Risk Measures and Management

VaR, TVaR, expected shortfall, coherent risk measures, and ERM frameworks.

Key Concepts

• •Value at Risk (VaR): the alpha-quantile of the loss distribution
• •Tail Value at Risk (TVaR): expected loss given that loss exceeds VaR
• •Expected shortfall and conditional tail expectation
• •Coherent risk measures: the four axioms (monotonicity, subadditivity, positive homogeneity, translation invariance)
• •VaR is not subadditive: the diversification problem
• •Risk-based capital: NAIC RBC formula components
• •Economic capital: internal models vs. standard formulas
• •Stress testing and scenario analysis for risk assessment
• •Enterprise risk management: identifying, measuring, and managing all risks
• •Capital allocation: Euler method and marginal contributions

Study Tips

• 1.Memorize the formulas for VaR and TVaR for common distributions.
• 2.Practice computing VaR and TVaR for both continuous and discrete distributions.
• 3.Understand why TVaR is preferred over VaR from a theoretical perspective.
• 4.Work through examples showing VaR violating subadditivity.
• 5.Connect risk measures to practical capital requirements.

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