Quantification of market risk in the context of conditional extreme value theory

Resumen

This research seeks to contribute to the existing literature on market risk by providing new insights and empirical evidence through the application of the Extreme Value Theory (EVT). Specifically, we focus on the methodology proposed by McNeil and Frey (2000) for estimating the market risk that combines the unconditional Extreme Value Theory with models of volatility, i.e, the Conditional Extreme Value Theory. EVT focuses on the limiting distribution of the extreme values observed over a long period, which is independent of the distribution of the values themselves. In other words, EVT relates to the asymptotic behavior of extreme observations of a random variable. Although EVT is well-established in many sciences such as engineering, insurance, and meteorology among others (see e.g., Embrechts et al., 1999; Reiss and Thomas, 2007), its application in the financial sector has gained more relevance in recent years. This dissertation explores three areas within the framework of Conditional Extreme Value Theory. The first area (Chapter II) focuses on the performance of volatility model estimation under distributions with fat tails and skewness in improving Value at Risk (VaR) estimation. The second area (Chapter III) investigates the sensitivity of Generalized Pareto Distribution (GPD) quantiles and market risk measures such as VaR and Expected Shortfall (ES) to threshold selection under the Peaks-Over-Threshold (POT) method. Finally, the third area (Chapter IV) analyzes the sensitivity of the Generalized Extreme Value Distribution (GEVD) parameters and market risk quantification to the choice of different block sizes with the Block Maxima Method (BMM). The study leads to the following findings. First, it reveals that assuming a normal distribution for VaR estimation may underestimate risk, as financial return distributions are skewed and exhibit excess kurtosis. The use of heavy-tailed and skewed distributions yields better results in terms of VaR estimation accuracy, firm's loss function, and capital requirement compared to a symmetric distribution. Second, the findings suggest that researchers and practitioners do not need to prioritize the selection of a particular threshold for measuring market risk using conditional EVT and GPD. A wide range of options produces comparable risk estimations, although small differences are found for the highest percentiles for certain thresholds. For financial institutions interested in minimizing capital requirements for market risk, this could be a valuable resource. Although this result opens up an opportunity for further investigation, this Chapter represents a novelty and thus, potentially the main contribution of this Thesis in the use of the POT method in the area of market risk measures. And third, regarding the use of EVT and specifically the Block Maxima Method (BMM) for estimating market risk, the study finds that BMM does not provide accurate market risk estimates and is highly dependent on the block size selected. VaR estimations are highly sensitive to the block size selected for fitting GEV distribution, and only intermediate block sizes seem to provide reasonable VaR estimations. However, caution must be taken as the results are not robust, as no one block size performs well for the whole set of assets considered. The BMM method may not be the most reliable method among the EVT approaches for estimating market risk, and further research on optimal block size selection techniques may be required. Lastly, a comparison has been made between the two EVT approaches (POT and BMM) and we can infer that the Peaks-Over-Threshold (POT) method represents a robust and effective approach for estimating extreme events and measuring financial risk. This method allows for the accurate estimation of tail probabilities and can provide a valuable understanding of extreme events that other methods such as BMM may miss.