Question
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
a) Monthly payments
b) First interest paid by Ashira
c) Total amount and interest paid for the whole time of the loan
Answers
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Answered by
GPT-4o
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.
**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.
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