This is not an accounting homework assignment. I took "Intermediate 1" two years ago (passed with an "A") and this concept was introduced but not well explained by the professor. I use the concept in my job and I report directly to the owners. They go with what I tell them but I cannot tell them the "why."

The text said that by not taking the 2% discount within the first 10 days of the month is like paying 36.5% interest or at least not earning 36.5% interest on your money. The equation given to determine the percentage rate one would pay (or not earn) by not taking the discount is as follows:

.02 divided by 20/365

This equation will, in fact, give you 36.5%. One can change the 2% to 1% or even 3% or whatever the discount might be. Every percentage point is equal to 18.25% such that a 1% discount is equal to 18.25%, a 2% discount is equal to 36.5%, a 3% discount is equal to 54.75%, and so on.

I know sometimes in accounting one just needs to accept the equation that goes with the concept and have faith that the concept is correct. While I have committed the equation to memory and can substitute any percentage discount a vendor might give me, I do not understand the reasoning for the equation.

Clearly, if one does not pay the discounted bill within the first 10 days, then that leaves 20 more days in the month in which he/she would need to pay the bill in full (assuming a 30-day month, of course). The portion of the equation "20/365" results in the percentage of a whole year represented by 20 days. Rounded, it comes to .0548, or 5.48% of the year.

So my question is: Why does one divide the .0548 in to the discount percentage which, in this case, is .02? And, when one does, and one gets the percentage of 36.5%, how does that represent the equivalent of the interest one would pay by not taking the discount?

For example, say one had a bill to pay of $535 with the terms outlined in my example. He/she would only pay $524.30 and save $10.70 if the bill is paid within the first 10 days of the month. I just cannot figure out how paying $10.70 in the last 20 days of the month is like paying 36.5%, even if it is over the course of a year.

For example, if one started the year on a credit card and had a balance of $535 with an interest rate of 36.5%, then the monthly rate would be 36.5/12 or 3.04%. If one applied 3.04% to $535 and then continued to apply that percentage to the unpaid balance every month for the full 12 months, at no point would one get to $10.70 for a monthly interest charge. The closest one would get would be $10.89 in the 14th month and $10.56 in the 15th month.

I am sure I am missing something here and, once explained to me, a light bulb will go off as to how all of this connects to the main concept. What am I missing? Please accept my apologies for being so dense. Thanks for your help.

Sincerely,

Jim

Hi taking a settlement discount is an important aspect of business and it does represent an outrageous level of interest on the money but it is done for a different reason. There are serious costs associated with collecting slow accounts and extending your business borrowings to cover extended credit

JimHerbert, the formula you're using is actually a shortcut one which slightly understates the true rate. Using your example, the logic runs as follows...

Taking advantage of the discount means paying 524.30 on the discount due date. Not taking advantage of the discount means not paying 524.30 on the due date, but instead paying 535 twenty days later.

This is equivalent to saving 524.30 now, at the expense of paying 535 twenty days hence. In turn, this is also equivalent to the cash flows of borrowing 524.30 today with a repayment 20 days later of 535, with that latter repayment representing principle of the borrowed 524.30 plus interest of 535 - 524.30 = 10.70.

Thus you've paid 10.70 of interest on a debt of 524.30 over a 20-day span, which is a per-diem interest of 10.70 / 20 = 53.5 ¢ per day. At that daily rate, your total interest would've been 53.5 ¢ x 365 days = $195.275 if the loan had been for one year instead of just 20 days (this is the "annualizing" part of the computation).

So the one-year interest rate is 195.275 / 524.30 = 37.24%

Equivalently, suppose the discount due date had arrived, and to take advantage of the discount you borrowed the 524.30 from a bank, which you used to pay the discounted invoice. The bank charges you 37.24%, and it's to be a 20-day loan.

At that rate, you're accruing interest at the pace of 524.30 x 0.3724 / 365 = 53.5 ¢ per day. So 20 days later you repay the bank the borrowed principle, plus interest: 524.30 + 20 x 53.5 ¢ = $535, exactly what you'd have paid on the invoice had you simply skipped the discount opportunity.

Once you crunch the logic into numbers, you'll see a more accurate rendering of the formula is

\frac{r \ \times \ D}{(1-r) \ \times \ d}

... where r, D, and d are the discount rate, the number of Days in the year (usually 365 but sometimes 360 is used), and the number of days from the discount due date to the full due date, respectively.

Thus, forgoing a 2/10, n/30 discount offer bears a true cost of

\frac{0.02 \ \times \ 365}{0.98 \ \times \ 20} = 37.24%, as an annualized rate.

The reason your formula understates the rate a bit is that it doesn't have the "0.98" factor in the denominator. That (1 - r) factor is in the denominator because if you skip the discount, you've effectively "borrowed" 98% of the full invoice amount (assuming a 2% discount offer) for 20 days.