 Another idea that is associated with annuities and loans is the notion of accumulated value. Suppose you make k deposits of $1 into an account at times 1, 2, 3, and so on. The total value at time k is designated, which we read as s-angle k. If we're assuming compound interest at a rate of i per time unit, the amount is s-angle k at interest rate i, and we have, let's think about that, the first dollar deposited at t equals 1 accumulates interest for k-1 periods, so its value at the end will be 1 plus i to power k minus 1. The second dollar deposited at t equals 2 accumulates interest for k minus 2 periods, so its value at the end will be 1 plus i to power k minus 2. The third dollar deposited at t equals 3 accumulates interest for k minus 3 periods, so its value at the end will be 1 plus i to the power k minus 3, and so on. Now the last dollar is deposited at t equals k. This won't accumulate any interest, so we could write it as 1, but we'll be consistent and write it as 1 plus i to power k minus k. Now note that 1 plus i to power k minus k is equal to 1, so we can use our geometric series summation formula. We can reverse the order of the series to make it look a little bit more familiar. And since this is a finite series, the sum will be 1 minus the next term we should have added divided by 1 minus the common ratio, and we simplify, and so we have a formula. So for example, beginning next month, you will deposit $100 a month into an account earning 3% annual interest compounded monthly. How much will be in the account after 5 years? So $1 a month for 5 years at 3% annual interest compounded monthly will have value s angle 60 at 312%. So if we deposit $100 every time, our total account value will be 100 s angle 60 at 312%. Now we have a handy formula for calculating this value, but remember, don't memorize formulas, understand concepts. So s angle 60 at 312%, well, that's just the accumulated value of the first dollar, which gains interest for 59 months. The second, which gains interest for 58, and so on. And this is a geometric series, so we can apply our geometric series summation formula and find which we can round down to 64.64 and 67 cents. This accumulated value provides us another way of finding the outstanding balance of a loan. Suppose you borrow an amount L and make regular payments of Q. What's the outstanding loan balance, O-L-B-K, at time K? There are two approaches. The retrospective method looks backwards and is based only on the payments already made. It also allows the loan period to be unspecified, and we go about it as follows. Imagine two accounts. The loan with no payments made with balance L times the accumulation function. We also have the K payments of Q, and we can regard those as an annuity with a value Q s angle K. Then the outstanding loan balance will be the difference between these two quantities, what the loan would have grown to if you made no payments minus the value of the payments you actually make. So you borrow $30,000 to pay for college at an interest rate of 3% per year compounded monthly. You make payments of $100 a month for five years. Find the outstanding loan balance. Now, since our payments are monthly, we'll measure time in months where five years is 16 months and our interest rate is 312.1%. Using the retrospective method, the outstanding loan balance is the loan amount as if we had never paid anything, that's L, A of 60, minus the total value of our payments Q s angle 60 at 312%. Now, since we're compounded the interest, our accumulation function will be the loan amount is 30,000 and T equals 60, so we can substitute those in and find. Meanwhile, our payments Q are 100 and we've already found 100 s angle 60 at 312%, that was the previous problem, so our outstanding loan balance will be or 28,383.33. Note that five years of payments have only reduced the balance by $1,616.17. This is something financial institutions would prefer you not know about it, so you... Unnecessary and irrelevant information has been removed according to our usage policy.