Geometric Series
A Dollars and Sense Application

We've all seen the ads --

-- and the prices seem right -- a brand new car for less than $200 per month. But there is lots of fine print and the deals are complicated. How can you decide what is a good deal? On one side, a lease offers a combination of relatively low out-of-pocket expenses -- a low down payment and small monthly charges -- and not-so-visible disadvantages at the end of the lease -- charges for extra miles and a large "balloon payment" if you want to keep the car. On the other side, buying a car often means larger monthly payments for a longer period of time but at the end you own the car and can continue driving it or it has some value if you trade it in as partial payment on a newer car.

Salespeople are well-armed. They deal with many customers and know what to say to make a sale.

"You can drive a new car home today. Only $189 per month and in three years you can do it all over again -- a brand new car every three years and only $189.00 per month"

"It's risk-free. With the usual purchase you're stuck with that car. We know how well our cars retain their value so we've figured this lease allowing a generous price at the end of the lease. You know now up-front how much it will cost and if you decide to keep the car, just pay the balance ... or... if you want a new car, just walk away. No risk. It's entirely up to you at the end of the lease."

It actually requires a fair amount of work to compare all the options you have as a car buyer. Dealers are reluctant to give you all the information that you need to make an intelligent decision -- they know that while you are thinking you aren't buying. In addition, there are a lot of possibilities and comparing them is complicated. The mathematics itself is simple -- all you need to know is addition and multiplication and something about geometric series -- but the details can get complicated. Here are some of the things that you need to consider.

What are your options?

If you have some money in the bank then one option is purchasing the car outright. This is often the best option because you are essentially borrowing money from yourself -- that is, from your savings -- and the interest charges are essentially the lost interest you would have earned if you had left the money in the bank. The advantage of "borrowing from yourself" is even more pronounced when you consider the tax consequences. You generally pay taxes on interest you receive from bank accounts and other investments. Thus, even if your savings were earning 6% interest, your after tax interest would be closer to 4%. The interest that you pay on most consumer loans is not tax deductible. So an interest rate of 10% really does cost you 10%.
If you do not have enough savings then you probably have only three options.
If you have a car that is in working condition then one option is postponing the purchase of a new car. Waiting and saving is often an excellent option.
If you have decided that now is the time to buy and you don't have enough money in the bank to buy the car that you want outright then you still have two options.
Taking out a loan and buying the car. This option has several advantages. Because you are buying the car, part of your car payments build up equity in the car -- that is, when you are done making payments you will own the car and can continue driving it or use it as a trade in on a newer car. How much equity you have will depend on the size of your down payment, the length of the loan, and the value of the car when you decide to sell it or trade it in. If you do purchase the car by taking out a loan you still have options -- you can finance the car through the dealer, or a bank, or possibly your credit union. The dealer often offers a choice of a rebate or another incentive or a low interest rate. You should compare whatever deal the dealer offers you on a package with financing with the dealer's best price and financing that you arrange yourself. When the dealer arranges everything it is often difficult to determine exactly what is going on. The dealer might, for example, offer you what appears to be a low price but then charge a higher interest rate than you might get elsewhere. Remember the dealer wants to make as much money as possible from this sale and has more experience than you do. Making the best decision requires some effort on your part but because cars are so expensive it is well worth the effort.
Leasing the car. This is becoming a popular choice. It has several very obvious advantages -- the down payment is often quite low and the monthly payments are often also low. In addition, the value of the car at the end of the lease period is guaranteed. The low payments are possible because the leasor agrees up-front to "buy back" the car at a fixed price. You won't get any money for the car at the end of the lease period but the money that you would have received from selling it is used to keep the monthly payments low. The big disadvantage is that all of your payments are going for the use of the car during the lease period. At the end of the lease you own nothing. In fact, you may find that you owe the leasor money either for excess miles or because the car is not in acceptable condition.

Finding out some of the information you need if you purchase the car using a loan is relatively easy. You need to know the purchase price of the car, the down payment, and the interest rate and term (length) of the loan. Subtract the down payment from the purchase price (don't forget to add any taxes that you finance) to get a number A called the amount financed. If the annual interest rate is R then the monthly interest rate is R/12 . If you make n payments then the amount financed is equal to the present value of the monthly payments -- that is, either

Missing equation

where P is the amount of each monthly payment. Use the first formula if the first payment is due at the time you purchase the car and the second formula if the first payment is due one month after you purchase the car. If the interest rate is 12% and the loan is for 48 months then these two formulas yield


A = 38.3537 P   If the first payment is due at the time of purchase 

A = 37.9740 P   If the first payment is due one month after the purchase.

Now you can use the appropriate one of these two formulas to determine your monthly payments.

Suppose that you want to buy a car by taking out a loan and have decided to borrow $15,000 -- the total cost of the car including everything minus all the money you must provide up front. Suppose that you will make 48 monthly payments with the first payment due one month after the purchase. Determine the amount of each monthly payment at the following interest rates -- 6%, 7%, 8%, 9%, 10%, 11%, and 12%.
Suppose that you want to buy a car by taking out a loan and have decided to borrow $15,000 -- the total cost of the car including everything minus all the money you must provide up front. Suppose that you can obtain a loan at an interest rate of 9%. Determine the amount of the monthly payments if the term of the loan is 24 months, 36 months, 48 months, or 60 months. Assume that the first payment is due one month after the car purchase.

Suppose that you can afford payments of $200 per month and that you have $3,000 in ready cash that you can use for a down payment and any other immediate expenses when you buy a car. How much can you pay for a car with each of the following loans. Assume that the first payment on each loan is due one month after the car purchase. Compute the total bill you can afford including purchase price, taxes, and everything else. Don't forget to deduct the $3,000 ready cash from the total bill to determine how much you need to finance.

Interest rate          Term

      6%            24 months
      6%            36 months
      6%            48 months
      6%            60 months

      7%            24 months
      7%            36 months
      7%            48 months
      7%            60 months

      8%            24 months
      8%            36 months
      8%            48 months
      8%            60 months

      9%            24 months
      9%            36 months
      9%            48 months
      9%            60 months

     10%            24 months
     10%            36 months
     10%            48 months
     10%            60 months

We haven't considered all the factors that go into a decision to purchase a car by taking out a loan. For example, you also need to have some idea of the value of the car at the end of the loan period. For a short loan the car may still be worth quite a bit (although probably less than you think) andthat value is like money in the bank. You can continue driving the car and build up savings instead of making car payments, or you can sell the car to get cash or a down payment for your next car. You have lots of options. You can sell the car privately or you can use it as a trade in with a dealer of your choice. Along with these options goes some uncertainty. It is difficult to predict what the car will be worth. You can, however, make some educated estimates based on past performance of similar models. If you are comparing purchasing a car with a loan to leasing the same car then you might want to use the dealer's figure for the value of the car at the end of the lease.

Now we suppose, for a concrete example, that you are interested in buying a car that sells for $20,000 including everything. Make sure, first, that the $20,000 is the real price. Talk to a dealer as if you are going to buy the car with cash. Suppose that you have $4,000 ready cash and that you can earn 5% interest on any money that you have. We will consider two options.

A lease at $400 per month with $1,000 down and 12,000 miles per year. Excess mileage will cost $0.15 cents per mile. The dealer says the residual value of the car at the end of the lease will be $10,000. This is the amount that you would have to pay to own the car. Suppose that you actually drive 15,000 miles per year. Thus, at the end of the lease you will have driven 45,000 miles which is 9,000 miles more than the allotment in your lease. This means that you will have to pay $1,350 if you do not buy the car. It does not affect the amount that you pay if you do buy the car. With this option you will keep $3,000 of your ready cash in the bank earning 5% interest compounded monthly, so at the end of the three year lease you will have $3,484.10 that you can use toward buying the car or paying off the extra mileage charge and making a down payment on your next car.
Purchasing the car with a 36 month loan at 8% interest. After putting $4,000 down you will have to borrow $16,000. That works out to payments of $501.38 per month. Notice that with this option your payments are higher than if you leased the car and you have not had the advantage of $3,000 earning interest for the three years but at the end of the three years you own the car -- no worries about excess mileage and you can do whatever you want. You could continue driving the car and saving money for your next purchase or you could use the car's value as a down payment on your next car.

One good way to compare a lease with an outright purchase is to assume that you would continue investing the difference between the monthly lease payments and the loan payments over the course of the loan. In this case the difference is

$101.38 = $501.38 - $400.00.

If you were to invest that at the same interest rate of 5% that we have been assuming for your other investments then over the course of 36 months the value would be

Missing equation

So by leasing the car your would have $3,928.81 + $3,484.10 = $7,412.91 in cash at the end of the 36 month lease and no equity in the car. If you turn in the car then you would be left with $6,062.91 after paying for the excess mileage. If you bought the car then you would have to come up with an additional $10,000 - 7,412.91 = $2,587.09 -- that is, the residual value of the car minus your savings.

For this particular example, it looks as if the better choice is purchasing the car with a loan but you need to look at your own individual circumstances. This was just one example and the figures we used were somewhat arbitrary. This example actually understates the advantage of purchasing the car in this example with a loan because we did not subtract taxes from the interest earned by the investments made under the leasing option.

Check Your Understanding

Suppose that in the example above the after tax rate of return on your investment is 3.5%. How does that affect the comparison made above.
Work out the details for a real example. First, find a car that you would like to buy and find out from the dealer the cash price of the car including everything and the terms of a lease. Be sure to get all the details of both options in writing. Then shop around for the best interest rate that you can get on a car loan. Determine the amount of ready cash that you have that could be used for a down payment or could be invested. Find the best interest rate that you might earn on any money that you invest. Estimate how many miles you usually drive each year so that you can compute any possible excess mileage charges. Finally, carry out calculations like the ones above in order to determine which option is the best. Be as realistic as possible. For example, it might be that your only possible option is the lease because of its lower down payment and lower monthly payments. If so, you might compare leasing with postponing a purchase while you save some money.

Geometric Series A Dollars and Sense Application

Check Your Understanding

Geometric Series
A Dollars and Sense Application