Question 14.11: Considering the case in Example 14.9.2 and assuming a loss r...
Considering the case in Example 14.9.2 and assuming a loss ratio of 80%, what are the total estimated claims for 2008 and 2009 under the Bornhuetter–Ferguson approach?
| Development year (d) | Earned premium | ||||
| 3 | 2 | 1 | |||
| 200 | 160 | 150 | 250 | 2007 | Year of claim (c) |
| – | 200 | 150 | 300 | 2008 | |
| – | – | 200 | 350 | 2009 | |
The link ratios l_{1,2} and l_{2,3} have already been calculated as 1.20 and 1.25 respectively. This means that the claims reported by the end of the third year – which are also the total claims – are 1.25 times the claims reported by the second year, and they are 1.25 ×1.20 = 1.50 the claims reported by the end of the first year. This means that 1/1.50=0.67 claims are paid by the end of the first year, whilst 1/1.25= 0.80 are paid by the end of the second year.
The first part of the Bornhuetter–Ferguson estimate for the claims arising in 2008 is therefore the value of claims already reported, 200. The second part is the product of the premiums earned, the loss ratio and the chain adder proportion of claims outstanding, 300×0.80×(1−0.80)=48. This means that the 2008 claims estimate is 200+48=248.
Using the same approach, the first part of the Bornhuetter–Ferguson estimate for the claims arising in 2009 is the value of claims already reported, again 200. The second part is the product of the premiums earned, the loss ratio and the chain ladder proportion of claims outstanding, 350×0.80×(1−0.67)=93. This means that the 2009 claims estimate is 200+93=293. The table can therefore be completed as follows:
| Development year (d) | Earned premium | ||||
| 3 | 2 | 1 | |||
| 200 | 160 | 150 | 250 | 2007 | Year of claim (c) |
| 248 | 200 | 150 | 300 | 2008 | |
| 293 | 240 | 200 | 350 | 2009 | |