Tables de Mortalité

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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

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• 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

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• 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

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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:

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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

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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

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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

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•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

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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


13

eeeeeee tk

x

tktxtr

x

tktxtrtk

x

ttxtrk

The prior equation also implies that:

x

txtr

lnwhere hence, finally

x

tx

tr

ln1


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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:


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 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


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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


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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.


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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


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


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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

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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).

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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.

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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:

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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

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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

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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)


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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

x

xxx qE

2xxx qE

2 x

xxx qE

xx

xxx

qE

qE

2

xEx xx qE x

x xx

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2

2

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

tx qE

n

xx

t

x

tx

tx

x xtx

tx

tx

tx

txt

xtx

x

tx

tx

x

tx

tx

x

tx

tx

qE

qE

qEqEqEqEnqEn

b

2

42222

1b2b

9 , 9 3 6 5 9 8 5

tx

tx

xtx

tx

tx

tx

txt

xco rr qEqE

qE

n

2

2 1 t

xtx qE

tx

tx

tx

tx

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

Documents

Tables de Mortalité