Understanding Decreasing CDS Curves

Abstract

One of the major input for evaluating a Credit Default Swap (CDS) position is the so-called CDS curve. This curve gives the term structure of the CDS: for some maturities (typically: 1 year, 2 years, 3 years, 4 years, 5 years, 7 years and 10 years) a market spread is given. The spread is the premium to pay to a counterparty to protect one unit of currency during one year. Most of the time, CDS curves are increasing: the larger the maturity (i.e. the longer the time period we want to protect against a credit event), the larger the spread. As from the beginning of the Credit crisis (Summer 2007), some CDS curves were reverted, meaning that they contained decreasing parts: the spread (premium given on an annual basis) to pay for having protection for m1 years is larger than the spread to pay for having protection for m2 > m1 years. In some pathological cases, when the reversion is quite important, these reverted curves caused the blocking of some CDS pricers. The goal of this report is to understand these pricer blockings from a practical and quantitative perspective. Our analysis is based on a market-driven model for pricing CDS products, assuming a piecewise linear cumulative density function for the implied default probabilities. In spite of its simplicity, the model is quite general because only few assumptions are made, and provides manageable closed-form expressions for some interesting quantities. In this paper, the following key results will be derived: • Closed-form solutions are obtained for some important quantities, allowing a deep understanding of the eects of the underlying parameters; • It is shown that there is no theoretical objection to the existence of decreasing parts in CDS curves; • However, indeed, there exists a threshold on the intensity of these reversions : the CDS curves cannot contain parts being arbitrarily decreasing. If the CDS curve contains too strongly reverted parts than allowed (according to the above-mentioned threshold), then it is natural that the pricer fails to return a valid cumulative default probability curve; • Finally, a business meaning of this “reverting threshold” is given in terms of arbitrage opportunities. Note: This paper suits to anyone being interested in credit derivatives product, and does not require any specific prerequise.