 Another approach to finding the balance is known as the Prospective Method. This does require knowing all future payments, or at least what they should be. Note that the final payment might be a balloon or a drop, so suppose that all but the last payments are Q with a final payment of R. Suppose a loan L is made at interest rate I for N time periods. At time K, the outstanding loan balance OLBK should be the present time K value of all remaining loan payments. Now we should be a little careful here. We'll interpret at time K to be immediately after the payment at T equals K. Also remember that the last payment is different, so there will be Q payments at time T equals K plus 1, K plus 2, all the way up to the next to last payment at N minus 1. So this will be a total of N minus 1 minus K, N minus K minus 1 payments of Q, so the value will be Q a angle N minus K minus 1 at interest rate I. The last payment, which might be a balloon or a drop, will be after another N minus K time periods. So its present value will be RV to power N minus K, whereas usual V is R reciprocal of 1 plus the interest rate. And this gives us the expression for the outstanding loan balance, the value of all the payments that we're going to make, including the last payment, which might be at a different value. So let's go back to that college loan, 30,000, with an interest rate of 3% compounded monthly to be paid back at $100 a year for 15 years with a final payment of R. Let's determine the final payment. Now there does seem to be a problem here in that the prospective method seems to require knowing how long we've made regular payments, but we don't know, so we can assume any amount. For example, in a previous problem we found that at $100 a month, the balance after five years would be 28383.8321, where we'll use the less rounded value for our calculations. So by the prospective method, the balance will be the value of the payments at Q plus the final payment at R. So after five years you'll continue to make payments for an additional 10 by 12 120 months. However, the 120th payment is the final payment that's R, R balloon or drop payments. So in fact, there's 119 payments of $100 with value 100 a angle 119 at 312%. Finally, that last payment R has a present value that's based on the fact that it's going to be paid off 120 months in the future. So that's R v to power 120. We'll compute the value of 100 a angle 119 at 312%, which will be Solving our equation for R gives us, which is a final payment of $24,425.68, which is very definitely a balloon payment. Of course, if we hadn't already solved the problem, we could also work with the fact that our balance at t equals zero is 30,000. And so we can write the equation, the outstanding loan balance at zero. Well, that would be 179 payments of 100 plus a final payment of R. And so that'd be 100 a angle 179 at 312% plus R v to power 180. And so we find and solve. And it's worth noticing that even after paying for 15 years that last payment is almost as large as the entire loan amount, which means you've been paying mostly interest for that entire time, which is something the financial institutions would prefer. Unnecessary and irrelevant information and persons have been removed according to our usage policy.