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## Optimal Risk Selection Using Cat Models

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**Thinking**The Box beyond Optimal Risk SelectionUsing Cat Models Lixin Zeng, Ph.D.CAS Seminar on Funding of Catastrophe RisksProvidence RIOctober 17, 2000**Optimal Risk Selection**Outline • Use and Misuse of Cat Model • Optimal Risk Selection • Example**Optimal Risk Selection**What a Cat Model Tells Us • Loss Probability Distribution • Expected Loss • Probable Maximum Loss (a.k.a. Value at Risk) • Relative Value • Deal A is riskier than Deal B • Correlation: Constructing a Portfolio with High Return on Risk Capital**Optimal Risk Selection**Great! Cat Problem Solved? • Underwriting Decisions • Rate Making • Reinsurance Purchasing • Securitization**Optimal Risk Selection**What’s Inside a Cat Model • State-of-the-Art Science in Meteorology and Seismology • Engineering Experts’ Opinions for Structure Damage • Modern Simulation Technology • Lack of Consensus in Scientific Community on Key Issues • Best Guesses Given Limited Data and Modeling • Computation Hurdles vs. Convergence**Optimal Risk Selection**User’s Responsibilities • Understand Key Assumptions • Appreciate Sources of Uncertainty • Independent Model Evaluation • Integrate Multiple Models**Optimal Risk Selection**What’s a Cat Model Good For? • Relative (Not Absolute) Indicators • Differentiate Good and Bad Risks/Areas • Risk Selection • Portfolio Optimization**“Bad” Risks**“Good” Risks Final Portfolio Maximum Return on Risk Capital Optimal Risk Selection Goal of Risk Selection Existing Portfolio**Optimal Risk Selection**Return on Risk Capital (RORC) • Return • Cat premium minus expected cat loss • Risk Capital • Probable maximum loss (or value at risk) • Expected policy holder deficit • Loss standard deviation • Applicable to Both Individual Risks and Portfolios**Optimal Risk Selection**RORC: Definition • A Simple Definition Cat Premium - Expected Cat Loss Cat X-Year PML • Different Definitions • Financial strength • Risk tolerance • etc.**Optimal Risk Selection**Identify “Bad” Individual Risks An Individual Risk Is the Worst in a Portfolio if (1) It has the lowest RORC among all risks (2) Removing it will increase the portfolio’s RORC the most vs. removing any other individual risk The right answer: (1) or (2)?**Optimal Risk Selection**A Sample Portfolio**Optimal Risk Selection**A Sample Portfolio**Optimal Risk Selection**Identify “Good” Prospective Risks: Same Idea An Individual Risk Is the Best Prospect for a Portfolio if (1) It has the highest RORC among all prospects (2) Including it in the portfolio will increase the portfolio’s RORC the most vs. including any other prospect The right answer: (1) or (2)?**Optimal Risk Selection**Real World: Computational Issues • Finding the X Worst (or Best) from N Risks • Requires CNX calculations • E.g. requires ~ 17,000,000,000,000 calculations to pick 10 worst (or best) out of 100 risks • Need a Faster, More Practical Approach**Remove #1 only**Remove #1 only Remove #2 only Remove #2 only ………. ……... ……... ……... Remove #N-1 only Remove #N-1 only Remove #N only Optimal Risk Selection A Real Solution: Discrete Steepest Descent Existing Portfolio Portfolio w/o worst risk Portfolio w/o X worst risks Remove #1 only ……….. Remove #N-X only**Optimal Risk Selection**Finding the X Worst (or Best) from N Risks • Requires O(N2) Calculations • E.g. requires 1,000 calculations to pick 10 worst (or best) out of 100 risks • Innovative algorithm to handle large portfolios • Stochastic Perturbation to Avoid Local Minimum**Optimal Risk Selection**Real-World Example: Portfolio of 1500 Risks Optimal Risk Selection Benchmark**Optimal Risk Selection**Cautions • Cat Model Relative Bias • Geographical and structural • Usually less than absolute bias • But cannot be ignored • Use of a Single Point on the PML Curve • Potentially misleading**Optimal Risk Selection**Conclusions • Cat Model • Relative indications more credible than absolute values • Portfolio Optimization • One of the best uses of cat models • Cat model relative bias must be evaluated and understood