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

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110 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGESProofOmitted.The Bayesian construction implies there is some parameteruncertainty about the true mean of any risk. Under one interpretation,the final charge value is the (prior) probability-weightedaverage of the dollar charge over all values of the true mean,divided by the expected value of the true mean. Under anotherinterpretation, we are dealing with a population of risks whosetrue overall mean we know, even though there is some parameteruncertainty regarding the mean of any particular risk in thepopulation. The prior then represents the spread in the populationand the formula arrives at the correct average charge forthe population. It is also important to note that, as in our example,the charge for an average risk is not the same as (and isusually lower than) the weighted average charge for the populationof risks. These two interpretations correspond to two typesof parameter risk. The first expresses our uncertainty about theoverall mean of a population, while the second expresses ouruncertainty about the parameter dispersion or heterogeneity of apopulation. While a hierarchical, “double-integral” model couldbe used to separately delineate their effects, Example 3 showsthat with respect to insurance charges, Equation (4.1) can beused to model both types of parameter risk together. For otherapplications such as in credibility theory, it may be importantto maintain a distinction between these sources of parameterrisk.Now let Q be a family of £ random variables and M(Q)the associated set of unconditional random variables. One quick,but important, result can be obtained assuming all the priors arescaled versions of a single distribution. Thus all the priors havethe same insurance-charge function. Assume we have a conditionalmodel in which charges decline with size. Then we showthat applying the scaled priors to this model generates an unconditionalmodel in which the insurance charge declines as afunction of the unconditional risk size.
WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES 1114.2. Unconditional Charge Declines with Risk Size in ScaledPriors ModelIfthen@' T()@· 0 and £ 2 =(1+c)£ 1 for c>0,' T(£2 ) (r) · ' T(£ 1 )(r): (4.2)ProofWe start by using Equation (4.1) to writewhereThen consider' T(£2 )(r)= 1 ¹ 2Z 10¹ 2 =E[£ 2 ]=¹ 2 =(1+c)¹ 1 r¹2d h 2 () ¢ ¢ ' T()Z 10d h 2 () ¢ :H 2 ()=Pr(£ 2 · ) = Pr((1 + c)£ 1 · )=Pr £ 1 · = H1+c 11+c 1h 2 ()=h 11+c 1+c :Substituting, rewrite the integral asZ' T(£2 ) (r)= 1 1 1d h¹ 1 (1 + c)10 1+c 1+c r¹1 (1 + c)¢ ¢ ' T() :
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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 49The las
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RISKINESS LEVERAGE MODELS 55We also
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RISKINESS LEVERAGE MODELS 57Mean Va
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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 753me keep my
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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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