Copula-free model for risk capital modelling

This graduate research project focuses on how to use various modeling techniques to calculate risk capitals, and in particular, how to improve the goodness-of-fit of the models and capture the dependence structure in high-dimensional settings. Results are illustrated using operational risk capital models.

Introduction

Historically, many risk models have been developed to deal with complicated high-dimensional data, one example is the Advanced Measurement Approach (AMA) model under Basel II for operational risk capital calculations. However, there are two major concerns about the these risk models:

1. Different risk models could have very different goodness-of-fit results especially for heavy-tailed data, and their predictive powers cannot be standardized.
2. Capturing the dependence structure in high-dimensional problems is very challenging. One common approach is to use some type of copula model. However, the pre-determined nature of a copula model can limit its ability to capture complicated dependence structure of high-dimensional data.

The purpose of this project is to deal with the above two questions by investigating two types of multivariate models.

Model 1 Multivariate Pascal Mixture and Mixture of Erlangs

The first model assumes that dependency can be captured at the frequency level of the data, and attempts to capture the dependence structure using a multivariate mixed frequency model - the Multivariate Pascal Mixture model. The severity in each marginal is fitted using an univariate Mixture of Erlangs model.

Desirable distributional properties of the Mixture of Erlangs model

• It is dense in the space of positive distributions, i.e. any continuous distribution can be approximated by a mixture of Erlangs to any accuracy. This will make the capital numbers much more stable from time to time.
• It is closed under mixture, convolution, and compounding, and this will make the estimation of certain statistical quantities much easier.
• Multimodal heavy-tailed data can be fitted easily .
• Censored/truncated data can also be handled.

Desirable distributional properties of the Multivariate Pascal Mixture model

• Any p-variate marginal is also a p-variate Pascal Mixture.
• Multimodal count data can be fitted easily .
• Censored/truncated data can also be handled.

Model Fitting
As both models are mixture-type latent variable models, the Expectation-Maximization (EM) Algorithm can be used to fit the model parameters.
Wikipedia's Page on EM Algorithm

Model 2 Multivariate Mixture of Erlangs

The second model assumes that dependency can be captured at the aggregate level of the data, and attempts to capture the dependence structure using a Multivariate Mixture of Erlangs. This is more challenging as it avoids fitting the frequency and severity data separately.

Closing Thoughts

In general, modelling high-dimensional heavy-tailed data can be very challenging, including both fitting the marginals and modelling the dependence structure across different marginals, and the problem can get even worse when there is not sufficient amount of data available. The AMA model which I have much experience working with is one example that can model such data, even though I personally believe it is too complex sometimes but not "reasonable" enough to model the data accurately and efficiently. The models in this project can be a good alternative, with the following advantages and disadvantages:

Advantages

• Computationally efficient. No need for Monte Carlo simulations and saves tones of time.
• Stable. The Mixture Erlangs model is much more stable and avoids many model selection procedures.
• Copula-free. This avoids the problem of copula selection and can potentially capture a much broader range of dependence structures.
• Overall, the models here are much more FLEXIBLE in nature using mixtures, which I really think is the major advantage compared to traditional insurance models.

Disadvantages

• Sometimes too good too be true. I have tried the models on various simulated dataset, one perception is that the MULTIVARIATE part of the models can make it very challenging to model the high-dimensional data when different marginals have very different TAIL BEHAVIOURS and when there is very limited data to train the model. I think in such case, using a COPULA model might be a better choice.

Last updated on Jan 1, 2019