In this short note we give some comments and general remarks on the methodology of IBNR computations, as presented at the workshop on IBNR computations at the 2000 ASTIN Meeting, Porto Cervo, Sardinia.Evaluation;

Dhaene, Jan

Goovaerts, Marc

Redant, Hendrik

Vanden, orre, E

English

Research Papers in Economics

KATHOLIEKE 
UNIVERSITEIT 
LEUVEN 
DEPARTEMENT TOEGEPASTE 
ECONOMISCHE WETENSCHAPPEN 
RESEARCH  REPORT 0050 
SOME REMARKS ON IBNR EVALUATION 
TECHNIQUES 
by 
M. GOOVAERTS 
J.DHAENE 
E.  VANDEN BORRE 
H. REDANT 
D/2000/2376/50 Some remarks on IBNR evaluation techniques* 
Marc J. Goovaerts\ Jan Dhaenet Eddy Vanden Borre~ Hendrik Redant~i 
November 14, 2000 
Abstract 
In this short note we give some comments and general remarkH  011 
the methodology ofIBNR. computations, as presented at thf! WOl'kHhop 
on IBNR.  computations at the 2000  ASTIN Meeting,  Porto C('.rVll, 
Sardinia. 
1  IBNR evaluations 
The past data (the upper triangle) are the key elements of IBNR cakulat.i0I1'3. 
In almost. any method, analysing the upper triangle is based on well-kllowu 
techniques from statistics, see e.g.  Neter, Kutner, Nachtsheirn & Wasserman 
(1995).  However,  the essential problem to be solved is  the  malla.g(~lllellt 
of the risk associated with the future (the lower t.riangle).  Most metho(h; 
estimate the lower triangle cell-by-cell,  and do not pay (enough)  a.t.teutioll 
to the structure describing the dependencies between these cells.  Indeed. 
each cell must be considered as a univariate random variahle being pa.rt  of 
the multivariate random variable describing the lower triangle.  Henn\ the' 
IBNR reserve must be considered as  a.  (univariate) random variable beillg 
the sum of the dependent components of the random vec:tor describing the 
lower triangle. 
-The computations of Section 2 have been performed with the Rnftw>U"e VACS-LHC'. 
tKU.Leuven, Universiteit V-dJl Amsterdam. 
~KU.Leuven, Universiteit van Amsterdam. 
§ Appointed actuary Fortis Bank Insurance. 
'KU.Leuven. 
1 Estimating the correlations from the past data, and using them for lllUl-
tivariate simulations of the lower triangle is a dangerous technique becamie 
the insurer is especially interested in the tail of the distribution flmctioll hy 
choosing his reserve as a percentile. In practice, the insurer will choose a  v(~ry 
high percentile as basis for  his reserve. From the viewpoint. of the  iIll';l1l"(~(l, 
the choice of a high percentile is a safe strategy, and will as sueh be favomd 
by the control authorities. The determination of the reserve as  a pen:f!lltijp 
makes it possible to compute an explicit safety loading (= re.'lerve  !Iliumi 
expectation of the payments).  Fiscal authorities tend to prefer explicit  (i.(~. 
visible) margins to implicit (i.e.  hidden) margins.  The choiee of a high pm'-
centile is also important for the insurers' rating. It  is a key element in a Risk 
Based Capital approach. In the Belgian and Dutch insuranee practice,  wp 
observe that insurers determine their reserves on pereentiles sueh <1.<;  99. 75'A , 
or even higher.  Hence, only very high time-eonsuming multivaria.te sinmlH-
tions will lead to a sufficient number of simulated values in sneh an extn~JlI(' 
tail.  Another disadvantage of a simulation teebnique is that there is uo way 
to measure the distanee between the "real" and the "simulated" distrilmtioll 
funetion.  Henee, there is no information available coneerning the error that-
is involved by using a simulation teehnique.  Of course, a multiva.ria.te silll-
ulation technique will only be possible if the whole dependeuey stmctmp of 
the lower triangle is known. In practiee, we encounter situatious when) (July 
the distribution functions of each eell ean be estimated with enough <1.C:Cll-
racy, but. where only limited informa.tion of the dependency structure Gau hp 
obtained (because not enough data are available).  We ean c:ondude that:  11 
multivariate simulation technique is not the appropriate way  to detenuill<' 
IBNR reserves. 
As mentioned above, the "true" multivariate distribution fuuc:tioll of tlw 
lower triangle eannot be determined in most eases,  because the mut.ual  <!o-
pendencies are not known,  or difficult to cope with.  The only eouc:eiv<l hIe 
solution is to find upper and lower bounds for this sum of depemlent I<11l(lolIJ 
variables which use as much as possible of the available iuformatioll.  H(~u<:(L . 
within a certain class of random vectors (with given margina.lH,  ~Ul(l ewmtll-
ally additional information), we propose to look for upper and lower bOllll<lH 
for the sum of the eells of the lower triangle. For details of Uris  teelllliqlle, 
we  refer to Redant &  Goovaerts  (1999),  Goova,erts,  Dhaell!)  &  De Sc:hqr 
per (2000)  and Kaas, Dhaene &  Goova.erts  (2000).  The upper and lower 
bounds presented in these papers are bounds in the sense of eOllV!:',x  on  1m', 
which means that the expectations are exact. and the stop-loss premiull1'i  an~ 
2 ordered.  The convex order can of course be interpreted in tenm.;  of utility 
theory. The "total variation distance" can be used as a measure bet.ween the 
true distribution function and each of the bounds. 
Moreover, the proposed terlmique leads to a solution of the IBNR proh-
lem which is similar to a."value at risk"  approach in finanee.  The tedllliqllc 
described in the above-mentioned papers also allows to ca.leulate the {:()udi-
tional tail expectation or the remaining tail risk for  a givell pen:ellt.ik  i.(~. 
for  a given level of t.he IBNR reserve. 
2  Numerical illustration 
The statistical model that will be used to describe the past.  and the fut.nrc 
claim amounts is a loglinear model whirll looks for trendR in t.he three clin)(:-
tions,  namely accident year, development year and calendm' yecu'.  Au  (,~arly 
reference to the use of such models in the actuariallitemtme is De Vyhb' 
&  Goova.erts  (1979).  Given the statistical model for  the claim amounts, 
the present value S  of the future IBNR payment.s follows  from the vadon; 
X = (Xl, ... ,X 71.) and Y  =  (Y b  ...  , Yn ) where the vector X dp.<lcribes t.he f11-
ture claim amounts and the vector Y  describes t.he discollnt procells: 
n 
S= 2:Xd/;. 
;=1 
We  assume that the vectors X  and Y  are mutually independent a.wl  thaj· 
both have lognormal marginals.  Hence, S  iR  a sum of dependent loguo1'lual 
random variables. 
In order t.o  illustrate the technique explained in Redant  &  Goovamts 
(1999),  Goovaerts,  Dhaene &  De  Schepper  (2000)  and Kaas,  Dhaello  & 
Goovaerts (2000) for determining IBNR reserves, we use the nUl-off triallgl(' 
of Table 1 in Mack (1993), see also Taylor & Ashe (1983) and Verral (IUDO, 
1991)  and references therein.  First, we will  assume that.  Y  ==  (1,···: L), 
hence we discount. at an interest rat.e equal to O. 
In Figures 1 and 2,  we use a loglinear model wit.h  2 paramet.fIT"S  ill the 
direction  of t.he  accident  years  (denoted by 0:;)  and 4  pammct.en;  in  the 
direction of the development years (denoted by fJ k)  as is charac:teristi(; fo]'  t 11(' 
chain-ladder model.  No parameters are used in the calendar yem'  clirectiou. 
This model is  obtained as a conceivable model (given the data.) , from  th(' 
software VACS-LRC.We call this model the"6 parameter model": 
3 j 
In Xij = Gti + L (3  k +  Cij, 
k=l 
where Xij is  the claim amount of accident year i  and development  ye~H], j 
and the Cij are mutually independent normally distributed random vm"iHhle~s 
(with zero mean and variance equal to 0.069).  The parameters Gti  awl  0h' 
are given by:  Gt1  = 12.514;  Gt2  = ... =  GtlO  = 12.838;  (31  =  0.938;  /14  = 
-0.579;  (35  =  (36  =  (37  =  -0.219;  (39  =  -1.089. The remaining  /fH  an' 
non-significant (equal to 0). 
Figure 1 shows the probability density function (pdf) of the optimal ap-
proximation bound, as explained in Kaas, Dhaene & Goovaerts (2000).  This 
approximation can be shown to be very close to the real distribution nmdion. 
The closeness can be illustrated by the fact that the first. moments are eqllal 
and the second moments are almost equal:  The "real"  Htandal"d  deviHticllJ 
equals 1,355,969, whereas the standard deviation of the lower bound f~qm1.b 
1,341,161.  An estimate for the 99,75% percentile is given by 22,1l1,049. 
Figure 2 shows the pdf of the comonotonic upper bound.  Here, the ollly 
information used to compute the distribution function of the SlUn  al"(~  t  hc~ 
marginal distribution functions of the respective cells.  Given t.he  margillal 
dist.ribution functions,  comonot.onicity is  the dependency struc:t.ure  of the' 
vector X which leads to the most. risky sum S  (in t.he sense of convex orcIn). 
The st.andard deviat.ion of the upper bound is given by 5,481,136 whic:h  is 
much higher t.han t.he  real st.andard deviat.ion,  as  could be pxpec:ted.  The 
est.imate for the 99.75% percentile now equals 39,779,075.  TIIis eHt.illlH1"('  i~ 
of course much higher than the est.imate in Figure 1. This come.'!  frOlll tlw 
fact. that in order t.o determine the best.  approximation, we make lise of tlu' 
(estimated values of t.he) correlat.ions between the cells of the lower triHllgle, 
whereas  in Figure 2,  the distribution function is  an upper hOllllcl  (ill  tIl(' 
sense of convex order)  for  any possible dependency struc:t.ure  bct.weell  Ow  . 
components of the vector X. 
In Figure 3, we show the pdf of the optimal approxima.tion of S, whtm we 
take a stochastic discounting process into account..  We asslmle that the ymrly 
returns are log  normally distributed (with parameterH J-L  and IT) and 11l1ltllally 
independent.  Three different scenarios are presented:  scena.rio 1 (p.  = (l.OG 
and IT  = 0.03), scenario 2  (J-L  = 0.08 and IT = 0.1) and finally,  t.he ease of uo 
discounting.  We obse.rve that increasing the expected yea.rly ret.urn shift:;; tlw 
4 pdf to the left, and increasing the variance of the yearly return maket; the pdf 
broader.  Remark that in scenario 1 the 95% percentile is given by 18,435,()(j3, 
whereas in scenario 2 this percentile is  given by 19,751,126 and in the em;e 
of no discounting the 95% percentile equals 20,274,672.  Hence, as  (:Oul(l  he 
expected, stochastic discounting will normally diminish the required reserve. 
Finally, remark that the second scenario leads to a higher reserve thau the 
first one, which means that the effect of the higher expected returu (which 
tends to decrease the reserve) is  overshadowed by the effect  of the h.igher 
variability (which tends to increase the reserve). 
3  REFERENCES 
De Vylder,  F.;  Goovaerts,  M.J.  (editors)  (1979).  Proceedings of tll()  fin;t 
meeting of the contactgroup "Actuarial Sciences",  K. U .Lellven,  nr 7904I3, 
Wettelijk Depot: D/1979/2376/5. 
Dhaene, J.; Denuit, M.  (1999).  "The safest dependeney structure HUlOUg 
risks" , Insurance:  Mathematics f1 Economics 25,  11-2l. 
Dhaene,  J.;  Goovaerts, M.  (1996).  "Dependeney of risks  and stop-loRH 
order", ASTIN Bulletin 26,  201-212. 
Dhaene, J.; Goovaerts, M.J. (1997).  "On the dependeney of riRks ill tll!' 
individual life model", Insurance:  Mathematics  f1 Economics 19, 243-253. 
Dhaene, J.; Wang, S.; Young, V.; Goovaerts, M  .. J.  (1998).  "Comonotmlle-
ity and maximal stop-loss premiums", Bulletin of the Swiss  Associat'ion of 
Actuaries, aceepted for publication. 
Goovaerts, M.J.; Dhaene, J. De Schepper, A.  (2000).  "Stochastic: Upper 
Bounds for Present Value Functions", Journal of Risk and Ins1J,mru:e  ThmTY, 
67.1,  1-14. 
GooV'aeri;s,  M.J.;  Redant, R.  (1999).  "On the distributioIl of IBNR  I'(-}- _ 
serves", Insurance:  Mathematics & Economics 25,  1-9. 
Mack, T. (1993).  "Distribution-calculation of the standard error of chain 
ladder reserve estimates" , ASTIN Bulletin 23(2), 213-225. 
Neter, J.; Kutner, M.H.; Nachtsheim, C.J.; Wasserman, W. (1095),  111)-
plied Linear' Statist'ical Models,  fourth edition, IRWIN, pp,  140K 
5 Taylor, G.C. and Ashe, F.R. (1983).  "Second moments of  estilIlat.(~:-; of 
outstanding claims", Journal of Econometrics 23,  37-61. 
Verral, R.J. (1990).  "Bayes and empirical Bayes estimatioll for th" dl<liu 
ladder model", ASTIN B'u.lletin  20,  217-243. 
Verral, R.J. (1991).  "On the estimation of reserves from loglinmr lll()d-
els", Insm-ance:  Mathematics  f1  Econom:ics  10, 75-80. 
Vyncke,  D.;  Goovaerts,  M.J.;  Dhaene,  J.  (2000):  "Collvex  upp(~r aud 
lower bounds for present value fWlctions", Applied Stochast'ic Models in IJ'/I,si-
ness and Ind'ustry,  accepted for publication. 
Wang,  S.;  Dhaene,  .J.  (1998).  "Comonot.onicit.y,  correlatioll  order aurl 
stop-loss premiums", Insurance:  Mathematics f1  Econom:ics,  22(3), 235-243. 
6 :r.k.m:  11.936.481 
StD~ound),  1.341.161 
StD"'(Re") ,  1.355.969 
C:oe1£oiV*I".  :  OfJ1 
Mod:  11.835.8.52 
Med:  11.936.083 
Pm~,;: 
'0,00%  19.096.415 
90,00%  19."732.:393 
95,00%  20.274.61::: 
99,00%  21.331.918 
99,50%  :H.132.164 
99,15%  :tUl1.049 
16.000.000  1.8.000.000  :20.000.000  22.000.000 
fig 1: 6-parameter-model:  optimal approximation 
:r.k.m:  11.986.481 
StDeo;.r(Bound) :  5.481.H6 
StDet'(Re") ,  1.355.969 
Coe1£oflhr. :  0,30 
1ufod:  15.161.461 
MIld:  11.200.696 
PtrterdilH: 
80,00%  22.102.390 
90,00%  25.203.503 
95,00%  28.092,433 
99,00%  34.444.102 
99,50%  31.116.112 
99,15%  39.119.015 
15.000,000  25.000.000  35.000.000  45.000.000 
fig2: 6-parameter model: comonotonic upper bound .; 
/' 
J 
I 
/  , 
I  , 
I  , 
/ 
11.000.000 
I 
•  I . 
, 
<  • 
. 
"....-.  .... , 
/  :  , 
.  , 
I  '\.  /  , 
I 
.  . 
•  . 
15.000.000 
Non,  95,00%  20.214.612 
Scm&:Iic,l  95,00%  12.435.063 
, .  $cmmo2  95,00%  19.7.51.126 
-= 
19.000.000 
fig3: 6-parameter model: optimal approximation, different discount processes 
Scenariol: 
Scenario2: 
None: 
1l=0.05, 0-=0.03 
1l=0.08, o-=O.l 
1l=cr=0 

111)-plied Linear' Statist'ical Models, fourth edition, IRWIN, pp,

Bayes and empirical Bayes estimatioll for th&quot; dl<liu ladder model&quot;,

Collvex upp(~r aud lower bounds for present value fWlctions&quot;, Applied Stochast'ic Models in IJ'/I,siness and Ind'ustry, accepted for publication.

Comonot.onicit.y, correlatioll order aurl stop-loss premiums&quot;,

Comonotmlleity and maximal stop-loss premiums&quot;, Bulletin of the Swiss Associat'ion of Actuaries, aceepted for publication.

Dependeney of risks and stop-loRH order&quot;,

Distribution-calculation of the standard error of chain ladder reserve estimates&quot;

On the dependeney of riRks ill tll!' individual life model&quot;,

On the distributioIl of IBNR I'(-}- _ serves&quot;,

On the estimation of reserves from loglinmr lll()dels&quot;,

Stochastic: Upper Bounds for Present Value Functions&quot;,

The safest dependeney structure HUlOUg risks&quot;

Some remarks on IBNR evaluation techniques.

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Some remarks on IBNR evaluation techniques.

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