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

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Author Info

  • 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))

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    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.

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    Bibliographic Info

    Paper provided by Institute for Monetary and Economic Studies, Bank of Japan in its series IMES Discussion Paper Series with number 10-E-10.

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    Date of creation: Jun 2010
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    Handle: RePEc:ime:imedps:10-e-10

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    Related research

    Keywords: stochastic recovery; default intensity model; affine diffusion; extended affine; survival probability; measure change;

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    1. Long Chen & Pierre Collin-Dufresne & Robert S. Goldstein, 2009. "On the Relation Between the Credit Spread Puzzle and the Equity Premium Puzzle," Review of Financial Studies, Society for Financial Studies, vol. 22(9), pages 3367-3409, September.
    2. Xin Guo & Robert A. Jarrow & Yan Zeng, 2009. "Modeling The Recovery Rate In A Reduced Form Model," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 73-97.
    3. Hui Chen & Scott Joslin, 2012. "Generalized Transform Analysis of Affine Processes and Applications in Finance," Review of Financial Studies, Society for Financial Studies, vol. 25(7), pages 2225-2256.
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