Finding the Optimal Number of Rate Paths and Seeds in a QRM System

This is the second of three articles on quantitative risk management. In the first article, the author discussed how to choose among different interest rate models. This article looks at regulations regarding validation and how to demonstrate adequate knowledge about vendor-supplied risk management systems.

Quantitative risk management (QRM) is widely used in large banks for assets/liability management purposes. The OCC and the Fed prudently require banks to document their testing results regarding the models they have used and the parameters they have chosen. This regulation requires banks to have adequate knowledge about the vendor-supplied risk management system and to be cautious when they choose models and parameters.

Any instrument for which the cash flow is rate path-dependent, such as mortgage-backed securities, is valued stochastically. Some are modeled by Monte-Carlo simulation and others use Lattice tree. To carry out a Monte-Carlo simulation, N independent random samples must be drawn from a normal distribution, which induces N corresponding rate paths. The products will be evaluated along each rate path and the final market value will be the average of the above N rate path evaluations; the greater the number of the rate path, the more accurate the results. However, the greater the number of rate paths chosen, the longer the time needed to calculate the results. The computing time is much more than a linear relationship to the number of rate paths chosen. Therefore, we need to find the optimal number of rate paths, given the error tolerance level that senior management has chosen.

The seed number signifies the starting point of the sequence where a random number generator begins. We found that the average results of using M number of seeds to run N rate paths is closer to the target results than the result from a single-seed run of M*N rate paths. The target result is determined by a very large number of runs, such as 10,000. However, running multiple seeds will create tremendous difficulties in the production process of assets and liability management. Numerous accounts need to be run and results need to be saved and averaged outside of the QRM system. Therefore, it is better to find a single seed to run a sufficient number of rate paths that will provide the results within the given error tolerance level.

The model we chose to test is the Black-Karasinski model. However, similar methods could be used to test the Hull-White or the Black-Derman-Toy models.

We chose to test mortgage-backed securities, since MBS is one of the major instruments in which cash flow is rate path-dependent. To avoid testing every MBS product, we tried to find the product with the highest standard deviation of market price for the same number of rate paths and seeds. The rationale is that when we put the most risky MBS instrument within the tolerance level, the rest of the MBS accounts should also stay within the boundaries. Naturally, the instruments with longer maturity bear higher interest-rate risks than others. Therefore, a GNMA 30-year mortgage was chosen instead of a product with a shorter maturity. We also found that a mortgage product with a low coupon bears more interest risk than a mortgage product with a high coupon. This proved to be true during all the testing we have done.

The instrument we chose to test is: GNMA 30 Year with Gross WAC (weighted average coupon) 6.50%; Net WAG 6.00%; WAM (weighted average maturity) 342 months; with Prepayment model--QRM (FMGV30).

We tested different seeds between seed 0 to seed 4 for MBS market price variation by number of rate paths between 250 to 10,000. We found that four seeds converge closely at 10,000 rate path. However, when the rate path is 1,000 or lower, the market price from seed 2 is consistently lower than the market price from the other three seeds (See Figure 1). There is up to a 56 basis point market price difference between different seeds for rate path of 250 as we tested. For rate path of 10,0 00, the maximum difference is 12 basis point among different seeds. If the market price variation is the binding criterion, you could then determine how many rate paths are sufficient to put your product within your error tolerance level.

There are several possible criteria that determine the error tolerance level. It could be determined by variation of market price, option-adjusted spread, duration, and/or some other variables. Therefore, testing has to be done regarding all these variables to make sure none of the criteria will be violated. One criterion is binding; as long as the binding criterion holds, other criteria will not be violated. Therefore, the final optimal number of rate paths will be determined by the testing results of this binding criterion.

On standard deviation of market price testing, we found that the results from Seed 2 are constantly higher than the other three seeds. (See Figure 2.) This signals that Seed 2 is not a good choice when a single seed is chosen for production. The standard deviation difference between four seeds is up to 12 bp for 250 rate path and reduced to 0.4 bp when rate path is 10,000.

There is not much difference between the option-adjusted duration for different seeds. The OA duration difference is up to 3.8 bp for a rate path of 250 and reduces to 0.5 bp for a rate path of 10,000. (See Figure 3.)

After getting the results on instrument-specific duration tests, you may need to convert the duration of the instruments into the duration of equity, if duration is one of the criterion that you need to comply. The conversion depends on the asset weight of MBS compared with overall assets on the A/L book. If duration of the equity, when compared with the price variation, is not binding, you may focus on bringing price variations within the tolerance level.

The final important step is to check how long it takes to run the A/L book using the new number of rate paths chosen. We found that the computer time tripled when we changed from 250 rate paths to 1000 rate paths. The actual optimal number of rate paths chosen depends on the error tolerance level, the size of A/L book, and the composition of different products in the A/L book.

Yan is vice president of Capita/Allocation & Quantitative Analysis, KeyCorp, Cleveland, Ohio.

Figure 1
The Market Price Convergence by Different Seeds
GNMA 30-Year Gross_WAC 6.5, Net_WAC 6.0, WAM 342
         250    500   1000   5000   9999
Seed 0  96.95  97.08  97.06  96.99  96.98
Seed 1  97.32  97.06  97.06  97.01  96.99
Seed 2  96.76  96.59  96.8   96.86  96.87
Seed 3  97.12  97.1   97.09  96.97  96.93
Note: Table made from line graph
Figure 2
The Standard Deviation of Market Price
GNM A 30-Year Gross_WAC 6.5, Net_WAC 6.0, WAM 342
         250    500   1000   5000
Seed 0  0.193  0.141  0.097  0.045
Seed 1  0.153  0.134  0.093  0.044
Seed 2  0.271  0.187  0.115  0.048
Seed 3  0.171  0.131  0.102  0.046
Note: Table made from line graph
Figure 3
The Duration of Instruments by Seeds
         250    500   1000   5000   9999
Seed 0  5.855  5.888  5.884  5.88   5.878
Seed 1  5.893  5.883  5.88   5.881  5.876
Seed 2  5.874  5.872  5.876  5.875  5.873
Seed 3  5.869  5.882  5.873  5.879  5.876
Note: Table made from line graph

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