Analytical Solution for Expected Loss of a Collateralized Loan: A Square-root Intensity Process Negatively Correlated with Collateral Value

Satoshi Yamashita and Toshinao Yoshiba
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Satoshi Yamashita: Associate Professor, The Institute of Statistical Mathematics (E-mail: yamasita@ism.ac.jp)
Toshinao Yoshiba: Director and Senior Economist, Institute for Monetary and Economic Studies, Bank of Japan (E-mail: toshinao.yoshiba@boj.or.jp)

No 10-E-10, IMES Discussion Paper Series from Institute for Monetary and Economic Studies, Bank of Japan

Abstract: In this study, we derive an explicit solution for the expected loss of a collateralized loan, focusing on the negative correlation between default intensity and collateral value. Three requirements for the default intensity and the collateral value are imposed. First, the default event can happen at any time until loan maturity according to an exogenous stochastic process of default intensity. Second, default intensity and collateral value are negatively correlated. Third, the default intensity and collateral value are non-negative. To develop an explicit solution, we propose a square-root process for default intensity and an affine diffusion process for collateral value. Given these settings, we derive an explicit solution for the integrand of the expected recovery value within an extended affine model. From the derived solution, we find the expected recovery value is given by a Stieltjes integral with a measure-changed survival probability.

Keywords: stochastic recovery; default intensity model; affine diffusion; extended affine; survival probability; measure change (search for similar items in EconPapers)
JEL-codes: G21 G32 G33 (search for similar items in EconPapers)
Date: 2010-06
New Economics Papers: this item is included in nep-ban, nep-fmk, nep-ore and nep-rmg
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Citations: View citations in EconPapers (2)

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