This is the third instalment of the risk in invoice-discounting series. Here we are going to take a look at a very specific view on how to set the advance, specifically as the strike of a put-option.

First, we introduce the idea, then we make some assumptions and, finally, we will be able to see how the advance rate affects the yield of the product!

### Introduction The idea is fairly simple. From our point of view, the customer payment is random, since we might not know about dilutions. In the above picture, assume the right pile is the amount the customer is supposed to pay and the middle and left payment include small and large dilutions. The blue line indicates the advance which the client receives in exchange for the customer payment.

Now consider that we only have to satisfy a claim in the size of the advance. Therefore, to the investors, there is no difference between full repayment and a small dilution. There is only a loss when there are large dilutions. The payoff profile is given by the blue line and coincides with the payoff profile of a put-option – a contract which gives the right, but not the obligation, to sell an asset at some point in the future for a previously agreed price.

### The maths part

In this part, we are going to make some assumptions about how exactly we think the customer payment is distributed and derive expressions for the expected payoff in this case. Feel free to skip this section to get the gist of this blog!

To make use of the observation explained in the introduction, we need some notation. Since the payment of the customer is never going to be bigger than his contractual obligation (the face value), it makes sense to model the payment as the percentage of the face value. Let C be this percentage. We can then calculate the expected payoff from a discounting and invoice as follows Where A is the advance rate and Ã is the amount the client has to repay, the advance and accrued interest. The first term is the expected payment of the customer, the second term is the amount which has to be spent to enter the contract. For C, a good and flexible choice of distribution, taking into account the severe skewness, comes from the following scheme: with probability p, the invoice defaults and the collection is beta distributed in this case. Everything is repaid with probability 1 – p. Some algebra yields where Fc, ƒc are the CDF and the density function of the underlying beta distribution.

### How does the advance affect the yield and collection rate?

To see how the advance affects the yield, let’s assume a contract with duration one month and a gross monthly interest rate of 1%, thus Ã = A(1 + 1%). We now find the expected net monthly yield r by solving the equation Fitting the values of p, α and β to our internal data (and neglecting some statistical issues), we find the following relation between the advance rate and the yield: A difference in yield coming from an advance rate of 50% and 90% seems to be around 0.09% in net monthly yield!

Further, we can look at the expected collection rate for different advance rates. This is essentially achieved by adjusting the value of p in the expression for to fit our definition of default and then plotting : So there is a difference of up to 6% in collection rate between an advance rate of 50% and 90%!

This shows how the advance rate influences yields and collection rates in a simple mathematical model. This also allowed us to get a quantitative feel for size of effects that the advance rate tends to have. However, the numbers in this blog are very crude averages and the yields/collection rates depend a lot on the specific cases, so don’t take them for granted!