Using Algebra To Predict Your Car's Future Value
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.
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 = Voert
where V0 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 $20,0001
Value After 5 Years - About $78732
Since 20,000 is the initial amount, we let Vo =
20000 to get
V = 20000ert
When t = 5, V = 7873. We plug in these values to get
7873 = 20000e5r
Solve this equation for r
7873/20000 = e5r
[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 trade-in
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
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
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.
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
2. This is the average of the trade-in value
and the retail value as obtained at
Kelly Blue Book, These
values were obtained in the year 2003 as part of another project.