**3. Methodology**

### **3.1 Introduction**

In the previous section, we outlined numerous articles and textbooks that discussed various aspects of operational risk using loss distributions. Those articles and textbooks formed a literature review that helped us figure out how to quantify operational risk using loss distributions which is the main objective of our research work. Firstly though, we provide detailed description of the EVT, VaR, and their corresponding properties.

Under EVT approach, [2] explained that the analysis of the tail area of the distribution is the main focus as well as using appropriate methods for modeling extreme losses and their impact in insurance, finance, and quantitative risk management. We fit classical distributions to the data, using the maximum likelihood criteria, starting from light-tailed distributions (e.g. Weibull distribution), to medium-tailed distributions (e.g. log-normal). Under this approach, the Kolmogorov-Smirnov (KS) test and the Anderson-Darling (AD) test are adapted to measure the distance between the empirical and theoretical distribution functions only in the tail area, after deciding on the desired quantile. Readers are referred to [25, 26] for the KS test and for the AD test. Note that mean-excess plots are used to assess the validity of modeling the tails, while the Hill method is also used to get rough estimates of the shape of the parameter of a distribution; see [2]. In addition, [2] stated that the KS and AD tests can be used to examine the goodness-of-fit of models that we want to fit on the data. These tests can be used to determine which loss distribution best fits our operational loss data. These tests use different measures of discrepancy between fitted continuous distributions and empirical distributions. KS test is the best at measuring the discrepancy around the median, while the AD test is good at measuring the discrepancy for the tails.

According to [2], VaR is the largest loss an investment portfolio might sustain over a specific length of time. The time frame can be a single day, a month, a quarter, or even an entire year. According to [2], it is the (1-*α*) th percentile of the loss distribution over a desired time frame, where (1-*α*) is the level of confidence and practitioners typically put it at 99.99%. Also note that [27] defined VaR as a number that indicates how much a financial institution can lose with probability over a specific time horizon, and that its measurements can be used in risk management, the assessment of risk takers' performance, and for regulatory requirements.

## *Quantifying Risk Using Loss Distributions DOI: http://dx.doi.org/10.5772/intechopen.108856*

The method of moments is a different analytical method for quantifying risk. In this method, the mean and variance of input distributions defined at the task level are utilized to calculate the moments of the probability distribution corresponding to the task completion date. The major moments of work breakdown structure simulations may be determined using this method almost instantly and precisely, and it is still utilized in the cost risk analysis community; see [28].

Now in this methodology section, we intend to use a dataset to illustrate to readers how to quantify real-life operational risk using loss distributions. In addition, we intend to measure the descriptive statistics such as the mean, skewness, and kurtosis to obtain a general idea of how each of the considered datasets are distributed. For instance, **Tables 3** and **4** below give a summary of how skewness and kurtosis are used to give an idea of how the dataset(s) may be distributed. As indicated in [2], there are important statistical approaches to consider when running a goodness-of-fit test to a dataset, i.e. the KS and AD test. More importantly, the maximum likelihood estimation (MLE) or method of moments shall be used to estimate the model parameters.

The pivotal role of our study is to determine the best loss distribution that fits the considered dataset. To complete this role, we ought to compare between Pareto, gamma, Weibull, log-normal, Burr, and exponential distributions and investigate using specific metrics which distribution best fits our data. We are going to perform these tasks with the help of R software (the dataset used and R codes can be requested from the authors) using packages, such as moments and fitdistrplus. Having determined which distribution best fits our data, we shall use the best-fitting loss distribution to calculate the probability of loss for that specific dataset. Consequently, it may happen that the data do not seem to fit well with any distribution; then in such an instance, we would conclude that using a nonparametric approach would be of better benefit.
