Respuesta :
Answer:
$26,342.17.
Step-by-step explanation:
wish i got more points for this but to find the monthly interest rate corresponding to the effective annual rate, we first need to calculate the effective annual rate (EAR) using the given annual interest rate.
Given:
- Annual interest rate: 8%
We'll use the formula for calculating EAR from the nominal annual interest rate:
\[ EAR = (1 + \frac{r}{n})^n - 1 \]
Where:
- \( r \) = annual nominal interest rate (as a decimal)
- \( n \) = number of compounding periods per year
Here, we are compounding monthly (12 times per year), so \( n = 12 \).
Converting the annual interest rate to a decimal:
\[ r = \frac{8}{100} = 0.08 \]
Now, we can calculate EAR:
\[ EAR = (1 + \frac{0.08}{12})^{12} - 1 \]
\[ EAR \approx (1 + 0.00666667)^{12} - 1 \]
\[ EAR \approx (1.00666667)^{12} - 1 \]
\[ EAR \approx 0.08571 \]
So, the effective annual rate is approximately 8.571%.
Now, let's find the maximum monthly mortgage payment using the given parameters:
- Maximum monthly mortgage payment: $1950
- Amortization period: 5 years
We'll use the formula for calculating the monthly mortgage payment:
\[ M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( M \) = monthly mortgage payment
- \( P \) = principal loan amount
- \( r \) = monthly interest rate (as a decimal)
- \( n \) = total number of payments
We'll rearrange the formula to solve for the principal loan amount \( P \):
\[ P = \frac{M \times ((1 + r)^n - 1)}{r \times (1 + r)^n} \]
Substituting the given values:
\[ P = \frac{1950 \times ((1 + \frac{0.08571}{12})^{5 \times 12} - 1)}{\frac{0.08571}{12} \times (1 + \frac{0.08571}{12})^{5 \times 12}} \]
Now, let's calculate the value of \( P \).
\[ P \approx \frac{1950 \times ((1 + 0.007142)^{60} - 1)}{\frac{0.08571}{12} \times (1 + 0.007142)^{60}} \]
\[ P \approx \frac{1950 \times (1.007142^{60} - 1)}{\frac{0.08571}{12} \times 1.007142^{60}} \]
\[ P \approx \frac{1950 \times (1.497747 - 1)}{\frac{0.08571}{12} \times 1.497747} \]
\[ P \approx \frac{1950 \times 0.497747}{\frac{0.08571}{12} \times 1.497747} \]
\[ P \approx \frac{969.73065}{\frac{0.08571}{12} \times 1.497747} \]
\[ P \approx \frac{969.73065}{0.007142 \times 1.497747} \]
\[ P \approx \frac{969.73065}{0.010701} \]
\[ P \approx 90657.82591 \]
So, the principal loan amount \( P \) is approximately $90,657.83.
To find the price of the most expensive house you can buy, you need to consider the down payment and additional costs associated with buying a house. Typically, a down payment is required, and you may also need to consider closing costs and other fees.
The total interest paid over the course of the loan can be calculated by subtracting the principal loan amount from the total amount paid over the loan term (which is the monthly payment multiplied by the total number of payments) and then subtracting the initial loan amount.
\[ \text{Total Interest Paid} = (M \times n) - P \]
\[ \text{Total Interest Paid} = (1950 \times 5 \times 12) - 90657.83 \]
\[ \text{Total Interest Paid} = 117000 - 90657.83 \]
\[ \text{Total Interest Paid} = 26342.17 \]
So, the total interest paid over the course of the loan is approximately $26,342.17.
