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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