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Model performance

Accuracy benchmarking

The unique combination of the versatile, heavy-tailed MGL distribution with asymmetric multivariate GARCH processes and the least-squares filter empowers the model exhibit the following key-features

  • Accomodate extreme market events
  • Dynamically and asymmetrically model risks and returns according to changing market regimes

Subsequently, we achieve more accurate forecasts, and our solution has a clear edge when measuring asset- and portfolio risks vis-a-vis the models commonly employed by the industry. The superiority of our approach is visible in the following table, which shows the Value-at-Risk violations at the 1% confidence level (i.e. percentage average when the market fell below the forecasted Value-at-Risk) for ten random assets from the SP500 over the last 15 years.

Ticker Gaussian Gaussian-GARCH StudentsT StudentsT-GARCH Laplace Laplace-GARCH
APC 0.018 0.015 0.013 0.014 0.013 0.012
AMZ 0.014 0.019 0.012 0.011 0.013 0.011
BRYN 0.019 0.016 0.013 0.013 0.013 0.013
JNJ 0.028 0.019 0.023 0.014 0.024 0.014
CMC 0.033 0.021 0.021 0.017 0.021 0.017
XONA 0.026 0.026 0.013 0.021 0.017 0.019
GOOGL 0.026 0.022 0.015 0.014 0.015 0.013
NCB 0.02 0.022 0.016 0.013 0.016 0.012
PFE 0.022 0.025 0.016 0.019 0.016 0.016
UNH 0.024 0.02 0.013 0.014 0.012 0.013
Avg. 0.023 0.0205 0.0155 0.015 0.016 0.014

Laplace-GARCH enotes our model. Note that values closer to 1%, are better; that is, a perfect model with an infinite amount of sample data would exhibit 1% or 0.01 violations.

Execution times and scalability

In addition to superior performance, a strategic and highly efficient implementation results in a low execution time, which allows us to apply our approach to large-scale portfolios:

Assets Model-estimation Value-at-Risk Total
1 8.14 1.09 9.23
5 16.7 1.45 18.15
10 24.5 1.45 25.95
50 107 1.53 108.53
100 240 2.49 242.49

The running times are given in 1/1000 seconds (i.e. it takes less than a 1/4 of a second to process a 100 asset portfolio). Note, we used the following lower mid-spec system: Core™ i3-8109U CPU @ 3.00GHz × 2 with 7.7 GB RAM.

Moreover, the following table shows that the computation times of our model is overall lower than that of available reference models:

# Estimate model paramter:
Gaussian GARCH (arch-package),  100 loops, best of 3: 15.4 msec per loop
Student-t GARCH (arch-package): 100 loops, best of 3: 23.6 msec per loop
Laplace GARCH (our model):      100 loops, best of 3: 14.6 msec per loop

# Compute Value-at-Risk:
Gaussian GARCH (arch-package):  10000 loops, best of 3:   90 usec per loop
Student-t GARCH (arch-package): 10000 loops, best of 3:  191 usec per loop
Laplace GARCH (our model):      10000 loops, best of 3: 1120 usec per loop

The reference implementations are from the well-respected arch Python package.

Some notes about the comparison:

  • These times refer to a single-asset portfolio, simply because there is no (publicly) available package in Python supporting GARCH in a multivariate setting.
  • It takes much longer (relatively) to compute the Value-at-Risk with our model because there is no closed-form expression available and we need to resort to numerical integration. However, since the Value-at-Risk computation takes much less of the total computation time and scales sub-linearly, this does not matter.
  • Note the significant difference vis-a-vis the Student-t model, which also features heavy tails is of similar complexity and computational requirements. While the difference in estimation time is small vis-a-vis the Gaussian model, this is expected as the estimation process of the Gaussian model is much less complicated than the ones of their heavy-tailed cases.