Math 563. Risk Modeling and Analysis Instructor Syllabus
Text
1. Denuit, Dhaene, Goovaerts and Kaas (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley.
2. McNeil, Frey, and Embrechts (2005). Quantitative Risk Management: Concepts, Techniques, Tools, Princeton University Press.
Chapter 1 Modeling Risks (3 hours)
Review basic properties of random variables, expectations, transforms, conditional distributions, comonotonicity and mutual exclusivity
Chapter 2 Measuring Risk (6 hours)
- Risk measures
- Value-at-Risk
- Tail-Value-at-Risk
- Risk measures based on expected utility
- Risk measures based on distorted expectation
Chapter 3 Comparing Risks (6 hours)
- Stochastic order relations
- Stochastic dominance
- Convex and stop-loss orders
Chapter 4 – Dependence between Risks (9 hours)
- Sklar’s representation theory
- Bivariate Copulas
- Properties of Copulas
- Archimedean family of copulas
- Multivariate copulas
Chapter 5 – Measuring Dependence (3 hours)
- Concordance measures
- Dependence structures
Chapter 6 – Comparing Dependence (3 hours)
- Correlation order
- Multivariate case using supermodular order
- Positive orthant dependence order
Chapter 7 – Dependence in Credibility Models Based on Generalized Linear Models (6 hours)
- Poisson credibility models for claim frequencies
- Static credibility model
- Dynamic credibility models
- Dependence induced by Bonus-Malus scales
Chapter 9 Integral Ordering and Probability Metrics (6 hours)
- Integral stochastic ordering
- Integral probability metrics
- Total-variation distance
- Kolmogorov distance
- Wasserstein distance
- Stop-loss distance
- Integrated stop-loss distance
- Compound Poisson approximation for a portfolio of dependent risks
If time permits, the following topics may be covered.
- Statistical inferences for copulas including maximum likelihood estimators, inference functions for margins, maximum pseudo-likelihood estimators, Kendall’s tau estimator, confidence intervals.
- Materials
- P. Embrechts and M. Hoffert. (2013) Statistical inference for copulas in high dimensions: a simulation study. ASTIN Bulletin. 43(2) 81-95.
- C. Genest and L.P. Rivest. (1993) Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88 (423) 1034-1043.
Midterm Exams (1 hour)
Total: 43 hours