a111Actuarial Mathematics #1 - Single Premium Life Insurance

8/17/2019 a111Actuarial Mathematics #1 - Single Premium Life Insurance

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Term Life and Whole Life © copyright THC 2003. All rights reserved by Thomas

re%erence n&mber ''' assi%ication ''0)0'*ersion '.0 level iintermediate instr&ctions

p&blication date +&ly, 2003 '. Change tha&thor d -et 2. The total p

a%%iliation T/AAC1

email address ebet$tiaacre%.orglast revised date A&g03

re%erences

Descriptions

%inancial model class present val&e modeliss&er i%e /ns&rer

model type bond modelris! so&rces mortality

ris! distrib&tion diversi%ied ris!economic ass&mptions 4A# modeltechnical ass&mptions m5' ann&al interest rate

!ey words liability, li%e ins&rance, premi&ms, val&ation

Links

data The #ociety

%inancial models

Test Case 66 ear old 7ale /nterest 1ate 8.69-ene%it i%e /ns&rance :;ays Diven 66Et Ass&mption :'Eint=F:t0.6

' 66 0.00@22 ' 0.00@22 0.?G@23'?@2 6) 0.00?0) 0.??'G@ 0.00@?@663 0.?3)'0G'63 6G 0.00??6 0.?@2G?88G 0.00?GG@@' 0.@?6G?)328 6@ 0.0'0?8 0.?G30'6)G 0.0'0)88G? 0.@6G22'3)

6 6? 0.0'206 0.?)23G0@@ 0.0''6?)6G 0.@2030G62) )0 0.0'32? 0.?60GG83' 0.0'2)36G? 0.G@8?@32GG )' 0.0'8)G 0.?3@'3@62 0.0'3G)28? 0.G6''@0'G@ )2 0.0')2) 0.?283G)03 0.0'603036 0.G'@@32)?? )3 0.0'@0) 0.?0?386)G 0.0')822G@ 0.)@G@G@'@

'0 )8 0.0200) 0.@?2?22@? 0.0'G?'203 0.)6@26))3'' )6 0.02226 0.@G60'0@6 0.0'?8)@?? 0.)2??'0)6

'2 )) 0.028)2 0.@6668'@) 0.02'0)388 0.)02G@63''3 )G 0.02G') 0.@388G@82 0.022))883 0.6G)@2@06

-owers, N, et al., Act&arial 7athematics, #ociety o% Act&aries. H&ebner, #.

http://www.soa.org/tablemgr/tablemgr.asp

&ration

%rom*al&ation

Attained

Age

;robability

o% eathbetweenage andage :E'=

;robability

o% s&rvivalto age

;robability

o% s&rvivalto age andeathbetweenage andage :E'=

/nterest

isco&nt

http://www.soa.org/tablemgr/tablemgr.asphttp://www.soa.org/tablemgr/tablemgr.asp


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'8 )@ 0.02??2 0.@''@'3?? 0.0282@?8G 0.66'?@@6)'6 )? 0.032?@ 0.G@G6286' 0.026?G26) 0.62@2'@G2') G0 0.03)G3 0.G)'66'?) 0.02G?G'@ 0.6068G28)'G G' 0.080G 0.G336@0'6 0.02?@6)G' 0.8@3G06G'@ G2 0.088?6 0.G03G2388 0.03')323G 0.8)2@G)2G'? G3 0.060'' 0.)G20?'0G 0.033)G@8@ 0.882?83@

20 G8 0.066G@ 0.)3@8'26? 0.036)'0)6 0.823@)?))2' G6 0.0)'@8 0.)02@0'?3 0.03G2GG2G 0.806)')?22 G) 0.0)@28 0.6)6628)) 0.03@6?'8 0.3@@'60'623 GG 0.0G8?3 0.62)?332) 0.03?8@3'' 0.3G'83666

28 G@ 0.0@'?6 0.8@G860'6 0.03??8)68 0.366880G226 G? 0.0@?62 0.88G603)' 0.0800)062 0.380'38))2) @0 0.0?G@@ 0.80G8830? 0.03?@@063 0.3268@GG'2G @' 0.'0G26 0.3)G6)26) 0.03?82'0@ 0.3''8G'8?2@ @2 0.''G@2 0.32@'8'8G 0.03@))')3 0.2?@06@@82? @3 0.'2?68 0.2@?8G?@6 0.03G8??22 0.2@6223GG30 @8 0.'82'@ 0.26'?@0)3 0.036@2))' 0.2G2?8'8'3' @6 0.'6686 0.2')'6802 0.033)0''8 0.2)''@G?6

32 @) 0.')?'@ 0.'@2662@@ 0.030@@83 0.28??80)233 @G 0.'@3') 0.'6'))@6@ 0.02GGG?)2 0.23?'GG)338 @@ 0.'?G33 0.'23@@@?) 0.02888G0' 0.22@@G@'236 @? 0.2''@? 0.0??88'?) 0.02'0G0G) 0.2'?022'23) ?0 0.22G06 0.0G@3G'2 0.0'GG?8'@ 0.20?6?0663G ?' 0.283') 0.0)06GG02 0.0'8G2??' 0.2006)6'23@ ?2 0.2)0@2 0.086@8G'' 0.0''?6G@8 0.'?'?2@383? ?3 0.2@'G6 0.033@@?2G 0.00?68@3 0.'@3))38@

note: hidden rows for 80 ?8 0.30?@3 0.028380?G 0.00G68'6) 0.'G6G6863

easier viewing 8' ?6 0.36'@) 0.0')G??8 0.006?''08 0.')@'@)'682 ?) 0.820?? 0.0'0@@@3G 0.0086@3@? 0.')0?83)?83 ?G 0.68' 0.00)3088G 0.0038'0G2 0.'680'3'

88 ?@ 0.G86'6 0.002@?3G6 0.002'6)2@ 0.'8G3@0?686 ?? ' 0.000G3G8G 0.000G3G8G 0.'8'0388'8) '00 ' 0 0 0.'38?)''6


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0.0'380G6' 0.'8?880380.0'3G'?'? 0.')3'6?630.0'8'3@?@ 0.'GG2?@6'0.0'888'@) 0.'?'G803G0.0'8)8'@G 0.20)[email protected]'8?'G)@ 0.22'2???2

0.0'60?82@ 0.23)3?8'?0.0'6'202? 0.26'6'88?0.0'8?G?2) 0.2))8?3G80.0'8))683 0.2@''6?'@

0.0'8'?@)3 [email protected]'3)26?G 0.30@[email protected]'2?@0)2 0.32'?)880.0'22G@68 0.338282?80.0''62388 0.386G))[email protected]'0)?6)G 0.36)8)2060.00?GG@6) 0.3))280)'0.00@GG)2' 0.3G60')@3

0.00GG'?28 0.3@2G3)0G0.00))882) 0.3@[email protected]?63? 0.3?8?G6G20.008)'8?) 0.3??6?0)@0.003G2?8? 0.803320'G0.002?683' 0.80)[email protected]?606 0.80@6)?620.00'G63)G 0.8'03232

0.00'3268) 0.8'')8@))

0.000??8'6 0.8'2)[email protected] 0.8'[email protected] 0.8'3?06@)

0.0003'GG? 0.8'8223))0.000'080' 0.8'832G))

0 0.8'832G)) Jhole i%e /ns&rance


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The simplest ins&rance model calc&lates the cost o% providing a level amo&nt o% ins&rance :in this case


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4rder i%e /ns&rance *al&ation TechniB&es 'i%e /ns&rance *al&ation TechniB&es 2i%e /ns&rance *al&ation TechniB&es 3i%e /ns&rance *al&ation TechniB&es 8i%e /ns&rance *al&ation TechniB&es 6

i%e /ns&rance *al&ation TechniB&es )i%e /ns&rance *al&ation TechniB&es Gi%e /ns&rance *al&ation TechniB&es @i%e /ns&rance *al&ation TechniB&es ?

i%e /ns&rance *al&ation TechniB&es '0i%e /ns&rance *al&ation TechniB&es ''i%e /ns&rance *al&ation TechniB&es '2i%e /ns&rance *al&ation TechniB&es '3i%e /ns&rance *al&ation TechniB&es '8

In progress:

'6')

'G'@'?202'

2223


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ile Name Topic /ll&strated Act&arial 7athematics (' #ingle ;remi&m i%e /ns&rance.ls Net single ;remi&ms % Act&arial 7athematics (2 #ingle ;remi&m /mmediate Ann&ity.ls Net #ingle ;remi&ms Act&arial 7athematics (3 Ann&al ;remi&m i%e /ns&rance.ls Net evel Ann&al ;re1eserves (' irst ;rinciples.ls 1eserves as C&m&lati1eserves (2 #imple *al&ation orm&las.ls 1eserve eB&al the ;re

1eserves (3 *al&ation orm&las That 1ecognie AcB&isiton pense.ls 1eserves &sing one y1eserves (8a 1eserve Acc&m&lation orm&la :Term=.ls 1eserves developed %1eserves (8b 1eserve Acc&m&lation orm&la :J=.ls 1eserves developed %1eserves (6a 1eserves in a Dain M oss hibit :Term=.ls 7ean 1eserves in as

1eserves (6b 1eserves in a Dain M oss hibit :J=.ls 7ean 1eserves in as;ricing ('a :Term=.ls 1ecogniing Ibest esti;ricing (2a :Term=.ls 1ecogniing Ibest esti;ricing (3a :Term=.ls 1ecogniing policyhol;ricing (8a :Term=.ls 1ecogniing operation

1ecogniing the need1ecogniing optional

1ecogniing optional1ecogniing that deat7odeling sec&lar tren7odeling sec&lar tren1ecogniing sec&lar tr

1ecogniing variation7odeling a b&siness li


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or i%e /ns&ranceor /mmediate Ann&ities

i&mse Cash lows

sent *al&e o% &t&re -ene%its ess &t&re ;remi&ms

ar preliminary term method om /ncome #tatement lements om /ncome #tatement lements

s&rrogate %or tr&e 1eserves

s&rrogate %or tr&e 1eservesmateI interest income vers&s val&ation interest in the income statementmate mortalityI vers&s val&ation mortality in the income statementer nonrenewal ratesepenses

%or capital in ecess o% 1eservesodes o% premi&m payment monthly, B&arterly semiann&ally

odes o% premi&m payment ad&sting death claim amo&ntss occ&r other than on ecember 3' e%%ect on reserves and interest income s> mortality improvements s> epense in%lation

ends by calendar year o% iss&e> Cohort de%ined

in ris! by iss&e age and other parametersne

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a111Actuarial Mathematics #1 - Single Premium Life Insurance