Estimating with the Gaussian, Clayton, and t Copulas An Exploration of Bond Defaults and Value of a Collateralized Debt Obligation

The use of copula analysis to estimate the lifetime of bonds is gaining popularity in the financial sector. Taking this concept a step further, the following research project explores the use of the copula method in estimating the joint occurrence of bond default as a method of predictive analysis of bond lifetime. This is accomplished by means of an indepth discussion on copulas followed by a series of illustrative experiments and examples. In particular, this text covers not only the definition of three different copulas, but also an explanation of how these copulas can be used to estimate the lifetimes of bonds, predict the probability of default rate, and simulate a the value of collateralized debt obligation (CDO). The three copulas explored within this text the Gaussian copula, the Clayton copula, and the t-copula. Illustrative examples included 100,000 portfolios that were randomly generated each containing 10 bonds. After the lifetimes were calculated, a toy model was implemented to simulate a CDO with two traunches: a Senior traunche and a Equity traunche. The paper concludes with simulations that utilize probability values gleaned from historical data as provided by Joshua Coval, Jakub Jurek, and Erik Stafford in their work entitled “The Economics of Structured Finance.”