On the White Board - February 2009

Feb 17, 2009

What happened to my correlation?

The recent incremental risk charge (IRC), suggested by the Basel II committee (IRC Comments), has led to an increased interest in the multi-step simulation of credit portfolio losses. While moving from a one-period to a multi-period model might seem a triviality, aside from growing computational costs, there are some mathematical complications awaiting.

Say we have two obligors, which default when their asset return falls short of certain thresholds. This is the setup in the CreditMetrics model, and the common assumption is that the asset returns are bivariate Gaussian distributed. The joint PD is governed by the individual PDs, which determine the default thresholds, and the asset return correlation. Suppose we want to move to a finer time resolution and set up a multi-period model, where one simulation step corresponds to one day, one month, one quarter or one semester. For scaling the one-year PDs to smaller period PDs, we assume constant hazard rates. In each step, an obligor either survives or defaults with the scaled PD. The resulting process can also be seen as a coin flip experiment, and the constant hazard rate approach ensures that the probability of not surviving all steps (i.e., defaulting within one year) coincides with the one-year PD we started with. The last element concerns the dependence of the two obligors for the smaller time interval. Here it seems at first sight correct to take the same asset return correlation as for one year; this is motivated by a Wiener process hypothesis for the logarithmic asset prices.

Obligor PDs being preserved, we ask ourselves whether the same holds true for joint PDs in the multi-step model. The answer given in Figure 1 is clearly no. The plot shows the ratio of one-year joint PD in the one-period model and the one-year joint PD produced by the multi-step setup. Various number of steps per year are compared; 250 steps correspond to daily, 12 steps to monthly simulations, and so on. The obligors are AA-rated (one-year PD=1 bp.). It is striking that the oneyear joint PD shrinks considerably when several steps are simulated. For monthly simulation and an asset correlation of 20%, which is a typical value used in practice, the joint PD is reduced by a factor four. Therefore the naive way of multi-step simulation described above leads to a dramatic underestimation of risks.

We call the correlation which recovers the multi-step joint annual PD in the one-step model implied correlation. The implied correlation vs. actual asset correlation used in the model are displayed in Figure 2. Not unexpectedly, implied correlation is smaller than the actual asset correlation. To give an example, for monthly simulations an actual asset correlation of 20% drops to an implied correlation of 8.7%.

More can be said. The implied correlation converges to zero when the number of simulation

Figure 1: Reduction of joint PD from multi-stepping

Figure 2: Implied correlation resulting from multi-stepping

steps n tends to infinity. By asymptotic expansions, we obtain for the multi-step joint annual PD (for obligors having identical one-year PD) that

where λ = – log(1 – PD) and ρ_d(η) is the default correlation in one step, i.e., the correlation of the two default indicator random variables. The probability of default in one step, which is 1 – exp(– λ/η), converges to zero as η → ∞, and therefore the corresponding default thresholds to – ∞. Consequently ρ_d(η) is governed by the lower tail probabilities of the bivariate Gaussian distribution. It is well known that the bivariate Gaussian distribution (with correlation parameter different from ± 1) has asymptotically independent marginals, and for this reason ρ_d(η) → 0 when η → ∞. The effects we observe are in other words caused by the tail behavior of the bivariate Gaussian distribution, which resembles that of independent Gaussian random variables the further out one is in the tails.

-Daniel Straumann