Capital Allocation for Reinsurance Pricing presentation by Ira Robbin

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

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. 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. 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. 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. 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. 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 = r0 + ( rm – ro) • 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. 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. 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. 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. 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. 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. 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. 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. 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. Reinsurance Loss Models • Attritional loss • May have lower truncation • 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. 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. 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. 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. 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!