Gemini Real Estate is offered a $2 million line of credit for four months at an APR of 9%. This loan has a loan origination fee of 1.5%. What is the actual four-month interest rate paid, expressed as an EAR

To calculate the actual four-month interest rate paid, expressed as an EAR (Effective Annual Rate), we need to consider both the APR (Annual Percentage Rate) and the loan origination fee.

Step 1: Calculate the monthly interest rate
The APR is given as 9%, so we divide it by 12 to get the monthly interest rate:
Monthly interest rate = APR / 12 = 9% / 12 = 0.09 / 12 = 0.0075 (or 0.75%)

Step 2: Calculate the loan origination fee
The loan origination fee is 1.5% of the $2 million line of credit:
Loan origination fee = 1.5% * $2,000,000 = 0.015 * $2,000,000 = $30,000

Step 3: Calculate the amount borrowed
The loan origination fee is subtracted from the line of credit amount to determine the actual amount borrowed:
Amount borrowed = $2,000,000 - $30,000 = $1,970,000

Step 4: Calculate the interest paid over four months
The monthly interest rate is used to calculate the interest paid over four months:
Interest paid = Amount borrowed * Monthly interest rate * 4 months
Interest paid = $1,970,000 * 0.0075 * 4 = $59,100

Step 5: Calculate the actual four-month interest rate paid (EAR)
The EAR takes into account the effect of compounding over a year, so we calculate it using the formula:
EAR = (1 + Monthly interest rate)^12 - 1

EAR = (1 + 0.0075)^12 - 1 ≈ 0.0916 (or 9.16%)

Therefore, the actual four-month interest rate paid, expressed as an EAR, is approximately 9.16%.

To find the actual four-month interest rate paid, expressed as an Effective Annual Rate (EAR), we need to take into account both the stated Annual Percentage Rate (APR) and the loan origination fee.

Step 1: Convert the APR to a monthly interest rate.
Since the APR is given as 9%, we need to divide it by 12 to get the monthly interest rate.
Monthly interest rate = APR / 12 = 9% / 12 = 0.75%.

Step 2: Calculate the interest expense for four months.
Multiply the line of credit amount ($2 million) by the monthly interest rate, and then multiply it by the number of months (4) to calculate the interest expense.
Interest expense = $2,000,000 * 0.75% * 4 = $60,000.

Step 3: Calculate the loan origination fee.
The loan origination fee is given as 1.5% of the line of credit amount.
Loan origination fee = $2,000,000 * 1.5% = $30,000.

Step 4: Calculate the actual amount received.
The actual amount received is calculated by subtracting the loan origination fee from the line of credit amount.
Actual amount received = $2,000,000 - $30,000 = $1,970,000.

Step 5: Calculate the actual interest rate paid, expressed as an EAR.
Divide the total interest expense ($60,000) by the actual amount received ($1,970,000) to calculate the actual interest rate paid.
Actual interest rate paid = $60,000 / $1,970,000 = 0.0305 or 3.05%.

Therefore, the actual four-month interest rate paid, expressed as an EAR, is 3.05%.