Time-series simulation of the income of
Insurance-Brokers, in Greece
Prof. Petros Kiochos* Dr.
Costas Kyritsis**
Abstract
In this paper with title “A time series model of the income
of insurance-Brokers, in Greece.”, we study quantitative models of the income of agents and brokers of insurance
policies. It is often said that their income grows in spectacular rates. We
describe quantitative models that are convolution filters. The numerical
calculations are best done with the application of the Discrete Fourier
Transform. We give also computer aided numerical simulation based on data of
the Greek Market which shows that the sales of the first years have significant
consequences to the income of later years.(JEL:G22,C3)
Key words: Insurance , commissions, Time Series, Discrete
Fourier Transform, Greek Insurance Market .
Introduction
In many situations, in which sales force is hired in agencies of
insurance companies, is claimed that the commissions of insurance products,
make a continually increasing income. In
each year is accumulated income by commissions from sales of previous years.
This is much better than the income of usual dealers of commercial products.
Would it be possible to give a
precise quantitative model of the growth
of the income by the commissions ? Is
such a model similar to other situations in finance and econometrics? Can we
have numerical estimations and computer
simulation , of the way that the income grows?
In this paper we describe a
deterministic model It is remarkable that such a common and familiar situation in
marketing of insurance products, involves the Discrete Fourier Transform.
We give a numerical simulation
with computer experiment based on data and
empirical constants from the
Greek insurance market .The experiment shows that the income of later years is
very sensitive to the sales of the first years.
1.The portfolio of insurance brokers and the way it grows.
As it is known insurance products are classified in to life
and wealth insurance policies . The former include health insurance and pensions
and the latter include car insurance
and of fixed or current, tangible (or
intangible) assets .
The commissions of pensions are not restricted to the first
year. They begin with a maximum percentage, that is gradually reduced for the
next five years and then it remains constant for all next years, till the end of the contract. In
this way we have an accumulation of income, because of the sales of the
previous years . It is almost like an arithmetic progression . Nevertheless the
increase of every year is not constant and it depends on the history of the sales of the past
years.
In the next tables we can see
the sequences of commissions of various
insurance products .We take as standard example the next sequence of
commissions:
Total Commissions of a portfolio with life insurance 60% and tangible assets 40%. Commission sequence for each year for life insurance CLIn =50%,15%,10% 5%,2%0% ,and for tangible assets CTA =22%. The formula to compute the total commissions per year is:
Cn =0.6*( CLIn )+0.4*(CTA) (1)
Then
the total commissions percentage per year for such a portfolio is the next
sequence
Table1
|
C1 |
C2 |
C3 |
C4 |
C5 |
C6 |
C7 |
C6+n |
|
0.388 |
0.178 |
0.148 |
0.118 |
0.1 |
0.88 |
0.88 |
0.88 |
Commission
rates of the insurance portfolio
2. The discrete Fourier transform,
convolution filters, and time series
model of the income of insurance brokers.
Let
us formulate in mathematical terms the
way the portfolio of an insurance broker grows
.
Let us
denote by cn the total commission percentage
after n-years that comes from an amount
of sales of the insurance broker at the
first year. Let us also denote by dn the remaining percentage of sales after
n –years of his initial amount of sales in the first year. We assume that these
sequences are constant and repeat for any new sales every year. And finally,
let us denote by un the amount of sales (production) at the
nth year. Then the total production sn at
the nth year is calculated by
sn= (2)
And the total income rn at the year n of the insurance broker is
rn=
(3)
If we denote by vn =
cn dn , then the last formula is written as
rn=
(4)
Both
formulas (2) ,(4) are recognised as discrete
convolutions of the sequences un dn
and un vn respectively . It is defined a linear convolution filter .
This
suggests, that the best way to compute the rate of return, is to use the
Discrete Fourier Transform DFT (or the Fast Fourier Transform FFT) . For the
definition of the DFT see (Firth J.M 1992) .One of the basic properties is that
the Fourier transform sends the convolution to the product. So a simple way to
compute the sequence rn is to transform the sequences un
vn with the DFT and then multiply them term by
term. By applying the inverse DFT we get back to the sequence rn.
We remark that «motivation-prim-bonuses » functions, between
commissions and volume of the portfolio production, may make the equations non-linear
3. Random coeficients time-series stochastic process version
In the previous formulations,
we considered the sequence sn without
reference to any stochastic process . As the phenomenon is random rather than
deterministic, we may consider the sequences un , sn ,dn rn as
time series. There is therefore a probability measure on the algebra of paths
and at each time n is defined a random
variable which is the un , sn ,dn rn. The equations that we described in
the previous paragraphs, are of the mean values of the random variables un
, sn ,dn
rn
(regression path) ,while there is an additional random or
innovation term εn , (noise) that we put
usually to be Brownian motion or white noise.
The equation (4) becomes a random coefficient time series :
(5)
It is
a random coefficients non-linear model (see (Tong H.Owell
1990)).
4.Computer aided
simulation for the Greek market .
The formulas of the previous paragraphs, permit numerical computations and
simulation. To simulate a normal random variable we make use of the Box-Muller
method. The next program in visual basic produces Gaussian random variables.
Sub BoxMulersimulationofnormalrandvar()
Dim M As Integer
Dim I As Integer
Dim U As Double
Dim R As Double
Dim p2 As Double
M = 1000
Randomize
p2 = 2 * 3.14159265
For I = 1 To M
R = Sqr(-2
* Log(Rnd))
U = Rnd
Workbooks("wind.xls").Worksheets("sheet2").Cells(I,
1).Value = R * Sin(p2 * U)
Workbooks("wind.xls").Worksheets("sheet2").Cells(I,
2).Value = R * Cos(p2 * U)
Next I
End Sub
In addition to the numerical data of the commissions that we can see in
the tables of the paragraph 2, we assume
that the perseverance of the contracts
in the sequence of years is the following sequence .
Table 2
|
D1 |
D2 |
D3 |
D4 |
D4+n |
|
100% |
80% |
70% |
60% |
60% |
Perseverance rates of the insurance
portfolio
These data are empirical for the Greek insurance market and are
suggested by insurance business consulting companies, in
Based on the previous numerical data, we can apply a numerical
simulation for 10 years and for
10,000,0000 of sales each year .
We estimate, at first, the
product of the commission sequence cn
and the perseverance sequence dn
.
Table 3
|
V1 |
V2 |
V3 |
V4 |
V5 |
V6 |
V7 |
v6+n |
|
0.388 |
0.1424 |
0.1036 |
0.0424 |
0.06 |
0.0528 |
0.0528 |
0.0528 |
Coefficients of the convolution filter of the income
from the insurance portfolio
And,
finally, we estimate through convolution as in formula (4), the average rate of
return, for a sequence of 10 years, in the next table:
Table 5
|
R1 |
R2 |
R3 |
R4 |
R5 |
|
|||||
|
3880000 |
5304000 |
6340000 |
6764000 |
7364000 |
|
|||||
|
R6 |
R7 |
R8 |
R9 |
R10 |
||||||
|
7892000 |
8420000 |
8948000 |
9476000 |
10004000 |
||||||
|
R10+k |
|
||||||||||
|
10004000+ k528,00 |
|
||||||||||
The rate of return from the insurance policies
portfolio for 10 or more years.
After
the first 10 years the sequence grows constantly by an amount of 528,000 .
The previous path is the regression
path of a random time series that has
Gaussian and i.i.d (independent ,identically
distributed) random term εn .
Using the previous Box-Muller method to produce with computer experiment,
normal random term we result to the next
random sample paths εn.
Figure 1
The
random sample paths (series1,2,4,5)and the regression path (series 3) for 10
years of the rate of return of the insurance policies portfolio.
The
regression path (path of average values) appears in this chart as the series
3.The series 1,2,4,5 are random sample paths derived from the simulation. The
x-axis is years and the y-axis is drachmas.
Although actuarial mathematics, are deductive statistics, we can suggest techniques of inductive statistics to infer the stochastic model for many different brokers, from the data (sample function) of one broker.
From
the diagram we notice that the final rate of return from commissions after 10
years is very sensitive to the random sales of the first years. The reason is
that the income of brokers grows recursively over the income of the first
years.
Conclusions
The previous analysis can be summarized with the next conclusions:
a)
The income from the sales of insurance brokers or insurance agents
increases much faster than the income of ordinary dealers of industrial
products in Marketing.
b)
The correct model is not that of an arithmetic
progression but of convolution filters. Therefore the Fast Fourier Transform is
very convenient for computations.
c)
The simulation with normal errors, shows that the
sales of the first years have significant consequences for later years.
References.
1.Berndt
E.R.1991 The practice of
Econometrics classic and contemporary Addison
Wesley .
2.Bowers ,N Gerber H.,Hickman J., Jones D.,
Nesbitt C.1986,Actuarial Mathematics
The Society of Actuaries ,Itasca
,Illinois.
3.Bracewell R.N. The Fourier Transform and its
applications McGraw
Hill
2nd edition
4.Feller W.1957, Probability Theory
and its Applications Vol
1 , Wiley.
5.Firth
J.M 1992, Discrete transforms Chapman and Hall
6.
7.Karlin S.,
8.Kiochos
P.1991.Techniques of Actuarial Studies
(In Greek)
Contemporary
Publications
9.Kiochos
P.1996. Actuarial
mathematics Interbooks,
10.Kiochos
P.1996. Methodology of
Actuarial Studies Interbooks.Athens
11.Kiochos P.1997. Methodology of the research Stamoulis P.Athens.
12.Kiochos P.1993. Statistics .Interbooks
13.Kloeden
P.E.Platen E.Schurtz
H.1997. Numerical Solution of SDE through Computer Experiments Springer.
14.Koopmans L.H.. The spectral Analysis of time series .Academic Press 1995.
15.Levy
H Lessman F.1992].Finite Difference Equations
16.Morgan
B.J.T.Elements of simulation Chapman-Hall 1984
17.Mood
A.M,Graybill F.A.Boes D.C
1974. Introduction to the theory of Statistics McGraw-Hill
18.Oksendal
B.1995. Stochastic Differential Equations Springer
.
19.Petauton
P.1991. Theorie et Pratique
de l’Assurance Vie, Dumond Paris .
20.Tong
H.Owell 1990.Non-linear Time Series A Dynamic System Approach Clarendon
Press .