Casualty Reinsurance Seminar, June 7th, 2004, Boston June 7, 2004 “Cat Bond Pricing Using Probability Transforms” published in Geneva Papers, 2004 Shaun Wang, Ph.D., FCAS

Shaun Wang, June 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

Shaun Wang, June 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.

Shaun Wang, June Example of Cat-bond transactions (Data Source: Lane Financial LLC)

Shaun Wang, June 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)

Shaun Wang, June 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

Shaun Wang, June 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?

Shaun Wang, June Ground-up Loss X has loss exceedence curve: S X (t) =1  F X (t) = Pr{ X>t }. Layer X(a, a+h);a=retention; h=limit Actuarial World

Shaun Wang, June Loss Exceedence Curve

Shaun Wang, June n Insurance prices by layer implies a transformed distribution –layer (t, t+dt) loss: S X (t) dt –layer (t, t+dt) price: S X *(t) dt –implied transform: S X (t)  S X *(t) Venter 1991 ASTIN Paper

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

Shaun Wang, June 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)

Shaun Wang, June 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).

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

Shaun Wang, June

Shaun Wang, June 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

Shaun Wang, June 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.

Shaun Wang, June 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:

Shaun Wang, June 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

Shaun Wang, June

Shaun Wang, June

Shaun Wang, June

Shaun Wang, June 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

Shaun Wang, June 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

Shaun Wang, June Cat-bond transactions (Data Source: Lane Financial LLC)

Shaun Wang, June Fit Wang transform to 1999 Cat bonds Date Sources: Lane Financial LLC Publications

Shaun Wang, June Use 1999 parameters to price 2000 Cat Bonds

Shaun Wang, June Corporate Bond Default: Historial versus Implied Default Frequency

Shaun Wang, June Fit 2-factor model to corporate bonds

Shaun Wang, June 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

Shaun Wang, June 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.

Shaun Wang, June Cat bond vs. Corporate Bond (after) During , 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.