Further ERM Session 1: Analysis of extreme events

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

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

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