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

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104 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESThis impliesUsingwe deriveG T¹+¢¹(t) ¡ G T¹(t)=E[T;t]=E[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]=Z t0Z t0Z t0dxdF T¹(s) ¢ G T¢¹(t ¡ s):dsG T (s)Z xSwitching orders of integration, we haveE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]==Z t0Z t00dF T¹(y)dF T¹(y) ¢ G T¢¹(x ¡ y):Z tydxG T¢¹(x ¡ y)dF T¹(y)E[T ¢¹ ;t ¡ y]:Next, we integrate by parts and evaluate terms to obtainE[T ¹+¢¹ ;t] ¡ E[T ¹ ;t]=F T¹(0) ¢ E[T ¢¹ ;t]Z t+ dy F T¹(y) ¢ G T¢¹(t ¡ y):0Applying Equation (3.3a), we can conclude that this decreaseswith ¹, thus proving our result.Note that these results apply to discrete as well as continuousdistributions and that the proof does not require F T¹(0) toequal zero. Exhibit 3 shows how Poisson, Negative Binomial,and Gamma limited expected values vary as the mean changes.See Figure 3 for a graph of Poisson limited expected values atthe limit 3.0 that vary as a function of risk size.Now inequality 2.4 will be used to show that decomposablemodels must have charges that decrease with risk size.
WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 105FIGURE 3Poisson Limited Expected Values E[T ¹ ;3]3.4. Charges Decrease with Risk Size in Decomposable ModelsSuppose M is a decomposable model. Then:¹ 1
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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 35where f
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RISKINESS LEVERAGE MODELS 37where a
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RISKINESS LEVERAGE MODELS 39Various
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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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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 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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