Casualty Excess Pricing Using Power Curves Ana Mata, PhD, ACAS CARe Seminar London, 15 September...

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Casualty Excess Pricing Using Power Curves Ana Mata, PhD, ACAS CARe Seminar London, 15 September 2009 Matβlas Underwriting and Actuarial Consulting, Training and Research

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  • Casualty Excess Pricing Using Power CurvesAna Mata, PhD, ACAS

    CARe SeminarLondon, 15 September 2009

    Matlas Underwriting and Actuarial Consulting, Training and Research

    Matlas

    *AgendaExcess pricing approachesUS vs. non-USProperty vs. casualtyPower curves Rationale and assumptionsThe alpha parameterGeneral formulaePractical issues and considerations

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    *Useful referencesSwiss Re publications on Pareto RatingMiccolis, Robert S. On the Theory of Increased Limits and Excess of Loss Pricing. Proceedings of the Casualty Actuarial Society 1977: LXIV 27-59 Paul Riebesell. Einfhrung in die Sachversicherungsmathematik, Berlin 1936.Thomas Mack and Michael Fackler. Exposure Rating in Liability Reinsurance, ASTIN 2003.

  • Excess pricing:US vs. Non-US ClassesUSA ClassesISO curves for property and casualty classesSalzmann (1963) and Hartford (1991) curves for propertyInternally developed (fitted) loss modelsBrokers curves and modelsNCCI ELPPF for Workers Comp

    Non-USA ClassesSwiss Re first loss scales or exposure curves - propertyPower curves casualtyInternally developed loss modelsMinimum ROLUnderwriters judgement

    *

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    *Excess pricingInsurance and reinsurance same rationaleCommonly used approaches Loss distribution ideal Exposure curves First loss scales Increase limits factors Excess factors Rates per million of coverageConsiderExpense handling and risk loads Currency and inflation adjustments

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    *Exposure curves Assumptions:Sum insured proxy for maximum possible lossNo incentive to under-insuredDirect relation between size of loss and amount at riskLoss cost distribution by % of IV independent of sum insuredCurves by peril or hazardFireWind Hurricanes

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    *Exposure curves as a loss elimination ratioLimited expected value and loss elimination ratio

    75% of losses are at 5% or less of Insured Value or Sum Insured

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    *Other considerationsConsider all policy sections:Building valueContentsOther structures (e.g. attached garage)Loss of useThus, loss can be > 100% of sum insuredConsider type of constructionCurves are inflation independentLoss increases in same proportion of sum insured

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    *Exposure curves vs. ILFsCommon (simplifying) assumption in propertyRatio is constant for risks within group

    Assumptions does not hold in liabilityTotal limit purchased chosen by p/hLimit maximum loss exposureDependence between limit and severity?Dependence between limit purchased and exposure base (turnover, payroll, fee income, etc)?

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    *Challenges when deriving ILFsClaim severity vs. limit purchasedClaim severity vs. exposure baseAssumption of independence between frequency and severityAggregate vs. any one claim limitsIBNER and claims reserving practicesLack of policy information at claim levelChanges of mix of business year on yearMultiple claim records for same underlying claim

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    *The power curves Commonly used in London MarketNon-US liability classesEL, GL, D&O, PI, FI, etcInsurance and reinsurance excess pricingOther names:Alpha curvesRiebesells curvesThe German methodIntroduced by Riebesell in 1936

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    *The rationale behind power curvesBasic limit = B, pure premium P(B)Pure premium for limit 2B = P(B)*(1+r), e.g. r=20%Pure premium for limit 4B = P(B)*(1+r)(1+r)Riebesells rule:

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    *The rationale behind power curvesFor any limit L, we can generalise

    where ld() is the binary log and 0 < z < 1

    ld(1+z) is usually called the alpha parameter

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    *The rationale behind power curvesThe loss elimination ratio is:

    The ILF formula

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    *Nice propertiesLER only depends on ratio of x to SIProperties of exposure curves applyScale invariantInflationCurrencyEasy closed form formulaPareto tail

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    *What is the distribution underlying the power curves?Mack and Fackler 2003: For any parameters of Riebesells rule we can construct a claims size distribution underlying such rule.The claim size distribution is a one parameter Pareto over a threshold u and parameter < 1.Below the threshold u? the distribution is constant (independent of x)The threshold u could be a problem if deductibles or reinsurance attachment are low

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    *Practical usesUnderwriters are primary users of these curvesDifficult to find an alpha that works ground upDifferent alphas depending on original policys excess or deductibleCommon to ignore deductibles and attachments from exposure rating formula

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    *Practical issuesExposure rating with co-insurance stacking of limitsDiscontinuitiesCurrency independent OKInflation independent realistic?Experience vs. exposure rating results may show significant differencesALAE or cost assumptionsCurves rarely adjusted for this

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    *Power curves with stacking of limits

    $25MM capacity spread over various layers

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    *Power curves with stacking of limitsExposure rate $5MM P/O $10MM xs $10MM in the 1st RI excess: $5MM xs $5MMAssume r = 30%, alpha = 0.3785$2MM are retained, $3MM are cededGround up basis:RI loss cost is 50% of $6MM xs $14MM P/O $10MM xs 10MM

    Ignoring original excessRI loss cost is 50% of $6MM xs $4MM P/O $10MM

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    *ALAE treatment and exposure ratingApply following adjustments:

  • Power curves:final remarksAdvantagesUnderwriters are comfortable Aligned with UW mindsetCommon alphas across the marketScale invariant (ok for currency)Closed form formula (no interpolation)Pareto tail (?)DisadvantagesDifficult to replaceValidity of assumptionsOdd results:High limits and excess on excessShare of limits and stacking of policiesLow original deductiblesInflation independentALAE/costs assumptions*