1. CARe Seminar, NYC February 28, 2002 Jonathan Hayes, ACAS, MAAA Uncertainty And Property Cat Pricing

2. Agenda n Models l Model Results l Confidence Bands n Data l Issues with Data l Issues with Inputs l Model Outputs n Pricing Methods l Standard Deviation l Downside Risk n Role of Judgment l Still Needed

3. “A Nixon-Agnew administration will abolish the credibility gap and reestablish the truth – the whole truth – as its policy.” Spiro T. Agnew, Sept. 21, 1973 The Search For Truth

4. Florida Hurricane Amounts in Millions USD

5. Florida Hurricane Amounts in Millions USD

6. Modeled Event Loss Sample Portfolio, Total Event

7. Modeled Event Loss By State Distribution

8. Modeled Event Loss By County Distribution, State S

9. Why Don’t The Models Agree?

10. Types Of Uncertainty (In Frequency & Severity) n Uncertainty (not randomness) l Sampling Error u 100 years for hurricane l Specification Error u FCHLPM sample dataset (1996) 1 in 100 OEP of 31m, 38m, 40m & 57m w/ 4 models l Non-sampling Error u El Nino Southern Oscillation l Knowledge Uncertainty u Time dependence, cascading, aseismic shift, poisson/negative binomial l Approximation Error u Res Re cat bond: 90% confidence interval, process risk only, of +/- 20%, per modeling firm Source: Major, Op. Cit..

11. Frequency-Severity Uncertainty Frequency Uncertainty (Miller) n Frequency Uncertainty l Historical set: 96 years, 207 hurricanes l Sample mean is 2.16 l What is range for true mean? n Bootstrap method l New 96-yr sample sets: Each sample set is 96 draws, with replacement, from original l Review Results

12. Frequency Bootstrapping n Run 500 resamplings and graph relative to theoretical t-distribution Source: Miller, Op. Cit.

13. Frequency Uncertainty Stats n Standard error (SE) of the mean: n 0.159 historical SE n 0.150 theoretical SE, assuming Poisson, i.e., (lambda/n)^0.5

14. Hurricane Freq. Uncertainty Back of the Envelope n Frequency Uncertainty Only n 96 Years, 207 Events, 3100 coast miles n 200 mile hurricane damage diameter n 0.139 is avg annl # storms to site n SE = 0.038, assuming Poisson frequency n 90% CI is loss +/- 45% l i.e., (1.645 * 0.038) / 0.139

15. Frequency-Severity Uncertainty Severity Uncertainty (Miller) n Parametric bootstrap l Cat model severity for some portfolio l Fit cat model severity to parametric model l Perform X draws of Y severities, where X is number of frequency resamplings and Y is number of historical hurricanes in set l Parameterize the new sampled severities n Compound with frequency uncertainty n Review confidence bands

16. OEP Confidence Bands Source: Miller, Op. Cit.

17. OEP Confidence Bands Source: Miller, Op. Cit.

18. OEP Confidence Bands n At 80-1,000 year return, range fixes to 50% to 250% of best estimate OEP n Confidence band grow exponentially at frequent OEP points because expected loss goes to zero n Notes l Assumed stationary climate l Severity parameterization may introduce error l Modelers’ “secondary uncertainty” may overlap here, thus reducing range l Modelers’ severity distributions based on more than just historical data set

19. The Building Blocks Policy Records/TIV

20. Data Collection/Inputs n Is this all the subject data? l All/coastal states l Inland Marine, Builders Risk, APD, Dwelling Fire l Manual policies n General level of detail l County/zip/street l Aggregated data n Is this all the needed policy detail? l Building location/billing location l Multi-location policies/bulk data l Statistical Record vs. policy systems l Coding of endorsements u Sublimits, wind exclusions, IM l Replacement cost vs. limit

21. More Data Issues n Deductible issues n Inuring/facultative reinsurance n Extrapolations & Defaults n Blanket policies n HPR n Excess policies

22. Model Output n Data Imported/Not Imported n Geocoded/Not Geocoded n Version n Perils Run l Demand Surge l Storm Surge l Fire Following n Defaults l Construction Mappings l Secondary Characteristics n Secondary Uncertainty n Deductibles

23. Synthesis/Pricing

24. SD Pricing Basics n Surplus Allocation v = z  L – r v = z  L – r u v is contract surplus allocation u r is contract risk load (expected profit) n Price P = E(L) +   L + expenses P = E(L) +   L + expenses n Risk Load or Profit  = [ y  z/(1+y)]  (C +  L /2S)  = [ y  z/(1+y)]  (C +  L /2S) u y is target return on surplus u z is unit normal measure u C is correlation of contract with portfolio u S is portfolio sd (generally of loss) With large enough portfolio this term goes to zero

25. SD Pricing with Variable Premiums   [Deposit*(1-Expense d %) + E(reinstatement)*(1-Expense r %)-EL]/  L   [Deposit*(1-Expense d %) + E(reinstatement)*(1-Expense r %)-EL]/  L n E(Reinstatement)= Deposit/Limit *E(1 st limit loss) * Time Factor n 2 or 3 figures define (info-blind) price l Aggregate expected loss l Expected loss with first limit (can be approximated) l Standard deviation of loss

26.  -Values (No Tax, C=1)

27. Tax & Inv. Income Adjustments n Surplus Allocation Perfect Correlation : v = z*  L – r Perfect Correlation : v = z*  L – r Imperfect Correlation: v = z*C*  L – r Imperfect Correlation: v = z*C*  L – r n After-tax ROE Start:  = [ y*z/(1+y)]*C Start:  = [ y*z/(1+y)]*C Solve for y: y =  /(z*C –  ) Solve for y: y =  /(z*C –  ) l Conclude:  y a = y*(1-T) =  *(1-T)/[z*C-r*(1-T)] +i f – T = tax rate – y a = after tax return – i f = after tax risk free return on allocated surplus

28.  -Values (adjusted for tax, inv. income)

29. Cat Pricing: Loss On Line & Risk Load

30. Select 2000 Cat Pricing Risk Load & Loss on Line

31. Loss On Line vs. Layer CV

32. Select 2000 Cat Pricing Risk Load & CV

33. SD Pricing Issues n Issues with C l Limiting case is C=1 l If marginal, order of entry problems for renewals Perhaps  book /  contract Perhaps  book /  contract u Need to define book of business u Anecdotally,C=0.50 for reasonably diversified US cat book u Adjust up for parameter risk, down for non-US cat business and non-cat business l Is it correlation or downside that matters? Issues with  Issues with  l Assumption of normality u On cat book, error is compressed u Further offsets when book includes non-cat u Or move to varying SD risk loads l Adjust to reflect zone and layer

34. SD Pricing Issues (Cont.) Issues with  L Issues with  L l Measure variability: Loss or result? l Variable premium terms u Reinstatements at 100% vs. 200% l Variable contract expiration terms u Contingent multi-year contracts with kickers  L : Downside proxy – can we get precise?

35. Investment Equivalent Pricing (IERP) n Allocated capital for ruin protection l Terminal funds > X with prob > Y (VaR) n Prefer selling reinsurance to traditional investment l Expected return and volatility on reinsurance contract should meet benchmark alternative

36. IERP Cash Flows Cedant Reinsurer Fund Premium = Risk Load + Discounted Expected Losses Fund = Premium + Allocated Surplus Return Actual Losses Net to Reinsurer Allocated SurplusFund Return - Actual Losses

37. IERP - Fully Funded Version Cedant Reinsurer Fund P = R + E[  ]/(1+f) F = P + A (1+r f )F  Expected return criterion: (1+r f )F - E[  ] = (1+y)A Variance criterion: Var[  ] <  y 2 A 2 Safety criterion: (1+r f )F > S

38. IERP, Q&D Example

39. Comparative Risk Loads SD –  L yz/(1+y) SD –  L yz/(1+y) n IERP – (y-r f )(S-L)/[(1+r f )(1+y)] l S is safety level of loss distribution l L is expected loss

40. SD vs IERP Pricing Price By Layer

41. SD vs IERP Pricing Loss Ratio By Layer

42. SD vs IERP Pricing Risk Load By Layer

43. Conclusions n Cat Model Distributions Vary l More than one point estimate useful l Point estimates may not be significantly different l Uncertainty not insignificant but not insurmountable l What about uncertainty before cat models? n Data Inputs Matter l Not mechanical process l Creating model inputs requires many decisions l User knowledge and expertise critical n Pricing Methodology Matters l But market price not always technical price n Judgment Unavoidable l Actuaries already well-versed in its use

44. References n Bove, Mark C. et al.., “Effect of El Nino on US Landfalling Hurricanes, Revisited,” Bulletin of the American Meteorological Society, June 1998. n Efron, Bradley and Robert Tibshirani, An Introduction to the Bootstrap, New York: Chapman & Hall, 1993. n Kreps, Rodney E., “Risk Loads from Marginal Surplus Requirements,” PCAS LXXVII, 1990. n Kreps, Rodney E., “Investment-equivalent Risk Pricing,” PCAS LXXXV, 1998. n Major, John A., “Uncertainty in Catastrophe Models,” Financing Risk and Reinsurance, International Risk Management Institute, Feb/Mar 1999. n Mango, Donald F., “Application of Game Theory: Property Catastrophe Risk Load,” PCAS LXXXV, 1998. n Miller, David, “Uncertainty in Hurricane Risk Modeling and Implications for Securitization,” CAS Forum, Spring 1999. n Moore, James F., “Tail Estimation and Catastrophe Security Pricing: Cat We Tell What Target We Hit If We Are Shooting in the Dark”, Wharton Financial Institutions Center, 99-14.

45. Q&A

46. APPENDIX A STANDARD DEVIATION PRICING Derivation Of Formulas

47. Risk Load As Variance Concept

48. The Basic Formulas P =  +  *  + E P =  +  *  + E P = Premium P = Premium  = Expected Losses  = Expected Losses  = Reluctance Measure  = Reluctance Measure  = Standard Deviation of Contract Loss Outcomes  = Standard Deviation of Contract Loss Outcomes E = Expenses E = Expenses  = y * z / (1 + y)  = y * z / (1 + y) y = Target Return on Surplus y = Target Return on Surplus z = Unit Normal Measure z = Unit Normal Measure

49. Initial Definitions V = z * S - R (1.1) given, per Brubaker, where V is that part of surplus required to support variability of a book of business with expected return R and standard deviation S given, per Brubaker, where V is that part of surplus required to support variability of a book of business with expected return R and standard deviation S R’ = R+ r (1.2) where R’ is expected return after addition of new contract with expected return r where R’ is expected return after addition of new contract with expected return r V’ = z * S’ - R’ (1.3) required surplus with new contract, as per (1.1) required surplus with new contract, as per (1.1)

50. Required Contract Marginal Surplus V’ - V = z *(S’ - S) - r (1.4) Proof, from (1.1) and (1.3): Proof, from (1.1) and (1.3): V’ - V = z*S’ - R’ - (z*S - R) V’ - V = z*S’ - R’ - (z*S - R) = z*(S’ - S) - (R’ - R) = z*(S’ - S) - (R’ - R) = z*(S’ - S) - r = z*(S’ - S) - r

51. Required Rate of Return r = y * (V’- V) (1.5) Given, but intuitively, required yield rate y times needed allocated surplus, V’ - V, given required return dollars Given, but intuitively, required yield rate y times needed allocated surplus, V’ - V, given required return dollars r = [y * z / (1 + y)] * (S’ - S) (1.6) Proof : Proof : r/y = (V’ - V) from (1.5) r/y = (V’ - V) from (1.5) r/y = z*(S’ - S) - r from (1.4) r/y = z*(S’ - S) - r from (1.4) r/y + r = z*(S’ - S) r/y + r = z*(S’ - S) r[(1+y)/y] = z*(S’ - S) r[(1+y)/y] = z*(S’ - S) r = [y*z/(1+y)]*(S’-S) r = [y*z/(1+y)]*(S’-S)

52. Marginal Standard Deviation

53. Reinsurer Reluctance ( 

54. Risk Load Simplification