Many believe that the calculation of exposure at default (EAD) on derivative contracts is fairly straightforward, so it can easily be done analytically. In many cases it is true, but not always.

Let us consider an easy case when EAD can be calculated analytically. By looking at how we do that, we will discover under which circumstances the method would not work.

Being a measure of our losses in case of the default of a counterparty, EAD is defined informally as “the amount of money that the counterparty owes us at the moment of default, at a certain confidence level.” More formally, if we have a contract C with counterparty Z, function MtM(C,T) is the mark-to-market value of C at time T, D is the event “Z defaults at time T” and L is the confidence level, then

where x is such that

Here, P[A|B] is the probability of A conditional on D.

For example, saying “with 95% confidence our exposure to X is â‚¬100” is equivalent to saying “we know with probability 0.95 that if X defaults, it will owe us not more than â‚¬100 at the moment of default.”

Suppose we have an FX forward contract with a counterparty: on October 3, 2010 we will pay them EUR 100 and receive USD 133. Today (October 3, 2009) the exchange rate is r(0) = 0.75 (that means USD 1 = EUR 0.75), and the risk-free interest rate in both currencies is 5% per annum [*Note: real rates are different, these numbers are only an example*]. Today (T=0) the contract is at par: MtM(C,0) = 0. We want to calculate EAD for T = 1 year with 95% confidence. That means that we have to find the maximum of MtM(C,1) with probability 0.95. We will calculate exposure in euros.

It is easy to see that MtM(C,1) = 133*r(1) – 100, where r(1) is the USD to EUR exchange rate at T = 1 year (October 3, 2010). The MtM is monotonic in r(1), so we can find EAD if we have the maximum of r(1). At this point we have to choose a model for the evolution of exchange rate r(t). Let us take geometric Brownian motion with annualised drift d = 2% and volatility Ïƒ = 8% [*again, these numbers are only an example; real parameters can be obtained from historical data*]. Then

where N is a standard normal random variable. In turn, this expression is monotonic in N, so we need to find the maximum of the standard normal random variable attained at probability 0.95. This is given by the inverse cumulative distribution function ; it is not available in closed form, but can be computed numerically. Using NORMSINV function of OpenOffice Calc, we find that

Substituting the values for r(0),Â d, Ïƒ and N, we get

(we drop t because t=1). Finally, r(1) = 0.87. In other words, with 95% probability US dollar will not be worth more than 87 eurocents on October 3, 2010 (remember, though, that we obtained this value from a model calibrated to arbitrary values of drift and volatility). This means that our exposure EAD = 133*0.87 – 100 =Â 15.71 in absolute value, or 15.71% of the principal amount of the transaction.

The above calculation was not difficult because:

- There was a single contract with a single underlying risk factor r.
- The MtM of the contract was monotonic in r.
- The chosen model for r(t) was analytically tractable.

However, if we had had many contracts with the counterparty or the contracts had depended on multiple risk factors, it would have been impossible (or prohibitively complicated) to find EAD analytically.

One way of calculating EAD on a complex portfolio (actually, the *only* way I am aware of) is market simulation, which is essentially a Monte-Carlo simulation. In brief, the method is as follows.

Let P be the portfolio of contracts with counterparty Z. P depends on underlying risk factors r1,…,rN. These risk factors might include interest rates, exchange rates, stock prices, etc. We have models for each risk factor.

We draw a set of random numbers and calculate simulated values of r1,…,rN at time T. Knowing the underlying risk factors, we calculate MtM(P,T). We repeat this process sufficient number of times (say, 10000) and thus generate 10000 simulated values of MtM(P,T). This gives us the distribution of MtM(P,T) implied by our models for r1,…,rN. Now, our EAD(Z, P, L) is the L*th* quantile of the MtM distribution.

This method scales to portfolios of any size, provided that we have enough processing power. Fortunately, it is fairly easy to run a Monte-Carlo simulation in parallel: if we need to do M iterations and have k processors to do the job, we let each processor do M/k iterations. From my experience, the aggregation of results and quantile calculation takes neglidgible time (compared to MtM calculation), so these can be done by a single processor (it is more convenient to implement).