1. Capital Allocation for Reinsurance Pricing presentation by Ira Robbin Casualty Actuaries in Reinsurance Seminar on Reinsurance, Boston May 19-20, 2008

2. 2 Ground Rules and Disclaimers  If anything I say gets me in trouble in the future, you are all witnesses that I never said it.  Nothing I say should be taken too seriously.  Don’t come near to violating Anti-trust guidelines!  No statements of corporate opinion will be made or should be inferred.  Ask questions to clarify the material anytime.  If you rely on ideas contained in this presentation and lose your shirt, remember I am not in the clothing trade.

3. 3 Overview of Discussion  Overall goal: raise awareness of issues that arise in allocating capital for reinsurance treaty pricing applications  Will provide opinions on answers to some key questions  RORAC context  Risk metrics  Capital for Property CAT treaties  Capital for Casualty treaties  Treaty features  Loss Models  Gauntlet of tests  Conclusion: It is harder than you think to get this right!

4. 4 Basic Context  Capital allocation in pricing  Hypothetical division: not an actual segregation of real funds  Nothing is actually allocated: total capital stands behind all contracts  Theoretical required amount for each deal  Use in corporate pricing process  Company may decline to write/ impose extra authority clearances on deals with pricing below indicated  Useful in price monitoring: follow indicated vs market price  Creates incentives – are these the ones that are intended?

5. 5 RORAC Pricing Approach  Return on Risk-Adjusted Capital (RORAC)  Theoretically required capital, not actual!  Indicated Price is price to achieve target ROE  Target ROE is set by management  Should be the same for all deals and LOBs  Should be sum of risk-free rate+ risk margin  Contrast with Risk-Adjusted Return on Capital (RAROC)

6. 6 Risk-Sensitive Pricing from RORAC  More risk  More capital  More capital  Higher theoretical price needed  Higher price needed to cover margin on larger amount of capital  Risk-sensitive capital leads to risk-sensitive theoretical target pricing  Actual market price driven by supply and demand

7. 7 Other Risk Pricing Approaches  Process versus parameter risk  Theory says “No charge for process risk”  Theory  CAT gets a small capital allocation  wrong  Non-Diversifiable risk pricing + CAPM  CAPM produces a target return: r = r 0 +  ( r m – r o ) ￚ Beta is Cov of outcome with stock market  A RAROC approach, not a RORAC approach.  Theory  Return on CAT should equal stock market return, on any amount of capital  wrong  Role of capital: amount of capital impacts insurance return, but is essentially irrelevant in CAPM stock pricing

8. 8 Capital Allocation Issues  Choice of risk metric  Calibration  Differing risks by LOB  Property – CAT risk  Casualty – Mass tort/reserve risk/capital duration  Treaty provision adjustments  Eg. reinstatements  Stand alone basis vs treaty impact on portfolio  Allocation of diversification benefits

9. 9 Risk Metric Classification Properties  Coherent or not  Coherence = Sensible scaling, shifting, and diversification benefit properties  Tail Focused vs Full Distribution  Capital consumption perspectives  Explicable  Can it be sold to management/ financial gurus?  Do the parameters have any intuitive meaning?

10. 10 Risk Metrics  Variance and Standard Deviation – not coherent  Includes favorable and adverse deviations  VaR – Value at Risk – not coherent  VaR(A+B) can exceed VaR(A) + VaR(B)  TVaR – Tail Value at Risk- is coherent  TCE = Tail conditional expectation- is coherent  Captures events in the extreme tail  Wang transform –is coherent  What does the power parameter represent?  Capital Allocation by Percentile Layer-Bodoff method  “Hold capital for the 250 year event” versus “Hold capital even for the 250 year event”

11. 11 Calibration  Total capital over all treaties should reflect management’s overall risk/return perspective  Regulatory and rating agency constraints  IRIS and RBC  S&P and Best’s  Eliminate capital for investment risk – assume risk-free rate in pricing model  Duration of capital for long-tail lines – need for capital to cover reserve inadequacy.

12. 12 Property CAT Treaty Capital  Per Occ Loss vs Annual Agg Loss  Capital needed to cover large loss event OEP or  Capital needed to cover a bad year AEP  Example: Katrina vs KRW  Treaty Loss vs Treaty Impact on portfolio  Stand-alone treaty – no credit for diversification  Should pricing be used to manage aggregation?  OEP Impact is sometimes $0 or negative  Danger with OEP Impact  Promotes writing of risky business in low PML zones

13. 13 Casualty Treaty Capital  Reserve risk  Total historic reserve inadequacy is not random  In concept, only need capital for inherent reserve uncertainty  Duration of capital  How to reflect long-term commitment of capital?  ROE = PVI/PVE is one solution  See “IRR, ROE, and PVI/PVE” paper by Robbin

14. 14 Treaty Provision Adjustments  How to measure downside risk by treaty  Treaty initial loss distribution  Treaty provisions  Reinstatements, Swing Rating, Sliding Scale Commission, Loss Corridor, Profit Commission, etc.  Provisions can impact commissions and premiums as well as losses  Some may not change expected amounts  Some reduce downside risk; others share gains

15. 15 Model UW Loss to Capture Net Treaty Risk  UW Loss = Loss + Expense - Premium  Loss alone does not fully describe treaty risk  Doesn’t capture impact of treaty features  UW Loss provides a more complete picture  General way to handle different features  Same risk for alternative deals with same UW Loss distribution  Note sign convention: negative UW Loss is a gain  See Robbin and DeCouto paper, “Coherent Capital for Treaty ROE Calculations”

16. 16 Reinsurance Loss Models  Attritional loss  May have lower truncation ￚ e.g no loss below 25% LR  Usually described via lognormal, gamma, and other well- known programmable distributions.  Excess loss  Low frequency/high severity potential  Focus on per risk XOL impact  CAT  Need to convert event lists to event frequency and severity per event

17. 17 Simulation Modeling of Loss  A brute-force adaptable approach  Model validation  Run single trials and extreme cases- check sample output  Black box syndrome  Confuse number of trials with accuracy of parameters  Neglect possibility structure is wrong  Practical concerns  Convergence issue - keep running till the answers stabilize?  Reproducibility – fix the random seed?  Pricing alternatives – is differential larger than error bar?

18. 18 Modeling Losses via Points and Probabilities (PnP)  Insurance loss distributions suitable for PnP modeling  Mass at zero  No mass below a truncation point  Conditional distribution described by a mix of tractable parametric models ( gamma, lognormal, pareto and so forth)  Technique  Choose 100 points of interest including zero  Compute Limited Expected Values (LEVs)  Derive Probs to match LEVs  Reproducible

19. 19 Gauntlet of Tests  LOB effects  Change in share  Does capital change in proportion to share?  Change in reinsurance rate adequacy  Should rate improvement decrease required capital?  Net rated deals  How much capital is needed for ceding commission?  Reinstatements  Do they reduce or increase reinsurer risk?

20. 20 Conclusions  Allocating capital is difficult  Presents major theoretical and practical challenges  Know before you go  Run all current treaties through any proposed model  Have line pricing actuaries look at pricing differentials – what incentives will it create?  Calibrate in advance  The proof of the capital method is in the pricing!