# Financial Mathematics

## Lecture Notes 01

In the first lecture we started to learn about credit cards. We will come back later to this subject, so don’t worry too much if you didn’t follow. I will present first some general information about credit cards, and then I will give more details about the example from class.

In general, a credit card owner will use the credit card to pay for various things: food, gas, some utilities, concert tickets, vacations, and so on. The billing cycle or billing period is a time interval. At the end of the billing cycle, the bank sends the owner of the credit card a bill and gives a deadline for paying the bill. Usually, if you pay on time you are not charged any interest.

• How does the bank make money if the owner pays her balance on time?
• How is the bank able to give incentives like cash back? Why do they do it?
• Who do you think pays for all the goodies the credit card issuer entices you with?

For example, Alice has an Amazon.com credit card. The first billing period is 07/13/11 – 08/12/11. This means that on 12 August 2011 a bill is issued and sent to her (either by snail mail or electronically).
In that statement she will see the payment information: new balance (how much she owes on 12 August 2011), payment due date (the latest time is is allowed to pay without penalties), minimum payment due (the minimum amount she must pay). Here is an example:
PAYMENT INFORMATION
New Balance \$5,351.23
Payment Due Date 09/04/11
Minimum Payment Due \$52.00
Late Payment Warning: If we do not receive your minimum payment by the due date, you may have to pay up to a \$39 late fee.

If Alice pays the entire balance (\$5,351.23) before the due date(4 September 2011), she does not pay any interest for that billing cycle.
If Alice pays only a part of the balance, but at least minimum payment, say \$52, then she is charged interest, but she does not pay finance charges. This is is when one of the methods for calculating interest presented in the first class are used.
If Alice does not pay the minimum payment(\$52), then she is charged finance fees(\$39) and interest. Her interest rate may increase considerably, her credit score will go down, which will make it harder to get a good interest rate.

If Alice payed her bill on time the previous billing cycle she will see something like this on her credit card statement:
Previous Balance \$5,600.56
Payment, Credits -\$5,600.56
Purchases +\$5,351.23
Balance Transfers \$0.00
Fees Charged \$0.00
Interest Charged \$0.00
New Balance \$5,351.23
Opening/Closing Date 07/13/11 – 08/12/11

Here is an example of the ‘goodies’ I mentioned earlier:
AMAZON REWARDS CARD POINTS SUMMARY
Previous Month’s Points Balance 7,785
+ 3x points on Amazon.com purchases 1,196
+ 2x points on gas, dining, office supplies 1,724
+ Points earned on all other Visa purchases 3,080
- Points redeemed this period 5,000
= Remaining points balance 7,589
This means that by making certain purchases she earned points. Each point is worth one cent, but she is only able to redeem them for cash on multiples of 5,000. By redeeming 5,000 points she got a \$50 check in the mail.

Now, lets see what happens if she makes only minimum payment. I used the calculator at http://www.bankrate.com/calculators/managing-debt/minimum-payment-calculator.aspx with interest rate 14%, minimum payment Interest+1%. It will take her 271 months (22.58 years) to pay her debt and she will pay \$5,742.90 in interest only, \$11,485.8 in total.

• Is it a smart decision not to pay your entire credit card balance?
• How costly is it to be late with your payment, even, for just one day?

Now, I will present in detail the example from class. The data for the example is:
Previous Balance: \$2,500
Expenses and credits:
Day 1: \$200
Day 10: \$1,000
Day 20: -\$1,000
Day 25: \$500
APR: 13.24%
Cycle length: 30 days
This means that Alice started the billing cycle by owing \$2,500. She spent \$200 on the first day, \$1,000 on day 10, and \$500 on day 25. She paid back \$1,000 on day 20. The daily balance is how much Alice owes in one particular day. The balance for days 1-9 is \$2,700, for days 10-19 is \$3,700, for days 20-25 is \$2,700, and for days 25-30 is \$3,200.
First, we will calculate the interest using the daily accrual method. Remember, the interest is the sum of daily interest. We calculate the daily interest by multiplying the day balance by the interest rate per day. In our case:
Days 1-9: daily interest= \$2,700 * 0.1324 / 365 = \$0.97
Days 10-19: daily interest= \$3,700 * 0.1324 / 365 = \$1.34
Days 20-24: daily interest= \$2,700 * 0.1324 / 365 = \$0.97
Days 25-30: daily interest= \$3,200 * 0.1324 / 365 = \$1.16
For the interest we obtain:
I = 9 * \$0.97 + 10 * \$1.34 + 5 * \$0.97 + 6 * \$1.16 = \$33.94
Next, we will calculate the interest using the average daily balance method. The average daily balance is calculated by adding the balance for each day in the cycle and dividing by the cycle length. In our case: