Memoire RAHARINOSY - Ressources actuarielles
Transcription
Memoire RAHARINOSY - Ressources actuarielles
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F ' ' 13 G ( )1 G 8 ' ( B ; % A 9 ' # H0 GI '' /G ' F ' A F ' 8 8 3./ I B H ' ' B F " 9 ) 7 • • •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∗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∗ ( ( ; ;9 9 4 6 ; ; ;Y ( ; 9 9 3 L L 3 L L 6 . / / / / / / / / (7 (: ( (D +6 + + +( ++ + +7 +: + +D 6 .// 2 2 . ). .2 0 3 3) 3)0 3 3 3 12 3 3 / 3 20 30.2 3 0 3. / 30.1 .2 3 2... .10 11 2 )310 0 / / . 31 . 233) 3 0 13)1 1 )0 0/ 0 .0 332 1 .3 1 2 10 2 0 ( + 7 : D 6 ( + 7 : D 6 ( + 7 : D (6 ( ( (( (+ ( ) . 0)) 03 102 )0 2) 231 .). . 3 .21 3 3)) 3 / 3 3 3) 3 1 3 3 /3. 2) 1 0 322 1 1 ) /0 0/31 1 0 3 21)0 .. 1 / 3 1.0 22 ).00 0.0) 1. 1 / / / / / / / / ( + / / 7 : D 76 7 7 7( 7+ 7 77 7: 7 7D :6 ) 1 0 2 3 2 2 ) 0 2 0 2 2 0 )/ ) 0/ )2 10 00 )2 0 01 12 )3 0) )2 )3 )2 )3 . )/ / 1 ) 0 3 0 + J, ,-*+4 6&* 5,!R C !5* /// &5 6&* 5,!R C !5* //1 66 7 66 + 66 !; < ! 66 !; < ! 666 6 66 !; * % 66 6 667 6 66+ 6 66 76 7 7+ 77 7 + 7 6 (6 ( (+ (7 ( +6 + ++ +7 + + 7 6 + 7 6 + 7 6 6 3 4 ( ( 8 /// ( ; N ) 10 ;( + * $ %% & ' H;I "% & ) & # ! % "8 " N !; &' < ! ; 8 8 ( ( " 8 8 " C 8 B ( 8 H;I ( 8 Z H;I ; B ( ) ( ∑ J H H;I β X/ ; ; β ∑ − H I− ' ' " 3/ G ' ' H I [ =/ H H II = − 9 3/G ; β 8 ( 8 ; H 7 IM H;I & 8 ' B 8 ( H I ' ; I ; ( 9 / / () H II = − H ' ; H I [ =/ & ( )= H I − H I− N N ( ) ( )= , ' 8 Z ' ( ; J I ( () +Z H;I '' Β H H;I ( N ;( ( )) '' ' & '' P H;I ; N $ %% & ## ; * # ) & 5 D6F % % # 66 7 66 + 66 !; < ! 66 9 ) 9 !# 66 6 66 6 667 6 66+ 6 66 6 D ( ( ( + ++ +: 6 3 )0 ( 7 D 7 7 $ %% 3 D (6 ( ( (( (+ ( (7 (: ( (D +6 + + +( ++ + +7 +: + +D 6 ( + 7 : D 76 7 7 7( 7+ 7 77 7: !; 9 ! / / 22G / /0.3G / /)01G / /10.G / / 1/G / /02 G / /00)G / /0 2G / /231G / / /)G / /20.G / /.3/G / /. 3G / 0 G / /.. G / 0/ G / 03G / 1 G / 3)G / /1)G / ..)G / 2 G / 1.G / 32 G / )0.G / 2/G /) 2 G / 231G / )01 G / ))0)G / )33)G / 0 3G /1 ) G / 02 .G / 1.0)G / 03. G / 3) 0G / 1 G / .30 G & ) & 5 D6F 9 / / 31G / / 3)G / / .G / / 12G / /))3G // G / / /.G // 3 G / /0./G / /)).G / /01 G / /1 2G / /1 /G / /.11G / /1 G / /33/G / /. G / 1G / /. G / 11/G / 0/1G / 0 G / 2 3G / 02/G / 2. G / /1 G / 3 /G / / G / ))G / 03)G / )/. G / )01 G /0 ) G / ))10G /0 / G / ))/ G / 02 G / ).1 G / 10/.G )1 ) 9 !# / )) G / / .G / /23/G / /)/G / )1G / /.. G / /.) G / /2.)G / 0 G / /33.G / 3G / ))0G / 1.G / 201G / ) G / 3)0G / 21 G / 3G / 3 G / 2 G / 02.G / )0 /G / 3/)G / 13)G / )/)2G / )0 )G / 0122G /) 1 G / 001/G / 0)3.G / 1)1 G / /3/G / 2)12G / 0 3G / 23 G / 2 0/G )//3G /)1/G )3.1G ) 8 " 9 B N 8 L 9 J N ; 8 ' ( ;' " 9 N ( " '' ; J ; ( ' 6 ( ( " (# N 8 D E ; )) - ; ; 8 ( ' B 8 * . " ( 8 ( 9 8 − 7 − −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• • L H . 1I 9 ' 8 ' N ( - ' 7 )/ 7 7 J I + > . * M( # V " 0/ ./ ( H 3 /I *' " # L ' 8 H 0/ 8 8 H ' ; I N B ' N " 8 V " ( V " ./ 8 (( % \ ( % 8 H 31 I 6 (8 ' # ; L B N ? <!%% M # *' 8 % 8 < 8 ' ' ' " + )3 ] ] B (+ X ) *!% H IY( / % 3* @ H;I %% M # *' B$ ' B ' = 9 ( ) + +$ − ( − $ J !) 3./ & ' + "# +Z 9 ( + $ − "# ) ( ' !) − + A! * ! # 8 ( ;N " % > . *J ! ' A % ; ; B ' 7 7 ' # ; ( ; 7 N ; B " ; ; ( 8 - ( P ( ' ( # ( J # ' # 1 )/ # + / 1 * ( # H I % $ %% ,' A! * J ' ( ' # N V " ;N ; H;I N ( 8 ( E D N Z 1 4 " " D # B ; ( E % ' 9 ; 0/ , 8 # '' 3 ( 2 0 U 3 . G 37 / 37 1 37 2 . 2)G 3/ ..G 3 1G U 3 . G D@7: D G F " " U)1 U0/ )17 1 0/7 1 / )1G )1 / G 2. 21G 0 /3G W 2 / 10G - - - - - 0 # # + ;# !; V " &' J ; ; * V " 8 Z 9 µ = + + 9 + = ; − ∫ µ % % + = ; − ∫ + % % + ( = ; − + − ) B ( − ) = ; (− ) ; − B ( ) 9 ( = − ) = ; H− I = ; − B ' # = − ( >/ >/ > +( . = − B " * ' / '' $ ( + 9 4 7 ; ' % 8 ) 8 ) B 3 " ^ " − ; ( ( 9 0 / / +) ) ∏ =∏ − /+ =/ ∏ /+ ∏) = ∏ /+ − ∏ = = )− = ( −) = ( − ) H /+ I = ( − ) H /+ ∏ ∏ + ∏) ∏ & 6 = = = /+ = / ( −) ∏ / I= + ( ) − − ) H I= H I= ( − ) " H ( / I 9 = ; & = / " H − ' 9 I=− − 9 X7 H I (XH 7 I H I X B " ^ $ % '' ( ; " 0) '' ; ( ++ * A! ; (: ) * # % * . ' ! * ; *!* V " ' # ( −) ; 8 8 H;I B(_) = ∏ 2 = 3 H;I ' H;I H I H I N " ; 8 ( H_I H − H_II H I ` ; ( 9 H I− H I % H − I lH I ; 9 B (_ ) = Z σ = ∏ ; − Π 2 = 3 σ ⋅ (θ ) − σ ∗ H − I lH I " ( 9 (B (_ )) = ∑ 2 = 3 Zσ ` ; ( ( B & J ( & *<% 9 H_I = − +Z _ = J ; $ < ; σ Π ∗ ∗ ; H −; I lH I 00 + ∑ 2 = 3 − ⋅ (θ ) − σ ∗ & ; ' ( ∑ 2 ∗ = 3 ( lH I ⋅ H − ∗I 8 ; (θ ) − ∗ 9 ) (8 J " θ V " ; B " ^ $ ( # % ; ( ( ( 6 ; ' ( % ( # ; # '' B '' % B ' # ( '' ( 9 ////// θX / 333 0/ /. )/3 ∑ 2 **= = = 3 lH I ∗ H − ∑ ∗ I 2 = 3 ∧ θ − lH I ∗ H − ∗ ∗ = 1 /0. / −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email protected]/ 56..73/ / /123G / / /G // 0 G // G / /2 )G / /. 1G / /3/.G / /3 3G / // G / /20G / 2G / 0 G / )01G / 0./G / G / 2 )G / 30 G / /03G / ) G / ) .G / 1) G / 212G / 331G /) 1 G / )033G / ) 0.G / ).31G / 0 2)G / 0 )0G / 1/0 G / 1)03G / 12 1G / 2G / 2 .G /2 G / 23/1G / .01 G / 3 /G / 1 G / /G / 20G / 2)3G / ./0G / 3 G / //3G / /2G / G / ) 2G / 1 /G / 0 G / .1/G /) G / )))0G / )2 G / 0/.1G / 0)13G / 0 13G / 1// G / 101 G / / G / .2G / 2) 3G / ./00G / .2.1G /3 G / / G 0 G 12G ))31G 00) G 1 1 G . G . G 30))G / 00G / G )).2G 1) /G + J, ,-*+4 6&* 5,!R C !5* &5 6&* 5,!R , !*5&* /G / 3G / .G / 2G !; A! !; 9 ! / G 5 ; / 1G / 0G / )G / G / G / /G 3 ) )) )1 )2 )3 0 0) 01 02 03 1 , 02 1) 11 12 13 ) 1 2 + J, ,-*+4 6&* 5,!R 6& B, 5,CB& 6& + 5,B-5& &5-=-&& &4 /// &5 6&* 5,!R , !*5&* &4 //1 # # !; 666 K ;# !; A! 66 & ! !; * % " <% $ % %& %" # ' & &% &! &$ ! !& 3 ' ; " L ( D E 0. 8 ( ( + J, ,-*+4 6&* 5,!R , !*5&* &5 6& B, +! C& 5@ ..73/ # # ( & ) ""*$ ! !; * % " $ %% %' # & $ 3 03 &% &' !# !& -45& @,BB& 6& +4=-,4 & , 31 G # $ " 9 ' !# !; A! 9 66 ) & !; * % ! % # $ %% %' # & $ &% &' !# !& 3 B " 5$&+ -& &5 J ,5-Q!& 6& B ,**! ,4 & @-& ; B " ' 1/ J&5,!5+4 ; " -45& @,BB& 6& +4=-,4 & &5 5,!R C !5* # $ 9 !# " !; 9 ! 9 66 ) ' & !; * % ! % # $ % %& %" # ' & &% &! &$ ! !& 3 B ;( 8 # 7 ' ' # 8 1 31G '' ; ! ! % -! % ' A! 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