I was contemplating a purchase where the store offered $6$ equal monthly payments with no interest. The catch was a \$$30$ service fee!!!
What formula would I use to determine the actual annual percentage rate cost of the fee.
Let's assume the purchase was \$$1,000.00$ so the monthly payments would be \$$166.67$.
Thank you.
On
In effect you only borrow $\$970,$ since you pay back $\$30$ immediately. The annual rate of interest (what is called "APR" in the United States) is the number $i$ that satisfies the equation $$ 970 = 166.67\sum_{k=1}^6(1+i)^{-k/6}. $$ I get $i=0.05384$ or $i=5.384\%$
Note that I'm not saying that the store should be advertising an APR of $5.384\%.$ There are, no doubt, laws about what is to be taking into account in calculating the APR, and I know nothing of these. I just know how to calculate the APR once the laws have been taken into account.
Call the annual percentage rate "r". For one month you have borrowed the entire \$1000 so the interest charged is 1000(r/12). Then you make a payment of 1000/6 so for the next month you are borrowing 1000(5/6). The interest charged is 1000(5/6)(r/12). Then you make another payment of 1000/6 so for the third month you are borrowing 1000(4/6). The interest charged is 1000(4/6)(r/12). Similarly the interest charged for the fourth, fifth, and sixth months is 1000(3/6)(r/12), 1000(2/6)(r/12), and 1000(1/6)(r/12). The total interest charged would be 1000(r/12)+ 1000(5/6)(4/12)+ 1000(4/6)(r/12)+ 1000(3/6)(r/12)+ 1000(2/6)(r/12)+ 1000(1/6)(r/12)= 1000(r/12)(1/6)(6+ 5+ 4+ 3+ 2+ 1)= 1000(r/12)(1/6)(21)= (21000/72)r= (875/3)r. That will be the same as the "service charge" of \$30 if (875/3)r= 30 or r= 15/875= 0.017 so 1.7%.
It does occur to me that you have to pay the service charge "up front" where the interest would be paid monthly. That won't be much difference but you could take it into account by calculating the "present value" of the interest.