Now that we know the value of the debt in three years, (the time we wish to have the debt paid off), we want to figure the monthly payments needed to bring the future value of the debt to zero. We now need to use the present value of an annuity formula to find the payments. The present value of an annuity formula is given as:
PV = CF * [(((1 + r)n) – 1) / r]
where PV is the present value, CF is the cash flow (or monthly payments), r is the interest rate per period, and n is the number of periods. For our example, we must now think of the future value figured above ($7153.84) as the present value of the debt because it is the amount we would have to pay in three years were we not to make any payments at all. Plugging our numbers into the present value of an annuity formula we have:
7153.84 = CF * [(((1 + .01)36) – 1) / .01]
7153.84 = CF * 43.0769
CF = $166.07
So, at 12% APR over three years with monthly compounding of interest, a cardholder would need to pay $166.07 per month over three years to pay off the $5,000 debt. Notice that the total amount paid would be $5,978.57 ($166.07 * 36). This gives an effective interest rate of about 19.57% ((5978.57 – 5000 / 5000)) over the course of three years.
The two steps given above can be used to figure the payments for any number of periods and any interest rate. For example, suppose we wanted to know the payments necessary to bring the debt to zero over one, two, three, four, and five years at the 12% APR interest rate. Using the two-step process above, we can figure out these payments as:
1 year = $444.24
2 years = $235.37
3 years = $166.07 (our example above)
4 years = $131.67
5 years = $111.22
Notice that the amount needed to pay off the debt as the number of periods increases does not follow a linear pattern. This is because of the term structure of interest rates. Essentially, cash flows far out into the future do not contribute much to the present value of debt (or an investment). Using this information wisely, a debt holder can figure out the payments that match what he/she is capable of paying.