That's a challenging question, I promise ;). I guess it will also take
some time to answer ...

Most of credit card issuers use average daily balance (ADB) to
calculate Finance Charge (FC). There are two ways:
1)average daily balance method, including new purchases;
2)average daily balance method, excluding new purchases;
Most of credit cards have Grace Period. A grace period is the time you
have before a credit card company starts charging you interest on your
new purchases -- usually a period of 20 to 25 days. There are two ways
of how to use Grace Period to calculate the Finance Charge:
1) A card with a typical grace period has an average daily balance
including new purchases as the balance calculation method. This means
you pay interest on all new purchases immediately, unless you have
paid your previous month's bill in full.
2)A card with a full grace period has an average daily balance
excluding new purchases as the balance calculation method. New
purchases are not included in figuring the amount of interest you owe
for the current month. Unlike the typical grace period described
above, you get the benefit of the grace period, whether or not you
paid off your balance in full the previous month.

  So, we are interested in two methods of calculating interests:

a)Average Daily Balance Method including new purchases
b)Average Daily Balance Method excluding new purchases

Now the question is how to calculate the following examples?Finance
Charge using both ways described above.

Specifications of the Card:
Grace Period(GP) = 20 days.
APR = 18.0%
No of Days in a Billing Cycle (BC)=30
Finance Charges(FC) = the interest calculated on the credit. We
disregard late fees, annual fees etc.


Case 1:
Billing Cyle 1:
Previous Balance =0$
day 1 : purchase = 100$  payment=0
da2-day 14 : No activity
day 15: purchase=0$  payment=20$
day 16:purchase = 0$ payment = 20$
day 17: purchase = 0$ payment = 20$
day 18-day 30: No activity

Billing Cyle 2:
Day 1-Day 20: payment=0$
(to show that there has been no payment towards the previous month?
purchases during the
grace period)


Questions:
1)So when the statement for BC 2 is made, it contains a statement of
the Financial Charges. Does the statement for BC2 contains the FC on
the balances of BC1 ??
2)How do we calculate (in a step-by-step manner) the FC for BC1 using
the two methods mentioned above  ?

Case 2:
Billing Cycle 1:
Previous Balance =0$
day 1: purchase = 100$ payment=0
day2 ~day 30: no activity

Billing cycle 2:
day 2 of Grace Period: payment 20$
day 10 of Grace Period: payment 20$
day 15 of Grace Period: payment 20$

Questions:

How do we calculate (in a step-by-step manner) the FC for BC1 using
the two methods mentioned above  ?
.

Case 3:
Billing Cyle 1:
Previous Balance =100$
day 1 : purchase = 100$  payment=0
da2-day 14 : No activity
day 15 of Grace Period: purchase=0$  payment=20$
day 16 of Grace Period: purchase = 0$ payment = 20$
day 17 of Grace Period: purchase = 0$ payment = 20$
day 18-day 30: No activity
Billing cycle 2:
day 2 of Grace Period: payment 20$
day 10 of Grace Period: payment 20$
day 15 of Grace Period: payment 20$
day 20 : purchase = 100$  payment=0

Questions:

How do we calculate (in a step-by-step manner) the FC for BC1 using
the two methods mentioned above?

There is some information available in following urls:
http://www.credit.com/slp/chapter4/Finance-Charges.jsp
http://www.hiwhy.com/finance/credit-cards.shtml
http://home3.americanexpress.com/corp/consumerinfo/grace.asp