Asymptotic Consistency and Inconsistency of the Chain Ladder

**Asymptotic** **Consistency** **and****Inconsistency** **of** **the** **Chain****Ladder**(O banálnom aktuárskom probléme, ktorý ignorujeme, a jeho nebanálnom, ale triviálnom riešení)Michal Pešta(joint work with Šárka Hudecová)Charles University in PragueFaculty **of** Ma**the**matics **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** consistency**and** inconsistency **of** **the** chain ladder. Insurance:Ma**the**matics **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)◮ o**the**r (aviation, travel insurance, legal protection,credit insurance, epidemic insurance, . . . )- Life insurance products are ra**the**r 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 **and**reporting) – 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** **the**contract 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 st**and**s 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 **of**right-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 r**and**om variable **of** which we have anobservation if i + j ≤ n + 1- Aim is to estimate **the** ultimate claims amount C i,n**and** **the** outst**and**ing claims reserveR i = C i,n − C i,n+1−i ,i = 2, . . . , n

Run-**of**f 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 **of**variability) Var ̂R i = 0- unbiased estimate, i.e., E ̂R i = ER i , or conditionallyunbiased◮ firstly, introduce a model with assumptions, **the**nconstruct 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** st**and**ard error**of** 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 **of**̂β (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** r**and**om variables {ζ n } ∞ n=1can be defined, because **the** concept **of** **the**se 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 sequence**of** r**and**om variables, but only given one r**and**omvariable ζ

**Consistency**Denote D (n)j= {C i,k : k ≤ j, i ≤ n − j + 1} **and**D 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→∞**and**respectively.̂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 than**the** 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** **the**estimate **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 **of**convergence- Complete characterization **of** **the** conditionalconvergence **of** development factors’ estimate- The slower (faster) divergence **of**∞∑i=1C i,jimplies **the** slower (faster) realization **of**consistency **of** **the** development factors’ estimates

On **the** necessary **and**sufficient 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)- Ei**the**rjis consistent for f j for all j ∈ N, ornone **of** **the**m is consistent̂f(n)j- The consistency **of** is equivalent to **the**condition ∑ 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 **the**first development year- For instance, **the** ratios C k+1,1 /C k,1 or **the**sequence 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 to**the** 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- Fur**the**rmore, splitting one existing line **of**business into several o**the**rs 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 **the**Poisson distribution with **the** parameter f j−1 C i,j−1for j = 2, . . . , N

Decreasing business- consider a fast decreasing business, i.e., **the**situation 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 **the**estimates are indeed not consistent- The same simulations were run also for C i,1 with**the** uniform distribution: **the** differences between(n)values **of** estimates ̂fj**and** **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 studies**the** total uncertainty in **the** claims developmentuntil **the** total ultimate claim is finally settled- Solvency II purposes- run-**of**f = opening balance – expenses – closingbalance- we predict **the** total ultimate claim at time n (with**the** 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 **the**se 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 **the**financial 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 for**the** observable claims development result at **the**end **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** **the**development factors’ estimate in **the**distribution-free chain ladder is investigated- necessary **and** sufficient condition is derived- weak, strong consistency, **and** consistency in **the**mean 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 **the**consequences is demonstrated

ReferencesMack, T. (1993)Distribution-free calculation **of** **the** st**and**ard error**of** 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** **the**chain ladder.Insurance: Ma**the**matics **and** Economics, 51(2),472–479.

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