You go to three different banks to borrow R10,000 for one year.Each says it will lend you the money at 10 percent, but their terms differ as follows: Banks A and C require a single payment at the end of the year.Bank B requires 12 equal monthly payments beginning at the end of the first month.What is the difference between the highest and lowest effective annual rate in this case?

Question

You go to three different banks to borrow R10,000 for one year.Each says it will lend you the money at 10 percent, but their terms differ as follows:  Banks A and C require a single payment at the end of the year.Bank B requires 12 equal monthly payments beginning at the end of the first month.What is the difference between the highest and lowest effective annual rate in this case?

Quizplus · Accepted Answer

For Bank A and Bank C, the effective annual rate (EAR) is simply 10%, since there is only one payment at the end of the year.

For Bank B, we need to find the monthly interest rate which is 10%/12 = 0.8333%. Using the formula for the present value of an annuity, we can find the monthly payment:

PV = PMT * (1 - (1/ (1+i)^n)) / i

where PV is the present value of the loan (R10,000), PMT is the monthly payment, i is the monthly interest rate (0.8333%), and n is the number of periods (12). Solving for PMT, we get R870.94 per month.

To find the total amount paid over the year, we need to multiply the monthly payment by the number of months and subtract the initial loan amount:

Total amount paid = (R870.94/month * 12 months) - R10,000 = R4,531.28

The EAR for Bank B is therefore:

EAR = (1 + i)^n - 1 = (1 + 0.1/12)^12 - 1 = 10.47%

So the highest EAR is from Bank B at 10.47%, and the lowest EAR is from Banks A and C at 10%. The difference between the two is 0.47%, which is approximately 0.475 or 9.5%. Therefore, the best choice for the lowest difference in EAR is B.

You Go to Three Different Banks to Borrow R10,000 for One