Final answer:
To choose the best bank to lend money to based on the interest rates provided, we need to calculate the effective annual rate (EAR) for each compounding interval. The highest EAR is for the 5.9 percent compounded continuously, followed by monthly, quarterly, and then annually compounded rates, which makes the continuously compounded option the best choice.
Step-by-step explanation:
To determine which bank to lend money to, we need to compare the effective annual rates (EAR) for the different compounding intervals offered by the banks: 6.0 percent compounded annually, 5.9 percent compounded continuously, 5.9 percent compounded monthly, and 5.9 percent compounded quarterly. Let's calculate the EAR for each option:
- For 6.0 percent compounded annually, the EAR is simply the stated rate: 6.0 percent.
- For 5.9 percent compounded continuously, we use the formula EAR = er - 1, where r is the annual rate. So, EAR = e0.059 - 1, which is approximately 6.06 percent.
- For 5.9 percent compounded monthly, we use the formula EAR = (1 + r/n)n - 1, where r is the annual rate and n is the number of compounding periods per year. In this case, EAR = (1 + 0.059/12)12 - 1, which is approximately 6.04 percent.
- For 5.9 percent compounded quarterly, EAR = (1 + 0.059/4)4 - 1, which is approximately 6.01 percent.
Comparing all the effective annual rates, the highest EAR is compounded continuously at approximately 6.06 percent, followed by compounded monthly, then compounded quarterly, and lastly, compounded annually.
Therefore, if you are the lender looking to choose a bank based on these interest rates, you would choose the bank offering compounded continuously at 5.9 percent because it has the highest effective annual rate and would maximize your returns on the loan.