Tables de Mortalité
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Transcript of Tables de Mortalité
1
Tables de Mortalité
Instituto de Seguros de Portugal Le 10 mars 2008
2
Calculation of mathematical provisions
• Carried out on the basis of recognised actuarial methods
• The mortality table used in the calculation should be chosen by the insurance undertaking taking into account the nature of the liability and the risk class of the product
• No mortality table is prescribed
3
Calculation of mathematical provisions
• Longevity risk is mainly important in annuities and in term assurance
• With respect to term assurance companies are very conservative in the choice of the mortality table used to calculate premiums and mathematical provisions (very high mortality rates compared to observed rates)
• In new life annuity contracts companies adequate the choice of mortality tables to the effects of mortality gains projected from recent experience
4
Calculation of mathematical provisions
• In old annuity contracts that were written on the basis of old mortality tables, actuaries regularly analyse the sufficiency of technical basis and reassess the mathematical provisions according to more recent mortality tables
• The relative weight of life annuity mathematical provisions represents about 2% of total mathematical provisions from the life business
5
Market Information to the Supervisor
Pure EndowmentsEndowments and Whole Life
Term Assurances
“Universal Life” types of policy“Unit-linked” and “Index-linked” types of policy
•Death Risk
•Survival Risk
Information on the Annual Mortality Recorded and on the Annual Exposed-to-Risk (broken down by age and sex) on the following types of Mortality Risk:
6
Market Information to the Supervisor
Annuities
•Annuitants Risk
Pension Funds Annuitant Beneficiaries
Information on the Annual Mortality Recorded and on the Annual Exposed-to-Risk (broken down by age and sex) on the following types of Mortality Risk: (follow up)
Number of Pension Fund Members
7
Supervisory process
• Responsible actuary report
• ISP’s mortality studies
• Static and dynamic mortality tables
• Publication of papers and special studies
• ISP analysis of suitability of mortality tables used
8
Supervisory process• Responsible actuary report
• The responsible actuary should:• comment on the suitability of the mortality tables
used for the calculation of the mathematical provision
• produce a comparison between expected and actual mortality rates
• Whenever significant deviations exist, he should measure the impact of using mortality tables that are better adjusted to the experience and the evolutionary perspectives of the mortality rates
9
•Feed-Back information from the Supervisor• Under the Life Business Risk Assessment, ISP conducts
independent research and runs various statistical methods (deterministic and stochastic) to ascertain the Trend and Volatility of the multiple variables and risk sources that affect the Life Business:
• Each year, ISP issues a Report on the Portuguese Insurance and Pension Funds Market in which it publishes Special Studies intended to feed-back information onto the Insurance Undertakings and their Responsible Actuaries on the above mentioned risk sources, their possible modelling techniques and the corresponding parameters.
ISP Supervisory Process
10
Mortality Table
0,000
0,100
0,200
0,300
0,400
0,500
0,600
0,700
0,800
0,900
1,000
0 5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
105
age (x)
Pro
b.
of
1 i
nd
ivid
ua
l o
f a
ge
x)
Dyi
ng
ov
er
1 y
ea
r =
q(x
)
tgx
tgx
tgx
xq
x
The cathets of the triangles should be taken
One should take into account the fact that In graph tg determine the value of To
In their correct scale
Mortality Table
0
100.000
200.000
300.000
400.000
500.000
600.000
700.000
800.000
900.000
1.000.000
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
105
age (x)
l(x)
= N
º in
div
idu
als
aliv
e at
ag
e (x
)
Mortality Projections for Life Annuities (example)
x
xh
h
xh
h
xx lh
d
h
q
dx
qd
00limlim hxxxh lld
xx
x
xh
hx
hxx
h
xhx
h
x llh
dl
h
ll
h
ll
dx
ld
000limlimlim
xx
x ldx
ld
with
The force of mortality (x) may be expressed as the first derivative of the rate of mortality (qx):
11
If a mortality trend follows a Gompertz Law, thentk
xtx e
hence tk
x
tx e
, then
x
txtk
ln and also
x
tx
tk
ln1
If mortality were static, then the complete expectation of Life would be
dzek
ee k
zk
x
o x
x
Z
1
1, or, in summary
k
kf
e
x
x
o
with
11
1
!ln
nZ nn
n
k
kdze
x
xkz
x
where ...5772157,0 Is the Euler constant
12
Let us suppose now, that for every age the force of mortality tends to dim out as time goes by, in such a way that an individual which t years before had age x and was subject to a force of mortality x , is now aged x+t and is subject to a force of mortality lower than x+t (from t years ago). The new force of mortality will now be: e
tr
txt
tx
Where translates the annual averaged relative decrease in the force of mortality for every age
er
If we further admit another assumption, that the size relation between the forces of mortality in successively higher ages is approximately constant over time, i.e.:
etk
xtx
and e
tktx
ttx
then eeee
trk
x
tktr
x
tktx
ttx
hence
etrk
xt
tx
John H. Pollard –“Improving Mortality: A Rule of Thumb and Regulatory Tool” – Journal of Actuarial Practice Vol. 10, 2002
Mortality Projections for Life Annuities (example)
13
eeeeeee tk
x
tktxtr
x
tktxtrtk
x
ttxtrk
The prior equation also implies that:
x
txtr
lnwhere hence, finally
x
tx
tr
ln1
Mortality Projections for Life Annuities (example)
14
Mortality Tables
0,000
0,100
0,200
0,300
0,400
0,500
0,600
0,700
0,800
0,900
1,000
0 5 10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
10
0
10
5
11
0
age (x)
Pro
b.
of
1 in
div
idu
al a
ged
(x)
d
yin
g i
n t
he
cou
rse
of
1 ye
ar
= q
(x)
tkxtx e
trx
tx e
tTqx
Tq x
tTx
Tx
The practical application of the theoretical concepts involving the variables k and r may be illustrated in the graph bellow:
Mortality Projections for Life Annuities (example)
15
In order to increase the “goodness of fit” of the mortality data by using the theoretical Gompertz Law model involving the variables k and r, it is sometimes best to assume that r has different values for different age ranges (we may, for example, use r1 for the younger ages and r2 for the older ages)
Mortality Projections for Life Annuities (example)
Mortality of insured lives of the survival-risk-type of life assurance in Portugal: 2000-2002 (Males )
-0,0100
0,0100
0,0300
0,0500
0,0700
0,0900
0,1100
0,1300
0,1500
20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98
xq
age
upper boundary
Observed mortality rates
lower boundary
mortality trend projected
from 1997 to 2001
ee
yearsrxyearskSplineGompertzx
199720015020205020199720
2
21
21
001
ee
yearsrxyearskSplineGompertzx
19972001100513610051199736
2
21
21
001
16
As may be seen, the previous graph illustrates several features related to the Portuguese mortality of male insured lives of the survival-risk-type of life assurance contracts (basically, endowment, pure endowment and savings type of policies) for the period between 2000 and 2002: The mortality trend for the period 2000-2002 (centred in 2001) is adequately fitted to
the observed mortality data and has been projected from the Gompertz adjusted mortality trend corresponding to the period between 1995 and 1999, with k=0.05 for the age band from 20 to 50 years and with k=0.09 for the age band from 51 to 100 years. The parameter r, which translates the annual averaged relative decrease in the force of mortality for every age assumes two possible values; r=0.05 for the age band from 20 to 50 years and r=0 for the age band from 51 to 100 years:
Some minor adjustments to the formulae had to be introduced, for example, the formula for the force of mortality for the age band from 51 to 100 years is best based on the force of mortality at age 36, multiplied by a scaling factorthan if it were directly based on the force of mortality at age 51:
eeyearsrxyearskSplineGompertz
x
19972001100513610051199736
2
21
21
001
199736 2
1 Spline
e
3650
Mortality Projections for Life Annuities (example)
17
Further to that, some upper and lower boundaries have also been added to the graph. Those boundaries have been calculated according to given confidence levels in respect of the mortality volatility (in this case and ) calculated with the normal approximation to the binomial distribution, with mean and volatility
%9,991 %1,01
xxx qEE xxxx qqE 1
10
1%9.99@
x
x
xxxxxvel forconfid. lex q subject to
E
qqEqEq
x
The upper boundary may, therefore, be calculated as:
10
1%9.99@
x
x
xxxxxvel forconfid. lex q subject to
E
qqEqEq
x
And the lower boundary may be calculated as:
Those approximations to the normal distribution are quite acceptable, except at the older ages, where sometimes there are too few lives in , the “Exposed-to-risk”
xE
Mortality Projections for Life Annuities (example)
18
As for the rest, the process is relatively straightforward:
From the Exposed-to-Risk ( )at each individual age, and from the observed mortality ( ) we calculate both the Central Rate of Mortality ( ) and the Initial Gross Mortality Rate ( ) and assess the Adjusted Force of Mortality ( ) using “spline graduation”
xE
x xmxq
Splinex 2
1
We then calculate the parameters for the Gompertz model that producein a way that replicates as close as possible the
Gompertzx 2
1 Splinex 2
1
The details of the process are, perhaps, best illustrated in the table presented in the next page;
This process has been tested for male, as well as for female lives, so far with very encouraging results, but we should not forget that we are only comparing data whose mid-point in time is distant only some 4 or 5 years from each other and that we need to find a more suitable solution for the upper and lower boundaries at the very old ages.
Mortality Projections for Life Annuities (example)
19
20
Mortality of insured lives of the survival-risk-type of life assurance in Portugal: 2000-2002 (Males)
-0,0050
0,0000
0,0050
0,0100
20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
Mortality Projections for Life Annuities (example)
xq
ages
21
Mortality of the Beneficiaries and annuitants of Pension Funds in Portugal2000-2002 (Males)
-0,0010
0,0040
0,0090
0,0140
0,0190
0,0240
0,0290
0,0340
20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
xq
ages
As may be seen in the graph below, between the young ages and age 50 there are multiple decremental causes beyond mortality among the universe of beneficiaries and annuitants of Pension Funds. That impairs mortality conclusions for the initial rates, which have to be derived from the mortality of the population of the survival-risk-type of Life Assurance
Mortality Projections for Life Annuities (example)
22
6. Mortality Projections for Life Annuities (example) In general, the mortality rates derived for annuitants have to be based on the
mortality experience of Pension funds’ Beneficiaries and Annuitants from age 50 onwards but, between age 20 and age 49 they must be extrapolated from the stable trends of relative mortality forces between the Pension Funds Population and that of the survival-risk-type of Life Assurance.
e
tttxttx T
2006008.005.0000144446.065.1
Ages 2040 :
Annuitants (Males)
etttx
ttx txT
2006008.005.0000144446.0
400654.065.1
Ages 4149 :
e
tttxttx T
20060075.009.0000044994.0
Ages 50 :
Where T is the Year of Projection and 2006 is the Reference Base Year
23
6. Mortality Projections for Life Annuities (example)
e
tttxttx T
2006008.0028.0000211957.0415.1
Ages 2034 :
Annuitants (Females)
etttx
ttx txT
2006008.0028.0000211975.0
3510495.0415.1
Ages 3544 :
e
tttxttx T
20060075.009.0000033095.0
Ages 45 :
Where T is the Year of Projection and 2006 is the Reference Base Year
The above formulae roughly imply (for both males and females) a Mortality Gain (in life expectancy) of 1 year in each 10 or 12 years of elapsed time, for every age (from age 50 onwards).
24
Mortality Projections for Life Annuities (example)
Annuitants
As was mentioned before, for assessing the mortality rates at the desired confidence level we may use the following formulae:
21
21
211
x
x
xq
levelconfidencedesiredthatsuchiswhere
E
qqEqEq
x
xxxxxlevelconfidencex
αα
α
Φ
0,1
max@
In our case ()=99,5% which implies that 2,575835
Now, to use the above formulae we need to know two things: The dynamic mortality trend for every age at onset, and the numeric population structure.
25
Mortality Projections for Life Annuities (example)
In order to calculate the trend for the dynamic mortality experience of annuitants we need to use the earlier mentioned formulae and construct a Mortality Matrix:
26
Mortality Projections for Life Annuities (example)
0
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
0 9 18 27 36 45 54 63 72 81 90 99 108
Population Structure of Pension Funds Beneficiaries and Annuitants (Males: Exposed-to-Risk) Projection for 2006
ages
3-year Exposed-to-Risk
20 year Moving Average for the 3-year Exposed-to-
Risk
1-year Exposed-to-Risk (stable
structure)
In order to calculate a Stable Population Structure we need to smoothen the averaged proportionate structures from several years experience
27
Mortality Projections for Life Annuities (example)We are now able to project the dynamic mortality experience for different
ages at onset and for different confidence levels
28
Supervisory process
ISP analysis of suitability of mortality tables used
• ISP receives annually information regarding the mortality tables used in the calculation of the mathematical provisions
• This information is compared with the overall mortality
experience of the market and with mortality projections • ISP makes recommendations to actuaries and insurance
companies to reassess the calculation of mathematical provisions with more recent tables whenever necessary
29
ee aa rxkSplineGompe rtz
x
19982004371998 37
0042 nos 5830nos 5830
21
21
ee asa rxkSplineGompertz
x
19982004571998 37
0042 nos 10059nos 10059
21
21
Idades (x)
qx
1998
2001
2004
-0,0100
0,0400
0,0900
0,1400
0,1900
0,2400
0,2900
0,3400
20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98
Ages (x)
Mortality Projections for Life Annuities (example)
30
YEARS: 1997/1999PENSION FUND PENSIONERS (MALES)
Males MalesIdade (2) (3) (4) (5)
Actuarial "Exposed to Risk" Observed Projected ProjectedEx Mortality Mortality Mortality
(X) Splines Gompertz50 1.630,0 8,0 13,9 15,0 ####51 2.062,5 16,0 17,1 19,2 ####52 2.411,0 24,0 19,9 22,6 ####53 2.820,5 23,0 23,6 26,7 ####54 3.355,0 33,0 29,2 32,1 ####55 3.985,0 27,0 35,3 38,5 ####56 4.681,0 34,0 41,8 45,7 ####57 5.010,0 35,0 46,1 49,4 ####58 5.278,0 65,0 51,2 51,5 ####59 5.595,0 41,0 58,3 56,4 ####60 5.809,5 70,0 63,3 64,0 ####61 6.115,0 67,0 68,7 73,7 ####62 6.015,0 69,0 72,2 79,2 ####63 5.926,0 87,0 78,5 85,4 ####64 5.552,5 83,0 83,1 87,5 ####65 6.369,0 121,0 105,7 109,7 ####66 6.413,0 131,0 115,5 120,7 ####67 6.244,5 127,0 123,2 128,5 ####68 5.951,8 124,0 129,6 133,9 ####69 5.434,0 115,0 131,5 133,6 ####70 5.226,5 150,0 139,6 140,5 ####71 4.956,5 138,0 144,9 145,6 ####72 4.693,5 154,0 150,5 150,6 ####73 4.485,5 137,0 158,2 157,3 ####74 4.137,0 139,0 160,8 158,4 ####75 3.844,5 157,0 166,5 160,8 ####76 3.512,5 184,0 169,1 160,5 ####77 3.190,5 168,0 168,9 159,1 ####78 2.678,5 155,0 154,4 145,8 ####79 2.190,0 160,0 136,3 130,1 ####80 1.955,0 136,0 133,9 126,8 ####81 1.837,5 152,0 140,5 130,0 ####82 1.665,0 125,0 140,1 128,4 ####83 1.428,5 141,0 130,6 120,1 ####84 1.220,5 147,0 119,7 111,8 ####85 1.063,0 136,0 114,8 106,1 ####86 872,0 117,0 106,4 94,8 ####87 721,5 116,0 98,1 85,4 ####88 601,0 113,0 89,7 77,4 ####89 481,5 100,0 77,1 67,4 ####90 364,5 74,0 63,5 55,5 ####
Sum 4.199,0 4.071,1 3.985,9 #####
M a l e sI d a d e S q u a r e d C o n t r i b u t i o n t o C o n f i d e n c e C o n f i d e n c e A p p r o x i m a t e A p p r o x i m a t e
A c t u a r i a l D e s v i a t i o n s D e v i a t i o n s t h e C h i - s q u a r e d L e v e l o f L e v e l o f S t d . D e v i a t i o n V a r i a n c es u m [ ( 3 ) - ( 5 ) ] o f M o r t a l i t y T e s t M o r t a l i t y a t M o r t a l i t y a t [ ( 3 ) - ( 4 ) ] / s q r ( 4 ) s u m [ ( 4 ) ]
( X ) G o m p e r t z 9 9 , 9 % 0 , 1 % G o m p e r t z G o m p e r t z5 0 - 6 , 9 9 5 5 2 - - 4 8 , 9 3 7 3 5 3 , 2 6 3 4 6 3 , 0 2 9 7 9 2 6 , 9 6 1 2 6 - 1 , 8 0 6 5 1 1 4 , 9 9 5 5 2
5 1 - 3 , 1 6 4 2 1 - 0 1 0 , 0 1 2 2 1 0 , 5 2 2 4 4 5 , 6 3 7 1 3 3 2 , 6 9 1 2 8 - 0 , 7 2 2 8 0 3 4 , 1 5 9 7 3
5 2 1 , 3 7 3 5 3 + + 1 , 8 8 6 5 9 0 , 0 8 3 3 8 7 , 9 2 8 1 8 3 7 , 3 2 4 7 6 0 , 2 8 8 7 6 5 6 , 7 8 6 2 0
5 3 - 3 , 7 3 4 2 6 - - 1 3 , 9 4 4 6 8 0 , 5 2 1 6 0 1 0 , 7 5 7 3 6 4 2 , 7 1 1 1 6 - 0 , 7 2 2 2 2 8 3 , 5 2 0 4 6
5 4 0 , 8 8 1 3 8 + + 0 , 7 7 6 8 4 0 , 0 2 4 1 9 1 4 , 6 0 6 5 7 4 9 , 6 3 0 6 6 0 , 1 5 5 5 2 1 1 5 , 6 3 9 0 7
5 5 - 1 1 , 5 3 1 3 9 - - 1 3 2 , 9 7 2 9 4 3 , 4 5 1 0 3 1 9 , 3 5 0 6 3 5 7 , 7 1 2 1 5 - 1 , 8 5 7 6 9 1 5 4 , 1 7 0 4 6
5 6 - 1 1 , 7 1 3 7 5 - 0 1 3 7 , 2 1 1 8 9 3 , 0 0 1 5 5 2 4 , 8 2 1 6 6 6 6 , 6 0 5 8 4 - 1 , 7 3 2 5 0 1 9 9 , 8 8 4 2 1
5 7 - 1 4 , 4 1 6 0 0 - 0 2 0 7 , 8 2 0 9 1 4 , 2 0 5 5 4 2 7 , 6 9 4 3 7 7 1 , 1 3 7 6 2 - 2 , 0 5 0 7 4 2 4 9 , 3 0 0 2 0
5 8 1 3 , 4 5 0 9 4 + + 1 8 0 , 9 2 7 6 7 3 , 5 0 9 8 1 2 9 , 3 6 3 5 8 7 3 , 7 3 4 5 5 1 , 8 7 3 4 5 3 0 0 , 8 4 9 2 7
5 9 - 1 5 , 3 5 1 3 7 - - 2 3 5 , 6 6 4 6 3 4 , 1 8 2 0 6 3 3 , 1 5 5 5 0 7 9 , 5 4 7 2 5 - 2 , 0 4 5 0 1 3 5 7 , 2 0 0 6 4
6 0 6 , 0 0 8 2 9 + + 3 6 , 0 9 9 4 9 0 , 5 6 4 1 3 3 9 , 2 7 3 3 1 8 8 , 7 1 0 1 1 0 , 7 5 1 0 8 4 2 1 , 1 9 2 3 6
6 1 - 6 , 6 6 1 8 7 - - 4 4 , 3 8 0 5 6 0 , 6 0 2 4 9 4 7 , 1 4 1 4 9 1 0 0 , 1 8 2 2 6 - 0 , 7 7 6 2 0 4 9 4 , 8 5 4 2 3
6 2 - 1 0 , 2 3 5 9 3 - 0 1 0 4 , 7 7 4 2 9 1 , 3 2 2 3 1 5 1 , 7 3 0 4 3 1 0 6 , 7 4 1 4 3 - 1 , 1 4 9 9 2 5 7 4 , 0 9 0 1 6
6 3 1 , 6 3 7 8 4 + + 2 , 6 8 2 5 3 0 , 0 3 1 4 3 5 6 , 8 1 3 1 4 1 1 3 , 9 1 1 1 8 0 , 1 7 7 2 7 6 5 9 , 4 5 2 3 2
6 4 - 4 , 4 5 4 9 4 - - 1 9 , 8 4 6 4 7 0 , 2 2 6 9 3 5 8 , 5 5 8 0 8 1 1 6 , 3 5 1 8 0 - 0 , 4 7 6 3 8 7 4 6 , 9 0 7 2 6
6 5 1 1 , 3 1 8 9 6 + + 1 2 8 , 1 1 8 9 3 1 , 1 6 8 1 0 7 7 , 3 1 9 8 6 1 4 2 , 0 4 2 2 1 1 , 0 8 0 7 9 8 5 6 , 5 8 8 2 9
6 6 1 0 , 2 5 8 6 5 + 0 1 0 5 , 2 3 9 9 5 0 , 8 7 1 6 1 8 6 , 7 8 7 7 0 1 5 4 , 6 9 5 0 0 0 , 9 3 3 6 0 9 7 7 , 3 2 9 6 4
6 7 - 1 , 5 2 6 9 2 - - 2 , 3 3 1 4 9 0 , 0 1 8 1 4 9 3 , 4 9 5 6 8 1 6 3 , 5 5 8 1 6 - 0 , 1 3 4 6 9 1 1 0 5 , 8 5 6 5 6
6 8 - 9 , 9 0 9 2 5 - 0 9 8 , 1 9 3 1 7 0 , 7 3 3 2 8 9 8 , 1 5 2 0 3 1 6 9 , 6 6 6 4 7 - 0 , 8 5 6 3 2 1 2 3 9 , 7 6 5 8 1
6 9 - 1 8 , 6 3 1 4 0 - 0 3 4 7 , 1 2 9 1 1 2 , 5 9 7 6 6 9 7 , 9 1 1 2 9 1 6 9 , 3 5 1 5 1 - 1 , 6 1 1 7 3 1 3 7 3 , 3 9 7 2 1
7 0 9 , 5 2 9 9 5 + + 9 0 , 8 1 9 9 0 0 , 6 4 6 5 4 1 0 3 , 8 4 7 3 5 1 7 7 , 0 9 2 7 5 0 , 8 0 4 0 8 1 5 1 3 , 8 6 7 2 6
7 1 - 7 , 5 7 4 4 4 - - 5 7 , 3 7 2 1 7 0 , 3 9 4 1 1 1 0 8 , 2 9 2 2 8 1 8 2 , 8 5 6 6 0 - 0 , 6 2 7 7 8 1 6 5 9 , 4 4 1 7 0
7 2 3 , 3 7 6 3 6 + + 1 1 , 3 9 9 7 8 0 , 0 7 5 6 8 1 1 2 , 7 0 0 4 4 1 8 8 , 5 4 6 8 5 0 , 2 7 5 1 1 1 8 1 0 , 0 6 5 3 5
7 3 - 2 0 , 2 6 7 1 1 - - 4 1 0 , 7 5 5 8 9 2 , 6 1 1 8 4 1 1 8 , 5 1 6 6 0 1 9 6 , 0 1 7 6 2 - 1 , 6 1 6 1 2 1 9 6 7 , 3 3 2 4 6
7 4 - 1 9 , 4 4 6 5 1 - 0 3 7 8 , 1 6 6 7 7 2 , 3 8 6 7 2 1 1 9 , 5 5 0 9 7 1 9 7 , 3 4 2 0 5 - 1 , 5 4 4 9 0 2 1 2 5 , 7 7 8 9 7
7 5 - 3 , 8 2 0 3 5 - 0 1 4 , 5 9 5 1 0 0 , 0 9 0 7 5 1 2 1 , 6 3 4 5 3 2 0 0 , 0 0 6 1 8 - 0 , 3 0 1 2 5 2 2 8 6 , 5 9 9 3 3
7 6 2 3 , 5 4 6 4 3 + + 5 5 4 , 4 3 4 3 0 3 , 4 5 5 4 2 1 2 1 , 3 1 2 4 6 1 9 9 , 5 9 4 6 8 1 , 8 5 8 8 8 2 4 4 7 , 0 5 2 9 0
7 7 8 , 8 7 2 5 3 + 0 7 8 , 7 2 1 8 1 0 , 4 9 4 7 1 1 2 0 , 1 4 8 4 4 1 9 8 , 1 0 6 5 0 0 , 7 0 3 3 6 2 6 0 6 , 1 8 0 3 7
7 8 9 , 1 7 0 3 5 + 0 8 4 , 0 9 5 4 1 0 , 5 7 6 6 7 1 0 8 , 5 1 4 8 2 1 8 3 , 1 4 4 4 7 0 , 7 5 9 3 9 2 7 5 2 , 0 1 0 0 1
7 9 2 9 , 8 7 1 3 8 + 0 8 9 2 , 2 9 9 1 0 6 , 8 5 7 0 5 9 4 , 8 7 9 7 8 1 6 5 , 3 7 7 4 7 2 , 6 1 8 6 0 2 8 8 2 , 1 3 8 6 4
8 0 9 , 2 4 9 8 2 + 0 8 5 , 5 5 9 2 2 0 , 6 7 5 0 2 9 1 , 9 6 1 9 1 1 6 1 , 5 3 8 4 4 0 , 8 2 1 6 0 3 0 0 8 , 8 8 8 8 1
8 1 2 2 , 0 4 5 3 4 + 0 4 8 5 , 9 9 7 0 8 3 , 7 3 9 7 4 9 4 , 7 2 9 3 8 1 6 5 , 1 7 9 9 3 1 , 9 3 3 8 4 3 1 3 8 , 8 4 3 4 7
8 2 - 3 , 4 1 6 6 6 - - 1 1 , 6 7 3 5 7 0 , 0 9 0 9 0 9 3 , 4 0 0 4 5 1 6 3 , 4 3 2 8 7 - 0 , 3 0 1 5 0 3 2 6 7 , 2 6 0 1 3
8 3 2 0 , 8 8 4 3 7 + + 4 3 6 , 1 5 7 0 1 3 , 6 3 1 1 4 8 6 , 2 5 0 0 7 1 5 3 , 9 8 1 1 9 1 , 9 0 5 5 6 3 3 8 7 , 3 7 5 7 6
8 4 3 5 , 1 5 2 2 0 + 0 1 2 3 5 , 6 7 7 3 2 1 1 , 0 4 7 8 5 7 9 , 1 6 8 5 4 1 4 4 , 5 2 7 0 6 3 , 3 2 3 8 3 3 4 9 9 , 2 2 3 5 6
8 5 2 9 , 8 6 9 6 9 + 0 8 9 2 , 1 9 8 5 0 8 , 4 0 6 6 3 7 4 , 2 9 7 2 6 1 3 7 , 9 6 3 3 5 2 , 8 9 9 4 2 3 6 0 5 , 3 5 3 8 7
8 6 2 2 , 1 8 6 0 5 + 0 4 9 2 , 2 2 0 9 3 5 , 1 9 1 4 4 6 4 , 7 2 5 8 6 1 2 4 , 9 0 2 0 4 2 , 2 7 8 4 7 3 7 0 0 , 1 6 7 8 1
8 7 3 0 , 5 9 9 4 4 + 0 9 3 6 , 3 2 5 6 1 1 0 , 9 6 3 9 3 5 6 , 8 4 5 1 2 1 1 3 , 9 5 6 0 0 3 , 3 1 1 1 8 3 7 8 5 , 5 6 8 3 7
8 8 3 5 , 5 9 4 5 5 + 0 1 2 6 6 , 9 7 1 7 9 1 6 , 3 6 7 9 9 5 0 , 2 1 9 5 2 1 0 4 , 5 9 1 3 9 4 , 0 4 5 7 4 3 8 6 2 , 9 7 3 8 3
8 9 3 2 , 5 5 4 3 4 + 0 1 0 5 9 , 7 8 4 8 7 1 5 , 7 1 3 1 7 4 2 , 0 6 8 9 4 9 2 , 8 2 2 3 8 3 , 9 6 3 9 8 3 9 3 0 , 4 1 9 4 9
9 0 1 8 , 5 0 0 8 0 + 0 3 4 2 , 2 7 9 6 8 6 , 1 6 7 2 9 3 2 , 4 7 9 3 8 7 8 , 5 1 9 0 1 2 , 4 8 3 4 0 3 9 8 5 , 9 1 8 6 9
S u m 2 1 3 , 0 8 1 3 1 V a l o r D i s t r i b u i ç ã o 1 3 0 , 4 8 5 7 5 6 3 , 1 3 4 1 3
0 9,3
2n
0 9,3
M a l e s M a l e sA c t u a r i a l S q u a r e d C o n t r i b u t i o n t o C o n f i d e n c e C o n f i d e n c e A p p r o x i m a t e
A g e D e s v i a t i o n s D e v i a t i o n s t h e C h i - s q u a r e d L e v e l o f L e v e l o f S t d . D e v i a t i o n V a r i a n c es u m [ ( 3 ) - ( 4 ) ] o f M o r t a l i t y T e s t M o r t a l i t y a t M o r t a l i t y a t [ ( 3 ) - ( 4 ) ] / s q r ( 4 ) s u m [ ( 4 ) ]
( X ) S p l i n e s 9 9 , 0 % 1 , 0 % S p l i n e s S p l i n e s5 0 - 5 , 8 5 7 5 6 - - 3 4 , 3 1 0 9 7 2 , 4 7 5 9 8 5 , 1 9 7 5 8 2 2 , 5 1 7 5 3 - 1 , 5 7 3 5 2 1 3 , 8 5 7 5 6 # # # #5 1 - 1 , 1 0 8 7 1 - 0 1 , 2 2 9 2 4 0 , 0 7 1 8 5 7 , 4 8 6 3 5 2 6 , 7 3 1 0 8 - 0 , 2 6 8 0 5 3 0 , 9 6 6 2 7 # # # #5 2 4 , 0 9 9 9 9 + + 1 6 , 8 0 9 8 8 0 , 8 4 4 7 2 9 , 5 2 2 3 4 3 0 , 2 7 7 6 9 0 , 9 1 9 0 8 5 0 , 8 6 6 2 8 # # # #5 3 - 0 , 6 4 5 2 6 - - 0 , 4 1 6 3 6 0 , 0 1 7 6 1 1 2 , 3 3 3 1 1 3 4 , 9 5 7 4 1 - 0 , 1 3 2 7 0 7 4 , 5 1 1 5 4 # # # #5 4 3 , 8 4 6 9 2 + + 1 4 , 7 9 8 8 1 0 , 5 0 7 6 2 1 6 , 5 9 2 3 3 4 1 , 7 1 3 8 2 0 , 7 1 2 4 8 1 0 3 , 6 6 4 6 2 # # # #5 5 - 8 , 3 1 7 3 7 - - 6 9 , 1 7 8 5 9 1 , 9 5 8 7 7 2 1 , 4 9 2 3 0 4 9 , 1 4 2 4 4 - 1 , 3 9 9 5 6 1 3 8 , 9 8 1 9 8 # # # #5 6 - 7 , 7 6 2 3 8 - 0 6 0 , 2 5 4 5 5 1 , 4 4 2 7 9 2 6 , 7 2 8 6 8 5 6 , 7 9 6 0 8 - 1 , 2 0 1 1 6 1 8 0 , 7 4 4 3 6 # # # #5 7 - 1 1 , 0 7 1 5 5 - 0 1 2 2 , 5 7 9 1 8 2 , 6 6 0 6 3 3 0 , 2 8 1 2 8 6 1 , 8 6 1 8 2 - 1 , 6 3 1 1 4 2 2 6 , 8 1 5 9 1 # # # #5 8 1 3 , 8 3 9 1 1 + + 1 9 1 , 5 2 1 0 5 3 , 7 4 3 5 1 3 4 , 5 2 1 3 1 6 7 , 8 0 0 4 6 1 , 9 3 4 8 1 2 7 7 , 9 7 6 8 0 # # # #5 9 - 1 7 , 3 1 3 7 1 - - 2 9 9 , 7 6 4 4 0 5 , 1 4 0 5 5 4 0 , 5 4 8 9 8 7 6 , 0 7 8 4 3 - 2 , 2 6 7 2 8 3 3 6 , 2 9 0 5 0 # # # #6 0 6 , 7 0 4 3 8 + + 4 4 , 9 4 8 7 6 0 , 7 1 0 1 4 4 4 , 7 8 7 5 9 8 1 , 8 0 3 6 4 0 , 8 4 2 7 0 3 9 9 , 5 8 6 1 2 # # # #6 1 - 1 , 6 9 2 4 2 - - 2 , 8 6 4 3 0 0 , 0 4 1 7 0 4 9 , 4 1 1 5 1 8 7 , 9 7 3 3 4 - 0 , 2 0 4 2 0 4 6 8 , 2 7 8 5 5 # # # #6 2 - 3 , 2 2 9 7 5 - 0 1 0 , 4 3 1 2 7 0 , 1 4 4 4 2 5 2 , 4 5 8 6 3 9 2 , 0 0 0 8 7 - 0 , 3 8 0 0 2 5 4 0 , 5 0 8 2 9 # # # #6 3 8 , 5 3 3 8 3 + + 7 2 , 8 2 6 1 8 0 , 9 2 8 1 2 5 7 , 8 5 9 1 9 9 9 , 0 7 3 1 6 0 , 9 6 3 3 9 6 1 8 , 9 7 4 4 7 # # # #6 4 - 0 , 0 5 1 6 8 - - 0 , 0 0 2 6 7 0 , 0 0 0 0 3 6 1 , 8 5 1 1 2 1 0 4 , 2 5 2 2 5 - 0 , 0 0 5 6 7 7 0 2 , 0 2 6 1 5 # # # #6 5 1 5 , 2 8 6 3 6 + + 2 3 3 , 6 7 2 9 4 2 , 2 1 0 4 3 8 1 , 7 9 4 8 7 1 2 9 , 6 3 2 4 0 1 , 4 8 6 7 5 8 0 7 , 7 3 9 7 9 # # # #6 6 1 5 , 4 7 2 5 5 + 0 2 3 9 , 3 9 9 8 8 2 , 0 7 2 2 3 9 0 , 5 2 3 0 9 1 4 0 , 5 3 1 8 1 1 , 4 3 9 5 3 9 2 3 , 2 6 7 2 4 # # # #6 7 3 , 7 7 6 2 1 + 0 1 4 , 2 5 9 7 3 0 , 1 1 5 7 2 9 7 , 3 9 9 9 7 1 4 9 , 0 4 7 6 1 0 , 3 4 0 1 8 1 0 4 6 , 4 9 1 0 3 # # # #6 8 - 5 , 6 4 2 9 0 - - 3 1 , 8 4 2 2 9 0 , 2 4 5 6 2 1 0 3 , 1 5 5 0 0 1 5 6 , 1 3 0 8 0 - 0 , 4 9 5 6 0 1 1 7 6 , 1 3 3 9 3 # # # #6 9 - 1 6 , 4 8 4 1 4 - 0 2 7 1 , 7 2 6 9 9 2 , 0 6 6 6 1 1 0 4 , 8 0 8 8 1 1 5 8 , 1 5 9 4 8 - 1 , 4 3 7 5 7 1 3 0 7 , 6 1 8 0 7 # # # #7 0 1 0 , 4 3 2 1 0 + + 1 0 8 , 8 2 8 7 2 0 , 7 7 9 7 5 1 1 2 , 0 8 4 7 8 1 6 7 , 0 5 1 0 1 0 , 8 8 3 0 4 1 4 4 7 , 1 8 5 9 7 # # # #7 1 - 6 , 8 6 4 2 2 - - 4 7 , 1 1 7 5 8 0 , 3 2 5 2 5 1 1 6 , 8 6 4 5 0 1 7 2 , 8 6 3 9 5 - 0 , 5 7 0 3 1 1 5 9 2 , 0 5 0 1 9 # # # #7 2 3 , 5 1 4 5 1 + + 1 2 , 3 5 1 7 8 0 , 0 8 2 0 8 1 2 1 , 9 4 7 6 9 1 7 9 , 0 2 3 2 9 0 , 2 8 6 5 0 1 7 4 2 , 5 3 5 6 8 # # # #7 3 - 2 1 , 1 5 6 5 0 - - 4 4 7 , 5 9 7 3 1 2 , 8 3 0 0 9 1 2 8 , 9 0 0 3 8 1 8 7 , 4 1 2 6 1 - 1 , 6 8 2 2 9 1 9 0 0 , 6 9 2 1 8 # # # #7 4 - 2 1 , 8 4 6 4 5 - 0 4 7 7 , 2 6 7 4 7 2 , 9 6 7 2 2 1 3 1 , 3 4 2 5 9 1 9 0 , 3 5 0 3 2 - 1 , 7 2 2 5 6 2 0 6 1 , 5 3 8 6 3 # # # #7 5 - 9 , 4 6 8 1 2 - 0 8 9 , 6 4 5 2 1 0 , 5 3 8 5 1 1 3 6 , 4 5 3 0 9 1 9 6 , 4 8 3 1 4 - 0 , 7 3 3 8 3 2 2 2 8 , 0 0 6 7 5 # # # #7 6 1 4 , 8 7 1 8 6 + + 2 2 1 , 1 7 2 1 2 1 , 3 0 7 7 2 1 3 8 , 8 7 4 2 6 1 9 9 , 3 8 2 0 3 1 , 1 4 3 5 6 2 3 9 7 , 1 3 4 8 9 # # # #7 7 - 0 , 8 6 8 0 3 - - 0 , 7 5 3 4 8 0 , 0 0 4 4 6 1 3 8 , 6 3 7 4 2 1 9 9 , 0 9 8 6 5 - 0 , 0 6 6 8 0 2 5 6 6 , 0 0 2 9 3 # # # #7 8 0 , 6 4 7 1 6 + + 0 , 4 1 8 8 1 0 , 0 0 2 7 1 1 2 5 , 4 5 0 6 7 1 8 3 , 2 5 5 0 2 0 , 0 5 2 0 9 2 7 2 0 , 3 5 5 7 7 # # # #7 9 2 3 , 7 3 9 6 1 + 0 5 6 3 , 5 6 9 2 4 4 , 1 3 5 9 7 1 0 9 , 1 0 4 8 7 1 6 3 , 4 1 5 9 0 2 , 0 3 3 7 1 2 8 5 6 , 6 1 6 1 6 # # # #8 0 2 , 0 8 6 1 3 + 0 4 , 3 5 1 9 4 0 , 0 3 2 5 0 1 0 6 , 9 9 3 1 9 1 6 0 , 8 3 4 5 5 0 , 1 8 0 2 7 2 9 9 0 , 5 3 0 0 2 # # # #8 1 1 1 , 5 4 5 9 2 + 0 1 3 3 , 3 0 8 2 9 0 , 9 4 9 1 2 1 1 2 , 8 8 3 8 5 1 6 8 , 0 2 4 3 1 0 , 9 7 4 2 3 3 1 3 0 , 9 8 4 1 0 # # # #8 2 - 1 5 , 0 9 9 6 0 - - 2 2 7 , 9 9 7 8 4 1 , 6 2 7 4 0 1 1 2 , 5 6 4 1 8 1 6 7 , 6 3 5 0 1 - 1 , 2 7 5 7 0 3 2 7 1 , 0 8 3 7 0 # # # #8 3 1 0 , 3 5 3 8 6 + + 1 0 7 , 2 0 2 4 4 0 , 8 2 0 5 6 1 0 4 , 0 5 5 9 5 1 5 7 , 2 3 6 3 3 0 , 9 0 5 8 5 3 4 0 1 , 7 2 9 8 4 # # # #8 4 2 7 , 2 8 7 0 5 + 0 7 4 4 , 5 8 3 3 4 6 , 2 1 9 7 4 9 4 , 2 5 9 6 7 1 4 5 , 1 6 6 2 3 2 , 4 9 3 9 4 3 5 2 1 , 4 4 2 7 9 # # # #8 5 2 1 , 1 6 2 5 0 + 0 4 4 7 , 8 5 1 2 2 3 , 8 9 9 8 7 8 9 , 9 0 7 9 2 1 3 9 , 7 6 7 0 9 1 , 9 7 4 8 1 3 6 3 6 , 2 8 0 2 9 # # # #8 6 1 0 , 5 6 0 7 6 + 0 1 1 1 , 5 2 9 5 9 1 , 0 4 7 8 2 8 2 , 4 3 8 5 3 1 3 0 , 4 3 9 9 5 1 , 0 2 3 6 3 3 7 4 2 , 7 1 9 5 3 # # # #8 7 1 7 , 8 6 2 5 2 + 0 3 1 9 , 0 6 9 7 1 3 , 2 5 1 2 5 7 5 , 0 9 1 7 4 1 2 1 , 1 8 3 2 2 1 , 8 0 3 1 2 3 8 4 0 , 8 5 7 0 1 # # # #8 8 2 3 , 3 2 9 3 5 + 0 5 4 4 , 2 5 8 6 0 6 , 0 6 9 5 3 6 7 , 6 4 1 4 7 1 1 1 , 6 9 9 8 3 2 , 4 6 3 6 4 3 9 3 0 , 5 2 7 6 6 # # # #8 9 2 2 , 9 0 0 4 7 + 0 5 2 4 , 4 3 1 3 6 6 , 8 0 2 0 0 5 6 , 6 7 2 7 9 9 7 , 5 2 6 2 7 2 , 6 0 8 0 7 4 0 0 7 , 6 2 7 1 9 # # # #9 0 1 0 , 4 9 7 9 9 + 0 1 1 0 , 2 0 7 7 4 1 , 7 3 5 5 0 4 4 , 9 6 3 8 4 8 2 , 0 4 0 1 9 1 , 3 1 7 3 8 4 0 7 1 , 1 2 9 2 1 # # # #
S u m 1 2 7 , 8 7 0 7 9 V a l u e o f t h e C h i - s q r 7 2 , 8 2 8 1 2 6 3 , 8 0 5 4 0 # # # # #
3263,2
2n
3263,2
4 1 3 3 8N u m b e r o f N u m b e r o f V a r i a b l e s N u m b e r o f D e g r e e s o f
e l e m e n t s i n p a r a m e t r i s e d i n t h e i n t h e C h i - s q u a r e d
t h e a g e b a n d a d j u s t e m e n t D i s t r i b u t i o n
S t a n d a r d D e v i a t i o n s ' T e s t
n
e
xkSplineGomper tz
xa n o s
571998
37
1998 1 0 05 9
21
21
Distribution at 38
Signs Quantile t k 0,0574% 1,332777438(+) 23( - ) 18 Value of the Chi-sqr 40,40533
Distribution between 28 1,338541933Groups age 50 and 80 at 6,0824% 3,09024
of signs the Quantile t k
(+) 11 3,11390( - ) 11 Prob.gr.desv.(+)>g(+)= 41,4287%
(Stevens Test)Standard Deviations' Test Standard Deviations' Test
Ranges (-oo, -3) (-3, -2) (-2, -1) (-1, 0 ) ( 0, 1 ) ( 1, 2 ) ( 2, 3 ) ( 3, oo)Observ.Dev. 0 0,82 5,74 13,94 13,94 5,74 0,82 0Observ.Dev. 0 1 8 9 11 8 4 0
3,26383 1,03960
xqE xSplinescorr
xSplinescorr
xGompertzcorr
32634,2338542,1%99338542,1 1
%9,991
kt
ktn
n2
n
2n
b
n
nbqE
bqE
x xx
xxx
2min
2
do Qui-quadrado no 38
Signs Quantil t k 0,0000% 1,783978488(+) 23
( - ) 18 Valor Distribuição 49,16566
do Qui-quadrado 28
Groups entre 50 e 80 anos 0,8001%
of signs no Quantil t k
(+) 10( - ) 10 Prob.gr.desv.(+)>g(+)= 66,1878%
(Stevens Test)Standard Deviations' Test
Ranges (-oo, -3) (-3, -2) (-2, -1) (-1, 0 ) ( 0, 1 ) ( 1, 2 ) ( 2, 3 ) ( 3, oo)Observ.Dev. 0 0,82 5,74 13,94 13,94 5,74 0,82 0Observ.Dev. 0 2 7 9 10 5 4 4
8,52283 1,06750
xqE xGompertzcorr kt
kt
n
n
2n
2n
b
n
nbqE
bqE
x xx
xxx
2min
2
Statistical Quality Tests for Mortality Projections
31
2 0 0 4Y E A R S : 2 0 0 3 / 2 0 0 5 T = 2 0 0 5 - 1 9 9 8 = 6
P E N S I O N F U N D P E N S I O N E R S ( M A L E S ) T I M E H O R I Z O N O F M O R T A L I T Y P R O J E C T I O N
( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) = ( 3 ) - ( 4 ) ( 6 ) = ( 5 ) 2 ( 7 )A g e " E x p o s e d t o R is k " O b s e r v e d E x p e c t e d D e s v ia t i o n s i n S q u a r e d C o n t r i b u t i o n t o
M o r t a l i t y M o r t a l i t y I d a d e M o r t a l i t y D e s v ia t i o n s i n t h e C h i - s q u a r e dG o m p e r t z M o r t a l i t y T e s t
5 8 7 . 9 1 8 , 5 9 8 , 0 7 3 , 8 5 8 2 4 , 2 5 8 3 , 7 7 , 95 9 8 . 2 8 4 , 0 1 1 8 , 0 7 9 , 8 5 9 3 8 , 2 1 . 4 6 0 , 7 1 8 , 36 0 8 . 3 5 3 , 0 1 4 2 , 0 8 8 , 0 6 0 5 4 , 0 2 . 9 1 8 , 0 3 3 , 26 1 8 . 1 5 0 , 5 1 7 6 , 0 9 3 , 9 6 1 8 2 , 1 6 . 7 4 2 , 7 7 1 , 86 2 7 . 7 6 2 , 0 1 4 9 , 0 9 7 , 8 6 2 5 1 , 2 2 . 6 2 3 , 7 2 6 , 86 3 7 . 4 6 4 , 0 1 2 5 , 0 1 0 2 , 8 6 3 2 2 , 2 4 9 2 , 0 4 , 86 4 7 . 3 0 6 , 5 1 7 0 , 0 1 1 0 , 1 6 4 5 9 , 9 3 . 5 9 3 , 3 3 2 , 76 5 8 . 2 6 9 , 0 2 0 1 , 0 1 3 6 , 2 6 5 6 4 , 8 4 . 2 0 0 , 8 3 0 , 86 6 9 . 0 8 1 , 0 1 9 5 , 0 1 6 3 , 5 6 6 3 1 , 5 9 9 1 , 1 6 , 16 7 8 . 6 2 5 , 0 1 7 9 , 0 1 6 9 , 8 6 7 9 , 2 8 4 , 8 0 , 56 8 8 . 0 3 9 , 5 1 8 1 , 0 1 7 3 , 0 6 8 8 , 0 6 3 , 9 0 , 46 9 7 . 3 5 3 , 0 1 8 0 , 0 1 7 3 , 0 6 9 7 , 0 4 9 , 6 0 , 37 0 7 . 8 2 5 , 0 2 0 4 , 0 2 0 1 , 2 7 0 2 , 8 8 , 0 0 , 07 1 8 . 7 4 7 , 0 2 2 6 , 0 2 4 5 , 8 7 1 - 1 9 , 8 3 9 0 , 4 1 , 67 2 8 . 3 5 1 , 5 2 7 9 , 0 2 5 6 , 4 7 2 2 2 , 6 5 1 0 , 6 2 , 07 3 7 . 7 7 2 , 5 2 3 8 , 0 2 6 0 , 7 7 3 - 2 2 , 7 5 1 6 , 3 2 , 07 4 6 . 8 8 3 , 0 2 4 8 , 0 2 5 2 , 2 7 4 - 4 , 2 1 7 , 9 0 , 17 5 5 . 9 1 8 , 5 2 0 7 , 0 2 3 6 , 9 7 5 - 2 9 , 9 8 9 4 , 2 3 , 87 6 5 . 1 4 4 , 5 2 2 3 , 0 2 2 4 , 9 7 6 - 1 , 9 3 , 6 0 , 07 7 4 . 5 2 4 , 5 1 9 8 , 0 2 1 6 , 0 7 7 - 1 8 , 0 3 2 2 , 9 1 , 57 8 4 . 1 2 7 , 0 2 2 6 , 0 2 1 5 , 1 7 8 1 0 , 9 1 1 9 , 6 0 , 67 9 3 . 6 7 4 , 5 1 9 6 , 0 2 0 9 , 0 7 9 - 1 3 , 0 1 6 9 , 1 0 , 88 0 3 . 1 9 8 , 0 2 1 8 , 0 1 9 8 , 5 8 0 1 9 , 5 3 8 0 , 3 1 , 98 1 2 . 8 2 9 , 0 2 1 5 , 0 1 9 1 , 6 8 1 2 3 , 4 5 4 8 , 9 2 , 98 2 2 . 4 9 0 , 5 2 0 6 , 0 1 8 3 , 9 8 2 2 2 , 1 4 8 6 , 4 2 , 68 3 2 . 1 2 9 , 5 1 9 6 , 0 1 7 1 , 5 8 3 2 4 , 5 6 0 0 , 4 3 , 58 4 1 . 7 6 9 , 5 1 4 3 , 0 1 5 5 , 3 8 4 - 1 2 , 3 1 5 2 , 2 1 , 08 5 1 . 4 6 3 , 5 1 5 7 , 0 1 4 0 , 0 8 5 1 7 , 0 2 8 9 , 2 2 , 18 6 1 . 2 4 2 , 0 1 6 4 , 0 1 2 9 , 4 8 6 3 4 , 6 1 . 1 9 6 , 3 9 , 28 7 1 . 0 4 4 , 0 1 4 4 , 0 1 1 8 , 4 8 7 2 5 , 6 6 5 3 , 1 5 , 58 8 8 3 8 , 0 1 4 9 , 0 1 0 3 , 5 8 8 4 5 , 5 2 . 0 7 2 , 6 2 0 , 08 9 6 7 0 , 5 1 1 1 , 0 9 0 , 1 8 9 2 0 , 9 4 3 8 , 3 4 , 99 0 5 3 7 , 5 9 3 , 0 7 8 , 5 9 0 1 4 , 5 2 1 0 , 2 2 , 79 1 4 1 3 , 5 8 6 , 0 6 5 , 6 9 1 2 0 , 4 4 1 5 , 0 6 , 39 2 3 1 0 , 5 7 2 , 0 5 3 , 5 9 2 1 8 , 5 3 4 1 , 5 6 , 49 3 2 4 2 , 5 6 0 , 0 4 5 , 4 9 3 1 4 , 6 2 1 4 , 1 4 , 79 4 1 7 1 , 0 4 6 , 0 3 4 , 7 9 4 1 1 , 3 1 2 7 , 7 3 , 79 5 1 2 2 , 0 3 8 , 0 2 6 , 8 9 5 1 1 , 2 1 2 4 , 7 4 , 6
S u m 6 . 2 5 7 , 0 5 . 5 6 6 , 5 6 9 0 , 5 3 5 . 0 0 7 , 8 3 2 7 , 93 8 5 3 3 C u m m u l a t i v e D e v i a t i o n s V a lu e o f t h e C h i - s q rV a lo r d a D is t r i b u i ç ã oV a lu e o f t h e C h i - s q r
N u m b e r o f N u m b e r o f V a r i a b le s N u m b e r o f D e g r e e s o f i n M o r t a l i t y D is t r i b u t i o n a td o Q u i - q u a d r a d o n oD is t r i b u t i o n a te le m e n t s i n p a r a m e t r i s e d i n t h e i n t h e C h i - s q u a r e d Q u a n t i l e t k Q u a n t i l t k Q u a n t i l e t k
t h e a g e b a n d a d ju s t e m e n t D i s t r i b u t i o n
0 , 0 0 0 0 %
1 , 0 3 4 7 20 , 9 5 6 7 2
x xxx qE
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2
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e a G o m p e r t
z x 0 0 4 2
n o s
1 0 0 2 1 - r s a n o s
1 0 0 k 5 9 e x 5 9 5 7 S p l i n e 1 9 9 8
3 7 2 1 1 9 9 8 2 0 0 4 kt
x
tx
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n
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x
tx
tx
x xtx
tx
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qEqEqEqEnqEn
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42222
1b2b
9 , 9 3 6 5 9 8 5
tx
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tx
tx
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txt
xco rr qEqE
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n
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xtx qE
tx
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txco rr qEqE 1522371,39365985,9
tx
tx
tx
tx
txcorr
tx
tx
tx qEqEqEt
x
1522371,31
tx
tx
tx
tx
txcorr
tx
tx
tx qEqEqEa t
x
1522371,31 1
( 8 ) ( 9 ) = ( 3 ) - ( 8 ) ( 1 0 ) = ( 9 ) 2 ( 7 )R e a d j u s t e d D e s v i a t i o n s i n S q u a r e d C o n t r i b u t i o n t o
E x p e c t e d M o r t a l i t y D e s v i a t i o n s i n t h e C h i - s q u a r e dM o r t a l i t y M o r t a l i t y T e s t
8 4 , 4 1 3 , 6 1 8 4 , 5 2 , 29 1 , 2 2 6 , 8 7 1 7 , 8 7 , 9
1 0 0 , 6 4 1 , 4 1 . 7 1 5 , 3 1 7 , 11 0 7 , 3 6 8 , 7 4 . 7 1 4 , 9 4 3 , 91 1 1 , 8 3 7 , 2 1 . 3 8 5 , 0 1 2 , 41 1 7 , 5 7 , 5 5 5 , 6 0 , 51 2 5 , 8 4 4 , 2 1 . 9 5 1 , 9 1 5 , 51 5 5 , 7 4 5 , 3 2 . 0 5 2 , 6 1 3 , 21 8 6 , 9 8 , 1 6 5 , 0 0 , 31 9 4 , 1 - 1 5 , 1 2 2 8 , 3 1 , 21 9 7 , 8 - 1 6 , 8 2 8 1 , 8 1 , 41 9 7 , 7 - 1 7 , 7 3 1 4 , 5 1 , 62 3 0 , 0 - 2 6 , 0 6 7 5 , 5 2 , 92 8 1 , 0 - 5 5 , 0 3 . 0 2 0 , 6 1 0 , 82 9 3 , 1 - 1 4 , 1 1 9 9 , 7 0 , 72 9 8 , 1 - 6 0 , 1 3 . 6 0 8 , 3 1 2 , 12 8 8 , 4 - 4 0 , 4 1 . 6 2 8 , 9 5 , 62 7 0 , 8 - 6 3 , 8 4 . 0 7 5 , 1 1 5 , 02 5 7 , 1 - 3 4 , 1 1 . 1 6 3 , 0 4 , 52 4 6 , 9 - 4 8 , 9 2 . 3 9 1 , 6 9 , 72 4 5 , 9 - 1 9 , 9 3 9 4 , 8 1 , 62 3 8 , 9 - 4 2 , 9 1 . 8 4 3 , 9 7 , 72 2 6 , 9 - 8 , 9 7 9 , 8 0 , 42 1 9 , 0 - 4 , 0 1 6 , 1 0 , 12 1 0 , 3 - 4 , 3 1 8 , 4 0 , 11 9 6 , 1 - 0 , 1 0 , 0 0 , 01 7 7 , 6 - 3 4 , 6 1 . 1 9 6 , 3 6 , 71 6 0 , 0 - 3 , 0 9 , 3 0 , 11 4 7 , 9 1 6 , 1 2 5 7 , 6 1 , 71 3 5 , 4 8 , 6 7 3 , 8 0 , 51 1 8 , 3 3 0 , 7 9 4 2 , 8 8 , 01 0 3 , 0 8 , 0 6 4 , 6 0 , 6
8 9 , 7 3 , 3 1 0 , 6 0 , 17 5 , 0 1 1 , 0 1 2 0 , 4 1 , 66 1 , 2 1 0 , 8 1 1 6 , 9 1 , 95 1 , 9 8 , 1 6 6 , 1 1 , 33 9 , 7 6 , 3 4 0 , 1 1 , 03 0 , 7 7 , 3 5 3 , 7 1 , 7
6 . 3 6 3 , 8 - 1 0 6 , 8 3 5 . 7 3 4 , 9 2 1 3 , 7V a l u e o f t h e C h i - s q r C u m m u l a t i v e D e v i a t i o n s V a l u e o f t h e C h i - s q r
D i s t r i b u t i o n a t i n M o r t a l i t y D i s t r i b u t i o n a tQ u a n t i l e t k
0 , 0 0 0 0 %
1 , 2 2 7 2 30 , 7 6 6 8 5
txx bqE t
xxx bqE 2txxx bqE
2 x
txxx bqE
t
xx
txxx
bqE
bqE
2
x xx
xxxk qE
qEtF
2
2
kt
61b
62b
Mortality Projections: Variance Error Correction
32
0
50
100
150
200
250
300
350
40058 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94
Nº
age
Observed values
Fitted values
confidence leval at 1% (readjusted)
confidence level at 99% (readjusted)
Non-adjusted Confidence Levels
Pension Fund Beneficiaries (Males 2003-2005)
33