Uncertainty AndProperty Cat Pricing CARe Seminar, NYC February 28, 2002 Jonathan Hayes, ACAS, MAAA
Agenda • Models • Model Results • Confidence Bands • Data • Issues with Data • Issues with Inputs • Model Outputs • Pricing Methods • Standard Deviation • Downside Risk • Role of Judgment • Still Needed
The Search For Truth “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
Florida Hurricane Amounts in Millions USD
Florida Hurricane Amounts in Millions USD
Types Of Uncertainty(In Frequency & Severity) • Uncertainty (not randomness) • Sampling Error • 100 years for hurricane • Specification Error • FCHLPM sample dataset (1996) 1 in 100 OEP of 31m, 38m, 40m & 57m w/ 4 models • Non-sampling Error • El Nino Southern Oscillation • Knowledge Uncertainty • Time dependence, cascading, aseismic shift, poisson/negative binomial • Approximation Error • Res Re cat bond: 90% confidence interval, process risk only, of +/- 20%, per modeling firm Source: Major, Op. Cit..
Frequency-Severity UncertaintyFrequency Uncertainty (Miller) • Frequency Uncertainty • Historical set: 96 years, 207 hurricanes • Sample mean is 2.16 • What is range for true mean? • Bootstrap method • New 96-yr sample sets: Each sample set is 96 draws, with replacement, from original • Review Results
Frequency Bootstrapping • Run 500 resamplings and graph relative to theoretical t-distribution Source: Miller, Op. Cit.
Frequency Uncertainty Stats • Standard error (SE) of the mean: • 0.159 historical SE • 0.150 theoretical SE, assuming Poisson, i.e., (lambda/n)^0.5
Hurricane Freq. UncertaintyBack of the Envelope • Frequency Uncertainty Only • 96 Years, 207 Events, 3100 coast miles • 200 mile hurricane damage diameter • 0.139 is avg annl # storms to site • SE = 0.038, assuming Poisson frequency • 90% CI is loss +/- 45% • i.e., (1.645 * 0.038) / 0.139
Frequency-Severity UncertaintySeverity Uncertainty (Miller) • Parametric bootstrap • Cat model severity for some portfolio • Fit cat model severity to parametric model • Perform X draws of Y severities, where X is number of frequency resamplings and Y is number of historical hurricanes in set • Parameterize the new sampled severities • Compound with frequency uncertainty • Review confidence bands
OEP Confidence Bands Source: Miller, Op. Cit.
OEP Confidence Bands Source: Miller, Op. Cit.
OEP Confidence Bands • At 80-1,000 year return, range fixes to 50% to 250% of best estimate OEP • Confidence band grow exponentially at frequent OEP points because expected loss goes to zero • Notes • Assumed stationary climate • Severity parameterization may introduce error • Modelers’ “secondary uncertainty” may overlap here, thus reducing range • Modelers’ severity distributions based on more than just historical data set
Data Collection/Inputs • Is this all the subject data? • All/coastal states • Inland Marine, Builders Risk, APD, Dwelling Fire • Manual policies • General level of detail • County/zip/street • Aggregated data • Is this all the needed policy detail? • Building location/billing location • Multi-location policies/bulk data • Statistical Record vs. policy systems • Coding of endorsements • Sublimits, wind exclusions, IM • Replacement cost vs. limit
More Data Issues • Deductible issues • Inuring/facultative reinsurance • Extrapolations & Defaults • Blanket policies • HPR • Excess policies
Model Output • Data Imported/Not Imported • Geocoded/Not Geocoded • Version • Perils Run • Demand Surge • Storm Surge • Fire Following • Defaults • Construction Mappings • Secondary Characteristics • Secondary Uncertainty • Deductibles
SD Pricing Basics • Surplus Allocation • v = z ´ sL – r • v is contract surplus allocation • r is contract risk load (expected profit) • Price • P = E(L) + Â´ sL + expenses • Risk Load or Profit • Â = [y ´ z/(1+y)] ´ (C + sL/2S) • y is target return on surplus • z is unit normal measure • C is correlation of contract with portfolio • S is portfolio sd (generally of loss) With large enough portfolio this term goes to zero
SD Pricing with Variable Premiums • Â = [Deposit*(1-Expensed%) + E(reinstatement)*(1-Expenser%)-EL]/ sL • E(Reinstatement)= Deposit/Limit *E(1st limit loss) * Time Factor • 2 or 3 figures define (info-blind) price • Aggregate expected loss • Expected loss with first limit(can be approximated) • Standard deviation of loss
Tax & Inv. Income Adjustments • Surplus Allocation • Perfect Correlation: v = z* sL – r • Imperfect Correlation: v = z*C* sL – r • After-tax ROE • Start: Â = [y*z/(1+y)]*C • Solve for y: y = Â /(z*C –Â) • Conclude: • ya = y*(1-T) = Â *(1-T)/[z*C-r*(1-T)] +if • T = tax rate • ya = after tax return • if = after tax risk free return on allocated surplus
SD Pricing Issues • Issues with C • Limiting case is C=1 • If marginal, order of entry problems for renewals • Perhaps sbook/Sscontract • Need to define book of business • Anecdotally,C=0.50 for reasonably diversified US cat book • Adjust up for parameter risk, down for non-US cat business and non-cat business • Is it correlation or downside that matters? • Issues with Â • Assumption of normality • On cat book, error is compressed • Further offsets when book includes non-cat • Or move to varying SD risk loads • Adjust to reflect zone and layer
SD Pricing Issues (Cont.) • Issues with sL • Measure variability: Loss or result? • Variable premium terms • Reinstatements at 100% vs. 200% • Variable contract expiration terms • Contingent multi-year contracts with kickers sL: Downside proxy – can we get precise?
Investment Equivalent Pricing (IERP) • Allocated capital for ruin protection • Terminal funds > X with prob > Y (VaR) • Prefer selling reinsurance to traditional investment • Expected return and volatility on reinsurance contract should meet benchmark alternative
IERP Cash Flows Cedant Premium = Risk Load + Discounted Expected Losses Actual Losses Reinsurer Fund = Premium + Allocated Surplus Return Fund Net to Reinsurer Allocated Surplus Fund Return - Actual Losses
IERP - Fully Funded Version Cedant P = R + E[L]/(1+f) L Reinsurer F = P + A (1+rf)F Fund Expected return criterion: (1+rf)F - E[L] = (1+y)A Variance criterion: Var[L] <sy2A2 Safety criterion: (1+rf)F >S
Comparative Risk Loads • SD – sLyz/(1+y) • IERP – (y-rf)(S-L)/[(1+rf)(1+y)] • S is safety level of loss distribution • L is expected loss
Conclusions • Cat Model Distributions Vary • More than one point estimate useful • Point estimates may not be significantly different • Uncertainty not insignificant but not insurmountable • What about uncertainty before cat models? • Data Inputs Matter • Not mechanical process • Creating model inputs requires many decisions • User knowledge and expertise critical • Pricing Methodology Matters • But market price not always technical price • Judgment Unavoidable • Actuaries already well-versed in its use
References • Bove, Mark C. et al.., “Effect of El Nino on US Landfalling Hurricanes, Revisited,” Bulletin of the American Meteorological Society, June 1998. • Efron, Bradley and Robert Tibshirani, An Introduction to the Bootstrap, New York: Chapman & Hall, 1993. • Kreps, Rodney E., “Risk Loads from Marginal Surplus Requirements,” PCAS LXXVII, 1990. • Kreps, Rodney E., “Investment-equivalent Risk Pricing,” PCAS LXXXV, 1998. • Major, John A., “Uncertainty in Catastrophe Models,” Financing Risk and Reinsurance, International Risk Management Institute, Feb/Mar 1999. • Mango, Donald F., “Application of Game Theory: Property Catastrophe Risk Load,” PCAS LXXXV, 1998. • Miller, David, “Uncertainty in Hurricane Risk Modeling and Implications for Securitization,” CAS Forum, Spring 1999. • 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.
APPENDIX A STANDARD DEVIATION PRICING Derivation Of Formulas
The Basic Formulas • P = m + Â*s + E P = Premium m = Expected Losses Â = Reluctance Measure s = Standard Deviation of Contract Loss Outcomes E = Expenses • Â = y * z / (1 + y) y = Target Return on Surplus z = Unit Normal Measure
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 R’ = R+ r (1.2) 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 Contract Marginal Surplus V’ - V = z *(S’ - S) - r (1.4) Proof , from (1.1) and (1.3): V’ - V = z*S’ - R’ - (z*S - R) = z*(S’ - S) - (R’ - R) = z*(S’ - S) - r