Discussion Paper Series
2007-E-21

# Analytical solutions for expected and unexpected losses with an additional loan

## Satoshi Yamashita, Toshinao Yoshiba

We evaluate expected and unexpected losses of a bank loan, taking into account the bank's 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's 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.

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

Views expressed in the paper are those of the authors and do not necessarily reflect those of the Bank of Japan or Institute for Monetary and Economic Studies.

Copyright © 2007 Bank of Japan All Rights Reserved.

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