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

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102 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESFirst, we observe:3.2. Decomposability Equivalence to Closure in Unique SizeModelM is decomposable , M is closed: (3.2)We prove one direction and leave the other as an ex-Proofercise.“ )” Omitted.“(” SinceM is complete, there exist T ¹ 2 M, T ¹12 M, andT ¹22 M. By assumption, M is closed under independent summation.So the independent sum, T ¹1+ T ¹2, is in M. Takingexpectations one has E[T ¹1+ T ¹2]=¹ 1 + ¹ 2 . In a unique sizemodel, we know T ¹1 +¹ 2has the unique distribution in M withE[T ¹1 +¹ 2]=¹ 1 + ¹ 2 .If we assume size differentiability in a decomposable model,we can obtain some results constraining the behavior of the cumulativedistribution and the limited expected value functionwhen these are viewed as functions of risk size.3.3. Inequalities for Risk Size Partials in Decomposable ModelsIf M is a continuously differentiable decomposable risk-sizemodel, then1 ¸ @E[T ¹ ;t]@¹@F T¹@¹ · 0¸ 0(3.3a)(3.3b)@ 2 E[T ¹ ;t]@¹ 2 · 0: (3.3c)
WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 103ProofApplying the definition of decomposability, we deriveF T¹(t)=Pr(T ¹ · t) ¸ Pr(T ¹ + T ¢¹ · t)=F T¹ +T ¢¹(t)=F T¹+¢¹(t):F is thus a decreasing function of the risk size and Equation(3.3a) follows.To prove the partial exceeds zero in Equation (3.3b), we usethe decomposability property to deriveE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]=E[T ¹ + T ¢¹ ;t] ¡ E[T ¹ ;t] ¸ 0:To prove the partial is less than unity, we similarly deriveE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]=E[T ¹ + T ¢¹ ;t] ¡ E[T ¹ ;t]· E[T ¹ ;t]+E[T ¢¹ ;t] ¡ E[T ¹ ;t]=E[T ¢¹ ;t] · ¢¹:It follows thatE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]· 1¢¹and this leads immediately to our result.As for Equation (3.3c), we claim that with our continuousdifferentiability assumption, it suffices to show that E[T ¹+¢¹ ;t] ¡E[T ¹ ;t] is a decreasing function of ¹ for any ¢¹ > 0. This issufficient because, if it is true, we can then use an argumentbased on the Mean Value Theorem to show that the first partialderivative with respect to risk size is decreasing. A decreasingfirst partial derivative forces the second partial to be less than orequal to zero.To show E[T ¹+¢¹ ;t] ¡ E[T ¹ ;t] is decreasing, we first use theadditivity and independence assumptions to write the convolutionformula:F T¹+¢¹(t)=F T¹ +T ¢¹(t)=Z t0dF T¹(s) ¢ F T¢¹(t ¡ s):
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VOLUME XCII NUMBERS 176 and 177PROC
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FOREWORDActuarial science originate
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2005 PROCEEDINGSCONTENTS OF VOLUME
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2005 PROCEEDINGSCONTENTS OF VOLUME
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Editorial Committee, Proceedings Ed
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2 AN EXAMINATION OF THE INFLUENCE O
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10 AN EXAMINATION OF THE INFLUENCE
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AN EXAMINATION OF THE INFLUENCE OF
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RISKINESS LEVERAGE MODELSRODNEY KRE
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RISKINESS LEVERAGE MODELS 33There a
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ADDRESS TO NEW MEMBERS 153You shoul
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ADDRESS TO NEW MEMBERS 155strength
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MINUTES OF THE 2005 CAS SPRING MEET
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MINUTES OF THE 2005 SPRING MEETING
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Volume XCII, Part 2 No. 177PROCEEDI
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WHEN CAN ACCIDENT YEARS BE REGARDED
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THE APPLICATION OF FUNDAMENTAL VALU
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A NEW METHOD OF ESTIMATING LOSS RES
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ARCHITECTURE FOR RESIDENTIAL PROPER
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ESTIMATING THE WORKERS COMPENSATION
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INCORPORATION OF FIXED EXPENSESGEOF
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INCORPORATION OF FIXED EXPENSES 681
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DISCUSSION OF PAPER PUBLISHED INVOL
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APPLICATION OF THE OPTION MARKET PA
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THE “MODIFIED BORNHUETTER-FERGUSO
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ADDRESS TO NEW MEMBERS-NOVEMBER 14,
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ADDRESS TO NEW MEMBERS 743Now to an
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PRESIDENTIAL ADDRESS-NOVEMBER 14, 2
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PRESIDENTIAL ADDRESS 749just lookin
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MINUTES OF THE 2005 CAS ANNUAL MEET
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MINUTES OF THE 2005 ANNUAL MEETING
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REPORT OF THE VICE PRESIDENT—ADMI
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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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PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

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