Micro–level stochastic loss reserving for general insurance Katrien Antonio * Richard Plat † October 1, 2012 Abstract 1 The vast literature on stochastic loss reserving concentrates on data aggre- gated in run–off triangles. However, a triangle is a summary of an underlying data set with the development of individual claims. We refer to this data set as ‘micro– level’ data. Using the framework of Position Dependent Marked Poisson Processes (as in Norberg (1993) and Norberg (1999)) and statistical tools for recurrent events (as in Cook and Lawless (2007)), a data set is analyzed with liability claims from a European insurance company. We use detailed information of the time of occur- rence of the claim, the delay between occurrence and reporting to the insurance company, the occurrences of payments and their sizes, and the final settlement. In contrast with the non–parametric Bayesian approach of Haastrup and Arjas (1996) our specifications are (semi)parametric and our approach is likelihood based. We calibrate the model to historical data and use it to project the future development of open claims. An out–of–sample prediction exercise shows that we obtain detailed and valuable reserve calculations. For the case study developed in this paper, the micro–level model outperforms the results obtained with traditional loss reserving methods for aggregate data. Key words: loss reserving, general insurance, Poisson process, recurrent events, survival analysis, prediction. * University of Amsterdam and KU Leuven (Belgium), email: [email protected]. Katrien Antonio ac- knowledges financial support from the Casualty Actuarial Society, Actuarial Foundation and the Com- mittee on Knowledge Extension Research of the Society of Actuaries, and from NWO through a Veni 2009 grant. † University of Amsterdam and Richard Plat Consultancy, email: [email protected]1 The authors would like to thank Jan–Willem Vulto and Joris van Kempen for supplying and ex- plaining the data. Please note that the original frequency and severity data have been transformed for reasons of confidentiality. 1
Transcript
Micro–level stochastic loss reserving for generalinsurance
Katrien Antonio ∗ Richard Plat †
October 1, 2012
Abstract
1 The vast literature on stochastic loss reserving concentrates on data aggre-
gated in run–off triangles. However, a triangle is a summary of an underlying data
set with the development of individual claims. We refer to this data set as ‘micro–
level’ data. Using the framework of Position Dependent Marked Poisson Processes
(as in Norberg (1993) and Norberg (1999)) and statistical tools for recurrent events
(as in Cook and Lawless (2007)), a data set is analyzed with liability claims from
a European insurance company. We use detailed information of the time of occur-
rence of the claim, the delay between occurrence and reporting to the insurance
company, the occurrences of payments and their sizes, and the final settlement. In
contrast with the non–parametric Bayesian approach of Haastrup and Arjas (1996)
our specifications are (semi)parametric and our approach is likelihood based. We
calibrate the model to historical data and use it to project the future development of
open claims. An out–of–sample prediction exercise shows that we obtain detailed
and valuable reserve calculations. For the case study developed in this paper, the
micro–level model outperforms the results obtained with traditional loss reserving
methods for aggregate data.
Key words: loss reserving, general insurance, Poisson process, recurrent events,
survival analysis, prediction.
∗University of Amsterdam and KU Leuven (Belgium), email: [email protected]. Katrien Antonio ac-knowledges financial support from the Casualty Actuarial Society, Actuarial Foundation and the Com-mittee on Knowledge Extension Research of the Society of Actuaries, and from NWO through a Veni2009 grant.
†University of Amsterdam and Richard Plat Consultancy, email: [email protected] authors would like to thank Jan–Willem Vulto and Joris van Kempen for supplying and ex-
plaining the data. Please note that the original frequency and severity data have been transformed forreasons of confidentiality.
1
1 Introduction
Figure 1 illustrates the run–off (or development) process of a general insurance claim.A claim occurs at a certain point in time (t1), consequently it is declared to the in-surer (t2) (possibly after a period of delay) and one or several payments follow untilthe settlement (or closing) of the claim. Depending on the nature of the business andclaim, the claim can re–open and payments follow until the claim finally settles. At thepresent moment (say τ) the insurer has to set reserves aside to fulfill his future liabili-ties. General insurers determine reserves for Reported But Not Settled (RBNS) claimsand Incurred Bot Not Reported (IBNR) claims. For an RBNS claim occurrence and dec-laration take place before the present moment and settlement occurs afterwards (i.e.τ ≥ t2 and τ < t6 (or τ < t9) in Figure 1). An IBNR claim has occurred before thepresent moment, but its declaration and settlement follow afterwards (i.e. τ ∈ [t1, t2)
in Figure 1). The interval [t1, t2] represents the so–called reporting delay. The interval[t2, t6] (or [t2, t9]) is the settlement delay.
Figure 1: Development of a general insurance claim
t1 t2 t3 t4 t5 t6 t7 t8 t9
Occurrence
Notification
Loss payments
Closure
Re–opening
Payment
Closure
IBNRRBNS
The measurement of future cash flows and their uncertainty becomes more impor-tant due to the introduction of new supervisory guidelines (Solvency 2) and reportingstandards (IFRS 4 Phase 2). The question naturally rises whether improvement of exist-ing techniques is possible. Reserving is nowadays based on data aggregated in run–offtriangles. A run–off triangle summarizes available information per arrival (‘AY’) anddevelopment year (‘DY’) combination. England and Verrall (2002) and Wuthrich andMerz (2008) give overviews of techniques for loss reserving based on triangles.
England and Verrall (2002) and Taylor et al. (2008) question the use of aggregateloss data. Due to aggregation, information from policy, policy holder or the past de-velopment process of a claim is not taken into account. Quoting England and Verrall
2
(2002) (page 507) “[. . . ] it has to be borne in mind that traditional techniques were developedbefore the advent of desktop computers, using methods which could be evaluated using penciland paper. With the continuing increase in computer power, it has to be questioned whetherit would not be better to examine individual claims rather than use aggregate data”. Manyproblems may arise with triangular data. Kunkler (2004) discusses the problem of zeroor negative cells in the triangle. Verdonck et al. (2009) put focus on robustness proper-ties and the influence of outliers on triangular methods. The number of observationsin a run–off triangle is typically small, and for recent accident years only few observa-tions are available. In that respect Wright (1990) and Renshaw (1994) discuss the over–parametrization of the chain–ladder method (a traditional technique applied to run–offtriangles). The separate assessment of true IBNR and RBNS claims in a run–off triangleis not straightforward, see Schnieper (1991) and Liu and Verrall (2009). Neither is thecombination of different sources of information, like paid and incurred losses (i.e. paidlosses plus case estimates set by experts), as demonstrated in Quarg and Mack (2008),Posthuma et al. (2008) and Merz and Wuthrich (2010). These references present usefuladjustments to the chain-ladder method, but have not been applied simultaneously.The existence of this substantial literature illustrates that a chain–ladder analysis notalways adequately captures the complexities of stochastic reserving for general insur-ance.
A small stream of literature has emerged with focus on stochastic loss reservingat individual claim level (i.e. micro–level). Arjas (1989), Norberg (1993) and Norberg(1999) formulate a mathematical framework for the development of individual claimsusing Position Dependent Marked Poisson Processes. Haastrup and Arjas (1996) con-tinue their work and implement a micro–level stochastic model for loss reserving.The use of non–parametric Bayesian statistics complicates the accessibility of their ap-proach. Furthermore, their case study is based on a small data set with fixed claimamounts. Recently, Larsen (2007) revisits the work of Norberg, Haastrup and Arjaswith a small case–study. Zhao et al. (2009) and Zhao and Zhou (2010) present a modelfor individual claims development using (semi–parametric) techniques from survivalanalysis and copula methods. However, a case study is lacking in their work.
Our work is an extensive case study developed in the probabilistic framework ofNorberg (1993) and Norberg (1999). We analyze a realistic data base from practice,motivate and verify our distributional assumptions and derive predictive distribu-tions for quantities of interest. Finally, we check the performance of the micro–levelmodel through an out–of–sample prediction exercise and compare our approach withtriangular methods. The micro–level approach circumvents many of the problems en-countered in a triangular analysis. Explicit quantification of the Incurred But Not Re-ported (IBNR) reserve as well as the Reported But Not Settled (RBNS) reserve follows
3
naturally. The large sample size allows flexible modeling of the claims developmentprocess. For example, covariate information (like deductibles, policy limits, calendaryear) can be included in the projection of the cash flows. The use of individual dataavoids robustness problems and over parametrization. Problems with negative or zerocells in the triangle are avoided, and small and large claims can be handled simultane-ously. Furthermore, individual claim modeling can provide a natural solution for thedilemma whether to use triangles with paid or incurred losses. We propose using theinitial case reserve as a covariate in the projection process of future cash flows.
2 Data
The data come from a general liability insurance portfolio (for private individuals)of a European insurance company. The exposure per month is given from January2000 till August 2009, as well as a detailed track record of each claim filed with theinsurer between January 1997 and August 2009. We are missing exposure informationbetween January 1997 and December 1999, but the impact of this lack on our reservecalculations is negligible. Below we present the data and focus on the different piecesof information available. The graphs and tables indicate the number of observationsavailable for each building block of the micro–level model.
Exposure. Exposure is expressed as ‘earned’ exposure. A policy covered duringthe whole month of January will contribute 31/365th to the exposure of that month,10/365th if it’s only covered during 10 days, and so on. Figure 2 shows the exposureper month. Note that the downward spikes correspond to the month February.
Type and number of claims. We distinguish two types of claims: material damage(‘material’) and bodily injury (‘injury’). Figure 3 shows the total number of claimsper arrival year (solid line), as well as the number of open claims at the end of theobservation period (i.e. end of August 2009). We analyze material damage and bodilyinjury claims separately, because their development pattern and corresponding lossdistributions behave differently (see further for some descriptive statistics).
Development processes. The claim file consists of 1,525,376 records correspondingwith 491,912 claims. Figure 4 shows the development of three random claims. We plotthe timing of events (namely: occurrence, declaration, payments and settlement) aswell as the size of payments (if any). Payments are indicated as jumps in the figure.Starting point of the development process is the accident date. This is indicated with
4
Figure 2: Available exposure per month from January 2000 till August 2009.
Year
expo
sure
Jan’00 Jan’02 Jan’04 Jan’06 Jan’08
4000
050
000
6000
070
000
8000
0
Figure 3: Number of open and closed claims of type material (left) and injury (right).
Number of Material Claims
Arrival Year
# cl
aim
s
1998 2000 2002 2004 2006 2008
2800
032
000
3600
040
000
# Claims# Closed Claims
Number of Injury Claims
Arrival Year
# cl
aim
s
1998 2000 2002 2004 2006 2008
600
700
800
900 # Claims
# Closed Claims
a sub-title in each of the plots and corresponds to the point x = 0. The x-axis is inmonths since the accident date. The y-axis represents the cumulative amount paid forthe claim.
Reporting and settlement delay. Important drivers of the IBNR and RBNS reservesare the reporting and settlement delays. Figure 5 (upper) shows the reporting delayregistered for material and injury claims. This delay is measured in months since oc-currence of the claim. Obviously, it is only available for claims that have been reported
5
Figure 4: Development of 3 random claims from the data set. The x-axis is in months since theaccident date. The y-axis represents the cumulative amount paid for the claim.
0 2 4 6 8
050
010
0015
00
time since origin of claim (in months)
Development of claim 327002
Acc. Date 15/03/2006
0 1 2 3 4
050
010
0015
00
time since origin of claim (in months)
Development of claim 434833
Acc. Date 01/10/2008
0 5 10 15
010
020
030
040
0
time since origin of claim (in months)
Development of claim 216542
Acc. Date 05/02/2003
Rep. DelayPaymentSettlement
to the insurance company before the end of the observation period. Figure 5 (lower)shows the settlement delay registered for injury and material claims. It is measured inmonths since the reporting of the claim and only available for closed claims. The distri-bution of observed reporting delays is similar for material and injury claims. However,the distribution of observed settlement delays is far more skewed to the right for injurythan for material claims.
Events in the development. We distinguish three types of events which possibly oc-cur during the development of a claim. ‘Type 1’ events imply settlement of the claimwithout a payment. A ‘type 2’ event is a payment with settlement at the same time.Intermediate payments (without settlement) are ‘type 3’ events. Figure 6 gives thecumulative number of events observed over the development of individual claims.Injury claims settle more slowly than material claims. They typically require moreintermediate payments before settlement than material claims do.
Payments. Events of type 2 and type 3 come with a payment. The distribution ofthese payments differs materially for the two types of claims. Figure 7 shows the dis-tribution of payments corresponding with material damage (left) and injury claims(right). The payments are discounted to 1-1-1997 with the Dutch consumer price in-flation, to exclude the impact of inflation on the distribution of the payments. Thegraphs suggest a lognormal distribution (see Section 4 for further discussion). Table 1gives characteristics of the observed payments for both material and injury losses. Asexpected, the distribution of injury payments has a heavier right tail than the distri-bution of material damage payments. The table reveals the presence of a large injurypayment (namely, 779,398 euro versus the empirical 99% quantile of 16,664 euro). Wewill discuss this further in Section 6.
6
Figure 5: Upper: reporting delay for material (left) and injury (right) claims. Lower: settle-ment delay for material (left) and injury (right) claims.
Initial case estimates. As explained in Section 1, the joint modelling of incurred andpaid losses is documented in the literature on stochastic loss reserving. It is worthwhileinvestigating the added value of case reserves when projecting future payments. Wecategorize the initial case reserve (for material damage and injury claims separately)and use it as an explanatory variable when modeling the distribution of payments.For each category, Table 2 shows the number of claims, the average settlement delay(in months) and the average cumulative paid amount for these categories. The tableclearly reveals that claims initially set as ‘large’, have large settlement delays and lead
7
Figure 6: Cumulative number of events over development years: material (left) and injury(right).
050
000
1000
0015
0000
2000
0025
0000
Number and Type of Events: Material
Development Year
Cum
. Num
ber
DY 1 DY 3 DY 5 DY 7 DY 9 DY 11 DY 13
Type 3 (paym.)Type 2 (settl. + paym.)Type 1 (settl.)
050
0010
000
1500
0
Number and Type of Events: Injury
Development Year
Cum
. Num
ber
DY 1 DY 3 DY 5 DY 7 DY 9 DY 11 DY 13
Type 3 (paym.)Type 2 (settl. + paym.)Type 1 (settl.)
Figure 7: Distribution of payments for material (right) and injury (left) claims. Normal den-sity reference line is included.
Material Payments
Log(payments)
Den
sity
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
0.5
Injury Payments
Log(payments)
Den
sity
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
to large ultimate cumulative amounts.
8
Table 2: Initial reserve categories: number of claims per category, average settlement delay andaverage cumulative paid amount.
Material InjuryInitial Average Average Cum. Initial Average Average Cum.
Case Reserve # claims settl. delay payments Case Reserve # claims settl. delay payments(months) (months)
We follow the framework of Arjas (1989), Norberg (1993), Norberg (1999) and Haastrupand Arjas (1996) and treat the claims process as a Position Dependent Marked PoissonProcess (PDMPP), see Karr (1991). The next paragraph repeats the essence of theirframework. Interested readers can find more mathematical details in their work.
The occurrence of claims is a Poisson process with non–homogeneous intensitymeasure λ(t) and associated mark distribution (PZ|t)t≥0 with T = t the occurrencetime of the claim. The mark distribution PZ|t is specified by the distribution PU|t of thereporting delay U (given occurrence time t) and the distribution PX |t,u of the develop-ment process X (given occurrence time t and reporting delay u). The development ofa claim consists of the timing and type of events and their corresponding severities.For claim i, Xi is short for (Ei(v), Pi(v))v∈[0,Vi]
where Ei(vij) := Eij is the type of the jthevent, which occurs at time Vij = vij (in time units) after notification of the claim). Vi isthe total waiting time from notification to settlement of claim i. If the event at time vij
includes a payment, its corresponding severity is given by Pi(vij) := Pij′ where j′
runsover all payments in the development of this claim. The process of reported claims isa Poisson process with measure
λ(dt)PU|t(τ − t)1(t∈[0,τ]) ·PU|t(du)1(u≤τ−t)
PU|t(τ − t)· PX |t,u(dx), (1)
on the set of reported claims Cr = {(t, u,x)|t ≤ τ, t + u ≤ τ}. Note that the intensityof the Poisson process driving the occurrence of claims is adjusted for the fact thatwe only consider reported claims. A similar remark applies to the distribution of the
9
reporting delay. The process of IBNR claims is a Poisson process with measure
λ(dt)(
1− PU|t(τ − t))
1(t∈[0,τ]) ·PU|t(du)1u>τ−t
1− PU|t(τ − t)· PX |t,u(dx), (2)
on the set of incurred but not reported claims C i = {(t, u,x)|t ≤ τ, t + u > τ}.
3.2 The likelihood
The observed part of the process consists of the development up to present time τ ofclaims reported before τ. We denote these observed claims as follows:
(Toi , Uo
i ,Xoi )i≥1. (3)
The likelihood of the observed claim development process is
Λ(obs) ∝
{∏i≥1
λ(Toi )PU|t(τ − To
i )
}exp
(−∫ τ
0w(t)λ(t)PU|t(τ − t)dt
)
·{
∏i≥1
PU|t(dUoi )
PU|t(τ − Toi )
}·∏
i≥1Pτ−To
i −Uoi
X |t,u (dXoi ). (4)
The superscript in the last term of this likelihood indicates the censoring of the devel-opment of this claim τ − To
i −Uoi time units after notification. w(t) gives the exposure
at time t.The statistical framework of multitype recurrent events (see Cook and Lawless
(2007), Chapter 6) is suitable to model the development of a claim. We specify haz-ard rates hse, hsep and hp, corresponding with type 1 (settlement without payment),type 2 (settlement with a payment at the same time) and type 3 (payment without set-tlement) events, respectively. Events of type 2 and 3 come with a payment. We denotethe severity distribution with Pp. The likelihood of observed claims (Λ(obs)) becomes{
∏i≥1
λ(Toi )PU|t(τ − To
i )
}exp
(−∫ τ
0w(t)λ(t)PU|t(τ − t)dt
)
·{
∏i≥1
PU|t(dUoi )
PU|t(τ − Toi )
}
· ∏i≥1
∏j
(h
δij1se (Vij) · h
δij2sep(Vij) · h
δij3p (Vij)
)exp
(−∫ τi
0(hse(u) + hsep(u) + hp(u))du
)· ∏
i≥1∏
j′Pp(dPij′ ). (5)
10
δijk is 1 if the jth event in the development of claim i is of type k (with k = 1, 2 or 3), and0 otherwise. j runs over all events registered in the observation period for claim i. Thisobservation period is [0, τi] with τi = min (τ − Ti −Ui, Vi). j
′runs over all payments
made during the development of this claim.
4 Distributional assumptions and estimation results
Section 2 presents an empirical investigation of the building blocks in likelihood (5).We now discuss distributional assumptions for each of its components. A (semi)parametricapproach is used, which is different from Haastrup and Arjas (1996) where non–parametricBayesian models were used. We motivate our distributional assumptions and show theoutcomes of the calibration process. Optimization of all likelihood specifications wasdone with the Proc NLMixed routine in SAS 2.
Reporting delay. The likelihood for reporting delays corresponds to the second linein (5). A large number of claims is reported in the days immediately following oc-currence of the claim, see Figure 5. We use a mixture of a Weibull distribution and 9degenerate components corresponding with settlement after 0, . . . , 8 days. Figure 8illustrates the fit of this mixture of distributions to the actually observed reporting de-lays.
Figure 8: Observed reporting delays for material (left) and injury (right) claims and fit obtainedwith 9 degenerate components combined with a truncated Weibull distribution.
Fit Reporting Delay − ’Material’
In months since occurrence
Den
sity
0.0 0.5 1.0 1.5 2.0 2.5 3.0
01
23
45 Weibull / Degenerate
Observed
Fit Reporting Delay − ’Injury’
In months since occurrence
Den
sity
0.0 0.5 1.0 1.5 2.0 2.5 3.0
01
23
4
Weibull / DegenerateObserved
2Parameter estimates and corresponding standard errors are available upon request from the firstauthor.
11
Occurrence process. We use the reporting delay distribution (as obtained in the pre-vious step). The part of the likelihood in (5) corresponding with claim occurrences (i.e.the first line) needs to be optimized over λ(t). We use a piecewise constant specifica-tion for the occurrence rate: λ(t) = λl for dl−1 ≤ t < dl, l = 1, . . . , m and d0 = 0.Hereby τ ∈ [dm−1, dm) and w(t) := wl for dl−1 ≤ t < dl. Let the indicator variableδ1(l, ti) be 1 if dl−1 ≤ ti < dl, with ti the occurrence time of claim i. The number ofclaims in interval [dl−1, dl) is then given by
Noc(l) := ∑i
δ1(l, ti). (6)
The likelihood corresponding with the occurrence times becomes
λNoc(1)1 λ
Noc(2)2 . . . λ
Noc(m)m
{∏i≥1
PU|t(τ − ti)
}
· exp(−λ1w1
∫ d1
0PU|t(τ − t)dt
)exp
(−λ2w2
∫ d2
d1
PU|t(τ − t)dt)
· . . . exp(−λmwm
∫ dm
dm−1
PU|t(τ − t)dt)
. (7)
Optimizing over λl (with l = 1, . . . , m) can be done numerically, or analytically. Thelatter approach leads to
λl =Noc(l)
wl∫ dl
dl−1PU|t(τ − t)dt
. (8)
For our data set, λ(t) is assumed to be constant on monthly intervals, ranging fromJanuary 2000 till August 2009. Figure 9 shows point estimates and corresponding 95%confidence intervals.
Development process. A piecewise constant specification is used for the hazard rates(hse, hsep and hp). For material claims, the hazard rate is constant on four month inter-vals: [0− 4) months, [4− 8) months, . . ., [8− 12) months and ≥ 12 months. For injuryclaims, the hazard rate is constant on intervals of six months: [0− 6) months, [6− 12)months, . . ., [36− 42) months and≥ 42 months. This piecewise specification can be in-tegrated in a straightforward way in (5) (third line). The resulting expression is similarto (7), but complex in notation. Similar to the occurrence process, the optimization ofthe likelihood can be done analytically or numerically. As an alternative for the piece-wise constant specification, we examine the use of Weibull hazard rates. Figure 10shows estimates for both approaches. As in Haastrup and Arjas (1996), separate haz-
12
Figure 9: Estimates of piecewise specification for λ(t): (left) material and (right) injury claims.
ard rates are specified for ‘first events’ and ‘later events’. In the rest of the paper weuse the piecewise constant specification. Section 5 includes brief discussion of the useof the Weibull hazard rates in the simulation routine.
Payments. We examine the fit of a Burr, gamma and lognormal distribution, includ-ing covariate information. Distributions for the payments are truncated at the coveragelimit of 2.5 million euro per claim. A comparison based on BIC shows that the lognor-mal distribution achieves a better fit than the Burr and gamma distributions. Whenincluding the initial reserve category as covariate or both the initial reserve categoryand the development year, the fit further improves. The latter approach is used in theprediction routine. We include covariate information in both mean (µi) and standarddeviation (σi) of the lognormal distribution for observation i:
µi = ∑r
∑s
µr,s IDYi=s Ii∈r
σi = ∑r
∑s
σr,s IDYi=s Ii∈r. (9)
Hereby r is the initial reserve category and DYi is the development year correspondingwith observation i. IDYi=s and Ii∈r are indicator variables referring to observation ibelonging to DY s and reserve category r, respectively. Figure 11 shows correspondingqqplots.
13
Figure 10: Estimates for Weibull and piecewise constant hazard rates driving the occurrence ofevents in the development of claims: (upper) injury claims and (lower) material claims. ‘Type1’ events represent settlement without payment, ‘type 2’ refers to settlement with payment and‘type 3’ is for intermediate payments.
0 40 80 120
0.00
0.04
0.08
0.12
Type 1 − Injury
t
h(t)
firstlaterWeibull
0 40 80 120
0.00
00.
010
0.02
00.
030
Type 2 − Injury
t
h(t)
0 40 80 120
0.05
0.10
0.15
0.20
0.25
Type 3 − Injury
t
h(t)
0 5 15 25
0.1
0.2
0.3
0.4
0.5
Type 1 − Mat
t
h(t)
0 5 15 25
0.0
0.1
0.2
0.3
0.4
Type 2 − Mat
t
h(t)
0 5 15 25
0.05
0.10
0.15
0.20
Type 3 − Mat
t
h(t)
Figure 11: Normal qqplots for the fit of log(payments) including initial reserve and develop-ment year as covariate information.
−10 −5 0 5
−4
−2
02
4
Normal QQplot Payments: Material
Emp. Quant.
The
or. Q
uant
.
−4 −2 0 2 4
−4
−2
02
4
Normal QQplot Payments: Injury
Emp. Quant.
The
or. Q
uant
.
14
5 Predicting future cash–flows
A distinction between IBNR and RBNS claims is necessary in the prediction routine.The following step by step approach is implemented.
Predicting IBNR claims. As noted in Section 3, an IBNR claim occurred already buthas not yet been reported to the insurer. Therefore, its occurrence time is missing at thetime of evaluation. The prediction process for the IBNR claims requires the followingsteps
(a) Simulate the number of IBNR claims in [0, τ] and their corresponding occur-rence times.IBNR claims are driven by a Poisson process with intensity:
w(t)λ(t)(1− PU|t(τ − t)), (10)
where λ(t) is piecewise constant. With NIBNR(l) being the number of IBNRclaims occurring in time interval [dl−1, dl), we know
NIBNR(l) ∼ Poisson(
λlwl
∫ dl
dl−1
(1− PU|t(τ − t))dt)
. (11)
Given the simulated number of IBNR claims nIBNR(l) for each interval [dl−1, dl),the occurrence times of the claims are uniformly distributed in [dl−1, dl). Theseare simulated as well.
(b) Simulate the reporting delay for each IBNR claim.Given the simulated occurrence time t of an IBNR claim, its reporting delay issimulated by inverting the distribution:
P(U ≤ u|U > τ − t) =P(τ − t < U ≤ u)1− P(U ≤ τ − t)
. (12)
Our mixture of a Weibull distribution and 9 degenerate distributions requiresnumerical evaluation of this expression.
(c) Simulate the initial reserve category.For each IBNR claim an initial reserve category has to be simulated for use in thedevelopment process. Given m initial reserve categories, we use the following
15
discrete probability density function:
f (c) =
pc for c = 1, 2, . . . , m− 1
1−∑m−1k=1 pk for c = m.
(13)
The probabilities used in (13) are the empirically observed percentages of claimsin a particular initial reserve category.
(d) Simulate the payment process for each IBNR claims.This step is common with the procedure for RBNS claims and is explained in thenext paragraph.
Predicting RBNS claims.
(e) Simulate the next event’s exact time.In case of RBNS claims, the time of censoring, say c, of a claim is known. ForIBNR claims this censoring time is c := 0. The next event – at time vnext – cantake place at any time vnext > c. To simulate its exact time we need to invert:(with p randomly drawn from a Unif(0, 1) distribution)
P(V < vnext|V > c) = p. (14)
From the relation between a hazard rate and cdf, we know
P(V ≤ vnext) = 1− exp
(−∫ vnext
0∑
ehe(t)dt
), (15)
with e ∈ {se, sep, p}. Numerical methods are required, for instance with a Weibullspecification for the hazard rates. With a piecewise constant specification for thehazard rates numerical routines as well as closed–form solutions are available.
(f) Simulate the event type. Given the exact time v of the next event, it is of typee ∈ {se, sep, p} with probability:
he(v)∑e he(v)
. (16)
(g) Simulate the corresponding payment (if any). Given a claim’s covariate infor-mation, payments are drawn from the appropriate lognormal distribution. Thecumulative payment cannot exceed the coverage limit of 2.5 million per claim.
16
(h) Stop or continue. Depending on the simulated event type in step ( f ), the predic-tion stops (in case of settlement) or continues.
In the next section, this prediction process will be applied separately for the materialclaims and the injury claims.
Comment on parameter uncertainty. To reflect uncertainty of predictions, process aswell as estimation or parameter uncertainty (see England and Verrall (2002)) should betaken into account. We cover process uncertainty by sampling from the distributionsselected in Section 4. To include parameter uncertainty bootstrapping or a Bayesianapproach can be used. A Bayesian approach would add significant complexity (seeHaastrup and Arjas (1996)), which reduces accessibility of the model and transparencytowards practicing actuaries. In our approach a bootstrap procedure is computer in-tensive, due to the large sample size (see 2) and the different stochastic processes used.We deal with parameter uncertainty through the asymptotic normal distribution of themaximum likelihood estimators. At every step in the routine, each parameter is sam-pled from its corresponding asymptotic normal distribution. Note that –due to ourlarge sample size– confidence intervals for parameters are narrow. This is in contrastwith run–off triangles where sample sizes are typically small and parameter uncer-tainty is an important point of concern.
6 Numerical results
We present the results of an out–of–sample prediction exercise. The micro–level resultsare compared with the results of a triangular analysis (based on aggregate data). Theout–of–sample test estimates reserves per 1-1-2005. The available data are summarizedin run-off triangles (see Table 3 for material damage and Table 4 for bodily injury). Theactual observations registered between January 2005 and August 2009 are in bold.
Output from the micro–level model. We consider the different types of output com-ing from the micro–level model. Figure 12 shows the results for injury payments madein calendar year 2006, based on 10,000 simulations. In Table 4 calendar year 2006corresponds to the diagonal going from 412, 268, . . . , up to 97. The first row in Fig-ure 12 shows (from left to right): the number of IBNR claims reported in 2006, the totalamount paid in this calendar year and the total number of events occurring in 2006.The IBNR claims are claims that occurred before 1-1-2005, but were reported to the in-surer during calendar year 2006. The total amount paid in 2006 is the sum of paymentsfor RBNS and IBNR claims, which are separately available from the micro–model. In
17
Table 3: Run–off triangle material claims (displayed in thousands), arrival years 1997-2004.
the second row of plots we take a closer look at the events registered in 2006 by lookingseparately at type 1, 2 and 3 events. In each of the plots the black solid line indicateswhat was actually observed. This figure shows that the predictive distributions fromthe micro–level model are realistic. However, the actual number of IBNR claims is farin the tail of the distribution, but note that this corresponds to a small number of IBNRclaims.
Comparing reserves. We compare the results obtained with the micro–level modelwith reserve calculations based on aggregate data. More specifically, we considerMack’s chain–ladder model, a stochastic overdispersed Poisson and lognormal chain–
18
Figure 12: Out–of–sample exercise per 1-1-2005, injury claims. Results are for calendar year2006, based on 10,000 simulations from the micro–level model. Top row (from left to right):number of IBNR claims, total reserve (i.e. IBNR plus RBNS reserve), total number of events.Bottom row (from left to right): number of type 1, 2 and 3 events. The black solid line indicatesactually observed quantities.
IBNR claims
Number
Fre
quen
cy
5 10 15 20 25 30 35
050
010
0015
00
IBNR + RBNS Reserve
Reserve
Fre
quen
cy
1000 2000 3000 4000 5000 6000
020
040
060
080
010
00
Total events
Number
Fre
quen
cy
700 800 900 1000 1100
020
040
060
080
010
00
Type 1 events
Number
Fre
quen
cy
100 120 140 160 180 200 220
050
010
0015
00
Type 2 events
Number of events
Fre
quen
cy
0 20 40 60 80
020
040
060
080
010
0012
00
Type 3 events
Number
Fre
quen
cy
500 550 600 650 700 750 800 850
020
040
060
080
010
00
ladder model. The model specifications for overdispersed Poisson (see (17)) and log-normal (see (18)) are given below, where Yij denotes cell (i, j) in a run–off triangle (andcorresponds to arrival year i and development year j). The models specified in (17)and (18) are implemented in a Bayesian framework. 3
Yij = φMij
Mij ∼ Poi(µij/φ)
log (µij) = αi + β j; (17)
log (Yij) = µij + εij
µij = αi + β j
εij ∼ N(0, σ2). (18)
3The implementation of the overdispersed Poisson model is in fact empirically Bayesian. φ is esti-mated beforehand and held fixed. We use vague normal priors for the regression parameters in bothmodels and a gamma prior for σ−1 in the lognormal model.
19
Figure 13 shows the reserves (in thousands) for material claims, as obtained with thedifferent methods (from left to right: micro–model, chain–ladder overdispersed Pois-son and chain–ladder lognormal). The histograms are based on 10,000 simulations ofthe total reserve. Corresponding numerical results are in Table 5. The total reservepredicts the complete lower triangle (i.e. all bold numbers, plus the three missing cellsin Table 3). The solid black line in each plot indicates what has really been observed,i.e. the sum of the numbers in bold in Table 3. We use the same scale on the x–axisof the histograms representing the micro–level and the overdispersed Poisson model.However, for the lognormal model a different scale on the x–axis is necessary, becauseof the presence of a long right tail. These unrealistically high reserves (see Table 5) are adisadvantage of the lognormal model for the portfolio of material claims. We concludefrom Figure 13 and Table 5 that the overdispersed Poisson as well as the lognormalmodel overstate the reserve; the actually observed amount is in the left tail of the cor-responding histogram. The predictive distribution obtained with the micro–model ismore realistic. The corresponding best estimate is closer to the true realization than thebest estimates from aggregate techniques.
Figure 13: Out–of–sample exercise per 1-1-2005, material claims. Results are for the totalreserve (i.e. IBNR + RBNS reserve), based on 10,000 simulations. From left to right: reservecalculations using the micro–level model, the aggregate overdispersed Poisson chain–ladder andthe aggregate lognormal chain–ladder model. The black solid line indicates the amount actuallypaid.
Micro−level: Total
Reserve
Fre
quen
cy
2000 3000 4000 5000 6000 7000
020
040
060
080
010
0012
0014
00
Aggregate Model − ODP: Total
Reserve
Fre
quen
cy
2000 3000 4000 5000 6000 7000
050
010
0015
0020
00
Aggregate Model − Lognormal: Total
Reserve
Fre
quen
cy
0 5000 10000 15000 20000 25000 30000
050
010
0015
0020
00
Figure 14 shows the distribution of the reserve (in thousands of euro) obtained forbodily injury claims (based on 10,000 simulations). In contrast to the plots in Figure 13the plots in Figure 14 use the same x–axis. Corresponding numerical results are inTable 5. The observed run–off triangle in Table 4 shows a large payment (779,383 euro)in occurrence year 2002, development year 8. This payment is much larger than allother payments in the data set (see the statistics in Table 1). The micro–level modelreflects this appropriately, i.e. the observed total amount is rather in the right tail of
20
the predictive distribution. The aggregate models again tend to overstate the reserve.
Figure 14: Out–of–sample exercise per 1-1-2005, injury claims. Results are for the total reserve(i.e. IBNR + RBNS reserve), based on 10,000 simulations. From left to right: reserve calcu-lations using the micro–level model, the aggregate overdispersed Poisson chain–ladder and theaggregate lognormal chain–ladder model. The black solid line indicates actually observed quan-tities.
Micro−level: Total
Reserve
Fre
quen
cy
4000 6000 8000 10000 12000 14000 16000
050
010
0015
00
Aggregate Model − ODP: Total
Reserve
Fre
quen
cy
4000 6000 8000 10000 12000 14000 16000
050
010
0015
00
Aggregate Model − Lognormal: Total
Reserve
Fre
quen
cy
4000 6000 8000 10000 12000 14000 16000
020
040
060
080
010
0012
00
We conclude that, for the case–study under consideration, the micro–model outper-forms the aggregate models under consideration and reveals a more realistic predictivedistribution of the reserve.
Table 5: Out–of–sample prediction per 1-1-2005: numerical results for material damage andinjury claims (in thousands), as obtained with Mack’s chain–ladder model, an overdispersedand lognormal stochastic chain–ladder model and the micro–level model. Real observed out-comes are also displayed.
Model Type Expected value Median s.e. VaR0.95 VaR0.99
Total ReserveChain–ladder Mack MD 2,865 349
BI 9,562 1,154Chain–ladder ODP MD 2,803 2,785 361 3,426 3,846
BI 7,386 7,209 1,259 9,721 11,725Observed MD > 1,861
BI > 7, 923
21
7 Conclusion
Continuing the work by Arjas (1989), Norberg (1993), Norberg (1999) and Haastrupand Arjas (1996) this paper demonstrates the usefulness of micro–level stochastic lossreserving as a way to quantify the best estimate of the reserve and its uncertainty.Stochastic models for the occurrence time, the reporting delay and the developmentprocess (including intermediate payments and settlement) of a claim are fit to a dataset with the development of individual claims. A micro–level approach allows muchcloser modeling of the claims process. The method is not restricted by limitations thatexist when using aggregate data.
We perform an out–of–sample test with respect to a general liability insurance port-folio from a European insurance company. The paper shows that micro–level stochas-tic modeling is feasible for real life portfolios. We compare prediction results from themicro–level model with results obtained by analyzing a run–off triangle. Conclusionof the out–of–sample test is that – at least for the comparisons made here – traditionaltechniques tend to overestimate the real payments. Predictive distributions obtainedwith the micro–model reflect reality in a more realistic way: ‘regular’ outcomes areclose to the median of the predictive distribution whereas pessimistic outcomes are inthe right tail.
The results obtained in this paper make it worthwhile to further investigate the useof this technique for loss reserving. Several directions for future research can be men-tioned. One could try to refine the performance of the individual model with respectto pessimistic scenarios by using a combination of e.g. a lognormal distribution forlosses below and a generalized Pareto distribution for losses above a certain threshold.Connected to this suggestion, we intend to explore the possibilities of the micro–levelapproach in a reinsurance context. Analyzing the performance of both the micro–levelmodel and techniques for aggregate data on simulated data sets and new case studieswill bring more insight in their performance. More careful modeling of inflation effectsand taking the ‘time value of money’ into account will be important in future research.Studying the micro–level approach in light of the new solvency guidelines, is anotherpath to be explored.
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