## Semi-parametric and non-parametric estimation of the operational risk and expected shortfall: simulation and empirical evidence

thesis

posted on 28.02.2017 by Tursunalieva, Ainura#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

Due to the large losses incurred in recent times by internationally active banks and other financial institutions in many industrialised countries, the recent literature has paid a considerable amount of attention to measuring and managing operational risk (OR). The OR is defined as the difference between the 99.9% quantile and the mean of the loss distribution. In other words, the OR is the unexpected loss, measured with a high degree of confidence. In their recent document, Basel Committee on Banking Supervision (2011) emphasised the need for a reliable OR estimate on which to base the calculation of the economic capital charge required to cover operational losses.
There are both pros and cons associated with the advanced measurement approach (AMA), which is currently widely used. Since the report by Basel Committee on Banking Supervision (2004), academics and practitioners have studied the applicability and flexibility of AMA for modelling the severity of operational losses, and have also contributed to the further development of AMA. However, Basel Committee on Banking Supervision (2011) recently provided new guidelines for estimating OR that can lead to an optimum level of capital requirement for banks. According to these guidelines, the tail of the severity loss distribution needs to be modelled adequately, capturing the high skewness and kurtosis of the loss distribution. In addition, scenario and sensitivity analyses have to be conducted, and a range of possible estimates of OR should be produced, rather than reporting only a single point estimate of OR.
The objective of the thesis is to adapt and study the performances of some recent advanced semi-parametric and non-parametric methods for modelling and estimating heavy-tailed severity distributions, which can be used under Loss Distribution Approach in compliance with the quantitative AMA standards. AMA allows banks and financial institutions develop their own methods to calculate OR. Therefore, the emphasis of this thesis is to provide banks and financial institutions with the detailed analysis of the performance of various non-parametric methods which have been proposed in the statistics literature recently in the context of a heavy-tailed distribution. The attractive feature of these non-parametric methods is that there is no need to estimate the entire loss distribution, or even its right tail - a key part of the distribution used in the estimation of OR. This thesis pays a considerable amount of attention to estimating the key parameters, such as the threshold loss and the tail index, which are used in estimating the economic capital requirement. The existing AMA for estimating OR has some weaknesses: the expected loss, computed as the simple sample average of operational losses, is biased and inconsistent, since the loss distribution is right heavy-tailed. It is difficult to assess the reliability of the OR estimate when only a point estimate of OR is reported. It is more informative to report either the standard error or an interval estimate of the underlying true OR. This thesis proposes improvements aimed at overcoming these weaknesses.
The objectives of this thesis are to: (i) propose some improvements to recently developed advanced semi-parametric and non-parametric methods for estimating OR; (ii) construct unbiased and consistent point and interval estimates for the mean of a heavy-tailed loss distribution; (iii) conduct a simulation study in order to assess the finite sample properties of the interval estimate of the mean of a right heavy-tailed distribution, in terms of coverage probabilities and lengths; (iv) construct point and interval estimates of the 99.9% quantile (operational value at risk, OpVaR) of a heavy-tailed distribution, and conduct a simulation study to assess the accuracy of interval estimates for the quantile; and (v) provide a step-by-step approach to estimating the two downside risks, OR and the expected shortfall, by using the methods adapted and studied in this thesis. These methods are illustrated by applying them to some business losses in the US.
This thesis finds that the sample mean estimate and the adjusted mean (unbiased and consistent) estimate of the expected loss (that is, the population mean of the loss distribution) can differ considerably, depending on the size of the tail index that measures the tail thickness. The heavier the tail of the loss distribution, the larger the difference between the two mean estimates, with the latter being larger than the former; the estimates are close to each other if the loss distribution is close to normal. These findings indicate that the regulatory capital requirement can be overestimated, if the expected loss is not estimated appropriately. The coverage and other properties of the empirical likelihood-based confidence interval estimate of the mean are good when the tail index is not close to one. On the other hand, when the tail index is very close to one, the sub-sampling bootstrap based interval estimate can be used. In addition, the results of the simulation study indicate that the data tilting method, which is the weighted empirical likelihood method, produces reliable confidence interval estimates for the 99.9% OpVaR. This method is attractive because it gives nearly zero weights to losses in the non-tail region and large weights to losses in the tail region - an important region when estimating OR.
The methods studied in this thesis are applied to the estimation of OR and the expected shortfall of some business losses in the US. To help practitioners who are working with heavy-tailed distributions, a step-by-step approach to constructing the point and interval estimates of operational risks is provided. The empirical results indicate that the operational risk and expected shortfall are close to each other when the tail index is close to two, which indicates that the loss distribution is close to normal. On the other hand, if the tail index is close to one, which indicates that the loss distribution is close to stable, then the expected shortfall can be noticeably greater than the OR. These findings have implications for risk management and regulators. Since the 2008 financial crisis, regulators would like to see large businesses and banks allocating large amounts of capital. The findings of this thesis show that if the tail index is close to one, then the expected shortfall provides a better cushion than does OR, while the economic regulatory capital can be based on either the expected shortfall or OR when the tail index is close to two.