Bornhuetter-Ferguson Formula Proof

Prove that the following 2 formulations of the reported Bornhuetter-Ferguson method are equivalent:

Formulation 1:

• Ult_r-BF   =   z x Ult_r-Dev + (1 – z) x Ult_ECR

where z   =   %reported   =   1/CDF

Formulation 2:

• Ult_r-BF   =   (reported claims) + (1 – 1/CDF) x Ult_ECR

Proof:

Starting with Formulation 1 and using the fact that Ult_r-Dev = (reported claims) x CDF, we have:

• Ult_r-BF

=   z x Ult_r-Dev + (1 – z) x Ult_ECR

=   1/CDF x (reported claims) x CDF + (1 – 1/CDF) x Ult_ECR

=   (reported claims) + (1 – 1/CDF) x Ult_ECR

And the last line is Formulation 2. Since these steps are reversible, we can conclude that Formulation 1 and Formulation 2 are equivalent.