In this short paper we discuss a new methodology for estimating reserves for IBNR (incurred but not reported) claims.

E. Vanden Borre

J. Dhaene

M. Goovaerts

English

Research Papers in Economics

Tijdsciirift voor Ecotiorilie ei1 Mailageme~it 
Vol. XLVI, 4, 2001 
Some Remarks sn 
IBNR Evaluation Techniquesl 
Maic J  Goovaerts 
K.U.LCLILLII,  Depa~tnieiit  of 
Applied Ecoiioinics and 
University of Ainsterdain, The 
Netherlands 
-  - 
Jan Dhaene 
K U Leuven, Department of 
Applied Ecoiloiuics and 
University of Amsteidarn, The 
Nethei lands 
Appointed  actuary  Fortis  Bank 
Iilsuraiice 
ABSTRACT 
In this short paper we discuss a ilew  iilethodology  for estiinating reserves for 
IBNR (incui-i-ed but not reported) claims. I.  IBNR EVALUATIONS 
Tlie past data (tlie upper triangle) are the key eleiiieiits of IBNR cal- 
culations. Iii alniost aiiy method, analysing the upper triangle is based 
on  well-known  techiiiques  from  stat~stics,  see  e.g.  Neter,  Kutner, 
Nachtsheiin and Wassermail  (1 995). However, the esselitial problem 
to be solved is the inanageineiit of tlie risk associated with tlie filture 
(the lower triangle). Most inethods  estiinate the lower triaiigle cell- 
by-cell, atid do iiot pay (enough) attention to tlie structure describing 
the dependencies betweeii +ese  cells. Indeed, each cell inust be con- 
sidered as a uiiivariate random variable being part of the inultivariate 
randoin  variable  describing  the  lower  triangle.  Hence,  the  IBNR 
reseive must be coilsidered as a (uiiivariate) random variable being 
the sum of the dependent coniponents of the random vector describ- 
ing the lower triangle. 
Estimating the correlations from the past data, and using thein for 
multivariate  simulatioiis of the  lower triangle  is a daiigerous tecli- 
nique because the insurer is especially interested in the tail of the dis- 
tribution function by choosing his reserve as a percentile. In practice, 
tlie insurer will clioose a vesy high percentile as basis for his reserve. 
Froin the viewpoint of the insured, tlie choice of a high percentile is 
a safe strategy, and will as such be favored by tlie control authorities. 
The detenninatioii of the reserve as a percentile maltes it possible to 
compute an explicit safety loading  (= reserve ininus expectatioii of 
the payments). Fiscal authorities tend to prefer explicit (i.e. visible) 
inargins to iiiiplicit (i.e. hidden) margiiis. Tlie choice of a liigli per- 
centile is also iinportant for the insurers' rating. It is a key eleinent in 
a Risk Based Capital approach. 111  tlie Belgian and Dutch iiisurance 
practice, we  observe that  insurers  determine  their reserves  on per- 
centiles such as 99.75%, or even higher. Hence, only vesy high time- 
consuming multivariate simulations will lead to a su.cient nuinber of 
siinulated values in sucli an extreine tail. Another disadvantage of a 
simulatioii teclinique is that there is no way to measure the distaiice 
betweeii the "real"  and the "sitnulated"  distribution function. Heiice, 
tliere is no infoimatioii available coiicerning the error that is involved 
by using a sirnulation techniqiie. Of course, a inultivariate simulation 
technique will only be possible if the whole dependency structure of 
the  lower  ti-iangle is  lmowii.  I11  practice,  we  encouilter  sitiiations 
where  only the distributioii filnctions of each cell caii be estimated 
with  enough  accuracy, but  where  only  limited  inforination  of  the depeiidency structure caii be obtained (because iiot enough data are 
available). We  caii coiiclude that a multivariate simulation technique 
is not the appropriate way to determine IBNR reseiyes. 
As mentioiied above, the "true"  in~iltivariate  distribution function 
of the lower triangle caniiot be deteriniiied in most cases, because the 
inutual dependencies are not kiiowii, os di.cult to cope with. The only 
conceivable solution is to find upper aiid lower bounds for tliis suin 
of dependent random variables which use as iiiuch as possible of the 
available inforiiiation. Hence, within a certain class of random vec- 
tors  (with given  marginals, and evenrually additional  information), 
we propose to look for uipper  and lower bouiids for the sum of tlie 
cells of tlie lower triailgle. For details of this technique, we refer to 
Redant and Goovaerts (1999), Goovaests, Dhaene aiid De Schepper 
(2000)  and  Kaas,  Dhaene  and  Goovaei-ts  (2000).  Tlie  uipper  and 
lower bouiids pi-eseiited in these papers  are boui~ds  in  the  sense of 
convex order, which means that the expectations are exact and the 
stop-loss preiniums are ordered. Tlie convex order can of coiirse be 
interpreted in temis of utility theoiy. 
The "total variation distance"  can be used as a measure betweeii 
the tiue distribution function and each of the bounds Moreover, tlie 
proposed technique leads to a solution of the IBNR probleniwhich is 
sinlilar  to  a  "value  at  risk"  approach  iii  finance.  The  techiiique 
described in the above-mentioned papers also allows to calculate the 
conditional tail expectation os tlie reiiiaining tail risk for a given per- 
centile, i.e. for a given level of the IBNR reserve. 
11.  NUMERICAL ILLUSTRATION 
Tlie siatistical znodel thai wil1 be used tû  describe thc past  and  Ihe 
future claim ainounts is a logliiiear model wliich looks for trends iii 
the three directions, naniely accident year, development year and cal- 
eiidar year. AII early referente to tlie use of sucli rnodels in the achi- 
arial literature is De Vylder and Goovaerts (1979). Given the statisti- 
cal model for tlie  claim ainounts, the present value S of the fiiture 
IBNR paymeiits follows  froin the vectors  X  =(Xl, . . ., X,,) and Y 
=(Yl, . . ., Y,,) wbere the vector X describes the future claim ainounts 
and the vector Y describes tlie discount process: We  assuine that the vectors X aiid Y are inutually independent and 
that both liave lognormal iiiarginals. Heiice, S is a sum of dependent 
lognorinal random variables. 
111  order  to  illustrate  tlie  technique  explaiiied  in  Redant  and 
Goovaerts (1 999), Goovaerts, Dhaene and De Schepper (2000) aild 
Kaas, Dhaene aiid Goovaerts (2000) for deteniiining IBNR reserves, 
we use tlie run-off triangle of Table 1 i11  Mack (1993), see also Taylor 
and Ashe  (1983)  aiid  Verral  (1990,  1991) aiid  references  therein. 
First, we wil1 assume tliat Y = (l, . . .,  l), hence we discount at an 
iiiteresl rale eclual io 0. 
111  Figures l aiid 2, we use a loglinear model witli 2 paraiiieters iii 
the direction of the accident years (deiloted by  a,)  and 4 paranieters 
in tlie direction of the developinent years (deiioted by D,)  as is cliar- 
acteristic for tlie chain-ladder inodel. No paraiiieters are used in tlie 
calendar year directioii. 
This niociel  IS  obtained as a conceivable inodel  (given the data), 
from tlie software VACS-LRC.We  cal1 this inodel tlie "6  parameter 
inodel": 
where X,  is the claiin aniount of accident  year  i and  developiiient 
year j  and the c, are m~itually  independent iionnally distributed ran- 
doin variables (with  zero mean  and variance  equal to  0.069). The 
parameters a, and p, are given by: 
a, = 12.514; a, =  = al,  = 12.838; fl, = 0.938; fl4  = -0.579;  = 
Pb = P7 = -0.219;  Pg = -1.089. 
The remaining p's are non-signi.caiit (equal to 0). 
Figure l sliows the probabiliiy density f~~iiction  (pdf) of the opti- 
mal  approxiination  bouiid,  as  explained  in  Kaas,  Dhaene  aiid 
Goovaerts  (2000). Tliis  approxiiiiation  can  be  shown  to  be  very 
close to the real distribution ftlilction. The closeness can be illus- 
trated  by the fact that the first inoinents are equaland the  second 
moments  are  alniost  equal: The "real"  standard  deviation  equals 
1,355,969,  whereas  the  standard  deviation  of  the  lower  bound 
equals 1,341,161.  Aii estiniate for tlie 99,75% percentile is given by 
22,111,049. FIGUUR 1 
6-Porati~efrr-holo~r'el  O~~trnlul  14ppin1  l~ncitroii 
Figure 2 shows tlie pdf of tlie coinonotonic upper bound. Here, tlie 
only  iiiformation  used  to  coinpute  the  distribution  function  of the 
suiii are the inarginal  distribution  hnctions of the respective  cells. 
Giveil  tlie  rnargiiial  distribution  functions,  comonotonicity  is  the 
dependeiicy structure of the vector X which leads to the most risky 
suili S (in the sense of convex order). 
The standard devirttion of the upper boud is giver, Sy 5,481,136 
whicli is iiiuch Iiiglier than the real standard deviation, as could be 
expected.  The  estiinate  For  tlie  99.75%  perceiitile  iiow  equals 
39,779,075. This estiinate is of course much higher than the estiinate 
in Figure  1. This coines froin tlie í'act  tlial in  order to deterniine the 
best approxiination, we make Lise  of tlie (estiinated values of tlie) cor- 
relations between the cells of the lower triangle, whereas in Figure 2, 
tlie distributioii f~iiictioi~  is an upper bound (111 the seiise of coiivex 
order) for  any  possible  depeiidency  structure  between  the  coinpo- 
iients of the vector X. In Figure 3, we show the pdf of the optililal  approximatioii of 
S, when we talte a stochastic discounting process into account. We 
assume that  the yearly returns  are lognorinally  distributed (with 
parameters p and a) and iilutually independent. Three di.ereiit sce- 
nario~  are preseilted: scenario l  (p  =-0.05 and a = 0.03), scenario 
2( p = 0.08 and  CJ  = 0.1) aild finally, the case of ilo discounting. 
We  observe that  increasing the expected yearly  return  shifts the 
pdf  to  the left, and increasing the variance  of the yearly  return 
iilaltes  the pdf broader. Reinark that  in  scenario  1 the 95% per- 
ceiitile  is  given  by  18,435,063, whereas  in  scenario  2  this  per- 
ceiitile is given by  19,751,126 and in  the case of no  discouiiting 
the  95%  percentile  eqiials  20,274,672.  Hence,  as  could  be 
expected,  stochastic  discounting  wil1  nornially  diininis11  the 
required reserve. FIGUUR 3 
6-Pmznneter Model  0/7f~rnaI  App~o~uwat~o~z 
Dz#er.eilt  D~scourit  Pr.ocesse~ 
Finally, remark that the second scenario leads to a higher reserve 
than  the  first  one,  which  means  that  the  e.ect  of  the  higher 
expected return (which teilds to decrease the reserve) is overshad- 
owed by the e.ect of the higher variability (which tends to increase 
the reserve). 
NOTES 
I  The coinputations of Section 11 have been performed with Lhe  software VACS-LRC. 
REFERENCES 
De Vylder, F.,  Goovaerls, M.J., editors,  1979, Proceedings  of the  First  Meeting of tlie 
Coiitactgro~ip "Actuarial  Sciences",  K.U.Le~iven, nr  7904B,  Wettelijk  Depot: 
D1 19791237615. 
Dhaene, J., Denuit, M., 1999. The Safest Dependency Struct~ire  Among Risks, lriszrr~~ir2ce: 
Mathelnatics & Ecor~on?ics  25, 11-21. 
Dhaene,  J., Goovaerts, M., 1996, Dependency  of Risks  and  Stop-Locs  Order, ASTIN 
B~rlletifi  26, 201  -2 12. Dhaene,  J., Goovaerts, M.J.,  1997, 011 tlie  Depeiidency  of Risks in tlie Individual Life 
Model, Inszrrol7c:  Matlzernatics & Ecoi7oinics 19, 243-253. 
Dhaene, J.. Wang, S., Young, V., Goovaerts, M.J., 1998, Coinoiiotonicity and Maxirnal 
Stop-Loss Preniiuins, B7rlletin of the S~viss  Associatioiz of Actz~aries,  accepted for pub- 
lication. 
Goovaerts, M.J., Dhaene, J. De Scliepper, A., 2000, Stocliastic Upper Bouiids for Preseiit 
Value Functions, Jolirnal ofRisk arld Instrrance Theoq~  67.1, 1-14. 
Goovaerts, M.J., Redant,  R., 1999, 011 the  Distribution  of 1BNR Reserves, In.~io.urzce: 
Mcrtl7enzatics & Ecoi7onzics 25, 1-9. 
Mack, T.,  1993, Distribution-Calculatioil of the Standard Error of Chaiii Ladder Reserve 
Estimates, ASTIN Bzrlletin 23(2), 2 13-225. 
Neter,  J.,  Kutnei;  M.H.,  Nachtslieim,  C.J.,  Wassennali,  W.,  1995,  Applied  Linrar 
Stafistical Model,s, fourth edition, (Irwin), pp.  1408. 
Taylor, G.C. and Ashe, F.R., 1983, Second Moinents of Estimates of Outstanding Claims, 
Jo~ir17~1i  qf  Econoïnerrics 23, 37-61. 
Venal, R.J., 1990, Bayes and Empirica1 Bayes Estimation  for tlie Chain Ladder Model, 
ASTIN Bzilletin 20, 2 17-243. 
Verral, R.J.,  1991, On the  Estiination of Reserves  froin  Loglinear Models, Inszirance: 
Mat/zematics & Economics  10, 75-80. 
Vyncke, D,, Goovaerts, M.J., Dhaene,  J., 2000, Convex Upper  aiid Lower Bounds for 
Present Value Fucictions, Applied Stochastie Models in Bzisiness and Ind71stty, accepted 
for publication. 
Wang, S., Dhaene, J.,  1998, Comonotonicity, Correlation Order and Stop-Loss Premi~ims, 
lilszirance: i21atlzt~nzntics  & Economics 22, 3, 235-243. 

Applied Linrar Stafistical Model,s, fourth edition,

Bayes and Empirica1 Bayes Estimation for tlie Chain Ladder Model,

Convex Upper aiid Lower Bounds for Present Value Fucictions, Applied Stochastie Models in Bzisiness and Ind71stty, accepted for publication.

Dependency of Risks and Stop-Locs Order,

Distribution-Calculatioil of the Standard Error of Chaiii Ladder Reserve Estimates,

On the Estiination of Reserves froin Loglinear Models,

Stocliastic Upper Bouiids for Preseiit Value Functions, Jolirnal ofRisk arld Instrrance Theoq~

Some Remarks on IBNR Evaluation Techniques

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Some Remarks on IBNR Evaluation Techniques

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