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The Dance on the Edge

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THE DANCE ON THE EDGE: A KALMAN-FILTERED MODEL OF CREDIT SPREADS
ANGELO CORELLI
ABSTRACT
The evidence on the relationship between observed spreads and their theoretical determinants is mixed. Using principal components analysis, Colin Dufresne et al. (2001) find that changes in individual bond spreads are driven by a single common systematic factor, unaccounted for by theoretical variables. A stream of recent literature demonstrates that credit risk accounts for a minor portion of spreads, with most variation due to alternative risk factors or a risk premium similar to that in the equity markets. A significant portion of spread levels has attributed to the positive difference between tax rates on corporate and treasury bonds.

Elton, Gruber, et al. (2001) find that expected loss accounts for less than 25% of the observed corporate bond spreads, with the remainder due to state taxes and factors commonly associated with the equity premium. Similarly, Delianedis and Geske (2001) attribute credit spreads to taxes, jumps, liquidity and market risk.
Liquidity risk itself has been found to be a positive function of the volatility of a firm assets and its leverage, the same variables that are seen as determinants of credit risk (Ericsson and Renault 2001). In Statistics and Economics, a filter is simply a term used to describe an algorithm that allows recursive estimation of unobserved, time varying parameters, or variables in the system. It is different from forecasting in that forecasts are made for future, whereas filtering obtains estimates of unobservables for the same time period as the information set. The Kalman Filter (KF) is a discrete, recursive linear filter, first developed for use in engineering applications and subsequently adopted by statisticians and econometricians.
The basic idea behind the filter is simple - to arrive at a conditional density function of the unobservables using Bayes’ Theorem, the functional form of relationship with observables, an equation of motion and assumptions regarding the distribution of error terms. The filter uses the current observation to predict the next period’s value of unobservable and then uses the realization next period to update that forecast. The Linear KF is optimal, i.e. Minimum Mean Squared Error estimator if the observed variable and the noise are jointly Gaussian. Else, it is best among the class of linear filters. This paper attempts to apply a KF to the well known multifactor Vasicek model for Credit spreads.In order to get the result the model has to be first expressed in state space form, so to allow the filter
to apply to the vector latent factors.

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