Function Construction based on Data Tables

by

Rob Walsh

It always intrigues me how a little change in an interest rate can drastically affect what someone can end up spending on a something that is financed. Take, for instance, cars. People commonly purchase cars over a 60-month period at rates from as low as 0% to well over 10%. This loan is designed to compound at a continuous rate, which means there is a constant refiguring of the amount being borrowed and interest is recalculated to reflect the initial amount (principal) and the interest rate over a certain amount of time (generally in years) as it is accrued. Luckily, Leonhard Euler delivered us the infamous Euler number (aptly known as e) and from it we are rewarded with such things as the Continuous Compounding Formula: A = Pert. Given a principal dollar amount (P), decimal interest rate (r), and length of time in years (t), one can quickly figure what they will make (or in this example) owe at the end of a given period of time.

Let's assume a five-year loan period and take the total purchase price of a vehicle to be \$15,000. I use an Excel Spreadsheet to figure out the actual amount that will be spent over the five years, depending on the rate at which the car is financed. Here is a snapshot of the findings:

We use our rate column (r) as our x values and our amount column (A) as our y values. It is no surprise that even by fixing time, there is clearly a difference in borrowing a specified amount of money at various rates. In this case, for instance, we see that if you borrow \$15,000 at 6% interest, you can expect to pay back \$20,247.88 by the end of five years. In comparison, if you borrow the same amount at 11.5% interest, you can expect to pay back \$26,656.96! That is a difference of more than \$6400! We can also learn by a little graph analysis that the increase in amount paid back is slightly more than linear. Rather, the higher the interest rate, the greater the proportion that is paid back. The slight upward curve tells us this. So, not only are you paying a higher rate, but you are paying that rate on top of a larger sum. Moral: pay cash!!!