7. Finding the Payment

Let's take a look at how to find the payment on an annuity. So we've already talked about how to do the present value, how to bring all four of these guys back to now. We do that using the present value of an ordinary annuity table. But that's when we're given the payments. What happens if we're not given these payments? How are we supposed to figure that out? Well, I've got a problem copied down here that we'll take a look at. It says, "You've just graduated and want to purchase a new car." And first off, don't do that. When you graduate, don't buy a new car.

You should never, ever buy a $29,000 car right when you're out of college, and you should never, ever buy a brand-new car. Buy a one- or two-year-old used car. Anyway, that's not part of the problem. The one you've picked out is $29,000. You decide to make a cash down payment of $5,000. Dealer will give you a trade-in of $7,000. The balance will be financed at 8% for three years. If you can't pay for the car, you can't afford the car. Anyway, enough of that. Okay, so we need to figure out how much we're actually financing. The car itself costs $29,000, but you're putting down $5,000 and the dealer is giving you $7,000.

So 29 less 12 leaves you with 17 owed. So when you see these problems, make sure that you're calculating how much is still left, how much you owe, and not the list price of the car. The $17,000 is how much you would owe today if you were to pay the rest today, which means this is the present value. This $17,000 is how much you would have to pay cash today to be free and clear on the car. So we know that the present value of this annuity is $17,000. And the payment, we don't know.

We do know that the present value equals the payment times the present value of an ordinary annuity. Right? This is the equation. The present value equals whatever the regular payments are times whatever the factor is from the table. So, so far we've got $17,000 equals X times something, this present value of an ordinary annuity. Now let's go figure out what this present value of an ordinary annuity is. So the balance will be financed at an 8% three-year note to be paid monthly. Okay, so this 8%, that is the annual percentage rate. That's for 12 months. So since we're compounding monthly, we need to divide this 8%...

We need to divide that by 12 months, and that gives us basically two-thirds percent. Now, we also have to adjust this three-year part to be monthly. So we have three years, and we compound it 12 times per year, which means we end up with 36 periods.

So you have to adjust both the n, and the n is the periods, and the i is the percentage rate. So now we know we're dealing with 36 and 0.67, which leads us to this 31 number. So now when we come back here, we've got $17,000 equals this payment that we're looking for times 31.9116. And now it's just algebra. Just move that guy to the other side. 31.9116 equals X, and that gives us $532.70, or $533. So our payment is $533. Now, they may ask you, "How much will you pay in interest?"

Well, that's pretty easy to figure out. So we've got $533. That's our payment. And we're making 36 payments, and that gives us $19,188. Now, on your test, I just rounded this up to $533 'cause it's easier. Now, on your test, your teacher may say, "Go to two decimal places." So if you did that, I think this comes out to $19,178. So whichever one, I would probably go with this one on the test just 'cause it's more accurate. So then you're paying a total cash of $19,188, or 78. But the present value, the amount you owed, was actually only $17,000.

So the difference is how much interest you're paying. So we take the payment amount times the number of payments, and then we subtract the present value. This is what you would have paid had you had the cash to pay up front. You would have paid an additional $17,000 and not had to borrow the money. But since you had to borrow the money, you have to pay $2,000 for the privilege of not paying immediately and instead paying over three years. And also, can you imagine having to pay $500 a month for your car? Don't do that.

Now, the thing you want to remember is the present value equals the payment times the present value of an ordinary annuity. Typically, you're given these two, but in this problem, we were given the present value and the factor, and so all we had to do was find the payment, and it just became an algebra problem. But you need to train yourself when you see word problems like this to recognize this as an annuity problem. And the way you can recognize that is you have to pay these payments monthly, so it's a stream of equal payments, and that is the definition of an annuity.