Calculating credit card charges based on provided APR, balance amount and monthly payment amount?

Question

Please first see the below sums and then the info below it to get a better understanding of my question:

My input:

Balance amount = $350
APR = 18%
Monthly Payment Amount = $60

The output:

Monthly Interest Amount = $5.25
Time to repay entire Balance Amount (months) = 7
Total Finance Charge Paid = $19.01

So I've been reading up on credit card APR, and I've used the many online calculators which gave me the above.

I know I could easily just use online calculators but I'd like to mathematically do so manually ie. whats the formulas based on my input to get the output?

I'd appreciate your responses to be in the "for dummies" style as you can say I'm not the brainiest of the bunch :-P (as I'm confused already!)

Appreciate all responses.

Answer 1

If you just want the formulas, check here.

Pretty scary...

For your type of problem, I think it would be best to just hack it out as follows (the computations are easily done in a spreadsheet):

Example: \$1000 is owed on a credit card that charges 3\% interest per month on outstanding balances. \$150 is paid on the card each month. Find the card balance for each month until the card is paid off. What is the final payment?

Solution: Note we have quite a bit to do here. We have to find the balance for each month, until the card is paid off. So, we first find the balance after one month. Then we find the balance after the second month. Etc...

This process will be simplified if we note that the change in balance between any two successive months is always $$\eqalign{ {\rm change\ in\ balance\ } &={\rm\ interest\ charged\ } - {\rm \ payment\ made\ }\cr &=(0.03)\cdot{\rm\ old\ balance\ } - 150.} $$ So, for any two successive months: $$ {\rm new\ balance\ }= {\rm old\ balance\ }+ (0.03)\cdot {\rm old\ balance\ } -150. $$ After one month, the old balance is 1000 and the new balance is $$ 1000 +(0.03)\cdot1000-150= 880. $$
After the second month, the old balance is 880 and the new balance is $$ 880 +(0.03)\cdot880-150= 756.40. $$
After the third month, the old balance is 756.40 and the new balance is $$ 756.4 +(0.03)\cdot756.4-150= 629.09. $$

Continuing in this manner we find:

The balance after month 5 is 362.90

The balance after month 6 is 223.79

The balance after month 7 is 80.50.

So, it takes 8 months to pay the card off and the final payment is $80.5*(1.03)=82.92$.

The total interest can be found by adding up the payments made and subtracting the initial balance.

Answer 2

David's answer covers the problem-solving method at work pretty well - I'll try to provide some of the financial and mathematical reasoning behind it, just to give you an idea of where the formulas on his link come from.

Essentially, this is time-value problem. You owe an amount to the card company today, and plan to pay it off in the future. Due to the potential interest the firm forgoes on money it loans to you, a dollar paid out a month from now is worth less than a dollar today. We can find the value of a dollar $n$ period from now today using the deflator: $$ \mbox{Present value} = \frac{1}{(1+i)^n}$$ where $i$ is the interest in that period. In your case, $i=\frac{.18}{12} = .015$. But, you're not making one payment in the future, you're paying out over a number of months. We can state the present value of a stream of payments in much the same way: $$ \mbox{PV} = P\frac{1}{(1+i)}+P\frac{1}{(1+i)^2}+P\frac{1}{(1+i)^3}+\cdots+P\frac{1}{(1+i)^n} = P\sum_{i=1}^{n}\frac{1}{(1+i)^i} $$ where $P$ is the payment, $PV$ is the loan balance in the 0th time period, and $i$ is your interest rate in every period. Solving for $n$ (a non-trivial task) gives you the time to repay. All of this assumes you begin payment in 1 time period (if you're making a payment in the 0th time perio,d just subtract it from the loan balance and continue).

You can then set up an amortization table with $n$ rows and columns for the current balance, that month's payment, interest accrued that month, etc, beginning with the balance of the loan as it stands in time period 0 and the specified payment running down that column. This is essentially what David has done in his answer.