2. Risk Modeling and Aggregation of Risks

2. Risk Modeling and Aggregation of Risks

1. 1. Extreme Value Theory

   • Generalized extreme value distr.
     • Formula + parameters, extreme values approach, major drawback
   • Generalized Pareto distr.
     • Formula + parameters, key + approach for choosing threshold
   • Modeling with a Hybrid Empirical /GPD Model

2. 2. Quantifying particular risks:

   • Market + Economic Risk
     • Interest rate risk
       • Foreign Exchange risk
         • Credit risk
           • Liquidity risk
             • Systemic risks
               • Demographic risk
                 • Non-Life Insurance risk
                   • Operational risk
                     • See OR in 2a
           • see counterparty risk in 2a See Credit risk in 3
       • See investment mngt in 2a

3. 3. Unquantifiable risks

   • ways to assess size and loss, risk map, very unquantifiable risks

4. 4. Computing VAR (defn, notations)

   • Nonparametric VAR ( relative, absolute, ETL)
   • Parametric VAR (relative, absolute)
   • 4 Properties for risk measures (coherency)
   • 5 uses of VAR + choices of factors: 1. benchmark, 2. potential loss measure, 3. equity capital (maxVar) 4. Backtesting 5. regulatory requirement. Note: conversion of VAR
   • VAR Precision (SE of estimates, CB, parametric vs nonparametric, comparison of methods)
   • Optional: EVT : quantiles, calc EVT estimators (VAR + ETL)
     • Variance of CTE estimator
       • sampe avg, distr of variance and CTE estimator

5. 5. VAR Delta-normal method (covariance matrix method)

   • Portfolio VAR (Individual VAR, Corr. pf risk, diversification benefit)
   • VAR tools: Marginal VAR, Beta, Incremental VAR (calc + approx), Best hedge, Component VAR, % Contribution (Drill-Down Exercises), VAR tools for general distributions
   • From VAR to Pf management: minimum risk pf, expected return, efficient frontier, Sharpe ratio, Global minim risk portfolio, optimal portfolio
   • See VaR shortcomings in 3

6. 6.Ways to model volatility

   • Distinguish time-varying risk from outliers
   • Time-Varying risk models
     • Moving Averages (defn, drawbacks)
     • GARCH Estimation (defn, adv, disav., parameter estimation, Long-horizon forecast
     • RiskMetrics Approach: EWMA (defn, decay, adv, disav.)
     • Modeling correlations using the 3 models. Crashes + correlations
   • Using Options Data with ISD ( risk-neutral vs real world, adv, disv, use of options-based data: adv vs disav.)

7. 7. Monte Carlo Methods

   • a. Dfn, Adv + Disadv.
     • b. Simulation w/ 1 variable: Simulating price path (GBM), create random nb, bootstrap, VAR, RM + pricing methods, speed vs accuracy + acceleration methods
     • c. Simulation w/ multiple variables, Independent to corr variables, nb of risk factors
   • d. Deterministic simulation
   • e. Choice of model: stochastic process, GBM, dynamics of interest rate, equilibrium vs arbitrage
   • f. Comparison of Var of diff methods

8. 8. Measurement and Modelling of Dependencies in Economic Capital

   • Different correlations
     • Dependency, Correlations (Pearson, Spearman, Kendall Tau, Properties & Deficiencies)
   • Risk aggregation methodologies
     • simple summation, fixed div factor, var-covar, copula, causal modeling

9. 9. 3 Aggregation techniques

   • Var-Covar approach
     • foundation, linear dependence, independence and diversification , methods to overcome weakness
   • Copula-based simulation
     • conditions, Gaussian , Archimedean, t-distirbution, size vs dependence, lower/upper bound, tail dependence, comparison
     • 14. Copula
       • Sklar's theorem, tail dependence
       • Families: Archimedean, Elliptical (Gaussian, student)
       • Extreme value copulas, archimax
       • Estimation of parameters: MLE, semiparametric, nonparametric, goodness-of fit testing (chi-square and univariate methods), simulation
   • Scenario-based simulation
     • scenario analysis, simulation, types of models

10. 10. Parameter risk

    • Definition, sources, principles of diversification, uncertain trend, parameter estimation methods (3)

11. 11. Economic Capital Modeling

    • Required EC vs Available EC, views of solvency : statutory vs economic, benefits of EC
    • Types of risks to consider + how to measure them
    • 7 Modeling considerations

12. 12. Aggregation of risks and allocation of capital

    • Risk Assessment techniques
      • Immediate stress, Projection Scenarios, Multivariate Stress Tests (+ESG)
        • Calibration: mkt-consistent (risk-neutral) vs real-world
    • Aggregation Techniques:
      • Top-down approaches: Correlations and Copulas. Multivariate methods.
    • Application of Capital Allocation
      • Pricing & Technical Provisions, Risk budgeting + Capital Allocation (mitigation, approaches)
        • Allocation of capital
          • Factors to determine, Approaches
            • 1. Marginal Approach: pro-rata/linear marginal contributions, discrete marginal contributions, continuous marginal contributions, Myers-Read allocation method
            • 2. Game Theory

13. 13. Stress Testing/Scenario Analysis

    • Defn of Scenario, Sensitivity and Stress test
      • Types of scenarios: reverse scenarios, historical sc., synthetic sc., company-specific sc., single-event sc., multi-event sc., global sc.

14. 15. Model Validation Principles

    • Key Model Validation Principles
    • Elements of Model Risk Management
    • Summary of Core Validation Principles
      • Model components
    • See 4 Model Governance