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This presentation is based on a part of an academic course on Advanced Enterprise Risk Management (Advanced ERM) titled ‘Risk aggregation and Extreme Events’ and covers topics such as: an introduction to the importance of these topics, modelling fat-tailed behaviour for individual risks, extreme value theory, modelling multiple risks, factor structures, copula based dependency structures and managing and mitigating (joint) fat-tailed risks. It also includes appendices on quantile-quantile plots and on possible active management selection effects

Slides

1	Session 5: Risk aggregation and Extreme Events
2	Session 5: Risk aggregation and Extreme Events
3	Introduction (1)
4	Introduction (2)
5	Session 5: Risk aggregation and Extreme Events
6	Modelling fat-tailed behaviour for individual risks
7	Many (most?) investment return series are ‘fat-tailed’
8	Skew(ness), kurtosis and Cornish-Fisher
9	Flaws in Cornish Fisher (and hence skew/kurtosis)
10	What causes fat-tailed behaviour?
11	Time-varying volatility
12	Explains some market index fat tails, particularly on upside
13	A longer term phenomenon too
14	Crowded trades and selection effects
15	Session 5: Risk aggregation and Extreme Events
16	Extreme Value Theory (EVT)
17	Extreme value theory results
18	Block maxima results
19	Generalised extreme value (GEV) distribution
20	Limiting behaviour
21	Main result for threshold exceedances (excesses)
22	Potential weaknesses
23	Using EVT to Estimate VaRs
24	Session 5: Risk aggregation and Extreme Events
25	Joint fat-tailed behaviour
26	Consider first multivariate Normal, i.e. Gaussian, case
27	MVaR in Gaussian Case
28	E.g. outcomes uncorrelated, equal weights
29	Central Limit Theorem
30	CLT can break down in the following ways:
31	Session 5: Risk aggregation and Extreme Events
32	Factor structure - notation
33	Factor structure - handling idiosyncratic risk
34	Advantages of introducing a factor structure
35	Identifying factor structures - 3 main model types
36	Loss distributions for credit portfolios
37	Single risk factor model for credit portfolios
38	Probability that a given fraction (k/n) default
39	Granularity
40	Analytical solution
41	Vasicek loss distribution
42	Session 5: Risk aggregation and Extreme Events
43	Copulas
44	Illustrative distribution (two risk factors) (1)
45	Illustrative distribution (two risk factors) (2)
46	Copulas: another illustration
47	E.g. bivariate copula (1)
48	E.g. bivariate copula (2)
49	Copula and copula density
50	Copulas and Sklar's theorem
51	Example Copulas
52	Tail dependence
53	Interpretation of tail index
54	Gaussian and Independence copula
55	Simulating random variables from Gaussian copula
56	Simulations with non-Gaussian copulas
57	Fitting copulas empirically
58	Risk aggregation
59	Risk aggregation using copulas (1)
60	Risk aggregation using copulas (2)
61	Risk aggregation using correlation matrix
62	Ranking copulas
63	Session 5: Risk aggregation and Extreme Events
64	Managing and mitigating joint fat-tailed risks
65	Creating multi-dimensional QQ plots
66	Characteristics of multidimensional QQ plots
67	Portfolio construction
68	Portfolio construction - sensitivities
69	Portfolio construction - impact of fat tails (1)
70	Portfolio construction - impact of fat tails (2)
71	Other approaches - (1) distributional mixtures
72	Other approaches - (2) lower partial moments
73	Estimating lower partial moments
74	Capital allocation: the Euler principle
75	Session 5: Risk aggregation and Extreme Events
76	Appendix A: Quantile-quantile plots
77	Example QQ-plot (versus Normal)
78	Quantile-quantile plots: other comments
79	Appendix B: Possible active management selection effects
80	Implications for modelling
81	PCA vs. ICA
82	Including ‘meaning’ as well as ‘noise’
83	Selection effects are potentially very important
84	Selection effects - Summary
85	Session 5: Agenda covered
86	Important Information

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