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This presentation is based on a part of an academic course on Further Enterprise Risk Management (Further ERM) titled ‘Analysis of extreme events’ and covers topics such as: typical univariate distributions used in practice, Extreme Value Theory (EVT) and its underlying distributions, generalised distribution fitting (e.g. to the tail of a distribution), adjusting for volatility, confidence levels, multivariate extreme value theory and implications for portfolio construction

Slides

1	Session 1: Analysis of extreme events
2	The problem
3	Session 1: Analysis of extreme events
4	Univariate probability distributions
5	Typical (univariate) distributions used in practice
6	Features shared by many distributions
7	Underlying mathematical formulation
8	(Sum) stable (aka Levy-stable) distributions
9	Session 1: Analysis of extreme events
10	Extreme Value Theory (EVT)
11	Restatement of EVT results
12	Main result for block maxima
13	Main result for threshold exceedances
14	Session 1: Analysis of extreme events
15	How should we choose between distributions?
16	Various ways of fitting a (univariate) distribution
17	A generalised least squares approach to tail fitting
18	Tail weighted maximum likelihood
19	C.f. estimating tail using GPD distributions
20	Estimate the quantile that starts the tail
21	Combined estimate
22	Choosing the threshold
23	Judging the start of the tail
24	Potential weaknesses
25	Alternative ways of estimating VaRs etc.
26	Some subtleties of EVT
27	Session 1: Analysis of extreme events
28	Adjusting for volatility
29	Session 1: Analysis of extreme events
30	Confidence levels
31	Session 1: Analysis of extreme events
32	Multivariate EVT, e.g. for block maxima
33	EV copulas
34	Example EV copulas
35	Domains of attraction and threshold exceedances
36	Session 1: Analysis of extreme events
37	Portfolio construction
38	Portfolio construction - sensitivities
39	Incorporating fat tails: Solution A, simplest
40	Incorporating fat tails: Solution B, more sophisticated
41	Or use lower partial moments?
42	Estimating lower partial moments
43	Euler capital allocation principle
44	Summary
45	Appendix A: Modelling fat tails for individual risks
46	Many (most?) investment return series are ‘fat-tailed’
47	Why are return series often fat-tailed?
48	Time-varying volatility
49	Explains some equity index fat fails, particularly upside
50	And over longer time periods
51	Crowded trades and leverage
52	Important Information

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