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

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

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

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

Model Correlation Model Assumptions Incremental Changes Price Adequacy

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

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

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

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

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

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?

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

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:

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:

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

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

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

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

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?

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

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

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?

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

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

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

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

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

Capital Allocation Average of Possible Combinations / Game Theory

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

Pricing / ROE Game Theory Allocation / Some Risk Adjusted ROE

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

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

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

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

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

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

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

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

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

Stop Loss Reinsurance Aggregate Excess Does Not Benefit Stop Loss

Stop Loss Reinsurance Aggregate Excess Does Benefit Stop Loss

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

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

Loss Portfolio Transfer Only Selected LOBs & Years

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

Questions?