SECURITIZATION AND DERIVATIZATION OF CATASTROPHE RISK

SECURITIZATION AND DERIVATIZATION OF CATASTROPHE RISK

Chiu-Cheng Chang

Abstract

In this paper, we use financial engineering of securities and derivative instruments to convey insurance risk directly to investors in the capital markets. It is shown how a reinsurer could form a bridge between traditional reinsurance and catastrophe-linked bonds. An equation is developed for the price of a pure catastrophe bond, which puts both principal and interest at risk, in terms of the probability of occurrence of the insured catastrophic event. Also, it is shown that catastrophe call spread contracts are economically equivalent to excess of loss reinsurance contracts and it is argued that such call spreads are likely to be the most efficient vehicle for transferring catastrophic risk exposure to investors.

Key Words

Catastrophic risks, excess of loss reinsurance, call spreads, catastrophe bonds, insurance derivatives, catastrophe risk securitization, catastrophe risk derivatization

Only in growth, reform, and change, paradoxically enough, is true security to be found.
Anne Morrow Lindbergh

1. Introduction
   The emergence of Lloyd's underwriting losses from late 1980's to 1990's had significantly reduced the reinsurance capacity on a world-wide basis. Lloyd's financial difficulties were caused mainly from its exposures to the risk of asbestos in North America. Claims from asbestosis are generally of long-tail type and so Lloyd's problem had lingered for some years.
   During these years, major catastrophes worldwide had occurred with more regularity and concentration. The most notable are Hurricane Andrew in 1992, Northridge Earthquake in 1994 and the Great Hannshin Earthquake Disaster in 1995. These natural disasters had further reduced world-wide reinsurance capacity to a considerable degree. As a result, catastrophe reinsurance prices had shot up dramatically. These phenomena had reinforced the need to search for ways to increase world-wide reinsurance capacity.
   Although modern securities and their derivatives have been around for many years, their growth had become explosive in the recent past. It is only natural to look into modern security market for possible solutions to this problem of reinsurance capacity shortage. As a consequence, financial approach to managing catastrophe risk, such as catastrophe bonds and catastrophe options, was introduced. This recent financial innovation in managing insurance risk can be seen as a specific response to the problem of insurance and reinsurance capacity. The convergence of insurance and financial markets has thus emerged.
   The continuation of this convergence is bolstered by a clear upward revision of estimates of probable maximum losses from major catastrophes worldwide. Recent earthquakes in Kobe and Northridge, as well as events such as hurricane Andrew, have shifted estimates of maximum potential losses by an order of magnitude. Moreover, the emergence of modeling firms using large technical and financial data bases, has provided the insurance marketplace with credible estimates of single events that could overwhelm the insurance industry. For example, the U.S. industry faces the real possibility of a $50 to $100 billion loss through a major Midwest or Western earthquake or from a hurricane hitting Miami. As a further example, studies have shown that total economic losses from a major earthquake in the Tokyo area could reach US $ 2 trillion.
   It is clear that the global insurance and reinsurance industry capital that will actually be available to cover insured losses from such events is woefully inadequate. On the other hand, the global capital markets are vast, estimated in the range of US $ 13 trillion to well over US $ 15 trillion in terms of marketable securities. With the proper initial market conditions and the proper securities and derivative structures, we believe that educated investors can become involved as part of a comprehensive long-term solution to dealing with global catastrophe risk exposure.


2. Pricing Catastrophe Reinsurance
   In general, catastrophe reinsurance pricing is more uncertain than virtually any other insurance pricing. It is often difficult, if not impossible, to get meaningful and credible loss experience relevant to the coverage being evaluated. Moreover, both low claim frequency and high severity nature of catastrophe reinsurance coverages cause more uncertainties, ranging from the lengthy time delays between the occurrence, reporting and settlement of covered loss events, and also from the leveraged effect of inflation upon excess claims. It is generally true that, in pricing catastrophe reinsurance, the lower the expected loss frequency, the higher the variance of results relative to expectation, and thus the higher the risk level.
   Other severe problems for catastrophe reinsurance pricing are IBNR emergence and case reserve development. Development beyond ten years can be large, highly variant, and extremely difficult to evaluate. Concomitant is the increased uncertainty for asset and liability matching, because of the very long tail and extreme variability of the distribution of loss payments over time. Future predictability is further decreased by greater uncertainty affecting loss severity inflation above excess cover attachment points.
   All these factors create a situation where the variance and higher moments of the loss process and its estimation are much more important relative to the expected value than is the case for other coverages. Historically, there were many ways to price catastrophe reinsurance coverages. For any given situation, there was no one right way. The pricing formula a reinsurance actuary would use depends upon the reinsurer's pricing philosophy and information availability. The probability models were selected to describe the real situation as best as possible given all the real statistical and analytical cost constraints. Because of the convergence of insurance and financial markets, it has become fashionable recently to price catastrophe reinsurance in the setting of economic equilibrium theory under uncertainty [1], [2], [3].
   Following an NBER Conference on the financing of catastrophe risk that was held in November, 1996, Ken Froot of Harvard University has observed the following two facts: (1) the failure of the global catastrophe risk distribution system to spread the risks of major catastrophes and (2) the high costs associated with the consequent inefficient risk sharing. He then offers different explanations for barriers that prevent high layers of risk from being spread, many of which lead to high prices for catastrophe reinsurance. Three of these barriers come to mind: (1) Insufficiency of capital within the global reinsurance industry (2) Inefficiency (i.e. high costs of capital) of the corporate form for reinsurance (3) Presence of moral hazard and adverse selection at the insurer level.
   Froot points out that if these barriers bear on the real-world problem of inadequate and inefficient spreading of catastrophe risk, then involving the capital markets via securitization or derivatization of insurance risk will improve the situation. With respect to barrier (1), we need only note the vast size of the global capital markets. As to barrier (2), catastrophe-linked instruments might provide a lower cost way for insurers and reinsurers to manage catastrophe risk than raising large amounts of equity capital. Finally, as to barrier (3), the solution depends on the creation of sound indexes of industry losses on which securities and derivatives can be based. It is thus believed that securitization and derivatization of catastrophe risk will ultimately develop into a meaningful market. We will now examine the early stages of the evolution of this market.


3. Transforming Catastrophe Reinsurance Into A Security
   We will now show how to link reinsurance pricing to bond pricing. Building a bridge between reinsurance and securities is essential to understanding how the capital markets will view and assess insurance risks.
   
   3.1 Building A Bridge Between Reinsurance And Securities
   Let us consider a reinsurance contract under which an insurer pays a premium to a reinsurer at the beginning of each period during the lifetime of the contract. Assume the contract lasts for periods and each period is of length years. Thus we have the total of years for the contract. Under the contract, the reinsurer agrees to pay the insurer a fixed indemnity at the end of the period in which an insured event first occurs. The contract terminates once the payment of is made by the reinsurer or at the end of years if the insured event has not occurred by that time. Since the pricing of a reinsurance contract is generally quoted as a rate on line, the rate on line in this example is simply .
   To build a bridge between this reinsurance contract and a security, we assume that an investor makes a contribution to capitalize a reinsurer so that it can underwrite the reinsurance contract with the insurer. It is assumed that the investor's capital contribution is made at the same time as the reinsurer is created which is also the same time as the underwriting of the reinsurance contract. Suppose the yield on a U.S. Treasury bill of maturity years is a fixed rate of per period throughout the lifetime of the reinsurance contract. Then a unit amount invested in a -year U.S. Treasury bill will mature to . The reinsurance contract is said to be fully collateralized by U.S. Treasuries if . This is an amount exactly sufficient to make the contractual fixed indemnity payment of if the insured event has occurred during the period.
   On the other hand, if the insured event does not occur during a -year period, the investor must keep his capital contribution invested in the reinsurer. However, he can collect from the reinsurer an excess fund equal in amount to or . The insurer will make another premium payment at the beginning of the next -year period. Again, an amount can be invested in a -year U.S. Treasury bill yielding over the next -year period. The equation remains valid and the reinsurance contract therefore remains fully collateralized.
   This process of periodic insurer premium payments and periodic investor collections of excess funds is repeated until the insured event occurs or until the -year expiration of the contract is reached without the insured event having occurred. In the former case where the reinsurer makes the loss payment to the insurer at the end of the period in which the insured event first occurs, the reinsurer is then dissolved, the insurer makes no further premium payments , the investor receives no further periodic returns and the investor loses his entire capital contribution . In the latter case where no insured event has occurred, the return to the investor's initial contribution is the sum of (1) a payment at the end of every -year period and (2) the repayment of the initial investment at the end of the -year period.
   
   3.2 Catastrophe Bond
   As described in 3.1 above, the periodic premiums from the insurer, the initial capital of the reinsurer and the periodic interest earnings on these amounts are sufficient to pay a periodic return to the investor so long as the insured event does not occur, and to pay the insurer the fixed-indemnity amount if the insured event does occur. The investor receives a contingent stream of payments equivalent to the coupon stream and principal repayment of a bond. In other words, the unit bond has a maturity of -years, makes coupon payments every -years at a rate of per unit face amount of the bond and repays the unit face amount at maturity. Bond default occurs when the insured event occurs. The coupon payments and principal repayment are made only if the bond does not default. For obvious reasons, we call this catastrophe bond.
   Because of the bridge built between the catastrophe reinsurance contract and the catastrophe bond, the probability of occurrence of the insured event is equivalent to the probability of bond default. The higher the probability, the higher the price of the reinsurance contract (i.e. the higher the rate on line ), and the higher the coupon of the bond (i.e. the higher the value ). These relationships can be further analyzed. Let the probability of default of the catastrophe bond be per period. For simplicity, we assume is a constant for all periods. We are interested in determining the per-period coupon rate for the catastrophe bond so that its fair price is equal to its unit par value. We assume that the investor sells the bond at the end of the period in which default occurs and recovers a fraction of the sum of the coupon then due and the unit par value of the bond.
   The following table lists all the possible default and no-default payments occurring during the -period and their associated probabilities of occurrence:
   
   Period Default Payment Probability No-default Payment Probability
   1 f(1+r) q r 1-q
   2 f(1+r) q(1-q) r (1-q)2
   3 f(1+r) q(1-q)2 r (1-q)3
   . . . . .
   . . . . .
   . . . . .
   n f(1+r) q(1-q)n-1 1+r (1-q)n
   
   To find appropriate discount factors, we let be the per-period yield of an -period default -free U.S. Treasury bill that matures for a unit amount in exactly periods and has a price of today. Thus can be considered as the present value of a unit amount to be received periods from today. The fair price of the catastrophe bond is then equal to the expected present value of the streams of payments shown in the table above, using the discount factors for the discounting purpose.
   Fair Price of Bond =
   The first term on the right-hand side of the equation is the expected present value of the stream of coupons, the second term is the expected present value of the principal repayment at maturity and the third term is the expected present value of the salvage recovered on default of the bond. Given the per-period probability of default , one can solve the bond-pricing equation for the per-period coupon rate that renders the fair price of the bond equal to its unit par value.
   
   3.3 Aggregate Excess of Loss Reinsurance Contracts
   Although reinsurance contracts with fixed-indemnity payments on the occurrence of an insured event do exist, they are not as useful to direct writers as are aggregate excess of loss contracts with maximum indemnity limits. It is not difficult to modify the analysis done in 3.1 and 3.2 to accommodate the additional features of aggregate excess of loss contracts. As long as the cumulative losses during the lifetime of an aggregate excess of loss contract are less than the lower level of the loss layer, the investor receives the coupon payments on his holding of catastrophe bonds. When cumulative losses reach the attachment point, the reinsurer is liable to make loss payment to the insurer under the reinsurance contract. This causes a full default on a portion of the investor's bond holdings. As incurred losses increase within the loss layer, more and more of the investor's catastrophe bonds default. When cumulative losses reach the upper level of the loss layer, all of the remaining bonds default.
   
4. Catastrophe Option
   We note first that an aggregate excess of loss reinsurance contract is basically an option. At inception of the contract, the insurer pays the reinsurer a premium in exchange for which the reinsurer must pay losses in excess of a specified attachment point . This is economically equivalent to the purchase of a catastrophe call option by the insurer from the reinsurer. The option premium is and the strike price of the option is .
   Most aggregate excess of loss reinsurance contracts carry only limited liability, i.e. they apply to a layer of risk. Such an arrangement can be considered as the superposition of two contracts: under the first contract, the insurer pays the reinsurer a premium to cover all losses in excess of incurred during a stated period, while, under the second contract, the reinsurer pays the insurer a premium to cover all losses in excess of incurred during the same stated period, where is greater than . The reinsurance price for the catastrophe cover, expressed as a rate on line, is . In this case, the insurer has purchased a call spread from the reinsurer. In other words, the insurer has purchased for premium a catastrophe call option with strike price and has written for premium a catastrophe call option with strike price .
   It seems clear that the most natural way to structure a risk transfer program directly between an insurer and an investor is via a catastrophe call spread contract. This is what the CBOT concluded in early 1990s and that is how the recently-revised CBOT contracts have been constructed [6]. In the case of the CBOT contracts, the strike prices are specific dollar levels of insurance industry losses arising from catastrophes as defined and measured by Property Claim Services. Instead of using industry-wide losses, one can apply the same concept to create call spread contracts that are based on an individual insurer's actual losses. These customized call spread contracts will have strike prices or expiration dates that differ from the standardized CBOT contracts.
   
5. Conclusion
   In this paper, we started by describing the background that leads to the convergence of insurance and capital markets. After reviewing traditional pricing practice for catastrophe reinsurance, we further point out the feasibility and potential advantages of using financial engineering of securities and derivative instruments to convey insurance risk directly to investors in the capital markets. It is shown how a reinsurer could form a bridge between traditional reinsurance and catastrophe-linked bonds. An equation is developed for the price of a pure catastrophe bond, which puts both principal and interest at risk, in terms of the probability of occurrence of the insured catastrophic event. Also, it is shown that catastrophe call spread contracts are economically equivalent to excess of loss reinsurance contracts and it is argued that such call spreads are likely to be the most efficient vehicle for transferring catastrophic risk exposure to investors.
   

References
[1] "Pricing Catastrophe Insurance Futures and Call Spreads: An Arbitrage Approach," J. David Cummins and Helyette Geman, The Journal of Fixed Income 4, No.4, (March 1995).
[2] "Pricing Excess-of-Loss Reinsurance Contracts Against Catastrophic Loss," J. David Cummins, Christopher Lewis, and Richard Phillips, NBER Conference of The Financing of Property/Casualty Risks (November 1996).
[3] "An Equilibrium Model of Catastrophe Insurance Futures and Spreads," Knut K. Aase, Norwegian School of Economics and Business Administration working paper (November 1996)
[4] "The Limited Financing of Catastrophe Risk: A Partial Diagnosis," Kenneth A. Froot, Harvard University and National Bureau of Economic Research working paper (January 1997)
[5] "Plan of Finance I-Issuance of Taxable Securities," California Earthquake Authority document (January 1996).
[6] "PCS Catastrophe Insurance Options - A User's Guide," Chicago Board of Trade (September 1995)