Mathematical Modeling and Statistical Methods for Risk Management
Lecture Notes
by Henrik Hult and Filip Lindskog

Preface

These lecture notes aims at giving an introduction to Quantitative Risk Management.We will introduce statistical techniques used for deriving the profit and-loss distribution for a portfolio of financial instruments and to compute risk measures associated with this distribution.The focus lies on the mathematical/ statistical modeling of market- and credit risk.Operational risk and the use of fnancial time series for risk modeling is not treated in these lecture notes.

Financial institutions typically hold portfolios consisting on large number of financial instruments.A careful modeling of the dependence between these instruments is crucial for good risk management in these situations.A large part of these lecture notes is therefore devoted to the issue of dependence modeling.

The reader is assumed to have a mathematical/statistical knowledge corresponding to basic courses in linear algebra, analysis, statistics and an intermediate course in probability.The lecture notes are written with the aim of presenting the material in a fairly rigorous way without any use of measure theory.


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