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Analytical Solution for the Loss Distribution of a Collateralized Loan under a Quadratic Gaussian Default Intensity Process

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Author Info

  • Satoshi Yamashita

    (Professor, The Institute of Statistical Mathematics (E-mail: yamasita@ism.ac.jp))

  • Toshinao Yoshiba

    (Director and Senior Economist, Institute for Monetary and Economic Studies, (currently Financial System and Bank Examination Department), Bank of Japan (E-mail: toshinao.yoshiba@boj.or.jp))

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    Abstract

    In this study, we derive an analytical solution for expected loss and the higher moment of the discounted loss distribution for a collateralized loan. To ensure nonnegative values for intensity and interest rate, we assume a quadratic Gaussian process for default intensity and discount interest rate. Correlations among default intensity, discount interest rate, and collateral value are represented by correlations among Brownian motions driving the movement of the Gaussian state variables. Given these assumptions, the expected loss or the m-th moment of the loss distribution is obtained by a time integral of an exponential quadratic form of the state variables. The coefficients of the form are derived by solving ordinary differential equations. In particular, with no correlation between default intensity and discount interest rate, the coefficients have explicit closed form solutions. We show numerical examples to analyze the effects of the correlation between default intensity and collateral value on expected loss and the standard deviation of the loss distribution.

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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 11-E-20.

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    Date of creation: Sep 2011
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    Handle: RePEc:ime:imedps:11-e-20

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    Keywords: default intensity; stochastic recovery; quadratic Gaussian; expected loss; measure change;

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    1. Li Chen & H. Vincent Poor, 2003. "Markovian Quadratic Term Structure Models For Risk-free And Defaultable Rates," Finance, EconWPA 0303008, EconWPA.
    2. Markus Leippold & Liuren Wu, 2002. "Design and Estimation of Quadratic Term Structure Models," Finance, EconWPA 0207014, EconWPA.
    3. Dong-Hyun Ahn & Robert F. Dittmar, 2002. "Quadratic Term Structure Models: Theory and Evidence," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 15(1), pages 243-288, March.
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