This is the second of three articles regarding a heuristic approach to measuring counterparty credit risk. In the first part, author Robert Garzottoexplained the approach broadly. In this part, he tests the approach using a static portfolio, that is, a portfolio with no new positions added over time. In the final part, he tests the approach using a portfolio in which new transactions added over the course of its life.
ASSESSING THE STABILITY OF THE TERM STRUCTURE OF EXPOSURE
In essence, the simulation approximation hinges on the assumption that exposure profiles are actually quite stable over time. More precisely, for a given portfolio, the term structure of potential exposure generated over the weekend should be very close to the exposure profile generated at the end of the following week despite the changes in risk factor values or the effects of instrument amortization (aging). We felt it would be instructive to first test this assumption for a static portfolio before attempting to model the arrival of new transaction data. This exercise also provided the answers to a number of other important questions:
* Would "re-centering" the exposure profile provide
better results than simply re-using the stored Net Present
Value vectors?
* How would the choice of instruments effect the results?
Would non-linear instruments introduce greater error?
* Would the approximation introduce any systematic bias, for
example, always underestimating the future potential exposure?
To answer these questions, the first phase of the analysis focused on very small static portfolios of a single instrument type.
The foundation for the analysis was a portfolio containing 3000 randomly generated positions in the following instruments: vanilla interest-rate swaps, swaptions and caps. For each instrument type, a small portfolio was constructed by choosing two transactions from the large portfolio such that:
* They each had a maturity greater that 41Ž2 years
* They each had exposure to a different currency yield
curve
* Their initial netted exposure was very close to zero.
In order to model potential changes in the initial risk factor values over the course of the following week, 10 Monte Carlo scenarios were generated over a four-day horizon. A four-day horizon was chosen since this represents the worst case-- combining new transactions generated on Thursday with delta-NPV vectors generated the previous weekend. See Table 1 for a detailed description of how the instrument parameters were randomized. Using this transaction and market data, four separate exposure profiles were generated for each of the 10 scenarios:1
1. Original exposure profile: based on a 2000 Monte Carlo
scenario simulation using initial risk factor values
(at time t). Here the delta-NPV vectors were calculated
and stored.
2. "Re-centered" exposure profile: based on adding the new
portfolio value (at time t+4 days) to the stored delta-NPV
vectors generated above.
3. Full simulation A: based on the results of a new 2000
scenario full simulation using the new risk factor values
(at time t+4 days). This was used as the benchmark.
4. Full simulation B: same as full simulation A above.
This additional simulation was performed to establish the
consistency of results between full simulations.
Clearly, the quality of the approximation depends on some measure of the "goodness-of-fit" between the approximated exposure profiles and the benchmark full simulation. The metric chosen for this analysis was "average unexpected exposure" (AUE), which is simply the average of the potential exposures (99th percentile, one-tailed) at each time step.
RESULTS
Table 2 summarizes the statistics on percentage error in AUE relative to the benchmark full simulation for each small portfolio. Recall that the statistics are based on 10 trials (four-day scenarios). Overall, the results indicate that the exposure profiles are quite stable. On average, the percentage errors in AUE between the original exposure profiles and the ones generated by the benchmark simulations are quite small--especially when you consider that the average error in AUE between independent full simulations is already on the order of 2.0%. The stability of the exposure profiles is particularly surprising when you examine the new risk factor values at the t+4 day horizon. The new (perturbed) term structures are quite "spikey" relative to the originals. This is because each node in the term structure2 evolves over the exposure horizon according to the following mean-reverting diffusion process:
where: a = constant between 0 and 1
b = forward rate for r at the exposure horizon
T as implied by the initial term structure
‡ = matrix of risk factor correlations
Since the risk factor dynamics are highly sensitive (via the mean-reversion target rate) to the initial term structure, it is instructive to examine how the forward-rate term structures change over the four-day horizon. Figure 1 compares the U.S. dollar forward-rate term structure implied by the original USD spot term structure to the new forward-rate term structure implied by the perturbed USD spot rates for one of the four-day scenarios. The marked difference in the curves is typical of all 10 four-day scenarios. Despite this, the original exposure profiles are very close to the exposure profiles generated by the benchmark simulations.
One important result indicated by the data is that re-centering the NPV vectors actually makes the results worse. One possible explanation for this surprising outcome is that re-centering the NPV vectors exaggerates the potential exposure which is actually constrained by the mean reversion in the risk factor diffusion process. Indeed, this hypothesis is supported by the data: the error in AUE for the re-centered data is worse when the portfolio value increases at the end of the four-day horizon. This has the effect of translating the re-centered exposure profile upwards too far (Figure 2).
Another interesting observation is that the exposure approximation for the small cap portfolio exhibits the smallest error in AUE relative to the benchmark. The quality of the approximation does not appear to be a function of the degree of non-linearity in the theoretical value of the instruments.
A final observation that is not revealed by the results reported in Table 2 is that the approximation does not exhibit a directional error bias. That is, the approximation was just as likely to overstate the AUE as understate it. Had the bias been systematic and repeatable, a corrective "haircut" could have been quantified.
1 99th percentile (one-tailed) exposure calculated at 10
six-month time steps (five-year horizon).
2 The DEM/USD spot exchange rate follows the same
diffusion process with the mean-reversion target rate equal
to the forward exchange rate implied by interest-rate parity.
