Operational Risk Capital Modeling for Extreme Loss Events

Hi, this is Heng Chen with HSBC. Today I'm going to share with you a research called Operational the Risk Capital Modeling for Extreme Loss Events Using Semi-Nonparametric Estimation

First, I start with research objective and summary. Then followed up by the challenge in the op risk capital estimation literature. Then, we'll talk about SNP model, using the extreme value theory - point over threshold approach to estimate capital.

To demonstrate the semi-nonparametric estimation model, we'll use two examples. The first one will use the simulation data sets. The second one will use the actual loss data sets.

On the research summary and objective. First, op risk modeling using the parametric models can lead to counter-intuitive capital estimates because of the extreme loss events. To overcome this challenge, the objective of this research is to propose a flexible SNP model using the change of variables technique, or draconian transformation. It enriches the family of distribution for modeling extreme events. Therefore overcoming the parametric model restriction or model misspecification.

Furthermore, the SNP model have the same main maximum domain of attractions as the parametric kernels, and therefore it follows that SNP models are consistent with the EVT-POT approach, but with different shape and scale parameters from the parametric model.

When we apply the SNP estimation to simulated data sets, we found that SNP model in Fréchet and the Gumbel MDAs can perform satisfactory by gradually increasing the number of model parameters. Based on the learning, we also applied estimator to the actual loss data set, and we found that the SNP model loss typical estimate is pretty stable and very intuitive. The estimate is around 2 to 2 and 1/2 times as large as the single largest or loss events in the data set, which is pretty intuitive.

So, the challenges in the op risk capital modeling-- here are a few examples in the literature. Back in 2004, a paper by Moscadelli found that the estimated Pareto tail shape often exceeded one, and the capital estimate is counterintuitive. And this is done by using Basel Committee collected data sets across seven different unit measures and eight line of business, and many of them leads to the counter-intuitive capital estimates.

In 2009 the paper by Cope found that by removing the top three loss events from the modeling data set, the quantile estimates at the 99.9% reduces by a 65%, reflecting the significant impact of extreme loss events in the capital estimation.

Colombo in 2015 used the weighted MRE by assuming the contaminated data points. However, we all know that for the op risk loss events, especially those big events, they are very carefully scrutinized by the modeling community as well as the business community. Therefore treating them as contaminated point and using weights and leads to arbitrary capital estimates.

Recently a paper by Abdymomunov and Curti proposed to rescale the bank loss by observable information to arrive at a more stable capital estimate. However, this approach ignored unobserved characteristics that might also influence their losses. Therefore, it is very ad hoc.

Neslova raised the concern about naive application of EVT-POT approach to op risk capital modeling without a careful understanding of the loss data set. The paper suggested that the mixed true data generating processes can turn out to be difficult to detect if one does not look for them.

Embrechts, in his book called Modeling External Events in 1997 commented about the challenge of making the body-tail cut off by following the EVT-POT approach. Basically if the cut off threshold is set too high, then the estimate could be volatile. On the other hand, if the cut off threshold is set too low, then the estimates could be biased. Therefore, it is difficult to set the right cut off point.

Gallant and Nychka back in 1987 first introduced the semi-nonparametric methodology by combining a normal kernel with Hermite polynomials. It claims that the approximation errors can make to be arbitrarily small by increasing the polynomial truncation points.

Chen in Handbook of Metrics indicated that another attractive feature of this estimator is its ease of implementation. This is because the SNP estimation is actually-- has a finite number of parameters once you make the truncation point. And therefore it can be estimated by maximum likelihood, generalized least squares, sieve minimum distance.

Chen and Randall in 1997 introduced SNP estimation using the Jacobian transformation in the context of binary choice model. They demonstrated that the estimated willingness to pay is substantially different from the initial parametric model. So in this research, we extend the SNP estimation to model tail events above EVT-POT threshold without treating extreme events as contaminated data points. We found that SNP distribution in the Fréchet and the Gumbel MDA can be used to model heavy tail loss events and results in a substantially different capital estimates from the parametric model as well.

In not in share the SNP estimation based on Jacobian transformation can be described as follows. For an unobservable f(x) density distribution for continuous variable x, we can pick a known density function called the g(v) through transformation v equal to h(x), where h is a monotone transformation. And by ensuring that gradient of this transformation be positive, then we're achieving what are a necessity in the CDF function.

For example, the following power series guarantees the gradient, or Jacobian, to be non-negative. That is by taking a power series of order k. And then take a square, right, it will be positive, and it will ensure the non-negativity.

Say g(v) be Pareto density function. The approximate true density function has the mixed form with the weight that is a power function of degree m, as you can see here that Am is the kernel, which is coming from the Pareto distribution with the polynomial series transformation. And therefore, the approximate density function, f , is the summation of these polynomials with weights that are a function of x and the parameters.

So it has a mixed form of distribution. If the kernel distribution belong to the Fréchet MDA, just like the Pareto distribution with shape parameter c, then the SNPGPD also belongs to the Fréchet MDA, but with shape parameter c equal to c divided by m. Well, m is the degree of SNP polynomial h(x).

A few comments. The SNPGPD model with K additional parameters or order m has the shape parameter c over m, or c over 1 plus 2K, in our case. So instead of a constant scale parameter b for the Pareto variable, the SNPGPD model variable x is transformed by the power series with parameters theta.

The SNPGPD model tail behavior is more stable, or it converged to 0 much faster than the GPD model as x goes to infinity asymptotically, were L(x) is a slow varying function. As a result, SNP distribution enriches the family of distributions that can be used to estimate the VaR capital model using the EVT-POT approach.

And these results can be proved using the Gnedenko condition. In general, the SNP model under the continuous monotonic transformation will not change the MDA of its kernel, and it follows that SNP model is consistent with the EVT-POT approach.

In this research we select a few examples to represent three MDAs. For the Fréchet MDA, we pick the Pareto and log-logistic distributions. For Gumbel MDA we picked a Lognormal distribution, and for Weibull MDA, we picked the Weibull distribution for maximum.

So for the first example simulation data set. First, we create a three simulated data set, each with 1,000 observations. As you can see for Weibull distribution, the shape parameter is 5 over 3. And its maximum is only 978. It's the smallest among the three. For Pareto, it has shaped parameter 4 over 3, which is bigger than 1, and the central maximum is 7856. It's almost like nine times as large as Weibull distribution. Now for log-logistic, it's a maximum actually doubled that of Pareto.

So from this sample, we actually applied three different body tail cut off, and we came up with three different exceedance data set for the model development approach.

In total, we estimated 63 models on the three simulation data set to evaluate model performance. For the parametric model, we estimate six different ones. The first one is Generalized Beta, distribution of Type 2, four parameters. And then second is Pareto, log-logistic, lognormal and Weibull. They have two parameters and exponential only for one.

So for the kernel, we picked five simple parametric distributions. As a result, we estimate 15 SNP models on five kernels with two, three, and four additional parameters.

Model Performance Assessment. The SNP model specification can be evaluated by gradually increasing the order of polynomial to find the best fit model to the data set. As a result, we can apply the simple likelihood ratio test and/or t statistic test for the additional parameters to see the significance, because now listed hypotheses.

Q-Q plots will also be evaluated to ensure that the model does not over-predict or under-predict the observed, especially for the extreme events. From the tail distribution and the quantile estimates at the 99.9 will be analyzed because of its influence on the capital estimates at 99.9% VaR.

Also, sensitivity of different body-tail cutoff will also be assessed on the quantile estimate at 99.9% because we have three different color point in our simulation data set.

For the parametric model, for GB2 model, it is the best-performing parametric model because it has four parameters that it operates. Its log-likelihood varies 47.83. Pareto and log-logistic also perform pretty well because our data-generating process contains the sample from them.

As you can see, across each of the three different data sets, they observe the pattern very similar. Let GPD, log-logistic, and the lognormal, their performance are quite similar. But when we look into the quantile estimate at the 99.9%, you can see that they actually change a lot. For GB2, we have 205,644, which is way bigger than that of GPD, which is also bigger than log-logistic and lognormal. So it indicates that the entire estimate is highly sensitive towards what parametric distribution you choose, even though the log-likelihood value may look very close.

For the SNP model, we found that SNPLGN3p performed the best among all the SNP models with three additional parameters. And SNP model with four additional parameters does not improve further a lot, and only by a little bit. It's not quite a bit. So, we think the SNPLGN3p is the best performing model that we can develop for this data set.

And you can see, the quantile estimate at the 99.9% by the SNPGPD, SNPLGT, and SNPLGN model are very stable. Unlike the parametric model, the quantile estimate is also very stable, either for three additional model or four additional model-- both of them.

Except that you don't look into Weibull MDA where it is. The model in that domain is not capable of modelling heavy data, tail data .

The Q-Q plots and tail distribution on the following pages also reconfirmed observation across estimated 63 models.

For example, we look into the first Q-Q plot here. We compare six different parametric models. You can see that none of them is able to be around 45 degree line, right. And when we look into SNP model with two additional parameters, then we find that it's in power with the best parametric model. This is across all three different data sets.

Now we look into SNPGPD with two, three, and four additional parameters, we found that once we had used additional three parameters, then the estimated-- then this Q-Q plot is very much around the 45 degree line.

Three additional and four additional model parameters does not change much, right. This is so for GPD model, for LGT, kernel SNP model, or SNPLGN model. Same picture, which is confirmed of our observation earlier.

And on the other hand, for Weibull or SMP Weibull with two, three, and four additional parameters, they're just simply not capable of modeling extreme events. So is exponential based SNP model.

For the second example, we use actual op-risk loss data set with 324 CPBP loss events. As you can see, that loss distribution has an extremely heavy tail.

For example, top 1% loss events constitute 90% of the total loss amount. The CPBP regulatory fines by misconducts is very heterogeneous. It can come from market manipulation, money laundering, antitrust violation, et cetera. The CPBP loss amount is typically in the magnitude of billions of dollars for the top loss events, and they can also vary across business units due to different business characteristics.

Following industry practice, we performed KS, CvM, AD tests on the data set to determine the body-tail threshold split. We found that at 43 tail events-- For the data set with 43 tail events, the model performs the best, the Pareto model performs the best with the lowest AD test statistics, which constitute 99% of the total loss amount.

For comparison purpose, we also estimate a log-logistic and a lognormal model, and we found that both model underperformed Pareto model with log-likelihood value of 41-ish.

For the SNP model with two and three additional parameters, both of them exhibit a significant improvement over the parametric model. You can see the right-most column of the second table here for SNPLGN3p. Its log-likelihood value is 45.93 compared to 41.33 lognormal one. It's highly significant with all the t-stats highly significant

For the Q-Q plot you can see here, for the SNP model with two additional parameters, they perform slightly better than parametric models, but it's only when we move to SNP with three additional parameters that it outperform the other models very clearly. In fact, for SNPGPD, SNPLGT, and SNPLGN-- all of them with three additional parameters-- they're all performing equally, fairly similar, around 45 degree lines in the Q-Q plot, which is way better than the parametric model.

From the tail distribution, you can see that Pareto tail is actually pretty thick, but when we move to SNPGPD with 3p, its tail is actually shorter, right. Same picture for SNPLGT and SNPLGN.

For the shape and scale parameter, the SNP model scale is determined by the polynomial transformation, a power function with estimated parameters theta hat. For the parametric GPD and the log-logistic models in the Fréchet MDA, the estimated shape parameters are 1.5858 and 0.8389, respectively. They suggest different tail behaviors. On the other hand, for the estimated shape parameter of SNPGPD3p and SNPLGT3p, they are 0.2158 and 0.1559, respectively. They are both substantially smaller than their parametric counterpart.

As a result, the SNP model tail behavior is more stable than the parametric model, because the tail value is way smaller as x goes to infinity asymptotically, where L(x) is a slow varying function. So as SNP becomes model, this whole value, the tail, becomes much smaller.

For the capital comparison, we just made a truncated lognormal model. And by combining this body model with the three, six different tail model, we simulated capital with hundreds of iterations. As you can see, for the GPD model, the capital is way bigger than the others when you compare to the largest loss event, which is only around 17.

So you can see here that the GPD model's capital estimate is more than 500 times larger than the single largest event, which is counterintuitive, right. And for the log-logistic model, the capital estimate is also very big, followed by lognormal.

On the other hand, for the semi-nonparametric model with three additional parameters, their estimates actually are quite close, which are only 2 to 2 and 1/2 times as large as the single largest or loss events.

If you look at that log-likelihood values, it's clear that SNPLGN3p performed the best. Therefore, we suggested that we should use this last model for estimating the capital.

So the conclusion is that, this research extends the SNP estimation to model the op-risk capital, leading to a more stable and intuitive capital estimate than the parametric counterpart.

The SNP model enriches the family of distribution to estimate heavy tails with shape perimeter as a function of order of polynomial series m, and the chosen kernel shape parameter c. The SNP model scale parameter is also a function of power series.

On the model performance, the SNP models nest any chosen parametric model as a special case. Therefore, the nested likelihood ratio test and the student t-test can be used to assess SNP model specification.

Q-Q plot can also be evaluated to ensure that incremental parameters are needed to accommodate the heavy tails. And tail distribution can also be compared to visualize the stability of economic estimates by looking how thick layer as x goes to infinity.

SNP model specification it's easy to implement, which yields the model parameter estimates in just one step. MATLAB enables this research to select a wide variety of parametric distributions and optimization algorithm in their toolbox to satisfy the precision requirements in SNP model estimation and the capital simulation.

Thank you.