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

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114 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESThen we perform integration by parts as follows to deriveZ 1=1d h()E[T();r¹]=¡(1 ¡ H())E[T();r¹]¯0The result follows since=+Z 10Z 101 ¡ H()= @E[£;] :@=0d(1 ¡ H()) @E[T();r¹]@d(1 ¡ H()) @E[(T();r¹] :@With this we can now show a smaller charge for the priorleads to a smaller charge for the unconditional distribution at acommon risk size.4.4. Smaller Charges for the Prior Lead to SmallerUnconditional Charges–Size FixedIfthen¹ =E[£ 1 ]=E[£ 2 ] and@ 2 E[T();r¹]@ 2 · 0' £2 · ' £1implies ' T(£2 ) · ' T(£ 1 ) : (4.4)ProofWe use Equation (4.5) to obtain' T(£1 )(r) ¡ ' T(£2 )(r)= ¡ 1 ¹Z 10Ã@E[£1 ;]d@!¡ @E[£2 ;] @E[T();r¹]:@ @
WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 1<strong>15</strong>We integrate by parts to obtain¹(' T(£1 ) (r) ¡ ' T(£ 2 ) (r))= ¡(E[£ 1 ;] ¡ E[£ 2 ;]) @E[T();r¹]@¯=1Z 1+ d(E[£ 1 ;] ¡ E[£ 2 ;]) @2 E[T();r¹]0@ 2 :Since E[£ 1 ]=E[£ 2 ]=¹ and the first partial of the limited expectedvalue is bounded by unity as per Equation (3.3b), it followsthat the first term vanishes and we have¹ ¢ (' T(£1 ) (r) ¡ ' T(£ 1 ) (r))Z 1= d(E[£ 1 ;] ¡ E[£ 2 ;]) @2 E[T();r¹]0@ 2 :We use the expectation formula for the insurance charge, A1, toarrive at the formula E[£;]=¹ 1 ¡ ' T(£) :¹We then use this to substitute into the previous integral to yield' T(£1 )(r) ¡ ' T(£2 )(r)Z 1 @ 2 E[T();r¹]= d ' £2¡ '0¹£1¹ @ 2 :The result then follows immediately from the assumptions of theproposition.To gain a better understanding of the formulas, consider thefollowing example.EXAMPLE 5: Poisson Conditionals and Exponential PriorsLet T() be Poisson. We leave it as an exercise for the reader to=0
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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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RISKINESS LEVERAGE MODELS 59APPENDI
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DISCUSSION OF A PAPER PRESENTED AT
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MINUTES OF THE 2005 SPRING MEETING
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Volume XCII, Part 2 No. 177PROCEEDI
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MODELING FINANCIAL SCENARIOS 179Bab
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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 747use of thes
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PRESIDENTIAL ADDRESS 749just lookin
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PRESIDENTIAL ADDRESS 751The quantif
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PRESIDENTIAL ADDRESS 753me keep my
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MINUTES OF THE 2005 CAS ANNUAL MEET
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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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