ON SOME MODELS OF ACCEPTABLE RISK. Maxim Finkelstein

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1 ON SOME MODELS OF ACCETABLE RISK Mxm Fkelse Derme of Mheml Sss Uversy of he Free Se O Box Bloemfoe Rebl of Soh Afr e-ml: FkelM.SCI@fs..z d Mx lk Ise for Demogrh Reserh Rosok Germy ABSTRACT We osder dsree d oos rsk dsrbo fos. Aeble rsk dsrbo fo s defed d dffere yes of sohs omrsos re dsssed. Aeble eble d ermede regos for he level of loss re deermed. The smlr hrerzo s sed for desrbg he loss for he oomes he seqee of hrmfl eves. The loss s osdered s eble f eher ll eves resl loss from he eble rego or o more h k of hem resl loss from he ermede level. The Lle rsform mehods re sed for obg he robbly of srvvl whe hrmfl eves from he osso roess re oo lose o eh oher. Keywords: Aeble rsk Sohs omrso osso roess Flre re. INTRODUCTION Rsk s slly dersood s dger h oelly hrmfl eves rerese o hm begs he evrome or he eoom vles. Nmerl oomes of hese eves be osdered s relzos of rdom vrble C whh oree los desrbe eoom loss mber of sles e. Therefore hs er rsk s mesred by he orresodg loss d we wll se hese erms erhgebly. Uslly see for exmle Ushkov d Hrrso 994 Vrjlg 995 he dsree seg s desrbed erms of robbles d oomes s he followg seqee:... where s he robbly of orree of relzo wheres s he vle of loss ssoed wh hs relzo. Some of hese relzos be hrmless whh mes h he loss hs se s eql o. The orresodg robbly s deoed by. Who loss of geerly ssme h re srly ordered: < < <... < <.

2 Therefore he mlve dsrbo fo Cdf of rsk G s smly defed v d s + G The exeed vle of loss s: < < < < 3 E[ C] 4 Relosh 4 be lredy osdered s some mesre of rsk lhogh slly s oo rde. I be lso sed for omrso of dffere rsks. Oher relev yes of sohs omrsos wll be dsssed he ex seo. Smlr o he dsree se he Cdf G r[ C ]. 5 s he robbly h he loss wll o exeed : <. Closely reled o he oo of rsk s he oo of srvvbly. I be erreed my wys. By srvvbly of some sysem we shll dersd s bly o erform he reqred fo who dmge d loss or wh dmge or loss o exeedg some resrbed vle. The robbly of erformg hs fo der sed odos defes he orresodg mesre of srvvbly. COMARISON OF RISKS As loss s slly evble s mor o orol d mmze. A my ses sefl oo of eble loss be defed d he sysem s erforme be qlfed s eble f for se: E[ C]. 6 The vle of slly reses rher rl reqreme.e. he me loss shold be boded by some resoble mo. Orderg 6 however s obvosly o lwys sffe bese he lrge flos of dmge ofe o be eed. The rl wy o del wh hs roblem from he heorel o of vew s o osder he geerlzo of 6 v he eble rsk fo. Le C deoe rdom vrble of eble loss wh G s s Cdf. I mes h we e he rel loss C f s less some resoble sohs sese h C. The orderg omrso of he mes of he orresodg Cdfs s he smles

3 3 orderg of hs ye. Defg G s mh more omlex sk h defg 6 d eed deled robbls lyss se sdes of smlr sysems or sos exer s oos e. If C does o exeed C he we ssme h he rel loss s eble. Ths he m ses he robbls rsk lyses of he desrbed ye re: o ob G o defe G d o erform he orresodg sohs omrso. I wh follows hs seo we osder some smle ses of sohs omrso for rsks. A well-kow Ross 996; Shked d Shkmr 6 d wdely sed ye of sohs omrso s he sl sohs orderg defed s follows. If G G [ ; G G G G 7 he C s C 8 d he loss s osdered s eble. Ieqly 8 s rher srog ye of orderg. O he oher hd eqly 6 desrbes he wekes verso of omrso. I s eresg o osder some relev ermede yes of orderg kg o o vrbly of C. Aordg o Ross 996 see lso Ks e l rdom vrble o be sohslly more vrble h rdom vrble C f for ll b : b C s sd G d G d. 9 Whe b eqo redes o he omrso of he mes. The Cdf G for he dsree se be defed by eqo 3 where shold be sbsed by. If eble vles of... re obed hs shold be erformed for eh sef so d be bsed o vros formo ldg exer oos e. he he obvos ee rle s: b... d s lerly see from 3 h hese eqles led o sohs orderg 7. Ths s rher ovee sffe odo for hs orderg. I s lso ler h s o eessry odo whh be llsred by he followg exmle. Exmle Le. The < G < <

4 4 d G s obed v sbsg by resevely. Ieqly 7 holds for hs se f + + I s obvos h eqles for mly b o vs vers. O he oher hd he omrso vrbly 9 leds o: As > eqles mly eqles 3 b o vs vers. Ths f llsres for he sef se der osdero h he omrso vrbly s ermede bewee he sohs omrso 7 d he omrso he me. The ler hs exmle s defed by he frs eqly 7. Usg robbles d smlr o 4 oe defe he eble me loss for he geerl dsree se s Ths f he E[ C] E[ C ]. E[ C]. 4 The verse s o eessrly re s llsred by Exmle : here exs seqees of robbles for whh eqly 4 s vld wheres eqly does o hold. 3. RECURRENT EVENTS 3. The roess of hrmfl shoks I hs seo we wll osder rsks sed by rerre eves. Assme h seqee of ossbly hrmfl seos eves s modeled by sohs o roess. Assme lso for smly h s o-homogeeos osso roess NH wh re λ. For oveee le s ll hese eves shoks. As revosly eh shok s sg rdom loss of mo C. Le C... be..d rdom vrbles wh he oos Cdf G. Or eres s osderg overll oseqees of shoks [. Dvde he xs o regos ˆ ] ˆ ˆ ]... ˆ. 5 The robbly h he loss does o exeed level ĉ s G ˆ d he robbly h s he rego ˆ ˆ ] j; j < ; ˆ s j <

5 5 G ˆ G ˆ G ˆ G ˆ G G ˆ. 6 j j The frs mor se s o derve he robbly j h ll eves h orred ] hd resled loss o exeedg ĉ. Ths robbly be defed s whh be esly see drely: where ex λ x dx 7 Λ Λ k k ex{ } ex λ x dx k k! Λ λ d. The orresodg roof for more geerl se whe re fos of me be fod Blok e l 985 Fkelse 3 Thomso 988 o me few. Smlr o 7 he robbly h ll eves hd resled loss he rge from o j be defed s Seflly for he 3 regos: ex j λ x dx. 8 j s ex s x dx ex s λ x dx 9 s ex λ x dx where s s he robbly h ll eves from he osso roess [ resl [ s sfe loss ; s deoes he robbly h ll eves resl loss. Evelly deoes he slemery o hvg rl more robbly h ll eves resl loss he rego [. Hee he sroges rero of he eble rsk s whe ll eves resl loss from he frs rego. The he erforme of sysem be osdered s eble. I s resoble o osder weker verso of he ee rero llowg for se o more h k... eves o resl loss from he ermede rego eve [ s o llowed ll. Ths we w o [ s k ssess he robbly s h ll eves resl loss o exeedg wheres o more h k of hem re llowed o resl loss. Le for smly [ s

6 6 he l roess be he H wh re λ. The be sl o 3 deede osso roesses wh res λ λ λ. s s De o or wekeed ee rsk rero he rsk [ s defed s eble f les oe eve from he roess wh re λ wll or or f more h k eves from he roess wh re λ wll or. These osderos led o he s followg eqo for he robbly of sfe wh eble rsk erforme: s k k λs ex{ λ }ex{ λs }.! Whe here s o ermede rego: we rrve s ex{ λ } ex{ } 3 s s s whh odes wh he frs eqo 9 for hs sef se. 3.. Oher ee rer Le ow exerl shok s m be desrbed by bry rdom vrbles wh oomes srvved or o srvved. The ler eve for oveee wll be lled flre lhogh rerese de dsser e. If eh eve from he osso roess wh re λ s srvved wh robbly d s o srvved wh omlemery robbly he smlr o 7 he srvvl robbly ll shoks re srvved [ s gve by ex λ x dx. 4 Cosder ow dffere rero of srvvl. Assme h shoks from he NH wh re λ re hrmless f hey re rher rre. A flre of sysem ors oly whe shoks re oo lose sysem dd o reover from he oseqees of revos shok. Le he reoverg me τ be rdom wh he Cdf R. The orresodg srvvl robbly be wre s Fkelse 7 ex + xex d + x d x d yex y dr y ˆ x y dydx 5 where he frs erm he rgh hd sde s he robbly h here ws o more h oe shok [ d he egrd defes he jo robbly of he followg eves

7 7 -he frs shok orred [ x x + dx -he seod shok orred [ x + y x + y + dy -he me bewee wo shoks y s sffe for reoverg robbly- R y -he sysem s fog who flres [ x + y. By ˆ 5 we deoe he robbly of sysem s fog who flres [ gve h he frs shok hd orred. Smlr o 5: x ˆ ex λ d + xex dr x ˆ x dx. 6 Eqos 5 d 6 be solved merlly. For he se of he homogeeos osso roess wh re λ hese eqos be esly solved v he Lle rsform Fkelse 7: s[ λr s + λ] λ R s + λ + λ s 7 s + λ [ λr s + λ] where s d R s deoe Lle rsforms of d R resevely. Exmle. Le τ be exoelly dsrbed: R ex{ µ }. Iverg he Lle rsform 7 for hs sel se: A ex{ s } + A ex{ s } 8 where s s d λ + µ ± λ + µ 4λ s s + λ + µ s + λ + µ A A. s s s s Aoher exmle of ee rero s s follows. Assme h we hve wo yes of shok roesses. The frs s he roess of oelly hrmfl shoks wh re λ d he seod s he roess of helg eves wh re λ. A flre ors whe les wo shoks of he frs ye or row. A helg eve h ors fer he hrmfl eve erlzes he oseqees of he revos shok. Smlr o 5-6 he srvvl robbly s obed from he followg eqos: ~ ex{ λ λ + ~ ex{ λ } + λ ex{ λ x}ex{ λ x} x ~ λ ex{ λ } x y dydx. The erreo of he erms hs eqo s smlr o or resog whe obg 5-6. Alyg he Lle rsform o he seod eqo resls

8 8 ~ λ λ + λ + s s. λ + λ + s λ + s λ λ λ + s Flly λ + λ + s λ + λ + s λ + s λ λ s. 9 The verse Lle rsform for 9 resls eqo 8 wh he sme vles of s d s. Ths s o srrsg s be esly see h boh segs re robblslly eqvle. Usg he Lle rsforms ehqe some oher reros of ee be lso osdered Fkelse d Zrdj. 4. CONCLUDING REMARKS robbls rsk ssessme s slly rher omled. The m roblem s o fd sble model h wll gve ossbly of resoble mheml desro d he sme me be rel d rl. I hs er we hve osdered some smle rohes for defg d modelg hrerss of eble rsk. The me eble loss s rl ee rero b my sos s obvosly o sffe. By defg he Cdf of eble loss G oe solve he roblem of lssfo of he rel loss heorelly v sble sohs omrso of G wh G. Defg G re s rher sbjeve d shold be jsfed by he deled lyses of he oomes of hrmfl eves der osdero. The oher wy of desrbg G s o erform dsrezo of he -xs by osderg regos e.g. wh eble loss ermede loss d eble loss. Alervely bry models for ee rer be lso osdered. I Seo 3. we del wh sef model whe he shoks from he osso roess re o llowed o be oo lose. Referees Blok H.W. Borges W. d Svs T.H Age deede mml rer J. Al. rob Cox D.R. d Ishm V. o roesses Chm d Hll Lodo 98. Fkelse M. d Zrdj V.. Lle rsform mehods d fs rer roxmos for mlle vlbly d s geerlzos. IEEE Trsos o Relbly Fkelse M. 7. Shoks homogeeos d heerogeeos olos. Relbly Egeerg d Sysem Sfey Klbflesh J.D. d ree R.L. The Ssl Alyses of Flre Tme D. Joh Wley & Sos 98. Ks R Coovers M. Dhee J. d De M. Moder Arrl Rsk Theory. Klwer.

9 9 Ross S.M. Sohs roesses. Joh Wley & Sos 996. Thomso W.A. Ir. o roess models wh los o sfey d relbly. Lodo Chm d Hll 988. Vrjlg J.K A frmework for rsk evlo. Jorl of Hzrdos Merls 43: Ushkov I. A. d Hrrso R.A. Hdbook of Relbly Egeerg. Joh Wley & Sos 994.