Difference between revisions of "Bornhuetter-Ferguson Formula Proof"
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'''Formulation 1''': | '''Formulation 1''': | ||
* Ult<sub>r-BF</sub> = z x Ult<sub>r-Dev</sub> + (1 – z) x Ult<sub>ECR</sub> | * Ult<sub>r-BF</sub> = z x Ult<sub>r-Dev</sub> + (1 – z) x Ult<sub>ECR</sub> | ||
| − | : | + | : ''where z = %reported = 1/CDF'' |
| − | |||
| − | |||
'''Formulation 2''': | '''Formulation 2''': | ||
* Ult<sub>r-BF</sub> = (reported claims) + (1 – 1/CDF) x Ult<sub>ECR</sub> | * Ult<sub>r-BF</sub> = (reported claims) + (1 – 1/CDF) x Ult<sub>ECR</sub> | ||
| − | Starting with Formulation 1: | + | '''Proof''': |
| + | |||
| + | Starting with Formulation 1 and using the fact that Ult<sub>r-Dev</sub> = (reported claims) x CDF, we have: | ||
* Ult<sub>r-BF</sub> | * Ult<sub>r-BF</sub> | ||
| Line 17: | Line 17: | ||
: = (reported claims) + (1 – 1/CDF) x Ult<sub>ECR</sub> | : = (reported claims) + (1 – 1/CDF) x Ult<sub>ECR</sub> | ||
| − | And the last line is Formulation 2. Since these steps are reversible, we can conclude that Formulation 1 | + | And the last line is Formulation 2. Since these steps are reversible, we can conclude that Formulation 1 and Formulation 2 are equivalent. |
Latest revision as of 14:05, 3 July 2020
Prove that the following 2 formulations of the reported Bornhuetter-Ferguson method are equivalent:
Formulation 1:
- Ultr-BF = z x Ultr-Dev + (1 – z) x UltECR
- where z = %reported = 1/CDF
Formulation 2:
- Ultr-BF = (reported claims) + (1 – 1/CDF) x UltECR
Proof:
Starting with Formulation 1 and using the fact that Ultr-Dev = (reported claims) x CDF, we have:
- Ultr-BF
- = z x Ultr-Dev + (1 – z) x UltECR
- = 1/CDF x (reported claims) x CDF + (1 – 1/CDF) x UltECR
- = (reported claims) + (1 – 1/CDF) x UltECR
And the last line is Formulation 2. Since these steps are reversible, we can conclude that Formulation 1 and Formulation 2 are equivalent.
