homeowener is considering refinancing his home. original amount of the 30 yr loan was 250,000 at 12% compounded monthly. The owner has made ten years of payments. how much is the remaining balance on the loan.
this is what I gotL
r=12%, m=12, i=12%/12=1%/month
I have to find out the annual payments he made so I did: A=P(A/P,i,n)
A=250,000(A/P,i,n)
using the percentage table, I get A.
Then how do I solve for the future value or also known as the balance.
An excel spreadsheet is very good for these kinds of problems. Unfortunately, this problem will contain an ugly 'sum' part.
(Finance guys are into these look-up tables, which appears to be the path you are heading. I prefer using math and an excel spreadsheet -- )
First thing I would do is solve for P. Find P such that the balance of the loan after 360 payments is zero.
We know B0 = 250000.
B1 would therefore be B0*(1+.12/12) - P = B0*1.01 - P
(That is, the balance changes by the interest hit, but is reduced by the constant payment). Continuing:
B2 = B1*(1.01) - P
+ B0*(1.01)^2 - P*1.01 - P
So, by extension
Bn = B0*(1.01)^n - P*(1.01^(n-1) + 1.01^(n-2) + ... 1.01^0)
We also know that Bn = B360 = 0 (the loan is paid off). So use the above equation to solve for P.
So, now you have everything you need.
(And if you used excel, your answer would appear in one of the cells)
B120 = B0*(1.01)^120 - P*(1.01^119 + 1.01^118 + ... 1.01^0)
To solve for the remaining balance on the loan, you can use the formula for future value of a loan, which takes into account the principal amount, interest rate, compounding frequency, and the number of periods. Here's how you can calculate the remaining balance:
1. Calculate the monthly interest rate: Since the annual interest rate is given as 12%, divide it by the number of compounding periods per year. In this case, there are monthly compounding periods, so divide 12% by 12 to get a monthly interest rate of 1% (0.01 as a decimal).
2. Calculate the total number of periods: A 30-year loan consists of 12 periods per year for a total of 30 years, resulting in 12 x 30 = 360 periods.
3. Calculate the annual payment (A): You mentioned that you already found the annual payment using the A=P(A/P,i,n) formula. Let's assume the annual payment is $X.
4. Calculate the remaining balance (FV): Now, use the formula for future value of a loan - FV = P(1 + i)^n - PMT[(1 + i)^n - 1] / i, where:
- P is the principal amount (original loan amount)
- i is the monthly interest rate
- n is the total number of periods
- PMT is the annual payment amount
Plug in the values: FV = 250,000(1 + 0.01)^360 - X[(1 + 0.01)^360 - 1] / 0.01
5. Subtract the payments made over the ten years from the remaining balance: The owner has made ten years of payments, so multiply the annual payment amount by ten (10X), and subtract this value from the FV calculated in the previous step to find the remaining balance.
By following these steps, you should be able to calculate the remaining balance on the loan after ten years of payments.