ON THE USP CALCULATION UNDER SOLVENCY II AND ITS APPROXIMATION WITH A CLOSED FORM FORMULA

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1 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA Filippo SIEGENHALER Zurich Insuranc Group Ld Valnina DEMARCO Univrsiy of Calabria Rocco Robro CERCHIARA Dparmn of Economics, Saisics and Financ, Univrsiy of Calabria Absrac: In his papr w rcall and commn, aiming o hlp undrsanding, h modl assumpions undrlying h undraking spcific paramr (USP mhod as dfind in h Solvncy II rgulaion. In addiion w propos an adjusd sandard dviaion simaor which also ak ino accoun h (simad porfolio volum of h solvncy yar undr considraion. Morovr w prsn a closd-form formula for approimaing h USP and provid som numrical ampls basd on ral daa showing h goodnss of h approimaion. his is a usful rsul for pracicing acuaris sinc i allows o approima h USP wihou prforming any numrical opimizaion. Kywords: Undraking spcific paramrs, Rsrv risk, Prmium risk, Solvncy II. BULLEIN FRANÇAIS D ACUARIA, Vol. 7, n 33, janvir juill 07, pp

2 64 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA. INRODUCION Wihin h dsign of h Sandard Formula undr Solvncy II framwork, insuranc and rinsuranc undrakings may rplac a subs of is paramrs (mark-wid paramrs for ach sgmn by spcific paramrs of h undraking (USP whn calculaing solvncy capial rquirmn (SCR for Non-Lif and Halh undrwriing risk modul. Sandard dviaions on on-yar im horizon for prmium and rsrv risks blong o his subs. On-yar volailiy modlling has rcivd wid covrag in acuarial and saisical liraur, bu in his papr w rfr only o EIOPA publicaions on USP (s Crchiara and Dmarco (06 for h ovrviw on scinific liraur in rspc of on yar volailiy. In Ocobr 00, EIOPA ( CEIOPS, has bn commid o carry ou a comprhnsiv rvision of h calibraion of h prmium and rsrv risk facors in h Non-Lif and Halh undrwriing risk modul of h SCR sandard formula (s EIOPA (0, EIOPA (04a and EIOPA (04b. Nw calibraion mhods for USP hav bn inroducd by h "Dlgad Ac Solvncy II" (DA (s Europan Commission (05. In paricular, ann XVII of DA shows hr nw calibraion mhods for USP, on for prmium risk (mhod and wo for rsrv risk (mhod and mhod. h horical modl undrlying mhod for prmium and rsrv risks is on of h four modls sd by h "Join Working Group - JWG - on Non-Lif and Halh NSL Calibraion" in h subscion Lognormal Modls wih Scond Varianc Paramrisaion (s EIOPA (0. According o mhod, w rcall ha h undraking spcific paramr (USP for a spcific sgmn (s is calculad as (,, s USP = c(, ( c (, s ( whr - c dnos h givn crdibiliy facor (varis according o h sgmn considrd and h numbr of yars 5 for which daa ar availabl, s CEIOPS (00. is a givn mark wid sandard dviaion and h symbol sands - (, s for ihr prm or rs.

3 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 65 whr - h undraking spcific sandard dviaion is dfind as Y = (, (, = p, = (, = (, = (. ( (3 - [0,] (calld miing paramr and (calld logarihmic variaion cofficin ar h soluions ha minimiz h following prssion Y Y = (, (,. (, (, = (, = = Rmarks: - h quaniis ( ar givn volum masurs. - For h rsrv risk h ind rprsns a financial yar and h quaniy Y Z = is dfind as prior yar dvlopmn ingoing rsrvs Z =, (5 ingoing rsrvs whr h prior yar dvlopmn is dfind as prior yar dvlopmn = paymns for prior accidn yars [ougoing rsrvs for prior accidn yars ingoing rsrvs ]. - For h prmium risk h ind rprsns an accidn yar and h Y Z = quaniy is dfind as simad ulima amoun afr h firs dvlopmn yar Z =. (6 arnd prmium (4

4 66 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA As dscribd abov, h compuaional procdur rquirs numrical opimizaion for simaing h paramrs and. A snsiiviy analysis (s Crchiara and Dmarco (06 prformd on h prssions involvd in USP calculaion shows, ha kping fid h logarihmic variaion cofficin, h valu assumd by h miing paramr dos no marially affc h final USP valu. In ohr words, h mpirical sudis in Crchiara and Dmarco (06 did show ha h USP valu is no marially influncd by h miing paramr valu. In his papr w will show ha by simply forcing h miing paramr o b qual o (valu which implis undrlying modl assumpions idnical o h cas whr h porfolio has consan volums (, s ( and (7, w ar abl o driv a closd-form formula (asily implmnd in a spradsh ha can b usd in ordr o approima h undraking spcific sandard dviaion wihou prforming any numrical opimizaion. h approimaion rsul will b (, p appro ( appro, (7 whr Y Y = = appro = p, (8 Y Y = = Y appro =p. = (9 h papr will b organizd as follows. In scion w provid h sochasic modl undrlying h mhod and show how h opimizaion rcis (4 for simaing h paramrs and is moivad. Morovr w provid som insighs why h undraking spcific sandard dviaion (, (and also h USP is no marially influncd by h miing paramr and w propos an adjusd undraking spcific sandard dviaion dfiniion givn by

5 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 67 Z Adjusd = (, = (, = p (. = (, = (, In scion 3 w considr h cas = and driv wihin his sup h abov prsnd closd-form formula for approimaing h undraking spcific sandard dviaion. Finally, in scion 4 w provid som numrical ampls.. HE MODEL ASSUMPIONS h sochasic modl undrlying h mhod can b formulad as follows. / (0 Modl Assumpions (Mhod hr iss consans, [0,] and such ha for {,, } - h pcd valu of - h varianc of Y Y is givn by Y E =, ( is givn by Y = Var = (, ( Y - h random variabls ar indpndn and lognormally {,, } disribud, i.. wih Y (,, (3

6 68 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA = = (, = (, =. Rmarks: - Assumpion is a saionary condiion rlad o h pcd raios which is ofn assumd in non-lif acuarial modls and is gnrally snsibl in pracic spcially whn looking a on-lvld variabls. - Assumpion allows for a flibl considraion of a spcrum of variancs assumpions (from linar in o quadraic in and hir miurs. - No ha assuming a compound Poisson modl for h random variabl Y would dlivr a varianc srucur of h following form Y Var =, (6 wih,. hrfor h varianc assumpion ( can b considrd o hav bn drivd from (6 by assuming h addiional srucur = (, = =, (4 (5 i.. nsuring ha h varianc of Y rsuls o b quadraic in for = and linar in for =0. - h log-normaliy Assumpion 3 is a gnrally wll accpd assumpion in non-lif insuranc, boh whn looking a an aggrga annual loss as wll as looking a prior-yar loss dvlopmns on rsrvs. h indpndnc assumpion can b considrd accpabl whn looking a porfolios wih limid paramr risk. - For an nsiv dscripion of possibl chniqus for sing h Modl Assumpion. w rfr o D Flic and Moriconi (05. - No ha for a porfolio wih consan volums w hav

7 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 69 = =, and hrfor h varianc assumpion ( bcoms Y Var =, (7 which coincids wih h varianc assumpions ( in cas h paramr is qual o.. Paramr simaion For simaing h paramrs, and w mak us of a maimum liklihood approach. Sinc h dnsiy of a lognormally disribud random variabl is givn by (( z f( z lognorm =, z w can asily driv ha h log liklihood o b maimizd is givn by ( Z (,, = (, (, which is quivaln as minimizing h prssion ( Z (,, = (, (, for, and. h opimizaion approach proposd according o h mhod is as follows. In a firs sp, for fid and, i is suggsd o minimiz h prssion (0 for. Doing his w g h condiion 0= ( Z (,. = (, (, hrfor, h simad paramr dos fulfill h following quaions (8 (9 (0 (

8 70 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA (, Z = 0, = (, ( Z = 0, = (, (, (3 Z = 0, = (, = (, (4 Z = (, ( =. = (, (5 In a scond sp, i is suggsd o plug in h las rsul from abov ino (0 and minimiz h rsuling prssion Z = (, (,, Z = (, = (, = (, for and. No ha his is acly prssion (4 mniond in scion. Rmarks: - h undraking spcific sandard dviaion (, as dfind in ( can b inrprd o b qual o h simad squar rood varianc for h n yar ( / Adjusd = (, := Var Z = ( / = =p ( ( (6

9 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 7 Z = (, = =p (, = (, = (, whn nglcing h scond facor from abov, i.. assuming = =. his fac sms o plain why h undraking spcific sandard dviaion (, dos acually no marially dpnd on h simad paramr, sinc nglcing h scond facor h abov formula dos no plicily dpnd on. - h simaor (, could b considrd o b unbiasd if i would hold ru E[ (, ]=. Unforunaly, in our viw, i is no possibl o acly compu h abov pcd valu and hrfor w ar no in a posiion nihr o chck h unbiasdnss propry nor o acly driv a bias adjusmn facor. Howvr, no ha for known paramrs and i holds ru Z = (, (, = (, = = ( and as a consqunc w hav /

10 7 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA = = (, [ (, ]= p (, Z E E = = =p ( ( (7 = = =p. ( hrfor, rplacing h unknown paramrs and wih hir maimum liklihood simaors and, w suggs o us h following bias adjusmn facor

11 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA p. = = ( 73 (8 As i appars in formula (, according o h EIOPA documnaion (s EIOPA (0 an appropria bias adjusmn facor is givn by. Howvr w ar no awar of a rigorous proof showing his facor bing rasonabl and hrfor w can jus inrpr i as a prudncy loading dfind by h rgulaor. 3. HE CASE = AND HE RELAED CLOSED FORM FORMULA In his scion w will analyz h spcial cas = wihin h Modl Assumpion oulind in scion. No again ha in his cas h implid varianc assumpion ( is approimaly h sam as undr h gnral Modl Assumpion whn considring a volum sabl porfolio. 3. Firs ordr condiions From (3 w s ha in his paricular sup w hav = ( = and h funcion o b minimizd is givn by (9

12 74 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA Z ( = = = (30 = Z ( =: g(,. As a consqunc w hav o solv h quaion 0= (, = g Z ( which implis = ( =, Z = = as wll as solv h quaion which implis (3 Z ( = Z (, (3 0= g(, = Z (, = Z ( =. (33 = (34 Insring quaion (34 ino quaion (3 w g Z Z = = Z Z = = =0 =, (35 which finally allows us o driv h following simaors for h wo paramrs and :

13 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 75 Z Z = = appro = p, (36 Z Z = = appro =p Z. = (37 h approimad undraking spcific sandard dviaion can b finally dfind as h simad squar rood varianc for h n yar (, i.. appro = appro Var Z =p appro ( appro (38 Z Z = = =pappro Z. = 3. Scond Ordr condiions In ordr o nsur ha our soluion ( appro, appro is a minimum for h funcion g(, w nd o chck h following scond ordr condiions: g( appro, appro > 0 -, g( appro, appro > 0 -, - ( appro, appro ( appro, appro g g g ( appro, appro >0. Compuing h scond parial drivaiv wih rspc o w g

14 76 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA g(, = Z ( = = = (, = and valuaing i a (, yilds appro appro Z (39 g( appro, appro = Z Z appro appro = = = > 0, appro appro (40 sinc i holds ru appro appro = ( = Z. (4 Compuing h scond parial drivaiv wih rspc o w g (s appndi A. for mor dails 4 4 = Z 4 ( = g(, = Z ( ( ( 4 ( 4 ( = 4 4 Z (, and valuaing i a (, yilds appro appro (4 4appro 4appro appro 4 g( appro, appro = >0 appro appro ( (43

15 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 77 sinc i holds ru Z Z = = appro Compuing h mid parial drivaiv w g =. g(, = Z ( = = = (44 = Z (, (45 and valuaing i a (, yilds appro appro appro appro g( appro, appro = Z appro Z appro = = Morovr i holds ru appro appro appro appro appro =. appro appro appro (46 g( appro, appro g( appro, appro g( appro, appro 4 4 appro appro appro 4 = appro appro appro appro (

16 78 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA 4 4 appro appro appro appro 4 appro 4appro appro 4appro appro = 3 appro appro appro ( appro appro 3 appro 4 8 appro = > 0 3 appro appro appro ( (47 and hrfor our soluion (, is a minimum for h funcion g(,. appro 3.3 On h approimaion rror appro In h abov scions w showd ha in h paricular cas = h sandard dviaion simaor is givn by (, = p appro ( appro. (48 In his scion w will show ha in h gnral cas h sandard dviaion simaor is linarly approimad by (, c(,, p (, whr c(,, is a funcion for which i holds appro appro appro c(,, appro. In fac, considring h linar approimaion ( for clos o 0, w hav (49 = appro = appro (, appro = ( (, (50 sinc from (36 i holds appro 0. = Z Z = Morovr, sinc i holds 0, w g

17 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 79 Z Z = = = (, appro ( p Z Z = = = (, (5 whr in h las sp w applid h linar approimaion p( for clos o 0. N, w can also driv h approimaion (s appndi 8. for mor dails ( ( = = = (, appro Z Z = = (5 and as a consqunc w g (s appndi 8.3 for mor dails (, p ( appro appro appro Z Z = = p ( ( = = Z p Z. = = ( ( = = = ( (53

18 80 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA Finally, as anicipad in (49, w g h blow linar approimaion (s appndi 8.4 for mor dails from whr on gs a viw on h magniud of h approimaion rror in cas h volums ar no consan or whn h miing paramr is no qual o : (, p appro ( appro ( p = = ( appro Z Z = = p ( ( = = Z p ( ( = = ( = p ( appro appro =: c(,, p (. appro appro appro = = Z (54

19 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 8 Rmark: Sinc w canno driv closd-form formulas for h gnral cas w canno analyically assss h valu of h approimaion adjusmn p, nor giv appro any bound. Howvr, rcalling ha h opimizaion rcis (4 for driving and appro jus diffr on h uilizd funcion (, and sinc w hav = (, (, = ( = = ( = ( ( = = 0 w can hurisically pc and appro o b wll clos o ach ohr. his will b confirmd in our numrical ampls in scion 4. (55 4. NUMERICAL EXAMPLES In his scion w rvisi h numrical ampls prsnd in Crchiara and Dmarco (06 which ar basd on daa from hr diffrn undrakings A (small siz, B (mdium siz and C (larg siz rgarding h sgmn moor vhicl liabiliy. Morovr w will considr an addiional moor vhicl liabiliy porfolio (undraking D for which h simad paramr is vry clos o. No ha for his sgmn h mark wid sandard dviaions in h "Dlgad Ac Solvncy II" ar givn by ( prm, s = 0% and ( rs, s =9% rspcivly. W g h following rsuls

20 8 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA Prmium Risk Undraking A Undraking B Undraking C Undraking D 0 c 74% 8% 8% 8% % 79.39% 83.85% 75.59% (, 6.86% 5.6% 4.08% 6.76% appro appro 68.6% 79.47% 84.4% 75.59% p ( 6.88% 5.% 4.6% 6.76% appro appro USP 8.% 6.48% 5.5% 7.90% USP closd-form formula 8.3% 6.53% 5.59% 7.90% Rsrv Risk abl : Prmium risk rsuls by undraking Undraking A Undraking B Undraking C Undraking D c 67% 74% 74% 74% % 99.7% 95.95% 5.48% (, 7.00% 4.33% 3.8%.40% appro appro 6.93% 99.% 96.5% 5.48% p ( 7.40% 4.39% 3.93%.40% appro appro USP 8.% 5.88% 5.47%.66% USP closd-form formula 8.5% 5.93% 5.56%.67% abl : Rsrv risk rsuls by undraking

21 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 83 whr h USP is givn by quaion (( and h USP closd-form formula is givn by cp appro ( appro ( c (, s. From h abov abls w can s ha in gnral h diffrncs bwn h undraking spcific sandard dviaion (, and is approimaion p ( as wll as bwn h final USP valus ar rahr small. appro appro Morovr, for ach daas w can valua h inrval = = I = min,ma rlad o h volum sabiliy of h porfolio and w g h following rsuls: Prmium Risk Undraking A Undraking B Undraking C Undraking D I [0.74,.7] [0.90,.7] [0.9,.5] [0.63,.7] (, 6.86% 5.6% 4.08% 6.76% p ( 6.88% 5.% 4.6% 6.76% appro appro abl 3: Prmium risk: volum sabiliy inrvals by undraking Rsrv Risk Undraking A Undraking B Undraking C Undraking D I [0.89,.6] [0.94,.07] [0.9,.9] [0.65,.33] (, 7.00% 4.33% 3.8%.40% p ( 7.40% 4.39% 3.93%.40% appro appro abl 4: Rsrv risk: volum sabiliy inrvals by undraking Looking a h abov abls w can vrify h following bhaviors: - if h valu is sufficinly clos o hn h approimaion is vry prcis indpndnly from h volums sabiliy (s h prmium risk for undraking A whr h approimaion is vry good dspi h poor volum sabiliy and also h prmium and rsrv risks for undraking D

22 84 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA - for porfolios wih similar valu h mor h abov inrvals ar concnrad around h br h approimaion is (compar h rsrv risk rsuls for undraking A and B Finishing his scion w would lik o - vrify how dos our approimaion prform in cas w fac a porfolio wih consan volums (s scion 4. - vrify how our adjusd undraking spcific sandard dviaion (s quaion (6 / Z Adjusd = (, = (, = p (, = (, = (, is acually influncd by h miing paramr valu (s scion Porfolio wih consan volums W considr h prmium risk daa for undraking B and w forc h volums ( o b consan, i.. w s = = =. Rcalculaing h paramrs w g Prmium Risk Undraking B c 8% % (, 9.45% appro.606 appro 90.6% p ( 9.45% appro appro USP 0.9% USP closd-form formula 0.9% abl 5: Prmium risk rsuls for undraking B in cas of consan volums and, as pcd, w s ha our approimaion is acually ac.

23 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA Influnc of on (, Adjusd W considr again h prmium risk daa for undraking B and assum an incras in prmium volum for h n yar compard o h obsrvd avrag prmium in h pas. In his rspc w choos for insanc = =0.95. hn, kping h simad logarihmic variaion cofficin fid, w calcula (, and (, Adjusd for diffrn valus: (, (, 0 5.6% 5.03% % 5.05% % 5.06% % 5.07% % 5.09% % 5.0% % 5.% % 5.3% % 5.4% % 5.6% 5.7% 5.7% abl 6: Influnc of on (, and on (, Adjusd Looking a h abl w s ha (, rsuls o b vry sabl whras (, Adjusd is much mor influncd by h miing paramr. 5. CONCLUSION In his papr w drivd a closd-form formula ha can b usd in ordr o approima h undraking spcific sandard dviaion (as dfind in h "Dlgad Ac Solvncy II" wihou prforming any numrical opimizaion. h main rsul was (, p (, appro appro

24 86 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA whr appro appro Y Y = = = p, Y Y Y = = = =p. No ha h abov approimaion (which has bn obaind by forcing h miing paramr o b qual o is ac in cas a porfolio wih consan volums is considrd, i.. a porfolio for which i holds ru = =, Y sinc in his cas h varianc assumpion for Z = coincids wih h varianc assumpion in h cas =. Morovr, our rsarch did highligh h fac ha, undr Modl Assumpions., h undraking spcific sandard dviaion should probably b bs dfind as / Z Adjusd = (, = (, = p (, = (, = (, i.. making us of h (simad porfolio volum for h n yar and no ha h adjusd undraking spcific sandard dviaion (, Adjusd and, as a consqunc, h rlad final USP valu will plicily dpnd on h simad paramr (as opposd o h findings in Crchiara and Dmarco (06 concrning (,.

25 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA ACKNOWLEDGEMEN W hank Agosino ripodi for hlpful commns on a prvious vrsion of his papr. Filippo Signhalr would lik o rmark ha his aricl rflcs h prsonal viw of h auhors and no ncssarily ha of Zurich Insuranc Group Ld or any of is subsidiaris. 7. REFERENCES CEIOPS (00: CEIOPS Advic for Lvl Implmning Masurs on Solvncy II: SCR sandard formula - Aricl j, k Undraking-spcific paramrs. CEIOPS-DOC- 7/0, 9 January 00 CERCHIARA, R. and DEMARCO, V. (06: Undraking spcific paramrs undr solvncy II: rducion of capial rquirmn or no?. Eur Acuar. J. (06 6: DE FELICE, M. and MORICONI, F. (05: Sulla sima dgli Undraking Spcific Paramrs la vrifica dll iposi. Working paprs of h Dparmn of Economics Univrsiy of Prugia (I EIOPA (0: Calibraion of h Prmium and Rsrv Risk Facors in h Sandard Formula of Solvncy II, Rpor of h Join Working Group on Non-Lif and Halh NSL Calibraion. EIOPA /63, Dcmbr 0 EIOPA (04a: h undrlying assumpions in h sandard formula for h Solvncy Capial Rquirmn calculaion. hps://iopa.uropa.u EIOPA (04b: chnical Spcificaions for h Prparaory Phas (Par I and II. hps://iopa.uropa.u Europan Commission (05: Commission Dlgad Rgulaion (EU 05/35 of 0 Ocobr 04 supplmning Dirciv 009/38/EC of h Europan Parliamn and of h Council on h aking-up and pursui of h businss of Insuranc and Rinsuranc (Solvncy II. Official Journal of h Europan Union, Volum 58, 7 January APPENDIX 8. SECOND PARIAL DERIVAIVE WIH RESPEC O Compuing h scond parial drivaiv wih rspc o w g

26 88 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA g(, = Z ( = ( Z = Z = = ( 4 ( 4 ( ( = Z ( ( 4 ( ( = Z ( 4 Z ( = 4 ( 4 = ( = Z ( 4 ( (

27 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 89 = Z ( 4 4 Z ( = Z 4 ( = = ( ( 4 ( 4 ( = 4 4 Z (. Evaluaing i a (, yilds appro appro 4 appro 4 g( appro, appro = Z Z appro appro ( = = 4 appro appro appro appro 8 4 Z Z 4 appro appro ( = = 4 appro ( appro appro 4 appro appro appro 4 4 Z Z appro appro ( = = 4 appro appro appro 8 4 = appro 3 appro appro ( 4 appro ( appro appro

28 90 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA sinc i holds ru 8. 4 appro appro appro 4 4 ( appro appro 4 4 appro appro appro 4 = > 0 appro appro ( Z Z = = appro = (, approimaion appro =. Sinc i holds = (, using h linar approimaion for clos o w hav (, appro Z Z = = = ( = ( = Z Z = = = = ( ( = Z Z = = = =

29 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 9 ( ( = = = Z Z = = 8.3 (, appro approimaion (, appro = p appro Z = (, appro (, (, = appro = appro Z = Z Z ( = = = p appro ( ( ( ( = = = = Z Z Z Z = = = = Z p = ( ( = = = ( =

30 9 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA = = appro = =p Z Z Z = = = = p ( ( Z Z = = = = = p ( ( ( Z Z appro appro =p ( = = = = p ( ( Z Z = = = = = p ( ( ( Z Z

31 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA (, approimaion = appro appro = (, (, = p (, Z = ( appro appro = appro appro = ( appro appro = ( p p ( Z ( appro = appro = appro appro = appro = ( ( =p p ( Z

32 94 F. SIEGENHALER V. DEMARCO R. R. CERCHIARA ( appro appro = appro = ( =p p ( = appro = appro = appro = ( p ( Z p appro ( appro = appro = ( p (

33 ON HE USP CALCULAION UNDER SOLVENCY II AND IS APPROXIMAION WIH A CLOSED FORM FORMULA 95 Z Z = = p ( ( = = Z p Z = = ( ( = = ( = p appro ( appro.