Mathematical Modeling and Statistical Methods
for Risk Management
Lecture Notes

c Henrik Hult and Filip Lindskog

Contents
1 Some background to nancial risk management 1
1.1 A preliminary example . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Why risk management? . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Regulators and supervisors . . . . . . . . . . . . . . . . . . . . . 3
1.4 Why the government cares about the bu er capital . . . . . . . . 4
1.5 Types of risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Financial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Loss operators and nancial portfolios 6
2.1 Portfolios and the loss operator . . . . . . . . . . . . . . . . . . . 6
2.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Risk measurement 10
3.1 Elementary measures of risk . . . . . . . . . . . . . . . . . . . . . 10
3.2 Risk measures based on the loss distribution . . . . . . . . . . . . 12
4 Methods for computing VaR and ES 19
4.1 Empirical VaR and ES . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Con dence intervals . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2.1 Exact con dence intervals for Value-at-Risk . . . . . . . . 20
4.2.2 Using the bootstrap to obtain con dence intervals . . . . 22
4.3 Historical simulation . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Variance{Covariance method . . . . . . . . . . . . . . . . . . . . 24
4.5 Monte-Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Extreme value theory for random variables with heavy tails 26
5.1 Quantile-quantile plots . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2 Regular variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Hill estimation 33
6.1 Selecting the number of upper order statistics . . . . . . . . . . . 34
7 The Peaks Over Threshold (POT) method 36
7.1 How to choose a high threshold. . . . . . . . . . . . . . . . . . . . 37
7.2 Mean-excess plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . 39
7.4 Estimation of Value-at-Risk and Expected shortfall . . . . . . . . 40
8 Multivariate distributions and dependence 43
8.1 Basic properties of random vectors . . . . . . . . . . . . . . . . . 43
8.2 Joint log return distributions . . . . . . . . . . . . . . . . . . . . 44
8.3 Comonotonicity and countermonotonicity . . . . . . . . . . . . . 44
8.4 Covariance and linear correlation . . . . . . . . . . . . . . . . . . 44
8.5 Rank correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8.6 Tail dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9 Multivariate elliptical distributions 53
9.1 The multivariate normal distribution . . . . . . . . . . . . . . . . 53
9.2 Normal mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.3 Spherical distributions . . . . . . . . . . . . . . . . . . . . . . . . 54
9.4 Elliptical distributions . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5 Properties of elliptical distributions . . . . . . . . . . . . . . . . . 57
9.6 Elliptical distributions and risk management . . . . . . . . . . . 58
10 Copulas 61
10.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10.2 Dependence measures . . . . . . . . . . . . . . . . . . . . . . . . 66
10.3 Elliptical copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
10.4 Simulation from Gaussian and t-copulas . . . . . . . . . . . . . . 72
10.5 Archimedean copulas . . . . . . . . . . . . . . . . . . . . . . . . . 73
10.6 Simulation from Gumbel and Clayton copulas . . . . . . . . . . . 76
10.7 Fitting copulas to data . . . . . . . . . . . . . . . . . . . . . . . . 78
10.8 Gaussian and t-copulas . . . . . . . . . . . . . . . . . . . . . . . . 79
11 Portfolio credit risk modeling 81
11.1 A simple model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
11.2 Latent variable models . . . . . . . . . . . . . . . . . . . . . . . . 82
11.3 Mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
11.4 One-factor Bernoulli mixture models . . . . . . . . . . . . . . . . 86
11.5 Probit normal mixture models . . . . . . . . . . . . . . . . . . . . 87
11.6 Beta mixture models . . . . . . . . . . . . . . . . . . . . . . . . . 88
12 Popular portfolio credit risk models 90
12.1 The KMV model . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
12.2 CreditRisk+ { a Poisson mixture model . . . . . . . . . . . . . . 94
A A few probability facts 100
A.1 Convergence concepts . . . . . . . . . . . . . . . . . . . . . . . . 100
A.2 Limit theorems and inequalities . . . . . . . . . . . . . . . . . . . 100
B Conditional expectations 101
B.1 De nition and properties . . . . . . . . . . . . . . . . . . . . . . . 101
B.2 An expression in terms the density of (X; Z) . . . . . . . . . . . 102
B.3 Orthogonality and projections in Hilbert spaces . . . . . . . . . . 103