 An annuity is one of the most common financial instruments. It consists of a sequence of payments made at specified intervals for a specified period of time. For example, a trust fund that disperses $10,000 monthly payments between the ages of 5 and 26, or a retirement fund that pays $559 a month after you turn 65 until you die. Note that making regular payments on the loan is mathematically indistinguishable from an annuity. So we'll introduce a few financial terms. If the payments are all the same, we say the annuity is level. If there are a predetermined and finite number of payments, the annuity is certain, otherwise the annuity is contingent. And these include things like life annuities, which are for the life of the recipient, and a special type of annuity is called a perpetuity, which are annuities that have unlimited terms. To begin with, let's consider an annuity that is level, so it pays out the same amount, certain, so it makes a predetermined number of payments, and immediate, so it pays at the end of each time unit. Since we can generally scale what happens to one unit of money, we can talk about a basic annuity immediate, which pays one at the end of each time period from t equals one to t equals n. Let's introduce some notation for writing about annuities. If v of t is our discount function, then the present value of a basic annuity immediate will be v of one plus v of two plus v of three, and so on, up to v of n. That's the present value of each dollar as it's paid out at times one through n. And the notation we use looks like this, where we read this as a angle n. Now our most commonly used discount function is the reciprocal of the t-th power of one plus the interest. So if we're using that, we'll let v equal this reciprocal, and we can express our present value, where we read this notation as a angle n at interest rate i. And then we can scale these by the amount q of the actual payments. While we don't generally encourage memorizing formulas, if you are useful enough to remember, but they're easiest to memorize if you use them repeatedly. In the context of annuities, a particularly useful one is the geometric series summation formula. And we use this when we want to add the powers of a starting with one. A quick derivation will help you remember the formula. Suppose this sum is s. Now if I multiply everything by a, and then subtract, then all but the first and last terms will cancel out, and then I can solve for s. Now since this is a finite series, the only thing we have to worry about here is a can't be equal to one. Otherwise we can always find the sum of this finite series using our formula. And one way you might remember this summation formula is that the sum of a finite geometric series, starting with one and adding the powers of a, is one minus the next term divided by one minus the common ratio. For example, you inherit an annuity, paying 1000 a year between the ages of 18 and 26, determine the type of annuity, and find the present value at age 17, assuming an interest rate of 5%. Also interpret your find again, translate into an action. Since the amount is the same 1000 every year, the annuity is a level. Since there are a fixed number of payments between the ages of 18 and 26, the annuity is certain. And if we tweet t equals zero corresponding to age 17, the payments are made at t equals one, two, three, and so on up to nine. So the annuity is immediate. And so this annuity is level, certain, and immediate. So the present value at age 17 will be 1000, a angle nine at 5% interest, or well it's really just the present value of all those payments from t equals one up to t equals nine. And so we compute. We almost have our series, but we need it to start at one. So we'll remove a factor of v. And now we have our geometric series. We apply the geometric series summation formula. And it might be useful to notice here that we have this ninth power of v, which is the same as the number of payments. And at an interest rate of 5%, v is a reciprocal of 1.05. So this sum will equal. And so we have a present value 710782. Now that's the present value, but what can we do with that? So what interpretation is that if you deposited this amount of money at age 17 into an account bearing 5% interest, you'd have the same value as the annuity. So we check bank rates. And if you check bank rates, you might not find anything making 5% interest. So you're better off keeping the annuity. Or let's consider another example. A bank agrees to pay you... Wait a minute, that never happens. Remember mathematically, there's no difference between an annuity and a loan. The only difference is the direction that the money moves. So suppose you agree to pay the bank 200 a month for 15 years. Assuming an annual interest rate of 5% per year convertible monthly, what's the present value of the loan? Since the payment period is in months, we'll find it easier to work in months. So 15 years will be 15 by 12, 180 months, and 5% per year will be 512% per month. We'll compute the value of a dollar, then scale it. So $1 paid monthly for 15 years at an interest rate of 512% per month will have present value, a, angle 180, 512%. Or the sum, v1 through v180. Our discount function, based on this interest rate of 512%, will be, and so our discount function will be the powers of v, will factor out a v, and then apply our geometric series summation formula. Again, a useful check is the power on v is the same as the term. So the present value, 126.98. But remember, this is the present value of $1. Since the payments are $200, the present value of this annuity should be 200 times the unrounded value, which we can then round now that we're done with the problem. So again, mathematically it makes no difference who pays whom. Since the present value of this payment plan is 25,396.43, this would also correspond to the amount a bank would lend if you agreed to the repayment plan. And note that you'd in fact be paying $200 for 180 months, $36,000. So the difference between the two represents the interest that you pay on the loan.