Aleš Ahčan Darko Medved Ermanno Pitacco Jože Sambt Robert Sraka Ljubljana, 11.-12-2011
Mortality data Slovenia Mortality at very old ages Smoothing mortality data Data for forecasting Cohort life tables Age shifting Selection effect Annuity tables Summary
Log death rate -3-2 -1 0 Log death rate -10-9 -8-7 -6 Log death rate -9-8 -7-6 Log death rate -3.5-2.5-1.5-0.5 Log death rate -3.0-2.0-1.0 0.0 Log death rate -9.5-8.5-7.5-6.5 Slovenia: male death rates (1975-2008) Slovenia: female death rates (1975-2008) 1975 1980 1990 2000 2008 75 80 85 90 95 100 age Slovenia: male death rates (1975-2008) 1975 1980 1990 2000 2008 1975 1980 1990 2000 2008 0 10 20 30 40 age Slovenia: total death rates (1975-2008) 1975 1980 1990 2000 2008 Population data were provided by the Statistical Office of the Republic of Slovenia. Data are available for the time span from 1971 to 2008 for 1 year age period 0 10 20 30 40 age Slovenia: female death rates (1975-2008) 1975 1980 1990 2000 2008 75 80 85 90 95 100 75 80 85 90 95 100 age Slovenia: total death rates (1975-2008) 1975 1980 1990 2000 2008 0 10 20 30 40 We can observe - decreasing log death rates by period - high volatility at young and old ages age age
Log death rate -7-6 -5-4 -3-2 Log death rate -8-7 -6-5 -4-3 -2 To build a life table in one cohort (say 1965) we must make an assumption of prior to 1971 From HMD average central mortality rates for the years 1930-1933, 1948-1952, 1952-1954, 1960-1962 were taken. Log linear interpolation to interpolate the missing m xt, in the 1945 to 1970 period Slovenia: male death rates (1945-2000) Slovenia: female death rates (1945-2000) x=20 x=40 x=60 x=20 x=80 x=40 x=60 x=80 1950 1960 1970 1980 1990 2000 1950 1960 1970 1980 1990 2000 Time
qx 2008 0.1 0.2 0.3 0.4 0.5 0.6 qx 2008 0.1 0.2 0.3 0.4 male female 75 80 85 90 95 100 x 75 80 85 90 95 100 x Slovenian population mortality data at very old ages have very low risk exposures This leading to large sampling errors and highly volatile crude death rates For 1971-1980 the data for age groups above 85 are not available at all. to make projections we need a method that can extrapolate a survival function at very old ages
Recent mortality studies suggested that the force of mortality is slowly increasing at very old ages, approaching a relatively flat shape Following this approach, the death rates for very old ages were estimated according to the logistic formula ln qˆ ( t) a b x c x q 2 x t t t xt ( t) 1 qx( t) x 0 x To ensure the concave behaviour of lnq x (t The log-quadratic regression model where one-year death probability at time t with ε xt is independent and normally distributed
R2 0.70 0.75 0.80 0.85 R2 0.85 0.90 0.95 R2 0.85 0.90 0.95 1.00 R2 0.95 0.97 0.99 R2 0.60 0.70 0.80 R2 0.75 0.85 0.95 R2 0.85 0.90 0.95 1.00 R2 0.93 0.95 0.97 0.99 This two constraints yield the following regression model ln qˆ ( t) c ( x) 2 x t xt R squared - male R squared - male R squared - male R squared - male 1970 1980 1990 2000 year 1970 1980 1990 2000 year 1970 1980 1990 2000 year 1970 1980 1990 2000 year R squared - female R squared - female R squared - female R squared - female startage: 75 stopage: 100 start.smooth: 75 stop.smooth: 100 startage: 75 stopage: 100 start.smooth: 75 stop.smooth: 110 startage: 75 stopage: 100 start.smooth: 75 stop.smooth: 130 1970 1980 1990 2000 startage: 60 stopage: 100 start.smooth: 75 stop.smooth: 130 1970 1980 1990 2000 1970 1980 1990 2000 1970 1980 1990 2000 year year year year we obtain the optimal fit (highest R2) at starting smoothing age of 75. We use 85 as an age for extrapolating mortality
mx 0.0 0.2 0.4 0.6 0.8 1.0 mx 0.0 0.2 0.4 0.6 0.8 1.0 mx 0.0 0.2 0.4 0.6 0.8 1.0 mx 0.0 0.2 0.4 0.6 0.8 1.0 Year 1985 startage: 75 stopage: 100 start.smooth: 75 stop.smooth: 130 0 20 40 60 80 100 120 age Year 1995 startage: 75 stopage: 100 start.smooth: 75 stop.smooth: 130 Year 2005 startage: 75 stopage: 100 start.smooth: 75 stop.smooth: 130 0 20 40 60 80 100 120 age Year 2008 startage: 75 stopage: 100 start.smooth: 75 stop.smooth: 130 the extrapolation for period 2008 in which the logistic nature of the extrapolated function at high ages can be seen this procedure is important to construct life annity tables, since we need rates for ages above 100 years 0 20 40 60 80 100 120 0 20 40 60 80 100 120 age age
Slovenian population death rates exhibit considerable variations also at young ages We therefore use smoothing techniques to obtain a better picture of the underlying mortality We use weighted penalised regression splines with a monotonicity constraint proposed by Hyndman and Ullah (m-splines) R y ( x ) f ( x ) ( x ) t i t i t i t, i
Log death rate -8-6 -4-2 0 Log death rate -8-6 -4-2 0 Log death rate -8.5-7.5-6.5 Log death rate -8.5-7.5-6.5 male 2008 female 2008 0 10 20 30 40 Age male 2008 0 10 20 30 40 Age female 2008 As one can see, a hump in mortality profiles for younger males is evident in contrast to the female mortality profile. This is mainly due to accidents which more often occur to males at younger ages. At higher ages a concave shape of mortality is evident 0 20 60 100 0 20 60 100 Age Age
The following procedure was implemented to derive mx () t ETR = size of the population at 1 July of each year, D xt, = the number of observed deaths 1. xt, 2. in year t at age x Dxt, mx () t ETR xt, 3. We used the following procedure to prepare basic raw data: a. replacing mx () t which are NA with zero b. smoothing mx () t at a very old age (above 85) a regression with a logistic function from 75 with q ( t) m ( t) / (1 0.5 m ( t)) at limit age 130 >R 2 0.97 x x x c. reverse back to m ( t) q ( t) / (1 0.5 q ( t)) d. cut to upper age 100 x x x e. where mx ( t) 0: interpolate mx () t with neighbouring values; i.e. mx ( t s) and m ( ) x t k, if mx ( t s) and mx ( t k) >0 for the first k and s, and predict if 0 at the beginning or end of the time series f. leave the population data as original g. fix Dxt, number as Dx, t ETRx, t mx () t for ages over 85, otherwise D xt, as observed h. this data set is then considered raw data for further research. 4. Data used for the Lee-Carter method: mx () t smoothed with m-splines, weights are D xt, and ETR xt, 5. Other methods: D xt, and ETR xt,
qx ( t min 0) qx ( t ) ( min n qx t min max) qx ( t max 0) qx ( t ) ( max n qx t max max) qx( t), t ( t0,, tn) qx t t tn tmax presents observed smoothed mortality data and ( ), ( 1,, ) represents projected mortality data. With n from which projections are made. t we denote the base year q ( t), q ( t 1), q ( t), q ( t), q ( t) The sequence x x1 is called a cohort table. The sequence x x 1 x 2 called a period table. is
for the period life table we calculate lx 1 ( t) (1 qx ( t)) lx ( t) d ( t) q ( t) l ( t) x x x e x () t x k1 lxk() t 1 l ( t) 2 x for the age cohort life table we must first choose the base cohort birth year τ then we calculate the life table to take diagonal probabilities from birth year τ lx 1 ( ) (1 qx ( x)) lx ( ) d ( ) q ( x) l ( ) e x x x x ( ) x k1 lxk( ) 1 l ( ) 2 x We can calculate an annuity of size 1 which is payable yearly at the beginning of each year while an insured is alive from a 1, k 0 x k1 k x ( ) (1 i) k0 (1 qx j( x j), k 0 j0
cohort 1945 1950 1955 1960 1965 1970 1990 2000 2010 2020 male central 60,8 65,4 69,0 72,0 74,8 76,8 82,8 84,8 86,7 88,0 upper 61,5 66,4 70,3 73,6 76,7 79,0 85,5 87,5 89,4 90,7 lower 60,1 64,4 67,6 70,3 72,7 74,5 79,7 81,5 83,3 84,7 female central 71,2 75,4 78,7 81,3 83,6 85,4 90,0 91,5 92,8 93,9 upper 72,3 76,7 80,2 83,0 85,4 87,3 92,0 93,5 94,8 95,7 lower 70,1 74,0 77,0 79,4 81,5 83,2 87,4 89,0 90,3 91,4
The annuity calculated with a cohort life table not only depends on age at entry but also on the individual birth year of the insured person. This would lead to the construction of a cohort life table for every generation. This is impractical and cannot be used in everyday actuarial calculations. The so-called Rueff method is adopted
first cohort period is chosen among a series of generation tables fundamental cohort by shifting the actual age, depending on the birth year, the exact actuarial values will be approximated by using the fundamental cohort with an age shift. as regards the type of criteria, usually the expected present value of a life annuity is used. Let us denote the chosen fundamental cohort year. Then the adjustment involves an age shift h( ) years (plus or minus). Assuming that mortality declines over time, the function h( ) must satisfy the following relations: 0, h( ) 0, 0,
For each birth year 1955,.., 2020 and for all ages xmin x xmax max 65, where xmin 55 and i x, the integer shift h (, x) is determined, which satisfies the following condition: a ( ) a ( ) a ( ) i i i i xh (, x) 1 x i xh (, x) We calculated an annuity with both a 2.75% interest rate and a 0% interest rate (i.e. i (2.75%,0%) ). The birth year =1965 was used as the fundamental cohort year because 1965, meaning those who turned 45 in 2010, is an age that can be considered intermediate between those who enter into a deferred annuity and insured persons who buy an immediate annuity. i For each period we obtain a set of h (, x) and we choose to select the average value of them to obtain a single value: x 2.75 1 2.75 h ( ) h (, x) x max max xmin 1 xxmin
leto rojstva do 1950 leto leto leto moški ženske moški ženske moški ženske moški ženske rojstva rojstva rojstva 2 2 1971-2 -1 1992-5 -4 2013-8 -6 1951 2 2 1972-2 -2 1993-5 -4 2014-8 -6 1952 2 2 1973-2 -2 1994-5 -4 2015-8 -6 1953 2 1 1974-2 -2 1995-5 -4 2016-8 -6 1954 1 1 1975-2 -2 1996-5 -5 2017-8 -7 1955 1 1 1976-2 -2 1997-6 -5 2018-8 -7 1956 1 1 1977-2 -2 1998-6 -5 2019-8 -7 1957 1 1 1978-3 -2 1999-6 -5 2020-8 -7 1958 1 1 1979-3 -2 2000-6 -5 1959 1 0 1980-3 -3 2001-6 -5 1960 0 0 1981-3 -3 2002-6 -5 1961 0 0 1982-3 -3 2003-6 -5 1962 0 0 1983-3 -3 2004-6 -5 1963 0 0 1984-4 -3 2005-6 -6 1964 0 0 1985-4 -3 2006-7 -6 1965 0 0 1986-4 -3 2007-7 -6 1966-1 -1 1987-4 -4 2008-7 -6 1967-1 -1 1988-4 -4 2009-7 -6 1968-1 -1 1989-4 -4 2010-7 -6 1969-1 -1 1990-4 -4 2011-7 -6 1970-1 -1 1991-5 -4 2012-8 -6
The birth year τ =1965 was used for the fundamental cohort year to generate a Slovenian base cohort population mortality table. We then used a smoothing procedure to obtain less variability in the data. At old ages we use logistic formula with start smoothing age at 95. Limit age is set to 120.
A life annuity purchaser is, with a high probability, a healthy person Particularly low mortality is observed in the first years of the life annuity payment An expected lifetime of annuity owner is higher than average. In order to take selection into account we have to adjust population mortality table
There are no adequate statistical data to make a conclusion regarding the selection effect in Slovenia. The idea is to use standardised mortality ratio (SMR) from another population which has similar characteristics Then we calculate life insurance market central death rates from LIM RC HMD m ( t) SMR ( t) m ( t) SMR stan x x x ETR mˆ () t xt d ETR mˆ () t xt x x
We used UK statistical data from 1999-2002 collected by the Continuous Mortality Investigation Bureau We used the mortality of those insured to deferred annuities PNM00 tables for men and PNF00 tables By comparing the mortality UK insured lives with that of the general population of the United Kingdom (taken from English Life Table No. 16, 2000-2002), it is possible to quantify SMR for deferred annuitants.
Percentage 0.5 0.6 0.7 0.8 UK Selection male female 20 40 60 80 100 age
To correct some irregular patterns we chose the following selection factors of mortality for males and females 42,5%, x 48 RCM ' SMRx, 48 x 64 61,5%, 65 x 71 RCM SMR RCM ' x SMRx, 72 x 77 RCM ' SMR78, 78 x 107 RCM ' RCM ' x 108 SMR78 (1 SMR78 ), 108 x 120 12 55%, x 50 RCF ' SMRx, 50 x 75 RCF SMR RCF ' x SMR75, 76 x 111 RCF ' RCF ' x 111 SMR75 (1 SMR75 ), 111 x 120 9 We obtain the projected and selected mortality table for the generation of insured persons born in 1965 for deferred annuitants: SDA65.
SDA65 is structured to represent the mortality of the insured's deferred annuity or pension insurance. In other cases, such as immediate annuity further selectivity should be added We applied a correction factor to the mortality rates of the SDA65 table for delayed commitments which includes the increased expected survival of recipients of immediate annuities, namely: K F x 1, x 54 1 0.02559( x 54), 55 x 59 IFL00 qx, 60 x 84 PNFA00 qx 1, x 85
selection in % 0.5 0.6 0.7 0.8 0.9 1.0 Selection factors males females 0 20 40 60 80 100 120 the mortality of deferred annuitants merges to the mortality of immediate annuitants after age 60 (Germany), so SIA65 is a table that may be considered an aggregate table which is recommended for use in the annuity business in Slovenia. Age
TABLE 1: Immediate annuity: Age at issue 60/birth year 1950 (annuity starts in 2010) net single premium R94 R04 Poisson model central rates Poissonlow mortality scenario R94/ Poisson male 18.10488 20.36943 18.50437 18.98097 0.978 female 20.49378 22.00024 20.98405 21.76160 0.977 TABLE 2: Deferred annuity: Age at issue 60/birth year 1980 (annuity starts in 2040) net single premium R94 R04 Poisson model central rates Poissonlow mortality scenario R94/ Poisson male 19.72056 22.94550 20.46949 21.28829 0.963 female 22.81581 24.42694 22.87369 23.58179 0.997 Abbreviations used in Tables 4 and 5: R94 DAV 1994 annuity life table R04 DAV 2004 annuity life table Poisson model Slovenian annuity life table based on the Poisson log-bilinear model
m f m f m f m f - 1950 2 2 1971-2 -1 1992-5 -4 2013-8 -6 1951 2 2 1972-2 -2 1993-5 -4 2014-8 -6 1952 2 2 1973-2 -2 1994-5 -4 2015-8 -6 1953 2 1 1974-2 -2 1995-5 -4 2016-8 -6 1954 1 1 1975-2 -2 1996-5 -5 2017-8 -7 1955 1 1 1976-2 -2 1997-6 -5 2018-8 -7 1956 1 1 1977-2 -2 1998-6 -5 2019-8 -7 1957 1 1 1978-3 -2 1999-6 -5 2020-8 -7 1958 1 1 1979-3 -2 2000-6 -5 1959 1 0 1980-3 -3 2001-6 -5 1960 0 0 1981-3 -3 2002-6 -5 1961 0 0 1982-3 -3 2003-6 -5 1962 0 0 1983-3 -3 2004-6 -5 1963 0 0 1984-4 -3 2005-6 -6 1964 0 0 1985-4 -3 2006-7 -6 1965 0 0 1986-4 -3 2007-7 -6 1966-1 -1 1987-4 -4 2008-7 -6 1967-1 -1 1988-4 -4 2009-7 -6 1968-1 -1 1989-4 -4 2010-7 -6 1969-1 -1 1990-4 -4 2011-7 -6 1970-1 -1 1991-5 -4 2012-8 -6 male female 0 83,35 89,32 20 65,50 71,51 40 46,24 51,89 50 36,77 42,25 51 35,84 41,31 52 34,92 40,36 53 34,00 39,42 54 33,09 38,48 55 32,19 37,55 56 31,30 36,61 57 30,41 35,67 58 29,52 34,74 59 28,65 33,80 60 27,79 32,86 61 26,93 31,92 62 26,09 30,99 63 25,26 30,05 64 24,43 29,11 65 23,62 28,16 66 22,80 27,22 67 21,98 26,28 68 21,17 25,35 69 20,36 24,41 70 19,55 23,48 remining life expectancy of annuity holder cohort 1965
Phase 1 Projecting mortality for Slovenian population we have tested different methods (deterministic and stochastic) criteria was best fit Poisson log-bilinear method was chosen to project mortality of Slovenian population
Phase 2 construction of aggregate and selected industry life annuity tables for Slovenia we have analyzed UK, German and Italian approach cohort 1965 was taken as basis age shifting table was constructed selection factor was built on the basis of UK experience (insured to deferred and immediate annuities) PN and IM 2000 tables Phase 3 recommendation regarding unisex life annuity tables
The net single premium based on the Poisson model is 2% to 4% higher than that calculated by the current minimum standard in Slovenia. Therefore the DAV 1994 R annuity tables are inappropriate for the best estimate valuation of annuity liabilities within the Solvency II Framework. In other words, technical provisions for annuities based on the DAV 1994 R tables are underestimated by 2% to 4%, which is not insignificant.
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