Understanding the Mathematics of Personal Finance


The ADB method of calculating finance charges that I presented above is not the only way that finance charges are calculated. Other approaches include the adjusted balance, the two-cycle ADB, the previous balance, and the ending balance. As best as I could ascertain, the ADB method is used by major credit card companies in the United States.

If you wish to learn more about the other approaches, try these websites:

1. www. finweb. com/banking-credit/how-credit-card-finance-charges-are- calculated. html;

2. http://money. howstuffworks. com/personal-finance/debt-management/credit- card8.htm;

3. http://www. bankrate. com/brm/green/cc/basics3-2a. asp.


A debit card looks almost exactly like a credit card. It is, however, completely different from a credit card. When you charge a purchase or get a cash advance with a credit card, you are borrowing money. When you pay for a purchase or get cash with a debit card, you are accessing your own money from a designated bank account. It’s pretty much the same as writing a check except that it’s more convenient, and the electronic fund transfers occur immediately. Also, many stores are wary of taking a check because the check might bounce (your account did not have enough money in it) or the check might be fraudulent or forged to start with. When you use a debit card, the store verifies the available funds in your account and gets its money immediately.

There are two common ways to use a debit card. You can pay for, say, groceries at the supermarket with your debit card. When you swipe your card and then enter your secret personal identification number (PIN) into the little machine, your pur­chase is debited from your bank account. Typically, at the same time, the machine asks if you’d like some cash back. If you say yes and then enter the amount you’d like, the store cashier will hand you cash along with your receipt—as if you had paid cash and were getting change. This cash is of course deducted from your account along with your purchase. There are usually no fees for purchases or pur­chases accompanied by cash back.

The second way of using a debit card is to go to an automatic teller machine (ATM). These machines let you swipe your card, enter your PIN, and request a cash disbursement. These disbursements are offered in discrete amounts such as $60, $80, $100, and so on. Because the machine is only loaded with certain denomination bills (in this case $20 bills), an arbitrary amount request cannot be handled. Often, ATMs will charge a fee for the transaction. This fee can be a combination of a fee from your bank and a fee from the owner of the ATM if it is not your bank’s ATM. Read everything you can find at an ATM—on the front of the machine, on the wall above the machine, on the screen as you enter information, and of course on your debit card to make sure that you understand what fees will be charged. These fees are not interest because there is no time element involved.

Remember that cash advances accrue interest starting from the day you take the cash advance, with a minimum interest (finance charge) applied. This means that the percentage cost of taking a cash advance varies directly with the amount of money you take and the length of time you take to repay this money (the minimum finance charge could skew this calculation a bit). On the other hand, the percentage cost of cash taken from an ATM not owned by your bank is high for small advances and low for large advances. Also, since it’s your own money that you’re using rather than borrowed money, you lose savings or checking account interest until you repay the money to your account.


1. Assume a purchase interest rate of 0.0333% per day. The current month has 30 days, and the billing period is from day 1 to day 30. You have no previous balances.

(a) You charge a $150 purchase on the first day of the month. What do you owe if you pay this fully before the due date?

(b) Your credit card company receives your payment of $100 on the third day of the next billing period. This payment is received before the due date so there are no penalties. What is your daily balance on the second day of the next billing period and on the third day of the next billing period?

(c) Assuming that this “next” billing period has 31 days, what is your purchase balance at the end of this month’s statement (assuming that you don’t use the card at all)?

2. You are always a little short of cash before your payday, which is the last day of each month. The last day of your credit card billing period is also the last day of each month.

On the twentieth day of each month, you get a cash advance of $250. Since you see your credit card bills electronically, your paycheck is automatically deposited into your check­ing account and you pay your bills from your checking account electronically; there are no unanticipated delays. On the first day of every month, you pay your credit card bill, and it reaches the credit card company on the third day of the month. Assume a cash advance interest rate of 0.0500% per day and that every month has 31 days. How much is your monthly credit card bill for your cash advances? Assume that you fully pay your bill each month.

3. Assume a purchase daily interest rate of 0.03% and a cash advance interest rate of 0.06%. You make a purchase of $150 on the first day of the month and take a cash advance of $250 on the twentieth day of the month (a 31-day month). You will pay your bill on the first day of the following month, and it will be received by the credit card company 2 days later (on the third day of that month). You do not have enough money to pay the entire bill when you first receive it, so you pay 50% of the bill. Assuming that you never use your card again and after the first payment you fully pay your bills, calculate the first two bills and payments.

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