You go to three different banks to borrow $10,000 for one year.Each says it will lend you the money at 10 percent,but their terms differ as follows: Bank A: Simple interest
Bank B: Add-on interest
Bank C: Discounted interest
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?
A) 13.0%
B) 9.5%
C) 9.0%
D) 8.5%
E) 8.0%

Question

Quizplus · Accepted Answer

For Bank A, we simply use the formula:

Interest = Principal x Rate x Time = $10,000 x 0.10 x 1 = $1,000

So the total amount due at the end of the year is $11,000.

For Bank B, we need to calculate the add-on interest. This means that the interest is added to the principal at the beginning, and then divided into 12 equal payments. So the total amount due is:

Total = Principal + Interest = $10,000 + ($10,000 x 0.10) = $11,000

Each of the 12 payments will then be:

$11,000 / 12 = $916.67

Using a financial calculator, we can solve for the effective annual rate (EAR) of this loan:

N = 12, PV = -$10,000, PMT = $916.67, FV = 0 
I/Y = 11.213%

For Bank C, we need to calculate the present value of the loan using the discount rate. This means that the interest is taken off the principal at the beginning. So the present value of the loan is:

PV = Principal / (1 + Rate x Time) = $10,000 / (1 + 0.10 x 1) = $9,090.91

The total amount due at the end of the year is still $10,000, but the interest has been taken off up front.

The effective annual rate (EAR) can be calculated using a financial calculator:

N = 1, PV = -$9,090.91, PMT = 0, FV = $10,000 
I/Y = 10.000%

So the highest EAR is from Bank B at 11.213% and the lowest EAR is from Bank C at 10.000%.

The difference between the two is:

11.213% - 10.000% = 1.213% = 12.13 basis points

However, the question is asking for the difference between the highest and lowest EAR, not the absolute difference. So the answer is:

Effectively, Bank A and Bank C are equally good choices, but Bank B should be avoided due to its higher EAR.

You Go to Three Different Banks to Borrow $10,000 for One