Crimini Foods is offered a $400,000 line of credit for six months at an APR of 10%. This loan has a loan origination fee of 2%. What is the actual six-month interest rate paid, expressed as an EAR
To calculate the actual six-month interest rate paid, expressed as an Effective Annual Rate (EAR), we need to consider both the Annual Percentage Rate (APR) and the loan origination fee.
Step 1: Calculate the total interest paid over the six-month period.
The APR of 10% represents the interest rate for the full year.
So, the interest paid over six months can be calculated as:
Total Interest = Principal × Interest Rate × Time
Total Interest = $400,000 × 10% × (6/12)
Total Interest = $400,000 × 0.10 × 0.5
Total Interest = $20,000
Step 2: Calculate the loan origination fee.
The loan origination fee is 2% of the line of credit ($400,000).
Loan Origination Fee = $400,000 × 2%
Loan Origination Fee = $400,000 × 0.02
Loan Origination Fee = $8,000
Step 3: Adjust the total interest paid by subtracting the loan origination fee.
Adjusted Total Interest = Total Interest - Loan Origination Fee
Adjusted Total Interest = $20,000 - $8,000
Adjusted Total Interest = $12,000
Step 4: Calculate the actual six-month interest rate paid (EAR).
The EAR considers compounding, and since this loan is for six months, we need to account for compounding semi-annually (twice in a year).
EAR = (1 + (APR / m))^m - 1
Where m is the number of compounding periods in a year.
In this case, we have semi-annual compounding, so m = 2.
EAR = (1 + (0.10 / 2))^2 - 1
EAR = (1 + 0.05)^2 - 1
EAR = (1.05)^2 - 1
EAR = 1.1025 - 1
EAR = 0.1025 or 10.25%
Therefore, the actual six-month interest rate paid (EAR) is 10.25%.