Actuarial Notes

Measurement and Modelling of Dependencies in Economic Capital

By R.A. Shaw, A.D. Smith and G.S. Spivak · Institute of Actuaries · 2010 · CFE101-114-25

Study guide: Exam CFE-101

A discussion paper presented to the Institute of Actuaries on 10 May 2010. It covers the measurement and modelling of dependency within economic capital models, from the variance-covariance matrix approach to copulas and causal models with feedback loops, together with data and model uncertainties and ways to communicate the impact of dependency to senior management. It is concerned with the aggregation part of an economic capital model, not the marginal risk distributions.

On the syllabus

  • Exam CFE-101 — CFE101-114-25: Measurement and Modeling of Dependencies in Economic Capital, Ch. 4 & 7; outcomes 2.2c, 2.2e, 2.2g, 3.3a, 3.3b.

1 Introduction

2 Executive Summary

3 Why Diversification is Important

  • 3.1 Economic Capital
  • 3.2 Regulatory Developments
  • 3.3 Solvency II – Internal Model Approval
  • 3.4 Diversifiers and Hedgers and the Importance of perspective

4 Correlation as the Simplest Type of Dependency

  • 4.1 Dependency Structures
  • 4.2 What do we mean by Dependency?
  • 4.3 Pearson Correlation Coefficient
  • 4.4 Spearman Coefficient
  • 4.5 Kendall Tau Correlation
  • 4.6 Numerical Example

5 Risk Aggregation

  • 5.1 Risk Aggregation Framework
  • 5.2 Risk Aggregation Methodologies
  • 5.3 Simple Summation
  • 5.4 Fixed Diversification Percentage
  • 5.5 Variance-Covariance Matrix
  • 5.6 Copulas
  • 5.7 Causal Modelling
  • 5.8 Natural Catastrophe vs Reinsurance Credit Risk Aggregation

6 Data and Model Uncertainties

  • 6.1 Spurious Relationships
  • 6.2 Correlation and Linearity (Anscombe's Quartet)
  • 6.3 Biases Introduced by the Way We Look at Data
  • 6.4 Wrong-Way Risk
  • 6.5 Multivariate Distributions: The Curse of Dimensionality
  • 6.6 Multivariate Distributions: Multivariate Model Selection
  • 6.7 Ten Properties of Good Calibration

7 Variance-covariance Matrix Methods

  • 7.1 Methodology
  • 7.2 Advantages and Disadvantages of Variance-Covariance Matrix Approach
  • 7.3 Risk Granularity
  • 7.4 Simpler Variants
  • 7.5 Parameterisation
  • 7.6 Variance-Covariance Matrix Cross-Terms
  • 7.7 Positive Semi-Definite Matrices
  • 7.8 Elliptical Distributions

8 Copula Modelling Methods

  • 8.1 Introduction
  • 8.2 Copula Mathematics
  • 8.3 Advantages and Disadvantages of Copula Approach
  • 8.4 Coefficient of Tail Dependence
  • 8.5 Different Types of Copula
  • 8.6 Copula Selection
  • 8.7 Parameterisation
  • 8.8 Monte Carlo Simulation
  • 8.9 Copula Aggregation through Vines

9 Tail Dependency

  • 9.1 What Do We Mean by Tail Dependency
  • 9.2 The So-Called 'Tail Correlation'
  • 9.3 Use of 'Tail Correlation's Instead of Tail Dependence

10 Causal Modelling Methods

  • 10.1 Methodology
  • 10.2 Some Examples
  • 10.3 Advantages and Disadvantages Causal Model Approach
  • 10.4 Causal Modelling Design Challenges

11 Dependency Communication

  • 11.1 Different Ways of Thinking About Dependency
  • 11.2 Economic Capital Aggregation
  • 11.3 Joint Probability Density Function
  • 11.4 Scatter Plot
  • 11.5 Joint Excess Probability
  • 11.6 Tail Concentration Function
  • 11.7 Kendall Tau Correlation
  • 11.8 Coefficient of Tail Dependence
  • 11.9 Implied Gaussian Correlation

12 Using Half-Space Probabilities to Capture Dependencies

  • 12.1 Combined Stress Tests and Two-Way Correlations
  • 12.2 Combined Events at a Given Confidence Level
  • 12.3 Two-Sided Correlations
  • 12.4 Half Space Definitions

13 Conclusions

Acknowledgements

References

Appendix A Positive Semi-Definite Matrices and Cholesky Decomposition

Appendix B Ranges for Missing Correlations

Appendix C Elliptical Distributions – Capital Aggregation Proof and Other Matters

Appendix D ABC Insurance Company – Other Distribution Parameters

Appendix E ABC Insurance Company – Lognormal Risks (CV = 25%, Corr = 10% to 50%)

Appendix F Canonical Maximum Likelihood – LIBOR 3M vs. FTSE All Share Total Return

Appendix G RJEP(0.95) & R(0.95) – t Copula 5 df, Gaussian Copula and Independence

Appendix H Kendall Tau – t Copula 5 d.f. and Gaussian Copula

Sources