To solve these problems, we need to break them down step by step.
**a) Monthly Payments**
To calculate monthly payments for a fixed-rate mortgage or loan, we use the amortization formula:
\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]
Where:
- \( M \) is the monthly payment
- \( P \) is the loan principal (\$200,000)
- \( r \) is the monthly interest rate (annual rate divided by 12)
- \( n \) is the total number of monthly payments (number of years times 12)
Here, the annual interest rate is 15%, so the monthly interest rate (\( r \)) is:
\[ r = \frac{15\%}{12} = \frac{0.15}{12} = 0.0125 \]
The loan term is 30 years, so the total number of monthly payments (\( n \)) is:
\[ n = 30 \times 12 = 360 \]
Plugging these values into the amortization formula:
\[ M = 200{,}000 \cdot \frac{0.0125(1 + 0.0125)^{360}}{(1 + 0.0125)^{360} - 1} \]
This is a complex calculation, typically done using a financial calculator or spreadsheet. Let's simplify it compartmentally:
1. Calculate \( (1 + r)^n \):
\[ (1 + 0.0125)^{360} \approx 58.584 \]
2. Now the numerator:
\[ 0.0125 \times 58.584 \approx 0.7323 \]
3. And the denominator:
\[ 58.584 - 1 = 57.584 \]
4. Finally, compute \( M \):
\[ M \approx 200{,}000 \times \frac{0.7323}{57.584} = 200{,}000 \times 0.01272 \approx 2{,}544 \]
So, Ashira's monthly payment is approximately \$2,544.
**b) First Interest Paid by Ashira**
The first interest payment is the interest on the initial loan amount for the first month. To find it:
\[ \text{First interest payment} = P \times r = 200{,}000 \times 0.0125 = 2{,}500 \]
So, Ashira's first interest payment is \$2,500.
**c) Total Amount and Interest Paid for the Whole Time of the Loan**
To find the total amount paid over the life of the loan, multiply the monthly payment by the total number of payments:
\[ \text{Total amount paid} = M \times n = 2{,}544 \times 360 \approx 916{,}000 \]
To find the total interest paid over the life of the loan, subtract the principal from the total amount paid:
\[ \text{Total interest paid} = \text{Total amount paid} - P = 916{,}000 - 200{,}000 = 716{,}000 \]
So, the total amount paid is approximately \$916,000, and the total interest paid is approximately \$716,000.
Ashira took a loan of $200000 from a bank which offers 15% interest compounded annually for 30 years.Calculate
a) Monthly payments
b) First interest paid by Ashira
c) Total amount and interest paid for the whole time of the loan
1 answer