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Reserve Variability Modeling: Correlation 2007 Casualty Loss Reserve Seminar San Diego, California September 10-11, 2007 Mark R. Shapland, FCAS, ASA, MAAA Consulting Actuary

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A “Range” is not the same as a “Distribution” A Range of Reasonable Estimates is a range of estimates that could be produced by appropriate actuarial methods or alternative sets of assumptions that the actuary judges to be reasonable. A Distribution is a statistical function that attempts to quantify probabilities of all possible outcomes. Ranges vs. Distributions

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How do we: correlate multiple ranges? correlate multiple distributions? incorporate different types of correlation? use correlation for ERM?

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Ranges vs. Distributions A Distribution can be used for: Risk-Based / Economic Capital – Reserve Risk – Pricing Risk – ALM Risk Pricing / ROE Reinsurance Analysis – Quota Share – Aggregate Excess – Stop Loss – Loss Portfolio Transfer Dynamic Risk Modeling (DFA) Strategic Planning / ERM Allocated Capital Risk Transfer

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Model Correlation Model Assumptions Incremental Changes Price Adequacy

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Correlation of Model Assumptions 10 d t = w+d Development Period Payment Period Accident Period w 1997 1998 2006 Future Past …

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Correlation of Incremental Changes Are the Incremental Payments Correlated? LOB A LOB B Are the Model Assumptions Correlated?

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Correlation of Incremental Changes Are the Residuals Correlated? LOB A LOB B Correlation: 26.3% P-Value: 5.7%

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Correlation of Premium Adequacy Are the Ultimate Loss Ratios Correlated? LOB A LOB B Is Premium Adequacy Correlated?

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Risk Based Capital Protection Against “Adverse Outcomes” Ruin Theory: “Risk of Bankruptcy < X%” Statutory Valuation / Stockholder Perspective Types of Risk Related to Bankruptcy: – Insurance Risks – Financial Risks – Operational Risks – Strategic Risks

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Risk Based Capital Insurance Risks: – Reserving Risk: Protect against “Adverse Development” Reserves Not Sufficient to Cover “All” Possibilities – Pricing Risk: Protect against “Future Losses” Will Combined Ratios Exceed Breakeven? Financial Risks: – Asset / Liability Matching Risk: Protect Against “Cash Flow Deficiencies” Will Timing of Investment Cash Match Claim Payments?

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Risk Based Capital Will Capital Cover 99% of Possible Outcomes? Reserving Risk: 99 th Percentile Required Capital

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Risk Based Capital Capital is a “Shared Asset” LOB “A” LOB “B” LOB “C” Aggregate Distribution with 100% Correlation (Added) Aggregate Distribution with 0% Correlation (Independent) Reserving Risk:

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Risk Based Capital Aggregate Distribution with 100% Correlation (Added) Aggregate Distribution with 0% Correlation (Independent) Expected Value 99 th Percentile Capital = 1,000M Capital = 600M Capital is a “Shared Asset” Reserving Risk:

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Risk Based Capital Capital is a “Shared Asset” Reserving Risk:

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Economic Capital Protection Against “Adverse Outcomes” How Bad if we Exceed Ruin Threshold? Market Valuation / Policyholder Perspective – Discounting / Tail Value at Risk Same Types of Risk Related to Bankruptcy Capital is a Shared Asset

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Economic Capital Is Capital Sufficient If We Exceed 99 th Percentile? Reserving Risk: Required Capital... 99% TVaR

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Economic Capital Capital is a “Shared Asset” Reserving Risk:

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Pricing Risk Protection Against “Adverse Outcomes” Focus on Future Losses – How Many Years into Future? Breakeven Loss Ratio – With or Without Investment Income? Assume Constant Expense Ratio – Some Expenses Vary with Loss Ratio Capital is a Shared Asset – Stronger Correlation than Reserve Risk?

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Risk Based / Economic Capital Capital Sufficient If We Exceed 99 th Percentile? Pricing Risk: Required Capital... 99% TVaR

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Risk Based / Economic Capital Multi-Year Approach Pricing Risk: Future Years

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Asset / Liability Matching Risk Protection Against “Cash Flow Deficiencies” Focus on Future Cash Flow – How Many Years into Future? – Timing of Current Liability Payments – Timing of Future Liability Payments Assume Constant Expense Ratio – Some Expenses Vary with Loss Ratio Capital is a Shared Asset – Blend Reserving & Pricing Correlation?

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Risk Based / Economic Capital Multi-Year Approach (Current Liabilities) Asset / Liability Matching Risk: Future Years

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Capital Allocation Total Capital is Based on Total Risks How Do We Allocate to LOB, SBU, etc.? – In Proportion to VaR or TVaR: Independent / “First In” Approach – In Proportion to Increase in VaR or TVaR: Marginal / “Last In” Approach – Average of All Possible Combinations: Game Theory (Shapley Values), Myers-Read, Covariance Share (Mango-Brehm-Kreps) Can Result in Negative Capital Not Intended to Completely Equalize ROE – DRM Approach to Equalize ROE

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Capital Allocation LOB “A” LOB “B” LOB “C” Aggregate Distribution with Model Correlation In Proportion to VaR / “First In” Total Capital Allocated Capital

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Capital Allocation LOB “A” LOB “B” LOB “C” Aggregate Distribution with Model Correlation In Proportion to Increase in VaR / “Last In” Total Capital Allocated Capital

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Capital Allocation LOB “A” LOB “B” LOB “C” Aggregate Distribution with Model Correlation Average of Possible Combinations / Game Theory Total Capital Allocated Capital

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Capital Allocation Average of Possible Combinations / Game Theory

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Pricing / ROE Allocate Capital “Equally” / In Proportion to Exposure – Strongest “Risk-Adjusted” Target ROE Allocate Capital Using “First In” or “Last In” Approach – Stong “Risk-Adjustment” for Target ROE Allocate Capital Using Covariance Share Approach – Some “Risk-Adjustment” for Target ROE DRM Approach to Allocate Capital / Equalize ROE – Constant Target ROE by LOB

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Pricing / ROE Game Theory Allocation / Some Risk Adjusted ROE

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Quota Share Reinsurance Reinsurer Assumes Constant Percentage of Both Premiums & Losses Ceding Commission Varies with Loss Ratio – With Maximum and Minimum Bounds Impact on Net and Ceded Results Expected Average Ceding Commission Future / Pricing Analysis

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Variable Ceding Commission: Gross / Ceded / Net All Different Quota Share Reinsurance Ceding Commission Gross Distribution Net Distribution Ceded Distribution

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Quota Share Reinsurance Gross / Ceded / Net All Different Variable Ceding Commission:

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Aggregate Excess Reinsurance Reinsurer Assumes Losses if Aggregate Exceeds Company Retention Aggregate Excess Layer Includes Maximum Limit Aggregate Layer Can Include Co-Insurance Percent Multiple Layers / Reinsurers Future / Pricing Analysis

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Aggregate Excess Reinsurance No Co-Insurance: Small Ceded Expected Value, Large Risk Gross Distribution Net Distribution Ceded Distribution Agg XS Layer

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Aggregate Excess Reinsurance No Co-Insurance: Small Ceded Expected Value, Large Risk

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Stop Loss Reinsurance Reinsurer Assumes Losses if Aggregate Loss Ratio Exceeds Company Retention Stop Loss Layer Includes Maximum Limit Stop Loss Layer Can Include Co-Insurance Percent Multiple Layers / Reinsurers Does Other Reinsurance Inure to Benefit of Stop Loss? Future / Pricing Analysis – Include Correlation

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Stop Loss Reinsurance LOB “A” LOB “B” LOB “C” Aggregate Distribution with Model Correlation Aggregate Excess Does Not Benefit Stop Loss Stop Loss Layer

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Aggregate Excess Does Benefit Stop Loss Stop Loss Reinsurance LOB “A” LOB “B” LOB “C” Aggregate Distribution with Model Correlation Stop Loss Layer

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Stop Loss Reinsurance Aggregate Excess Does Not Benefit Stop Loss

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Stop Loss Reinsurance Aggregate Excess Does Benefit Stop Loss

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Loss Portfolio Transfer Reinsurer Assumes All Losses for Specific LOBs and Years LPT Could Include Maximum Limit Multiple Layers / Reinsurers Are Liabilities Discounted? Historical Analysis – Include Correlations Similar to Commutations

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Loss Portfolio Transfer LOB “A” LOB “C” Aggregate Distribution with Model Correlation Only Selected LOBs & Years

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Loss Portfolio Transfer Only Selected LOBs & Years

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Enterprise Risk Management DRM is the Quantifying Risk Portion of ERM Start Small to Build a Complete Management System Capital Management Strategic Financial Planning Performance Management Reinsurance Optimization

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Questions?

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