Validating Risk Estimation Models

Abstract

We compare five expected shortfall estimation models applied to 25 equally weighted two-stock portfolios. The relevance of the topic comes from the need to measure the minimum capital requirements for market risk for banking institutions. The choice of risk measure is based on the requirements of the Basel III international regulatory framework for banks. We used normal and general Pareto distribution for the marginals and Gauss, Clayton, and Gumbel copulas to model the dependency between the stocks. Gauss copula models linear correlation between the stocks while Clayton and Gumbel copulas result in losses occurring together more frequently. The Clayton and Gumbel copulas are defined by the equations (4) and (5), respectively.

The distribution and dependency parameters were estimated from the time series and the Expected Shortfall risk measure was calculated using Monte Carlo simulations in two steps. First, we simulated unified distributions according to the selected copula. The second step was to apply the selected marginals. Finally, we calculated the Expected Shortfall from the resulting 10 000 simulated return values. The models were evaluated using (3), the back-testing methodology proposed by Acerbi and Székely (2014).

The stocks were chosen randomly from FTSE 100 index constituents, the calculations are based on daily return percentages on a 16-year long time series that ranges from 2000 to 2015. Our empirical analysis confirms that the choice alone of using Expected Shortfall as the risk measure is not enough to remedy all the problems of the risk measure Value at Risk. The model choice for estimating Expected Shortfall also has great importance. Our back-testing results show that the Pareto distribution is a good choice for modelling marginals. We have observed the best results for the model with Pareto marginals and Clayton dependencies. We can also observe in our empirical analysis that the Pareto-Clayton model tends to be overly cautious, often overestimating the risk of the portfolio. This is especially a problem for stable periods. Finally, we conclude that the choice of marginal distribution seems to be more important than the choice of dependency structure considering estimation accuracy. 

The research was financed by the Higher Education Institutional Excellence Programme of the Ministry of Human Capacities in Hungary, within the framework of the 4th thematic programme ,,Enhancing the Role of Domestic Companies in the Reindustrialization of Hungary" of the University of Pécs (reference number of the contract: 20765-3/2018/fekutstrat)