Asymptotic Consistency and Inconsistency of the Chain Ladder

Asymptotic Consistency andInconsistency of the ChainLadder(O banálnom aktuárskom probléme, ktorý ignorujeme, a jeho nebanálnom, ale triviálnom riešení)Michal Pešta(joint work with Šárka Hudecová)Charles University in PragueFaculty of Mathematics and PhysicsActuarial Seminar, Prague5 October 2012

Overview- Claims reserving in non-life insurance- Chain ladder model- Open problem: consistency of the developmentfactors- Simulations and real data example• Based on:Pešta and Hudecová (2012). Asymptotic consistencyand inconsistency of the chain ladder. Insurance:Mathematics and Economics, 51(2): 472 – 479.• Support:Czech Science Foundation project “DYME DynamicModels in Economics” No. P402/12/G097

Non-life insurance- Operates on the lines of business (LoB):◮ motor/car insurance (motor third party liability,motor hull)◮ property insurance (private and commercialinsurance against fire, water, flooding, businessinterruption, . . . )◮ liability insurance◮ accident insurance◮ health insurance◮ marine insurance (including transportation)◮ other (aviation, travel insurance, legal protection,credit insurance, epidemic insurance, . . . )- Life insurance products are rather different, e.g.,terms of contracts, type of claims, risk drivers

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

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

Notation- C i,j . . . cumulative payments in origin year i after jdevelopment periodsC i,j =j∑k=1X i,k- C i,j . . . a random variable of which we have anobservation if i + j ≤ n + 1- Aim is to estimate the ultimate claims amount C i,nand the outstanding claims reserveR i = C i,n − C i,n+1−i ,i = 2, . . . , n

Run-off triangleAccidentDevelopment year jyear i 1 2 · · · n − 1 n1 C 1,1 C 1,2 · · · C 1,n−1 C 1,n2 C 2,1 C 2,2 · · · C 2,n−1. ..... C i,n+1−in − 1 C n−1,1 C n−1,2n C n,1

Reasonable Estimatefor Reserves- nonsense estimate, e.g., ̂Ri = 10 6 or ̂R i = −1◮ why bad? the most precise estimate (in terms ofvariability) Var ̂R i = 0- unbiased estimate, i.e., E ̂R i = ER i , or conditionallyunbiased◮ firstly, introduce a model with assumptions, thenconstruct an estimate◮ have you ever heard of a reserving model withunbiased estimates of reserves?- consistent estimate (but where’s n ?)̂R i(stochastically)−−−−−−−−−→ R in→∞

Chain ladderMack (1993)[1] E[C i,j+1 |C i,1 , . . . , C i,j ] = f j C i,j[2] Var[C i,j+1 |C i,1 , . . . , C i,j ] = σ 2 j C i,j[3] Accident years [C i,1 , . . . , C i,n ] are independentvectors

Development factors f j∑ n−ĵf (n)j=i=1 C i,j+1∑ n−ji=1 C , 1 ≤ j ≤ n − 1i,ĵf (n)n ≡ 1 (assuming no tail)

Properties- Ultimate claims amounts C i,n are estimated byĈ i,n = C i,n+1−i ×(n)(n)̂fn+1−i× · · · × ̂f n−1- Under the assumptions [1], [3], and∑ n−j[4]i=1 C i,j > 0̂f (n)jare unbiased and mutually uncorrelated- Assumption [2] is essential for the standard errorof Ĉi,n

Unbiasedness- unbiasedness of development factors unbiasedness of reserves’ estimate !- given data D j = {C i,k : k ≤ j, i ∈ N} or {C i,1 , . . . , C i,j },where j = n + 1 − iE [R i |D n+1−i ] = E [C i,n − C i,n+1−i |D n+1−i ]= E [C i,n |D n+1−i ] − C i,n+1−i= E [E {C i,n |C i,1 , . . . , C i,n−1 } |D n+1−i ] − C i,n+1−i= E [f n−1 C i,n−1 |D n+1−i ] − C i,n+1−i = . . .= f n−1 . . . f n+2−i E [f n+1−i C i,n+1−i |D n+1−i ] − C i,n+1−i= C i,n+1−i (f n+1−i × . . . × f n−1 − 1) , a.s.

(Un)biasedness- ̂R( )i = Ĉi,n(n)(n)− C i,n+1−i = C i,n+1−i ̂fn+1−i× · · · × ̂f n−1 − 1[ ] ( [ ] )(n)(n)E ̂Ri |D n+1−i = C i,n+1−i E ̂fn+1−i× · · · × ̂f n−1 |D n+1−i − 1= . . . = E [R i |D n+1−i ] , a.s.- how much is unbiasedness important? and can wealways achieve it? do we need to?

(In)consistency- disadvantage:◮ asymptotic property (data history sufficiently long)◮ only a qualitative property- advantages:◮ easier to verify (?)◮ characterize the accuracy (and, hence,meaningfulness) of an estimate◮ can be quantified (e.g., rate of consistency)◮ is retained by algebraic operations

Open problem- From some point of view, consistency is moreimportant than unbiasedness(n)(n)- E.g., ̂fn+1−i× · · · × ̂fmethod usesn−1- The unbiasedness of̂β (n)jin any senseor Bornhuetter-Ferguson̂β (n)n−1∏j=̂f(n)jk=j- Ex: Y 1 , . . . , Y n iid with finite EY1̂f (n)kdoes not “transfer” to- T 1 (Y 1 , . . . , Y n ) = Y 1 vs T 2 (Y 1 , . . . , Y n ) = 1 n∑ ni=1 Y i + 1 n

StochasticConvergence- deterministic: one convergence (of real numbers)- almost sureP- in probability- L p , p ≥ 1-X n[ω ∈ Ω :∀ε > 0 :(stochastically)−−−−−−−−−→ Xn→∞]lim X n(ω) = X(ω) = 1n→∞lim P [|X n − X| ≥ ε] = 0n→∞lim E|X n − X| p = 0n→∞a.s.−−−→ ⇒ P−−−→ ⇐ −−−→Lp;n→∞ n→∞ n→∞P−−−→⇒ −−−→Dn→∞ n→∞

Conditionalconvergenceξ nξ n[P ζ ]-a.s.−−−−−→ χ, [P]-a.s. meansn→∞{} ]P[P ζ lim ξ n = χ = 1 = 1n→∞P ζn−−−→ χ, [P]-a.s. meansn→∞[]∀ε > 0 : P lim P ζn→∞ n{|ξ n − χ| ≥ ε} = 0 = 1L p(P ζn )ξ n −−−−−→ χ, [P]-a.s. (p ≥ 1) meansn→∞[]P lim E ζn→∞ n|ξ n − χ| p = 0 = 1

Conditioning- Conditional convergence in probability and in L palong some sequence of random variables {ζ n } ∞ n=1can be defined, because the concept of these twotypes of convergence comes from a topology- Despite of that, the almost sure convergence doesnot correspond to a convergence with respect toany topology and, hence, it is not metrizable- Thereafter, the conditional convergencealmost surely cannot be defined along a sequenceof random variables, but only given one randomvariable ζ

ConsistencyDenote D (n)j= {C i,k : k ≤ j, i ≤ n − j + 1} andD j = {C i,k : k ≤ j, i ∈ N}. Then (i)–(iv) are equivalent:(i)̂f (n) [P Dj ]-a.s.j−−−−−−→ f j, [P]-a.s.;n→∞(ii)(iii)(iv)̂f (n)jn−j∑i=1̂f (n)jP D(n)j−−−→n→∞ f j, [P]-a.s.;L 2(P (n) Dj)−−−−−−−→ f j, [P]-a.s.;n→∞C i,j −−−→n→∞∞, [P]-a.s.

Remark IDue to the independence of the different accidentyears (assumption [3]), the statements (ii) and (iii) canbe equivalently replaced bŷf (n)jP Dj−−−→ f j, [P]-a.s.n→∞andrespectively.̂f (n)jL 2 (P Dj )−−−−−→ f j, [P]-a.s.,n→∞

Remark II- Unconditional consistency in case of the L 2convergence-̂f(n)j→ f j in L 2 (unconditionally) as n → ∞ iff[ ]1E ∑ n−ji=1 C → 0,i,jn → ∞- This condition is obviously more complicated thanthe condition (iv), and it is practically unverifiable- Thus, the conditional convergence is not onlymore natural one in this case, but even moreconvenient one

Rate of convergence- Consistency of an estimator is a very importantbut only qualitative property- Measure consistency – quantitative way- Denote the conditional mean square error of theestimate of development factor f j asMSE( ) { [(n)(n)̂fj:= E ̂fj− E(̂f(n)j)] 2 ∣ ∣∣D (n)j}.Then, with probability one holds⎛[ ( )n−j] −1⎞(n)MSE ̂fj= O ⎝ C i,j⎠ , n → ∞.∑i=1

On the rate ofconvergence- Complete characterization of the conditionalconvergence of development factors’ estimate- The slower (faster) divergence of∞∑i=1C i,jimplies the slower (faster) realization ofconsistency of the development factors’ estimates

On the necessary andsufficient conditionLet j ∈ N be fixed. Then the following conditions areequivalent.2.1. The condition (iv) holds.∞∑EC i,1 = ∞. (1)i=13. The condition (iv) holds for j 0 ∈ N, j ≠ j 0 .

Practical aspectŝf(n)- Eitherjis consistent for f j for all j ∈ N, ornone of them is consistent̂f(n)j- The consistency of is equivalent to thecondition ∑ ni=1 C i,1 → ∞, [P]-a.s. as n → ∞- Denote S k = ∑ ki=1 C i,1, k = 1, . . . , n the cumulativesums of the cumulative claims C i,1 in thefirst development year- For instance, the ratios C k+1,1 /C k,1 or thesequence k√ C k,1 can be studied- Artificial data set Taylor and Ashe (1983)

Real data exampleS k500000 2000000 3500000●●●●●●●●●●2 4 6 8 10k

Inconsistency- What kinds of business behavior corresponds tothe violation of consistency?- For instance, condition (iv) can be violated if oneobserves a decreasing trend (decreasing fastenough) in payments across the accident years- So to speak, the corresponding line of business isworsening maybe due to new insurance companiesentering the market or changing (decreasing)prices of such insurance product- Furthermore, splitting one existing line ofbusiness into several others can also causeinconsistency in the estimation of developmentfactors

Simulations- C i,1 was generated such that C i,1 ∈ L 2 and C i,j ≥ 0- f j was set to f j = j/(j + 1)- C i,2 , . . . C i,N were generated successively such thatC i,j satisfies [1] and [2]- Hence, for each j, C i,j is drawn from a distributionwith mean f j−1 C i,j−1 and variance σj 2C i,j−1 for someσj 2 ∈ (0, ∞)- Since the accident years are assumed to beindependent (assumption [3]), the rows of the datasets were generated separately, using the sameapproach- C i,1 were drawn from the exponential distribution,and the C i,j was generated from thePoisson distribution with the parameter f j−1 C i,j−1for j = 2, . . . , N

Decreasing business- consider a fast decreasing business, i.e., thesituation where condition (1) does not hold- C i,1 was generated from the exponentialdistribution with the mean i −2 × 10 6-̂f(n)jdo not converge to the true value f j- Their values are close to f j in this setting, but theestimates are indeed not consistent- The same simulations were run also for C i,1 withthe uniform distribution: the differences between(n)values of estimates ̂fjand the true values f j aremore noticeable

Example Ij = 1j = 2f j0.498 0.499 0.500 0.501 0.502●^(n) ^(n)● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●f j10 20 30 40 5010 20 30 40 50nn0.665 0.666 0.667 0.668j = 3j = 40.748 0.749 0.750 0.751 0.752^(n)f jf j● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●10 20 30 40 50n0.798 0.799 0.800 0.801 0.802^(n)10 20 30 40 50n

Slowly decreasingbusiness- situation where EC i,1 decreases with increasing i,but in a slow manner such that condition (1) holds- EC i,1 = i −1/2 × 10 6 in the exponential distribution- clear convergence pattern can be observed,(n)confirming that the estimates ̂fjare consistentin this case

Example IIj = 1j = 20.498 0.499 0.500 0.501 0.502^(n) ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●f j● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●10 20 30 40 5010 20 30 40 50nn0.665 0.666 0.667 0.668^(n)f jj = 3j = 40.748 0.749 0.750 0.751 0.752●● ● ● ●●● ● ● ● ● ● ●^(n) ●● ●● ● ●●●f ^(n)● ●j● ● ●f j●● ● ● ● ● ● ●● ● ● ● ●●● ●10 20 30 40 50n0.798 0.799 0.800 0.801 0.80210 20 30 40 50n

Growing business- EC i,1 increases with increasing i- The parameters of the exponential distributionwere set such that EC i,1 = √ i × 10 6- The figure obviously confirms that the estimatesare consistent(n)- Moreover, ̂fjconverges to the true values f jmuch faster than in previous case

Example IIIj = 1j = 20.498 0.499 0.500 0.501 0.502^(n) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●f j● ●f ● ● ● ●●j^(n)● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●10 20 30 40 5010 20 30 40 50nn0.665 0.666 0.667 0.668j = 3j = 40.748 0.749 0.750 0.751 0.752^(n) ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●f ● ● ● ●j● ● ● ● ● ● ● ●●f j●● ●10 20 30 40 50n0.798 0.799 0.800 0.801 0.802^(n)● ● ● ● ●10 20 30 40 50n

One Year Prospective- in the classical claims reserving, one usually studiesthe total uncertainty in the claims developmentuntil the total ultimate claim is finally settled- Solvency II purposes- run-off = opening balance – expenses – closingbalance- we predict the total ultimate claim at time n (withthe available information up to time n), and oneperiod later at time n + 1 we predict the same totalultimate claim with the updated informationavailable at time n + 1- difference between these two successivepredictions is the claims development result (CDR)for accounting year (n, n + 1]

CDR- a direct impact on the P&L statement and on thefinancial strength of the insurance company- CDR for accident year i and accounting year(n, n + 1]CDR i (n + 1) = E[C i,n |D (n) ] − E[C i,n |D n+1 ]where= E[R (n)i|D (n) ] − (X i,n+2−i + E[R n+1i|D n+1 ])D (n) = {C i,j : i + j ≤ n + 1}D n+1 = {C i,j : i + j ≤ n + 2 & i ≤ n + 1}R (n)iRin+1= D (n) ∪ {C i,n+2−i : i ≤ n + 1}= C i,n − C i,n+1−i= C i,n − C i,n+2−i

Merz-Wüthrich(1i)- implied Chain Ladder assumptions (time series):C i,j = f j−1 C i,j−1 + σ j−1√Ci,j−1 ε i,j ,where ε i,j are iid with Eε i,j = 0 and Varε i,j = 1(2i) Accident years [C i,1 , . . . , C i,n ] are independentvectors

Reasonable results?- using the martingale propertyE[CDR i (n + 1)|D (n) ] = 0- prediction uncertainty in the budget value 0 forthe observable claims development result at theend of the accounting periodMSEĈDRi(n+1)|D (n) (0) = E[(ĈDR i (n + 1) − 0) 2 |D (n) ]- in the solvency margin, we need to hold risk capitalfor possible negative deviations of CDR i (n + 1)from 0? are the “results” by MW reasonable (e.g.,consistent estimate of MSEĈDRi(n+1)|D (n) (0))?

Conclusions- conditional consistency and inconsistency of thedevelopment factors’ estimate in thedistribution-free chain ladder is investigated- necessary and sufficient condition is derived- weak, strong consistency, and consistency in themean square are equivalent- convergence rate is provided- practical recommendations, how to check thisnecessary and sufficient condition, are discussed- real data example and numerical simulations toillustrate the performance of the estimates- possible violation of the condition with theconsequences is demonstrated

ReferencesMack, T. (1993)Distribution-free calculation of the standard errorof chain ladder reserve estimates.Astin Bulletin, 23(2), 213–225.Merz, M. and Wuthrich, M. V. (2008)Modeling the claims development results forSolvency purposes.CAS E-Forum, Fall 2008, 542–568.Pesta, M. and Hudecova, S. (2012)Asymptotic consistency and inconsistency of thechain ladder.Insurance: Mathematics and Economics, 51(2),472–479.

Thank you !pesta@karlin.mff.cuni.cz