Mortality in the Nordic countries Sweden. IAA Life 2007, Stockholm Ellinor Forslund

Transcription

1 Moraliy in he Nordic counries Sweden IAA Life 2007, Sockholm Ellinor Forslund

2 Ouline Background Mehodology Daa Resuls Conclusions

3 Background Mehodology Daa Resuls Conlusions The need of a new common moraliy invesigaion Lagislaion IOPR-direcive Solvency II IFRS Phase II Include moraliy rends in life insurance reserves Laes common moraliy invesigaion in 1989

4 Background Mehodology Daa Resuls Conlusions M90 Tradiionally, he Makeham formula has been used μ a b 10 c -f Laes, public moraliy invesigaion resuled in he M90-parameers a b c females f 0 males Flip around age 60 Higher moraliy in younger ages Lower moraliy in older ages

5 Background Mehodology Daa Resuls Conlusions The Lee-Carer model μ e α κ β Developed by Lee and Carer in 1992 The mehod is erapolaive and makes no effor o incorporae knowledge abou medical, behavioral, or social influences on moraliy change

6 Background Mehodology Daa Resuls Conlusions The Lee-carer model - α μ e α κ β The average of he logarihm of he force of moraliy over he observed period of ime The general shape of he moraliy

7 Background Mehodology Daa Resuls Conlusions The Lee-carer model - κ μ e α κ β The ime rend

8 Background Mehodology Daa Resuls Conlusions The Lee-carer model - β μ e α κ β The age-specific paerns of he moraliy change Shows he sensiiviy of he logarihm of he force of moraliy a age o he ime rend The higher value, he higher response of he ime rend

9 Background Mehodology Daa Resuls Conlusions Esimaing he parameers Basic conceps Eposure R N 1 1 N 2 Number of deahs D Force of moraliy μ D R Assume ha he number of deahs are approimaely Poission disribued, i.e. D is Po R μ E[ D ] R μ

10 Esimaing he parameers Maimum Likelihood Mehod I Esimaing he parameer vecors Seing up and maimizing he likelihood sum:...,,,,... 3, 2, 1,,...,,,, β β β β β κ κ κ κ κ α α α α α d D P L,, β κ α Background Mehodology Daa Resuls Conlusions

11 Background Mehodology Daa Resuls Conlusions Esimaing he parameers Maimum Likelihood Mehod II The maimizaion is done by he following hree seps Taking he logarihm of he likelihood sum Taking he derivaive of he logarihmed likelihood sum Solving he equaion by using a Newon-Raphson mehod, ieraing one erm a a ime

12 Esimaing he parameers Ieraion [ ] [ ] [ ] [ ] [ ] v v v v v v v v v v v v v v v v e R R d D E d h d h d h a a β κ α μ λ κ λ κ λ β β β λ β λ κ κ λ λ Background Mehodology Daa Resuls Conlusions

13 Esimaing he parameers Conains The following consrains are used The consrains are used in each ieraion b 1 1 κ n n a a β β β κ κ β κ κ 1 1 Background Mehodology Daa Resuls Conlusions

14 Background Mehodology Daa Resuls Conlusions Trend esimaion daa Long, homogeneous ime series need o esimae rends in moraliy Changes in moraliy among insured individuals due o The moraliy has acually changed? The characerisics of he insured has changed and herefor also he moraliy? Only homogeneous ime serie available for rend esmaion is he moraliy of he enire Swedish populaion

15 Background Mehodology Daa Resuls Conlusions Swedish populaion daa

16 Background Mehodology Daa Resuls Conlusions Insurance daa Number of insured Number of deceased Volunarily insured Compulsory insured

17 Background Mehodology Daa Resuls Conlusions Classificaion of insurance daa Swedish populaion Females Males The insured Volunarily insured Compulsory insured Females Males Females Males Insured professional employees Females Males

18 Background Mehodology Daa Resuls Conlusions Daa specifics Boh moraliy and longeviy risks Healh eaminaion/able-bodied Insurance based and no individual based, i. e. individuals can be found in more han one group

19 Background Mehodology Daa Resuls Conlusions Lee-Carer parameers - α

20 Background Mehodology Daa Resuls Conlusions Lee-Carer parameers - κ

21 Background Mehodology Daa Resuls Conlusions Lee-Carer parameers - β

22 Background Mehodology Daa Resuls Conlusions Populaion moraliy - by calender year

23 Background Mehodology Daa Resuls Conlusions Populaion moraliy - by birh cohor

24 Background Mehodology Daa Resuls Conlusions Converion from populaion o insurance moraliy Raion beween populaion and insurance force of moraliy Smoohing i, P k The smoohed raio is applied o he projeced populaion force of moraliy μ μ i P i μ k i, P μ

25 Background Mehodology Daa Resuls Conlusions Converion from populaion o insurance moraliy - raios Volunarily insured Compulsory insured

26 Background Mehodology Daa Resuls Conlusions Riskinde

27 Background Mehodology Daa Resuls Conlusions Epeced remaining life ime all ages, volunarily insured

28 Background Mehodology Daa Resuls Conlusions Epeced remaining life ime all ages, compulsory insured

29 Background Mehodology Daa Resuls Conlusions Epeced remaining life ime a age 65

30 Makeham formula Makeham formula for each cohor Manual adjusmens for beer fi in high and low ages ω ω μ ω μ ω k b a F c,, 10 * Background Mehodology Daa Resuls Conlusions

31 Background Mehodology Daa Resuls Conlusions Cohor Makeham formulas volunarily insured

32 Background Mehodology Daa Resuls Conlusions Conclusions Epeced remaining life ime depends on age, se, year of birh and group of insured The laer year of birh, he longer epeced remaining life ime Volonarily insured individuals live longer han compulsory insured individuals born in he same in year Makeham is no good approimaion a all ages a he same ime