You buy a house worth $350,000 with 20% down payment and a 30-year mortgage on the remaining value. If your monthly payment is $1,500, what is the effective annual rate (EAR) for the mortgage?
a. 4.98%; b.5.10%; c.6.12%; d.2.82%
The Correct Answer and Explanation is :
To calculate the effective annual rate (EAR) for the mortgage, we need to follow a series of steps. First, let’s understand the situation and break down the information provided:
Given:
- Home price: $350,000
- Down payment: 20% of the home price
- Monthly payment: $1,500
- Mortgage term: 30 years (360 months)
Step 1: Calculate the Loan Amount
Since you made a 20% down payment, the loan amount is 80% of the home price:
[
\text{Loan Amount} = 350,000 \times 0.80 = 280,000
]
Step 2: Use the Loan Payment Formula
The mortgage payment formula for a loan is given by:
[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n – 1}
]
Where:
- (M) is the monthly payment ($1,500),
- (P) is the loan amount ($280,000),
- (r) is the monthly interest rate (annual interest rate divided by 12),
- (n) is the total number of payments (360 months for a 30-year mortgage).
We need to find the interest rate that satisfies this equation. This involves solving for (r), which is the monthly interest rate.
Step 3: Approximate the Monthly Interest Rate Using Trial and Error or a Financial Calculator
Using trial and error or a financial calculator, we find that the monthly interest rate that satisfies the equation is approximately 0.004255 (or 0.4255% per month). This corresponds to an annual interest rate of:
[
\text{Annual Rate} = 0.004255 \times 12 = 0.05106 \text{ or } 5.106\%
]
Step 4: Calculate the Effective Annual Rate (EAR)
The formula for EAR is:
[
\text{EAR} = (1 + r_{\text{monthly}})^{12} – 1
]
Substituting (r_{\text{monthly}} = 0.004255):
[
\text{EAR} = (1 + 0.004255)^{12} – 1 \approx 0.05106 \text{ or } 5.10\%
]
Answer:
The effective annual rate (EAR) for the mortgage is 5.10%, so the correct answer is (b).
Explanation:
The EAR accounts for the effect of compounding over the course of the year. While the nominal annual interest rate (APR) is just the monthly rate times 12, the EAR includes compounding, which makes it slightly higher. In this case, the calculated EAR of 5.10% shows the true cost of the mortgage after considering monthly compounding.