sth type building being in kth damage state and the probability of Sth type building being in kth damage state PS(Dk). PS(Dk) is referred to as the engineering seismic risk and can be described as the product of the conditional probability of Sth type building being in kth damage state given intensity I, or so-called vulnerability of Sth type building PS(Dk|I) and the possibility of intensity I occurred P(I). P(I) called seismic hazard depends on the regional seismic environment and attenuation relationship of ground motion. It is generally referred to as seismic hazard curve and it can be derived from P(I≥i). In nature, earthquake intensity is a sequential classified variable, so P(I≥i) is not really a continuous curve. Of course, the intensity I can be substituted with other ground motion parameter Y, so that P(Y>y) is a continuous curve.In general, P(I) is the product of P(I≥i) minus P(I≥i+1). However, the result of the subtraction will not be reasonable, if I is low. For example, the result of seismic hazard analysis in a region with low seismicity shows P(I=i) is less than P(I=i+1), and one can see there must be something wrong. In nature, P(I) must be a monotone decreasing function, i.e. P(I=i) must be greater than P(I=i+1), even if intensity is low. To deal with the procedure of seismic hazard, one can understand that P(I≥i) is contributed by earthquakes in many potential sources with various magnitudes and frequencies. It means that P(I≥i) consists of not only P(I=i) and P(I≥i+1), but also P(I=i and I>i). The later cannot be ignored in seismic zone with strong activity for intensity less thanⅦand long evaluate period, like 50 years or 100 years. The general seismic hazard analysis assumes that the occurrence of earthquake is independent each other. Therefore a solution is that the exceeding probability in a short time period t can be firstly calculated from the hazard in long period T. Obviously, P(I=i and I>i) can be ignored when the period is short. In general, one month is short enough for a region with generic seismic activity; it should be shorten to days for region with high activity. The occurrence probability in short period can be obtained, and that in long period can be obtained. Hazard curves from the traditional method and the modified one are compared. Another example is given to show that the similar underestimation may happen for higher intensity in a region with high seismicity. Exceeding probability curve of loss ratio is drawn and compared with that from the traditional way to show the big difference.(2) The result of seismic hazard analysis is the occurring or exceeding probability of an earthquake action in a city or region for a specified exposure time. However, high intensity areas are general small and cannot cover the whole metropolis because of the fast attenuation in epicenter area. And it never happens that several earthquakes occur in a given period to cause the same intensity and cover the whole metropolis one by one. In general, high intensity areas are always limited and the whole city will not be an intensity 8 or greater area, so there will be an overestimation if the exceeding probabilities of high intensity events are applied directly in loss estimation. Scenario earthquake approach can be adopted as the solution to this overestimation. One or more scenario earthquakes can characterize the hazard with the same exceeding probability. It must be consistent with the regional seismic environment and be derived from the regional attenuation relationship of ground motion.To make it clear, a case study is demonstrated in this dissertation. High intensities and their probabilities of occurrence are converted into three scenario earthquakes by probabilistic analysis based scenario earthquake method to represent the probabilistic meaning in the deterministic way. This method was presented in the mid of 1990s and has been adopted in many microzonations, seismic hazard analysis of critical project sites and some codes, like NRC Regulatory Guide.165. By this way, the isoseismals and loss from the earthquake can be estimated explicitly, and design spectra can also be determined very well. Since then, many processes to determine scenario earthquakes have been worked out. For example, Ishikawa and Kameda recommended the mean magnitude and distance should be determined separately for each natural frequency of interest and for each seismic zone that contributes to the hazard on the site. The dominant earthquake was recommended to be the mean magnitude and distance of the seismic event that caused a ground-motion exceedence at the chosen return period by McGuire. Luo emphasized that the ground motion on a site should be equal to the ground motion with the corresponding exceeding probability. Chen suggested choosing a scenario earthquake from several ones satisfying conditions with the principle of maximum probability.Concerning with the principle that the peak ground acceleration (PGA) from the scenario earthquake must be kept the same as the corresponding design parameter, the supervisor of the author developed a process, although there is strict principles, the steps depend on the conditions and the process never be published. So in this dissertation, the steps are summarized and discussed in details. In the process, the most dominant potential source area (MDPSA) is selected from PHSA result firstly. According to the design PGA corresponding to three design levels (exceeding probability 63%, 10% and 2-3% in 50 years, respectively), and attenuation relationship, the distances of major and minor axes given magnitudes no bigger than the upper bound magnitude of the MDPSA, can be calculated. The scenario earthquakes are determined finally from the constrain condition of that the distance matches the spatial area of the MDPSA.It is not prescribed clearly in the Chinese code of"Evaluation of seismic safety for engineering sites (GB17741-2005)"if the scenario earthquake method is applicable to construct design spectra. In fact, there are different opinions for improving the code. Some compilers worried about whether the scenario earthquake determined from attenuation relationship of peak ground acceleration (PGA) can contribute properly to the amplitudes at long period. In order to show the applicability of scenario earthquake method to construct design spectra, scenario earthquake determined from attenuation relationship of peak ground acceleration is examined to see if it can contribute properly to the amplitudes at long period. In some cases, the spectra from the scenario earthquakes are quite close to the corresponding uniform spectra, else the spectral amplitudes are not very different from the amplitudes of uniform spectra at short period range, and those determined by attenuation relationship of spectral amplitude at period 1.0 second are closer to uniform one at long period range. The key point in the process is to determine the MDPSA. The contribution of each PSA on the motion at the site depends on the seismicity parameter, area and location of each PSA, the location of the site, and so on. As a rule, the ground motion at short period range attenuates with distance quickly, the contribution of near PSA is obvious; and the ground motion at long period range attenuates relatively slowly, the contribution of near PSA fades and that of far one increases. Therefore, the hazard curves for two natural periods (0.0 and 1.0 second) must be examined carefully to see if one source dominates the hazard at both periods, to make sure whether it is reasonable to represent the hazard with one or more scenario earthquake(s). In a word, spectra from scenario earthquakes can be adopted as design spectra directly if the most dominant areas are the same for spectral amplitudes at short and long period ranges, and more than one spectrum or enveloped spectra of those from scenario earthquakes are suggested to be adopted for design, if the areas are quite different.In order to produce the exceeding probability curves of loss rate, it is necessary to analyze the damage from the scenario earthquakes. Many spatial operations must be involved in the calculation, and the amount of data available for each event is in tens of thousands of items. A GIS based decision-making support system for earthquake disaster reduction can be adopted for its spatial operating capacity.GIS based systems for damage evaluation are developed for many cities and regions worldwide. This kind of systems is a powerful tool to perform spatial analysis and mapping of losses due to a scenario earthquake. The damaged areas of buildings, damaged length of highways, bridges or pipelines, damaged number of electric power or communicate facilities in 5 damage states from scenario earthquake can be assessed very fast. So spatial distributions of casualties, the caused shaking, damage and loss of engineering structures from these three scenario earthquakes are estimated by a GIS based system. By this way, the overestimation of losses in metropolis is avoided. Finally, the exceeding probability curves of loss rate for the traditional and improved ways are drawn and compared.In the second part, it is emphasized how to price a catastrophe insurance derivative, CAT bond, as a supplement of earthquake insurance, that is, how to build a pricing model based on the improved engineering seismic risk assessment method.Although"derivatives"is a new instrument in insurance industry, it has a long history in financial market, including forward contracts, futures, swaps and options etc. As derivatives become more and more active and provide lower-cost solutions to investor and entity, option-pricing theories have been adopted in a wide variety of financial instruments and contracts. All of these pricing models can be distributed into two classes, complete market models and incomplete market ones.The vast majority of these models are assumed in complete market, where all sources of risks can be hedged perfectly, because of the possibility of replication of all assets. In other words, all derivatives can be perfectly replicated using a finite number of specified bonds. In incomplete market, such replication by dynamic hedging is not possible, which means that there are sources of risks that cannot be perfectly hedged. Assets'prices are supposed to be ruled by a dynamic process based on a constant drift, jumps and stochastic volatility. Incomplete market is due to the fact that jumps and stochastic volatility are not traded in market. The incomplete market modeling leads to some drawbacks, so we cannot set a portfolio perfectly hedged with derivatives written on these assets. Some pricing models of financial derivatives are reviewed emphatically as the foundation of next section, including Louis Bachelier (1900), Markowitz (1952), Sprenkle (1961), Boness (1964), William Sharpe (1964) and J. Lintner (1965), Samanelson (1965), Kassouf (1969), Fischer Black and Myron Scholes(1973), Merton (1976), Madan D. B. and E. Seneta (1990) etc. Some pricing models of CAT bonds in two types of markets, evolved from the models of financial derivatives, are reviewed, including Cummins and Geman (1995), Litzenberger et al (1996), Briys (1997), Henri Loubergéet al (1999), Morton N. Lane et al (1998-2004), Wang (2002,2004), Christofides (2004), Angelika Sch?chlin (2001), Yu. Baryshnikov et al (2001), Lee and Yu (2002), Cummins and Phillips (1999), Samuel H. Cox et al (2000), Li Yong (2005). In these models, the stochastic processes described the prices of subject matters is adopted to describe catastrophe loss, loss index or issuer's loss which are not traded in market.However, for earthquake disaster, direct losses are from the damage of engineering structures mainly, the statistic data of historical loss cannot present the engineering characters, like the type, age, purpose, with or without seismic design, etc. So it is not reasonable to estimate the losses from future earthquakes and price CAT bonds through historical data only, the engineering characters should be represented in pricing models.Obviously, both earthquake insurance and its derivatives widen the mitigation channel against earthquake disasters. There might be a complementary relation between the parameters of these instruments to balance the issuer's seismic risk. To show the complementarity, our setup in the final chapter makes an insurer who provides with earthquake insurance policy also, the issuer of a CAT bond. If the insured amount is high, the circulation of the CAT bond needs to be increased to handle high claim payments; otherwise, the circulation to be low to reduce interest payment. Seismic hazard assessment is characterized by scenario earthquakes, and the probability of earthquake catastrophe occurrence estimated by the improved seismic risk assessment method mentioned above. A pricing model for the CAT bond, according to the equilibrium between the incomes of investors and the issuer of the CAT bond, is built. In this model, a stochastic process, Geometric Brownian Motion, is adopted to describe the earning of earthquake insurance premium and the payout of reinsurance premium with two different sets of parameters respectively. The probability of earthquake catastrophe occurrence is an input of the model, ratio and yield of reinvestment, principal protected ratio, issuance fee, circulation and maturity are designed as factors. A partial equilibrium circumstances among the cash flows of the issuer on earthquake risk is built to price the CAT bond in complete market firstly.However, for earthquake risk, the market is incomplete because the insurance premium will jump after a destructive earthquake or effected by some social factors, not a catastrophe. The jump of premium is proved in some cases, like several earthquakes in Japan. In this dissertation, it is also proved from a certain angle by the influence of fire losses on property insurance premium in the next year, the correlation coefficient is 0.9083. It is supposed that the market is arbitrage-free, and Jump-Diffusion process with two different sets of parameters is adopted to describe the earning of insurance premium and the payout of reinsurance premium in incomplete market. The jump size is supposed as standard log-normal distribution, and the jump times follow an independent Poisson process.For explanation, a region in Northeast of China is adopted as an example. Since earthquake insurance has not been launched in large-scale in China now, related data and information is not enough. We have to suggest that the insurance rate in this field is 5% and the policy-holders are distributed uniformly in space. Then, the probable maximum loss of buildings from a probable maximum earthquake (M=6), assessed by the improved method, is RMB 1.556 billion (2004 value), which is discounted from 998.2 million (1994), considering the influence of interest rate and inflation. So the probable maximum claim payment is 77.79 million.In recent years, several insurers covered earthquake risk as extraneous risk of enterprise property insurance, and the premium rate is 10% of main risk, which is limited strictly. It is supposed that 50% policy-holders of enterprise property insurance purchase the additional earthquake insurance, then the data is converted to those in the region by the ratio of annual national GDP to the affiliated city's, in which 90% are adopted as the earning of earthquake insurance premium in the region. Earthquake reinsurance premium is adopted as 20% of the insurance premium. The expected volatility of the earning of insurance premium is 0.0544, the standard deviation is 0.0084, and those of the payout of reinsurance premium are 0.0109 and 0.0017 respectively.According to the improved engineering seismic risk assessment method and the definition of catastrophe in this dissertation, the loss from the maximum considered earthquake is not a catastrophe, and the occurring probability of a defined earthquake catastrophe in 3 years p is 0.0003114%. If a catastrophe occurs, the loss should be 15.436 billion at least, so the claim payment of earthquake insurance is 771.8 million, and that of reinsurance is 154.4 million. If it does not occur, the claim payment of earthquake insurance is 77.79 million, and that of reinsurance is 15.56 million. Let the period of validity T be 3 years, the proportion of reinvestment k 10%, the yield r1 5%, the circulation of the CAT bond M 100 million, the risk-free interest rate r0 3.96%, the principal protected ratio n 50% and the issuance fee B 1% of the circulation. Then, the earning of insurance premium and the payout of reinsurance premium in complete and incomplete market can be obtained. The jumps of insurance and reinsurance premium from destructive earthquakes are considered in incomplete market. The annual coupon rates of the CAT bond r are 5.52% and 6.67% in complete and incomplete markets respectively. An investor will be refunded 118.01 yuan (complete market) and 122.48 yuan (incomplete market) for a coupon bond at maturity if the defined catastrophe does not occur, and the prices of a zero coupon bond, at t=0, are 84.74 yuan (complete market) and 81.64 yuan (incomplete market). In other words, the investor purchases the CAT bond at price of 84.74 yuan or 81.64 yuan and will get 100 yuan back at maturity. If the catastrophe occurs, half of the principal will be lost, that is, the investor will get 50 yuan, 42.37 yuan or 40.82 yuan back. The sensitivity of the annual coupon rate, the interest at maturity and the price on four factors are analyzed, the order, from high to low, is the period of validity, circulation, the yield of reinvestment and the proportion of reinvestment.By comparing with the pricing model in the author's Master thesis, this model is improved in the following areas: (1) M, r1, n, B and p are the factors in the primary model. In this model, the character of CAT bonds as a catastrophe insurance derivative is emphasized, the cash flows of earthquake (re)insurance premium are considered. (2) There are two states in the primary one, the catastrophe occurs or not. The influence of devastating earthquakes, not catastrophes, on the cash flows is considered in this one. (3) All parameters in the primary one are constant. This model is time-depended, although the interest rate of the CAT bond is calculated at maturity. For pricing in T, this model can be referred. For example, if a catastrophe occurs early in the period, the premium is not accumulated enough, M needs to be increased with other conditions unchanged. (4) The proportion of reinvestment is improved. 10% of funds financing from the CAT bond and the initial premium is venture investment, 90% is risk-free investment. The stability of insurance funds is assured and proceeds are increased. Let r1 is 5%, 6% or 7%, r is calculated by these two model. The comparison shows the rate from the primary one is underestimated if the yield is low. For example, r is just higher than r0 0.04% if r1 is 5%. So the attraction of the CAT bond will be lost if the factors are considered simply.By comparing with current models, this model is characterized as: (1) Combined with engineering seismic risk assessment. The expected losses from future earthquakes are estimated by seismic hazard analysis and vulnerability estimation. The exceeding probability curve of loss rate is constructed to obtain the occurring probability of a catastrophe. The expected claim payment of insurers can be estimated by the assessment, if the earthquake insurance information is sufficient. (2) Geometric Brownian Motion and Jump-Diffusion process are adopted to describe the cash flow of insurance premium in complete and incomplete markets respectively, since it is irrational to describe catastrophic losses, insurers'loss or loss index etc by stochastic processes in current ones. (3) The complementarity between CAT bond and earthquake insurance is built to balance the capital of the issuer on earthquake risk. (4) A partial equilibrium circumstances among the cash flows of the issuer on earthquake risk is built in complete market. In incomplete market, a jump process is added in the circumstances. (5) According to the definition, the loss or occurrence of a catastrophe is not correlated with the operation of insurers directly, the moral hazard of this model can be avoided to a certain extent. The basis risk can be reduced also, since the loss of the issue in a catastrophe can be estimated by seismic risk assessment and the distribution of the policyholders.