A time change strategy to model reporting delaydynamics in claims reserving
4th EAJ Conference, Leuven
Katrien AntonioLRisk - KU Leuven and ASE - University of Amsterdam
February 8, 2018
K. Antonio, KU Leuven & UvA 1 / 26
Acknowledgement
The talk is based on joint work with:
Gerda Claeskens, Jonas Crevecoeur (present!) and Roel Verbelen.
K. Antonio, KU Leuven & UvA 2 / 26
IntroductionDevelopment of a single claim
Time
IBNR RBNS Closed
Occurrence Reporting Closure
PaymentsReporting delay
K. Antonio, KU Leuven & UvA Introduction 3 / 26
IntroductionAggregated approach
We aggregate the data from the time line into a run-off triangle or claimsdevelopment triangle:
Time
Occurrence Reporting Closure
PaymentsReporting delay
All claims in portfolio
Compress data
Run-off time
Occ
urr
ence
Yea
r
K. Antonio, KU Leuven & UvA Introduction 4 / 26
Research focusIBNR claim counts
Time
IBNR RBNS Closed
Occurrence Reporting Closure
PaymentsReporting delay
K. Antonio, KU Leuven & UvA Research focus 5 / 26
Research focusIBNR claim counts
Time
IBNR RBNS Closed
Occurrence Reporting Closure
PaymentsReporting delay
K. Antonio, KU Leuven & UvA Research focus 5 / 26
Research focusIBNR claim counts
Time t
Time sinceoccurrence claim
K. Antonio, KU Leuven & UvA Research focus 6 / 26
Research focusIBNR claim counts
Time t
Time sinceoccurrence claim
Evaluation date τ
The insurance company is not aware (yet) of claims related to pastexposures that are not (yet) reported!
K. Antonio, KU Leuven & UvA Research focus 6 / 26
Research questions
I Research questions with focus on IBNR?
• How many claims occurred but are not yet reported, because reportingdelay is subject to right truncation?
• When will these IBNR claims be reported?
I Pioneering work by Ragnar Norberg (1993, 1999).
I Recent contributions (a.o.) by Avanzi, Wong & Yang (2017), Badescu,Lin & Tang (2016), Verrall & Wuthrich (2016).
K. Antonio, KU Leuven & UvA Research questions 7 / 26
Case studyStructure of the data
I Large European dataset of liability claims (from private individuals).
I Three essential variables:
• Occurrence date
• Reporting date
• Exposure.
I Observation window: July 1, 1996 to August 31, 2009.
K. Antonio, KU Leuven & UvA Case study 8 / 26
Case studyClaim occurrence process
0
100
200
300
400
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
occurrence date in days
obse
rved
cla
im c
ount
K. Antonio, KU Leuven & UvA Case study 9 / 26
Case studyReporting process
0
100
200
300
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
reporting date in days
repo
rted
cla
im c
ount
K. Antonio, KU Leuven & UvA Case study 10 / 26
Case studyReporting delay
Declining pattern in reporting delay + intra-week pattern, depending on theoccurrence day of the week.
All claims
0.0
0.1
0.2
0 7 14delay in days since occurrence
repo
rtin
g pr
obab
ility
Monday
0.0
0.1
0.2
0 7 14delay in days since occurrence
repo
rtin
g pr
obab
ility
Thursday
0.0
0.1
0.2
0 7 14delay in days since occurrence
repo
rtin
g pr
obab
ility
K. Antonio, KU Leuven & UvA Case study 11 / 26
Case studyHolidays
0 25 50 75 100
Liberation Day
Easter
Christmas
Day after Christmas
Pentecost
New Year
King’s Day
Easter Monday
Pentecost Monday
Ascension Day
New Year’s Eve
Good Friday
Daily Average
average number of reports
K. Antonio, KU Leuven & UvA Case study 12 / 26
Case studyTotal IBNR counts
1600
1800
2000
2200
Oct 2003 Jan 2004 Apr 2004 Jul 2004evaluation date
unre
port
ed c
laim
s
1600
1700
1800
1900
2000
Oct 01 Oct 15 Nov 01 Nov 15 Dec 01evaluation date
unre
port
ed c
laim
s
K. Antonio, KU Leuven & UvA Case study 13 / 26
The statistical model
I Two contributions:
1. Verbelen, R., Antonio, K., Claeskens, G & Crevecoeur, J. 2018.
• joint estimation of occurrence process and reporting delay distribution
• regression approach
2. Crevecoeur, J., Antonio., K. & Verbelen, R. 2018.
• incorporate calendar day effects in reporting delay distribution
cfr. national holidays and during weekend, reporting at specific delays(e.g. 14 days, 1 year)
K. Antonio, KU Leuven & UvA The statistical model 14 / 26
Notations
I Nt : the (total) number of claims that occurred on day t.
I Nt,s : the number of claims from day t that are reported on day s.
I Each claim gets reported eventually, thus
Nt =∞∑s=t
Nt,s .
K. Antonio, KU Leuven & UvA The statistical model 15 / 26
The statistical modelAssumptions
(A1) The daily total claim counts Nt for t = 1, . . . , τ are independentlyPoisson distributed with intensity λt
Nt ∼ POI(λt).
(A2) Conditional on Nt , the claim counts Nt,s for s = t, t + 1, . . . aremultinomially distributed with probabilities pt,s .
Combining (A1) and (A2), the Nt,s are independent and
Nt,s ∼ POI(λt · pt,s).
K. Antonio, KU Leuven & UvA The statistical model 16 / 26
The statistical modelThe likelihood
I We observeNR = {Nt,s | t ≤ s ≤ τ}
where t ≤ τ indicates claim occurrence and t ≤ s ≤ τ reporting of theclaim.
I Log-likelihood of observed data:
`(λ,p;NR) =τ∑
t=1
τ∑s=t
(−λt · pt,s︸ ︷︷ ︸
(?)
+ log(λt) · Nt,s + Nt,s · log(pt,s)− log(Nt,s !))
difficult to optimize (due to ?).
K. Antonio, KU Leuven & UvA The statistical model 17 / 26
Time change strategyNon parametric occurrence process
I Estimate the occurrence process non-parametrically.
I Likelihood is maximal when
λt =
∑τs=t Nt,s∑τs=t pt,s
=NRt (τ)
pRt (τ).
I Replacing λt
`(p;NR) =τ∑
t=1
τ∑s=t
Nt,s · log(pt,s)−τ∑
t=1
NRt (τ) · log(pRt (τ)) + constants.
K. Antonio, KU Leuven & UvA The statistical model 18 / 26
Time change strategyThe idea pictured!
αthu,0
s − tThu Fri Sat Sun Mon Tue Wed
pt,s
u
αthu,0
Thu Fri Sat Sun Mon Tue Wed
fUt
ϕt(u)
αThu
ThuFri
SatSun
MonTue
Wed
fU
αSat
pt,s =
∫ s−t+1
s−tfUt (u)du
= FUt (s − t + 1)− FUt (s − t).
K. Antonio, KU Leuven & UvA The statistical model 19 / 26
Time change strategyThe idea pictured!
αthu,0
s − tThu Fri Sat Sun Mon Tue Wed
pt,s
u
αthu,0
Thu Fri Sat Sun Mon Tue Wed
fUt
ϕt(u)
αThu
ThuFri
SatSun
MonTue
Wed
fU
αSat
pt,s =
∫ s−t+1
s−tfUt (u)du
= FUt (s − t + 1)− FUt (s − t).
K. Antonio, KU Leuven & UvA The statistical model 19 / 26
Time change strategyThe idea pictured!
αthu,0
s − tThu Fri Sat Sun Mon Tue Wed
pt,s
u
αthu,0
Thu Fri Sat Sun Mon Tue Wed
fUt
ϕt(u)
αThu
ThuFri
SatSun
MonTue
Wed
fU
αSat
pt,s = FU(ϕt(s − t + 1))− FU(ϕt(s − t))
where ϕt(d) =d∑
i=1
αt,t+i−1 with αt,s reporting exposure
K. Antonio, KU Leuven & UvA The statistical model 19 / 26
Time change strategyStructuring the reporting exposures
I Use a standard distribution for U.
I Explain the daily reporting exposures as a function of covariates:
αt,s = exp (x′t,s · γ).
I Joint estimation of distribution U and regression parameters tostructure αt,s .
K. Antonio, KU Leuven & UvA The statistical model 20 / 26
Case studyResults - first evaluation
How many IBNR claims (on August 31, 2004) will be reported byAugust 31, 2009?
• Observed: 2049 claims
• Granular: 2012.7 claims
• Chain ladder: 2043.2 claims
K. Antonio, KU Leuven & UvA Back to the case study 21 / 26
Case studyResults - second evaluation (only granular)
When are the claims that are IBNR (on August 31, 2004) reported?
0
50
100
Sep 01 Sep 15 Oct 01 Oct 15 Nov 01reporting date
repo
rted
cla
ims
0
300
600
900
Sep2004
Oct2004
Nov2004
Dec2004
Jan2005
Feb2005
Mar2005
Apr2005
May2005
Jun2005
Jul2005
Aug2005
reporting monthre
port
ed c
laim
s
methoddatamodel
K. Antonio, KU Leuven & UvA Back to the case study 22 / 26
Case studyResults - third evaluation
1500
1750
2000
2250
Oct 2003 Jan 2004 Apr 2004 Jul 2004evaluation date
unre
port
ed c
laim
s
1600
1700
1800
1900
2000
Oct 01 Oct 15 Nov 01 Nov 15 Dec 01evaluation date
unre
port
ed c
laim
s
data granular
K. Antonio, KU Leuven & UvA Back to the case study 23 / 26
Case studyResults - third evaluation
1600
1800
2000
2200
Oct 2003 Jan 2004 Apr 2004 Jul 2004evaluation date
unre
port
ed c
laim
s
1600
1700
1800
1900
2000
Oct 01 Oct 15 Nov 01 Nov 15 Dec 01evaluation date
unre
port
ed c
laim
s
data chain ladder year chain ladder 28 days
K. Antonio, KU Leuven & UvA Back to the case study 24 / 26
Conclusion
I The message is not that chain-ladder should disappear!
I Take home messages:
• the presented methods increase insight in the available data and thedynamics in claim development patterns
(fits within the increasing interest in data analytics)
(characteristics can be taken into account)
• caution: many choices involved, should be done with care!
K. Antonio, KU Leuven & UvA Conclusion 25 / 26
More information
For more information, please visit:
LRisk website, www.lrisk.be
my homepage, www.econ.kuleuven.be/katrien.antonio.
Thanks to
Ageas Continental Europe Argenta
K. Antonio, KU Leuven & UvA More information 26 / 26
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