On CAT Options and Bonds Hanspeter Schmidli University of Cologne Advanced Modeling in Finance and Insurance Linz, 24th of September 2008
Introduction Models Pricing and Hedging Introduction 1 Why are Catastrophes Dangerous? CAT Products CAT Products from the Investor’s Point of View Models 2 Models for the CAT Futures Models for the PCS Option A Model Based on Individual Indices Pricing and Hedging the PCS Option 3 Pricing the PCS Option Hedging the PCS Option Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? Catastrophic Claims: In Mil. USD Event Date Claim Hurricane Katrina 25.08.05 68 515 Hurricane Andrew 23.08.92 23 654 Terrorist attacks in US 11.09.01 21 999 Northridge Earthquake 17.01.94 19 593 Hurricane Ivan 02.09.04 14 115 Hurricane Wilma 19.10.05 13 339 Hurricane Rita 20.09.05 10 704 Hurricane Charley 11.08.04 8 840 Typhoon Mireille 27.09.91 8 599 Hurricane Hugo 15.09.89 7 650 Storm Daria 25.01.90 7 413 Storm Lothar 25.12.99 7 223 Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? Catastrophic Claims (80 Events) 35 30 25 20 15 10 5 10 20 30 40 50 60 70 Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? Empirical Mean Residual Life 40000 30000 20000 10000 5000 10000 15000 20000 Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? The Hill Plot 20 40 60 80 0.9 0.8 0.7 0.6 0.5 Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? The Index of Regular Variation Estimation of the index of regular variation α = 1 . 14027. Variance does not exist! Mean value of the estimated distribution: 8 441 Mil Empirical mean 4 653 Mil. (-45%) Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? New Record Mean value of a new record: 556 968. (8.1 times Hurricane Katrina, 23.5 times Hurricane Andrew) Reinsurance? Risk is too large for the insurance industry. Financial market could bear risk without problems. Need for new financial products that take over the rˆ ole of reinsurance. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? Largest Possible Claim — Financial Market Largest possible claim Daily standard deviation Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging Why are Catastrophes Dangerous? The Use of CAT Products Insurers use the CAT product as a substitute for reinsurance. Insurance claims are supposed to be nearly independent of the financial market. Kobe earthquake? 9/11? Counterparty risk is reduced because the credit risk is spread amongst investors. Investors can use CAT products for diversification. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The ISO Index The underlying index for the CAT future is the ISO (Insurance Service Office, a statistical agent) index. About 100 American insurance companies report property loss data to the ISO. ISO then selects for each of the used indices a pool of at least ten of these companies on the basis of size, diversity of business, and quality of reported data. The ISO index is then I t = reported incurred losses . earned premiums The pool is known at the beginning of the trading period. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The CAT Futures � I T 1 � F T 1 = min Π , 2 × 25 000$ Π: Premium volume I : ISO index Apr May Jun Jul Aug Sep Oct Nov Dec Jan ✻ ✻ Event Quarter Interim Report Final Settlement Reporting Period Hurricane Andrew: I = 1 . 7Π. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The CAT Futures No success because poorly designed. Reasons: Index only announced twice Information asymmetry Lack of realistic models Moral hazard problem Index may not match losses (slow reporting) The insurer cannot define a layer for which the protection holds. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The PCS Indices A PCS index is an estimate of (insurance) losses in 100 Mio$ occuring from catastrophes in a certain region in a certain period. There are 9 indices: National Eastern Northeastern Southeastern Midwestern Western California Florida Texas Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The PCS Option A PCS option is a spread traded on a PCS index F T 2 = min { max { L T 2 − A , 0 } , K − A } = max { L T 2 − A , 0 } − max { L T 2 − K , 0 } . L T 2 is PCS’s estimate at time T 2 of the losses from catastrophes occurring in (0 , T 1 ] in a certain region. The occurrence period (0 , T 1 ] is 3,6 or 12 months. The development period ( T 1 , T 2 ] is 6 or 12 months. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The PCS Option The value of a basis point is $200. When a catastrophe occurs, PCS makes a first estimate and then continues to reestimate the claim. The option expires after a development period of at least six months following the occurrence period. The index is announced daily which simplifies trading. Moreover, there is no information asymmetry. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products How Does the PCS Option Work An insurer chooses first the layer. Then he estimates the market share and its loss experience compared to the whole market. From that the strike values and the number of spreads is calculated. In this way one gets the desired reinsurance if the estimates coincide with the incurred liabilities. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products How Does the PCS Option Work Example An insurer wants to hedge catastrophes. The layer should be 6 Mio in excess of 4 Mio, i.e. with strike values 4 Mio and 10 Mio. He estimates the market share to 0.2%. He estimates his exposure to 80% of the industry in average. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products How Does the PCS Option Work Example (cont) Lower strike: 0 . 002 × 1 1 4 × 0 . 8 = 2500 = 25 pt . Upper strike: 0 . 002 × 1 1 10 × 0 . 8 = 6250 = 62 . 5 pt . Strikes are only available at 5 pt intervals, thus 25/65 are chosen as strike prices (10 . 4 Mio upper strike value). Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products How Does the PCS Option Work Example (cont) The number of spreads needed is 6 000 000 200(65 − 25) = 750 . Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The Act-of-God Bond A bond with coupons, usually at a high rate. The coupons and/or principal are at risk, i.e. if a well-specified event occurs the coupon(s)/principal will not be paid (back). Possible variant, principal will be paid back with a delay. For the insurer, the coupons (and the principal) serve as a sort of reinsurance payment. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The Act-of-God Bond: An Example Riskless interest rate: 2% Coupon rate: 4% Probability of the event: 5% Price of the Act-of-God bond: 1 1 . 02(0 . 95 × 104 + 0 . 05 × 0) = 96 . 86 . Price of a riskless bond with coupon rate 4%: 1 1 . 02104 = 101 . 96 . Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The Act-of-God Bond: An Example ✟ ✯ 104 ✟ ✯ 104 ✯ ✟ 0 ✟✟✟✟✟ ✟✟✟✟✟ ✟✟✟✟✟ ❍❍❍❍❍ ❍❍❍❍❍ ❍❍❍❍❍ 96.86 101.96 5.1 ❥ ❍ ❥ ❍ ❍ ❥ 0 104 104 Hanspeter Schmidli University of Cologne On CAT Options and Bonds
Introduction Models Pricing and Hedging CAT Products The Act-of-God Bond: An Example Suppose the principal is (in case of the event) paid back in ten years, but the coupon is not valid. The price becomes then 0 . 95 1 1 1 . 02104 + 0 . 05 (1 . 02) 10 100 = 100 . 96 . That is the insurer gets a reinsurance of 4 and an interest-free loan of 100 in the case of the event. The price of this agreement is 1. Hanspeter Schmidli University of Cologne On CAT Options and Bonds
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