Risk measurement

Risk measurement with mathematics computing Introduction, including motivation, statement of the problem, maybe, some literature review (not strictly necessary for projects of your level), main results and conclusion. So Introduction is a kind of a gist of the note. Hence the double name: introduction and main results. Motivation for your research is that a financial risk is modelled by a probability distribution (on positive numbers), while decision making is done on the ground of numbers, not distributions. Hence statement of the problem or aim: how to measure a risk, that is, how to represent a distribution by a single number called a risk measure. The properties these functionals (by this definition, a risk measure is a functional, since it converts a distribution into a number) ought to possess is subject of your study, the extended Gamma distribution family (including the Pareto subfamily) and several principal premiums and VaR and, optionally, TVaR, are your objects. You test risk measures properties of your choice on couples of the form: a risk + a premium principle. For some cases you can consider fairly general risks, for others – confine yourself to an extended Gamma distribution. Normal distribution does not fit you as it is equally distributed on positive and negative numbers and money are never negative. In finance they use log-Normal distribution, so that log(Risk) is distributed normally. However, this model is rather difficult for computation. Next section is the background which can be nicely called extended (or transformed) Gamma distribution family. Not a bad reference is Loss Models by Klugman, Panjer and Willmott (Appendix A), but you can use any other source, including the internet. In this section you define the distribution and bring its main statistical parameters: expectation, variance, characteristic or moment generating function, percentiles etc. Next, you discuss desirable properties of principles of premium calculations (the latter are typical risk measures). You will bring a motivation for every such property, its extreme form (like additivity on independent risks) and relaxed form (like subadditivity) if there are several forms. Then you list some premium principles. Do not take to many and too complicated. However, loaded expectation and variance principles, as well as VaR should be there. You test each of them on every property listed above. Then, the result section will be the table. Columns will contain the premium principles (risk measures), rows – desirable properties. On the intersection there will be your conclusion on this property for this risk measure, like “always”, “always for Gamma”, “never” etc. Finally, the bibliography. you are to explain what a financial risk is, what we model it with, why we need to measure it in order to make decision, bring a list of properties an ideal risk measure to posses, bring a list of risk measures actually in use and study which of the measures has which of those properties. So that the conclusion can be written in form of a nice table. must include section with subsections divided into paragraphs, and for every paragraph you should have a list of definitions, propositions (including lemmas and theorems), examples, figures and tables to be there, for examples and applications it will suffice to consider Gamma and Pareto distribution.

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