Ma inek, László
A icle
Analysis o s ochas ic ese ing models by means o NAIC
claims da a
Risks
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Ma inek, László (2019) : Analysis o s ochas ic ese ing models by means o
NAIC claims da a, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 7, Iss. 2, pp. 1-27,
h ps://doi.o g/10.3390/ isks7020062
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/257900
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by/4.0/
isks
A icle
Analysis o S ochas ic Rese ing Models By Means o
NAIC Claims Da a
László Ma inek 1,2
1Depa men o P obabili y Theo y and S a is ics, Eö ös Lo ánd Uni e si y, Pázmány Pé e sé ány 1/C,
1117 Budapes , Hunga y; [email p o ec ed]
2NN G oup, P inses Bea ixlaan 35, 2595 AK The Hague, The Ne he lands
Recei ed: 26 Feb ua y 2019; Accep ed: 28 May 2019; Published: 4 June 2019


Abs ac :
In he pas wo decades inc easing compu a ional powe esul ed in he de elopmen
o mo e ad anced claims ese ing echniques, allowing he s ochas ic b anch o o e come he
de e minis ic me hods, esul ing in o ecas s o enhanced quali y. Hence, no only poin es ima es,
bu p edic i e dis ibu ions can be gene a ed in o de o o ecas u u e claim amoun s. The signi ican
expansion in he a ie y o models equi es he alida ion o hese me hods and he c ea ion o
suppo ing echniques o app op ia e decision making. The p esen a icle compa es and alida es
se e al exis ing and sel -de eloped s ochas ic me hods on ac ual da a applying compa ison measu es
in an algo i hmic manne .
Keywo ds:
s ochas ic claims ese ing; p obabilis ic o ecas ; compa ison me ics; c edibili y;
Mon e Ca lo
1. In oduc ion
Insu ance and einsu ance ins i u ions, pa icula ly p ope y and casual y insu e s, pu a
conside able amoun o e o in o he unde s anding o ou s anding claims ese es. These amoun o
he mos ma e ial p opo ion o echnical p o isions, hence, hei olume and unce ain y a e c i ical
o be con olled well by ac ua ies and managemen . No only he measu e and pa e n o u u e cash
ou lows and me ics o associa ed isks play a ole in he insu ance business, bu also managemen
decisions a e igge ed by he ou come o calcula ions.
Schola s and indus y p o essionals ha e been s udying di e en es ima ion models in he pas
decades ex ensi ely. In e es in s ochas ic models has ou g own he in e es in de e minis ic ones,
shi ing om simple poin es ima ions o app oxima ion o p obabili y dis ibu ions, enabling he
calcula ion o ea u es o he examined objec wi h mo e insigh in o he na u e o he unde lying
phenomenon. The demand o o ecas s embodied in dis ibu ional o ms a he han poin es ima es
has g own apidly along wi h he g ow h o compu a ional powe , simul aneously allowing o he
p agma ic implemen a ion o Mon e Ca lo ype algo i hms. This inc easing in e es has eme ged no
only in insu ance bu in se e al o he disciplines, such as me eo ology o inance, demanding a mo e
meaning ul p edic ion o u u e ou comes. England and Ve all (2002); Wü h ich and Me z (2008)
con ain comp ehensi e o e iews o ese ing me hods. In ou iew, he alida ion o he models
on ac ual indus ial da a and he compa ison o hese models’ app op ia eness is a c ucial ques ion.
In spi e o he ele ance o model sui abili y, p opo ionally o he size o exis ing li e a u e on models,
e en mo e a en ion has o be gi en o he subs an ia ion o model quali y and o he compa ison o
me hodologies. P o essionals who a e o e ed coun less di e en models need guidelines ha can
suppo an op imal selec ion. A mo e ecen wo k, Meye s (2015) pe o ms in es iga ion on boo s ap
and Bayesian models using publicly a ailable claims da a om Ame ican insu ance companies.
The wo k also p oposes new me hods p ac ically sol ed h ough MCMC simula ions.
Risks 2019,7, 62; doi:10.3390/ isks7020062 www.mdpi.com/jou nal/ isks
Risks 2019,7, 62 2 o 27
A case s udy is pe o med in Wü h ich (2010) in o de o analyse he accoun ing yea e ec s
in he iangles. This s udy compa es Bayesian models wi h mean squa e e o o p edic ion
(MSEP) and de iance in o ma ion c i e ion (DIC). Shi and F ees (2011) and Shi e al. (2012) p o ide
ano he compa ison al e na i e wi h QQ-plo s and PP-plo s. Ne e heless, he i s one ocusses on
unde s anding he dependency among he iangles o di e en business lines wi h a copula eg ession
model, and he second one desc ibes e ospec i e es s on he models p oposed. E en mo e ocus is
pu on he alida ion o me hods in Ma ínez-Mi anda e al. (2013), e alua ing which me hodology
should be p e e ed. Th ee me hods, he double chain ladde , he Bo nhue e –Fe guson and he
incu ed double chain ladde me hods a e compa ed h ough wo eal da a se s om p ope y and
casual y insu e s, and he me ics used a e call e o , calenda yea e o and o al e o . Suppo ed by
eal-li e claims da a, Tee e al. (2017) compa es h ee models wi h di e en esidual adjus men s using
he Dawid–Sebas iani sco ing ule (DSS).
This pape analyses di e se s ochas ic claims ese ing me hods by means o se e al
goodness-o - i measu es. In a game- heo e ic in e p e a ion o o ecas s, i se s up a anking amewo k
selec ing om compe ing models. Ce ainly, he e is ha dly any manne o anking me hodology which
all ac ua ies would unanimously ag ee wi h, as a pe emp o y selec o o he mos p ope p edic ion
models. Howe e , i is easonable o de ine and obse e he impo an cha ac e is ics o es ima ions,
which pu oge he may suppo he decision-making p ocess and he alida ion o he applied me hods.
In he assessmen o ese ing models, he e is a s ong in en ion o p omo e measu es o iginally
used in s ochas ic o ecas ing. Ano he objec i e o he pape is o suppo he me hodological
backg ound and pe o m assessmen s o di e se se s o models on ac ual da a. P obabili y in eg al
ans o m (PIT) p o ides mo e jus i ica ion on he p edic i e dis ibu ion app op ia eness, while he
Kolmogo o –Smi no o C amé – on Mises s a is ics would ail o shed ligh on wha exac ly goes
w ong wi h he hypo hesis. Es ablished sco es compa e and e i y quali ies o i al p obabilis ic
o ecas ing models on he basis o es ima ion and eal ou comes.
F om he wide ange o sco ing ules, we apply he con inuous anked p obabili y sco es (CRPS)
due o hei lexible applicabili y on di e ing dis ibu ions, see Gnei ing and Ra e y (2007). Co e age
shows he cen al p edic ion in e al o a p edic ion gi en a eal go e ning dis ibu ion. Sha pness,
a ela ed me ic is he wid h as expec ed di e ence be ween lowe and uppe
p
-quan iles, he na owe
he be e exp essed in paymen , see Gnei ing e al. (2007). Al e na i ely, sha pness is also called
a e age wid h. Fo back es ing he s ochas ic ese ing models we apply hese i e me ics on he ull
quad angles, i.e., on he un-o ingles comple ed wi h he lowe pa . In se e al cases, when he
p edic ion model is dis ibu ion- ee, he empi ical o ecas has been d awn h ough boo s apping.
This makes an empi ical p edic i e dis ibu ion uni o mly a ailable.
In o de o measu e acco ding o eal scena ios, he da abase published in
Meye s and Shi (2011)
has been used. Paid and incu ed claims da a o igina e om he Na ional Associa ion o Insu ance
Commissione s (NAIC), and con ain ables o six di e en lines o business, encompassing (1)
comme cial au o and uck liabili y and medical, (2) medical malp ac ice, (3) p i a e passenge au o
liabili y and medical, (4) p oduc liabili y, (5) wo ke s’ compensa ion and (6) o he liabili y. Lines o
business a e homogeneous g oups o policies wi h iden ical co e age. Da a a e segmen ed in o hese
clus e s in o de o a oid he amalgama ion o claim paymen un-o s wi h signi ican ly di e en
cha ac e is ics. Leong e al. (2014) e alua es back es ing on he e e ed da a wi h espec o he
applica ion o boo s ap o e dispe sed Poisson model.
Simula ions ha e been ca ied ou wi h
R
, using packages
ChainLadde
Gesmann (2018) and
jags
o he MCMC simula ions. Besides he sel -w i en p og am codes, sc ip s published in
associa ion wi h Meye s (2015) ha e been embedded in o he calcula ions.
The p ima y objec i e o he p esen a icle is o suppo decision making among se e al a ailable
models applied on un-o iangles, by de ining and calcula ing measu es o he ac ual and p edic i e
dis ibu ions. Gi en ha ac ual dis ibu ions can ha dly be ex ac ed, we ha e used empi ical
dis ibu ions om eal ul ima e claims da a.
Risks 2019,7, 62 3 o 27
To he au ho s’ knowledge, nei he he c edibili y boo s ap me hod in Sec ion 3.3, no he
collec i e semi-s ochas ic model in Sec ion 3.5 ha e e e been discussed in pee - e iewed jou nals.
Two o he models inco po a e expe ience a emaking om he claims his o y o an en i e communi y
o companies. One s ep u he is exploi ing collec i e da a o imp o e indi idual (insu ance company
le el) p edic ion eliabili ies, equi ing he coo dina ion o egula o y au ho i ies as da a collec o s
and p ocesso s.
To summa ise he no el ies communica ed by he p esen pape : (1) Me ics in ac ua ial ese ing
such as CRPS, co e age and sha pness o se e al models o analyse hei pe o mance and de e mine
an o de o app op ia eness ha e been p esen ed by A a ó e al. (2017) on simula ed da a. He e we
apply all he calcula ions on ac ual iangles om mul iple isk g oups. (2) PIT has al eady been
applied by Meye s (2015) on s ochas ic models, he e we con inue p esen ing he calcula ions in ol ing
u he me hods no co e ed elsewhe e (c edibili y boo s ap, boo s ap Munich, semi-s ochas ic). (3)
Two new models a e in oduced, c edibili y boo s ap in Sec ion 3.3 and collec i e semi-s ochas ic
in Sec ion 3.5. (4) We emphasise he impo ance o an algo i hmic way o model selec ion om
compe ing pee s in Sec ion 4. (5) Models based on in e nal in o ma ion only (single iangle) a e also
compa ed wi h collec i e ones (mul iple iangles and c edibili y), and he a icle in ends o con ey
he po en ial o o e sigh da a collec ion and possible applica ion on mul iple iangles. (6) Sc ip s
published by Meye s (2015) a e de eloped u he wi h new code chunks and made a ailable in he
pape ’s supplemen .
The a icle is s uc u ed as ollows: Sec ion 2con ains he exposi o y desc ip ion o insu ance da a
published by he NAIC and used o compa a i e analysis, consis ing o obse a ions o claims and
p emiums om hund eds o insu ance ins i u ions. Sec ion 3enume a es o me hodologically dis inc
and di e se ese ing models, a numbe o which a e applied widely in he insu ance indus y. Ha ing
app oached he o iginal, claims ese ing p oblem as a p obabilis ic o ecas , Sec ion 4p o ides insigh
in o i e measu es. The sec ion includes he alida ion o indi idual models om he angle o he i e
indica o s. Sec ion 5concludes he pape .
2. Da a
Open sou ce da a enables he alida ion o me hodologies on eal loss igu es. The Na ional
Associa ion o Insu ance Commissione s (NAIC) published da a ables consis ing o he names o
insu ance ins i u ions, incu ed and paid loss pe acciden yea and pe de elopmen yea , and ea ned
p emiums pe con ac yea . Meye s and Shi (2011) published hese ables along wi h he a icle.
His o ical alues applied in he p esen pape conce n he un-o iangles buil up by paid
and incu ed losses. Six di e en lines o business can be dis inguished; (1) comme cial au o and
uck liabili y and medical, (2) medical malp ac ice, (3) p i a e passenge au o liabili y and medical,
(4) p oduc liabili y, (5) wo ke s’ compensa ion and (6) o he liabili y, wi h a a iable numbe o
co po a ions con ibu ing o he da a se . Business lines co espond o homogeneous segmen s o
insu ance po olios, which a e add essed sepa a ely o he eason ha hey gene ally show dis inc
un-o beha iou . Hence, clus e s on he basis o co e age ype a e made in o de no o amalgama e
di e en un-o cha ac e is ics. Le one obse a ion mean he loss iangle associa ed o one insu ance
company, see Table 1.
Table 1. Numbe o obse a ions (insu ance ins i u ions) in he da a se s.
Business Line Numbe o Obse a ions
(1) comme cial au o and uck liabili y and medical 158
(2) medical malp ac ice 34
(3) p i a e passenge au o liabili y and medical 146
(4) p oduc liabili y 70
(5) wo ke s’ compensa ion 132
(6) o he liabili y 239
Risks 2019,7, 62 4 o 27
In ac , acciden yea s co e a 10-yea ime span be ween 1988 and 1997, wi h a 10-yea
de elopmen lag o each acciden yea . In o he wo ds, no only he iangle alues abo e
(and including) he an i-diagonal a e a ailable (Table 2), bu he en i e ec angle in each case. F om a
alida ion pe spec i e, i is c ucial ha he ac ual ul ima e claim alues, i.e., he lowe iangles a e
known (Table 3).
Table 2.
Cumula i e paid loss iangle obse ed in he pas (comme cial au o da a se , g oup code 2712).
1 2 3 4 5 6 7 8 9 10
1988 5407 14422 19063 22447 24142 25404 26829 27202 27443 27449
1989 6279 15031 21203 25697 27807 28726 29173 29375 29444
1990 7256 15923 20701 24963 27847 29274 30163 30656
1991 5028 10345 15042 18837 21708 22808 23465
1992 5712 11809 18198 22000 26306 27168
1993 7413 16798 24570 30420 33803
1994 10868 23205 31171 39702
1995 10143 24336 32406 ?
1996 9596 21831
1997 9076
Table 3.
Cumula i e paid loss iangle obse ed in he u u e (comme cial au o da a se , g oup
code 2712).
1 2 3 4 5 6 7 8 9 10
1988
1989 29459
1990 30691 30749
1991 24243 25020 25061
1992 27525 27888 27951 28042
1993 34881 35984 36313 36509 36524
1994 43225 45450 46662 47034 47027 47186
1995 38533 42552 44730 45197 45362 45516 45765
1996 27594 31228 33710 36683 36417 37068 37086 37141
1997 17689 23270 29846 33532 35205 35410 35443 35501 35540
Ul ima e claim alues ange om ze o o millions in ex eme cases, see Table 4 o paid losses,
implying magni udinal di e si y in he se o companies in e ms o ese es. In ac , only ew ou lie s
can be ound wi h nega i e o al claims, which we conside he less eliable pa o he da a se . These
ins ances ha e been aken ou o he analysis. Hence, a na u al and a no i ial ques ion is whe he
o no o apply a no malisa ion on he un-o iangles, in o de o make ese ing models easonably
compa able wi h each o he by mi iga ing he he e ogenei y o he unde lying igu es. Fo ins ance, his
can be achie ed by mul iplying each iangle by di e en cons an s o make ul ima e ese es equal o a
uni alue. Se e al pi alls accompany he scaling: applying a disc e e model such as he o e dispe sed
Poisson model ( amily) on iangles consis ing o small numbe s, he es ima ion will be useless i
he Poisson pa ame e is close enough o ze o o make u u e claim inc emen s equal o ze o wi h
high p obabili y. As a ma e o ac , his issue can be emedia ed by choosing an app op ia ely la ge
no malising cons an . The s anda disa ion o such o e dispe sed Poisson da a has been ex ensi ely
discussed in he pas in connec ion wi h s ochas ic ese ing. Each o he un-o iangle elemen s
a e no malised by a olume measu e ela ed o he acciden yea , i.e., each inc emen al o cumula i e
claim in ow
i
is di ided by a weigh
wi>
0. This exposu e olume can be he numbe o epo ed
claims in acciden yea
i
, see Wü h ich (2003). Ano he con en ion is o choose he ea ned p emium
olume o he numbe o policies, see Shi and F ees (2011).
The second and mo e con adic o y a gumen agains scaling is embedded in he da a: la ge
companies likely p o ide mo e obus claim eco ds han hei smalle coun e pa s, i.e., i is a ional

Risks 2019,7, 62 5 o 27
o ake hem in o accoun wi h la ge weigh s, which is ensu ed by he la ge ese e alues. Hence,
he ques ion is whe he o allow ins i u ions o con ibu e o he o al loss alues acco ding o hei
ese e olumes, o compose a democ a ic agg ega e obse a ion se wi h a simila con ibu ion om
each ins i u ion in e ms o ul ima e claim. An in e media e solu ion can be a noncons an escaling o
da a, which migh be conside ed by he eade . In loss ese ing calcula ions, he au ho in Shi (2015)
applies no malisa ion in o de o mi iga e he he e ogenei y o he da a. P esen calcula ions lea e
o iginal igu es as hey a e, as a consequence o ou en i ely a bi a y choice. No malised calcula ions
migh be eplica ed easily based on he supplied sc ip s.
Table 4. Ranges o paid losses pe business line.
Min. Median Mean Max.
comme cial au o and uck liabili y and medical −1 3906 50,820 2,227,000
medical malp ac ice 0 15,600 95,370 883,900
p i a e passenge au o liabili y and medical 0 19,810 818,100 91,360,000
p oduc liabili y 0 316 19430 750,300
wo ke s’ compensa ion 0 8828 101,900 1,837,000
o he liabili y −115 913 20,460 2,191,000
3. Claims Rese ing Models
In his sec ion i e concep ually dis inc modelling app oaches a e enume a ed in claims ese ing,
whe e in some o he cases, he model e e s o a me hod amily a he han a single one. These
a e he (1) boo s ap models wi h Gamma and o e dispe sed Poisson backg ound, (2) Bayesian
models using MCMC echniques, (3) c edibili y models, including a newly in oduced one combined
wi h boo s apping, (4) o iginal Munich Chain Ladde and i s boo s apped modi ica ion and (5) a
semi-s ochas ic model.
No a ions he eade equen ly encoun e s in his sec ion a e he ollowing:
I
and
J
deno e he
numbe o occu ences and de elopmen yea s in he iangles (and quad angles), i.e., hey s and o
he dimensions. Le
CI
and
CP
deno e he incu ed and paid iangles in Sec ion 3.4. A oid con using
he supe sc ip in
CI
, which s ands o ’Incu ed’, wi h he
I
numbe o ows in he iangle. I he
paid o incu ed indica i es a e no ele an om a echnical pe spec i e, hey will no be ma ked.
Supe sc ip
(k)
in connec ion wi h cumula i e iangle elemen
Ci,j
means ha he alue is ela ed o
company
k
.
Dj
s ands o he uppe un-o iangle o he
j
h company, i.e., he claims da a acqui ed
un il he ime o ese e calcula ion.
3.1. Boo s ap Models
Boo s apping in he ma hema ical sense has a p ope li e a u e and has been s udied o almos
ou decades, well be o e applica ions in insu ance eme ged. The o iginal in oduc ion da es back
o E on (1979b,1979a) as a gene alisa ion o jackkni e, enhancing he powe o a ailable sample by
esampling. In oducing an applica ion o boo s apping in insu ance, Ashe (1986) was among he i s
pape s, es ima ing dis ibu ion e o . La e , England and Ve all (1999) analyses he p edic ion e o in
conjunc ion wi h gene alised linea models (GLMs) wi h boo s apping, whils
Pinhei o e al. (2003)
p oposes an al e na i e boo s ap p ocedu e o he p e ious one, using co ec ed esiduals. The
capabili y o e o p edic ion was he p ima y ea u e o he concep which has d i en he de elopmen
o such models in he ac ua ial ield. Con a y o he simple chain ladde model, i allows o cap u e
he a iabili y o he ou come. Mo e ecen achie emen s a e Bjö kwall e al. (2009); Leong e al. (2014)
and a mo e p ac ical guide is Shapland (2016). Thus, models using boo s apping ha e become widely
applied in ac ua ial p ac ice, and s udied in nume ous wo ks. In his pape we apply he o e dispe sed
Poisson and gamma boo s ap models. Fo mo e comp ehensi e wo ks ha desc ibe he unde lying
GLM and esiduals, he eade is ad ised o see England and Ve all (2002); Wü h ich and Me z (2008).
Risks 2019,7, 62 6 o 27
3.2. Bayesian Models Using MCMC
Two me hods based on Ma ko Chain Mon e Ca lo simula ion ha ollow a Bayesian concep a e
p esen ed by Meye s (2015). The au ho made he sel -p epa ed R codes public in o de o acili a e he
eplica ion o esul s. These code chunks ha e been embedded in o he se o codes suppo ing he
analysis in he p esen a icle. Models wi h MCMC sampling a e he mos compu a ion-in ensi e ones
among he modelling p inciples he eade encoun e s he e.
3.2.1. Co ela ed Chain Ladde Model
In he co ela ed chain ladde (CCL) model incu ed claims a e he basis o calcula ion, in he
o m o cumula i e losses. The mo i a ion is o add ess he possible unde es ima ion o ul ima e
claim a iabili y in he o iginal Mack model Mack (1993). The unde lying assump ion is ha he
unknown losses
˜
Ci,j
a e go e ned by he log-no mal dis ibu ion. See Meye s (2015) o he de ailed
model assump ions.
3.2.2. Co ela ed Inc emen al T end Model
The second model is buil on he inc emen al paid loss amoun s a he han he incu ed claims,
and has a dis ibu ion skewed o he igh . Fo he in oduc ion o skew-no mal dis ibu ion see
F ühwi h-Schna e and Pyne (2010).
Meye s (2015) poin s o he issue ha skew-no mal dis ibu ion has a skewness o a unca ed
no mal a iable in he ex eme case, which s ill may no e lec he eal skewness s emming om he
loss da a, c ea ing he demand o an e en mo e skewed dis ibu ion o be applied ins ead o he
unca ed no mal.
No e ha ano he model in he e e ed monog aph, called changing se lemen a e model, may
add ess he phenomenon o accele a ing claim se lemen s, d i en by echnological changes.
3.3. C edibili y Models
The p esen subsec ion con ains he basic idea o c edibili y heo y and i s connec ion wi h claims
ese ing. By combining his idea wi h he me hodology o boo s apping, a new ese ing model
is in oduced.
Pape s Bühlmann (1967,1969) con ain he o iginal concep o expe ience a emaking. The co e
p inciple is o exploi he a ailable in o ma ion om sou ces ou side o he sample, bu somehow
ela ed o i , and combine he wo da a se s in o de o ge a mo e eliable app oxima ion o unknown
cha ac e is ics. Conside ing one business line, in o de o c ea e he claim o ecas o one pa icula
iangle, he o he un-o iangles o he same g oup a e also aken in o accoun . F om ano he angle,
he model consis s o 2 u ns, whe e we pick he isk pa ame e
ϑ
om he i s one, which de e mines
he alue sampled om he second u n. Shi and Ha man (2016) p oposes c edibili y based s ochas ic
ese ing d i en by he idea ha da a om pee coun e pa y insu e s can lead o an imp o emen o
p edic ion eliabili y.
To he analogy o he Mack Chain Ladde me hodology Mack (1993), cons uc he ollowing
model assump ions in a Bayesian hinking.
Assump ions 1. (C 1)
Le each unknown chain ladde ac o be a posi i e andom a iable
Fj
o
∀j∈
{1, . . . , J−1}, Fiindependen o Fj o ∀i6=j.
(C 2) C1,j, . . . , CI,ja e condi ionally independen o F.
(C 3)
The condi ional dis ibu ion o
Ci,j+1
Ci,j
unde he cons ain
σ{F1, . . . , Fj,Ci,1, . . . , Ci,j}
depends only on
σ{Fj,Ci,j}. Fu he mo e, condi ional expec a ion and a iance a e
E"Ci,j+1
Ci,j|Fj,Ci,j#=Fj
Risks 2019,7, 62 7 o 27
and
Va "Ci,j+1
Ci,j|Fj,Ci,j#=σ2
j(Fj)
Ci,j
.
Recall om Bayesian s a is ics ha o an a bi a y andom a iable
ξ
and a ay o obse a ions
X
, he linea Bayesian es ima o sa is ies
a gmin
ˆ
ξ:ˆ
ξ=∑
i
aiXi+cons
E(ˆ
ξ−ξ)2|X
. Also ecall om Gisle and
Wü h ich (2008) he De ini ion 2o he c edibili y based p edic o and a ele an Theo em 3.
De ini ion 2. The c edibili y based p edic o o he ul ima e claim Ci,Jgi en DIis
Cc ed
i,J=Ci,I−i+1
J−1
∏
j=I−i
Fc ed
j,
whe e
Fc ed
j=a gmin
ˆ
Fj:ˆ
Fj=
I−j
∑
i=1ai,jYi,j+cons
Eh(ˆ
Fj−Fj)2|B(j)i
and Yi,j=Ci,j+1
Ci,j,B(j) = {Ci,k:i+k≤I+1, k≤j} ⊂ DI he subse o uppe iangle in o ma ion.
Gi en he mul iplica i e s uc u e o he ul ima e claim es ima o i may no be app op ia e o call
i simply a c edibili y es ima o , which is by de ini ion a linea unc ion o he obse a ions, hence he
c edibili y based appella ion.
Theo em 3. The c edibili y es ima o s o he de elopmen ac o s a e gi en by
Fc ed
j=αjˆ
Fj+ (1−αj) j,
whe e
ˆ
Fj=
I−j
∑
i=1Ci,j+1
I−j
∑
i=1Ci,j
,
j=E[Fj]
,
αj=
I−j
∑
i=1Ci,j
I−j
∑
i=1Ci,j+σ2
j
τ2
j
,
σ2
j=E[σ2
j(Fj)]
and
τ2
j=Va [Fj]
. The la e wo a e he
s uc u al pa ame e s (o c edibili y ac o s and hei quo ien , κj=σ2
j
τ2
j
is he c edibili y coe icien ).
Fo he mean squa e e o o p edic ion i is also ue ha msep(Fc ed
j) = (1−αj)τj, see De ini ion 17.
P oo : See Gisle and Wü h ich (2008).
Da a conce ning he c edibili y ac o in pa icula a e no a ailable in gene al. In he
p esen a icle hese pa ame e s a e app oxima ed on he basis o claim iangles published by
se e al companies.
F om egula o y pe spec i e i is ex emely impo an o unde s and how he in lowing da a
can be exploi ed in o de o suppo he insu ance ins i u ions wi h eliable in o ma ion. Financial
egula o y au ho i ies end o collec an inc easing amoun o de ailed da a o he pu pose gaining
insigh in o he insu ance ins i u ions’ sol ency. In Eu ope, o ins ance, he Eu opean Insu ance and
Occupa ional Pensions Au ho i y (EIOPA) shows guidance o local egula o s and collec s submissions
o s a is ical and inancial da a om se e al coun ies. Besides anspa ency, he in o ma ion enables
he adequa e suppo o co po a ions by p o iding hem wi h p ocessed da a o hei bene i . This is
whe e c edibili y models ha e an un apped po en ial. The ques ion whe he o no o use collec i e
expe ience o imp o e indi idual app oxima ions is pa icula ly ele an due o he ac ha egula o y
au ho i ies collec as amoun o in o ma ion om insu ance companies. Thus, he p ocessed da a
migh be o alue o sha e wi h he con ibu o s, enabling mo e p ecise sol ency e alua ions.
Risks 2019,7, 62 8 o 27
Le
C(k)
i,j
s and o he cumula i e paymen o incu ed claim alue wi h occu ence yea
i
and
de elopmen yea
j
wi h espec o company
k
. In gene al, o simplici y’s sake i is supposed ha o
each insu ance ins i u ion he iangle dimensions a e equal, mo eo e ,
I=I(1)=I(2)=. . . =I(n)
.
n
deno es he numbe o companies obse ed in a homogeneous isk g oup and
I(k)
he dimension
o he
k
h iangle. The pa ame e es ima ion o c edibili y ac o s is cons uc ed in acco dance wi h
Sec ion 4.8 in Bühlmann and Gisle (2006). Le index
j
be ixed and le
S(k)
jk∈ {
1,
. . .
,
n}
be de ined
o each iangle as
S(k)
j=1
I−j−1
I−j
∑
i=1
C(k)
i,j




C(k)
i,j+1
C(k)
i,j−
I−j
∑
=1C(k)
,j+1
I−j
∑
=1C(k)
,j





2
.
Obse e ha
S(k)
j=1
I−j−1
I−j
∑
i=1
C(k)
i,j




C(k)
i,j+1
C(k)
i,j−Fj+Fj−
I−j
∑
=1C(k)
,j+1
I−j
∑
=1C(k)
,j





2
=
=1
I−j−1
I−j
∑
i=1






C(k)
i,j

C(k)
i,j+1
C(k)
i,j−Fj

2
−
I−j
∑
=1
C(k)
,j




I−j
∑
=1C(k)
,j+1
I−j
∑
=1C(k)
,j
−Fj




2





,
which implies ha
E[S(k)
j|Fj] = σ2
j(Fj)
in line wi h Assump ion 1. Hence,
E[S(k)
j] = E[E[Sk|Fj]] =
E[σ2
j(Fj)] = σ2
j
o each
j
, i.e.,
S(k)
j
p o ides an unbiased es ima o o
σ2
j
. Taking he a e age o
S(k)
j
alues o all he companies esul s in an unbiased es ima o o σ2
j:
ˆ
σ2
j=1
n
n
∑
k=1
1
I−j−1
I−j
∑
i=1
C(k)
i,j




C(k)
i,j+1
C(k)
i,j−
I−j
∑
l=1C(k)
l,j+1
I−j
∑
l=1C(k)
l,j





2
. (1)
I can also be shown wi h u he calcula ions ha ˆ
ˆ
τ2
jis an unbiased es ima o o τ2
j:
ˆ
ˆ
τ2
j=cj





n
n−1
n
∑
k=1
I−j
∑
i=1C(k)
i,j
n
∑
l=1
I−j
∑
i=1C(l)
i,j





I−j
∑
i=1C(k)
i,j+1
I−j
∑
i=1C(k)
i,j
−
n
∑
l=1
I−j
∑
i=1C(l)
i,j+1
n
∑
l=1
I−j
∑
i=1C(l)
i,j





2
−n·ˆ
σ2
j
n
∑
k=1
I−j
∑
i=1C(k)
i,j






(2)
wi h cj=n−1
n


n
∑
k=1
I−j
∑
i=1C(k)
i,j
n
∑
l=1
I−j
∑
i=1C(l)
i,j·

1−
I−j
∑
i=1C(k)
i,j
n
∑
l=1
I−j
∑
i=1C(l)
i,j





−1
.
Pa ame e
τj
needs ex a a en ion ha ing obse ed ha he es ima o below can a ain nega i e
alues, no only in an ex emely heo e ical sense, bu on he eal wo ld ajec o ies, as well. Fo ha
eason, le he app oxima ion be capped by 0 om below.
ˆ
τ2
j=max 0, ˆ
ˆ
τ2
j. (3)
Risks 2019,7, 62 15 o 27
o in highe dimension, his p ope y will always be alid, excep ha in he la e case ans o ma ion
has o be ca ied ou wi h condi ional dis ibu ions on he p e ious coo dina es, see
Rosenbla (1952)
.
Now le
ˆ
Fi
be he p edic ion gi en o
Fi
. Rega dless o he ques ion whe he he s ochas ic me hod has
a dis ibu ion o i is dis ibu ion- ee, he empi ical p edic i e dis ibu ion can always be gene a ed by
d awing andomly o boo s apping a su icien amoun o samples. Fo a ixed ese ing me hod, each
quad angle is associa ed wi h one
ˆ
Fi
and he combina ion o hese is used o back es ing. Coinciding
wi h he eal dis ibu ion
Fi
has a necessa y condi ion such ha
ˆ
Fi(xi)∼U(
0,1
)
. In i s analysis o
anking his og ams Hamill (2001) in oduced a coun e example wi h biased p edic ion and uni o m PIT
a he same ime, disp o ing he uni o m p ope y as a sa is ying condi ion. The pape highligh s he
possible allacies and misin e p e a ions o quali ies ha he ank his og am ensembles may conceal.
P oceed o he implemen a ion o he PIT concep in o he claims ese ing model amewo k.
A ce ain se o companies ela ed o one business line has
n
claims his o y quad angles, e.g., he 132
ins i u ions o wo ke s’ compensa ion. Fix an a bi a y ese ing model and pe o m he ul ima e
claim alue es ima ion o each o he iangles, ollowed by he obse a ion o ac ually occu ed o al
claims om he lowe iangles. The la e s and o he ealisa ion om he eal unknown dis ibu ion,
whe e he alue is p ac ically unknown o u u e es ima ion, bu known o pas da a enabling
alida ion. The esul is
n
pai o
{ˆ
Fi
,
xi}
alues, de e mining he PIT alues
ˆ
F1(x1)
,
ˆ
F2(x2)
,
. . .
,
ˆ
Fn(xn)
and hence, he his og am. Should he se consis o an ex emely low numbe o da a poin s, hen
he applica ion o a andomised PIT o a non- andomised uni o m e sion o PIT is mo e p ope ,
see Czado e al. (2009).
Gene ally, he de ia ion o he PIT his og am om uni o mi y e lec s he dispe sion o he
p edic i e model. A
∩
-shaped his og am can be ansla ed as an o e dispe sed p edic ion wi h
excessi ely wide p edic ion in e al, i.e., o e ly hea y ailed dis ibu ion. By con as ,
∪
-shaped PIT
sugges s ha he p edic ion shall be unde dispe sed wi h na ow p edic ion in e al, i.e., ligh e ail
han he unde lying dis ibu ion would imply. In he la e case, a iabili y o he eal go e ning
dis ibu ion exceeds he a iabili y o he model, whils i is he o he way a ound in he o me
case. Going o wa d, eal-li e da a and models esul in a his og am o less pu e shapes, which a e
combina ions o he men ioned wo ins ances: skewed
∩
-shaped PIT o en i ely biased owa ds 0
(o 1), o ins ance.
Each igu e in he ollowing subsec ions uses consis en abb e ia ions o indica e ese ing
me hods, see Table 9.
Table 9. Legends o ese ing models.
Abb e ia ion Model Subsec ion
boo .gamma boo s ap model wi h gamma dis ibu ion 3.1
boo .od.pois boo s ap model wi h o e dispe sed Poisson dis . 3.1
boo s ap.munich Munich Chain Ladde wi h boo s apping 3.4
CCL co ela ed chain ladde model 3.2
CIT co ela ed inc emen al end model 3.2
c ed.boo s ap.od.pois c edibili y boo s ap wi h o e dispe sed Poisson dis . 3.3
munich Munich Chain Ladde (o iginal) 3.4
SemiS collec i e semi-s ochas ic model 3.5
Resul s o he wo business lines on Figu es 2and 3sugges simila in e ences. I becomes ins an ly
ob ious ha none o he ese ing models p o ide unbiased es ima ion o he ul ima e claim. In ac ,
he ques ion is wha exac ly goes w ong wi h each one o hem.
The Munich chain ladde (MCL) is an odd one ou , he only model discussed in he p esen
a icle, which is no sui able o p oducing p edic i e dis ibu ion, and wo ks only o a ac ion
o unde lying un-o iangles, hus he lowe amoun o equencies. Since MCL esul s in one
single
ˆ
UC1,i
p edic ion, he
ˆ
Fi(z) = (1, z>ˆ
UC1,i
0, o he wise
equencies a e e lec ed on he MCL his og ams.

Risks 2019,7, 62 16 o 27
Besides, bo h ela ed his og ams p o e ha in each case, MCL consis en ly unde es ima ed he
ac ual ou come. The co ela ed inc emen al end (CIT) model has a simila de iciency, esul ing in
unde dispe sed p edic ions wi h one-sided biasedness.
The boo s apped e sion o MCL and co ela ed chain ladde (CCL) models a e bo h on he
o e dispe sed spec um. The o me ends o esul in a symme ic PIT his og am, sugges ing ha
he expec ed alue o he ul ima e claim o ecas is close o he expec a ion om he eal dis ibu ion,
which implies a signi ican imp o emen compa ed o he o iginal MCL. PIT alues o CCL model a e
biased o he le , as a sign o unde es ima ion o ul ima e claims.
The hi d g oup ha ing simila esul s consis s o boo s ap gamma and o e dispe sed Poisson
and c edibili y boo s ap o e dispe sed Poisson models, ha ing
∪
-shaped PIT, i.e., na ow p edic ion
in e als. Fu he mo e, biasedness can be obse ed o he le , indica ing an unde es ima ion o he
eal ul ima e claims. The collec i e semi-s ochas ic app oach pe o ms ela i ely well in e ms o PIT
uni o mi y. We may conclude ha he la e ou models ha e he bes quali ies om a PIT pe spec i e.
munich
SemiS
CCL
CIT
c ed.boo s ap.od.pois
boo .gamma
boo .od.pois
boo s ap.munich
0.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00
0
20
40
60
0
20
40
60
0
20
40
60
alue
equency
His og ams o PIT alues
Figu e 2. His og ams o PIT alues om he comme cial au o da a.
Risks 2019,7, 62 17 o 27
munich
SemiS
CCL
CIT
c ed.boo s ap.od.pois
boo .gamma
boo .od.pois
boo s ap.munich
0.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00
0
20
40
60
0
20
40
60
0
20
40
60
alue
equency
His og ams o PIT alues
Figu e 3. His og ams o PIT alues om he p i a e passenge au o liabili y da a.
4.2. Con inuous Ranked P obabili y Sco e
Sco es suppo he quali y e i ica ion o p obabilis ic o ecas s based on he dis ibu ion
es ima es and obse ed ou comes. The e a e sco es wi h a wide spec um o ypes used o bo h
disc e e and absolu ely con inuous dis ibu ions, such as B ie sco e, loga i hmic sco e, sphe ical
sco e, con inuous anked p obabili y sco e, ene gy sco e, e c. Fo an ex ensi e in oduc ion see
Gnei ing and Ra e y (2007)
, including a me eo ological case s udy. In spi e o he applicabili y in
o he disciplines, o ou knowledge, sco es ha e been esea ched o a limi ed ex en in pee - e iewed
jou nals in he con ex o echnical ese ing in insu ance. A simula ion-based me hodology is
cons uc ed in A a ó e al. (2017) o he selec ion om compe ing models. In he ex ension o
eg ession models in non-li e a emaking o gene alised addi i e models o loca ion, scale, and shape
(GAMLSS),
Klein e al. (2014)
compa es a ious models h ough hei sco e con ibu ions. B ie sco e,
loga i hmic sco e, sphe ical sco e and de iance in o ma ion c i e ion (DIC) is used o Poisson,
ze o-in la ed Poisson and nega i e binomial assump ions, whils CRPS is also calcula ed o h ee
ze o-adjus ed models. Using a eal-li e da a se , Tee e al. (2017) compa es he o e dispe sed Poisson,
gamma and log-no mal models in he boo s ap amewo k and hei esidual adjus men s using he
Dawid-Sebas iani sco ing ule (DSS). In modelling o claim se e i ies and equencies in au omobile
insu ance Gschlössl and Czado (2007) conside s sco es o model compa ison, which ei he apply o
exclude spa ial and ce ain claim numbe componen s.
De ini ion 11
(Sco e)
.
Gene ally, le
S(F
,
x):P ×Ω→R
be a eal alued unc ional wi h he wo possible
excep ions o
−∞
and
+∞
, whe e
P
s ands o a amily o p obabili y measu es and
Ω
o a sample space.
The i s a gumen can be in e p e ed as a p edic ion, whils he second one as a ealisa ion.
De ini ion 12 (Expec ed sco e).Le he expec ed sco e be S(P,Q) = RS(P,ω)dQ(ω).
Risks 2019,7, 62 18 o 27
Wi hou loss o gene ali y, suppose ha o ecas
P1
is no wo se han
P2
, i
S(P1
,
x)≥S(P2
,
x)
in
expec a ion, whe e
x
is go e ned by p obabili y measu e
Q
. Le a sco ing ule be p ope i
S(P
,
Q)≤
S(Q
,
Q)
o
P
,
Q∈ P
amily o dis ibu ions, see Be na do (1979); S aël on Hols ein (1970), o ins ance.
Fu he mo e, le a sco ing ule be s ic ly p ope i S(Q,Q) = S(P,Q)i and only i Pd
=Q.
Di e en dis ibu ions abo e a e analogous o di e en o ecas e s, o using insu ance claims
p edic ion e minology, he compe ing models o ese ing. Gi en ha hese models may ei he esul
in disc e e o in absolu ely con inuous p edic i e dis ibu ions, i is o high p ac ical ele ance o selec
an app op ia e sco e unc ional lexible enough o cope wi h bo h cases. The ollowing sco ing ule is
mo e obus han he loga i hmic o B ie sco es, and equi es p ac ically no assump ion wi h ega ds
o he dis ibu ion obse ed, le i be ei he disc e e o no .
De ini ion 13 (Con inuous anked p obabili y sco e (CRPS)).
CRPS(F,x) = −
∞
Z
−∞F(u)−χ{x≤u}2du,
whe e indica o unc ion χ{x≤u}equals 1 i x ≤u and 0 o he wise.
Some o he a icles de ine posi i e CRPS, howe e , he e we will use i s nega i e coun e pa .
CRPS can be conside ed as gene alisa ion o he B ie sco e (BS); i is he in eg al o BS o e he domain
o all h eshold alues, see He sbach (2000). In o he wo ds, he e is a di ec connec ion be ween he
CRPS and an e en -no-e en sco e. Vice e sa, he concep o ene gy sco e (ES) can be hough o as
he gene alisa ion o CRPS.
De ini ion 14
(Ene gy sco e)
.ESβ(F
,
x) = 1
2EF|X−X0|β−EF|X−x|β
wi h an a bi a y cons an
β∈(
0,2
)
. Le
X
and
X0
be independen copies om p obabili y dis ibu ion
F
. Fo
β=
1,
ESβ(F
,
x) =
CRPS(F,x), see Székely and Rizzo (2005).
On a se o obse a ions and co esponding p edic i e dis ibu ions, he goal is o maximise
he mean sco e, esul ing in a anking o compe ing p edic i e models h ough maximising he
expec ed u ili y:
Smodel =1
n
n
∑
i=1
S(Pmodel
i h company,xi h company). (9)
Le
Pmodel
i h company =ˆ
Fj=Pmodel
i h company ˆ
UC1,i, . . . , ˆ
UCM,i
s and o he empi ical p edic i e
dis ibu ion de i ed o company
i
on he basis o a ixed ese ing model, whe e
ˆ
UCk,i
deno es he
k
h
andomly gene a ed o al ul ima e claim o company
i
(
i=
1,
. . .
,
n
). We ha e seen in he discussion o
PIT ha dis ibu ion- ee models can also be used o gene a e p edic i e dis ibu ion by boo s apping.
Fu he mo e, ull quad angles ha con ain ac ual ul ima e claims enable back es ing. Analy ical
o mulae can a ely be de i ed o CRPS, no o men ion he p ac ical models o claims p edic ion,
al hough, i is easible i he dis ibu ion
F
is no mal, see Gnei ing and Ra e y (2007). A easonable
ques ion is how sensi i ely he mean sco e is exposed o ex emely inapp op ia e models, i.e., i he
sample size is ela i ely small and an ou s anding sco e alue is in ol ed. Fo ha eason he comple e
scale o sco e ou comes is p oposed o be analysed in he o m o a boxplo , he
−log (−sco e)
plo ed
o he sake o be e isual unde s anding, see Figu es 4and 5. The highe he boxplo , he be e he
pe o mance o o ecas acco ding o he sco ing ule.
CRPS is no de ined in ela ion o he MCL model due o he lack o p edic i e dis ibu ion.
On Tables 10 and 11 he mean CRPS alues a e demons a ed, which de e mine he anking o
compe ing models. In o de o see whe he an ex eme alue has in luenced he mean ou come
(de ined in Equa ion (9)), he median sco es a e added o he second column. Rese e calcula ions in
Risks 2019,7, 62 19 o 27
acco dance wi h he CIT model on bo h comme cial and p i a e passenge po olios show sco es o
ou s andingly la ge absolu e alue, implying ha o ecas s on some o he companies pe o med poo ly.
−15
−10
−5
boo .gamma
boo .od.pois
boo s ap.munich
CCL
CIT
c ed.boo s ap.od.pois
munich
SemiS
me hod
sco e alues (on loga i hmic scale)
Boxplo s o CRPS alues
Figu e 4. Boxplo s o CRPS alues om he comme cial au o da a.
Table 10. A e age and median CRPS alues om he comme cial au o da a.
Mean.CRPS Median.CRPS SampleSize
CIT −1,805,000 −5082 71
CCL −11,880 −2260 71
boo .gamma −2990 −662 71
boo .od.pois −9404 −655 71
munich 0
boo s ap.munich −20,970 −2094 71
SemiS −4573 −1073 71
c ed.boo s ap.od.pois −2698 −733 71
−20
−15
−10
−5
boo .gamma
boo .od.pois
boo s ap.munich
CCL
CIT
c ed.boo s ap.od.pois
munich
SemiS
me hod
sco e alues (on loga i hmic scale)
Boxplo s o CRPS alues
Figu e 5. Boxplo s o CRPS alues om he p i a e passenge au o liabili y da a.
Risks 2019,7, 62 20 o 27
Table 11. A e age and median CRPS alues om he p i a e passenge au o liabili y da a.
Mean.CRPS Median.CRPS SampleSize
CIT −11,410,000 −10,350 73
CCL −247,900 −7163 73
boo .gamma −22,620 −780 73
boo .od.pois −23,200 −831 73
munich 0
boo s ap.munich −132,800 −2108 73
SemiS −31,760 −1644 73
c ed.boo s ap.od.pois −101,400 −1253 73
In he calcula ion on he comme cial au o da a, he bes pe o ming model has been he c edibili y
boo s ap o e dispe sed Poisson one, using expe ience a emaking, whils applied on he p i a e
passenge au o da a i has pe o med behind he o he boo s ap me hods. The semi-s ochas ic claims
ese ing echnique becomes he hi d one applied on each o he da a se s. Boo s ap MCL and CCL
can be anked behind hese ou models, and he CIT model yields signi ican ly lowe mean sco e
alues han he p e ious ones.
4.3. Co e age and A e age Wid h
The in en ion o he ollowing de ini ion is o g asp he consis ency be ween he p obabili y o
alling ou o a gi en in e al assuming a p edic i e dis ibu ion, and he eal dis ibu ion. In o he
wo ds, o ind he likelihood ha a andom a iable o measu e
Q
coincides wi h a cen al p edic i e
in e al de e mined by
F
. Me eo ology ela ed discussion can be ound in Ba an e al. (2013). Fo an
applica ion om he inancial sec o see Ch is o e sen (1998), add essing condi ional in e al o ecas s
and asymme ic in e als, whils he closes one o s ochas ic claims ese ing can be ound in
A a ó and Ma inek (2015); A a ó e al. (2017)
. Bo h on co e age and a e age wid h he mos de ailed
s udy is belie ably p o ided by Gnei ing e al. (2007).
De ini ion 15
(Co e age
α
)
.
Le
Q
s and o he p obabili y measu e go e ning he eal dis ibu ion o he
ul ima e claim, and
F
he o ecas dis ibu ion.
QF−11−α
2,F−11+α
2
is he cen al
α
p edic ion in e al
o F gi en Q.
The de ini ion abo e esul s in he obse a ions coinciding wi h he in e al bounded by he
lowe and uppe quan iles o he p edic i e dis ibu ion. In o de o gi e he concep meaning in he
con ex o un-o iangles and ul ima e claims, condi ional dis ibu ions ha e o be de ined, gi en he
uppe ingles. Suppose ha
Dj
is an uppe iangle associa ed wi h he
j
h company. Fix an a bi a y
model discussed in Sec ion 3, o be applied on each iangle o claim o ecas ing pu poses. Le
Qηj|Dj
s and o he ul ima e claim dis ibu ion esul ed by he chosen model gi en
Dj
, whils
Qξj|Dj
is he
ac ual condi ional dis ibu ion. Wi h he p e ious no a ions, he de ini ion o co e age con e s in o
PQξj|DjQ−1
ηj|Dj1−α
2<ξj<Q−1
ηj|Dj1+α
2. (10)
I is easy o see ha i
ηj
has iden ical dis ibu ion o
ξj
, which means a pe ec p edic ion,
exp ession Equa ion (10) equals o
α
o any
α
alue in
(
0,1
)
. Now assume ha he model de e mines
he p edic i e dis ibu ion gi en
Dj
in he o m o a andom sample
η1,j
,
. . .
,
ηM,j
o
j∈ {
1,
. . .
,
n}
and
a bi a ily la ge posi i e in ege
M
. Le
Qu(η•j
,
p)
s and o he
p
-quan ile o he empi ical dis ibu ion
de e mined by sample
η1,j
,
. . .
,
ηM,j
. Fo
α∈(
0,1
)
he cen al p edic ion in e al’s app oxima ion is
1
n
n
∑
j=1χ{Qu(η•j,1−α
2)<ξj<Qu(η•j,1+α
2)}
, using
χA
o he no a ion o he indica o unc ion o e en
A
. Tha is

Risks 2019,7, 62 21 o 27
gi en by gene a ing an ul ima e claim andom sample on he basis o he ixed model, condi ionally on
Dj
o each
j∈ {
1,
. . .
,
n}
. In o de o achie e con e gence, inc ease he sample size
M
a bi a ily la ge.
As an ancilla y measu e besides co e age, a e age wid h o p edic ion co e s he expec ed
di e ence be ween he lowe and uppe
p
-quan iles, a alue exp essed in ac ual paymen . Al e na i ely
i is called he sha pness o he p edic i e e alua ion. The na owe he wid h, he be e he p edic ion.
De ini ion 16
(A e age wid h (sha pness))
.
Le
Qξj|Dj
be he condi ional p obabili y measu e o he ul ima e
claim based on a ixed model, p o ided ha he uppe iangle is
Dj
. Suppose he e is an unde lying mul i a ia e
dis ibu ion QDgo e ning uppe iangle D. The a e age wid h o he model is
EQDQ−1
ξj|Dj1+α
2−Q−1
ξj|Dj1−α
2|Dj.
Simila ly o he p ac ical e alua ion o co e age, gene a e o each uppe iangle
Dj
a
su icien ly la ge amoun o andom ul ima e claim alues, whe e
M
deno es an in ege la ge
enough. Hence, he sha pness o he model gi en he se o un-o iangle obse a ions is
1
n
n
∑
i=1Qu(η•j,1+α
2)−Qu(η•j,1−α
2).
In he calcula ions wi h NAIC da a, each wid h in he a e age calcula ion o mula abo e is
no malised in e e y iangle wi h he ealised incu ed bu no epo ed (IBNR) alue. Tha no malising
alue s ands o he lowe iangle sum in case o an inc emen al poin o iew, o , in o he wo ds,
he ul ima e claim educed by he paymen al eady a ailable in he uppe iangle. Hence, i e lec s
he a e age span in e al as a uni o ealised IBNR alue.
In he ideal case o coinciding p edic i e and ac ual p obabili y measu es
Pd
=Q
, co e age
α
equals o
α
o any gi en
α∈(
0,1
)
. Tables 12 and 13 calcula ed on he basis o wo
α
alues p o e ha
he applied models p oduce co e ages ha a e a om ideal. The o iginal MCL me hod does no
ha e any co e age o a e age wid h ou pu due o lack o p edic i e dis ibu ion. CIT and boo s ap
MCL show he mos inapp op ia e cha ac e is ics, in essence wi h degene a e co e ages, ei he equal
o close o 0 o 1. CCL pe o ms be e in he sense ha he lowe
α=
67% co e age is 84% and 94%
in he wo cases. The c edibili y boo s ap and o iginal boo s ap gamma and o e dispe sed Poisson
me hods esul in simila co e age and a e age wid h: Measu es a e balanced among hese h ee
models, and ha e he na owes sha pness. The collec i e semi-s ochas ic me hod esul s in co e ages
closes o iden i y, howe e , a he cos o ha ing wide a e age wid h alues.
Table 12. Co e age and a e age wid h om he comme cial au o da a.
67% Co e 90% Co e 67% Wid h 90% Wid h SampleSize
CIT 0.00 0.00 0.01 0.02 71
CCL 0.84 1.00 4.96 10.60 71
boo .gamma 0.45 0.79 1.04 2.43 71
boo .od.pois 0.45 0.78 1.00 2.13 71
munich 0.00 0.00 0.00 0.00 56
boo s ap.munich 1.00 1.00 37.83 113.90 71
SemiS 0.73 0.99 1.51 3.55 71
c ed.boo s ap.od.pois 0.51 0.75 1.13 2.34 71
Risks 2019,7, 62 22 o 27
Table 13. Co e age and a e age wid h om he p i a e passenge au o liabili y da a.
67% Co e 90% Co e 67% Wid h 90% wid h SampleSize
CIT 0.00 0.00 0.00 0.01 73
CCL 0.94 1.00 5.38 10.30 73
boo .gamma 0.30 0.59 0.59 1.14 73
boo .od.pois 0.32 0.57 0.58 1.12 73
munich 0.00 0.00 0.00 0.00 61
boo s ap.munich 0.97 1.00 98.43 411.20 73
SemiS 0.59 0.93 0.97 2.33 73
c ed.boo s ap.od.pois 0.37 0.59 0.58 1.03 73
4.4. Mean Squa e E o o P edic ion
Measu ing he expec ed squa ed dis ance be ween he p edic o and he ac ual ou come has
been pa o he con en ional way o ac ua ial ese ing. We shall dis inguish he condi ional e o
gi en he
D
uppe iangle and he uncondi ional one. E en ually, in he judgmen o he speci ic
model, he uncondi ional e sion is assessed in o de o measu e he a e age pe o mance o he
model wi hou cons aining i on a ixed un-o iangle. Se e al a icles b eak down he de ini ion on
occu ence yea s, i.e., inspec ing
Ci,J
eal and
ˆ
Ci,J
es ima ed ul ima e claims o occu ence yea
i
, o he
u u e ( ese e) pa o he claims
Ci,J−Ci,J−i+1
eal and
ˆ
Ci,J−Ci,J−i+1
. Wi hou loss o gene ali y,
he de ini ion in he p esen pape is o malised o o al ul ima e claims
UC =I
∑
i=1Ci,J
. Fo he sake o
aceabili y, he de ini ion con ains he no a ion o
ξi∼Qi
ul ima e claim o company
i
and
ηi∼Fi
ul ima e claim p edic ion. Fu he mo e,
Di
s ands o he
σ
- ield gene a ed by he uppe iangle,
as al eady used p e iously.
De ini ion 17
(Mean squa e e o o p edic ion (MSEP))
.
The condi ional mean squa e e o o p edic ion
o es ima o ηi o ξigi en Diis
msepξi|Di(ηi) = Eh(ξi−ηi)2|Dii.
The uncondi ional MSEP is
msepξi(ηi) = Eh(ξi−ηi)2i=EhEh(ξi−ηi)2|Diii.
I is easy o see ha MSEP can be spli in o
E(ξi−ηi)2|Di=Va [ξi|Di] + (ηi−E[ξi|Di])2
,
whe e he i s e m is he a iance o he p ocess, whils he second e m e lec s he es ima ion e o .
Simila ly o he condi ional e sion,
E(ξi−ηi)2=E[Va [ξi|Di]]+E[ηi−E[ξi|Di]]2
. In conjunc ion
wi h some o he pa ame e ic models, MSEP can be de i ed in an analy ical o m, see Mack (1993) o
he o iginal Mack model and Buchwalde e al. (2006) in a ime se ies me hod e isi ing he esul o
he p e ious a icle.
Resul s calcula ed he e di e om he o iginal de ini ion in he sense ha each ou come is
no malised by he ul ima e ese e. The eason co esponds o he one discussed in Sec ion 2,
i.e., he magni udinal disc epancies among he claims in dis inc companies. Hence, ins ead o
E(ξi−ηi)2|Di
es ima e
Eh(ηi
ξi−1)2|Dii
. D aw a andom sample om he dis ibu ion o
ηi
de e mined by he o ecas ing model, and he eal obse ed ealisa ion o
ξi
;
ˆ
UC1,i
,
. . .
,
ˆ
UCM,i
and
UCi
.
S a emen 18. 1
M
M
∑
j=1
(ˆ
UCj,i−UCi)2
UC2
i
is an unbiased es ima o o E h(ηi
ξi−1)2|Dii.
Risks 2019,7, 62 23 o 27
P o ing he s a emen wo ks by aking expec a ion
E"1
M
M
∑
j=1
(ˆ
UCj,i−UCi)2
UC2
i|Di#=E"(ˆ
UC1,i−UCi)2
UC2
i|Di#=E"(ηi−ξi)2
ξ2
i|Di#.
Finally, he MSEP es ima o o he model, uncons ained on he uppe iangle is he a e age o
he elemen s calcula ed o each company
i
. Howe e , should he mean be domina ed by any ex eme
alue, he median o condi ional MSEPs is included in he calcula ion esul s. Obse e he di e ing
alues on Tables 14 and 15, suppo ing he ac ua y wi h insu icien backg ound in o de o de e mine
eliable me hods on he da a se s. Ex eme alues may easily occu whe e e y high squa es a e
possible wi h a low p obabili y. Taking exclusi ely he MSEP in o accoun in model decisions is clea ly
no he p ope way o anking hem and does no p o ide in o ma ion conce ning he app op ia eness
o p edic i e dis ibu ion.
Table 14. Mean squa e e o o p edic ion om he comme cial au o da a.
Mean.Msep Median.Msep SampleSize
CIT 127.7 1.0 71
CCL 445.1 6.2 71
boo .gamma 352.6 0.2 71
boo .od.pois 6137.0 0.1 71
munich 1.9 0.0 52
boo s ap.munich 6235000.0 16.5 71
SemiS 4.3 1.7 71
c ed.boo s ap.od.pois 3112.0 0.2 71
Table 15. Mean squa e e o o p edic ion om he p i a e passenge au o liabili y da a.
Mean.Msep Median.Msep SampleSize
CIT 61800000.0 1.0 73
CCL 25.9 12.9 73
boo .gamma 38450.0 0.1 73
boo .od.pois 874.0 0.1 73
munich 2.1 0.0 59
boo s ap.munich 2791000.0 3.1 73
SemiS 14.0 6.5 73
c ed.boo s ap.od.pois 7.7 0.1 73
4.5. Ranking Algo i hm
We summa ise he algo i hmic s eps o he anking amewo k. Suppose ha he iangles s em
om one homogeneous isk g oup.
1.
S ochas ic o ecas phase. Fo
me h ∈
{ boo s ap gamma, boo s ap ODP, ...}, o
j∈
{se o
companies}, gene a e Mul ima e claim alues.
Resul : ˆ
UC1,j,me h, . . . , ˆ
UCM,j,me h ∀j∀me h.
2.
Back es phase. Fo
me h ∈{ boo s ap gamma, boo s ap ODP, ...}
,
j∈
{se o companies}
calcula e PIT, CRPS, co e age, sha pness, MSEP om ˆ
UC1,j,me h, . . . , ˆ
UCM,j,me h and eal UCj.
Resul : (a)
PITj,me h ∈(
0,1
)
, (b)
CRPSj,me h ∈R−
, (c)
co e j,me h,p∈(
0,1
)
, (d)
sha pj,me h,p∈R+
,
(e) MSEPj,me h ∈R+∀j∀me h ∀p∈ {67%,90%}.
3.
Ranking phase. Sepa a e compa ison o me ics (a)-(e). Combined compa ison o me ics ( ). (We
assume o compa e 7 s ochas ic me hods, excluding MCL.)
(a)
Calcula e he en opy
PIT·,me hi
o each se
{PITj,me h :∀j}
and o de
PIT·,me h1>. . . >
PIT·,me h7. Assign ank i o me hi, he lowe he ank he be e he pe o mance.
Risks 2019,7, 62 24 o 27
(b)
Calcula e a e age CRPS and o de
CRPS·,me h1>. . . >CRPS·,me h7
. Assign ank
i
o
me hi
.
(c)
Calcula e co e age alues
co e ·,me hi,p
and o de
(co e ·,me h1,p−p)2<. . . <
(co e ·,me h7,p−p)2
o each
p
and assign ank
i
o
me hi
. Fo each me hod, ake he
a i hme ic a e age o he wo anks.
(d)
Calcula e sha pness alues
sha p·,me hi,p
and o de
sha p·,me h1,p<. . . <sha p·,me h7,p
o
each
p
and assign ank
i
o
me hi
. Simila ly o co e age ake he a e age o he wo anks
o each me hod.
(e) Calcula e MSEP alues and ank as o sha pness.
( ) Fo me h ∈{ boo s ap gamma, boo s ap ODP, ...} de e mine
ank o al
me hi= ankPIT
me hi+ ankCRPS
me hi+ ankco e
me hi+ anksha p
me hi+ ankMSEP
me hi
. Me hod
k
pe o ms
be e han li ank o al
me hk
< ank o al
me hl.
Obse e ha he me ics ha e iden ical weigh s in anking, which is an a bi a y choice. These
s eps desc ibe a combined anking based on di e en cha ac e is ics. Howe e , his anking should
no be applied wi hou sc u inising PIT, CRPS, e c. sepa a ely in o de o see he exac weakness o a
ese ing me hod. The anking esul s pe business line can be ound on Table 16. Obse e ha in
con as o all o he models, he boo s ap gamma one ne e anked wo se han 3.
Table 16. Combined ankings o s ochas ic ese ing me hods pe business line. (Excluding MCL.)
Comau o Medmal Ppau o P odliab Wkcomp O hliab
CIT 5 4 7 5 7 6
CCL 6 6 5 6 5 4
boo .gamma 2 1 3 1 2 1
boo .od.pois 4 3 4 2 3 2
boo s ap.munich 7 7 6 7 6 7
SemiS 1 5 2 3 1 3
c ed.boo s ap.od.pois 3 2 1 4 4 5
5. Conclusions
Rapidly inc easing compu a ional powe has been gene a ing a shi om de e minis ic claims
ese ing models o s ochas ic ones. Simul aneously, he alida ion o model app op ia eness has o
ecei e su icien a en ion om esea che s. In ou iew i is c ucial o unde s and he pe o mance o
di e en me hodologies o he calcula ion o emaining u u e paymen s in an insu ance po olio,
and o compa e hem om se e al pe spec i es. We ha e in e p e ed claims ese ing as a p obabilis ic
o ecas , as al eady done by o he disciplines, such as me eo ology o inance. Da a se s o six business
lines om Ame ican insu ance ins i u ions suppo ed calcula ions in o de o emain in con ac wi h
ac ual eal-li e claim ou comes.
Eigh di e en models ha e been used wi h key pa ame e es ima ion de ails, ou o which i e
p incipally di e en me hod amilies can be dis inguished. Two o he models a e i s in oduced in
he p esen a icle, using no only he indi idual insu e s’, bu collec i e claims obse a ions om
o he companies o calib a ion. See expe ience a emaking embedded in o he c edibili y boo s ap
o e dispe sed Poisson model. Semi-s ochas ic and c edibili y boo s ap models ha e been among
he bes pe o ming ones, howe e , esul s lack signi ican e idence ha hey would conside ably
ou pe o m hei egula boo s ap coun e pa s.
Goodness-o - i measu es desc ibing he na u e o p edic i e dis ibu ion a e clea ly mo e
in o ma i e han exclusi ely obse ing he mean squa e e o o he p edic ion. P obabili y in eg al
ans o m is be e han Kolmogo o –Smi no o C amé – on Mises in he sense ha i highligh s
wha goes w ong wi h he hypo hesis. Con inuous anked p obabili y sco es can widely be applied on
dis ibu ions wi h no cons ain on absolu e con inui y, de ining a anking among compe ing models.
Fu he cha ac e is ics such as co e age and sha pness explain he cen al p edic ion in e al and
i s expec ed wid h. Models di e signi ican ly in e ms o hese wo me ics. Me hodologies wi h