PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

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58 RISKINESS LEVERAGE MODELSREFERENCES[1] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath, “CoherentMeasures of Risk,” Math. Financie 9, 3, 1999, pp. 203—228,www.math.ethz.ch/ » delbaen/ftp/prepreints/CoherentMF.pdf.[2] Gogol, Daniel, “Pricing to Optimize and Insurer’s Risk-Return Relation,” PCAS LXXXIII, 1996, pp. 41—74.[3] Kreps, Rodney, “Mini dfa.xls,” Spreadsheet presented atthe 2005 Casualty Actuaries in Reinsurance (CARe) LimitedAttendance Seminar–Risk and Return in Reinsurance,http://www.casact.org/coneduc/care/0905/handouts/dfa.xls;INTERNET.[4] Mango, Donald, “An application of Game Theory,” CasualtyActuarial Society Forum, Spring 1997, pp. 31—35.[5] Meyers, Glenn, “Coherent Measures of Risk: An Expositionfor the Lay Actuary,” Available through the CASat http: // www.casact.org / pubs / forum // 00sforum / meyers /Coherent%20Measures%20 of%20Risk.pdf.[6] Myers, Stewart, and James Read, “Capital Allocation for InsuranceCompanies,” Journal of Risk and Insurance 2001 68,4, pp. 545—580.
RISKINESS LEVERAGE MODELS 59APPENDIX ASOME MATHEMATICAL ASIDES(x) is the step function: zero for negative argument and 1 forpositive. It is also referred to as the index function.±(x) is the Dirac delta function. It can be heuristically thoughtof as the density function of a normal distribution with mean zeroand standard deviation arbitrarily small compared to anythingelse in the problem. This makes it essentially zero everywhereexcept at zero but it still integrates to 1.The index function can also be thought of as the cumulativedistribution function of the same normal distribution, and it isin this sense that the delta function can be thought of as thederivative of the index function. All the usual calculus rules aboutderivatives apply without modification.Always, we are implicitly taking the limit as the standard deviationof this distribution goes to zero. This whole usage canbe justified in the theory of linear functionals, but the author hasno idea where.These notions lead to some fundamental properties of the deltafunction. For any continuous function f(x)Zf(a)= f(x)±(x ¡ a)dx,(A.1)and for c>bZ cf(x)±(x ¡ a)dx = (c ¡ a)(a ¡ b)f(a):bIf h(a)=0thenZf(x)±(h(x))dx = f(a)jh 0 (a)j :(A.2)(A.3)
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VOLUME XCII NUMBERS 176 and 177PROC
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FOREWORDActuarial science originate
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WHEN CAN ACCIDENT YEARS BE REGARDED
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REPORT OF THE VICE PRESIDENT-ADMINI
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2005 EXAMINATIONS—SUCCESSFUL CAND
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OBITUARIESROBERT GRANT ESPIECLYDE H
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OBITUARIES 825SIDNEY M. HAMMER1931
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OBITUARIES 827J. GARY LaROSE1947—
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OBITUARIES 829PAUL J. SCHEEL SR.193
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OBITUARIES 831LEO M. STANKUSCirca 1
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INDEX TO VOLUME XCIIPage2005 EXAMIN
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835GRAVES, CLYDE H.INDEX—CONTINUE
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837INDEX—CONTINUEDREPORT OF THE V
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