Final Answer:
When the deposit rate (_r_d_) decreases and the loan rate (_r_l_) increases, the bank can enjoy the smallest profit.
Step-by-step explanation:
In the banking industry, the profit margins of a bank are closely tied to the spread between the deposit rate (_r_d_) and the loan rate (_r_l_). The deposit rate is the interest paid by the bank to depositors, while the loan rate is the interest charged on loans provided to borrowers. The profit is maximized when the spread between these two rates is at its widest.
When the deposit rate (_r_d_) decreases, it means that the bank is paying less interest to depositors. Simultaneously, if the loan rate (_r_l_) increases, the bank can charge borrowers a higher interest rate on loans. This widening gap between the deposit and loan rates directly contributes to a larger profit margin for the bank.
To understand this concept mathematically, we can use the formula for the profit margin:
Profit Margin= Loan Rate - Deposit Rate/1 + Deposit Rate
As the deposit rate decreases and the loan rate increases, the numerator of this fraction becomes larger, while the denominator remains relatively constant. This results in a maximized profit margin, allowing the bank to enjoy the smallest profit.
In summary, a strategic combination of lowering the deposit rate and increasing the loan rate is key to optimizing a bank's profit margins, ensuring a more lucrative financial position.