Asymptotic Consistency and Inconsistency of the Chain Ladder

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Timeline of a claimAccident dateClaim closingClaim closingReporting dateReopeningClaim paymentsPayments✞✝❄☎✆❄❄❄❄❄❄ ❄❄❄✲Insurance periodTime
Settlement of a claim- Reporting delay (between occurrence andreporting) – can take several years (liabilityinsurance: asbestos or environmental pollutionclaims)- After being reported to the insurer – severalyears may elapse before the claim is finally settled(fast in property insurance, liability or bodily injuryclaims: long time before the total circumstancesare clear and known)- Reopening – (unexpected) new developments, or ifa relapse occurs
Page 1 and 2: Asymptotic Consistency andInconsist
Page 3: Non-life insurance- Operates on the
Page 7 and 8: Terminology- X i,j . . . claim amou
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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