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Decisions about Cars New or Used Of course new cars are nice. They have the latest gadgets in the car world. They have a distinct smell. The floor doesn’t have very much stuff on it.

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New or Used

Of course new cars are nice. They have the latest gadgets in the car world. They have a distinct smell. The floor doesn’t have very much stuff on it.

Financial items to consider on any car include taxes, title and insurance. The miles per gallon the car will travel also will help you see the advantages in terms of gas expenses. Newer cars typically get better mpg’s because of government regulation.

Another cost to consider on the purchase of any asset, but here in the context of the car, is depreciation. Here we mean the lose of value in the car due to driving it. Used cars will typically depreciate less because the most depreciation occurs in the first year or so. What I really mean is depreciation occurs fastest the first year and then the depreciation slows down.


The Odometer

On a used car check the odometer. If it seems low relative to the way the car looks, maybe you should pass on the car. By law, odometers are not supposed to be messed with.


Be sure the seller has the title to the car and is the rightful owner. In the USA if you buy a car from a person who does not legally have title, the car could be return to the original owner. You then have to get your money back from the crook.



Say you have determined you can afford to pay 375 a month on a car loan. How much car can you buy?

Let’s assume you have no down payment, you are looking at a 5 year loan with a nominal rate of 6.9%. Car loans are compounded monthly, although I do not think this is made clear to the consumer. In Excel we can find the present value of the uniform series, or annuity, we will pay by the following

=PV(0.069/12,5*12,-375) = PV(interest rate, time frame, annuity)

= 18,983

Now, if just the interest rate is higher, you can afford less car. If just the time frame is higher you can afford more car. If the amount you can pay a month is higher you can afford more car.


As a consumer, when you walk into a car dealer and the sales person asks you how much can you pay each month, should you lie?

Back to the new or used decision. Once last detail about any car is there is a possibility you will buy a lemon. If it is new you can get the dealer to work with you because of manufacturer problems. Take a used car to a mechanic before you buy to have them check it out.

Some folks think that used cars sold by private citizens sell for less than at dealers because the owner has to offer a discount as an “insurance policy” against the car being a lemon. The buyer then takes the car as is.

We can not say here if it is better to take used or new. Some folks are willing to make trade-offs others aren’t. So there is no iron clad decision rule.


Lease or Buy?

Lease ideas to consider:

Closed-end lease means you walk away with no obligation at the end of the lease period, unless you abused car or went over preset mileage limits.

Open-end lease means if the value of the car at the end of the lease is less than the estimated value, then you pay the difference.

On a lease you may be asked to make a down payment and a security deposit.



The first diagram on the left is the one we have become accustomed to. We have an A value at the end of each of two periods and the F value occurs at the end of the second period.





Say A = 1 and i = 10, then

F2 = 1[appen b page 691 column 10% row 2 value]

= 1[2.100] = 2.10

On the previous screen the graph on the right would have

F3 = 2.10 + the A at time zero taken as a single payment to time 2 in a single payment. In other words

F3 = 2.10 + 1[appen a page 690 column 10% row 2 value]

= 2.10 + 1[1.21]= 3.31 = 1[3.31]

So F3 has only two time periods but three A’s. So long as the last A occurs at the same time as F we have a new story. Look at appendix b page 691 10% column row 3 value. We have 3.31. WOW, what does this mean?


This means if we have an A at time zero, then we can just imagine we had a problem that started one period before.




Now that we have more formally introduced compounding a section or two back, let’s consider a case where payments are made more often than compounding occurs. Lets’ consider a case where payments are made quarterly, but compounding is semi-annual. Say we have a two year deal at 10 percent nominal interest.

a a a a a a a a

0 1 2 3 4 5 6 7 8


To find the future value of the annuity we need to first recognize that the interest is only compounded every other quarter. If we take the a’s in the 2nd, 4th, 6th, and 8th periods we have F = a [value in appendix B page 691 column 5% row 4].

= a [4.310]

Now, when we look at the a’s in the 1st, 3rd, 5th and 7th quarters we could find the F at the 7th period as

F = a[4.310]. Since we need a half year for the interest to be earned the F at the end of the 7th period does not have time to earn interest by the end of the 8th quarter. If we want to know the total value at the end of the 8th period we simply move the F at the end of the 7th period over to the 8th period and add it to the other value.


We have

F = a[4.310] + a[4.310] = 2a[4.310]

Do you see the significance of this result? If funds are deposited more often than the compounding period, add the values up to the end of the compounding period. Thus if a is made quarterly and interest is compounded semi-annually, just assume 2a is made semi-annually.

In fact, the only time dollar values should be moved across time without making an interest adjustment is when the money does not have time to earn interest.

Now, back to our story about comparing a lease to buying a car.


Say with a lease you have

At time 0 a $1500 down payment and a $300 security deposit.

Then over the next 36 months a $300 monthly payment will be made.

Say if you buy you have

At time 0 a $2500 down payment and a 5% sales tax payment on a $15000 car of $750.

Then over the next 36 months you have a payment of 392 (financing 12500 at 8% nominal over 3 years)

At the end of three years the car is still worth $8000.


The authors say in order to compare the two first do this for the lease:

1500 down payment +

300 month times 36 months = 10800 +

(1500 down pay + 300 security dep)times 3 years time .04 interest earned on savings ( an opportunity cost calculation) = 216

For a grand total of $12,516. This is the cost of the lease.

For the car the authors say:

2500 down payment +

750 sales tax +

392 a month for 36 months = 14112 +

Opportunity cost of down payment 2500times 3 times.04 = 300 –

8000 in car value at end of loan, for a total cost of the car being


So the authors say buy the car.


I say buy the car, but for different reasons. The authors, I believe, violated a rule of finance. They added values across time without adjusting for interest. You can only when there is not enough time for interest to accrue.

They added apples and oranges. You can only do this when you want to make a fruit salad! They added values at time 0 to values at time n to values each period. This is very bad.

What they should have done. Pick a time frame – either the present at time 0, the annuity time frame, or the end of the story.

Let’s do an end of the story at the end of the 36 months.


The lease would be

1800 in a single payment using 4% interest compounded annually = 1800[append a page 690 row 3 in between 3 and 5%]

=1800[1.124 this is a guesstimate] = 1800[1.124864 from excel]

= 2024.76 from excel


300 times F/A factor at .08/12 for 36 months

= 12,160.67 from Excel


300 back from the down payment

= 2024.76 + 12,160.67 – 300 = 13885.43


Note on the lease I used an opportunity cost value of the 1800. Opportunity cost means what do I give up when I make a payment. The down payment was not required and the security deposit will be given back, so what does it cost to give up these values.

I took the 300 monthly and used the same rate that the car loan will occur at because I want to compare to the car loan.

I subtracted out 300 at the end because the security deposit is given back at time 36.


The car would be

[2500 down payment + 750 tax] [1.124864 from excel] = 3655.81


392 monthly payment times F/A factor at .08/12 for 36 months = $15,889.94


8000 value of car at time 36.


So, the car is the better deal.


Note, the emphasis on the problem we just did was the future. Most folks use the present as the emphasis. When we consider costs, the option that is chosen is the one with the LOWEST NET PRESENT COST.

The lease would be

Net present cost = 1800 + 300PV(.08/12,3*12,,-300) –300/power(1.04,3)

=1800 + $9,573.54 - 266.698908 = 11106.84

The car would be

Net present cost = 15000 + 750 – 8000/power(1.04,3) = 8638.02913