Advanced ERM Session 5: Risk aggregation and Extreme Events

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

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

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