Notional Defined Contribution Pension Method and the Construction of

Annuitization Coefficients

Janssen Jacques, JACAN and EURIA Université de Bretagne Occidentale,

6 avenue le Gorgeu, CS 93837, 29238 BREST Cedex 3, France

E-mail: jacan@belgacom.net

Manca Raimondo Università “La Sapienza”, Dipartimento di Matematica per le

Decisioni Economiche, Finanziarie ed Assicurative, via del Castro Laurenziano,

9, 00161 Roma, ITALY, Telephone: +390649766507, Fax: +390649766765, E-

mail: raimondo.manca@uniroma1.it

ABSTRACT

In this paper, we show how it is possible to construct efficient annuitization

coefficients of the cumulated sum in the corresponding pension annuity in a

Notional Defined Contribution pension scheme.

The proposed method will be compared with the Italian method by means of

which the transformation coefficients of the NDC Italian rules were constructed.

Numerical examples will be provided.

Keywords: Notional defined contribution system, annuitization coefficients,

pension annuity.

Acknowledgements The authors are grateful to the following friends and

colleagues: above all Sandro Gronchi who introduced the authors to the logic of

NDC schemes and explained the many social advantages of this pension system

but also Sergio Nisticò for many discussions on the NDC argument.

1. INTRODUCTION

Notional Defined Contribution (NDC) pension system is a pay as you go defined contribution

method that was developed independently in the early 1990s in Italy and in Sweden. The

interested reader should refer in particular to Gronchi (1992), Niccoli (1992), Ministero del

Tesoro (1994), Gronchi (1995a) (1995b) for Italian experience and to Broms (1990), Olsson

and Schubert (1991), Persson (1991), Ackerby (1992), Palmer Sherman (1993) for Swedish

experience.

The idea has its roots in X papers by De Finetti (1956), Samuelson (1958), Aaron (1966),

Buchanan (1968) and Castellino (1969). For a more detailed bibliography the reader should

refer to Gronchi Nisticò (2004).

For a full description of the properties of the NDC pension scheme and a comparison between

it and the pay as you go defined benefit method, the reader should refer to the papers by

Lindbeck Persson (2003) and Gronchi Nisticò (2004).

We wish only to outline that the NDC environment insures the actuarial fairness.

The concept means that there is an actuarial equivalence between the contributions paid and

the benefits received (Lindbeck Persson (2003), in contrast to the pay as you go defined

benefit system which is unfair, as proved in Gronchi Nisticò (2004). Gronchi Nisticò’s paper

shows, by means of a four generation scheme, that employees that retire early and/or those

that have had a more successful career get proportionately greater benefits with regard to the

value of the contributions they have paid compared to employees who remain in service for

longer and have less successful careers whose benefits are inferior with regard to the value of

the contributions they have paid.

Considering all these aspects and taking into account that the NDC system is more suitable

way of achieving an intergenerational equilibrium (see Lindbeck Persson (2003)), we think

that this pension scheme will be adopted in the near future in all European Union countries

(see Holzmann (2003)) as it is already the case in Italy, Latvia, Poland and Sweden.

In a defined contribution method, there are two aspects that assume great relevance. The first

is the financial management of the contribution and the pension annuity that will be given at

the time of retirement.

We think that annuitization (conversion from capital to pension annuity) is the most relevant

actuarial problem, and that assets management is more of a financial problem.

Furthermore, we are discussing the main amount of money that a retired person will get after

his working life because it represents the sum obtained from the first pillar. For this reason, it

is evident that the fund manager should not be involved in risky investment and the

investment should rather guarantee a minimum rate or return. In our opinion, this kind of

pension fund should be managed by an organization which is at least connected to central

government.

In this paper, we do not consider the financial aspects of money management but we discuss,

instead, the problem of how to construct good unitary coefficients that permit a fair

transformation from the accumulated capital to the pension annuity.

In the next section of the paper we briefly describe how the annuitization coefficients for the

so called Dini reform pension (which came into practice in 1996 Italy) were computed in

1995.

The third section will introduce precise actuarial tools useful for the construction of

transformation coefficients.

The fourth section will describe in detail the so called “direct method”.

The fifth section will present numerical results and comparisons between the two different

methods of construction and will also explain the problems that are highlighted by numerical

results.

The final section is devoted to our conclusions and to future lines of research.

2. THE DINI REFORM TRANSFORMATION COEFFICIENTS

The coefficients of the Dini reform were described in the document published by Ministero

del Lavoro e delle Politiche Sociali (2001) which gave the formula used for their construction.

In this section, we will report this formula and the meaning of the involved variables (for

more details see the above reference).

The calculation formula is given by

1

x

x

TC = ,

( )( ) ( )

, ,

,

2

t t

x s x s

s m f

x

a A

k=

+

= ,

where:

xTC is the transformation coefficient at age x (age of retirement);

x can be seen as the present value of a unitary mean pension annuity; it is a mean because it

is not possible to give different pensions to males and females; ( )

,

t

x sa and ( )

,

t

x sA represent, respectively, the mean present value of the instalments paid to the

pensioner and the mean present value of the pensions that will be paid to the widow/ widower

of age x and sex s (from now on we will use widow to refer to both sexes);

k is a corrector taking into account the way in which the pension instalment is distributed

(monthly, bimonthly and so on); in the Italian case it is bimonthly and 0.423k = .

Furthermore we have:

( ) ,

,

0 ,

1

1

tx

t x t s

x s

t x s

l ra

l

+

=

+=

+

,( ) ,

, , ,

0 1, 1 ,

1 1

1 1

s

s

s

t widx txx t st x t s

x s s x t s x t s widt x s x t s

ll r rA q

l l

++ ++

+ +

= = + +

+ +=

+ +

where:

,x sl is the number of survivors at age x and of sex s;

is the max reachable age (in this paper 110);

,x sq is the probability of dying at age x;

,x s is the probability of leaving family at age x;

,

wid

x sl is the number of surviving widows at age x and of sex s;

s is the age difference between spouses; in this model,

s has the value to 3 ? for male and

-3 for female;

is the percentage of pension that tooks to the widow without offspring, 0.6 in Italy;

s is the percentage of widows that have the right to the survivor pension;

s a value of 0.9

for widowers and 0.7 from widows;

r is the internal rate of return;

is the inflation rate;

1

1

r+

+is supposed constant in the time and is considered to be equal to 1.015.

The data used to obtain the transformation coefficients were produced partly from ISTAT (the

Central Italian Institute of Statistics) and partly from INPS (1989) (the National Italian

Institute of Social Securityà. The input data were given by the Ministero del Lavoro e delle

Politiche Sociali (2001).

The results are related to an age range of 57 to 65 and with our own Mathematica program,

we obtained exactly the same results that are given in the cited report.

3. THEORETICAL PRELIMINARIES

It is possible to construct the mean present value of any kind of unitary pension annuity and

its inverse that is the transformation (annuitization) coefficient.

The problem how to forecast the course of a pension scheme is very old one; indeed the first

results were published by Manly (1902) who offered the possibility of computing the amount

of the liability of a pension scheme. It is a fact that the method used for the computation of the

Italian transformation coefficients described in the previous section can be considered as a

direct consequence of this approach. A good description of the method can be found in

Tomassetti (1994).

This problem is relevant and the tools used to solve it have been evolving continuously from

and thus we can identify three fundamental approaches.

The first main results in a stochastic process environment were given by Balcer and Sahin

(1983) (1986), Carravetta et al (1981), De Dominicis et al (1991), Janssen Manca (1994),

(1997), (1998). These models are able to simulate all the aspects of a pension scheme but

remain theoretical constructions because they were not yet totally applied.

Another approach is by means of the Monte Carlo method. This approach, if applied in the

right way, that is with at least 100000 considered cases to obtain one result, gives very

reliable results and in the authors’ opinion, it is currently the best way to study the evolution

of liabilities of a pension scheme.

The method was introduced in Tomassetti (1973) and was fully established in Tomassetti

(1991).

The study of the evolution of the liabilities of a pension scheme with for example 2000

workers in the fund needs, to carry out at least 50 repetitions in order to obtain reliable values,

for each fund member and then to compare the results obtained with 51 repetitions and only

then, and only if the results differs for less than a fixed error, is it possible to conclude the

iterative process.

In the case of computation of the mean present value of a pension annuity, it is necessary, for

each value to be found, to begin with at least 100000 repetitions and to stop the iterative

process when we obtain an error that is less than the one we fixed.

It is clear that in order to obtain reliable results a very long and time consuming algorithm

would be necessary

We must also state that Monte Carlo methods can also be applied in another way (see

Bacinello (1988)), but the number of repetitions necessary in order to obtain reliable results

does not change.

A third approach is also a simulation model but, instead of working with the reconstruction of

the random variable distribution, as in the Monte Carlo method, it works on the mean values

of the variables to be reconstructed. This method is named the direct method and is described,

for example, in Volpe Manca (1988). In this case too, it is possible to study the variance and

therefore the risk (see Micocci(1998)).

The direct method will follow all the different trajectories that a person with unitary pension

can have during her/his life. Each time that there are sums of money for the pensioner, the

present value at time 0 of these sums will be computed and will be added to give a variable

that at the end of the process, will give the mean present value of all the sums of money that

the pensioner under consideration has received directly or indirectly by means of offspring.

This method is reliable and allows results to be obtained very quickly.

The difference between this method and Monte Carlo methods is that Monte Carlo follows

only one possible trajectory at each repetition and computes the cost of this trajectory.

Repeating the process a great number of times, the MC method permits the reconstruction of

the random variable given by each expense level and the related probability.

The direct method visits the tree of all possible trajectories and the method, knowing its

probability, can compute its actuarial present value when there is a payment of money.

The computer time necessary in the last case is comparable with the time necessary to find the

results using the relations described in the previous section.

It is possible to find more details and a study of computational complexity of the general

direct method in Volpe Manca (1988).

4. THE CONSTRUCTION OF UNITARY PENSION MEAN PRESENT VALUE

In the construction of the mean present value of a unitary direct pension it is, firstly, necessary

to compute the mean present value of the unitary pensions of a widow, of the offspring and of

a family consisting of a widow and offspring. Once these values are known, the computation

of the mean present value of the direct pensions is simplified because the most part of the

branches of the tree of the trajectories can be avoided.

In this section, we will, consider the four following

(i) the widow alone,

(ii) the offspring,

(iii)the complete family

(iv) the direct pensioner case

4.1 The case of the widow

This is the case in which the pensioner does not have children or the children are too old to

have a right to a pension. We fixed this age at 23 years.

The graph that describes this case is shown in Figure 1

Figure 1: case of only widow

Possible states are:

W: widow state

E: absorbing state representing the elimination by the fund.

This is a really particular case and represents the present value of a life annuity. We wish to

show, starting from this very simple case, the recursive method that we will use to compute

the present values.

We suppose that the widow is of age x at time t. Going from the time t to the time 1t + , the

widow will remain in the state W with probability , ( )W W

p x or will go to the absorbing state

with probability , ( )W E

p x . Clearly it results:

, ,( ) ( ) 1.W W W E

p x p x+ =

In the year t the fund will pay for the unitary pension:

,

,

( )( )

2

W E

W W

p xp x + ,

because we suppose that any state change will happen in the middle of the year. To simplify,

we will also suppose that all the payments will be made in the middle of the year. In this way,

the mean present value at the beginning of the considered period is given by

( )0.5,

,

( )( ) 1 .

2

W E

W W t

p xp x r+ + (4.1)

withtr representing the interest rate at time t.

All the expenses and the related present values represent mean values and we will not repeat

mean in the following.

At time 1t + the unitary pension gives the following payment:

,

, ,

( 1)( ) ( 1)

2

W E

W W W W

p xp x p x

++ +

having as present value at time 0:

( ) ( )1 0.5,

, , 1

( 1)( ) ( 1) 1 1 .

2

W E

W W W W t t

p xp x p x r r+

++ + + +

In general, we can write the following relation:

,

, , , ,

( )( , ) ( ) ( 1) ( 1) ( )

2

W E

W W W W W W W W W

p x hP x h p x p x p x h p x h

+= + + + +L . (4.2)

giving the disbursement that the fund will have at h t+ year related to the unitary pension.

Its present value is given by: 1

1 0.5( , ) ( , ) (1 ) (1 )t h

W W i h

i t

V x h P x h r r

+

=

= + + . (4.3)

So, the value of a unitary pension reserve of a widow is:

( ) ( , )c

W W

h x

C x V x h

=

= . (4.4)

What we described is really simple and represents the present value of a unitary life annuity

with variable interest rate and instalments paid at the middle of the period.

Now, starting from the previous relations, we will show how to construct the algorithm.

The following triangular matrix gives all future expenses of the fund in correspondence with

the widow’s unitary pension.

0 1 2 - -1 - - 1

1 ( 1,0) ( 1,1) ( 1,2) ( 1, 1) ( 1, ) ( 1, 1)

( ,0) ( ,1) ( ,2) ( , 1) ( , )

1 ( 1,0) ( 1,1) ( 1,2) ( 1, 1)

3 ( 3,0) (

W W W W W W

W W W W W

W W W W

W W

x x x

etinzc

x P x P x P x P x x P x x P x x

x P x P x P x P x x P x x

x P x P x P x P x x

P P

+

+

+ + + + +

L L

M

L

L

L

M M M M N

3,1) ( 3,2) ( 3,3)

2 ( 2,0) ( 2,1) ( 2,2)

1 ( 1,0) ( 1)

( ,0)

W W

W W W

W W

W

P P

P P P

P P

P

Table 1: yearly fund disbursements for a widow unitary pension

where etinzc represents the initial age of the widow.

Each table row represents the pensioner’s age at computation time and each column the

number of years spent from the beginning of the computation. Each row contains all the

expenses that the fund should pay for the widow of a given age.

The main algorithm problem is the construction of Table 1.

Once we get this table, the reserve for each age can be constructed by first computing the

present value at time 0 all the table elements and after summing all the elements of the row

corresponding to each specified age.

As it results from (4.1), the disbursement of the fund at time 0 for each age x is given by:

, ,

1( ,0) ( ) ( ), , ,

2W W W W E

P x p x p x x etinzc= + = K .

In particular, we have:

, ,

1( ,0) ( ) ( )

2W W W W E

P p p= +

and furthermore :

, , , ,

1( 1,1) ( 1) ( ) ( ) ( 1) ( ,0)

2W W W W W W E W W W

P p p p p P= + = .

In the following we report some of the relations of the last three rows of the Table 1.

, , , ,

1( 2,1) ( 2) ( 1) ( 1) ( 2) ( 1,0),

2W W W W W W E W W W

P p p p p P= + =

, , , , ,

1( 2,2) ( 2) ( 1) ( ) ( ) ( 2) ( 1,1),

2W W W W W W W W E W W W

P p p p p p P= + =

, , , ,

,

1( 3,2) ( 3) ( 2) ( 1) ( 1)

2

( 3) ( 2,1),

W W W W W W W W E

W W W

P p p p p

p P

= +

=

, , , , ,

,

1( 3,3) ( 3) ( 2) ( 1) ( ) ( )

2

( 3) ( 2,2).

W W W W W W W W W W E

W W W

P p p p p p

p P

= +

=

Starting from the unique element of the last row and going back, it is possible to compute

Table 1.

4.2 The case of offspring

The graph in Figure 2 represents the considered model used to find the present value of the

disbursements of the fund for the family formed by only offspring.

Figure 2: case of only offspring

where O represents the state of only offspring.

We ignore the case of a disabled child. This case could anyway be considered similar to the

case of the widow because the pension to the disabled will be until death.

The pension will be paid up to a given age represented by mxao.

As the family could be more than one child in the family, we have to follow the life of each of

the children up to mxao in order to exactly compute the present value of the disbursements.

This fact would mean that a person with more children should have a smaller pension, and

that the pensioner that does not have children will have the highest pension.

Clearly this is socially wrong for a first pillar pension and so, for this reason, we will consider

the mean number of children at a given age without distinguishing the sex of the child. In this

way, all the people are considered in the same way and there is no social discrimination.

This kind of ideal offspring has a mean age and this data can be found in INPS (1989).

Taking into account the simplification and the fact that, for young people, the probability of

dying is very low, we will consider this pension as a unitary annuity paid at the middle of the

period and discounted at time 0.

For each age , 0, ,x x mxao= K , we construct the present value of the related unitary annuity.

It results:

( )( ) ( )

O mxao ao x rC x a no x= &

where:

( )O

C x is the reserve of the pensioner that dies at age x and leaves only offspring with a

unitary pension;

n ra& : is the present value of a unitary annuity of n instalments paid in the middle of the year;

ao(x): is the children mean age of a person of age x;

no(x): is the mean number of offspring of age x.

Clearly we do not report the computation of the unitary annuity.

4.3 The complete family case

Figure 3 gives the graph of the behaviour of a survivor family formed by widow and

children.

Figure 3: case of a complete family

where F represents the state of family formed by widow and children.

To explain the Figure 3 graph, let us recall that the children do not die, so given ao(offspring

age), until they will be at an age less than mxao.

When the widow dies, the transition from F to O is activated. When the children reach mxao

then the transition to W is done for the part of widow that is still living. In the same year, we

must also activate the transition from F to E for the widow who dies in that year.

In this case, the computations will be done until mxao is reached. At this point, the children

can not receive the pension and we enter into the case of only the widow. To compute the

subsequent disbursements it is sufficient to multiply the ( )W

C x obtained in (4.1) for the

probability of the widow surviving from the beginning of the period up to the moment when

the children leave home.

In this year, the widow will be age 1x .

In the last examined period for half year, the system is considered to be in X state F and the

other half in X state W. For this reason, it results that the probability of changing state in this

year is divided into four parts, one quarter is considered to belong to F, one quarter to O, one

quarter to W and the last X quarter goes to E. This fact is shown in Figure 4.

Figure 4: behaviour of the last year of a complete family

In the following the relations useful for constructing the unitary reserve of a family formed by

widow and children are given.

The spouse is of a given age, and we suppose that, in function of her/his age she/he has a

mean number of children and that these children have a mean age.

The cost the first year for a family in which the widow at time t is of age x is given by:

( )0.5

, , ,

1 1( ,0) ( ) ( ) ( ) 1

2 2F F F F O F O t

PensOP x p x p x p x r

PensF= + + +

where:

, ,( ) ( ) 1F F F O

p x p x+ =

and

PensO

PensF represents the percentage of the pension that goes to the children.

The expense at time 1t + is given by:

( ) ( )

, , , , ,

1 0.5

1

1 1( ,1) ( ) ( 1) ( 1) ( 1) ( )

2 2

1 1 .

F F F F F F O F O F O

t t

PensO PensOP x p x p x p x p x p x

PensF PensF

r r+

= + + + + + +

+ +

That can be rewritten in the following way:

( ) ( )

, , , , ,

1 0.5

1

1 1( ,1) ( ) ( 1) ( 1) ( ,0) ( 1,1)

2 2

1 1 ,

F F F F F F O F O F O

t t

PensOP x p x p x p x PR x PR x

PensF

r r+

= + + + + + +

+ +

where:

1

, , ,

0

( , ) ( ) ( )h

F O F F F O

i

PR x h h p x i p x h=

+ = + + .

Generalizing, the following is obtained:

1

, , ,

1

, ,

0

1( , ) ( ) ( ) ( )

2

1( , ) ( , ) ,

2

x h

F F F F F F O

j x

h

F O F O

j

P x h p j p x h p x h

PensOPR x j j PR x h h

PensF

+

=

=

= + + +

+ + + +

where 1

,

0

( , )h

F O

j

PR x j j=

+ considers all the previous transitions done to the offspring state.

Denoted by h’ the year in which the children reach mxao, it results that the disbursement for

this year is given by:

( )

' 1

, ,

0

' 1

, , , , ,

1 1( , ') ( , ) ( ', ')

2 2

1 1 1 1( ) ( ') ( ') ' ( ') ,

2 4 2 4

h

F F O F O

j

x h

F F F F F O F W F E

j x

PensOP x h PR x j j PR x h h

PensF

PensWp j p x h p x h p x h p x h

PensF

=

+

=

= + + +

+ + + + + + + +

where PensW

PensF represents the percentage of the pension that goes to the widow.

It should be outlined that , ,( ') ( ')F F F Wp x h p x h+ = + because it represents the probability of a

widow surviving to age 'x h+ . Different symbols were used because of the different meaning

that they have as can be seen from Figure 4. The same can be said of , ( ')F Op x h+ and

, ( ')F Ep x h+ , both representing the probability of dying.

Now we introduce the following expression:

' 1

, ,( ', ) ( ) ( ') ( ' 1)x h

F F F W W

j x

PensWE h x p j p x h C x h

PensF

+

=

= + + + . (4.6)

This expression represents, having arrived at state W, the reserve mean present values of the

reached state avoiding further computations.

The widow reaches this situation at age 'M x h= + .

Now defining:

( ) ( )1

1 0.5( , ) ( , ) 1 1 , 0, , '

t h

F F j t h

j t

V x h P x h r r h h+

+

=

= + + = K

the following present value of the reserve is obtained:

' '

1

0

( ) ( , ) ( ', ) (1 ) .h t h

F F j

h j t

C x V x h E h x r+

= =

= + +

To construct the related algorithm, as in the case of the widow, we first consider the triangular

matrix where the unitary mean disbursements are contained. The fund will sustain these

expenses for the F members with a unitary initial pension. The calculations will be done until

age M representing the theoretical maximum age to have a child yet with age less or equal to

mxao.

0 1 2 1 1

1 ( 1,0) ( 1,1) ( 1,2) ( 1, 1) ( 1, ) ( 1, 1)

( ,0) ( ,1) ( ,2) ( , 1) ( , )

1 ( 1,0) ( 1,1) ( 1,2) ( 1, 1)

3 ( 3,0) (

F F F F F F

F F F F F

F F F F

F F

M x M x M x

etinzc

x P x P x P x P x M x P x M x P x M x

x P x P x P x P x M x P x M x

x P x P x P x P x M x

M P M P

+

+

+ + + + +

L L

M

L

L

L

M M M M N

3,1) ( 3,2) ( 3,3)

2 ( 2,0) ( 2,1) ( 2,2)

1 ( 1,0) ( 1,1)

( ,0)

F F

F F F

F F

F

M P M P M

M P M P M P M

M P M P M

M P M

Table 2: yearly fund disbursements for a complete family unitary pension

As in Table 1 row index represents age at the pensionable moment and the column index the

number of years that have passed from the beginning of the pension. ( , )F

P x h represents the

fund disbursement at time t h+ for a unitary pension of a widow with offspring that at time t

is of an age x.

Each row represents all the expenses that the fund will sustain for a member that is of age x at

time 0.

To construct the elements of Table 2 in a recursive way, we begin with the unique element of

the last row, i.e.

( ) ( ), , , , ,

1 1( ,0) ( ) 1 ( ) ( ) 1 ( )

2 4F F W W F F E O F W F

P M p M RP M p M RP M RP= + + + + ,

where:

, ( )W F

PensWRP M

PensF= represents the percentage of the pension that goes to widow. This time

we need to take into account the age because it is a function of the child mean number and so

function of the age;

, ( )O F

PensORP M

PensF= represents the percentage of pension that goes to offspring.

Once that the mean number of children is known, the two percentages are constant. In fact, all

the children will exit at the same time because it is supposed that area all the same age.

The other elements in the 0 column are given by:

( ), , ,

1( ,0) ( ) ( ) 1 ( )

2F F F F O O F

P x p x p x RP x= + + .

In the following, some elements of the row 1M and 2M are reported as examples:

( ), , ,

1( 1,0) ( 1) ( 1) 1 ( )

2F F F F O O F

P M p M p M RP x= + + ,

, , ,

1( 1,1) ( 1) ( ,0) ( 1) ( 1)

2F F F F F O O F

P M p M P M p M RP M= + ,

, , ,( 2,1) ( 2) ( 1,0) ( 2) ( 2)F F F F F O O F

P M p M P M p M RP M= + .

By extension, we get:

, , ,( , ) ( ) ( 1, 1) ( ) ( ), 1, , 1F F F F F O O FP M x h p M x P M x h p M x RP x h x= + + = K

, , ,

1( , ) ( ) ( 1, 1) ( ) ( )

2F F F F F O O F

P M x x p M x P M x x p M x RP x= + +

From our knowledge of Table 2, it is possible to compute the expressions (4.6) and then all

the reserve mean present values.

4.4 The direct pensioner case

The graph in Figure 5 shows the model useful for describing the problem.

Figure 5: case of direct pensioner

where D represents the direct pensioner state.

First of all, the amount paid at year 0 by a person of age x is given by

( )

( )

, , , , ,

, , , , , ,

1( ,0) ( ) ( ) ( ) ( ) ( )

2

1( ) ( ) ( ) ( ) ( )

2

D D D D F D W D O D E

D F F D D W W D D O O D

P x p x p x p x p x p x

p x RP x p x RP p x RP x

= + + + + +

+ + +

(4.7)

where:

, , , , ,( ) ( ) ( ) ( ) ( ) 1D E D F D W D O D E

p x p x p x p x p x+ + + + = ;

, ( )F D

RP x represents the percentage of pension that goes to a complete family of survivors.

Also a function of the number of offspring and so of age;

,W DRP represents the percentage of pension that goes to the widow alone, here not a function

of age but fixed by rules ( for example in Italy, it has a value of 60%);

, ( )O D

RP x represents the percentage of pension that goes to the survivor family formed by

only children, and also a function of age.

The amount that will be given, in year 1, to a pensioner of age x at year 0, has the following

value:

, , ,

, , , ,

( ,1) ( ) ( 1,0) ( ) ( ) ( 1)

( ) ( 1) ( ) ( ) ( )

D D D D D F F D F D

D W W D W D D O O D O x

P x p x P x p x RP x C x

p x RP C x p x RP x C n

= + + +

+ + + (4.8)

where:

( )F D

C x represents the mean present value of a unitary pension of the survivor complete

family. The value of this reserve is function of the age of the widow, and so of the age of the

pensioner;

( )W D

C x represents the mean present value of a unitary pension of the widow alone. Also

here, this value is function of the widow’s age;

( )O x

C n represents the mean present value of a unitary pension of a family formed only by

children, a function of the mean number and mean age of the children.

A time h, a person that was at time 0 age x will receive:

,( , ) ( ) ( 1, 1)D D D DP x h p x P x h= + (4.9)

The recursive approach is strictly connected inside the relation (4.9). In fact, the disbursement

is obtained by multiplying the probability of remaining direct pensioner at time 0 for a person

of age x, all the other information is given by ( 1, 1)D

P x h+ .

Let us precise that for:

(

)

,

, , , ,

, , , , 1

( ,2) ( ) ( 1,1)

( ) ( 1) ( 2,0) ( 1) ( 1) ( 1 1)

( 1) ( 1 1) ( 1) ( 1) ( ) .

D D D D

D D D D D D F F D F D

D W W D W D D O O D O x

P x p x P x

p x p x P x p x RP x C x

p x RP C x p x RP x C n +

= +

= + + + + + + +

+ + + + + + +

(4.10)

relation (4.10) could be exploded once more substituting ( 2,0)D

P x + using (4.7) with the

right age index.

The way in which the algorithm works is shown in the Table 3.

0 1 2

( ,0) ( ,1) ( ,2)

1 ( 1,0) ( 1,1)

2 ( 2,0)

D D D

D D

D

x P x P x P x

x P x P x

x P x

+ + +

+ +

L

M M M M N

L

N

N

M M N

Table 3: the algorithm path

The pensioner that starts at age x at time 0 will get (4.7) i.e. ( ,0)D

P x ; at time 1 the algorithm

solves the problem of the survivors of time 0 giving the related reserves, but the survived part

of pensioner will be in the situations ( 1,0), ( 1,1), ( 1,2), , ( 1, ),D D D D

P x P x P x P x h+ + + +K K

with probability , ( )D D

p x and so on.

The recursive nature of this algorithm reduces minimally the number of operations necessary

to construct the mean present values of unitary pension annuity.

Finally, we obtain:

1

1 0.5( , ) ( , ) (1 ) (1 )t h

D D j t h

j t

V x h P x h r r+

+

=

= + + ,

and the related mean present value of the unitary direct pension is given by

( ) ( , ).D D

h x

C x V x h

=

=

5. NUMERICAL EXAMPLE AND RESULTS COMPARISON

In this section, the results obtained by means of the construction of the transformation

coefficients obtained by the Italian rules described in section 2 are presented, as well as the

same transformation coefficient constructed by means of the algorithms described in the

previous section .

After, a comparison between the two kinds of results is given, the problems that arise from the

risk of mortality will be outlined.

The survival data useful for the implementation of algorithms were downloaded by Human

Mortality Database (see in the references)

The reported results go from age 57 to age 65 as in the work of the Ministero del Lavoro e

delle Politiche Sociali (2001). The time horizon goes from 1981 up to 2001. So we give

results for 21 years and so we can see how the risk mortality influences the unitary annuity

present values.

In Tables 4 and 5 present the transformation coefficients obtained by the Italian rules and by

the direct method respectively.

1981 0.048832 0.050301 0.051874 0.053549 0.055356 0.057232 0.059317 0.061536 0.063953

1982 0.048438 0.049878 0.051416 0.053066 0.054817 0.056698 0.058675 0.060876 0.063219

1983 0.048763 0.050236 0.051809 0.05348 0.055283 0.057195 0.059258 0.061422 0.063852

1984 0.048025 0.049445 0.050958 0.052575 0.054295 0.056145 0.05811 0.060242 0.062457

1985 0.048052 0.049473 0.050994 0.052611 0.05434 0.056186 0.058185 0.060301 0.062603

1986 0.047766 0.049176 0.050674 0.052276 0.053986 0.055807 0.05776 0.059876 0.062125

1987 0.047295 0.048667 0.050135 0.051694 0.053361 0.055133 0.057031 0.059062 0.06126

1988 0.047147 0.048515 0.049978 0.051537 0.053192 0.054959 0.056842 0.058868 0.061045

1989 0.046737 0.04809 0.049522 0.051052 0.052691 0.054428 0.056286 0.058269 0.060392

1990 0.046684 0.04803 0.049464 0.050991 0.052618 0.054359 0.05622 0.058215 0.06035

1991 0.046581 0.047921 0.04935 0.050872 0.052494 0.054227 0.05608 0.058053 0.06018

1992 0.046228 0.047552 0.048956 0.050452 0.052046 0.053749 0.055573 0.057531 0.059621

1993 0.046087 0.047399 0.048801 0.050298 0.051889 0.05359 0.055409 0.057352 0.059441

1994 0.045905 0.047219 0.048608 0.05009 0.051672 0.053363 0.05517 0.057097 0.059162

1995 0.045642 0.046934 0.04831 0.049778 0.051343 0.053002 0.054781 0.056703 0.05875

1996 0.045391 0.046672 0.048027 0.049471 0.05101 0.052657 0.054407 0.0563 0.05833

1997 0.045271 0.046543 0.047899 0.04934 0.050882 0.052523 0.054279 0.056147 0.058166

1998 0.045276 0.046548 0.047907 0.049355 0.050896 0.052544 0.054309 0.056197 0.058213

1999 0.044964 0.046224 0.04756 0.048985 0.050508 0.052123 0.053873 0.05574 0.057742

2000 0.044622 0.045848 0.047161 0.048563 0.050062 0.051657 0.053355 0.055192 0.057149

2001 0.044359 0.045566 0.046856 0.048233 0.049701 0.051274 0.052955 0.054748 0.05669

Table 4: annuitization coefficients obtained by Italian rules

57 58 59 60 61 62 63 64 65

1981 0.049926 0.051462 0.053107 0.054857 0.056745 0.058703 0.060879 0.063207 0.065745

1982 0.049515 0.051021 0.052629 0.054352 0.056181 0.058144 0.060205 0.062514 0.064972

1983 0.049854 0.051394 0.053039 0.054785 0.056668 0.058666 0.060817 0.063087 0.065638

1984 0.049083 0.050567 0.052148 0.053837 0.055632 0.057564 0.059612 0.061847 0.064169

1985 0.049112 0.050597 0.052186 0.053874 0.05568 0.057607 0.059692 0.061911 0.064326

1986 0.048812 0.050285 0.05185 0.053522 0.055307 0.057207 0.059244 0.061462 0.063821

1987 0.048321 0.049754 0.051286 0.052913 0.054653 0.056501 0.058479 0.060607 0.062912

1988 0.048166 0.049595 0.051121 0.052749 0.054475 0.056319 0.058281 0.060403 0.062685

1989 0.047739 0.04915 0.050645 0.052241 0.053951 0.055763 0.057699 0.059775 0.061999

1990 0.047683 0.049087 0.050583 0.052176 0.053873 0.055689 0.057628 0.059717 0.061955

1991 0.047574 0.048972 0.050463 0.05205 0.053742 0.05555 0.057481 0.059546 0.061775

1992 0.047207 0.048588 0.050052 0.051612 0.053274 0.05505 0.05695 0.059 0.061189

1993 0.047059 0.048428 0.049889 0.05145 0.053109 0.054882 0.056778 0.058812 0.061

1994 0.04687 0.04824 0.049688 0.051232 0.052882 0.054645 0.056528 0.058545 0.060708

1995 0.046596 0.047942 0.049376 0.050907 0.052537 0.054267 0.056121 0.058132 0.060276

1996 0.046334 0.047669 0.049081 0.050586 0.05219 0.053907 0.05573 0.05771 0.059836

1997 0.046208 0.047534 0.048947 0.050449 0.052055 0.053766 0.055595 0.057549 0.059663

1998 0.046214 0.047539 0.048955 0.050464 0.05207 0.053788 0.055627 0.057602 0.059712

1999 0.045889 0.047201 0.048593 0.050077 0.051664 0.053347 0.055171 0.057123 0.059219

2000 0.045534 0.046811 0.048178 0.049638 0.0512 0.052862 0.054631 0.056552 0.0586

2001 0.045261 0.046518 0.047861 0.049294 0.050823 0.052461 0.054213 0.056087 0.058119

Table 5: annuitization coefficients obtained by Direct Method

As can be easily seen, the transformation coefficients constructed by direct method are

slightly bigger than the ones constructed by Italian rules. To have a better understanding we

report :

57 58 59 60 61 62 63 64 65

1981 20.47843 19.8805 19.27742 18.67462 18.06484 17.47268 16.85864 16.25063 15.63641

1982 20.64512 20.04881 19.44902 18.8444 18.24246 17.63743 17.04314 16.42684 15.8181

1983 20.50717 19.90608 19.30168 18.69859 18.08882 17.48395 16.87542 16.2808 15.66123

1984 20.82249 20.22455 19.62401 19.02029 18.41793 17.81108 17.20874 16.59972 16.01101

1985 20.81074 20.21297 19.61031 19.00761 18.40275 17.79796 17.1866 16.5834 15.97368

1986 20.9353 20.33513 19.73396 19.12927 18.52331 17.91897 17.31295 16.70112 16.09652

1987 21.14404 20.54791 19.9462 19.34476 18.74038 18.1381 17.53435 16.93143 16.32386

1988 21.21013 20.6121 20.0089 19.40345 18.79999 18.19528 17.59251 16.9871 16.38128

1989 21.39624 20.79452 20.19309 19.58803 18.97855 18.37286 17.76629 17.16183 16.55856

1990 21.42042 20.82011 20.21681 19.61124 19.00495 18.39614 17.78724 17.17782 16.56988

1991 21.46798 20.86778 20.26363 19.65734 19.0498 18.44107 17.83158 17.22568 16.61693

1992 21.63187 21.02965 20.4266 19.82097 19.21365 18.60496 17.99439 17.38196 16.77268

1993 21.69796 21.09736 20.4915 19.88154 19.27175 18.6603 18.04764 17.43603 16.8233

1994 21.78411 21.1779 20.57285 19.96424 19.35303 18.73959 18.12594 17.51415 16.90272

1995 21.90957 21.30665 20.69975 20.08915 19.47697 18.86734 18.25451 17.6358 17.02118

1996 22.03097 21.42632 20.82181 20.21395 19.60389 18.99086 18.38004 17.76198 17.14391

1997 22.08942 21.48546 20.87739 20.26747 19.65334 19.03933 18.42338 17.81048 17.1922

1998 22.08678 21.48314 20.87381 20.26144 19.6478 19.03165 18.41308 17.79439 17.17836

1999 22.23984 21.63377 21.02593 20.41454 19.7988 19.18522 18.56219 17.94043 17.31842

2000 22.41054 21.81122 21.20399 20.59181 19.97511 19.35855 18.74235 18.11849 17.49801

2001 22.54351 21.94609 21.34193 20.73278 20.12023 19.50324 18.88385 18.2654 17.63994

Table 6: unitary annuity mean present value obtained by Italian rules

57 58 59 60 61 62 63 64 65

1981 20.02984 19.43192 18.82987 18.22928 17.62257 17.03478 16.42601 15.821 15.2102

1982 20.19605 19.59978 19.00101 18.39851 17.79968 17.19856 16.61002 15.99654 15.3912

1983 20.05866 19.45761 18.85421 18.25331 17.6466 17.04579 16.44271 15.85123 15.23497

1984 20.37357 19.77573 19.17616 18.57451 17.97516 17.3721 16.7751 16.16888 15.58375

1985 20.36176 19.76405 19.16237 18.56176 17.95988 17.35893 16.75262 16.15227 15.54581

1986 20.48671 19.88666 19.2865 18.68385 18.0808 17.48025 16.87938 16.27024 15.66893

1987 20.69503 20.09907 19.49833 18.8989 18.29736 17.69877 17.10001 16.49978 15.89533

1988 20.76135 20.16347 19.56124 18.95774 18.35711 17.756 17.15821 16.55545 15.95276

1989 20.94733 20.34581 19.74537 19.14221 18.53537 17.93315 17.3313 16.72943 16.12921

1990 20.97191 20.37182 19.76952 19.16586 18.56226 17.95684 17.35257 16.74568 16.14072

1991 21.01971 20.41972 19.81657 19.21217 18.60729 18.00193 17.39702 16.79369 16.1879

1992 21.18326 20.58132 19.97928 19.3755 18.77077 18.16535 17.55919 16.94916 16.34274

1993 21.24971 20.64939 20.04452 19.43637 18.82911 18.22083 17.61248 17.00328 16.39334

1994 21.3357 20.72981 20.12576 19.51895 18.9102 18.29985 17.6904 17.081 16.47234

1995 21.46117 20.8586 20.25267 19.64382 19.03404 18.4275 17.8188 17.20233 16.59039

1996 21.58221 20.97796 20.37442 19.76831 19.16062 18.55054 17.9438 17.32796 16.71247

1997 21.64104 21.03746 20.43039 19.82219 19.21036 18.59926 17.98721 17.37659 16.76087

1998 21.6385 21.03524 20.4269 19.81622 19.2049 18.59162 17.97687 17.36042 16.74699

1999 21.7916 21.18592 20.57904 19.96931 19.35578 18.74503 18.12561 17.50599 16.88642

2000 21.96147 21.36256 20.7563 20.14571 19.53113 18.91725 18.30466 17.68283 17.06474

2001 22.09426 21.49723 20.89405 20.28649 19.67608 19.06168 18.44573 17.82935 17.20617

Table 7: unitary annuity mean present value obtained by Direct Method

also the unitary annuity present values obtained by the two methods respectively in Tables 6

and 7.

In Table 8 the division of each element of X Table 6 with the corresponding element of

Table 7 is reported.

57 58 59 60 61 62 63 64 65

1981 1.0223959 1.0230845 1.0237678 1.02443 1.0250966 1.025706 1.026338 1.0271559 1.0280215

1982 1.0222355 1.0229101 1.0235781 1.0242353 1.0248757 1.0255178 1.0260758 1.026899 1.0277363

1983 1.0223597 1.0230487 1.023733 1.0243945 1.0250598 1.0257052 1.0263161 1.0271 1.027979

1984 1.0220341 1.0226959 1.0233547 1.0239998 1.0246319 1.0252688 1.0258502 1.0266462 1.0274169

1985 1.0220501 1.0227141 1.0233757 1.0240199 1.0246588 1.0252912 1.0259052 1.0266915 1.0275228

1986 1.0218968 1.0225513 1.023201 1.0238403 1.024474 1.0250979 1.0256865 1.0264823 1.0272887

1987 1.0216962 1.0223314 1.0229693 1.0235919 1.0242125 1.0248231 1.0254 1.0261609 1.0269594

1988 1.021616 1.0222498 1.0228851 1.0235106 1.0241258 1.0247397 1.0253116 1.0260726 1.0268621

1989 1.0214306 1.022054 1.0226743 1.02329 1.0239101 1.0245191 1.0250984 1.0258469 1.0266195

1990 1.0213862 1.0220055 1.0226253 1.0232383 1.0238488 1.0244645 1.0250498 1.0258064 1.0265889

1991 1.0213261 1.0219427 1.0225597 1.0231712 1.0237815 1.0243941 1.0249788 1.0257235 1.0265033

1992 1.0211778 1.0217833 1.0223894 1.0229914 1.0235943 1.0242006 1.0247849 1.0255348 1.0263078

1993 1.0210948 1.0216938 1.022299 1.022904 1.0235084 1.0241195 1.0247079 1.0254511 1.0262274

1994 1.0210168 1.0216159 1.0222151 1.0228133 1.0234172 1.0240295 1.0246203 1.0253588 1.0261274

1995 1.0208935 1.0214805 1.0220751 1.0226706 1.0232708 1.0238686 1.0244524 1.0251983 1.0259659

1996 1.020793 1.0213729 1.0219584 1.0225433 1.0231341 1.023736 1.0243117 1.0250476 1.0258153

1997 1.0207189 1.0212953 1.021879 1.0224639 1.0230596 1.0236605 1.0242492 1.0249699 1.0257344

1998 1.0207169 1.0212927 1.0218785 1.0224674 1.0230621 1.0236685 1.0242649 1.0249978 1.0257579

1999 1.0205696 1.0211391 1.0217158 1.0222959 1.0228879 1.0234827 1.0240861 1.0248169 1.0255823

2000 1.0204479 1.0210021 1.0215692 1.0221436 1.0227318 1.0233282 1.0239112 1.0246375 1.0253898

2001 1.0203329 1.0208795 1.0214359 1.0219991 1.0225732 1.0231647 1.0237515 1.0244569 1.0252101

Table 8: relations between Direct Method and Italian rules

57 58 59 60 61 62 63 64 65

1982 1.00814 1.008467 1.008902 1.009092 1.009833 1.009429 1.010944 1.010843 1.011619

1983 1.001403 1.001287 1.001259 1.001283 1.001327 1.000645 1.000995 1.001856 1.001587

1984 1.016801 1.017306 1.017979 1.01851 1.019546 1.019367 1.020767 1.021482 1.023957

1985 1.016227 1.016723 1.017268 1.017831 1.018706 1.018617 1.019453 1.020478 1.021569

1986 1.02231 1.022868 1.023683 1.024346 1.025379 1.025542 1.026948 1.027721 1.029425

1987 1.032503 1.033571 1.034692 1.035885 1.037396 1.038084 1.040081 1.041894 1.043964

1988 1.03573 1.0368 1.037945 1.039028 1.040695 1.041356 1.04353 1.045319 1.047637

1989 1.044819 1.045976 1.0475 1.048912 1.05058 1.051519 1.053839 1.056072 1.058974

1990 1.045999 1.047263 1.04873 1.050155 1.052041 1.052852 1.055082 1.057056 1.059698

1991 1.048322 1.049661 1.051159 1.052623 1.054524 1.055423 1.057711 1.060001 1.062707

1992 1.056325 1.057803 1.059613 1.061386 1.063594 1.064803 1.067369 1.069617 1.072668

1993 1.059552 1.061209 1.062979 1.064629 1.06681 1.06797 1.070528 1.072945 1.075905

1994 1.063759 1.06526 1.0672 1.069057 1.071309 1.072508 1.075172 1.077752 1.080985

1995 1.069886 1.071736 1.073783 1.075746 1.07817 1.079819 1.082798 1.085238 1.08856

1996 1.075814 1.077756 1.080114 1.082429 1.085196 1.086889 1.090244 1.093003 1.096409

1997 1.078668 1.080731 1.082997 1.085295 1.087933 1.089663 1.092815 1.095987 1.099498

1998 1.078539 1.080614 1.082812 1.084972 1.087627 1.089224 1.092204 1.094997 1.098612

1999 1.086013 1.088191 1.090703 1.093171 1.095985 1.098012 1.101049 1.103984 1.10757

2000 1.094349 1.097116 1.09994 1.102663 1.105745 1.107933 1.111735 1.114941 1.119055

2001 1.100842 1.1039 1.107095 1.110212 1.113779 1.116213 1.120128 1.123981 1.128132

Table 9: Italian rules increasing values because of mortality decreasing

The Italian pensioners have a lower pension because of the way in which the transformation

coefficients are computed. The difference is not really big but always more than 2% and tends

to increase with the age.

For example, with the data of 1990 that was the year considered for the computation of Italian

coefficients, the difference fluctuates from 2.14% for an age of 57 years up to 2.66% for 65

year-old pensioners.

Finally, the divisions among the elements of the years 1982, 1983,…, 2001 with the elements

of the 1981 for each of the two considered cases are reported in Tables 9 and 10 respectively

57 58 59 60 61 62 63 64 65

1982 1.008298 1.008638 1.009089 1.009283 1.01005 1.009614 1.011202 1.011096 1.0119

1983 1.001439 1.001322 1.001293 1.001318 1.001363 1.000646 1.001017 1.001911 1.001628

1984 1.017161 1.017693 1.01839 1.018938 1.020008 1.019802 1.021252 1.021989 1.024559

1985 1.016571 1.017092 1.017658 1.018239 1.019141 1.019029 1.019884 1.020939 1.022065

1986 1.02281 1.023401 1.02425 1.024936 1.026003 1.026151 1.027601 1.028396 1.03016

1987 1.03321 1.034333 1.0355 1.036733 1.038291 1.038978 1.041032 1.042904 1.045044

1988 1.036521 1.037647 1.038841 1.039961 1.041682 1.042338 1.044575 1.046423 1.04882

1989 1.045806 1.04703 1.04862 1.050081 1.051797 1.052737 1.055113 1.05742 1.06042

1990 1.047033 1.048369 1.049902 1.051378 1.053323 1.054128 1.056407 1.058446 1.061177

1991 1.04942 1.050834 1.052401 1.053918 1.055878 1.056775 1.059114 1.061481 1.064279

1992 1.057585 1.05915 1.061042 1.062878 1.065155 1.066368 1.068986 1.071308 1.074459

1993 1.060902 1.062653 1.064507 1.066217 1.068465 1.069625 1.072231 1.074729 1.077786

1994 1.065196 1.066791 1.068821 1.070747 1.073067 1.074264 1.076974 1.079641 1.08298

1995 1.07146 1.073419 1.075561 1.077597 1.080094 1.081757 1.084791 1.08731 1.090741

1996 1.077503 1.079562 1.082027 1.084426 1.087277 1.08898 1.092401 1.095251 1.098767

1997 1.08044 1.082624 1.084999 1.087382 1.090099 1.09184 1.095044 1.098325 1.101949

1998 1.080313 1.08251 1.084814 1.087055 1.08979 1.091391 1.094415 1.097303 1.101037

1999 1.087957 1.090264 1.092893 1.095453 1.098352 1.100398 1.10347 1.106503 1.110204

2000 1.096438 1.099354 1.102307 1.105129 1.108302 1.110507 1.11437 1.117681 1.121927

2001 1.103067 1.106285 1.109623 1.112852 1.116527 1.118986 1.122959 1.126942 1.131226

Table 10: Direct Method increasing values because of mortality decreasing

As can be seen, mortality assumes a very important role in the value of coefficients. In twenty

years the increase, greater in the case of the direct method, is, in both cases, more than 10%.

The Italian rules were constructed 10 years ago with 1990 data (that means 15 years

difference). Considering that the people in Italy that retired in this year get 7% more pension

than they should receive.

Fortunately, the transformation coefficients were undervalued and this can compensate in a

certain sense for the big increase. But we think that if the government would give the new

transformation coefficients it will be really difficult to convince a worker that will go in

pension in the next future that a person with the same seniority and age that retired the year

before has the right to have a 7% higher pension than him.

But the worst think will happen if the government does not compute the new coefficients. In

fact it could be a disaster for the Italian public budget.

6. CONCLUSIONS

From our results, it is clear that the transformation coefficients are a very important factor in

the social security politics of a country using the NDC pension scheme.

We think that, theoretically, this pension method can provide both actuarial fairness and can

help avoid the expected explosion in the cost of state pensions. Great care must be taken,

however, in the construction of a transformation coefficient by adopting a truly reliable

method for the coefficients.

In Italy the rules make provision for the updating of the transformation coefficients every 10

years. In Sweden, for example, it makes provision for updating each year. This fact implies

that the differences between pensions from one year and another are very low.

In our opinion this behaviour is not sufficient to warrant the account equilibrium.

To the recommendations given above, we must add what is, in our opinion, a very important

task and that is, taking into account the mortality risk with a view to constructing reliable

coefficients.

As is well known from actuaries, the fact that pensions are a long term problem gives a strong

importance to the risk of mortality in social security accounts.

This is the line of research that we will take in the future in order to construct transformation

coefficients by means of direct methods. Another future line of research will be the

computation of variability of the obtained results.

In a certain sense, these two research lines intersect because the best way of forecasting the

mortality risk derives from Lee and Carter’s method (1992) and which uses particular time

series instruments that are connected to stochastic process to forecast future mortality and so

in a certain sense, it is possible to introduce variability into the problem of mortality.

REFERENCES

Aaron H., (1966). The social insurance paradox. Canadian Journal of Economics and

Political Science. 23, 371-374.

Ackerby S., (1992). Pensionfragan, bilaga 12 till SOU 1992:12, Finansdepartementet,

Stockholm.

Bacinello A.R. (1988). A stochastic simulation procedure for pension schemes. Insurance:

Mathematics and Economics. 7, p. 153-161

Balcer Y, Sahin I. (1983). A stochastic theory of pension dynamics. Insurance: Mathematics

and Economics. 2, p.179,197.

Balcer, Y, Sahin, I (1986). Pension accumulation as a semi-Markov reward process, with

applications to pension reform. In Semi-Markov models, Janssen J. (ed).. Plenum N.Y.

Broms J., (1990). Ur askan av ATP. SACO, Stockholm.

Buchanan J., (1968). Social insurance in a growing economy: a proposal for radical reform.

National Tax Journal, 21, 386-395.

Carravetta M., De Dominicis R., Manca R. (1981) Semi-Markov stochastic processes in

soscial security problems. Cahiers du C.E.R.O., 23, 333-339.

Castellino O., (1969). Un system di pensioni per la vecchiaia commiserate ai versamenti

contributivi effettuatie alla dinamica dei redditi medi da lavoro. Giornale degli Economisti e

Annali di Economia, 28, 1-23.

De Dominicis R., Manca R., Granata L., (1991). The dynamics of pension funds in a

stochastic environment. Scandinavian Actuarial Journal. 118-128.

De Finetti B., (1956)Sviluppo della popolazione e sicuezza sociale. In Atti del Convegno di

Studi su “I Problemi Attuariali e Statistici della Sicurezza Sociale”, Ordine Nazionale degli

Attuari, Roma.

Gronchi S., (1992). Rendimento implicito e pensionamento flessibile. In Atti delle Giornate di

studio su “Le contradizioni del sistema pensionistico italiano”, Rivista Italiana di Economia,

1-2, 69-87.

Gronchi S., (1995a). Sostenibilità ed equità del sistema pensionistico italiano. Economia

Politica: Rivista di Teoria ed Analisi, 12, 3-21.

Gronchi S., (1995b). I rendimenti impliciti della previdenza obbligatoria: un’analisi delle

iniquità del sistema. Economia Italiana, 1, 41-93.

Gronchi S., Nisticò S., (2004) Fair and sustainable pay-as-you-go pension systems:

theoretical models and pratical realizations. Documenti CNEL 27.

http://81.208.28.44/portale/pubblicazioni.nsf/

Holzmann R., (2003). Toward a reformed and coordinante pension system in Europe:

rationale and potential structure. In Proceedings of Conference on NDC Pensions. Sandhamn,

Sweden. Riksforsakrings Verket

Human Mortality Database. University of California, Berkeley (USA), and Max Planck

Institute for Demographic Research (Germany). Available at www.mortality.org or

www.humanmortality.de

INPS, (1989). Il modello INPS e le prime previsioni al 2010. Roma

Janssen J., Manca R. (1994). Semi-Markov modelization for the financial management of

pension funds. Proceedings of the IV AFIR Colloquium. Orlando

Janssen J., Manca R. (1997), A realistic non-homogeneous stochastic pension funds model

on scenario basis. Scandinavian Actuarial Journal 113-137

Janssen J., Manca R. (1998), Non-homogeneous stochastic pension fund model: an

algorithmic approach. ICA Transactions vol. V.

Lee R. D., Carter L. R. (1992). Modeling and Forecasting U.S. Mortality.Journal of the

American Statistical Association 87.

Lindbeck A., Persson M., (2003). The gains from pension reform. Journal of Economic

Literature. 41, 74-112.

Manly H.W. (1902). On the evaluation of staff pension funds. Journal of the Institute of

Actuaries, 36, p. 193-244.

Micocci M., (1998). On the measure of the variability of yearly financial flows in Pension

Funds; the comparison of two different calculation methods. Proceedings of the XXVI

International Congress of Actuaries. Birmingham

Ministero del Lavoro e delle Politiche Sociali, (2001). Commissione per la valutazione degli

effetti della legge n. 335/95 e successivi procedimenti. Verifica del sistema previdenziale ai

sensi della legge 335/95 e successivi provvedimenti, nell’ottica della competitività, dello

sviluppo e dell’equità. Roma

Ministero del Tesoro – Ragioneria Generale dello Stato (1994). I rendimenti impliciti della

previdenza oblligatoria. Conti Pubblici e Congiuntura Economica. n. 2.

Niccoli A., (1992). Un approccio di equilibrio finanziario di lungo periodo ai problemi di un

sistema pensionistico a ripartizione. Moneta e Credito, 45, n. 177, 23-47.

Olsson H., Schubert G., (1991). Det framtida pensionsystemet, ESO, Ds 1991:27 Stockholm.

Palmer E., Sherman K.G. (1993). En ny socialforsakrin, Stockholm: National Social

Insurance Board.

Persson M., (1991). Vad ar det for fel pa ATP-systemet?. Ekonomisk Debatt, n. 3, Stockholm.

Samuelson P., (1958). An exact consumption loan model of interest with or without the social

contrivance of money. Journal of Political Economy, 68, 467-482.

Tomassetti A. (1973). Sulle valutazioni attuariali dei salari e degli oneri annui con estrazione

a sorte per il passaggio di stato degli assicurati e pensionati. Quaderni dell’Istituto di Scienze

Attuariali dell’Università di Roma.

Tomassetti A. (1991). Sulle distribuzioni di probabilità delle riserve matematiche, premi,

oneri, salari nei fondi pensione, metodi di calcolo. Conferenza tenuta all’Istituto Italiano degli

Attuari e non ancora pubblicata.

Tomassetti A. (1994) Tecnica attuariale per collettività. Kappa

Volpe di Prignano E., Manca R. (1988). Simulation models for dynamic management of

pension funds. Transaction of the 23rd International Congress of Actuaries. Helsinki.