IDEAS home Printed from
   My bibliography  Save this paper

Analytical solutions for expected and unexpected losses with an additional loan


  • Satoshi Yamashita

    (The Institute of Statistical Mathematics (E-mail:

  • Toshinao Yoshiba

    (Institute for Monetary and Economic Studies, Bank of Japan (E-mail:


We evaluate expected and unexpected losses of a bank loan, taking into account the bank fs strategic control of the expected return on the loan. Assuming that the bank supplies an additional loan to minimize the expected loss of the total loan, we provide analytical formulations for expected and unexpected losses with bivariate normal distribution functions.There are two cases in which an additional loan decreases the expected loss: i) the asset/liability ratio of the firm is low but its expected growth rate is high; ii) the asset/liability ratio of the firm is high and the lending interest rate is high. With a given expected growth rate and given interest rates, the two cases are identified by two thresholds for the current asset/liability ratio. The bank maintains the current loan amount when the asset/liability ratio is between the two thresholds. Given the bank fs strategy, the bank decreases the initial expected loss of the loan. On the other hand, the bank has a greater risk of the unexpected loss.

Suggested Citation

  • Satoshi Yamashita & Toshinao Yoshiba, 2007. "Analytical solutions for expected and unexpected losses with an additional loan," IMES Discussion Paper Series 07-E-21, Institute for Monetary and Economic Studies, Bank of Japan.
  • Handle: RePEc:ime:imedps:07-e-21

    Download full text from publisher

    File URL:
    Download Restriction: no

    More about this item


    Probability of default (PD); Loss given default (LGD); Exposure at default (EaD); Expected loss (EL); Unexpected loss (UL); Stressed EL (SEL);

    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:07-e-21. 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.

    We have no references for this item. You can help adding them by using 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.