# Computing Annual Percentage Rates (APRs) when buying invoices

1. How to compute the annual percentage rate (APR) of interest when investing in invoices?
Suppose that an invoice of 10 000 € is to be paid by a firm in 50 days time. Assume that there are transaction fees of 0.2% per month and that the investor is willing to pay right now an amount of 9 900 €. How much will be the effective APR of that investment?
First, one should compute the total amount to be paid immediately (including transaction costs). In this case, since there are 50 days until maturity, the total amount of transaction costs are 0.333% (=(0.2% /30) x 50 = 0.333%) of the price to be paid for the invoice. This means that if such price is 9 900 €, the total amount to be paid by the investor will be 9 900 x (1+ 0.333%) = 9 933.00 €.
Next, the implicit 50-days interest rate can be computed as 10 000 / 9933.00 – 1 = 0.675%
Finally, to reach the effective APR one assumes periodic compounding of interest for identical periods (there are in the year 365/50 periods of 50 days, that is, 7.3 periods). Therefore, the APR will be
APR = (1+0.675%)7.3-1 = 5.03%

2. How to define a maximum price for the invoice in order to reach a desired APR?
Looking at the example above in 1., this now will require a little reverse engineering!
Suppose now that the investor wants to obtain, after fees, an APR of 5.03% and faces an invoice of 10 000 € which is due in 50 days while the transaction fees are 0.2% per month.
First, one should compute the corresponding interest rate for a 50 day-period. Since there are 365/50=7.3 periods of 50 days in a year, we will have, assuming periodic compounding, that such interest rate will be (1+5.03%)1/7.3 -1 = 0.675%
Then, since the Price (P) to be paid will also bear transaction fees of 0.2% per month, the total price to be paid will be
P (1+(0.2%/30)x50)=Px(1+0.333%)
Finally, in order to ensure a periodic interest rate of 0.675%, it must be that
0.675% = 10 000 / [P (1+0.333%)] – 1.
Thus,
P= 10 000 /[(1+0.333%)x(1+0.675%)]
therefore finally yielding the desired maximum price P = 9 900 €