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

Author

Listed:
  • 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))

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.

Suggested Citation

  • Satoshi Yamashita & Toshinao Yoshiba, 2010. "Analytical Solution for Expected Loss of a Collateralized Loan: A Square-root Intensity Process Negatively Correlated with Collateral Value," IMES Discussion Paper Series 10-E-10, Institute for Monetary and Economic Studies, Bank of Japan.
    • Handle: RePEc:ime:imedps:10-e-10
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    References listed on IDEAS

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    Cited by:

    1. Satoshi Yamashita & Toshinao Yoshiba, 2011. "Analytical Solution for the Loss Distribution of a Collateralized Loan under a Quadratic Gaussian Default Intensity Process," IMES Discussion Paper Series 11-E-20, Institute for Monetary and Economic Studies, Bank of Japan.
    2. Satoshi Yamashita & Toshinao Yoshiba, 2013. "A collateralized loan's loss under a quadratic Gaussian default intensity process," Quantitative Finance, Taylor & Francis Journals, vol. 13(12), pages 1935-1946, December.

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    More about this item

    Keywords

    stochastic recovery; default intensity model; affine diffusion; extended affine; survival probability; measure change;
    All these keywords.

    JEL classification:

    • G21 - Financial Economics - - Financial Institutions and Services - - - Banks; Other Depository Institutions; Micro Finance Institutions; Mortgages
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
    • G33 - Financial Economics - - Corporate Finance and Governance - - - Bankruptcy; Liquidation

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