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 Effective Annual Rate (EAR), we need to consider both the APR (Annual Percentage Rate) and the loan origination fee.
Step 1: Calculate the interest charged over four months based on the APR.
To do this, we need to find the simple interest for four months using the formula:
Interest = Principal x Rate x Time
In this case, the principal amount is $2 million, the rate is 9% (APR), and the time is four months.
Interest = $2,000,000 x 0.09 x (4/12) (since the time is given in months)
Step 2: Calculate the loan origination fee.
The loan origination fee is 1.5% of the loan amount, which is $2 million.
Loan Origination Fee = $2,000,000 x 0.015
Step 3: Calculate the effective loan amount.
To find the effective loan amount, we subtract the loan origination fee from the principal amount.
Effective Loan Amount = Principal - Loan Origination Fee
= $2,000,000 - ($2,000,000 x 0.015)
Step 4: Calculate the EAR (Effective Annual Rate).
The EAR is the annualized interest rate that takes into account compounding over the year. We can convert the four-month interest payment into an EAR using the formula:
EAR = (1 + Periodic Rate)^n - 1
where the Periodic Rate is the interest rate over the period (four months).
Periodic Rate = (Interest + Loan Origination Fee) / Effective Loan Amount
EAR = (1 + Periodic Rate)^n - 1
where n represents the number of compounding periods in a year.
In this case, since the loan is for four months (1/3 of a year), we have n = 3.
Step 5: Plug in the values and calculate the EAR.
Periodic Rate = (4-month interest + Loan Origination Fee) / Effective Loan Amount
= (Interest + Loan Origination Fee) / Effective Loan Amount
EAR = (1 + Periodic Rate)^3 - 1
Using the calculated values from earlier, plug them into the formulas to find the actual four-month interest rate paid, expressed as an EAR.