How much interest will I pay over the life of the balance?

Question

I am looking to solve an equation:

How much total Interest will be accrued on the whole balance of $250,000 calculated daily at 7%, knowing that $800 will be deducted every 7 days until the balance reaches 0.00?

1) Total Interest paid

2) How long in months or years?

Answer 1

The formula for number of periods is:

N = (-ln(1-(PV/pmt)*i))/(ln(1+i)), where PV is the loan amount, pmt is the recurring payment amount, and I is the interest rate.

The tricky thing to calculate is the interest rate. The APR is 7%, as stated in the problem, with daily compounding. However, payments are only made every 7 days. First we need to come up with the EAR of a 7% APR compounded daily, and then we need to turn that into a 7-day rate. The formula for EAR is:

EAR = (1+APR/n)^n-1, where n is the number of compounding periods, daily in this problem.

This gives an EAR of 7.25%. We can now rearrange the above equation to calculate the APR, but with (365/7) compounding periods instead of 365. That formula is:

APR = n*(1+EAR)^(1/n)-n, which gives us a new APR of 7.004%. We can use this ARP to plug into the original equation:

N = (-ln(1-(250000/800)*7.004%))/(ln(1+7.004%)) = 405.5 periods, which means we will make 405 payments of 800, and then the 406^th payment will only be 400, a half-payment to clear the balance.

To calculate the # of days, we should round the periods up to 406, which gives us 2,842 days, which is ~7.8 years or ~93 months. (We rounded up to 406 because the final payment will still be made 7 days after the 405^th payment, even though it is only half the amount).  

To calculate total interest paid, we know we make 405.5 payments of 800, which is $324,400. Since the principal was $250,000, that means $74,400 of interest was accrued (324,400-250,000).