ADONGO'S CENTRAL CREDIBILITY THEORY

CENTRAL CREDIBILITY

The Central Credibility is a theory I have had developed using the basic idea of my ”Central Point Theory”. The Central Credibility is a linear estimate that produced exact credibility estimate without random errors. The Central Credibility is used to estimate risks that are independent and identically distributed or not identically distributed for any time t without error.

In Central Credibility, it is the size of the error that matter, so that a 0.00 error in 0.10 rate gets the same weight as 0.00 error in 100.00 rate. The formula for Central Credibility is given as;

µ^*(ѳ_i) = α_ix_im+ β_ix_m

Where;

µ(ѳ_i)=Credibility estimate per risk

x_im=mean claim per risk

x_m=total mean claims of all risks

α_i=variable coefficient  per risk

β_i=variable coefficient per risk

The variable coefficient α_i and β_i is defined as;

α=[z_ix_im+(1-z_i)x_m]/2x_im

β=[z_ix_im+(1-z_i)x_m]/2x_m

where z_i is a credibility assigned to the estimator x_m and is defined as;

z=[Variance of The Hypothetical Means/Total Variance of The Estimator x_m]

In symbolically form ,we have

Z = n/(n+k)→Homogeneous in time is assumed.

Z=m/(m+k)→Heterogeneous in time is real.

 n =number of observations

m=total exposure.

k=[Expected Process Variance/ Variance of The Hypothetical mean].

The error of the estimate using the weight µ^* as the complement of credibility is given as;

e=x_m-µ^*

Where;

µ^*=Σm_ix_im/2Σβ_im_i

EXAMPLE

SIC offers a janitorial services policy that is rated on per employee basis. The two measured shown in the table below were randomly selected from SIC Policyholder database. Over a four-year period the following was observed.

INSURED		Y	Y+1	Y+2	Y+3
A	No. of claims	3	2	3	1
	No. of employees	2	2	2	1
B	No. of claims	0	1	1
	No. of employees	4	4	4

1)      Estimate the exact annual claim frequency per employee in each insured using the Central Credibility theory.

2)      Is the model (or theory) actually perfect?

OLUTION

m_A=7 and m_B=12

X_Am = 9/7

X_Am = 1/6

X_m=11/19

σ_A=0.1429

σ_A =0.0833

EPV^*=0.1191

VHM^*=0.617

K=0.1944

Z^*_A=0.9730

Z^*_B=0.9841

  

INSURED A

 α_A=0.4926

 β_A =1.0939

INSURED B

α_B=0.5196

β_B  =0.1496

ence, the expected annual claim frequency for Insured A and B are;

µ(ѳ)A=0.4926(9/7) + 0.5182(11/19)=0.9333

µ(ѳ)B=0.5196(1/6) + 0.1496(11/19)=0.1732

2)  e=x_m=µ^*

   

     µ^*=[(7(9/7) + 12(1/6))/2(1.0939(7) +0.1496(12))]

    

       µ^*=[10.9999992/18.905]=0.58

        e=0.58-0.58=0

         

      Yes: zero error indicates perfect model.





REFERENCE
*Adongo Ayine William(Me). Transcript(2008), Posted(EMS-Bolgatanga Branch) to Mathematical Association of Ghana in the Year 2008.
*Buhlmann, Hans, "Experience Rating and Credibility" ASTIN Bullitin, Vol.4, No.3, 1967, PP.199-207.