A bridge from ruin theory to credit risk |
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Authors: | Cho-Jieh Chen Harry Panjer |
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Institution: | (1) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada;(2) Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada, N2L 3G1 |
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Abstract: | The structural model uses the firm-value process and the default threshold to obtain the implied credit spread. Merton’s (J
Finance 29:449–470, 1974) credit spread is reported too small compared to the observed market spread. Zhou (J Bank Finance
25:2015–2040, 2001) proposes a jump-diffusion firm-value process and obtains a credit spread that is closer to the observed
market spread. Going in a different direction, the reduced-form model uses the observed market credit spread to obtain the
probability of default and the mean recovery rate. We use a jump-diffusion firm-value process and the observed credit spread
to obtain the implied jump distribution. Therefore, the discrepancy in credit spreads between the structural model and the
reduced-form model can be removed. From the market credit spread, we obtain the implied probability of default and the mean
recovery rate. When the solvency-ratio process in credit risk and the surplus process in ruin theory both follow jump-diffusion
processes, we show a bridge between ruin theory and credit risk so that results developed in ruin theory can be used to develop
analogous results in credit risk. Specifically, when the jump is Logexponentially distributed, it results in a Beta distributed
recovery rate that is close to market experience. For bonds of multiple seniorities, we obtain closed-form solutions of the
mean and variance of the recovery rate. We prove that the defective renewal equation still holds, even if the jumps are possibly
negative. Therefore, we can use ruin theory as a methodology for assessing credit ratings.
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Keywords: | Credit risk Ruin theory Jump-diffusion process Structural model Reduced-form model |
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