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