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