Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability

390 Chapter 9

h

i

N

: the number of members present in the fund at time h in state i,

h

N

: the total number of members present in the fund at time h.

By summation, we clearly get.

1

,, ,

1

()

m

hh h

i

si

N

Ns N

ω

ττ τ

α

−

==

∑∑

, (1.80)

1

,,, ,

1

(), () ()

m

hhh h

ii i

si

N

Ns Ns Ns

ω

τττ τ

α

−

==

∑∑

, (1.81)

1

,

01

wm

hhh

i

N

NN

τ

−

==

∑

. (1.82)

To go further, we have to introduce a scenario, often called the central scenario-

also sometimes called the basic strategy- concerning the number of active people

the firm wants to have at any time. In other words, this means that the value of

,0,1,,

h

i

N

hT= … 1, 2, , 5im

=

−… , (1.83)

where

,0,1,,

h

i

N

hT= …

represents the total number of people present in the fund

at time h in state 1,2, , 5im=−

… , T being a fixed time horizon.

From now on, we will work conditionally according to the observations resulting

from this selected central scenario. In other words, this means that at time h, we

know exactly the past counting evolution which may be used for computing

probabilities of future events.

Now, we are to study the population fund on any time interval

[

)

,,hh t s+−

()

s

t< ; so let us define:

,

(,)

tsh

N

st

τ

+−

: the number of members present in the fund initially of age s and

seniority

τ

at time h still present at time ( )htst s

+

−>and so of age t,

,

(,)

tsh

i

N

st

τ

+−

: the number of members present in the fund initially of age s, and

seniority

τ

at time h still present at time ( )htst s

+

−> and so of age t but in

state i.

With such counting observations, one could estimate some basic probabilities.

For example, let us consider the event that a member should definitively leave

the system. More precisely, let

,

(,)

tsh

Lst

τ

+−

represent the probability that a

member in state i with seniority

τ

at time h will go to the absorbing state m at

time h+t

−s.

Using the bi-dimensional associated semi-Markov process (1.29) with transition

probabilities (1.30), we get:

Generalized NHSMP models

391

1

,

1

,

(,) ()

(,) .

()

m

h

im i

ts

i

h

s

tNs

Lst

Ns

ττ

τ

φ

−

+−

=

∑

(1.84)

It is now possible to compute the distribution of the counting variables

ν

+−tsh

st

N

,

()

in the following way:

()()()

,

()

,

()

(,) 1 (,) (,) ,

0,1, , ( ).

h

gNsg

tsh ts ts

h

Ns

P N st g Lst Lst

g

gNs

τ

τττ

τ

−

+− +− +−

⎛⎞

== −

⎜⎟

⎝⎠

= …

(1.85)

More generally, using the multinomial distribution, we get for the joint

distribution of the variables

,

(,)

tsh

i

N

st

τ

+−

, i=1,...,m, the following result:

()

()( )

1

,,

11

,

1

,,

1

(,) , , (,)

()!

(,) (,) ,

!!

m

tsh tsh

mm

m

h

i

kk

i

tsh tsh

m

PNstk Nstk

Ns

qst q st

kk

ττ

τ

ττ

+− +−

=

+− +−

==

⎛⎞

⎜⎟

⎝⎠

=

∑

…



…

(1.86)

where

,

1

,

(,) ()

(,) , 1, , .

()

m

h

ij i

tsh

i

j

h

st N s

qst j m

Ns

ττ

τ

φ

+−

=

==

∑

… (1.87)

Let us recall that we are using a central scenario; also let us denote, at every time

h, by

h

Ψ all the information available including the past observations and also

the scenario.

So, it is possible to compute conditional means at chosen times for some

interesting counting variables. For example let us take the following one:

(

)

,,

(,) (,) .

tsh tsh

jjh

Nst E Nst

ττ

+− +−

=Ψ (1.88)

From result (1.87), we get:

,,

(,) () (,).

tsh h ts

jj

N

st N s q st

τττ

+− +−

= (1.89)

Another interesting conditional mean concerns the number of members being in

state i, seniority

τ

and age s at time h and still not in the absorbing state at time

tsh−+ . If we note this conditional mean by

,

(,)

h

i

N

st

τ

, we immediately have:

1

,,

1

(,) (,) ().

m

hh

iiji

j

N

st st N s

τττ

φ

−

=

∑

(1.90)

This mean represents at any time

h the mean number of workers at time h+t −s

we can predict at time

h.

392 Chapter 9

1.7 Financial Equilibrium Of The Pension Funds

Before studying the equilibrium of the fund, let us remark that the formulas we

wrote before are useful for solving the equations in the general case but, if we

want to face the problem of pension applications, there are some differences. In

fact, as we said before, the age horizon of the fund member must be at least 85

years and the seniority horizon at most fifty years. If in writing formulas we were

to be more precise we would take into account the maximum seniority. For this

reason the formulas of evolution equations should be adapted.

Let us define:

s

K = maximum reachable seniority.

In this light the equations (1.43) and (1.44) can be written in the following way:

1

()

1

((,)) (,)(,); ,

((,)) ()(,)

(, ) ( ,) ; .

s

t

s

Ks

st

s

t

K

s

Ks

IHst Bs ttKs

IHst Bs t

Bs P t t K s

τττθ

θ

τ

ν

τττθ

θ

τ

θτ

θ

φθ τ

φ

θφθ

θθ τ

+−

=+

+−

=+

=++−

⎧

−+ ≤+−

⎪

=

⎨

−+

⎪

+>+−

⎪

⎩

∑

(1.91)

Also in this case there is no matrix inversion and the system can be solved in the

same way as presented previously.

The presented model allows the management of a pension fund in which the

considered rewards change because of the state, seniority and time and

furthermore allows one to follow the

dynamic financial evolution of the fund and

therefore its financial equilibrium, the central objective of the fund managers.

Generalized NHSMP models

393

1

()

1

()

11

,

11

,

(1 ( , )) '( , )

()'(,)

() (,)(, ),

,

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

(,)

s

hts

kh

iikh

kh

tm hs

kh

is i

skh

tm

sh s

i

s

hK

kh

ii

kh

h

i

Hst hk

bkhk

bs V t hh s

tK s

Hst k hk

Vst

ττ

θ

ττ

βθ

θβ

ττθθ

ββ

θβ

τ

ττ

τ

ψν

θθνθ

τ

ψν

+−−

+−

+

=

+−−

+−

=+ = =

+− +−

=+ =

+−−

+−

=

−

+

++−

≤+−

−

=

∑

∑∑ ∑

∑∑

1

11

1

11

1

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

(, ) ( ) '(, )

(, ) () '(, )

()'(,)

(

s

hts

K

ii

khK

Ks

mh

kh

ii

skh

hK

tm

kh

ii

Ks kh

hs

K

i

khK

i

Hst k hk

bs k hk

khk

bs

τ

θτ

ττ

β

θβ

τ

ττ

β

θτβ

θ

τ

β

ψν

θψν

ψν

+−−

=+ −

+−

+−−

+−

=+ = =

+−−

+−

=+−+= =

+−−

=+ −

+−

+

⎛

+

⎜

⎝

⎞

⎟

⎠

+

∑

∑∑ ∑

∑

(

)

,

11

,

11

,

,) (,)(, )

(, ) ( ,)(, )

(,)(, ); .

s

Ks

m

sh s

s

tm

sh s

i

Ks

Kh s

s

Vthh s

bs V t hh s

Vthh s tKs

τ

τθ θ

β

θβ

ττθθ

ββ

θτβ

θ

β

θθνθ

θν θ τ

+−

+− +−

=+ =

+− +−

=+−+=

+−

⎧

⎪

⎨

⎪

+−

⎪

++−

⎪

++−>+−

⎪

⎩

∑∑

(1.92)

First of all let us define:

E: equity of the fund at time 0

A: total salary mean present value

O: total outlays mean present value

M: mean equilibrium rate (i.e. the percentage of the salary that is necessary for

the equilibrium of the fund) defined as:

OE

M

A

−

=

. (1.93)

To find the dynamic pension fund evolution, we need to know the

M value and

so it is necessary to make a static study of the fund.

Of course

E is known and it suffices to compute O and A.

To obtain these values we need to solve twice the (1.92) and (1.93) evolution

equations, the first time putting equal to 0 all the outlay rewards and equal to the

394 Chapter 9

salary amount the positive (for the fund) rewards. We denote the reward vector

obtained in this way '

ψ . The second time it is necessary to put equal to 0 all the

positive rewards, giving the right values to the outlays. The related reward vector

will be denoted "

ψ . These vectors are given in the following formulas:

1

5

'( )

'()

'( ) ; 0,1, , ; 0, , ,

0

m

s

h

hhwK

τ

ψ

τ

−

⎡⎤

⎢⎥

===

⎢⎥

⎣⎦

ψ



……



(1.94)

4

0

";0,1,,;0,,.

()

ks

m

hw K

h

τ

ψ

−

⎡⎤

⎢⎥

===

⎢⎥

⎣⎦

ψ



……



(1.95)

Now we can solve twice the evolution equations (1.93) and we get respectively

,

'( , )

h

i

Vst

τ

, the mean present value of the salaries paid by the firm to a class of

people that at time h had a seniority

τ

, an age s and was in the state i in the time

period from h up to h+t

−s and

,

"( , )

h

i

Vst

τ

, the mean present value of the

pensions that the fund paid to a class of people that at time h had a seniority

τ

,

an age s and was in the state i in the period from h up to h+t

−

s.

Now we can obtain the values of A and O as follows:

min( , )

,,

001

'( , ) ( , )

s

sK

wt m

hh

ii

ht s i

AVstNst

α

ω

ττ

αα τ

−

== = = =

=

∑∑∑∑∑

, (1.96)

min( , )

,,

001

"( , ) ( , )

as

KsK

wt m

hh

ii

ht s i

OVstNst

α

ττ

αα τ

−

== = = =

=

∑∑∑ ∑ ∑

, (1.97)

and finally the mean equilibrium rate M from relation (1.93).

By means of relations (1.96) (1.97) and (1.93) we get the static study of the fund

in the sense of a knowledge of total outlay mean present value (technical

reserve), of total salary mean present values and equilibrium rate.

Once we get the mean equilibrium rate M, we can follow the dynamic

development of the pension fund.

Now let:

() '() ,

jj

hhM

ττ

ψψ

=

1,..., 5, 0,..., , 0,...,

s

jm Kh w

τ

=

−= =

, (1.98)

then, we have to solve (1.91) and we can obtain the

,

(,)

tsh

j

N

st

τ

+−

.

Generalized NHSMP models

395

If

k

I

and

k

O

represent respectively the total mean input and total mean output

for year k, we have:

5

,

001

(, ) (),

s

K

km

h

i

ki

hs i

INsskhh

ω

ν

τ

ατ

ψ

−

== = =

=+−

∑∑∑∑

(1.99)

,

004

(, ) ().

s

K

km

h

i

ki

hs im

ONsskhh

ω

ν

τ

ατ

ψ

== ==−

=+−

∑∑∑ ∑

(1.100)

By means of these last two results, we get respectively the annual entrances and

outlays of the fund and we can follow its dynamic development.

1.8. Scenario And Data

The DTNHSM pension fund model presented here has been seen to be

mathematically tractable, though it is a very general model, i.e., able to take into

account a lot of possibilities such as age, seniority, salary line inflation rate,...

and of course with the help of strong computer technology.

However, due to the long time horizon on which we study any pension fund, we

must clearly distinguish between scenario and other relevant data.

A scenario is always represented by a set of data selected by the society or other

authorities to study their influence in the future. There may be some changes in

the rules of the pension scheme or in the future manpower planning of the

society, thus such a scenario is never definitive and at any time it is possible to

adapt it.

But there still remains the basic problem of the statistical estimation of the two-

dimensional NHSM kernel given by (1.19). By (1.21), it suffices to estimate

the

(,)

ij

bst

τ

. To do so in a usual statistical way, we need not only a set of

historical data big enough to accomodate classical statistical estimators but also

relevant data concerning mortality experience tables, survivor distributions, etc.

Furthermore, we also need data concerning salary lines, seniority, inflation rate

and so on.

Here too, let us mention that the model is very useful to project in the future

some consequences related to changes in such data.

For example, we can suppose that the seniority of a new member always begins

at 0, as we did above; we can also suppose some relation between the salary line

and the inflation rate. Indeed it is possible that the increasing of salaries will just

compensate the inflation rate modification or will just give 1% more.

To summarize, we may distinguish between three types of data sets:

(i) partial data given by a selected internal scenario,

(ii) historical data for estimation of the two-dimensional non-homogeneous

semi-Markov kernel,

396 Chapter 9

(iii) economic, financial and demographic data selected as an economic scenario

predicting changes in a global economic environment.

This ability of the model to consider such possibilities is very important as it will

give by simulation, in a completely rigorous way, the fund evolution on the

selected time horizon.

Moreover, we can also proceed to quantitative comparisons for data as

Khorasanee (1994) did.

These potentialities of the GDTNHSM pension fund model show that it

constitutes a realistic approach to pension funding (see Thornton and Wilson

(1993)).

Let us now give more details about the estimation of these data.

1.8.1 Internal Scenario

The first thing is to refer to the rules of the pension scheme. Of course some rules

must be in keeping with the law and some other rules are special for the society

under scrutiny.

At time 0, these are all known and this constitutes the given central internal

scenario, but they may change in the sequel and some changes may be included

and then constitute another internal scenario.

As said above, this type of scenario can also include future manpower planning.

This means that it depends on strategic decisions of the society for its future

development and not only for survival of the pension fund. This strategy will

give the future counting values for all active states.

The total choice at time h constitutes the information

h

Ψ

given in section 1.6.

Remark 1.1 More theoretically the sequence

(

)

,0

h

Ψ

≥ constitutes a filtration

that must be added to a basic probability space

(

)

,,

Ω

ℑℜ but for simplicity we

will not develop these more theoretical aspects here.

1.8.2 Historical Data

The general methodology for estimating the ( , )

ij

bst

τ

is to go from s to s+t with

successive unit steps so that we can find the estimation values in related

statistical tables.

For example, let us consider the element ( , )

ai

bst

τ

where a is one of the 5m −

active states and i the disability state.

Generalized NHSMP models

397

Figure 1.8: times of statistical estimation

By means of classical statistical methods,, functions ( , )

ij

F

st

τ

and ( )

ij

p

s

τ

can be

estimated. Taking into account relation (1.27) it is then possible to estimate the

(,)

ij

Qst

τ

and at last we obtain:

0,

(,)

(,) (, 1) .

ij

ij ij

s

t

bst

Qst Qst s t

τ

ττ

≥

⎧

⎪

=

⎨

−

−<

⎪

⎩

(1.101)

Of course analog formulas may be easily found for the other possible transitions.

Usually the necessary probabilities are available from actuarial tables.

Special attention is necessary to mortality tables as it is now well known that, at

the present time, for most Western countries the mean lifetime increases by three

months every four years! It is clear that this fact already explains the great

anxiety of pension fund managers concerning the financial equilibrium of their

funds.

Once more, this fact shows that the management of pension funds must be

continuous to take into account new reliable information.

1.8.3 Economic Scenario

The simplest assumption, often made in the study of pension funds, is that the

real interest rate is constant over all time horizon. However, we prefer to create

scenarios both for the inflation rate and the market rate on the time horizon.

This is really the most difficult problem, but we can always use Fisher's relation

to find the real interest rate.

It is also possible to find help in e.g. the Wilkie model (Wilkie (1994)) and in

recent evolution in interest rate models in stochastic finance.

Anyway we have to choose and this choice is a factor in our economic scenario.

For the salaries we propose to adopt the following strategy: let us construct a

three-way matrix where one index represents the hierarchical rank and the other

two respectively time and seniority.

We emphasize that the model includes the change in hierarchical ranks with the

transitions between states 1 to m

−

5. So we only need to construct the salary line

for each rank, which can be seen as a sheet of our three-way salary matrix.

Clearly, salary lines for each rank at time 0 are known. This gives the following

matrix:

[]

(,)

ii

mt

τ

=M 1,..., 5, 1,..., , 1,...,

s

im Ktw

τ

=

−= = (1.102)

398 Chapter 9

where

(, )

i

mh

τ

represents the salary rank in state i, (i=1,...,m

−

5) and seniority

τ

at time h. This spreadsheet can be used as a Lexis diagram for each active

member of the fund.

Now we have to adapt an index evolution of the salary to go from row 0 to the

following ones, and so to know the general elements of matrix

i

M .

An important part of an economic scenario is to select indexes

01

' , ' ,..., ' ,....

h

rr r

representing the rates at which we decide the salary will increase.

So we have the following relations:

(, 1) (1 ') (, )

ihi

mh rmh

τ

+

=+

(1.103)

solving the problem of salary line evolution.

The same problem remains for the evolution of pension amount but fortunately in

a simple way.

In fact we can suppose that all the pensions grow with only one index rate. This

means that this evolution is independent of the obtained hierarchical rank of the

member just before getting a pension, but it is time dependent. So we need just to

select a vector

01

" , " ,..., " ,....

h

rr r as another part of our economic scenario.

Let us finally remark that this last choice is also very important from the political

point of view, particularly for relations between governments, trade unions and

management representatives.

Remark 1.2.

(i) All the parameters we introduce in the economic scenario will in fact define

solidarity inside the fund.

Also, large simulations can give a financial measure of matching increases of

lower pensions and decreases or stability of higher pensions.

(ii) The evolution of disability and survivor pensions can be treated in the same

way.

1.9 Usefulness Of The NHSMPFM

(i) As we said in the introduction, to our knowledge, the GDTNHSM pension

fund model presented here gives for the first time a general, rigorous and

tractable framework for studying the time evolution of a fund, taking into

account economic, financial and demographic possibilities of changes.

(ii) Its use as a simulation model will provide a very powerful internal tool, for

society or for legal authorities, to measure the influences of these modifications.

(iii) The presentation of our GDTNHSM model given here is microeconomic, i.e.

related to one society, but its extension as a macroeconomic model to be used by

regional or national authorities or also big insurance companies is

straightforward. The only new problem is that of how to aggregate the data

Generalized NHSMP models

399

necessary to start with the model and it may be judicious not to consider too

many states.

(iv) Let us also mention that, at least in our opinion, in the future more and more

members of pension funds will take out private pensions contracted with

insurance companies. The GDTNHSM model can be useful to fix the amount of

premium to be paid for this supplementary pension in connection with the

"usual" pension amount.

(v) From the computational point of view, it is clear that this model cannot be

used as a simulation model without a good computer environment giving an easy

way to measure the influence of the selected scenario. Interactive software is now

being prepared by the authors in collaboration with some private pension funds.

2. GENERALIZED NON-HOMOGENEOUS

SEMI-MARKOV MODEL FOR MANPOWER

MANAGEMENT

2.1 Introduction

In the preceding section, we have seen that the evolution of salary lines is one of

the most important aspects in the study of the dynamic evolution of pension

funds, so we need to construct a model giving a good forecast of future salary

lines. Of course, this model can also be used for other aims, in manpower

management for example when a firm decides to change the rules of the rank

promotions or its pyramidal job organisation. In this case, it is interesting to

evaluate the cost differences between the new and the former rules in view of

improving the manpower planning of the firm (Bartholomew (1982),

Bartholomew, Forbes, McClean (1991), Vajda (1978).)

In this section, we solve the problem by giving a generalisation of non-

homogeneous semi-Markov processes slightly different from the one used in the

previous section.

This problem was treated by Volpe (1997), Janssen and Manca (1997b) (2002)

Janssen, Manca, Volpe (1997) Manca (2004a) (2004b), using different kinds of

stochastic models.

Let us remember that the evolution of salary lines has a strong influence on the

behaviour of pension funds.

Indeed, if the fund is with defined contribution (that means the pension is a

function of the paid contributions) then it is important to know the expected

evolution of the salary lines to know what the entrances into the fund will be for

each member and in this way to compute the expected pensions for the working

people in the fund.

Furthermore if the fund is with defined benefit (performance) (that means the

pension is a function of the last salaries), it is important to know the evolution of

Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability

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