Economic Catastrophe Bonds

In this paper, the authors, Josh Coval, Jakob Jurek and Erik Stafford, analyze the pricing of Collateralized Debt Obligation (CDO) senior tranches. They show that the spreads paid on these tranches do not sufficiently compensate investors for the risk of economic catastrophe. By focussing narrow the expected default component of the tranche credit spreads, rating agencies and investors ignore the systematic risk exposure of these tranches.

The authors combine the Merton credit default model and the CAPM to price credit portfolios and portfolio tranches. In the Merton framework, default of a firm occurs when the firm asset value (driven by a market factor and an idiosyncratic factor) falls below the face value of its debt. The authors can then calculate the default probabilities and the expected losses at the single firm level but also at the portfolio level.They first focus on the pricing of digital tranches that pays 1 if the portfolio losses are less than a given level (the attachment point) and 0 otherwise. They can price any tranches using these digital ones.

They manage to calculate closed-form valuation formula when the number of assets is very large (with the Vasicek formula). They find that the tranche spread is increasing as the number of assets rises because the exposure to systematic risk is larger.To price the tranches, the authors use the state price densities estimated from European equity index options (5-year options on the S&P500 since they are interested in the 5-year CDX portfolio). It is well known that the risk neutral probability is equal to the discounted value of the second order derivative of a call option with respect to the strike price.

They check the validity of their calibration by examining the predicted changes in spreads and the coverage ratios (ratio of risk neutral default intensity over the actual default intensity).

They later find that the more senior tranches provide with the highest risk premium.They then price the CDX trances by simulation of the market and point out that the payoff of a tranche is very similar to the payoff of a put spread (purchase of a put and writing of another put with a higher strike price) on the market.

They calculate the price of a put spread that should replicate the index tranche in combination with a risk less position in a bond. The replicating put portfolio pays a higher yield than the equivalent tranche.

Reference:

Coval, Joshua D., Jurek, Jakub W. and Stafford, Erik, "Economic Catastrophe Bonds" (July 2007). HBS Finance Working Paper No. 07-102 Available at SSRN: __http://ssrn.com/abstract=995249__

See also {ln:nw:An Empirical Analysis of the Pricing of Collateralized Debt Obligations} and {ln:nw:Measuring Default Risk Premia from Default Swap Rates and EDFs}