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

Calculation of IBNER and Pure IBNR This article gives a brief about the different forms of IBNR component and its estimation that we generally come across in the General Insurance industry. Hopefully everyone is already familiar with the other terms of GI before reading this article. It will be useful to familiarize yourself with the following terms before reading this article any further (Ultimate Claims, Reported Claims, Incurred, Accident/Underwriting/Reporting Year Cohort, Chain Ladder Estimate, BF Estimate). Before we get to the estimation, let us first clear our understandings of the various terms. Incurred but not reported (IBNR): For a particular year the actuaries estimate the Ultimate Cost (generally referred as the Ultimate Claims) for all the business that has been written. This ultimate cost can be divided into Incurred Claims (Reported claim amount) and IBNR. IBNR can be further split down to two categories: 1)Incurred but not enough reported (IBNER): This portion of the IB…

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:
https://www.linkedin.com/pulse/what-exactly-bayesian-statistics-its-role-pricing-general-sardana/

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 loss ratio with your Earned Premium then you get your initial Ultimate claims.
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.

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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

Follow us on LinkedIn : Actuary Sense


Follow me on LinkedIn: Kamal Sardana

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