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

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400 Chapter 9

salary from the point of view of fund entrances, but in this case the pension is

directly a function of the last salaries paid to the members, so it is possible to

understand the relevance of salary lines in this case.

Here, we face the problem of making a generalisation of a model useful to

forecast the development of pension funds (Janssen and Manca (1997a)),

presented in the preceding section, which is an extension of the discrete time

non-homogeneous semi-Markov processes (see for example Janssen and De

Dominicis, (1984)) presented in Chapter 4.

Markov processes and semi-Markov processes were already used in the man-

power planning problems, see for example Bartholomew (1982).

2.2 GDTNHSMP For The Evolution of Salary Lines

A model for salary line evolution uses a state space with m

−

1 possible ranks in

the active state and an absorbing state m representing the state of leaving the job

for retirement, for death or for any other reason.

As usual, let us denote this state space by I with

{

}

1,...,Im= . (2.1)

Let us introduce now a discrete time scale: we observe the state at times

0,1,2,...,k,....and so, clearly, at any time k, each member of the company is in one

and only one of these m states.

A quadruple of random variables

(

)

,, ,

nn n n

TH K is now introduced, where

represents the state of the member at transition n,

T the time in which there is

the transition n,

S the seniority of the member at transition n and

K the age of

the member at transition n.

Remark 2.1 Here seniority means the effective number of years that the member

is in the company.

The seniority

and the age

are usually defined by the relations:

nnnn

SSTT

+− (2.2)

nnnn

KKTT

+− (2.3)

Now, at each state transition, the considered member of the company is

characterised by the quadruple

(

)

,,,

nn n n

TSK . (2.4)

This (J,T,S,K) process may in fact also be considered as a three-dimensional non-

homogeneous Markov renewal process (J,(T-S),(T-K)) with kernel

111 1

,, ,

(,) .

,, ,

nnn n

nn n n

jT tS t sK t s

Qst P

JiTsS K

μτ

τμ

+++ +

=≤≤+− ≤+−

⎛⎞

⎜⎟

== = =

⎝⎠

(2.5)

Generalized NHSMP models

401

Now, we define the matrix B having as general element:

111 1

,, ,

(,)

,, ,

nnn n

nn n n

jT tS t sK t s

bst P

JiTsS K

μτ

τμ

+++ +

==+− =+−

⎛⎞

⎜⎟

== = =

⎝⎠

(2.6)

so that:

0, ,

(,)

(,) (, 1) ,

ij ij

bst

Qst Qst s t

μτ

μτ μτ

≥

⎧

⎪

⎨

−

−<

⎪

⎩

(2.7)

or equivalently:

(,) (, ).

ij ij

Qst bsh

μτ μτ

∑

(2.8)

Similarly, it is possible to write:

()

() , , , ,

ij n n n n n

ps PJ jJ iT sS K

μτ

======

(2.9)

representing the general element of the kernel of the process.

We also have:

() (, ),

ij ij

ps Qs

μτ μτ

∞ (2.10)

(,) (,)

iij

Hst Qst

μτ μτ

∑

, (2.11)

where:

(

)

(,) , , , ,

innnnn

Hst PT tJ iT sS K

μτ

=≤==== (2.12)

(, ) 1,

μτ

∞

= (2.13)

(,)

(,) ,

()

Qst

Fst

μτ

= (2.14)

where

(

)

(,) , , , ,

ij n n n n n n

Fst PT tJ iJ jT sS K

μτ

=≤=====, (2.15)

i.e. the sojourn time conditional distribution entering in state i at transition n

with a seniority

at age

Finally, it is also possible to relate the associated semi-Markov process in

keeping with the three-dimensional non-homogeneous Markov renewal process

(J,(T-S),(T-K)) noted as

(

)

;0, 0, 0

μτ

τμ

≥≥ ≥ (2.16)

having as transition probabilities:

(

)

,,,

(,) .

ts ts

ij t s

tP ZjZi

μτ μ τ μτ

+− +−

== (2.17)

These transition probabilities satisfy the following system:

,, ,,

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

ij ij i ih hj

tHstbs t

μτ μτ μτ μ θ τ θ

δθφθ

+− +−

=− +

∑∑

. (2.18)

402 Chapter 9

It should be emphasized that the probabilities (2.9) are related to the generalized

non-homogeneous Markov process embedded in the generalized non-

homogeneous semi-Markov process (2.16).

Furthermore

(,)

μτ

represents the probability of staying in state j at time t

once the process was in state i at time s with a seniority

at age

2.3 The GDTNHSMRWP For Reserve Structure

To apply this model to the computation of present value of the current salary

cost, it is necessary to consider a reward structure connected to the semi-Markov

process.

Clearly the salary value changes because of time and seniority but it does not

change as a function of age, however the probability of being promoted and the

transition to the absorbing state can be related to age.

For this reason it is necessary to write the reward equation taking into account all

these aspects. The equation will be slightly different from relation (1.39) because

time is the main temporal variable.

In this light, relation (1.39) becomes:

()

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

(, ) ( )1

(, ) ( ,)(1 ) .

ii i

ss s

Vst Hst r

bs s r

bs V t r

μτ μτ τ θ

μτ τ ϑ

θβ ϑ

μτ μ θ τ θ θ

ββ

θβ

ψθ

θψϑ

θθ

−

+−

−

== =

+− +− −

⎛⎞

=− +

⎜⎟

⎝⎠

+++

∑

∑∑ ∑

∑∑

(2.19)

In the case of the salary cost evaluation model, the financial meaning of the

equations (2.19) can be given.

The

(,)

Vst

μτ

are the discounted expected values of the salaries that were paid

from s to t when an employee was in rank i at time s with a seniority

and an

age

. These formulas are compounded from the following parts:

()

(1 ( , )) ( ) 1

Hst r

μτ τ θ

ψθ

−

+−

⎛⎞

−+

⎜⎟

⎝⎠

∑

, (2.20)

()

(, ) ( )1

bs s r

μτ τ ϑ

θβ ϑ

θψϑ

−

== =

∑∑ ∑

, (2.21)

(, ) ( ,)(1 ) .

ss s

bs V t r

τμθτθ θ

ββ

θβ

θθ

−+− −

∑∑

(2.22)

In relation (2.20), the term

(1 ( , ))

Hst

μτ

− represents the probability of

remaining in state i once a member has arrived at time s with a seniority

and

Generalized NHSMP models

403

an age

. So this member for each time period gets the salary ( )

τθ

+−

depending on the time and seniority and so (2.20) represents the expected related

value.

Relation (2.21) gives the expected present value of the salaries that a member

arrived at i at time s with seniority

and age

got in this state before it

changed.

Finally, the expression (2.22) represents the expected value of the salaries that a

member, having arrived in state i at time s with a seniority

and an age

and

having changed his situation at time

, has to get in the new state.

These values are paid at time

−

, so it is necessary to discount them.

In this way, the probabilities of changing states differ because of seniority, but it

is also possible to consider different rewards as a function of different seniorities.

2.4 Reserve Structure With Stochastic Interest Rate

A stochastic interest rate structure is introduced in this part.

The structure will be constructed by means of DTNHSMP as in Chapter 6.

In this case the state of the process will be:

{

}

,, ,

rr r= … (2.23)

where the

r represents all the possible implied stochastic interest rates and k

gives the number of the implied interest rates.

Now

),( ts

represents the probability that at time t the implied interest rate

will be

r , given that the implied interest rate was

at time s and

(, )

represents the mean discounting factor for a time (h

−

s) given that at time s the

interest rate was

r .

Now, the evolution equation (2.19) becomes:

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

(, ) ( ) (, )

(, ) ( ,) (, ),

iii

Vst Hst s

bs s ss

bs V t s

μτ ε μτ τ θ

μτ τ ϑ

βε

θβ ϑ

μτ μ θ τ θ ε

ββε

θβ

θν θ

θψϑν ϑ

θθνθ

+−

−

== =

+− +−

⎛⎞

=−

⎜⎟

⎝⎠

+++

∑

∑∑ ∑

∑∑

(2.24)

assuming that

will be the known interest rate at time s.

From (7.8) of Chapter 10 it results that:

(,) (,) (,)

Vt s Vt

μτ ε μτ

βεβ

φθ θ

∑

(2.25)

so that we obtain the following results:

404 Chapter 9

(,) (,) (0,)

Vst V st s

μν μν η

ηε

∑

(2.26)

where r

is the known rate of interest at time 0.

The solution of the GDTNHSMP and the GDTNHSMRWP can be obtained by a

backward substitution process similar to the one described in section 1 for the

pension model.

2.5 The Dynamics Of Population Evolution

Also the population evolution can be studied by means of the same relations

given in section 1.6.

By means of relations given in that part, it is possible to evaluate the number of

people in the ranks of a given company and furthermore taking into account the

possibility of new additions to the workforce.

We report only the definitions because of the different uses of temporal variables

as the related formulas can be obtained in the same way as given in the previous

part.

()

μτ

: the number of members present in the company in rank i at time s with

seniority

and age

()

μτ

: the number of members present in the company at time s with seniority

and age

()

: the number of members present in the company in rank i at time s with

seniority

()

: the number of members present in the company at time s with seniority

()

s : the number of members present in the company in rank i at time s,

()

s : the total number of members present in the company at time s.

Here too, we can introduce a scenario

concerning the number of active people

the firm wants to have at any time in each rank; the values at time 0 being

known:

(), 0,1, ,

ss T

… ;

1, 2, , 1im

−…

, (2.27)

where, as usual, T is the fixed time horizon.

All the relations given previously hold and will not be repeated.

Taking into account the different meanings of the temporal variable and the

given scenario, we can evaluate the

()

μτ

values by means of the relations

given in section 1.6. These values represent the mean number of people present

at time s with seniority

and age

in state i.

Generalized NHSMP models

405

Then we get the mean charge of future work at any time t that can be predicted at

present time s (s<t):

,,,

(,) (,) ().

iiji

st st N s

μτ μτ μτ

−

∑

(2.28)

The seniority and the age are also known and it is clear that all these results are

fundamental for the future labour policy of the firm.

2.6 The Computation Of Salary Cost Present Value

Once the systems (2.18) and (2.19) are solved, given a person that is at time s

with seniority

and age

in state i, the probabilities:

,, , ,

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

ij ij ij ij

ss ss ss sw j m

μτ μτ μτ μτ

φφ φ φ

++ + =−… (2.29)

are known.

They represent the probabilities of being in rank j after one year, two years and

so on; these probabilities will be equal to 0 if they represent some impossible

cases.

By means of results (2.28) and (2.29), we can obtain all the

(,)

μτ

that are

necessary.

Furthermore for each year it is possible to compute the salary line, if the salary

that will be paid to a person with a given seniority in a given rank for each year

is known, which in general is the case.

If we suppose that the rates of salary evolution are known for the given company

(we suppose we are in a given economic scenario), then it is possible to

construct, for each rank, a matrix the elements of which will give the expected

salary in the firm for each seniority and each year in our time horizon.

In this way we will obtain a three-dimensional matrix with as elements:

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

jm Ks w

ψτ

=−= =……… (2.30)

representing the salary in state j, of a person with seniority

at time s in the

given scenario.

The mean salary line of a given situation in the time horizon, as specified in

Janssen Manca Volpe (1997), will be given by:

(,) ();

(,) ; 1, , .

(,) ();

ij j s

st t t s K

Ast t s w

st t t s K

μτ τ

μτ

φψτ

−

+−

−

⎧

+−≤

⎪

==+

⎨

⎪

+−>

⎪

⎩

∑

… (2.31)

To apply the model it is necessary to know all the

(,)

bst

μτ

. To obtain these

probabilities it is necessary to know all the functions

(,)

Qst

μτ

, obtained, taking

into account relation (2.12), knowing

(,)

μτ

and

()

μτ

406 Chapter 9

Though not easy to be found, these data may be obtained by observation and

there are some models that can be used (see for example De Dominicis and

Manca (1984a)).

Once the data are known, it is possible to solve (2.18) and (2.19) and to obtain

the results.

Finally we get the following formula:

,0 ,0

(0, ) (0, )

(, ) (, )(1 )

smwa

mwa

AVwNw

Vsw Nsw r

μτ μτ

τμα

μμ

μα

===

−

===

∑∑∑

(2.32)

giving the present value of salary cost, where the first part represents the cost for

the people that are in the company at time 0 and the second part the salary cost of

the people that will enter into the company.

As already specified, the number of the new employees is a function of the

scenario in which the model develops.

Finally, let us remark that the salary line forecasting problem can be solved also

by means of other models based on the generalizations of the Bernoulli stochastic

process (Manca (2004a)).

However in this case the probabilities

(, 1)

μτ

become inputs for the

generalized binomial process.

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