Deriving the credit card debt formula
Deriving the Credit Card Debt Formula
We will approximate a formula to calculate our account balance over time 10000 +
10000*(.12/12) assuming we have a $10,000 balance on the card. We will assume a 12%
APR, and will also assume that we are only paying 2% of our monthly balance. This is because credit cards ask you to only pay 2% of your debt at a time so that you keep your balance high and they get more interest from you!
Month 1:
10000 + 10000*(.12/12)
This is our original $10000 we borrowed, plus the monthly interest that they charge.
Now I need to subtract 2% of this total balance to represent making a payment for Month
1:
10000 + 10000*(.12/12) – [10000 + 10000*(.12/12)]* (.02)
So above is my formula for Month 1. Let’s make it prettier:
10000 + 10000*(.12/12) – [10000 + 10000*(.12/12)] *(.02)
Highlighted in pink above is a common term in both parts of the equation. I pull it out to get: (10000 + 10000*(.12/12))(1-.02)
Highlighted in green above is a common term in the first set of parentheses. I pull it out to get:
10000*(1+.12/12)*(1-.02)
So this is my formula for Month 1. Now onto Month 2! I need to take the balance from
Month 1 and add interest on that balance, then subtract 2% of that balance:
10000*(1+.12/12)*(1-.02) +10000*(1+.12/12)*(1-.02)*(.12/12)
This represents my balance from Month 1 plus that balance times interest.
Now I need to subtract 2% of that whole gigantic thing to represent making my minimum payment! It looks like this:
10000*(1+.12/12)*(1-.02) + 10000*(1+.12/12)*(1-.02)*(.12/12) - 10000*(1+.12/12)*(1.02) + 10000*(1+.12/12)*(1-.02)*(.12/12)*(.02).
Pull out the gray common terms to get:
(10000*(1+.12/12)*(1-.02) + 10000*(1+.12/12)*(1-.02))*(.12/12)*(1-.02)
Pull out the red common term to get:
10000*(1+.12/12)*(1-.02)*(1+.12/12)*(1-.02)
Finally, combine the matching sets of parentheses to get:
10000*(1+.12/12)2*(1-.02)2.
Hence, the formula for remaining debt over time is:
A = P(1+r/n)nt(1-.02)nt
A =
We will approximate a formula to calculate our account balance over time 10000 +
10000*(.12/12) assuming we have a $10,000 balance on the card. We will assume a 12%
APR, and will also assume that we are only paying 2% of our monthly balance. This is because credit cards ask you to only pay 2% of your debt at a time so that you keep your balance high and they get more interest from you!
Month 1:
10000 + 10000*(.12/12)
This is our original $10000 we borrowed, plus the monthly interest that they charge.
Now I need to subtract 2% of this total balance to represent making a payment for Month
1:
10000 + 10000*(.12/12) – [10000 + 10000*(.12/12)]* (.02)
So above is my formula for Month 1. Let’s make it prettier:
10000 + 10000*(.12/12) – [10000 + 10000*(.12/12)] *(.02)
Highlighted in pink above is a common term in both parts of the equation. I pull it out to get: (10000 + 10000*(.12/12))(1-.02)
Highlighted in green above is a common term in the first set of parentheses. I pull it out to get:
10000*(1+.12/12)*(1-.02)
So this is my formula for Month 1. Now onto Month 2! I need to take the balance from
Month 1 and add interest on that balance, then subtract 2% of that balance:
10000*(1+.12/12)*(1-.02) +10000*(1+.12/12)*(1-.02)*(.12/12)
This represents my balance from Month 1 plus that balance times interest.
Now I need to subtract 2% of that whole gigantic thing to represent making my minimum payment! It looks like this:
10000*(1+.12/12)*(1-.02) + 10000*(1+.12/12)*(1-.02)*(.12/12) - 10000*(1+.12/12)*(1.02) + 10000*(1+.12/12)*(1-.02)*(.12/12)*(.02).
Pull out the gray common terms to get:
(10000*(1+.12/12)*(1-.02) + 10000*(1+.12/12)*(1-.02))*(.12/12)*(1-.02)
Pull out the red common term to get:
10000*(1+.12/12)*(1-.02)*(1+.12/12)*(1-.02)
Finally, combine the matching sets of parentheses to get:
10000*(1+.12/12)2*(1-.02)2.
Hence, the formula for remaining debt over time is:
A = P(1+r/n)nt(1-.02)nt
A =