in insurance : IFRS / Solvency II

Transcription

1 Impac es of ormes he IFRS asse jumps e assurace i isurace : IFRS / Solvecy II 15 h Ieraioal FIR Colloquium Zürich Sepember 9, 005 Frédéric PNCHET Pierre THEROND ISF Uiversié yo 1 Wier & ssociés Sepember 9, h Ieraioal FIR Colloquium Page 1

2 Coex (1/) The aim of his paper is o propose a operaioal saisical model wih jumps for a risy asse, by geeralizig he Blac & Scholes model. The B&S model is a curre referece for expers i isurace, bu i he ew coex geeraed by : - he accouig sadards chages (IFRS) ad - prudeial chages (Solvecy II), his model resuls i a uderesimae of he ris of placeme ad herefore of he provisios or he solvecy required capial of he compay. Sepember 9, h Ieraioal FIR Colloquium Page

3 Coex (/) The saisical procedures for esimaig he parameers of he seleced model (Mero s model) are preseed ad illusraed o real daa. The he model is applied i problems of: - pricig of opios (> IFRS 4), - deermiaio of a SCR (> Solvecy II) i a very simple framewor. We show ha he resuls may be very differe wih or wihou jumps, uder a fixed level of global volailiy. Sepember 9, h Ieraioal FIR Colloquium Page 3

4 The model The asse dyamic is expressed as follows : S( ) N B U 1 S( 0) exp µ B ( B ) ( 0) N ( N ) ( 0) Where is a browia moio, a Poisso process wih iesiy. U ( U ) ( ) 1 λ is a sequece of i.i.d. radom variables, ormally disribued wih 0 mea ad volailiy. U This model is very racable ad calculaios are easy. Refereces : Mero [1976], Ramezai & Zeg [1998] Sepember 9, h Ieraioal FIR Colloquium Page 4

5 15 h Ieraioal FIR Colloquium Sepember 9, 005 Page 5 Saisical iferece Two-sep parameers esimaio: - mome mehod : - maximum lielihood (he mome esimaors are used o iiialize he algorihm) ( ) ( ) ( ) λ λ 0 u e m r E m!!! ( ) ( ) ( ) λ µ λ π λ µ i u i u u x e x x exp!,,,,,...,

6 Numerical illusraio : lcael (1/) The x i x ( ) i S l S i i 1 are i.i.d. r.v. Probabilié 0,08 0,07 0,06 0,05 0,04 0,03 0,0 0,01 The expeced value of he uderlyig soc is 9,07, wih sadard deviaio,97 ; he asymmery equals 0,53>0. The reur is NOT gaussia. 0-5% -0% -15% -10% -5% 0% 5% 10% 15% 0% Redeme Sepember 9, h Ieraioal FIR Colloquium Page 6

7 Numerical illusraio : lcael (/) Desié Mero Blac e Scholes 50% of he volailiy comes from he jump compoe! (expeced reur 5%, volailiy 45%) 0-0,0-0,16-0,1-0,08-0,04 0,00 0,04 0,08 0,1 0,16 0,0 Valeur Mome esimaors ielihood esimaors µ 0, , , , λ 0, , u 0, , Sepember 9, h Ieraioal FIR Colloquium Page 7

8 Fair value of a call opio (1/) I he case of he Merom s model, marigale measure is o uique ad he mare is a icomplee mare. Various approaches ca be adoped o jusify he choice of ris measure used o price he opio (see BOTT [004]). We adop here he iiial soluio of Mero cosisig i cosiderig ha he ris associaed wih he jump compoe is o-sysemaic (i.e. specific o he asse) ad may hus be diversified: oe does o associae ris premium o i. Tha hus resules i simply evaluaig he expeced value of associaed flows. Sepember 9, h Ieraioal FIR Colloquium Page 8

9 Fair value of a call opio (/) If he exercise probabiliy is very close i boh models, he opio fair value is very differe. For example, wih S100, K110, T1, a global volailiy equals o 5% ; he volailiy of he Browia moio par is 15% ad 0% for he jump par. I his case, he call opio price is 9,15 i he B&S model, 5,0 i he Mero model. The exercise probabiliy is 4% i boh cases. I a IFRS 4 perspecive, his could have a big impac i pricig ad reservig a guaraeed miimum deah beefis opio i a ui-lied isurace corac for example. Sepember 9, h Ieraioal FIR Colloquium Page 9

10 15 h Ieraioal FIR Colloquium Sepember 9, 005 Page 10 Solvecy II : arge capial (1/4) The model comes from Deelsra & Jasse [1998] : The liabiliy : The asse : d he mismach process : B µ 0 exp N U B 1 0 µ exp a l N U B a a 1 0 µ µ

11 Solvecy II : arge capial (/4) I a coex of he ype Solvecy, a isurace compay mus have a level of ow capial (he arge capial) which corols he oal ris of he compay a a predeermied horizo. e us cosider here ha he measureme of he oal ris of he compay is he probabiliy of rui which is a quesio of corollig a horizo 1 year wih a probabiliy of 1 %. We suppose ha a he ed of he year, he compay was o have eough capial correspodig o a liabiliy which has o be 100. The isurer has a iiial ime his amou i provisios, i hus acs o deermie he amou of he arge capial γ such as: 1 Pr N ( 100 γ) exp µ B , 01 U 1 Sepember 9, h Ieraioal FIR Colloquium Page 11

12 Solvecy II : arge capial (3/4) We suppose moreover : µ l 0, 08 λ 1, 5 λu 0, 16 α U λ λ U Sepember 9, h Ieraioal FIR Colloquium Page 1

13 Solvecy II : arge capial (4/4) I comes ou from his illusraio, ha he use of he Blac & Scholes model leads o uderesimae he level of he arge capial i a proporio which ca be impora. s a example if half of oal volailiy is explaied by he jumps, he arge capial resulig from he B&S model uderesimaes of 13,5 % he rue required capial. Sepember 9, h Ieraioal FIR Colloquium Page 13

14 Coclusio (1/) The Blac & Scholes model is a sadard used i may pracical siuaios i isurace: opio pricig, M, deermiaio of he probabiliy of rui, ec. The pricipal qualiy of his model lies i is simpliciy of implemeaio (calculaio of he associaed fucioals, esimae of he parameers, ec), is adequacy wih he daa beig i geeral of quie poor qualiy. Sepember 9, h Ieraioal FIR Colloquium Page 14

15 Coclusio (/) The model ha we prese here, iiially suggesed by Mero preserves he ease of use, while improvig he sesiiviy ad he adequacy o he daa. I is possible o iegrae io he asse modellig properies such as asymmery ad a ail of disribuio hicer ha ha of a ormal law. These properies are o wihou cosequece o he appreciaio of he level of he capial of solvecy wihi he meaig of Solvecy II. Thus, he reflexios o he sadard model of Solvecy II (see for example Djehiche ad Hörfel [004] or Plache ad Thérod [005]) mus iegrae his ype of model o guaraee a sufficie appreciaio of solvecy i his ew referece framewor. Sepember 9, h Ieraioal FIR Colloquium Page 15

16 Coacs Frédéric PNCHET Pierre THEROND Wier & ssociés hp:// ISF hp:// 9, rue Beaujo F Paris 18, aveue Félix Faure F yo 50, aveue Toy Garier F yo Cedex 07 Sepember 9, h Ieraioal FIR Colloquium Page 16