# Syllabus Math 563

## 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