IDEAS home Printed from
   My bibliography  Save this paper

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


  • Satoshi Yamashita

    (Professor, The Institute of Statistical Mathematics (E-mail:

  • Toshinao Yoshiba

    (Director and Senior Economist, Institute for Monetary and Economic Studies, (currently Financial System and Bank Examination Department), Bank of Japan (E-mail:


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.

Suggested Citation

  • 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.
  • Handle: RePEc:ime:imedps:11-e-20

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Dong-Hyun Ahn & Robert F. Dittmar, 2002. "Quadratic Term Structure Models: Theory and Evidence," Review of Financial Studies, Society for Financial Studies, vol. 15(1), pages 243-288, March.
    2. Markus Leippold & Liuren Wu, 2003. "Design and Estimation of Quadratic Term Structure Models," Review of Finance, European Finance Association, vol. 7(1), pages 47-73.
    3. Markus Leippold & Liuren Wu, 2002. "Design and Estimation of Quadratic Term Structure Models," Finance 0207014, EconWPA.
    4. Li Chen & H. Vincent Poor, 2003. "Markovian Quadratic Term Structure Models For Risk-free And Defaultable Rates," Finance 0303008, EconWPA.
    Full references (including those not matched with items on IDEAS)

    More about this item


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

    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

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:ime:imedps:11-e-20. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Kinken). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.