Perturbation Approximations to the Price of Default Risk

Tibiletti (2006) uses the zero utility principle to develop insight into the market for default risk. Using general utility functions and a first-order Taylor series approximation to the price of default she develops a shortcut way for the default risk price for both the bargaining counter-parties. If higher order Taylor expansion is carried out, the results are not so simple, then we suggest a different approach, based on perturbation theory to generalize the results. Initially we assume that the probability of default is known, as Tibiletti did. The results for the seller’s price of default involve the same groupings of variables found in the original paper, but higher orders of risk aversion parameters also appear in higher order approximations. Then we relax the assumption so that uncertainties in both estimated probability of default as well as the uncertainty in the outcome of the transaction are taken into account. Thus we deal with the unconditional expected utility, including the sources of uncertainty in real problems, rather than the more limited conditional expectation which assumes that estimated probabilities of insolvency are known with no uncertainty. The results of the first part extend the findings of the original paper, those of the second part show that the solutions when estimates of default probabilities are uncertain differ substantially from those when the default probabilities are known.