Trasrm apprach r peraial risk mdellig: VaR ad TCE Jiwk Jag Deparme Acuarial Sudies Divisi Ecmic ad Fiacial Sudies Macquarie Uiversiy, Sydey 2109, Ausralia jjag@es.mq.edu.au Geyua Fu PricewaerhuseCpers Ceer 202 Hubi Rad Shaghai 200021, Peple s Republic Chia paul.u@c.pwc.cm Thisversi:April1,2007 Absrac T quaiy he aggregae lsses rm peraial risk, we emply acuarial risk mdel, i.e. we csider cmpud Cx mdel peraial risk deal wih schasic aure is requecy rae i realiy. A sh ise prcess is used r his purpse. A cmpud Piss mdel is als csidered as is cuerpar r he case ha peraial lss requecy rae is deermiisic. As he lss amus arisig due mismaageme peraial risks are exremes i pracice, we assume he lss sizes are Lggamma, Fréche ad rucaed Gumbel. We als use a expeial disribui r he case-exremelsses. Emplyiglss disribui apprach, we derive he aalyical/explici rms he Laplace rasrm he disribui aggregae peraial lsses. The Value a Risk (VaR) ad ail cdiial expecai (TCE, als kw as TailVaR) are used evaluae he peraial risk capial charge. Fas Furier rasrm is used apprximae VaR ad TCE umerically ad he igures he disribuis aggregae peraial lsses are prvided. Numerical cmpariss VaRs ad TCEs baied usig w cmpud prcesses are als made respecively. Keywrds: Operaial risk; al lss; he cmpud Piss/Cx prcess; sh ise prcess; lss disribui; VaR; ail cdiial expecai (TCE); Fas Furier rasrm. 1
1. Irduci A capial charge r peraial risk is required he iacial isiuis. The Basel Cmmiee r Bakig Supervisi (2006) deies peraial risk as llws: The risk lsses resulig rm iadequae r ailed ieral prcesses, peple ad sysems r rm exeral eves. A lis lss eve ypes (level 1) peraial risks is shw i Table 1.1 ha is adped rm Aex 9 Basel Cmmiee Bakig Supervisi (2006). Table 1.1 Eve-Type Caegry (Level 1) Ieral raud Exeral raud Emplyme Pracices ad Wrkplace Saey Clies, Prducs & Busiess Pracices Damage Physical Asses Busiess disrupi ad sysem ailures Execui, Delivery & Prcess Maageme Deiii Lsses due acs a ype ieded deraud, misapprpriae prpery r circumve regulais, he law r cmpay plicy, excludig diversiy/ discrimiai eves, which ivlves a leas e ieral pary Lsses due acs a ype ieded deraud, misapprpriae prpery r circumve law, by a hird pary Lsses arisig rm acs icsise wih emplyme, healh r saey laws r agreeme, rm payme persal ijury claims, r rm diversiy/discrimiai eves Lsses arisig rm a uieial r eglige ailure mee a pressial bligai speciic clies (icludig iduciary ad suiabiliy requiremes), r rm he aure r desig a prduc Lsses arisig rm lss r damage physical asses rm aural disaser r her eves Lsses arisig rm disrupi r sysyem ailures Lsses rm ailed rasaci prcessig r prcess maageme, rm relais wih rade cuerparies ad vedrs The cllapse Briai s Barigs Bak i February 1995 is perhaps he quiesseial ale peraial risk maageme ge wrg. A similar eve mre severe ailure came ligh i he las ew weeks a he Frech bak Sciee Geerale. Bh ailures were cmpleely uexpeced. Over he curse days, Barigs, Briai s ldes mercha bak, we rm appare sregh bakrupcy. I bh cases he ailure ad upheaval was caused by he acis a sigle rader. The esimaed lss r Barigs was 700milli while irs esimaes he Sciee Geerale lss are arud $US 7b. T quaiy he aggregae lsses rm peraial risk, i his paper we use a acuarial risk mdel (Cramér 1930; Bühlma 1970; Gerber 1979; Gradell 1976, 1991; Beard e al. 1984 ad Asmusse 2000). Csiderig e lie busiess, le X i,i=1, 2,, be he lss amus rm ype k peraial risk, which are assumed be idepede ad ideically disribued wih disribui uci H (x) (x >0), he he al lss arisig rm ype k peraial risk up ime is deied by = XN 2 i=1 X i, (1.1)
where k =1, 2,,d ad N is he al umber lsses up ime. We assume ha he prcess N ad he sequece {X i } i=1,2, are idepede each her. The grad al lss is hece give by L = dx k=1. (1.2) Accrdig he Basel II Advaced Measureme Apprach (AMA) guidelies, he iacial isiuis may use he Value a Risk (VaR r he q-quaile) as a risk measure decide he capial amu required r ex years peraial risk, i.e. VaR 99.9% (L ). Hwever ³ bai he VaR 99.9% (L ), i requires derive he ji disribui he al lss radm vecr L (1),L (2),, L (d), which is a challegig ask. Accrdigly, he Basel II AMA guidelies prpse use dx VaR 99.9% ( ) (1.3) k=1 r a capial charge ad csider a diversiicai eec uder apprpriae crrelai assumpis, i.e. Ã dx! dx VaR 99.9% (L )=VaR 99.9% VaR 99.9% ( ). (1.4) k=1 k=1 This assumpis mus be made persuadable he lcal regulars. Numerus papers have lked a he mdellig peraial lsses arisig rm several surces ad heir depedece. The wrk by Nešlehvá e al. (2006) ad he paper by Chavez-Demuli e al. (2006) cai umerus mdels his eec. The hree issues hey address are: Issue 1: The peraial lss disribui is exremely heavy-ailed. Issue 2: The peraial lss arrival ime is irregular ad here exiss a edecy icrease ver ime. Issue 3: The prblem mdellig he depedece bewee varius peraial risk surces ha may lead a reduci he calculaed risk capial. Fr simpliciy, i his paper, we igre he crrelai assumpis, i.e. we assume ha,k= 1, 2,,d are idepede each her bu ideical. I rder calculae he each cmpe ), we eed calculae he disribui he al lss, i.e. P l. Hwever he calculai P l i geeral is diicul ad i ca be derived explicily. S i Seci 2 we derive he explici ad aalyical expressis he Laplace rasrms he disribuis (1.3), i.e. VaR 99.9% ( he al lss Seci 4. ad iver heir Fas Furier rasrms calculae VaR 99.9% ( We als calculae he ail cdiial expecai deied by E VaR 99.9% ( ) as a chere risk measure (Arzer e al. 1999) ad calculae ) umerically i (1.5) dx E VaR 99.9% ( ) k=1 (1.6) 3
as capial amu required r ex years rm all ypes peraial risk. As examied i Mscadelli (2004) ha lsses arise rm he mismaageme peraial risk are heavy-ailed i pracice, i Seci 2 we emply Lggamma, Fréche ad rucaed Gumbel as lss size disribuis deal wih his issue. We als use a expeial disribui r he case -heavy-ail lsses. A discussi he echiques exreme value hery; see r isace, Embrechs e al. (1997). T ccer irregular arrival peraial lsses ad is edecy icrease ver ime, we use he Cx prcess wih sh ise iesiy λ r he lss arrival prcess N. A hmgeeus Piss prcess wih lss requecy λ is als examied as is cuerpar. I Seci 3, we prese he expressis r iiial prbabiliies he al lss ad he expressis r iiial value is desiies, which are required imprve he accuracy he disribuis he al lss iverig he Fas Furier rasrms. We cmpare simulaed umerical values VaRs ad TCEs baied usig cmpud Piss ad cmpud Cx mdel respecively i Seci 4. Seci 5 cais sme ccludig remarks. 2. The Laplace rasrm he disribui al lss I rder evaluae he risk measures VaR ad TCE, i is ecessary r us calculae he disribui al lss. Hwever i is diicul derive i explicily. Hece r ha purpse, we csider usig he Laplace rasrm as i ca be ivered calculae releva risk measures (1.3) ad (1.6) umerically. 2.1. Hmgeeus Piss prcess As we ca see i Table 1.1, raud, busiess disrupi, execui errr ad sysem ailure ec. are primary eves. I rder measure he ccurrece peraial lsses u hese primary eves, we eed a cuig prcess deal wih deermiisic r schasic aure heir arrival raes i pracice. Therere i is aural use pi prcesses csider series peraial lsses. The simples e is usig a hmgeeus Piss prcess ha has deermiisic requecy. Assumig ha he lss arrival prcess N llws a hmgeeus Piss prcess wih lss requecy λ ad ha 0 =0, he Laplace rasrm he he disribui al lss is give by h i E e νl(k) =exp λ 1 m(ν), (2.1) where ν 0 ad m(ν) = Z 0 e νx dh(x) <. (2.2) As i has bee kw ha lsses arise rm he peraial risk are exremes i pracice (Mscadelli, 2004), i his paper we csider hree heavy-ailed disribuis, i.e. a Lggamma, h(x) = ½ µ ¾ βα x α 1 µ x β 1 l +1 +1,x>0, σ 2 > 0, β>0 ad α>0, (2.3) σ 2 Γ (α) σ 2 σ 2 a Fréche, h(x) = ς µ ( x ς 1 µ ) x ς exp,x 0, σ 3 > 0 ad ς>0, (2.4) σ 3 σ 3 σ 3 ad a rucaed Gumbel, h(x) = exp {exp (ζ/η)} 1 exp {exp (ζ/η)} 1 η exp ½ x ζ µ exp η 4 x ζ η ¾,x 0, ζ>0 ad η>0. (2.5)
I he lss amus arisig due mismaageme peraial risk are exremes, we may csider usig a expeial r lss size disribui, i.e. h(x) = 1 µ exp 1 x, x 0, σ 1 > 0. (2.6) σ 1 σ 1 Usig (2.2)-(2.6), we ca easily bai he crrespdig expressis r he Laplace rasrm he disribui al lss,i.e. E e νl(k) =exp λ + λ Z ³ exp νσ 2 exp(β 1 z 1/α ) 1 z 1/α dz, (2.7) αγ (α) where z = where z = β l ³ x σ 2 +1 α, ς xσ3, E E e νl(k) e νl(k) where η = σ 4,c= e ζ/η, Γ(φ; ϕ) =exp λ + λ 0 Z 0 ³ exp νσ 3 z 1/ς z dz, (2.8) ½ ¾ exp {exp (ζ/η)} = exp λ + λ exp ( νζ) Γ(νη +1;e ζ/η ) exp {exp (ζ/η)} 1 ½ ¾ = exp λ + λc νσ 4 1 e c Γ(νσ 4 +1;c) E φr 0 ³ z φ 1 e z dz, z =exp e νl(k) =exp ½ λ x ζ η ad µ σ1 ν 1+σ 1 ν (2.9) ¾. (2.10) 2.2. Sh-ise Cx prcess T deal wih schasic aure peraial lss arrival i pracice, we csider a Cx prcess as a aleraive pi prcess. The Cx prcess prvides lexibiliy by leig he iesiy ly deped ime bu als allwig i be a scasic prcess. Therere he Cx prcess ca be viewed as a w sep radmisai prcedure. A prcess λ is used geerae aher prcess N by acig is iesiy. Tha is, N is a Piss prcess cdiial λ which isel is a schasic prcess. Lsses arisig rm he mismaageme peraial risks deped he iesiy primary eves. Oe he prcesses ha ca be used measure he impac primary eves is he sh ise prcess. Sme wrks isurace applicai usig sh ise prcess ad a Cx prcess wih sh ise iesiy ca be ud i Klüppelberg & Miksch (1995), Dassis & Jag (2003) ad Jag & Krvavych (2004). The sh ise prcess is paricularly useul i lss arrival prcess as i measures he requecy, magiude ad ime perid eeded deermie he eec primary eves. As ime passes, he sh ise prcess decreases as mre ad mre lsses are igured u. This decrease ciues uil aher eve ccurs which will resul i a psiive jump i he sh ise prcess. Therere he sh ise prcess ca be used as he parameer a Cx prcess measure he umber peraial lsses, i.e. we will use i as a iesiy uci geerae a Cx prcess. We will adp he sh ise prcess used by Cx & Isham (1980): XM λ = λ 0 e δ + Y i e δ( S i) 5 i=1
Figure 1: Graph illusraig sh ise prcess where: λ 0 is he iiial value λ ha is carried rm primary eves icurred previusly; {Y i } i=1,2, is a sequece idepede ad ideically disribued radm variables wih disribui uci G (y) (y > 0) ad E (Y ) < (i.e. magiude cribui primary eve i iesiy); {S i } i=1,2, is he sequece represeig he eve imes a Piss prcess wih csa iesiy ρ; δ is he rae expeial decay. Sme eves such as ieral raud, may ake much lger maerialise ha hers s he decay rae may be expeial. I is assumed be his rm r a maer cveiece, i.e. clsedrm expressis ial resuls are easily derived. We als make he addiial assumpi ha he Piss prcess M ad he sequeces {Y i } i=1,2, ad {X i } i=1,2, are idepede each her. Figure 1 illusraes sh ise prcess. Nw le us assume ha he lss arrival prcess N llws a Cx prcess wih is iesiy λ. Figure 2 illusraes a Cx prcess wih sh ise iesiy. Similar a hmgeeus Piss prcess r N, he Laplace rasrm he he disribui al lss is give by h i E e νl(k) λ 0 = E exp 1 m(ν) Λ λ 0, (2.11) where λ 0 is assumed be kw. The equai (2.11) suggess ha he prblem idig he Laplace rasrm disribui, is equivale he prblem idig he Laplace rasrm disribui Λ = R 0 λ s ds, he aggregaed prcess. Assumig ha jump size primary eve llws a expeial disribui, i.e. g (y) =b exp( by), y>0, b>0 ad λ is saiary, he explici expressi (2.11) is give by 6
Figure 2: Graph illusraig he Cx prcess wih sh ise iesiy 7
E e νl(k) = δb + 1 m(ν) 1 e δ δbe δ bρ δb+ 1 m(ν) ρ δ. (2.12) Fr deails he abve expressi, we reer he reader Dassis ad Jag (2003). We mi he crrespdig expressis r he Laplace rasrm he disribui al lss usig (2.3)-(2.6) as hey ca be easily baied. I {Y i } i=1,2,, which are he magiude cribui primary eve iesiy λ,arehigh, we eed csider heavy-ailed disribuis r jump size primary eve G (y). I causes higher umber peraial lss csequely ad eveually he iacial isiuis eed prepare higher peraial risk capial charge as he risk measures VaR ad TCE becme higher. This primary evejumpsizemeasureg (y) als ca be relaed wih lss size measure H (x) i here exiss depedece bewee hem, e.g. he higher he magiude cribui primary eve is, he higher lsses rm he peraial risk arise. Cmpared (2.1), he abve Laplace rasrm prvides he iacial isiuis wih mre lexibiliy i peraial risk mdellig as i cais schasic iesiy wih hree parameers δ, ρ ad G (y). 3. Tal lss disribui via he Fas Furier rasrm I rder calculae he risk measures (1.3) ad (1.6), we iver he Fas Furier rasrms rm he Laplace rasrms baied i Seci 2. Fr deails hw iver he Fas Furier rasrm, we reer yu Hes (1993), Duie e al. (2000), Caslema (1996), Gzalez ad Wds (2002) ad Gzalez e al. (2004). Bere we shw he calculais risk measures i Seci 4, we prese he expressis r iiial prbabiliies al lss ad he expressis r iiial value is desiies. These are required imprve he accuracy he disribuis he al lss iverig he Fas Furier rasrms. I we le ν i (2.1), we have he expressi r iiial prbabiliy al lss, i.e. P =0 = e λ. (3.1) Regardless lss size disribuis, we have he same iiial prbabiliy al lss whe he lss arrival prcess N llws a hmgeeus Piss prcess wih lss requecy λ. I we se h i lim ν exp λ 1 m(ν), ν we have he expressi r iiial value he desiy al lss, i.e. =0 = λe λ h (0), (3.2) where is he desiy uci al lss. Based (3.2), we ca easily bai he expressis r iiial prbabiliies al lss, i.e. r a expeial lss size, r a Lggamma lss size =0 = λe λ, (3.3) σ 1 =0 = 8 0, α > 1 λβe λ,α=1, α < 1 σ 2, (3.4)
r a Fréche lss size ad r a rucaed Gumbel lss size =0 =0, (3.5) =0 = λc (e c 1) e λ σ 4. (3.6) Similarly, i we le ν i (2.11), we have he expressi r iiial prbabiliy al lss, i.e. µ P =0 = δbe δ 1 e δ + δb ρ δ(1+δb), (3.7) Regardless lss size disribuis, we als have he same iiial prbabiliy al lss whe he lss arrival prcess N llws he Cx prcess wih sh ise iesiy λ. I we se lim ν ν δb + 1 m(ν) δbe δ 1 e δ bρ δb+ 1 m(ν) we have he expressi r iiial value he desiy al lss, i.e. ρ δ, =0 = h (0) ρ µ δbe δ ρ δ(1+δb) 1+δb 1 e δ + δb ½µ µ b 1 e δ + δb l 1+δb δbe δ + 1 e δ ¾ δ (1 e δ. (3.8) + δb) Based (3.8), we ca easily bai he expressis r iiial prbabiliies al lss, i.e. r a expeial lss size, =0 = µ ρ δbe δ σ 1 (1 + δb) 1 e δ + δb ½µ µ b 1 e δ + δb l 1+δb δbe δ ρ δ(1+δb) + 1 e δ ¾ δ (1 e δ, + δb) (3.9) r a Lggamma lss size =0 = r a Fréche lss size ad r a rucaed Gumbel lss size ρ ½ βρ δbe δ ³ δ(1+δb) b σ 2 (1+δb) 1 e δ +δb 1+δb 0, α > 1 ¾ l 1 e δ +δb + 1 e δ,α=1 δbe δ δ(1 e δ +δb), α < 1, (3.10) =0 =0, (3.11) 9
Figure 3: The disribui al lss wih respec Piss/Cx prcess wih Expeial lss size disribui =0 = µ cρ δbe δ σ 4 (e c 1) (1 + δb) 1 e δ + δb ½µ µ b 1 e δ + δb l 1+δb δbe δ + ρ δ(1+δb) 1 e δ ¾ δ (1 e δ. + δb) (3.12) Figure 3-6 are he disribuis al lss wih respec a Piss prcess ad a Cx prcess r N respecively, where lss size disribuis are Expeial, Lggamma, Fréche ad rucaed Gumbel. I shws ha he disribuis al lss wih respec a Cx prcess have heavier ail ha heir cuerpars wih respec a Piss prcess. I will becme appare by umerical values VaRs ad TCEs i Example 4.1-4.4. Sice we derive he prbabiliy desiies r al lss umerically via he Fas Furier rasrm, all ³ values he prbabiliy desiies i Figure 3-6 are apprximaed values excep he irs pi, =0 ad P =0. These w values are calculaed usig he explici rmulae abve. The irs pi, =0 is usually disred aer he Fas Furier rasrm s we replace hese disred values wih he values baied rm he explici rmulae =0. 10
Figure 4: The disribui al lss wih respec Piss/Cx prcess wih Lggamma lss size disribui 11
Figure 5: The disribui al lss wih respec Piss/Cx prcess wih Fréche lss size disribui 12
Figure 6: The disribui al lss wih respec Piss/Cx prcess wih rucaed Gumbel lss size disribui 13
4. Calculaig risk measures Nw wih w risk measures, i.e. VaR q ( )=i l R : P ( >l) 1 q ad h i E TCE q ( )=E VaR q ( I VaR q ( ) ) = (4.2) (1 q) where I ( ) is he idicar uci, le us illusrae heir umerical values rm he iversi he Fas Furier rasrms. The parameer values used simulae N ad calculae he abve risk measures are λ =10,ρ=4,b=1,δ=0.4 ad =1. We use he abve parameer values ha prvide us wih he same meas al lss regardless he speciicai he lss arrival prcess N see he diereces he VaRs ad TCEs due he ails he lss size disribuis, i.e. E Piss I rder make he cmpuig easier, we als chse i.e. ad = E Cx. E Expeial (X) =E Lggamma (X) =E Fréche (X) =E rucaed Gumbel (X) = π, ½µ β α σ 1 = σ 2 1¾ β 1 µ = σ 3 Γ 1 1 ½ = σ 4 (l c) ς 1 1 e c σ 1 = π ad σ 2 = σ 3 = σ 4 =1. Frm (4.3), we have he relaiship r he parameers, i.e. Z c 0 (4.1) ¾ (l y) e y dy = π (4.3) µ β α = π +1, β > 1 ad α 1, β 1 µ Γ 1 1 ς = π, ς > 1, (l c) 1 1 e c Z c 0 (l y) e y dy = π. Usig Malab, he VaRs ad TCEs r each lss size disribui wih respec a Piss/a Cx prcess are shw i Table 4.1-4.8. Example 4.1: Expeial The calculais he w risk measures as capial charges rm ype k peraial risk up ime whe lss size llws a expeial are shw i Table 4.1 ad 4.2, where Var(X) =π. 14
Table 4.1: Piss prcess q VaR q ( ) TCE q ( ) 0.999 49.5368 53.3579 0.99 39.8691 44.1164 0.95 32.1210 36.8966 0.9 28.3286 33.4675 0.5 16.8305 23.9688 where E =10 π, Var =20π Table 4.2: Cx prcess q VaR q ( ) TCE q ( ) 0.999 57.9834 63.0779 0.99 45.2875 50.8550 0.95 35.3129 41.4676 0.9 30.5114 37.0697 0.5 16.4034 25.1592 where E =10 π, Var =90.45 Table 4.1 ad 4.2 shw ha here is sigiica icrease i w risk measures respecively by chagig N rm a hmgeus Piss prcess a sh-ise Cx prcess as lss size measure H (x) is a expeial which is a heavy-ailed disribui. I als shws ha TCEs are slighly higher ha VaRs regardless he lss arrival prcess N. Example 4.2: Lggamma The calculais he w risk measures as capial charges rm ype k peraial risk up ime whe lss size llws a Lggamma are shw i Table 4.3 ad 4.4, where α =1,β= π+1 π ad Var(X) =. Table 4.3: Piss prcess Table 4.4: Cx prcess q VaR q ( ) TCE q ( ) 0.999 337.2020 1017.4245 0.99 99.0571 246.1027 0.95 43.7821 97.4792 0.9 31.3680 67.0061 0.5 11.9483 28.2826 where E =10 π, Var = q VaR q ( ) TCE q ( ) 0.999 378.9434 1018.9412 0.99 100.9566 247.8994 0.95 45.5266 99.3334 0.9 32.7244 68.7072 0.5 11.7146 28.9562 where E =10 π, Var = Table 4.3 ad 4.4 shw ha here is sigiica icrease i w risk measures respecively by chagig N rm a hmgeus Piss prcess a sh-ise Cx prcess as lss size measure H (x) is a Lggamma which is a heavy-ailed disribui. I als shws ha TCEs are much higher ha VaRs regardless he lss arrival prcess N. Example 4.3: Fréche The calculais he w risk measures as capial charges rm ype k peraial risk up ime whe lss size llws a Fréche are shw i Table 4.5 ad 4.6, where ς =2ad Var(X) =. Table 4.5: Piss prcess Table 4.6: Cx prcess q VaR q ( ) TCE q ( ) 0.999 118.9029 218.5851 0.99 51.6413 82.8136 0.95 33.4748 48.0499 0.9 28.0027 39.2087 0.5 15.8570 24.2760 where E =10 π, Var = q VaR q ( ) TCE q ( ) 0.999 121.1252 220.5757 0.99 55.1842 85.8063 0.95 36.5327 51.3762 0.9 30.2885 42.1960 0.5 15.5337 25.4611 where E =10 π, Var = Similar Lggamma case, we ca see i Table 4.5 ad 4.6 ha w risk measures icrease respecively by chagig N rm a hmgeus Piss prcess a sh-ise Cx prcess. I als shws ha TCEs are higher ha VaRs regardless he lss arrival prcess N. 15
Example 4.4: Trucaed Gumbel The calculais he w risk measures as capial charges rm ype k peraial risk up ime whe lss size llws a rucaed Gumbel are shw i Table 4.7 ad 4.8, where c =2.97957 ad Var(X) =1.51625. Table 4.7: Piss prcess q VaR q ( ) TCE q ( ) 0.999 43.5250 46.4355 0.99 36.0213 39.3241 0.95 29.8658 33.6550 0.9 26.7942 30.9191 0.5 17.1443 23.1380 where E =10 π, Var =46.58 Table 4.8: Cx prcess q VaR q ( ) TCE q ( ) 0.999 53.0242 57.3668 0.99 42.1127 46.9002 0.95 33.4607 38.7967 0.9 29.2596 34.9745 0.5 16.6829 24.4921 where E =10 π, Var =74.19 Table 4.7 ad 4.8 shw ha here is sigiica icrease i w risk measures respecively by chagig N rm a hmgeus Piss prcess a sh-ise Cx prcess. I als shws ha TCEs are slighly higher ha VaRs regardless he lss arrival prcess N. Ieresigly, he values w risk measures are lwer ha heir cuerpars calculaed usig expeial lss size disribui i Example 4.1 whe q 0.9. Whe q =0.5, he VaRs/TCEs are slighly higher/lwer ha heir cuerpars calculaed usig expeial lss size disribui i Example 4.1. 5. Cclusi We used a cmpud Cx prcess mdel al lsses arisig rm peraial risk accmmdae schasic aure heir requecy raes i pracice. The sh ise prcess was used as a iesiy a Cx prcess as he umber lsses arisig rm peraial risk depeds he requecy ad magiude primary eves ad ime perid eeded deermie he eec primary eves. We als examied a cmpud Piss prcess as i cuerpar. T deal wih a issue raised by Mscadelli (2004) ha he lsses arise rm he mismaageme peraial risk are heavy-ailed i pracice, we csidered Lggamma, Fréche ad rucaed Gumbel as lss size disribuis. We als used a expeial disribui r he case -heavy-ail lsses. As i is diicul calculae he disribuis al lss, we derived heir Laplace rasrms ad ivered heir Fas Furier rasrms umerically calculae releva risk measures, i.e. VaR ad TCE. We preseed he expressis r iiial prbabiliies he al lss ad he expressis r iiial value is desiies, which were used imprve he accuracy he disribuis he al lss iverig he Fas Furier rasrms We als cmpared simulaed umerical values VaRs ad TCEs baied usig cmpud Piss ad cmpud Cx mdel respecively. We examied ur diere lss size disribuis wih w cuig prcesses rea he issues aced by he praciiers i bak ad iacial isiuis. Risk measures csidered bai he peraial risk capial charge were VaRs ad TCEs. We hpe ha wha we preseed i his paper prvides he praciiers wih easible mdels measure peraial risk capial charge wih lexibiliy usig real daa available. There are several appraches mdel ierdepedece bewee peraial lss prcesses, e.g. liear crrelai r cpula-based -liear crrelai. Fr simpliciy, we assumed depedece bewee peraial risk ypes s we leave i as a urher research. Reereces 16
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