I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S (I S B A)

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1 I S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E C E S A C T U A R I E L L E S (I S B A UIVERSITÉ CATHOLIQUE DE LOUVAI D I S C U S S I O P A P E R 0/5 SOLVECY REQUIREMET FOR LO TERM UARATEE: RISK MEASURE VERSUS PROBABILITY OF RUI P. DEVOLDER

2 SOLVECY REQUIREMET FOR LO TERM UARATEE: RISK MEASURE VERSUS PROBABILITY OF RUI Pi DEVOLDER ( (Vsion: Jun 0 ABSTRACT Solvncy quimnts a basd on th ida that isk can b accptd if nough capital is psnt. Th dtmination of this minimum lvl of capital dpnds on th way to consid and masu th undlying isk. Apat fom th kind of isk masu usd, an impotant facto is th way to intgat tim in th pocss. This topic is paticulaly impotant fo long tm liabilitis such as lif insuanc o pnsion bnfits. In this pap w study th makt isk of a lif insu offing a fixd guaantd at on a ctain tim hoizon and invsting th pmium in a isky fund. W dvlop and compa vaious isk masumnts basd ith on a singl point analysis o on a continuous tim tst. Dynamic isk masus a also considd. Kywods: Solvncy capital, isk masu, pobability of uin, dynamic isk masu ( Institut of Statistics, Biostatistics and Actuaial Scincs (ISBA / Univsité Catholiqu d Louvain (UCL, Blgium / Pi.Dvold@uclouvain.b

3 . Intoduction: Solvncy quimnts fo insuanc companis a mo than v of fist impotanc in th contxt of th cnt financial cisis and th achivmnt of Solvncy uls. On of th most impotant aspcts of Solvncy is to cogniz that isks affcting insuanc companis a of diffnt natus and that in paticula, financial isks a at th hat of th qustion, spcially fo lif insuanc companis. Amongst th vaious financial divs of isk, quity isk plays a vy spcial ol. In dd, vn if taditionally insuanc companis invst th biggst pat in isk f instumnts (bonds, cash,, a significant popotion of pmiums a invstd in quity in od to boost th tuns. Pnsion funds hav also impotant positions in quity. At th sam tim, invsting in quity is iski and will qui mo capital. A good quilibium has to b found btwn th safty of th liability poposd to th clints and th pfomanc of th invstmnt statgy. A ptinnt masumnt of this makt isk is thfo cucial. In paticula th influnc of th tim hoizon is a classical qustion in financ ( s fo instanc Bodi(995,Campbll/Vicia(00,L(990,Samulson(994, Tholy(995. Solvncy uls a only basd on an annual masu (annual valu at isk at 99.5%. Th pupos of this pap is to compa vaious instumnts of solvncy masu of th makt isk, intgating th tim dimnsion and th hoizon of th liabilitis. In od to intgat tim in th pocss, w hav considd 3 diffnt ways to chck th solvncy of th contact. Fist, w can consid classical isk masus such as valu at isk o tail valu at isk but applid to th long tm isk. Th ida is thn to analyz th fom of th ndd capital in lation with th matuity of th liability. A scond appoach is basd on a continuous chck of th solvncy thoughout th duation and not only at matuity o at th nd of th ya. Th classical actuaial concpt of pobability of uin has bn usd and compaisons can b don with static isk masus. Finally th tchniqu of dynamic isk masu can b poposd to obtain tim consistnt masumnt. Th pap is oganizd as follows. In sction w dscib th gnal famwok of th modl in tms of asst and liability stuctu. Sction 3 is basd on a static isk masumnt and analyzs th influnc of th tim hoizon on th pobability of dfault at matuity and on th solvncy capital using a valu at isk o a tail valu at isk appoach. In sction 4, w consid a continuous tim masumnt basd on pobability of uin duing a whol tim intval (and no mo at a singl point using a simila philosophy as in classical actuaial uin thoy. In sction 5 w mov to dynamic isk masus and w comput solvncy quimnts basd on itatd valus at isk. Sction 6 concluds th pap.

4 . nal famwok: In od to captu th makt isk of a lif insuanc contact with guaant, w consid a unit amount of mony paid at tim t=0 ( singl pmium assumd to b invstd on th asst sid in a fund divn by a classical gomtic Bownian motion. On this amount paid initially by th policyhold w assum th insu will guaant a fixd tun at a fixd matuity t = (liability sid. Bing only intstd in th makt isk, w will consid in this pap only pu financial poducts without any motality ffct. Th mthodology can b asily adaptd intoducing a dtministic lif tabl. Fo =, w hav a yaly guaant (hoizon of on ya as in Solvncy. Th guaant can b nominal (0% guaant cosponding to th guaant to pay back at matuity th initial pmium o basd on a positiv tun dnotd by. In gnal w assum: 0. So th liability at matuity is givn by: L ( = (. On th asst sid, w dnot by A(t th valu of th isky asst at tim t, givn by a gomtic Bownian motion: A(t xp(( δ / t + w(t (0 t = (. wh: δ = man tun of th fund = volatility of th fund w= standad Bownian motion. W can asily intoduc an asst allocation in od to masu th influnc of th invstmnt policy on th isk. W consid in this contxt a continuous balancing btwn th isky asst modld by (. and a isklss asst, assuming a dtministic constant isk f at. Dnoting by: = isk f at β = popotion in isky asst (0 < β thn th nw mixd asst bcoms: A (t = xp(( βδ + ( β β / t + βw(t β (.3 3

5 3. Static isk masumnt W stat ou solvncy analysis of this poduct by applying static isk masus basd on th vntual dficit at matuity of th contact (at tim. 3.. Pobability of dfault: A fist intsting qustion is to look at th pobability of dfault at matuity without any xta capital (i.. th isk to hav not nough assts at matuity to pay th quid liability. In paticula w can consid this pobability as a function of th tim hoizon. This pobability can b asily computd: ϕ( = P(A( < L( = P(xp(( δ With : = P(( δ = P(w( < / ( ( ( δ = Φ( = Φ(a ( δ / a = t / Φ(s = dt π s + w( < ( δ / / + w( < / (3. (3. (Th numb a psnting th spad btwn th guaantd at and th man tun nomalizd by th volatility. Fo a cohnt lvl of guaant (compad to th man tun of th asst, this spad is ngativ: < δ Thn th pobability of uin dcass with th tim hoizon. W can conclud h that invsting in isky assts such as quitis fo th long un givs btt chanc to achiv a fixd guaantd tun. This could lad to mo aggssiv statgis fo long duation ( Bodi Z. (995, Campbll /Vicia (00. / 4

6 Exampl 3. : W will wok with th cntal following scnaio: - guaantd at: = % - isk f at : = 4% - man tun of th quity fund : δ = 7% - volatility of th quity fund : =6% Figu shows th volution of th pobability of dfault as a function of th tim hoizon. Fo this scnaio, th pobability of dfault on on ya is qual to 40.8% but on 30 yas this pobability dcass to 0.4%. Poba 45,00% 40,00% 35,00% 30,00% 5,00% 0,00% 5,00% 0,00% 5,00% 0,00% Poba Figu : Pobability of dfault Rmak 3.. : W can also look at th influnc of th asst allocation on this pobability. Using a mixd invstmnt statgy, th pobability of dfault, considd now as a joind function of th tim hoizon and th asst allocation bcoms: ϕ(, β = with : P(A β ( < L( = Φ(a( β. a( β = ( βδ + ( β β β / 5

7 3.. Valu at isk : Th Solvncy famwok is basd on a quantil masumnt. So w will us in this sction th valu at isk mthodology. W intoduc th following notations: = initial solvncy capital using a valu at isk mthodology VaR = valu at isk = isk f at α( = chosn safty lvl fo a hoizon of yas ( 99.5% on on ya in Solvncy. Fo this safty lvl w can fo instanc choic th following valu basd on yaly indpndnt dfault pobabilitis (pobability of dfault of ( α indpndntly ach ya: α (3.3 = (α Assuming fist that th solvncy capital is invstd in th isklss asst, w can dfin th solvncy capital as th solution of: P{A( +. < L(} = α This pobability is givn by: ( δ / +w( ln{. } + ( δ P{ +. < } = P{w( < ln{. } + ( δ = Φ( So w obtain in this cas th following valu fo th solvncy capital: / } / wh ( δ / + z α = (3.4 z β = β quantil of th nomal distibution such that : Φ(zβ = β 6

8 W can no mo conclud h in absolut tms that th isk masu is systmatically a dcasing function of th tim hoizon vn if a dcasing fom appas on long tm as showd by th following xampl: Exampl 3.. : - safty lvl on on ya : α=99.5% - safty lvl on yas : α = (α Figu shows thn th volution of th valu at isk as a function of th tim hoizon with sam assumptions as in xampl 3.. Valu at isk 0,4 0,3 0, 0, Valu at isk 0-0, , Figu : Valu at Risk Rmak 3.. : W could also assum that th initial solvncy capital is invstd in th isky asst (cf. Biys /D Vann ( 997. Thn th solvncy condition bcoms: P{A( +.A( < L(} = α Simila dict computations lad to th following altnativ valu fo th solvncy capital : = ( ( δ / z α (3.5 Rmak 3.3 : as in sction 3.., w could intoduc a mixd invstmnt statgy fo th pmium and/o th solvncy capital. 7

9 3.3. Tail Valu at isk: Valu at isk is a classical tool in isk managmnt and is th fnc fo Basl and Solvncy. But fom a thotical point of viw it is wll known that valu at isk psnts som impotant poblms (Atzn t all ( 997. Th mthodology psntd abov can b asily xtndd using tail valu at isk instad of valu at isk. Assuming that th solvncy capital is invstd in th isklss asst and dnoting by: = initial solvncy capital using a tail valu at isk mthodology thn w obtain: with = ( E{L( A( L( A( > V α } V α = ( δ / +.z α Thn using th poptis of th log nomal distibution w gt: = ( E{A( A( < ( δ / + z α } = ( δ α Φ(z α Finally: ( ( δ = Φ(z α α (3.6 Exampl 3.3. : Figu 3 shows thn th volution of th tail valu at isk as a function of th tim hoizon with sam assumptions as in xampls 3.. and 3.. Th fom is simila to th valu at isk (fist incasing thn dcasing on a long tm basis. 8

10 Tail VaR 0,5 0,4 0,3 0, Tail VaR 0, Figu 3 : Tail valu at isk 4. Continuous tim isk masumnt : Th diffnt appoachs psntd in th pcding sction show undoubtdly that a hoizon ffct xists in tms of masumnt of quity isk whn basd on solvncy tools. W usd fo this a static isk masumnt consisting on a point masu only at matuity. Anoth classical actuaial appoach should b to chck th solvncy not only at matuity of th contact but at any tim. Th pupos of this sction is to analyz this philosophy inspid by th actuaial classical uin thoy. 4.. Pobability of uin without capital: W fist look at th pobability of uin without solvncy capital givn by: ( = P{A(t L(t, t [ 0, ]} Ψ (4. In od to comput this pobability, w hav to dfin th lvl of liability L(t not only at matuity but also at any tim, th asst pocss bing still dfind by its makt valu ( Accounting viw : W will fist us an accounting viw (mathmatical sv appoach : t L (t = (4. 9

11 Thn th pobability (4. can b sn as a pobability latd to th minimum of a omtic Bownian motion along an intval : Ψ( = P(min = P(min 0 s 0 s ( δ ( δ / s+w(s < / s+w(s s < (4.3 W can thn obtain th following sult : Poposition 4.. Without solvncy capital and using th accounting valu (4. fo th liabilitis, th uin is su whatv th man tun of th isky asst is. Poof: This is a dict consqunc of th law of th minimum of a omtic Bownian motion ( s fo instanc Back(005 : If th pocss S is givn by : and if : S(t = z min0 0 < L ( µ = s t / t+w(t {S(s} µ P Thn: (z L = Φ(d + L. Φ(d with: d, ln L ( µ = t / t Taking L= in (4.4, w obtain fo th pobability of uin (4.3 : ( δ Ψ ( = Φ( 4... Sund viw : / ( δ / + Φ( = (4.4 Instad of using fo th liability valuation th mathmatical sv, w could intoduc a classical sund pnalization ( s fo instanc osn / Jognsn (00, and altnativly substitut to (4. th following fom fo th liability : 0

12 L(t = ρ(t. wh th function ρ is a non dcasing function such that : ρ(t fo ρ( = t 0 t (- ρ (t psnting th sund chag in cas of sund at tim t. As xampl w will us h an xponntial fom fo this pnalty (usd in som lif insuanc poducts: ρ(t = λ( t lading to th following liability function: ( λ > 0 L(t = = λ( t λ. t ( +λt. (4.5 Thn th pobability of uin (4.3 bcoms: Ψ (, λ = P(min ( δ λ / s+w(s 0 s < Using th gnal fomula (4.4 w obtain th following sult: Poposition 4.: Without solvncy capital and using a ngativ xponntial sund pnalty of th fom (4.5, th pobability of uin is givn by : λ ( δ Ψ(, λ = Φ( = Φ(a + ( λ / ( δ + ( λ λ ( δ λ Φ(( a λ / ( δ Φ( λ (4.6 / Rmak 4.: th fist pat of this pobability cosponds to th pobability of dfault (3. only at matuity. Th scond tm flcts th isk of insolvncy bfo matuity.

13 4..3. Fai valuation of liability: Anoth possibility fo th liability function is to intoduc a fai valuation concpt. Thn th liability at tim t can b sn as th psnt valu of th final liability discountd at th isk f at: L(t = ( t. o: L(t =. ( ( t t So this xpssion can b sn as a spcial cas of sction 4.. with: λ = This paamt is stictly positiv as pscibd in sction 4... if th isk f at is stictly high than th guaantd at of th poduct. W can thn obtain th following coollay of poposition 4. : Coollay 4. : If th liability is masud at its fai valu and if th guaantd at is lss than th isk f at, th pobability of uin without solvncy capital is givn by : ( δ ( Ψ( = Φ(a + ( Φ(( a ( / ( Pobability of uin with isklss capital: W assum now that an initial solvncy capital is injctd in th businss at tim t=0 and invstd in th isklss asst till matuity. Thn th pobability of uin (4. bcoms: Ψ (, = P{A(t +. t L(t, t [ 0, ]} Using as liability valuation th sund viw of sction 4.., this pobability can b wittn as: Ψ(,, λ = P{A(t +.., t t λ( t t [ 0, ]}

14 This pobability can b onc again xpssd as a minimum but unfotunatly not as th minimum of a omtic Bownian motion: Ψ(,, λ = P(min ( /(. < ( δ / s+w(s λ ( +λt t 0 s (4.8 vthlss w can obtain an xplicit fom of this pobability if w us a fai valuation appoach fo th liabilitis : Poposition 4.3. If th liability is masud by its fai valu and if th guaantd at is lss than th isk f at, th pobability of uin with a isklss solvncy capital stictly ( lss than is givn by: ln( Ψ(, = Φ( + ( ( ( ( δ ( δ ln( Φ( ( + ( δ / / (4.9 Poof : Th fai valuation appoach cosponding to th cas (4.8 bcoms : P(min 0 s = P(min ( 0 s ( δ ( / s+w(s ( δ /( / s+ w(s λ < ( +λt ( W can thn us dictly fomula (4.4 with : L = (. t < λ = thn pobability which satisfis th condition 0 < L taking into account th assumptions of th poposition. Rmak 4.. : Th condition on th solvncy capital just mans that th initial capital is lss than th psnt valu of th total liability at matuity, which sms tivial. 3

15 4.3. Pobability of uin with isky capital: Assuming finally that th initial solvncy capital is invstd in th isky asst A givn by (. till matuity, w obtain fo th pobability of uin : Ψ(,, λ = P{A(t +.A(t λ ( t t., t [ 0, ]} In this cas it is possibl to obtain an xplicit valu of this pobability whatv th valu of λ is: Poposition 4.4: With a isky solvncy capital and using a ngativ xponntial sund pnalty of th fom (4.5, th pobability of uin is givn by : Ψ(, λ, = Φ(a + ( + λ ( δ λ Φ( a ln( + ln( + + λ (4.0 Poof : Th pobability of uin can b wittn in this cas : Ψ(,, λ = P{A(t +.A(t = P(min = P(min 0 s 0 s ((A(s +.A(s / ( ( δ λ( t λ / s+w(s. t <, t ( +λs λ < [ 0, ] λ } /( + W can thn apply dictly fomula (4.4 to obtain th sult. Rmaks 4.3 : Fo λ = 0 w obtain th accounting viw ( sction 4.. and fo λ =, th fai valuation ( sction Fomula (4.0 shows th influnc of th two possibl tools availabl fo th 4

16 insu to dcas its pobability of uin: putting a sund pnalty on th poduct and/o injcting an initial solvncy capital Computation of th solvncy capital : Fo a fixd lvl of solvncy capital, th fomulas dvlopd in sctions 4. and 4.3 pmit us to comput th cosponding pobability of uin. W could also us a vs mthodology and ty to dtmin th lvl of capital cosponding to a maximum accptd pobability of uin (quivalnt to sction 3. in static masumnt. Taking into account th fom of ths pobabilitis of uin (s fo instanc (4.9 o (4.0, no xplicit solution of th capital can b hopd. umical pocdus must thn b applid. Fo instanc, in th cas of a isky solvncy capital, using (4.0, th solvncy capital cosponding to a fixd lvl of safty α is th solution of th following implicit quation: Φ(a λ ln( + + ( + ( δ λ Φ( a ln( + + λ = α This quation can also b wittn as : = λ Φ(d *( (d( α Φ / Κ with : ( δ λ K = ln( + d( = a and d *( = a ln( + + λ suggsting th following cusiv algoithm fo th computation of : n = λ Φ(d *(n α Φ(d(n / Κ (4. 5

17 5. Itatd isk masumnt : Instad of using only a point appoach as in sction 3 o a continuous isk masumnt though a pobability of uin as in sction 4, w could intoduc an intmdiat quimnt by asking an initial solvncy capital such as to main solvnt only at som fixd disct tims (fo instanc at th nd of ach ya. Dynamic isk masus must b thn considd; in paticula w can wok with th concpt of itatd isk masus (s fo instanc Pflug / Romisch(007 o Hady/Wich ( nal pincipl of itatd masumnt: W will us th following notations in this sction: X = isk = stochastic cashflow tobpaidat matuity t = (X = solvncy capital at tim t to cov th isk X at t, tim ρ = static isk masu (fo instanc valu at isk { It,0 t } = filtation ( pogssiv aival of inf omation on X = iskf at Thn th initial ndd capital at tim t=0 using a classical static isk masumnt is givn by th psnt valu of th isk masu (cf. sction 3: 0, (X. ρ(x I0 = (5. With dynamic isk masumnt w a now intstd to comput th ndd capital not only at tim 0 but fo instanc at th nd of ach ya. Two appoachs can b immdiatly intoducd: - th accumulatd appoach: succssiv solvncy capitals a just th accumulation with intst of th initial capital (assumd to b invstd in a isklss asst: t ( t t, (X =.0, (X =. ρ(x I0 (5. As suggstd by fomula (5. no additional infomation is intgatd in th succsiv capitals. 6

18 - Th calculatd appoach: th static isk masumnt is applid at ach tim t and givs thn: t, (X = ( t. ρ(x I This appoach sms to b mo tim consistnt, taking into account th pogssiv aival of nw infomation on th isk X. t (5.3 But it could gnat duing th pocss unxpctd vaiations of th solvncy capital. Thn a thid appoach calld th itatd isk masumnt, tis to anticipat ths possibl futu capital quimnts; in this vision, th succssiv isks a no mo th futu cash flow X but th ndd capital at th nxt piod of tim. In this cusiv mthod w fist comput th ndd capital on piod bfo matuity (w will assum h annual units of tim following th accounting uls but off cous fomula can b dvlopd using oth tim units. At tim t=- w will us th calculatd fomula (5.3 :, (X =. ρ(x I Thn at tim t=-, th solvncy capital bcoms : In gnal, at tim t :, (X =. ρ(, I t, (X. ρ( = t+, 5.. Application to solvncy quimnt: I t (5.4 Lt us adapt h th valu at isk computations of sction 3. in this itatd appoach. Mo pcisly w would lik to gnaliz, in a dynamic itatd nvionmnt, fomula (3.4 wittn as: ( 0, = ( δ / + z α 7

19 Poposition 5.. Th itatd solvncy capital at tim t (t=0,,, - using a valu at isk appoach is givn by : ( t ( δ / ( t +( tz α t, =. A(t (5.5 wh A(t is th asst valu at tim t. In paticula th initial solvncy capital at tim t=0 is givn by: ( ( δ / +z α = (5.6 0, Ths valus can b compad with a simpl calculatd appoach at ach tim: ( t ( δ / ( t + t z α t =. A(t (5.7 Poof : t, W stat th cusiv calculation with th capital at tim t=-. This fist computation givs by dfinition a sam valu as in th calculatd appoach:, =. A(. ( δ Thn at tim t=-, this capital is stochastic bcaus of th psnc of th futu valu of th asst. This andom vaiabl, sn fom tim t=-, can b wittn: / +z α A(t = A(t. ( δ / +(w(t w(t So th ndd solvncy capital at tim t=- sn fom tim t=- is a andom vaiabl X givn by : X =. A(. ( δ / Th ndd capital at tim t=- is thn th amount +(w( w( +z, α such that : O : P(. X = α ( / (w( w( z δ + + α,. (. A(. P( = α 8

20 O: ( / z δ + α P((w(t w(t (ln(.,. ln(a(. So that: = α ln(.,. ln(a(. ( δ / +z α =.z α And finally:, =. A( ( δ / + z α Th sam pocdu can b usd cusivly to gnat all th valus (5.5. Coollay 5. : If w us a sam safty lvl fo any tim hoizon and high than 50% : α s = α > / thn th itatd appoach ( 5.5 givs high capital quimnts than th classical appoach (5.7. It is possibl to hav a sam capital lvl in th appoachs if w adjust th safty lvl accoding to th quilibium lation: z α = s. z α S 6. Conclusion : Stating fom th classical poblm of solvncy linkd to a guaantd intst at in lif insuanc and assuming th invstmnt in som isky asst divn by a gomtic Bownian motion, w hav dvlopd vaious mthodologis to masu th isk and dtmin th solvncy capital quimnt. Mo pcisly w hav compad 3 diffnt appoachs: - a static appoach basd on a isk masu on th final position of th insu; - a continuous appoach basd on th pobability of uin duing all th lif of th contact; - a dynamic appoach basd on itatd isk masus on annual succssiv isks. 9

21 Ths diffnt sults illustat th impotanc of considing in th capital quimnt computation, not only th kind of isk masu usd but also th way to intgat th tim hoizon. Rfncs Atzn P., Dlban F.,Eb J.M.,and Hath D. (999: Cohnt Masus of Risk, Mathmatical Financ 9, pp Back K.(005 : A Cous in Divativs Scuitis, Sping Bodi,Z.( 995: On th Risks of Stocks in th Long Run. Financial Analysts Jounal 5(3,pp Bodi, Z, Mton and Samulson ( 99: Labo Supply Flxibility and Potfolio Choic in a Lifcycl Modl, Jounal of Economic Dynamics and Contol 6(3, pp Biys E.,d Vann F. (997: On th Risk of Lif Insuanc Liabilitis : Dbunking Som Common Pitfalls, Jounal of Risk and Insuanc 64(4, pp Campbll J., Vicia L. (00: Statgic Asst Allocation Potfolio Choic fo Long tm Invstos, Oxfod Univsity Pss osn A., Jognsn P. ( 00 : Lif Insuanc Liabilitis at Makt Valu : An Analysis of Insolvncy Risk, Bonus Policy and Rgulatoy Intvntion Ruls in a Bai Option famwok, Jounal of Risk and Insuanc, 69,,pp Hady M., Wich J.L. ( 004 : Th Itatd CTE : a Dynamic Risk Masu, oth Amican Actuaial Jounal, 8, pp L,W. (990: Divsification and Tim,Do Invstmnt Hoizons matt?, Jounal of Potfolio Managmnt 4, pp Mcil A.,Fy R.,Embchts P. (005 : Quantitativ Risk Managmnt, Pincton Mton R. ( 99: Continuous Tim Financ, Wily Pflug., Romisch W.(007: Modling, Masuing and Managing Risk, Wold Scintific 0

22 Samulson,P.(994: Th Long Tm Cas fo Equitis. Jounal of Potfolio Managmnt (, pp 5-4 Samulson P. ( 963 : Risk and unctainty : A fallacy of Lag umbs, Scintia, 98, pp Sandstom A. (006 : Solvncy, Chapman & Hall Tholy S. ( 995: Th Tim Divsification Contovsy, Financial Analyst Jounal 5(3, pp.8- Wich J.L.,Hady M.R. (999 : A synthsis of isk masus fo capital adquacy, Insuanc : Mathmatics and Economics, 5, pp

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