# Mathematical definitioln of Potential Future Exposure

I have come across a risk measure called "Potential Future Exposure" and I have not really understood the meaning of it. Knowing that this has to do with counterparty credit risk, I read different pages, for example here: http://www.thetaris.com/wiki/PFE that for in continious time PFE is defined by:

$PFE(t)=\int_{-\infty}^{+\infty}max(x,0)f(x)ds$.

How should I intrepate this? Is $x$ an underlying asset and $f(x)$ its probability distribution? Also I heard that PFE's are usually simulated by monte carlo methods. Are the assumptions to use standard geometric brownian motions? Also, I have really tried to find som literature on the topic covering PFE, but it seems that it is very limited. Is there anyone who knows or have recommendations on literature/articles in this topic?

Thank you.

When you enter into a derivative trade, such as a swap, the intial value is zero; as interest rates change the value may become positive or negative. If it is positive and the counterparty defaults you could be out a big sum of money (imagine that you are trading with Lehman Brothers in 2008). The idea of PFE is to estimate at some time in the future the typical positive value (hence max(x,0) ) to estimate how big is this risk: that the trade has gone in your favor but the counterparty can't pay.

The way PFE is found is to simulate the derivative (swap) value by MonteCarlo methods over all possible future interest rate paths.

PFE is a hot topic with a lot of discussion currently; it is mentioned in almost any treatment of credit risk under Basel III and Dodd Frank.

Let's discuss what a PFE is before looking at the equation.

PFE is a common statistical measure for the amount of money you'll lose if your counterparty defaults. Let's give an example, say if you were to long 1000 far-in-the-money call options with a bank. Those options worth a lot to you because they're all in-the-money, it's something that you want to exercise upon maturity. However, what if the bank defaults (assume zero recovery)? You'd lose everything! In this example, your future exposure would be the value of your options, discounted back to today.

Now, what if you were to short those 1000 call options? Your portfolio would be negative. Now if the bank defaulted, you shouldn't be unhappy because you weren't supposed to get anything from the bank anyway. In credit risk, we say there is zero exposure.

Thus, counterparty exposure can be defined mathematically as: max(x, 0). We'll also need a probability distribution for the future. f(x) in your equitation defines the probability distribution. The integration gives you the average. You can think of the equation like: "average all the possible future exposure".

PFE is indeed commonly estimated by Monte-Carlo methods. How it's simulated depends on what it's simulated. For example, if I'm interested in how the interest rate would affect my portfolio. I would simulate the interest rates (eg: LIBOR) by modelling the change in basis points. I would set the time horizon to say, 2 years. Too short horizon would give unrealistic PFE as the default probability is usually quite low. Then I would take the 95% quartile of the simulated distribution as my PFE.