Credibility Premium Calculation in Motor Third-Party Liability Insurance

Abstract: - Improving the quality of premium calculation methods is an effective factor in reducing the insurance technical risk of insurance company. With the development of the competitive insurance market in motor third-party liability insurance in former socialist countries the Bühlmann-Straub credibility model has a wide range of possibilities to be used. In this article we deal with its application based on real data of five Slovak insurance companies.

Keyword: -Credibility premium, Motor third-party liability insurance, Empirical Bayes credibility model II, Credibility factor, Credibility pure premium

1 Introduction

The original only one state insurance company in the Czech and in the Slovak Republics had a monopoly position in motor third-party liability insurance until 2005. With the emergence of a competitive market in this important type of insurance a typical situation for the use of empirical Bayes credibility models was created.

Empirical Bayes credibility theory is the collective name for the vast literature which has developed since Bühlmann and Straub`s (1970). Although this model is a basis for other more specific models such as hierarchical, multidimensional or regression credibility models, in this article we deal with one-dimensional Bühlmann-Straub credibility model and its application.

The problem for the insurance company is to determine the pure premium for the coming year and there are two extreme positions:

1. the insurance company do not has any own data or could decide to ignore the past data from the policy itself and base the pure premium equal to µ on the almost certainly larger amount of data from similar policies;

2. the insurance company could decide to ignore the data from similar policies and charge a pure

premium equal to x based solely on past data from the policy itself.

A credibility premium represents a compromise between the two extreme situations given above. The credibility premium formula is:

(

Z

)

µ

x Z

Pc = + 1− (1)

where Z is the credibility factor. This factor is the weight put on the data from the risk itself and depends on the amount of past data available from the policy itself. The value of Z is between zero and one and should increase from year to year as more data are obtained.

2 The Empirical Bayes Credibility

Model II (EBCT II)

Our problem is to estimate a pure premium for a risk, given some data. Let Y1,Y2,...,Yj,...,Yn denote the aggregate claims in successive year from this risk and be a corresponding sequence of known constants

n

j P

P P

P1, 2,..., ,..., . Constant Pj, j=1,2,...,n is interpreted as a measure of „the amount of business“ in the j-th year, for example the premium income or the number of policies issued in year j. Then

Credibility Premium Calculation in Motor Third-Party

Liability Insurance

BOHDAN LINDA, JANA KUBANOVÁ

Department of Mathematics and Quantitative Methods

University of Pardubice

Studentská 95, 532 10 Pardubice

CZECH REPUBLIC

j j j

P Y

X = , j=1,2,...,n (2) are standardised variables by removing the effect of different business levels.

We assume that the distribution of each Xj depends on a fixed, but unknown parameter θ.In accordance with Bayesian approach to estimation we consider θ as random variable.

Assumptions of the EBCT II model:

1. Variables X1/θ,X2 /θ,...,Xn /θ are independent but not necessarily identically distributed,

2. E

(

Xj θ

)

and D

(

Xj θ

)

⋅Pj does not depend on j.

With these two assumptions we can define:

( )

θ E

(

Xj θ

)

m = (3)

( )

D

(

Xj

)

Pj

s2 θ = θ ⋅

(4)

2.1 Derivation of the Credibility Premium

We define a pure premium in year j, j=1,2,...,n, as

(

Y θ

)

=P ⋅E

(

X θ

)

=P ⋅m

( )

θ

E _{j j j j} (5)

Because of P_j is known, we need estimate m

( )

θ as a linear combination

n nX a X a X a

a0 + 1 1 + 2 2 +...+ (6) with constants a0,a1,a2,...,an which minimize:

( )

[

]

(

2

)

2 2 1 1

0 a X a X ... anXn

a m E

Q= θ − − − − −

By solving equations

0 ... , 0 , 0 1 0 = ∂ ∂ = ∂ ∂ = ∂ ∂ n a Q a Q a Q

and using the relations (proof in Waters, 1994)

( )

(

)

(

( )

)

[

(

( )

)

]

2

θ +

θ =

θ D m E m

m X

E _k

(

)

(

( )

)

[

(

( )

)

]

2

θ θ E m m

D X X

E j k = +

( )

2

(

2

( )

)

(

( )

)

[

(

( )

)

]

2 θ θ

θ P D m E m

s E X

E j = j + +

we get a0 and aj, j=1,2,...,n in the forms

( )

(

)

φ + φ ⋅ θ =

∑

= n k k P m E a 1

0 (7)

φ + =

∑

= n k k j j P P a 1

, j=1,2,...,n (8)

where

( )

(

)

( )

(

θ

)

θ = φ m D s E 2 .

Putting (7) and (8) into (6), the estimate of the pure premium per unit of volume of risk is

( )

(

)

(

( )

)

( )

(

)

( )

(

)

( )

(

)

( )

(

θ

)

θ + + θ θ ⋅ θ = θ

∑

∑

= = m D s E P Y m D s E m E m E n j j n j j 2 1 1 2

X (9)

which can be rewritten as credibility premium

( )

(

mθ

)

=ZX +

(

−Z

)

E

(

m

( )

θ

)

E /X 1 (10)

where

∑

∑

= = = n j j n j j j P X P X 1 1 ,

( )

(

)

( )

(

θ

)

θ + =

∑

∑

= = m D s E P P Z n j j n j j 2 1 1 .

2.2 Parameter’s Estimation

To calculate the pure premium for i-th risk we need to estimateE

(

m

( )

θi

)

, D

(

m

( )

θi

)

, E

(

s

( )

θi

)

2

from suita-ble set of data.

For the purposes of these parameters estimation we regard the particular i-th risk as one of a set of N risks. We assume that for each of the N risks we have n observed values Yij of the aggregate claims in past years j=1,2,...,n. Furthermore, we know the values of weights Pij, i=1,2,...,N, j=1,2,...,n.

We assume that for each i=1,2,...,N the distributions of

ij ij ij

P Y

X = depends on the unknown parameter θi, which is fixed for each j=1,2,...,n.

Variables Xi1 θi,Xi2 θi,...,Xin θi for each

N

i=1,2,..., are independent, but not necessarily identically distributed.

(

Xij i

)

m

( )

i

E θ = θ

(

)

( )

ij i i

ij

P s X

D θ θ

2

=

are the same for all i and E

(

m

( )

θ_i

)

, D

(

m

( )

θ_i

)

,

( )

(

s i

)

E 2 θ

are independent of i so we shall to denote them simply as E

(

m

( )

θ

)

D

(

m

( )

θ

)

(

2

( )

θ

)

s

E .

We will use the following notation: ,

1

∑

=

= n

j ij

i P

P ,

1

∑

=

= N

i i

P P

∑

=

      ₋ ⋅ −

= N

i

i i

P P P

Nn P

1 *

1 1

1

(11)

The proposed estimators are shown below:

( )

(

m

)

X

estE θ = (12)

( )

(

)

₍

₎

_∑∑

(

)

= =

− −

=

θ N

i n

j

i ij

ij X X

P n

N s

estE

1 1

2 2

1 1

(13)

( )

(

mθ

)

=

estD (14)

(

)

_{( )}

(

)

   

 

− −

− − −

=

∑∑

∑∑

= =

= =

N i

n j

i ij ij N

i n j

ij

ij P X X

n N X X P Nn

P 1 1

2 1 1

2 *

1 1 1

1 1

The credibility pure premium for i-th risk is

( )

(

m

)

Z X

(

Z

)

X

E θ /X = i i+ 1− i (15)

where credibility factors Zi are different for each risk i, i=1,2,...,N, calculated by

( )

(

)

( )

(

θ

)

θ +

=

m D

s E P

P Z

i i

i 2 (16)

3 Application of the EBCT II Model

in Motor Third-Party Liability

Insurance

The data below (Table 1) show the aggregate claims for motor third-party liability insurance in six Slovak insurance company in years 2006-2010.

Table 1 Total claims Yij (in thousands of €)

Insurance Years j

comp. i 2006 2007 2008 2009 2010 Allianz 45,82 43,75 51,48 52,21 47,4 ČSOB 4,88 3,81 2,66 2,31 2,19 Generali 2,44 3,08 12,60 13,1 17,19 KOOP 43,40 55,10 67,74 72,09 65,91 Uniqa 2,18 3,66 6,01 7,99 9,21 Wusten 4,88 4,40 4,88 2,4 3,98

Source: Annual Reports 2006-10, Slovak Insurance Assoc.

Table 2 contains the numbers of policies Pij for this type of insurance business for each risk i, i=1,2,...,6 and each year j, j=1,2,...,5. These constants we will use as the weights to calculate the empirical Bayes premiums for all the companies in the coming year base on the data in Table 1.

Table 2 Number of underwriting policies Pij (in

thousands)

Insurance Years j

comp. i 2006 _{2007 2008 2009 2010}

Allianz 731,0 736,4 769,1 745,2 701,7 ČSOB 100,9 67,8 56,0 51,6 55,8 Generali 59,5 57,2 192,0 182,1 185,7 KOOP 442,5 518,6 603,6 595,5 608,8 Uniqa 35,2 59,2 92,8 106,9 114,3 Wusten 52,9 68,2 78,1 41,8 106,4

Source: Annual Reports 2006-2010, Slovak Insurance

Association

Supporting calculations based on the data in Table 2 and in Table 4 of the standardized variables Xij by

relation (2) contains Table 3.

Table 3 Supporting calculations

Insurance

company i Pi Xi

Allianz 3683394 0,065324

ČSOB 332021 0,047201

Generali 676513 0,065000

KOOP 2768979 0,109194

Uniqa 408467 0,068773

Wusten 347482 0,062847

Source: Own calculations

Totals of columns in Table 3 we get characteristics

∑

=

= = 6

1

856 216 8

i i

P

P and X =0,0697231.

Table 4 Standardized values of Xij Insurance Years j

comp. i 2006 2007 2008 2009 2010 Allianz 0,063 0,059 0,067 0,07 0,068 ČSOB 0,048 0,056 0,047 0,045 0,039 Generali 0,041 0,054 0,066 0,072 0,093 KOOP 0,098 0,106 0,112 0,121 0,108 Uniqa 0,062 0,062 0,065 0,075 0,081 Wusten 0,092 0,064 0,062 0,058 0,037

In Table 5 there are values Pij

(

Xij−Xi

)

2 for each combination of i=1,2,...,6, j=1,2,...,5. Last column of this table contains the sums Ai of these

values for each row.

Table 5

Values Pij

(

Xij −Xi

)

2 and Ai

i 2006 2007 2008 2009 2010 Ai 1 5,14 25,74 2,00 16,71 3,44 53,03

2 0,14 5,43 0,00 0,29 3,60 9,47

3 34,02 7,17 0,07 8,77 141,18 191,21 4 54,70 4,28 5,57 83,99 0,52 149,06 5 1,57 2,89 1,52 3,75 15,93 25,66 6 46,04 0,17 0,01 1,19 68,73 116,14

Source: Own calculations

As the sum of the values

Ai

in last column we get

(

)

544,58

6

1 6

1 5

1

2

= =

−

∑

∑∑

=

= = i

i

i j

i ij

ij X X A

P

Table 6

Supporting

values

Insurance

comp. i _

   

−

P P 1

P i

i

Allianz 2032228

ČSOB 318605

Generali 620814

KOOP 1835867

Uniqa 388162

Wusten 332787

Source: Own calculations

Table 6 contains the

supporting

values to

calculate

P*

by (11). Using the sum of the values

in last columns of table 6, that is

5 528 464,

we

get

184282,1 5528464

29 1 1

1 1

1

* _{= ⋅ =}

     ₋ ⋅ −

=

∑

=

N

i

i i

P P P

Nn P

Table 7

Values P_ij

(

X_ij −X

)

2and Bi

i 2006 2007 2008 2009 2010 Bi

1 36,3 78,3 6,0 0,1 3,3 124,0

2 45,9 12,5 27,7 32,0 52,1 170,2 3 48,8 14,5 3,3 0,9 97,0 164,4 4 355,8 694,6 1090,7 1570,0 904,6 4615,6

5 2,1 3,7 2,3 2,6 13,5 24,2

6 27,1 1,9 4,1 6,2 111,0 150,3

Source: Own calculations

In Table 7 there are values Pij

(

Xij −X

)

2 and the sums Bi of these values for each row. Total sum

of values in table 7 will be

(

)

5248,7

6

1 5

1

2

= −

∑∑

= =

i j

ij

ij X X

P

Now we can estimate parameters of EBCT II model by relations (12), (13), (14):

( )

(

m

)

= X =0,0697231 estE θ

( )

(

)

₍

₎

*544,58 22,69

1 5 * 6

1

2 ₌

− =

θ

s estE

( )

(

)

000859 , 0

58 , 544 24

1 7 , 5248 29

1 1 , 184282

1

=

=   

 _{⋅ − ⋅}

⋅ =

θ

m estD

Table 8 contains for each insurance company values of factor credibility and value of credibility pure premium per unite of risk, which we have calculated by relations (16) and (15).

Table 8

Values of credibility pure premiums

Insurance

company i

Credibility factor Zi

Credibility pure premium

(in €) Allianz 0,992880 65,36 ČSOB 0,926303 48,86 Generali 0,962421 65,18 Koop 0,990550 108,82 UNIQA 0,939258 68,83 Wüstenrot 0,929351 63,33

Source: Own calculations

6 Conclusions

Formula (15) is a simple matter to calculate the empirical Credibility premium for each insurer if we have estimatedE

(

m

( )

θ

)

, D

(

m

( )

θ

)

,

(

2

( )

θ

)

s

E . The

credibility factor is a measure of how much reliance we are prepared to place on data from the risk itself. It is an increasing function of the number of years n, for which data are available and is asymptotically equal to 1.

Estimated

(

2

( )

θ

)

s

E is a measure of the mean of the variance of the data from each risk and D

(

m

( )

θ

)

is a measure of variance between risks. The larger are these values, the less reliable are data from risk itself or from other risks.

References:

[1] Bühlmann, H., Straub, E, Glaubwürdigkeit für Schadensätze. Mitteilungen der Vereinigung Schweizerischer Versicherungs-mathematiker 70, 1970, pp.111–133.

[2] Bühlmann,H., Gisler, A., Course in Credibility Theory and its Applications, Berlin: Springer, 2005.

[3] Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M., Modern Actuarial Risk Theory, Boston: Kluwer Academic Publishers, 2001.

[4] Gerber, H. U., An Introduction to Mathematical Risk Theory, S.S. Huebner Foundation for Insurance Education, Monograph number 8, Richard D. Irwin Inc. Homewood, Illinois 1979. [5] Herzog, T. N., Introduction to credibility theory.

3rd. edition, Winsted, Connecticut: ACTEX Publications, 1999.

[6] Pacáková, V., Aplikovaná poistná štatistika (Applied Insurance Statistics), Bratislava: Iura Edition, 2004.

[7] Pacáková, V., Šoltés, E., Šoltésová, T.,Kredibilný odhad škodovej frekvencie (Credibility Estimation of Claim Frequency). Ekonomie a Management E+M, XII, 2/2009, pp. 122-126. [8] Straub, E., Non-Life Insurance Mathematics,

Zőrich: Springer-Verlag, 1988.

[9] Šoltés, E, Pacáková, V., Šoltésová, T., Vybrané kredibilné regresné modely v havarijnom poistení (Selected Credibility Regression Models in Accident Insurance), Ekonomický časopis, Vol. 54, No. 2, 2006, pp. 168-182.

[10]Tse Y. K., Nonlife Actuarial Models, Theory, Methods and Evaluation, Cambridge: Cambridge University Press, 2009.

[11]Waters, H. R., Credibility Theory, Edinburgh: Heriot-Watt University, 1993.

[12]Waters, H. R., An Introduction to Credibility Theory, London and Edinburgh: Institute of Actuaries and Faculty of Actuaries, 1994.