### How can you calculate Pure IBNR and IBNER from the IBNR within 5 minutes?

Reserving of Insurance Products using Bayesian Statistics:
Before reading this article, I am assuming that you should know what Bayesian Statistics is and What its role is in case on insurance industry. Here is the link for both of the article:

Believe me there is no use to read that article if you haven't read above two articles that i mentioned.
Okay so in last article I ask about what will be the value of Z in following case:
1.
In Case of Chain Ladder Method: Z = 1 because we calculate our projected ultimate claim amount just on the basis of data mentioned in our triangle. So, we are relying 100% on the available data we have.

2.
In Case of Expected Loss Ratio Method: Z = 0 because we are relying on Loss Ratios and earned premium to calculate our projected ultimate claim amount as our projected amount will not be changed by changing data in cohorts.
Earned Premium we use as a exposure for commercial insurers but for self-insurers it may change. But for beginners just remember till earned premium.
Example: Suppose this is my triangle based on accident year.

Loss Development Triangle
 12 24 36 2016 100 120 150 2017 120 160 2018 140

Now I am going to write my Loss ratios assumptions and earned premium data

 Accident Year Loss Ratios Earned Premium Projected Ultimate Amount 2016 75% 200 150 2017 75% 250 187.5 2018 80% 270 216

Projected Ultimate Amount = Loss Ratio * Earned Premium. So you can see that to calculate my projected ultimate claim data, I have not used data mentioned in original loss development triangle.

Now the journey Begins for Bornhuetter Ferguson Method. I am assuming that you all have done Bornhuetter-Ferguson method. I am here to tell you what those things represent actually.
 Reported Claims Amount 12 24 36 48 60 72 84 2002 12,811 20,370 26,656 37,667 44,414 48,701 48,169 2003 9,651 16,995 30,354 40,594 44,231 44,373 2004 16,995 40,180 58,866 71,707 70,288 2005 28,674 47,432 70,340 70,655 2006 27,066 46,783 48,804 2007 19,477 31,732 2008 18,632 dev fac 1.774526 1.368305 1.184769 1.059779 1.049963 1.00614 1 f-cumu fac 3.220674 1.814949 1.326422 1.119561 1.05641 1.00614 1 AY 2008 2007 2006 2005 2004 2003 2002 F 3.220674 1.814949 1.326422 1.119561 1.05641 1.00614 1 1/f 31.05% 55.10% 75.39% 89.32% 94.66% 99.39% 100.00% 1-1/f 68.95% 44.90% 24.61% 10.68% 5.34% 0.61% 0.00% Initial UL 38237.6 43706.6 69925.7 82890.6 71511.84 44963.8 48169 EL 26365.05 19625.15 17208.15 8852.125 3818.551 274.371 0 RL 18,632 31,732 48,804 70,655 70,288 44,373 49,000 UL 44,997 51,357 66,012 79,507 74,107 44,647 49,000

I am assuming that you have read first 2 articles otherwise there is no use to proceed further.
Okay so lets see over here:
so here you f represents cumulative development factor. As you can see for most recent accident years it is very high (3.2206 for 2008) because it will be multiplied by the reported claim mentioned in cohort to reach at ultimate claim cost.

1/f here basically your Z. That I mentioned in second article. So as I told you at that time that your Z represents that how much trust do you have in your own data. So for most mature years say 2002 or 2003, Z should be high. Lets see then
1/f for 2002 = 100% ( because if you see the triangle then its fully run off so we have used all our own data)
1/f for 2004 = 94.66% ( so as long as we reaches our most recent data our trust is keep on losing on our data and we are looking for something else)

So what that else is?
is.
is.

is your Loss Ratios. Which is going to act as Collateral data that I am going to use for projection.
Please note that 1/f represents how much claim has developed till now. That’s why for year 2002 its 100% because it has developed fully.

So 1-1/f represents how much is your claim going to develop in future which is dependent on Collateral data i.e. Loss ratios.

 Earned Premium 2002 61,183 2003 69,175 2004 99,322 2005 1,38,151 2006 1,07,578 2007 62,438 2008 47,797 Loss Ratios 2002 78.73% 2003 65.00% 2004 72.00% 2005 60.00% 2006 65.00% 2007 70.00% 2008 80.00%

So when you multiply your initial ultimate claims with your remaining percentage that is yet to be developed ( i.e. 1-1/f) you will get your emerging liability( which is yet to be developed). I mentioned emerging liability as EL.

Now we have RL i.e. reported liability ( claims of every accident year of latest development year).

So you ultimate Claim amount is RL + EL.

So you can see that We use both data to refine our estimate that I mentioned in my first article.
That is the role of Bayesian statistics that how you can refine your estimate by using information from outside source(which is loss ratio over here).

Still seems messy.
Do Let me know.

Thanks and Regards

Kamal Sardana