Analytical Solutions for Expected Loss and Standard Deviation of Loss with an Additional Loan

We evaluate the expected loss and the standard deviation of loss of a bank loan, considering 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 the expected loss and the variance of loss with bivariate normal distribution functions. Using a given expected growth rate and interest rates, two thresholds for the asset/liability ratio can specify three cases. The bank supplies an additional loan to decrease the expected loss in two cases: (i) the asset/liability ratio of the firm is low, and its expected growth rate is high; and (ii) the asset/liability ratio of the firm is high, and the lending interest rate is high. The bank maintains the current loan amount when the asset/liability ratio lies between the two thresholds. Depending on the bank’s strategy, the bank can decrease the initial expected loss of the loan. Conversely, the bank would face a greater risk of the standard deviation of loss.