2019.Fall 17
I am totally confused as to why in the 2nd year we can arbitrarily add 5% to the LRs for purposes of calculating the avg commission within each range. ??
I am totally confused as to why in the 2nd year we can arbitrarily add 5% to the LRs for purposes of calculating the avg commission within each range. ??
Comments
In our opinion this is a weak spot in the Clark reading. A carryforward provision means if the actual loss ratio, y, for the previous year exceeds the loss ratio at the minimum commission, x, for that year then the difference, y-x may be carried forward to the next loss year. The uncertainty arises from how to do this.
Looking at the loss distribution given in the question, the minimum commission occurs at an 80% loss ratio. There is a 15% probability that the loss ratio will exceed 80% and when it does it is 85% on average. So x =80% and y=85% here which gives the carryforward percentage as 5%.
Clark provides two ways of pricing a carryforward provision. The first is to adjust the next year only for the carryforward provision, while the second is to model the provision using a multi-year loss distribution. To avoid additional assumptions we will use the first method.
There are two main ways we can apply the first method. We know we have a 5% carryforward provision when a carryforward provision exists. We can add the expected carryforward provision uniformly to all loss ratios to model the next year's loss distribution. (Remember the carryforward provision is a penalty for poor loss performance to incentivize the insured to improve their record.)
The expected carryforward provision is 0.15*5% + (100%-15%)*0% = 0.75% as the actual loss ratio is expected to exceed the minimum commission loss ratio only 15% of the time. From here, we can proceed as in the first part of the question. This is sample answer 3 in the CAS examiner's report.
The second approach is to realize the commissions paid and associated loss distribution in year 2 will be the same as in year 1 85% of the time. This is because the actual loss ratio exceeds the minimum commission loss ratio only 15% of the time, so 85% of the time we don't trigger the carryforward provision.
For the 15% of the time when the actual loss ratio exceeded the minimum commission loss ratio, we adjust the commission loss ratios down by 5% to get a minimum commission of 12.5% at a 75% loss ratio, sliding 1:1 to 27.5% at a 60% loss ratio, then sliding 0.5:1 to 35% at a 45% loss ratio. From here we calculate the expected ceding commission and get the technical ratio = 87.2%.
However, we're not done for the second approach because this technical ratio will only occur the 15% of the time when the actual loss ratio exceeded the loss ratio at minimum commission. So calculate the weighted average of the technical ratios, i.e. 15%*87.2% + 85%*90.075%. This is sample answer 1 in the CAS examiner's report.
Finally, note that with a bit of algebra you can show reducing the loss ratios associated with the ceding commissions by (y-x) is equivalent to increasing the average loss ratios used in the distribution by (y-x). This is sample answer 2 in the examiner's report.
Hope this helps
Yes, very helpful - thank you! I never caught that the avg loss ratio in the range above 80% was 85%. I failed to make the connection to use that figure - makes sense! Your answer also helped to illuminate the examiner report solutions which I appreciate greatly.
I agree that the Clark paper did not make the solution to this problem obvious.