You have just purchased a new warehouse. To finance the purchase, you’ve arranged for a 30-year mortgage loan for 70 percent of the $3,300,000 purchase price. The monthly payment on this loan will be $16,300.

What is the APR on this loan?

Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

What is the EAR on this loan?

Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

To find the APR (Annual Percentage Rate) and the EAR (Effective Annual Rate) for this loan, let's go through the following steps:

### Step 1: Find the Loan Amount
The loan covers 70% of the purchase price:
\[
\text{Loan Amount} = 0.70 \times 3,300,000 = 2,310,000
\]

### Step 2: Monthly Interest Rate Calculation
Since the monthly payment (\$16,300) is known, we use the formula for the monthly mortgage payment of an amortizing loan:
\[
P = \frac{r \times PV}{1 - (1 + r)^{-n}}
\]
Where:
- $ P $ = monthly payment = 16,300
- $ PV $ = loan amount = 2,310,000
- $ r $ = monthly interest rate (unknown)
- $ n $ = total number of payments = 30 years × 12 months/year = 360

We need to solve this equation for the monthly interest rate $ r $. This equation doesn't have a simple algebraic solution, so we'll use numerical methods to find $ r $.

### Step 3: Finding APR
The monthly interest rate $ r $ needs to be converted to an annual percentage rate (APR):
\[
\text{APR} = r \times 12 \times 100
\]

### Step 4: Finding EAR
The Effective Annual Rate (EAR) is given by:
\[
\text{EAR} = \left(1 + r\right)^{12} - 1
\]
We convert this to a percentage:
\[
\text{EAR} = \left( (1 + r)^{12} - 1 \right) \times 100
\]

Let's calculate these values.The APR (Annual Percentage Rate) on this loan is **7.59%**.

The EAR (Effective Annual Rate) on this loan is **7.86%**.

Would you like more details on how these values were computed, or do you have any other questions?

Here are 5 related questions to consider:
1. How does the APR differ from the nominal interest rate?
2. What impact does the loan term have on the APR and EAR?
3. How would changes in monthly payments affect the APR?
4. How does compounding frequency affect the EAR?
5. What would the monthly payment be if the loan term were shorter?

**Tip:** When comparing loans, always consider both the APR and EAR to understand the true cost of borrowing.

You have just purchased a new warehouse. To finance the purchase, you’ve arranged for a 30-year mortgage loan for 70 percent of the $3,300,000 purchase price. The monthly payment on this loan will be $16,300. What is the APR on this loan? What is the EAR on this loan?

Learn how to calculate the APR (Annual Percentage Rate) and EAR (Effective Annual Rate) for a 30-year mortgage loan, given the loan amount and monthly payments. This guide covers key financial formulas for amortizing loans, suitable for students of financial mathematics or anyone learning about loans and interest rates.

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