Generalized Loss Variance Bounds

Transcription

1 Int. J. Contem. ath. Scences Vol. 7 0 no Genealzed Loss Vaance Bounds Wene Hülmann FRSGlobal Swtzeland Seefeldstasse 69 CH-8008 Züch Swtzeland wene.huelmann@fsglobal.com whulmann@bluewn.ch Abstact A stuctue of fnancal loss atton on a obablty sace s consdeed. To the events of a atton and to each fnancal loss t assocates event based losses and gans. Event based loss vaance bounds ae obtaned whch genealse evous nequaltes by Keme and the autho. Shaness and the maxmum of the loss vaance bounds unde dffeent constants ae also detemned. athematcs Subect Classfcaton: Pmay 60E5 Seconday 6P05 Keywods: fnancal loss atton vaance bounds. Intoducton As a measue of maxmum dseson fom the mean ue bounds on vaance have alcatons n all aeas of theoetcal and aled mathematcal scences (e.g. Seaman and Odell [4] Agawal et al. [] Banett et al. []). Snce the cng and valuaton of actuaal and fnancal ss often deends on vaance aoate bounds ae of consdeable actcal nteest. In evous wo the autho has consdeed a stuctue of fnancal loss on a obablty sace by eesentng the fnancal loss as dffeence between loss and gan (see [6] [7]). In ths settng the nequalty of Bowes [4] on the mean loss as well as nequaltes by Keme [] Hülmann [5] and Bel [3] on the loss vaance ae smle consequences of the non-negatve oety of vaance functons. Shaness and extemal oetes of the vaance bounds have been analysed followng the ognal contbuton by Schmtte [3]. These esults have been aled n ensuance mathematcs to ce sto-loss contacts (see [8]) and to show that the lnea combnaton of ootonal and sto-loss ensuance s not otmal unless t s a ue sto-loss contact at least n case the vaance emum ncle s used to set nsuance ces ([9] oof of Theoem []). In the esent follow-u the ntoduced fnancal loss stuctue s extended to the fnancal loss atton stuctue alluded to n [7] Remaque.. Fst the

2 560 W. Hülmann samle sace of a obablty sace ( Ω ) AP s attoned nto dsont events E. Then to a fnancal loss X that s a measuable E S such that Ω S eal-valued andom vaable on ths obablty sace one assocates E -losses X I( E ) and E -gans X X I( E ) X S whee I() s the ndcato functon and E s the comlement event belongng to E. Fo examle a smle atton nto thee events E { X < l } E { l X < h} E { X h} 3 s able to model vaous egmes of low nomal and hgh losses accodng to whch event s evealed when the fnancal loss s ealsed. The detaled constucton s esented n Secton. Genealsed loss vaance bounds fo fnancal loss attons ae obtaned n Secton 3. Shaness and maxmum loss vaance bounds unde dffeent constants ae deved n Secton 4.. A fnancal loss atton stuctue Let ( Ω ) AP be a obablty sace. We consde measuable eal-valued andom vaables X on t that s mas X : Ω R. Each X s nteeted as a fnancal loss such that fo ω Ω the eal numbe X ( ω ) s the ealzaton of a loss and oft functon wth X ( ω) 0 fo a loss and X ( ω ) < 0 fo a oft. It s assumed that the mean μ and vaance σ of X exst. To each event E A one assocates ts comlement event E A such that PE ( ) PE ( ). Fo a fnte subset S N of ndces consde a atton of events E S such that Ω E E E fo all whch yelds S a stuctue of fnancal loss atton as follows. Let I : A { } ndcato functon and defne fo S the andom vaables X X ( ) ( ) 0 be the X I E : the amount to be ad f the event E occus called E -loss X I E : the amount ganed f the event E occus called E -gan X X I E + X I E X X whch as a dffeence between loss and gan ustfes the nteetaton of X as a fnancal loss. The negatve value G X X X s called fnancal gan. Snce I( E ) + I( E ) one has ( ) ( ) Examles. () Wth S { } the atton E { X } > 0 E { X } 0 yelds the ostve loss X X+ and the ostve gan X G + whch s the smlest fnancal loss stuctue studed n [6] [7].

3 Genealzed loss vaance bounds 56 a atton of the tye E { X < l } E { l X < h} E { X h} 3 allows one to dstngush between a egme of nomal losses evealed when the event E occus and two egmes of extemal losses consstng of a egme of low losses evealed when E occus and a egme of hgh losses evealed when E 3 occus. () Wth S { 3} () It s also ossble to secalse to nsuance layeed ss of degee m that s non-negatve andom vaables X such that thee exsts a atton nto dsont events E... m of the samle sace whch s of the tye { } { } { } E X d E d < X d... m E X d m m wth 0 < d < d <... < d m. Snce emums fo a gven laye may vay consdeably on the ensuance maet t s of modal motance fo a cedent to now the extent of ossble vaaton of the emums n ode to otmze hs laye stuctue and the ce to ay fo ensuance. We le to note that such nds of chans of ensuance layes often occu n actcal wo on (e)nsuance catves and have been consdeed n [0]. Fo alcaton n the actuaal and fnancal context we wll assume that the loss and gan events E E occu wth non-zeo obablty that s ( ) PE ( ) 0< PE < fo all S. In geneal snce E E ae dsont fo S and E E ae dsont fo one has the followng loss and gan denttes of an abtay ode n N : n n n n X ( ) + X X S (.) X n n X. (.) S At the usual elementay level one s only nteested n fst and second ode moments that s fo each S defne [ ] E[ X ] E X : the mean E -loss : the mean squaed E -loss [ ] V Va X : the E -loss vaance [ ] E[ X ] E X : the mean E -gan : the mean squaed E -gan [ ] V Va X : the E -gan vaance

4 56 W. Hülmann The denttes (.) and (.) mly the followng elementay elatonshs. Lemma.. A fnancal loss atton stuctue satsfes the loss and gan aty elatons fo each μ (.3) μ + σ (.4) S and the denttes V + V σ (.5) μ (.6) S μ + σ (.7) S S V σ. (.8) S Poof. The elatons (.3)-(.5) ae (.5)-(.7) n [6]. The denttes (.6) and (.7) follow mmedately fom (.) wth n. Usng the latte elatons one obtans V S σ S + < μ + S. σ S + < (.9) Fom ths one obtans fo fxed S usng (.6) and (.3) that σ + + σ + ( μ ) + V S < < σ + <. Summng ove S usng (.9) one gets (wth S the cadnalty of the set S) S V S σ + S ( ) S S < S σ + ( S ) V σ S whch mles mmedately (.7). S

5 Genealzed loss vaance bounds 563 One obseves that (.5) exesses the second ode total E -loss and E -gan vaance n tems of the vaance and the fst ode mean E -loss and E -gan. Smlaly the dentty (.8) decomoses the total loss vaance defned as the sum of all E -loss vaances n tems of the vaance and all mean E -losses and E -gans. In secal case S the elatons (.5) and (.8) ae equvalent. 3. Vaance bounds fo fnancal loss attons. The followng vaance nequaltes genealze Theoems. and. n [6]. Theoem 3.. If the loss obablty PE ( ) S s unnown then the E -loss vaance satsfes the ue bound V σ (3.) S Poof. Snce vaances ae non-negatve ths follows dectly fom (.8). Theoem 3.. If the loss obabltes PE ( ) S ae nown then the E - loss vaance satsfes the lowe and ue bounds PE ( ) ( ) V σ PE S S PE PE ( ) ( ) Poof. Condtonng on the event E wth ( ) PE [ ] [ ] [ ] PE ( ) PE ( ) Va X E E X E E X E whch mles that V PE PE PE ( ) ( ) ( ). (3.) 0 by assumton we have 0 that s the lowe bound n (3.). The ue bound follows fom the dentty (.8) usng the lowe bounds fo V.

6 564 W. Hülmann 4. Shaness and maxmum loss vaance bounds. It s natual to as when the obtaned loss vaance bounds n Theoem 3. ae sha that s attaned fo some fnancal loss atton stuctue. We show that ths s the case fo fnte S -atomc fnancal losses whch ae defned as follows. Let x S be the atoms of a standadsed dscete andom vaable X wth mean zeo and vaance one and obabltes P( E ) abtay mean μ and vaance σ the elatonshs ( μ + σ x ) ( ) 0 S. Wth an a fnte S -atomc fnancal loss satsfes μ + σ x S hence also V S. (4.) A comason of (.8) and (3.) shows mmedately that the equaltes n (3.) ae attaned fo fnte S -atomc fnancal losses and one has fo S the dentty V μ + σ S. (4.) It s now ossble to obtan maxmum loss vaance bounds n (3.) by solvng the followng mnmsaton oblem S mn! (4.3) whch may be constaned by some sde condton. In actcal alcatons seveal altenatve stuatons may be of nteest. Snce a detaled dscusson of the secal case S s found n [6] we assume hee that S 3. Fo llustaton we analyse two man cases. Fst we as fo the maxmum E -loss vaance bound ove the sace D D( μσ S) of all fnancal loss andom vaables wth fxed mean μ vaance σ loss P E and mean E -losses S. Then we consde obablty ( ) the maxmum of V ove the sace D D( S) μσ of all fnancal loss andom vaables wth fxed mean μ vaance σ mean E -loss and P E S. Two dstngushed esults ae obtaned. loss obabltes ( )

7 Genealzed loss vaance bounds 565 Theoem 4.. The maxmum E -loss vaance bound ove the sace of fnancal μσ S S 3 s gven by losses D D( S) max{ V } μ + σ + D (4.4) S and s attaned at the S -atomc fnancal loss wth atoms μ + σx S and obabltes / / ( ) (4.5) whee 0 fo some. Poof. The statonay ont of the Lagange functon ( λ) λ ( ) L (4.6) yelds the elatonsh s s s s. (4.7) Wth an ndex such that 0 one obtans. Usng the sde condton one gets (4.5). Insetng nto the ghthand sde of (4.) the best ue bound (4.4) follows. Theoem 4.. The maxmum E -loss vaance bound ove the sace of fnancal losses D D( μσ S) S S 3 does not deend on and s gven by { V } ( μ ) max μ + σ. D (4.8)

8 566 W. Hülmann It s attaned at the S -atomc fnancal loss wth atoms μ + σx obabltes S whee and μ. (4.9) Poof. The statonay ont of the Lagange functon ( λ ) λ ( μ ) L (4.0) yelds the elatonsh (4.) fo some. Usng the sde condton μ one gets (4.9) whch nseted nto the ght-hand sde of (4.) yelds the best ue bound (4.8). Refeences [] R.P. Agawal N.S. Banett P. Ceone and S.S. Dagom A suvey on some nequaltes fo exectaton and vaance Comut. ath. Al. 49(-3) (005) [] N.S. Banett P. Ceone and S.S. Dagom Inequaltes fo Random Vaables ove a Fnte Inteval Nova Scence Publ. New Yo 008. [3] T. Bel Elementay ue bounds fo the vaance of a geneal ensuance teaty Blätte de DGVF (3) (994) [4] N.L. Bowes An ue bound fo the net sto-loss emum Tansactons of the Socety of Actuaes XIX (969) -6. [5] W. Hülmann Slttng s and emum calculaton Bull. Swss Assoc. Actua. (994) [6] W. Hülmann An elementay unfed aoach to loss vaance bounds Bull. Swss Assoc. Actua. (997)

9 Genealzed loss vaance bounds 567 [7] W. Hülmann Fonctons extémales et gan fnance Elem. ath. 5 (997) [8] W. Hülmann On the loadng of a sto-loss contact: a coecton on extaolaton and two stable ce methods Blätte de DGVF 4() (999) [9] W. Hülmann Non-otmalty of a lnea combnaton of ootonal and non-ootonal ensuance Insuance ath. Econom. 4(3) (999) 9-7. [0] W. Hülmann Otmzaton of a chan of excess-of-loss ensuance layes wth aggegate sto-loss lmts Bull. Swss Assoc. Actua. (006) 5-6. [] W. Hülmann Otmal ensuance evsted: ont of vew of cedent and ensue ASTIN Bulletn 4() (0) [] E. Keme An elementay ue bound fo the loadng of a sto-loss cove Scand. Actua. J. (990) [3] H. Schmtte An ue lmt of the sto-loss vaance Poceedngs 6 th Intenatonal ASTIN Colloquum (995) Büssels. [4] J.W. Seaman and P.L. Odell Vaance ue bounds In: S. Kotz and N.L. Johnson (Eds.) Encycloeda Statst. Scences vol. 9 (988) Receved: Febuay 0