# Credit card pay back period calculation

What is the industry standard equation for calculating the payback period of a credit card? For example: Mark has a credit card with a rate of 17%. He has a balance of 5500 and a limit of 5500. If he just paid the minimum, how long would it take him to pay it back?

• Minimums vary by credit card apparently. What is the minimum in this case, or should we just pick one? (e.g. \$20 or 2% whichever is higher) – xirt Apr 5 '18 at 5:45
• Yeah let's pick one – Jeremy Bray Apr 5 '18 at 5:46

Without considering fees, the present value of a loan is equal to the sum of the repayments discounted to present value

``````s is the principal
n is the number of periods
d is the periodic repayment
r is the periodic interest rate
`````` ``````s = (d - d (1 + r)^-n)/r

∴ d = r (1 + 1/(-1 + (1 + r)^n)) s

and n = -(Log[1 - (r s)/d]/Log[1 + r])
``````

Nominal vs. Effective Interest Rates

APR is the USA is quoted in nominal terms, so if the APR is 17% nominal compounded monthly the monthly rate is

``````r = 0.17/12 = 0.0141667
``````

In Europe APR is quoted in effective terms, so with 17% effective annual interest the monthly rate is

``````r = (1 + 0.17)^(1/12) - 1 = 0.0131696
``````

Proceeding with your example as if you are in Europe

``````r = 0.0131696
s = 5500
``````

If the minimum monthly repayment `d = 100`

``````n = -(Log[1 - (r s)/d]/Log[1 + r]) = 98.4852
``````

It would take 99 months to repay the loan, with a smaller final repayment.

In the USA, with `r = 0.0141667` it would take 108 months.

Edit

Further to Dilip's post, noting that the minimum payment would decrease...

With `a` representing the fraction of the balance (prior to interest) to be paid, say 30%, the principal remaining at the end of month `x` after the repayment has been made is derived from

``````p[x + 1] = p[x] (1 + r) - a p[x] where p = s

∴ p[x] = (1 - a + r)^x s
``````

So with

``````s = 5500
r = 0.17/12 = 0.0141667
a = 0.3
``````

The principal remaining at the end of month 9 is

``````p = 265.802
``````

If the minimum repayment is \$100 the next two repayments will be \$100

``````p = p (1 + r) - 100 = 169.568
p = p (1 + r) - 100 = 71.9701
p = p (1 + r) - 72.9897 = 0
``````

With 30% minimum repayment capped to a minimum of \$100 the credit card debt is repaid in 12 months.

Up until the cap comes into play the repayment `d` at the end of month `m` is given by

``````d[m] = a p[m - 1]

E.g. d = 1650
d = 1178.38
`````` • When I learned in university that Americans and Canadians quote APR in nominal terms I was baffled. Why give the percentage in yearly terms if that's not the interest over that year? (I guess that is the cost to continually service the loan but nonetheless it is strange.) – Lan Apr 5 '18 at 15:15
• @Lan how else would it one refer to an interest rate for something like credit cards? – quid Apr 5 '18 at 17:50
• @quid Monthly rate. – Lan Apr 5 '18 at 17:57
• @chrisdegnen, sorry mate, what is APR? – Jeremy Bray Apr 5 '18 at 21:58
• @JeremyBray APR is annual percentage rate. – Chris Degnen Apr 5 '18 at 22:08

In the US, minimum monthly payments of credit card debt are required by law to consist of the sum of all interest and service fees charged that month plus a fraction of the statement-closing_day balance (before interest and fees are tacked on) so as to ensure that the balance is paid off in a "reasonable time". (There is also a de minimus out: when the minimum monthly payment gets to be under \$X (something like \$5 or \$10), the entire balance can be made due right away). Most US credit-card companies take reasonable time to mean 8+ years since most of them set the minimum payment to be 1% of the outstanding balance. If you began with \$5500 owed, and paid all interest and fees due + \$55 each month, you would pay off the debt in exactly 100 months or 8 years and 4 months. In actuality, it would take longer because the minimum required payment would decrease from "interest + fees + \$55" each month as the statement-closing-date balance would decrease month after month. The minimum required payment would also decrease because the interest charged each month decreases because the outstanding balance decreases. It takes iron will to choose to pay extra when the first option on the "Make a payment" tab on the credit-card web page is "Pay minimum required amount".

So much so for the theory. In practice, what the US credit-card companies count on is that most cardholders will continue charging purchases (or even better for the company, cash advances) to the card so that the balance is back to the \$5500 limit each month and so making minimum monthly payments will never reduce the month-end balance below \$5500. By law, minimum monthly payments must be applied to interest and fees first, and the excess can be applied to any part of the balance, including the 0%-rate balance, if any. The law requires only that the amount in excess of the minimum monthly payment be applied to the part of the balance being charged the highest interest rate.

• thanks for taking the time Dilip. this is really helpful knowledge to add to the equation – Jeremy Bray Apr 5 '18 at 22:48

It depends on what the minimum is. If it is less than or equal to the interest (at 17% on \$5,500 that comes to \$77.916 per month) then he would never pay it off.

If it is just \$20 of principal per month (plus the interest), e.g. \$97.92, then \$5,500 divided by \$20, which is 275 months, or 22 years, 11 months.

• I am more after the equation that includes the compounding interest – Jeremy Bray Apr 5 '18 at 5:53
• Assuming the principal is being paid down, the interest is \$77.916 only in the first month. In subsequent months it becomes progressively less. – Chris Degnen Apr 5 '18 at 9:11