Asymptotic Consistency and Inconsistency of the Chain Ladder

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Reserving- Claims reserves represent the money which shouldbe held by the insurer so as to be able to meet allfuture claims arising from policies currently inforce and policies written in the past- Most non-life insurance contracts are writtenfor a period of one year- Only one payment of premium at the start of thecontract in exchange for coverage over the year- Reserves are calculated byforecasting future losses from past losses
Terminology- X i,j . . . claim amounts in development year j withaccident year i- X i,j stands for the incremental claims in accidentyear i made in accounting year i + j- n . . . current year – corresponds to the mostrecent accident year and development period- Our data history consists ofright-angled isosceles triangles X i,j , wherei = 1, . . . , n and j = 1, . . . , n + 1 − i
Page 1 and 2: Asymptotic Consistency andInconsist
Page 3 and 4: Non-life insurance- Operates on the
Page 5: Settlement of a claim- Reporting de
Page 9 and 10: Run-off triangleAccidentDevelopment
Page 11 and 12: Chain ladderMack (1993)[1] E[C i,j+
Page 13 and 14: Properties- Ultimate claims amounts
Page 15 and 16: (Un)biasedness- ̂R( )i = Ĉi,n(n)(
Page 17 and 18: Open problem- From some point of vi
Page 19 and 20: Conditionalconvergenceξ nξ n[P ζ
Page 21 and 22: ConsistencyDenote D (n)j= {C i,k :
Page 23 and 24: Remark II- Unconditional consistenc
Page 25 and 26: On the rate ofconvergence- Complete
Page 27 and 28: Practical aspectŝf(n)- Eitherjis
Page 29 and 30: Inconsistency- What kinds of busine
Page 31 and 32: Decreasing business- consider a fas
Page 33 and 34: Slowly decreasingbusiness- situatio
Page 35 and 36: Growing business- EC i,1 increases
Page 37 and 38: One Year Prospective- in the classi
Page 39 and 40: Merz-Wüthrich(1i)- implied Chain L
Page 41 and 42: Conclusions- conditional consistenc
Page 43: Thank you !pesta@karlin.mff.cuni.cz

Asymptotic Consistency and Inconsistency of the Chain Ladder

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