Classification and Ordering of Portfolios and of New Insured Unities of Risks

Transcription

1 Classification and Odeing of Potfolios and of New Insued Unities of Risks Augusto Feddi, Giulia Sagenti Univesity of Rome La Sapienza Depatment of Actuaial and Financial Sciences 36th Intenational ASTIN Colloquium. Zuich, 4 7 Septembe 2005

2 Intoduction The classical definitions of classification and odeing of isks. The classification of isks is used to goup individual isks to which it must be applied the same pemium. The aim is the potection of the insuance system s financial soundness. The odeing of isks is a compaison of isks belonging to two diffeent classes. The aim is to establish to which isk it must be applied the geate pemium. Both the classification and the odeing ae based on the isks measues. of New Insued Unities of Risks 2

3 Intoduction The basic ideas of ou model. The classification and the odeing ae made afte the isks ae insued (the pupose is the outline of a einsuance stategy). The classification and the odeing ae based on the changes poduced in the state of the business in the passage fom a geneic potfolio to the new potfolio managed by a popety-casualty insuance company (i.c.). potential policyholde(s) i.c. unity of isk potfolio t = 0 t = = i.c. potfolio { ; } time of unity of isk s tansfe time of New Insued Unities of Risks 3

4 Intoduction Tools used in ou model. The (actuaial) business of the i.c.: Business={Potfolio, Opeative stuctue}. The opeative stuctue: it is the set of the constaints and ules imposed on the potfolio s management by the insuance maket and the egulatoy authoity and of the citeia adopted by the i.c. The state of the business: it is descibed by using the loss exceedance pobability (LEP) cuve and some vectos defined as state vaiables. The state vaiables measues: they ae used to establish the odeing citeia. of New Insued Unities of Risks 4

5 Outline of the pape Potfolio of isks un by the i.c. at time The desciption of the business The state of the business The gaphic epesentation of the state of the business The classification of The odeing of The new unity of isk intoduced at time The new potfolio.. The state of the business The gaphic epesentation of and The classification of The odeing of... B( ). B( ). B( ). B( ) t = 0. t B( ). = 0. B ( ). of New Insued Unities of Risks 5

6 Potfolio of isks time t = 0 un by the i.c. at Notations and assumptions. = {, K, }, l >>, independent components. l The isk s analysis is ove a time hoizon of one yea. The composition of does not change ove The total claim amount in one yea is epesented by F S i is known and defined on [ 0,]. S. [, M ( S )], M ( ) (0, ). 0 The i.c. opeates out of a continuous time economic envionment. In paticula the model does not include taxes, commissions, investment incomes, dividend payout to shaeholdes, inflation. S of New Insued Unities of Risks 6

7 The desciption of the business B( ) The business elative to potfolio : Θ( ) is the opeative stuctue elative to potfolio Citeia fo defining B( ) = (, Θ( )). Θ( ).. Insuance maket s constaint. 2. Citeia adopted by the i.c. 3. Conditions imposed on the i.c. by the egulatoy authoity (.a.).. of New Insued Unities of Risks 7

8 The desciption of the business B( ) Independently of the pinciples used to assess the single pue pemiums P( i ), the pemium income ove one yea can be expessed by [ S ] + c( ), c( ) = η( ) E[ S ]. P( ) = E. Insuance maket s constaint. M η > 0 η( ) η (0, Mη ], η [ 0,]. whee is the maximum of in the competitive insuance maket (i.m.) duing of New Insued Unities of Risks 8

9 The desciption of the business B( ) 2. Citeia adopted by the i.c. ε 0 (0,) The i.c. fixes as the maximum acceptable uin pobability pe yea. The i.c. selects the pemium calculation pinciples. Having expessed the fee eseve popotional to the pue pemium, u( ) = αp( ), α α( ) > 0, α, > 0, the i.c. fixes the maximum fo than a value linked to the i.m. M M M α lowe of New Insued Unities of Risks 9

10 The desciption of the business B( ) 3. Conditions imposed on the i.c by the.a. ( I ). The i.c. must opeate within the maximum acceptable uin pobability pe yea ε ( 0, ε to which 0] coesponds the maximum acceptable loss MAL ( ). The i.c. must have a minimum fee eseve u = α + η ) E[ S ]. ( The minimum acceptable capital stuctue is CS * * ( ) = u + ( + η ) E h = ( + α )( + η ) [ ] S = h E[ ], S whee is the minimum acceptable actuaial capitalization facto. of New Insued Unities of Risks 0

11 The desciption of the business B( ) 3. Conditions imposed on the i.c by the.a. ( II ). The only authoized state of the business is the acceptable state: CS( ) CS ( ), ( α, η) As paticula case, it includes the stable state: CS( ) + ϕ, ( α, η) CS ( ) whee ϕ > 0 0 CS ( ) = u( ) + P( ) [ m, M ] [ m, M ]. α α [ m, M ] [ m, M ], 0 α α η η is independent on the potfolio and is the capital stuctue of the i.c. η η of New Insued Unities of Risks

12 The desciption of the business B( ) The acceptable and the stable states of the business ae equivalent to a non negative and to a stictly positive capacity, espectively, having defined the capacity of the i.c. elative to the potfolio by whee C ( α, η) = CS( ) CS ( ), [ m, M ] [ m, M ]. ( α, η) α α η η of New Insued Unities of Risks 2

13 The desciption of the business B( ) Θ( ) The opeative stuctue expesses all the pevious constaints, citeia and ules and it can be epesented by the following set { Θ( ) = ε, Mα, Mη, ϕ0; α( ), η( ), h( ), ε (0, ε ]}. of New Insued Unities of Risks 3

14 The gaphic epesentation of the state of the business F S To each d.f. coesponds a loss exceedance pobability (LEP) cuve. We tace the LEP elative to on the -plane, whee H = S E[ ]. S We epesent the state of the business on the LEP cuve though the points ( = h, ε ), B( ) ( H,0,ε ) (, Θ( )) B( ) = A B = ( h,ε ), Q = ( 0, ε ). of New Insued Unities of Risks 4

15 The gaphic epesentation of the state of the business A, B, Q B( ) ( H,0,ε ) define on the -plane the state vaiables, that is, the following six vectos: ( v = h,0), ( v 2 = h h, ε ε ), ( ( v v 3 = h, ε ε ), 4 = 0, ε ε ), ( v 5 = h h,0), v ( 6 = h,0). of New Insued Unities of Risks 5

16 The gaphic epesentation of the state of the business B( ) LEP ε ε 0 Q v 4 v A v v 3 2 v v 5 6 h h B H of New Insued Unities of Risks 6

17 The gaphic epesentation of the state of the business B( ) One of the thee possible systems of basic state equations is v v v + v2 v3 = 2 + v4 v5 = 3 + v4 v6 = 0, 0, 0. of New Insued Unities of Risks 7

18 The classification of The classification is based on the state of the business and is gaphically expessed only though the vecto Definition. The management of potfolio. is authoized if eithe sgn v 4 > 0 (the stable state) o v = 0 (the acceptable state), 4 2. It is not authoized if sgn v 4 < 0. v 4. of New Insued Unities of Risks 8

19 The odeing of The odeing citeia fo potfolio ae defined by the measues ρ( v i ) of the vectos v i, i =, K The measues of the vectos ae: ρ( vi ) = vi, i =,6, ρ( vi ) = sgn v sgn v 4 vi, i = 4 > 0 o ρ v ) = sgn v, if eithe whee If ( 5 5 v5 v 4 = 0, v > [ ] 0 when h > [ < ] h. sgn 5 < then v oc i oc 2,3,4, sgn v 4 < 0. = i (Θ) = v3 = v6, v2 = v5 = 0.,6. of New Insued Unities of Risks 9

20 The odeing of Let be two potfolios whose states of the business ae defined by the vectos v ), v ( ), i =,,6. Definition. pecedes in the - ode, i.e. iff, 2 2 oc ( Θ( )) pecedes 2 i 2, i ( i 2 K oc i ( Θ( )) ρ( vi ( )) ρ( vi ( 2 )), i =, K,6. in the Θ ( ) -ode, i.e., Θ ( ) 2 iff ρ( vi ( )) ρ( vi ( 2 )), i =, K,6. of New Insued Unities of Risks 20

21 The new unity of isk intoduced at time t = 0 Notations and assumptions. = {, K, }, m, dependent components. S F S m h is the total claim amount of pe yea. is known and defined on [ 0, M ( S )], M ( S ) > 0. of New Insued Unities of Risks 2

22 The new potfolio The i.c. uns the potfolio in The state of the business is { } = ; B( ) = (, Θ), Θ = Θ( ). [ 0,]. The opeative stuctue elative to potfolio is epesented by Θ = { ), ( ), ( ), ε, M, M, ϕ ; α( η h ε (0, ε ]}. α η 0 of New Insued Unities of Risks 22

23 The state of the business B( ) The state of the business is epesented on the - plane though the points ( H,0,ε ) by which we define the new state vaiables v = v, v 3 = v + AA 3 + BF + BG, v = v + BF A, B( ) = ε A 5 5 A ( ) ( ) h, ε, B = h, ε, Q = ( 0, ), v A, 2 v2 + BF + BG A = v 4 = v 4 BG, v 6 = v 6 + BF. of New Insued Unities of Risks 23

24 The gaphic epesentation of B( ) B( ) and LEP Q ε ε v 3 ε v 4 3 B 4 0 v v v A h h v 2 v h A v 2 h B H of New Insued Unities of Risks 24

25 The classification of Assumption. The state of the business is authoized and is completely etained by the i.c. The classes and sub-classes of ae B( ) nomal isk capacity geneato isk capacity isk geat isk dangeous isk catastophic isk mega-catastophic isk of New Insued Unities of Risks 25

26 The classification of Definition. B( ) is a nomal isk iff is authoized, i.e., is a dangeous isk iff Definition. Let be a nomal isk. is a capacity geneato isk iff is capacity isk iff whee h h < h. > h + τ ( h h ), h h < h + τ ( h h ). τ = E [ S ] E[ ]. S h h. of New Insued Unities of Risks 26

27 The classification of Definition. Let is a geat isk iff is a catastophic isk iff h h < be a dangeous isk. h ( M α, M η ). is a mega-catastophic isk iff h( Mα, Mη ) < h h( M M, Mη ). h( M M, Mη ) < h. of New Insued Unities of Risks 27

28 The odeing of Let be a state of the business. Let be two unities of isk and let be the coesponding new potfolios. = Definition. pecedes in the - ode, i.e. iff B( ), 2 { } 2 ; pecedes 2 iff 2, 2 oci ( B( )) 2 = oc i ( B( )) oc ( Θ( )) 2, i =, K,6. i in the B( ) -ode, i.e. ρ( vi ( )) ρ( vi ( 2 )), i =, K,6. { ; },, B ( ) 2 of New Insued Unities of Risks 28

29 The odeing of Poposition. If then that is, ρ oc ( Θ( )) 2, i =,4,5,6, i ( )) ρ( vi ( )), i =, K,6, ( vi 2 B ( ) 2. of New Insued Unities of Risks 29