Using Algebra To Predict Your Car's Future Value
Introduction
Exponential growth or exponential decay occurs when a fixed percentage
is added or subtracted each time period. For example, money
earning interest in a bank grows exponentially since it earns a fixed
amount of interest each year. I theorized that a car would lose it's
value in an exponential manner since it's components lose a percentage
of their value due to weathering, wear and tear, and age.
Mathematical Model
Let V = value of car at time t years of age
Let r = rate of depreciation per year (ex. 5% yearly
depreciation implies r = .05)
We get, via calculus, V = V_{o}e^{rt }
where V_{0} is the value of the car when new.
Obtaining a Mathematical Model
I had obtained the following data 5 years ago on a 1997 Dodge Grand
Caravan SE minivan:
Retail New Price  About $20,000^{1}
Value After 5 Years  About $7873^{2}
Since 20,000 is the initial amount, we let V_{o} =
20000 to get
V = 20000e^{rt }
When t = 5, V = 7873. We plug in these values to get
7873 = 20000e^{5r }
Solve this equation for r
7873/20000 = e^{5r }
[LN(7873/20000)]/5 = r =  0.18645, which equates to about an
18.6% loss in value each year.
So our equation becomes V = 20000e^{ 0.18645t }
Now, Test Out The Mathematical Model
In the year 2007, t=10 and our model predicts that the value will be
V = 20000e^{ 0.18645●10 } = $3099
The value I obtain from
Kelly Blue Book by averaging the retail value and the tradein
value is $2965, assuming the vehicle is in "Good Condition". If
the car is in "Excellent Condition", the private party value is listed
as $3100. The estimate provided by our
model is right on!
A graph displaying the results is shown here
Letting a Computer Do The Math  The Car Value Calculator
The Car Value Calculator below uses input data to find the constant
r of the mathematical model and then calculates a future value.
Let's say we were to use this calculator back when the vehicle was 5
years old. We would have entered 20000 for the Value New and a Value
of 7873 at 5 years of age. We would enter 10 for the future age
of the car and we would get $3099
as shown in this
screen shot.
This Car Value Calculator predicts the future value
of your auto.
Enter the top 4 values on the right. NO dollar signs!
Click on CALCULATE.
How This Calculator is Made
This calculator uses a coding language called
Javascript, a coding language that is imbedded into the web page
and executed by the browser of the person visiting the page. A portion
of the code used to run this calculator is shown in
this screen shot.
As you can see from the screen shot, the Javascript code makes use of
variables and functions.
1. Actually, the sticker price that includes
destination charges was about $21335 but it is assumed that the
average buyer could get a reduced price of about $20000. This sticker
price was obtained at
http://edmunds.com.
2. This is the average of the tradein value
and the retail value as obtained at
Kelly Blue Book, These
values were obtained in the year 2003 as part of another project.
