Reserve Variability Modeling: Correlation 2007 Casualty Loss Reserve Seminar San Diego, California...

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


How do we: correlate multiple ranges?

correlate multiple distributions?

incorporate different types of correlation?

use correlation for ERM?


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

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:

Risk Based Capital


Economic Capital


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

Economic Capital


Pricing Risk


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


Multi-Year ApproachPricing 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?


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”


In Proportion to Increase in VaR / “Last In”

Total Capital

Allocated Capital

Allocated Capital

Allocated Capital

Capital Allocation

LOB “A”

LOB “B”

LOB “C”


Average of Possible Combinations / Game Theory

Total Capital

Allocated Capital

Allocated Capital

Allocated Capital

Capital AllocationAverage 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 / ROEGame 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


Ceding Commission

Gross Distribution

Net Distribution

Ceded Distribution


Gross / Ceded / Net All DifferentVariable 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 ReinsuranceNo Co-Insurance:

Small Ceded Expected Value, Large RiskGross Distribution Net DistributionCeded Distribution

Agg XS Layer

Aggregate Excess ReinsuranceNo 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


Does Other Reinsurance Inure to Benefit of Stop Loss?

Future / Pricing Analysis– Include Correlation


LOB “A”

LOB “B”

LOB “C”


Aggregate Excess Does Not Benefit Stop Loss

Stop Loss Layer

Aggregate Excess Does Benefit Stop Loss


LOB “A”

LOB “B”

LOB “C”


Stop Loss Layer

Stop Loss ReinsuranceAggregate Excess Does Not Benefit Stop Loss

Stop Loss ReinsuranceAggregate Excess Does Benefit Stop Loss

Loss Portfolio Transfer

Reinsurer Assumes All Losses for Specific LOBs and Years

LPT Could Include Maximum Limit


Are Liabilities Discounted?

Historical Analysis– Include Correlations

Similar to Commutations

Loss Portfolio Transfer

LOB “A”

LOB “C”


Only Selected LOBs & Years

Loss Portfolio TransferOnly 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?

Reserve Variability Modeling: Correlation 2007 Casualty Loss Reserve Seminar San Diego, California...

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