Disney World Debt

Classwork

You want to know when you are ever going to use this stuff? Okay, I will spell out a scenario for you. But, since you asked for true life, this may not be pretty. Suppose you are trying to impress a potential mate. This person is a single parent of three children. You decide that all five of you should take a week's vacation to Disney World. Of course, you can't actually afford this, so you charge everything (all $4725) to your credit card. Foolishly, the bank has extended a $20000 credit line to you, so you can make the charge without even maxing out the card.

Once you charge something, you have a month's grace period before they start charging you interest on your debt. After that, you are required to make a minimum payment each month, while interest continues to accrue on the original debt (also known as the principal). A typical interest rate on a credit card is 16% per annum. We'll use that figure for our experiments.

1) Suppose your bank requires you to pay at least $80 each month (minimum payment). How long will it take you to pay off the full $4725? (You must take into account the interest they are charging you.) Keep track of the total amount of money you spend, so you can see just how much that vacation cost you. (By the way, the potential mate dumps you because you can't be trusted with money.)

2) If you have not done so already, write a recursive equation to model the situation in #1. Show how next month's debt level is derived from this month's debt level.

3) Generalize your equation so that the interest rate, the principal and the minimum payment are all variables.

4) Gee maybe that bank wasn't so foolish after all when they gave you such an unrealistic line of credit. Assuming you live long enough to pay it off, they make a tidy sum by lending you money. Look at the same situation as before, only this time, we'll say that the bank is nicer to you and requires just $60 as a minimum payment. Now what happens?

5) Experiment with different interest rates and minimum payments. See how those two parameters influence the situation. Try to generalize a formula for minimum payments (as a function of interest rate and principal) that allows the bank to maximize total profit.

6) So far, we have done all of these problems recursively. An explicit formula also exists, but it is very difficult to find. See how far you can get. You want to express what the level of debt is at any given month, without needing to know the previous month.