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COLLECTIVE RISK MODEL

(MODEL RISIKO KOLEKTIF)

Faihatuz Zuhairoh,

M.Sc.

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1.1 Pengenalan

1.2 Distribusi Mae!u" (CompoundDistribution)

1.# Ru!us K$n%$lusi

1.& Si'atsi'at ari Compound PoissonDistribution 

M*TERI

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M$el risi"$ +aa risi"$ ini%iual,!e!+erti!bang"an setia+ +$lis ini%iu an"lai! +aa setia+ +$lis asuransi. Seang"an!$el risi"$ "$le"ti' -aitu !engu!+ul"anse!ua "lai! -ang i+er$le engan!enu!la"an se!ua +$lis ala! suatu+$rt'$li$.

Dala! !$el risi"$ "$le"ti', iasu!si"anba/a suatu "u!+ulan "lai! ala! +$rt'$li$ari +$lis !eru+a"an suatu +r$ses ran$!.

PENGENALAN

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*nai"an N  in$tasi"an sebagaiban-a"n-a "lai! -ang iasil"an $lesuatu +$lis ala! +eri$e tertentu.

Misal"an  X 1 in$tasi"an sebagai u!la"lai! -ang +erta!a,  X 2  sebagai u!la

"lai! -ang "eua, an seterusn-a,

!a"a0

!enunu""an "u!+ulan "lai! -ang

iasil"an ari +$rt'$li$ sela!a +eri$e

DEFINISI

1 2   N S X X X  = + + +L

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ntu" !e!buat suatu !$el -ang bai",u!u!n-a iguna"an 2 asu!si asarberi"ut.

1. !eru+a"an %ariabel ran$!-ang beristribusi ienti".

2. Variabel ran$! saling bebas.

Prinsi+ ala! +enge!bangan te$ri+aa !$el risi"$ "$le"ti' aala 'ungsi+e!bang"it !$!en (!.g.') .

1 2, , X X  L

1 2, , , N X X    L

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Misal"an in$tasi"an sebagai 'ungsiitribusi ari  X i  -ang ienti" an

ine+enen. *!bil 3 sebagai %ariabel

ran$! engan 'ungsi istribusi, !a"a

in$tasi"an sebagai !$!en "e" an

in$tasi"an sebagai !.g.' ari 3.

 

s r us umpu anKlaim

( ) P x

k k  p E X  =

( )   tX 

 x M t E e =

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in$tasi"an sebagai !.g.' ariban-a"n-a "lai!, an

in$tasi"an sebagai !.g.' ari

"u!+ulan "lai! -ang a"an itulis"anengan

( )   tN  N  M t E e =

( )  tS 

S  M t E e

=

( )S  F s

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[ ]

[ ]

[ ]

[ ]

[ ] [ ]

[ ]

[ ]

0

1 20

1 2

0

1 20

0

 N N 

 N 

n

 N n

n

n

nn

n

 E S E E S N 

 E S N n P N n

 E X X X N n P N n

 E X X X N n P N n

 E X X X P N n

n P N n

 E N 

 µ 

 µ 

∞

=

∞

=

∞

=

∞

=

∞

=

=

= = =

= + + + = =

= + + + = =

= + + + =

= =

=

∑

∑∑

∑∑

L

L

L

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( )   ( )   ( )( ) ( )

[ ]   ( ) ( )

[ ]   ( ) ( )2

 N N N Var S E Var S N Var E S N  

 E N Var X Var N 

 E N E Var X Var N 

 E N Var x Var N 

 µ 

 µ 

 µ 

= +

= +

= +

= +

( )

( )   ( )

( )

log

log

 X 

tS tS  

S 

 N    N M t 

 X 

 N X 

 M t E e E E e N 

 E M t E e

 M M t 

= =     = =

=  

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Diasu!si"an ba/a 4 beristribusi ge$!etri",engan 'ungsi "e+aatan +eluang ari 4aala

Di!ana an . Tentu"an

( )Pr   n

 N n pq= =

CO4TO5

( ) N  M t 0 1q< <   1 p q= −

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E!IMA KASI"