# Computing Annual Percentage Rates (APRs) when a firm is selling invoices

1. How to compute the Annual Percentage Rate (APR) cost when resorting to the sale of invoices?
Let’s take the following example: Suppose a firm plans to “sell” an invoice that has a face value of 70 000 € and a maturity of 100 days. An investor offers to pay for this a price of 69 900 €. The transaction fees of 0.2% per month. How much will be the implied annual percentage interest rate (APR) in this form of financing?
From the standpoint of the invoice issuing firm, the APR of this financing can be computed by first comparing the net proceeds from the sale of the invoice, 69 900€, with the face value that would be received after 100 days (70 000 €).
Accordingly, we first compute the net amount after transaction fees, which will be:
69 900 x (1-(0.2%/30)x90)=69 434 €.
Next, the implicit 100-days interest rate can be computed as 70 000 / 69 434 – 1 = 0.815%
Finally, to reach the effective APR one assumes periodic compounding of interest for identical periods (there are in the year 365/100 periods of 100 days, that is, 3.65 periods). Therefore, the APR will be
APR = (1+0.815%)3.65-1 = 3.01%
This will be the implicit APR that can now be compared with the all-in cost of similar short-term loans with identical maturities!

2. How to define a minimum price for the invoice in order not to exceed a maximum APR cost?
Looking at the example above in 1., this now will require a little reverse engineering!
Suppose now that the firm wants to define a minimum price that ensures an indifference between the cost, after fees, of selling the invoice with the APR cost of short-term loans that currently stands at 3.01%.
First, one should compute the corresponding interest rate for a 100 day-period. Since there are 365/100=3.65 periods of 100 days in a year, we will have, assuming periodic compounding, that such interest rate will be (1+3.01%)1/3.65 -1 = 0.815%
Then, since the minimum Price (P) will also bear transaction fees of 0.2% per month, the minimum total price to be received by the firm will be computed as
P (1-(0.2%/30)x100)=Px(1-0.667%)
Finally, in order to ensure a periodic interest rate of 0.815%, it must be that
0.815% = 70 000 / [P (1-0.667%)] – 1.
Thus,
P= 70 000 /[(1-0.667%)x(1+0.815%)]
therefore finally yielding the desired minimum price P = 69 900 €