Step 2: Now that we know our original Loan Amount = 19,072.10, we need the balance of the loan at time 5 after the 5th payment is made.
The formula for outstanding balance on a standard amortization loan is:
Outstanding Balance at time t = Payment * (1 - v^(n-t))/i where v = 1/(1+i).
Using your series formula, n - t = 5, so do a series with a term of 1/1.0525 and use your 2500 as your common term. We only want to do a term of 5.
That is the outstanding balance formula, but for your series, you want to match the 10,749.30.
2500/1.0525 + 2500/(1.0525^2) + 2500/(1.0525^3) + 2500/(1.0525^4) + 2500/(1.0525^5) = 10749.30
Let me know if that makes sense, and you are ready for Step 3.
We have our Balance at time 5 of 10749.3. We need to figure out how to pay this loan off in 5 years to make our 10 year deadline from what you said in your problem. Therefore, we have 5 more payments, and interest has now increased to 7.75%
=10479.30/[1/1.0775 +1/(1.0775^2) + 1/(1.0775^3) + 1/(1.0775^4) + 1/(1.0775^5)] = 2674.52
2674.52 - 2500 = 174.52
I fixed it above, I hope you saw it.
This time, we are not calculating a present value, but rather, a payment.
Loan = Payment * Present Value factor. Rearranging this, we get:
Payment = Loan / PV Factor.
Make sense?
Step 1 and Step 2 calculated PV's, Step 3 calculates a payment.
Yes, I went back and corrected that.
Forget the numerator for now of 10749.30. Let's take denominator.
1/(1.0775^1) <----- 6th payment discounted back to time 5
1/(1.0775^2) <----- 7th payment discounted back to time 5
1/(1.0775^3) <----- 8th payment discounted back to time 5
1/(1.0775^4) <----- 9th payment discounted back to time 5
1/(1.0775^5) <----- 10th payment discounted back to time 5
Add these 5 terms up. That is your Present Value factor.
Divide your Present Value of 10749.30 your sum above.