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# 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 = 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 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
.

Value Type Enter Top 4 Values
Value of New Car
Value of Car Now
Age of Car Now in Years
Future Age of Car in Years
Future Value of Car Calculated

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 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.