“ Cat Bond Pricing Using Probability Transforms ” published in Geneva Papers, 2004

1. “Cat Bond Pricing Using Probability Transforms”published in Geneva Papers, 2004 Shaun Wang, Ph.D., FCAS

2. What is CAT bond? • A high-yield debt instrument: if the issuer (insurance company) suffers a loss from a particular predefined catastrophe, then the issuer's obligation to pay interest and/or repay the principal is either deferred or forgiven. • Covered events: CA Earthquake, Japan Earthquake, FL Hurricane, EU Winter Storm; Multi-Peril & Multi-territory • Actual-dollar trigger or Reference-index trigger

3. Why CAT bond? • For bond issuers: • Alternative source of capital/capacity for insurance companies with large risk transfer needs • Not subject to the risk of non-collectible reinsurance • For investors: • High yield coupon rate • CAT bond performance is not closely correlated with the stock market or economic conditions.

4. Example of Cat-bond transactions(Data Source: Lane Financial LLC)

5. State of the Cat-bond Market • In the past, unfamiliar class of assets to investors, led to limited number of transactions • Phenomenal performance of CAT bond portfolios, led to recent surge of interest by institutional investors • Cat Bond Market Grew 42% in 2003 • Total bond issuance $1.73 billion • Reduced cost of issuing (coupon interest and transaction costs)

6. Cat-bond offers a laboratory for reconciliation of pricing approaches • Capital market pricing is forward-looking: • prices incorporate all available information • No-arbitrage pricing (Black-Scholes Theory) • Actuarial pricing is back-forward looking • Using historical data to project future costs • Explicit adjustments for risk

7. Financial World • Black-Schole-Merton theory for pricing options and corporate credit default risks • A common measure for fund performance is the Sharpe ratio:  ={ E[R] r }/[R], the excess return per unit of volatility •  also called “market price of risk” • How can we relate it to actuarial pricing?

8. Actuarial World • Ground-up Loss X has loss exceedence curve: • SX(t) =1 FX(t)= Pr{ X>t }. • Layer X(a, a+h); a=retention; h=limit

9. Loss Exceedence Curve

10. Venter 1991 ASTIN Paper • Insurance prices by layer implies a transformed distribution • layer (t, t+dt) loss: SX(t) dt • layer (t, t+dt) price: SX*(t) dt • implied transform: SX(t)  SX*(t)

11. Insight of Gary Venter (91 ASTIN ): “Insurance prices by layer imply a transformed distribution” S(x)=1F(x), or Loss Exceedence Curve

12. Attempt #1 by Morton Lane (Hachemeister Prize Paper) • Morton Lane (2001) “Pricing of Risk Transfer transactions” proposed a 3-parameter model: • EER = 0.55 (PFL)0.49 (CEL)0.57 • PFL: Probability of First Loss • CEL: Conditional Expected Loss (as % of principal) • EER: Expected Excess Return (over LIBOR)

13. Attempt #2: Wang Transform(Sharing 2004 Ferguson Prize with Venter) • Let  be standard normal distribution: (1.645)=0.05, (0)=0.5, (1.645)=0.95 • Wang introduces a new transform: • F(x)=0.95, =0.3,  F*(x) = 1(1.6450.3) =0.91 • Fair Price is derived from the expected value under the transformed distribution F*(x).

14. WT extends the Sharpe Ratio Concept • If FX is normal(), FX* is normal(+ ): E*[X] = E[X] +  [X] • If FX is lognormal( ), FX* is lognormal(+ ) • The transform recovers CAPM & Black-Scholes (ref. Wang, JRI 2000) •  extends the Sharpe ratio to skewed distributions

16. Unified Treatment of Asset / Loss • The gain X for one party is the loss for the counter party: Y = X • We should use opposite signs of , and we get the same price for both sides of the transaction

17. Baseline Sampling Theory • We have m observations from normal(,2). Not knowing the true parameters, we have to estimate  and  by sample mean & variance. • When assessing the probability of future outcomes, we effectively need to use Student-t with k=m-2 degrees-of-freedom.

18. Adjust for Parameter Uncertainty • Baseline: For normal distributions, Student-t properly reflects the parameter uncertainty • Generalization: For arbitrary F(x), we propose the following adjustment for parameter uncertainty:

19. A Two-Factor Model • Wang transform with adjustment for parameter uncertainty: • where •  is standard normal CDF, and • Q is Student-t CDF with k degrees-of-freedom

23. Insights for the second factor • Explains investor behavior: greed and fear • Investors desire large gains (internet lottery) • Investors fear large losses (market crash) • Consistent with “volatility smile” in option prices • Quantifies increased parameter uncertainty in the tails

24. Empirical Studies • 16 CAT-bond transactions in 1999 • Fit well to the 2-factor Wang transform • Better fit than Morton Lane’s 3-parameter model (in his 2001 Hachmeister Prize Paper) • 12 CAT bond transactions in 2000 • Use 1999 estimated parameters to price 2000 transactions, remain to be the best-fit

25. 1999 Cat-bond transactions(Data Source: Lane Financial LLC)

26. Fit Wang transform to 1999 Cat bondsDate Sources: Lane Financial LLC Publications

27. Use 1999 parameters to price 2000 Cat Bonds

28. Corporate Bond Default:Historial versus Implied Default Frequency

29. Fit 2-factor model to corporate bonds

30. Risk Premium for Corporate Bonds • Use 2-factor Wang transform to fit historical default probability & yield spread by bond rating classes • Compare the fitted parameters for “corporate bond” versus “CAT-bond” •  parameters are similar, • “CAT-bond” has lower Student-t degrees-of-freedom, • In 1999, CAT-bond offered more attractive returns for the risk than corporate bonds

31. Cat bond vs. Corporate Bond (before) • Before Sept. 11 of 2001 fund managers were less familiar (or comfortable) with the cat bond asset class. • Fund managers were reluctant to expose themselves to potential career risks, since they would have difficulties in explaining losses from investing in cat bonds, instead of conventional corporate bonds. • At that time, because of investors’ weak appetite for cat bonds, cat bonds issuers had to offer more attractive yields than corporate bonds with comparable default frequency & severity.

32. Cat bond vs. Corporate Bond (after) • During 2002-3, fund managers' interest in cat bond has growth significantly, due to superior performance of the cat bond class. They now complaint about not having enough cat bond issues to feed their risk appetite. • In the same time period, the perceived credit risk of corporate bonds increased, in tandem with the general broader market. Investors began to value more the benefit of low correlation between cat bond and other asset classes. • It has been reported that the yields spreads on cat bonds have tightened while the yields spreads on corporate bonds have widened (cross over) – Polyn April 2003.