Exam MLC Flashcards. Samuel Broverman

Transcription

1 Exam MLC Flashcards Samuel roverman

2 S. roverman 2007 Samuel roverman

3 Table of Contents Introductory Topics Life Contingencies Poisson Processes Multi-State Transition Models IN-1 to IN-11 LC-1 to LC-135 PP-1 to PP-9 MC-1 to MC-15

4

5 Uniform Distribution of Deaths in Each Year of Age (Linear Assumption) LC-14

6 j > is a linear function for > on the interval! Ÿ > Ÿ ": j > œ j >. for!ÿ>ÿ". For!Ÿ>Ÿ", ; œ> ; > 0XÐÑ Ð>Ñ œ > :. Ð >Ñ œ ;,. Ð >Ñ œ " > ;, > ; ; œ if! Ÿ = > Ÿ ", > = " = ; 7 œ ; 7 ß ; œ / " " " œ / " ; " 7 # # #,. UDD is related but not equivalent to De Moivre's Law. DeMoivre's Law is a stronger assumption and implies UDD. ; LC-14

7 Constant Force of Mortality in Each Year of Age (Exponential Assumption) LC-15

8 . Ð >Ñ œ. ÐÑ for! > ". ÐÑ : œ /,. ÐÑ œ 68:. : œ : œ/ œð: Ñ if!ÿ> =Ÿ" > ÐÑ > > > = 7 œ. ÐÑ LC-15

9 Hyperbolic Interpolation in Each Year of Age (alducci's Hypothesis) LC-16

10 Under the hyperbolic assumption, for!ÿ>ÿ", we have the following relationships: ; œ Ð" >Ñ ;, " > >.Ð >Ñ œ > = ; " Ð" >Ñ; > ; " Ð" > =Ñ; ; œ if! Ÿ > = Ÿ ", " j > is a linear function on the interval Ò!ß "Ó, LC-16

11 Insurance Under DeMoivre's Law LC-35

12 E _ œ + œ E œ : 8 " " " = 8l = 3, = = l #8 #Ð= Ñ # " # " E1 _ œ E 8 : = #3 3#, œ = #3 3# 8 " E œ + " " _ œ E œ : = 8l = $, = = l # $ 8 # $ Ð= Ñ # " " / # " " / E1 _ œ E 8 : = # $, œ = # $ LC-35

13 Insurance Under The Constant Force Model. ÐCÑ œ. LC-36

14 . / œ : and ; œ " : E _ œ, E œ, E œ : 8 8 : Ñ ; # ; ; 3 ; 3 ; 3 3# 8Ð. $ Ñ. Ð" / Ñ # E _ œ, E œ, E œ 1 8 :... $. $. # $ LC-36

15 The Normal Approximation LC-37

16 If [ has a normal distribution with mean IÒ[Ó and variance ZÒ[Ó, then [ IÒ[Ó ÈZÒ[Ó Probabilities for [ can be found from has a standard normal distribution. [ IÒ[Ó + IÒ[Ó + IÒ[Ó Ò Ó Ð Ñ TÒ[ Ÿ+ÓœT Ÿ œf ÈZÒ[Ó ÈZÒ[Ó ÈZÒ[Ó The central limit theorem of probability shows that if [ œ \" \ # â \ 8, then [ approaches a normal distribution as 8p. This justifies using the normal approximation. LC-37

17 Fully Discrete Policy enefit Premiums Whole Life and Endowment Insurance LC-60

18 Whole life insurance of 1 with premiums for life: E ".E T œ.. + œ.. +. œ " E ß ÞÞ " T + œ ß E œ T. T. 8-year endowment insurance with premiums for 8 years: T œ ÞÞ œ ÞÞ œ T T À8l E E E À8l À8l À8l + + " " À8l À8l À8l À8l E À8l ".EÀ8l ÞÞ ÞÞ + À8l + À8l " EÀ8l " T À8l TÀ8l. À8l TÀ8l. T œ œ. œ ß À8l ÞÞ À8l + œ ß E œ LC-60

19 Fully Discrete Policy enefit Premiums For Other Insurances LC-61

20 8-year term insurance: T œ " À8l E year pure endowment: T œ À8l 1 " À8l À8l E À8l À8l 5-payment whole life policy: 5T œ E À5 l.. 8l 8l À8l 8-year deferred annuity: TÐ + Ñ œ, premiums are paid during the 8-year deferral period LC-61

21 Variance of Issue Date Loss Random Variable For Whole Life and Endowment Insurance LC-62

22 Fully continuous whole life: TÐE Ñ # # # Z+<ÒPÓœ " Ò E E Ó " " E œ Š # $ # # Ò E E Ó Similar for fully discrete whole life. Fully discrete endowment insurance: TÀ8lÑ. Z+<ÒPÓœ " Ò E E Ó # " " E À8l œ Š Ò E E Ó # Similar for fully continuous endowment insurance. LC-62 # # # À8l À8l #

23 Contingent Probability 8 ; " C LC-109

24 ; œ Probability that ÐÑ dies before ÐCÑ and ÐÑ dies before 8 8 " C years have elapsed œtòðxðñÿxðcññ ÐXÐÑŸ8ÑÓ 8 œ ' '! = 0XÐÑXÐCÑÐ=ß>Ñ.>.= 8 œ '! T ÒX ÐCÑ =lx ÐÑ œ =Ó = :. Ð =Ñ.= ; œ ' ' 0 Ð=ß >Ñ.>.= œ TÒXÐÑ XÐCÑÓ " C! = XÐÑXÐCÑ If XÐÑand XÐCÑare independent, then 8 8 " C! = = C ; œ ' : :. Ð =Ñ.= and " C! = = C ; œ ' : :. Ð =Ñ.= LC-109

25 Contingent Probability 8 ; # C LC-110

26 8; # C œ Probability that ÐCÑ dies after ÐÑ and ÐCÑ dies before 8 years have elapsed œtòxðñ XÐCÑŸ8Ó 8 >!! œ ' ' 0 Ð=ß>Ñ.=.> XÐÑXÐCÑ If XÐÑand XÐCÑare independent then 8 8 ; # œ ' C! > ; > : C. ÐC >Ñ.> œ 8; " C 8: C 8; ; œ ; # C " C LC-110

27 Additional Contingent Probability Relationships LC-111

28 ; œ ; ;, ; ; œ ; œ " 8 8 " 8 # " C C C # C ; œ ; ;, ; ; œ ; œ " 8 C 8 " 8 " " C C C " C C ; œ ; ;, ; ; œ " 8 C 8 # 8 # C C # C # C LC-111

29 Uniform Distribution of Decrement in the Multiple Decrement Table LC-125

30 Within each year of age we have Ð4Ñ Ð4Ñ Ð7Ñ Ð7Ñ > > ; œ > ; and ; œ > ; for! Ÿ > Ÿ ". > Ð 7 Ñ Ð4Ñ Ð4Ñ > Ð 7 Ñ Ð 7 Ñ Ð 7 Ñ For! > " :. Ð>Ñ œ ; and :. Ð>Ñ œ ;. 691 : wð4ñ Ð4Ñ Ð7Ñ 7 7 ; Î; 691 : Ð Ñ 7 Ð4Ñ Ð Ñ wð4ñ Ð Ñ ; œ ; and : œ Ð: Ñ wð4ñ Ð Ñ For!Ÿ=Ÿ", : œ Ð : Ñ. = = Ð4Ñ Ð7Ñ 7 ; Î; LC-125

31 UDD in the Associated Single Decrement Tables Two Decrement Model LC-126

32 wð4ñ > wð4ñ > wð4ñ Ð4Ñ. wð4ñ Within each year of age ; œ> ; for!ÿ>ÿ". Then for! > ", : Ð>Ñœ;. We assume this for each decrement. For # decrements we have Ð"Ñ " Ð Ñ Ð"Ñ " wð"ñ wð#ñ Ð"Ñ ; œ ' : 7. Ð>Ñ.> œ ' : :. Ð>Ñ.>! >! > > wð"ñ " wð#ñ œ; Ò" # ; Ó Ð#Ñ " Ð Ñ Ð#Ñ " wð"ñ wð#ñ Ð#Ñ! >! > > wð#ñ " wð"ñ œ; Ò" # ; Ó ; œ ' : 7. Ð>Ñ.> œ ' : :. Ð>Ñ.> LC-126

33 UDD in the Associated Single Decrement Tables Three Decrement Model LC-127

34 Ð"Ñ " Ð Ñ Ð"Ñ " wð"ñ wð#ñ wð$ñ Ð"Ñ! >! > > > ; œ ' : 7. Ð>Ñ.> œ ' : : :. Ð>Ñ.> wð"ñ " wð#ñ wð$ñ " wð#ñ wð$ñ # $ œ; Ò" Ð; ; Ñ ; ; Ó Ð#Ñ Similar relationships hold for ; and ; (we reverse the roles of decrements 1 and 2 or 1 and 3 in the equation) Ð$Ñ LC-127

35 One-Step Transition Probability Matrix MC-5

36 It is usually convenient to represent the one-step transition probabilities in the form of a one-step transition probability matrix. As a simple illustration, suppose the Markov Chain has three states Ö"ß #ß $. Then at time 8, the chain can be in any one of those states, and it can transfer to any one of the states at time 8 ". The one-step transition probability matrix at time 8 will be a $ $ Ð"ß"Ñ Ð"ß#Ñ Ð"ß$Ñ Ô U8 U8 U8 matrix of the form U 8 œ Ö Ð#ß"Ñ Ð#ß#Ñ Ð#ß$Ñ U U U Ù Õ Ð$ß"Ñ Ð$ß#Ñ Ð$ß$Ñ U U U Ø MC-5