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Reserve Variability Modeling:Correlation
2007 Casualty Loss Reserve SeminarSan Diego, California
September 10-11, 2007
Mark R. Shapland, FCAS, ASA, MAAAConsulting 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
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
Correlation of Model Assumptions
100d
t = w+d
Development Period
Payment Period
Accident Period
w
19971997
19981998
20062006Futu
re
Past
…
Correlation of Incremental ChangesAre the Incremental Payments Correlated?
LOB A LOB B
Are the Model Assumptions Correlated?
Correlation of Incremental ChangesAre the Residuals Correlated?
LOB A LOB B
Correlation: 26.3% P-Value: 5.7%
Correlation of Premium AdequacyAre 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:
99th 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
Expected Value
99th Percentile 99th Percentile
Capital = 1,000M Capital = 600M
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 99th Percentile?Reserving Risk:
Required Capital
...
99% TVaR
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 99th Percentile?Pricing Risk:
Required Capital
...
99% TVaR
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
Allocated 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
Allocated 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
Allocated Capital
Allocated Capital
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
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
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 ReinsuranceNo Co-Insurance:
Small Ceded Expected Value, Large RiskGross Distribution Net DistributionCeded Distribution
Agg XS Layer
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
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
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