Sophia Is Looking To Take Out A 15-year Mortgage From A Bank Offering A Monthly Interest Rate Of $0.475 %$. Using The Formula Below, Determine The Maximum Amount Sophia Can Borrow, To The Nearest Dollar, If The Highest Monthly Payment She
Introduction
Sophia is considering taking out a 15-year mortgage from a bank with a monthly interest rate of 0.475%. To determine the maximum amount she can borrow, we need to use the formula for calculating the monthly payment on a fixed-rate loan. In this article, we will walk through the steps to calculate the maximum amount Sophia can borrow, given the monthly interest rate and the loan term.
Understanding the Formula
The formula for calculating the monthly payment on a fixed-rate loan is:
M = P[r(1+r)n]/[(1+r)n – 1]
Where:
- M = monthly payment
- P = principal loan amount (the amount borrowed)
- r = monthly interest rate (in decimal form)
- n = number of payments (the number of months the loan is for)
Given Values
- Monthly interest rate (r) = 0.475% = 0.00475 (in decimal form)
- Loan term (n) = 15 years = 180 months
Calculating the Maximum Amount
To calculate the maximum amount Sophia can borrow, we need to rearrange the formula to solve for P. We can do this by multiplying both sides of the equation by [(1+r)^n – 1] and then dividing both sides by [r(1+r)^n].
P = M[(1+r)^n – 1]/[r(1+r)^n]
We can plug in the given values and calculate the maximum amount Sophia can borrow.
Step 1: Calculate (1+r)^n
First, we need to calculate (1+r)^n.
(1+0.00475)^180 ≈ 2.989
Step 2: Calculate r(1+r)^n
Next, we need to calculate r(1+r)^n.
0.00475(2.989) ≈ 0.0142
Step 3: Calculate [(1+r)^n – 1]
Now, we need to calculate [(1+r)^n – 1].
2.989 - 1 ≈ 1.989
Step 4: Calculate P
Finally, we can plug in the values and calculate P.
P = M[(1+r)^n – 1]/[r(1+r)^n] P = M(1.989)/0.0142
Assuming a Maximum Monthly Payment
To determine the maximum amount Sophia can borrow, we need to assume a maximum monthly payment. Let's assume the maximum monthly payment is $1,000.
Calculating the Maximum Amount
Now, we can plug in the value of M and calculate the maximum amount Sophia can borrow.
P = 1000(1.989)/0.0142 P ≈ 139,511
Rounding to the Nearest Dollar
Finally, we can round the result to the nearest dollar.
The maximum amount Sophia can borrow is approximately $139,511.
Conclusion
Introduction
In our previous article, we walked through the steps to calculate the maximum amount Sophia can borrow, given the monthly interest rate and the loan term. We used the formula for calculating the monthly payment on a fixed-rate loan and rearranged it to solve for P. In this article, we will answer some frequently asked questions related to calculating the maximum amount that can be borrowed.
Q: What is the formula for calculating the monthly payment on a fixed-rate loan?
A: The formula for calculating the monthly payment on a fixed-rate loan is:
M = P[r(1+r)n]/[(1+r)n – 1]
Where:
- M = monthly payment
- P = principal loan amount (the amount borrowed)
- r = monthly interest rate (in decimal form)
- n = number of payments (the number of months the loan is for)
Q: What is the monthly interest rate (r) and how is it calculated?
A: The monthly interest rate (r) is the interest rate charged on the loan per month. It is calculated by dividing the annual interest rate by 12.
For example, if the annual interest rate is 6%, the monthly interest rate would be:
r = 6%/12 = 0.005
Q: What is the number of payments (n) and how is it calculated?
A: The number of payments (n) is the number of months the loan is for. It is calculated by multiplying the number of years the loan is for by 12.
For example, if the loan is for 15 years, the number of payments would be:
n = 15 years x 12 months/year = 180 months
Q: How do I calculate the maximum amount that can be borrowed?
A: To calculate the maximum amount that can be borrowed, you need to rearrange the formula to solve for P. You can do this by multiplying both sides of the equation by [(1+r)^n – 1] and then dividing both sides by [r(1+r)^n].
P = M[(1+r)^n – 1]/[r(1+r)^n]
Q: What is the maximum amount that Sophia can borrow?
A: In our previous article, we calculated the maximum amount that Sophia can borrow, assuming a maximum monthly payment of $1,000. The result is approximately $139,511.
Q: Can I use this formula to calculate the maximum amount that can be borrowed for a variable-rate loan?
A: No, this formula is only applicable for fixed-rate loans. For variable-rate loans, the interest rate may change over time, so the formula would need to be adjusted accordingly.
Q: Can I use this formula to calculate the maximum amount that can be borrowed for a loan with a balloon payment?
A: No, this formula is only applicable for loans with a fixed monthly payment. For loans with a balloon payment, the formula would need to be adjusted to account for the balloon payment.
Conclusion
In this article, we answered some frequently asked questions related to calculating the maximum amount that can be borrowed. We provided explanations and examples to help clarify the concepts. We hope this article has been helpful in understanding how to calculate the maximum amount that can be borrowed.