### 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 IBNR refers to the not reported amount for the already reported claims. To simply this let us take an example: A company received a claim for Motor Insurance of Rs. 10,000 which will be recorded as the Incurred Claim amount. However based on experience the Actuary estimates that the final cost of the claim would be Rs. 15,000. Hence the difference of Rs. 5,000 is treated as the IBNER component of the claim.
2)      Pure IBNR: This portion of the IBNR is the component that refers to the claims cost for the claims which have not been reported till now. We can also understand this as new claims which are not in the book but are expected to be experienced. For ex – A company has written some business in an area against which it has received 10 claims. However based on experience the Actuary estimates these claims to be 15 in total. Hence the cost of the additional 5 claims is considered as Pure IBNR.
When we estimate the Ultimate Claims for General Insurance classes on a Reporting Year cohort we will be estimating the final cost of the Claims which have already been reported to the company. Hence any IBNR suggested under this methodology would all be IBNER as we would not be expecting to incur any further reported claims under this basis.
However, when we perform projections on an Underwriting Year/Accident Year basis we face the issue of IBNER and Pure IBNR. Please note that in general there is no need for the Actuary to split his results between IBNER and Pure IBNR but we still need to have an understating to explain the IBNR component in totally to other stakeholders.
A general method used to separate out the IBNER and Pure IBNR component can be based on the Average Cost per Claim Method (ACPC).

When we estimate the ultimate claims based on Incurred/Paid amount in general for a Run-Off triangle we do not go into the separation of the Claims Cost by each Claim. We can also estimate the ultimate claims cost by the ACPC method. The estimation would require estimate the ultimate claim count to be the expected and also estimating the ultimate average claim cost expected.
Let us understand this with an example.
Suppose we have the reported claims and incurred amounts data for a few Accident Years. Using the standard techniques we have estimated the ultimate number of claims for each of the AYs. These can be seen below:
 AY Reported Claims Ultimate Claims Unreported claims 2015 140 140 0 2016 150 150 0 2017 140 145 5 2018 135 150 15 2019 100 155 55

Further, based on the Incurred Amounts we have estimated the current Average Cost per Claim (Incurred Amounts/Reported Claims). Again using the standard estimation techniques we will estimate the ultimate ACPC for each of the AYs.
 AY ACPC Ultimate ACPC 2015 1000 1000 2016 1100 1100 2017 1050 1100 2018 900 1150 2019 700 1145

Based on the above two tables we can estimate the IBNR component for these claims.
 AY Incurred Amount Ultimate Amount IBNR 2015 ₹       1,40,000.00 ₹        1,40,000.00 ₹                               - 2016 ₹       1,65,000.00 ₹        1,65,000.00 ₹                               - 2017 ₹       1,47,000.00 ₹        1,59,500.00 ₹               12,500.00 2018 ₹       1,21,500.00 ₹        1,72,500.00 ₹               51,000.00 2019 ₹           70,000.00 ₹        1,77,475.00 ₹            1,07,475.00

Incurred Amount = Reported Claims * ACPC
Ultimate Costs = Ultimate Claims * Ultimate ACPC
IBNR = Ultimate Costs – Incurred Amount
Please note that the IBNR estimated in the table above is the total IBNR (IBNER + Pure IBNR). Based on the information above we can estimate the IBNER and Pure IBNR components:

 AY IBNER Pure IBNR IBNR 2015 ₹                          - ₹                           - ₹                               - 2016 ₹                          - ₹                           - ₹                               - 2017 ₹             7,000.00 ₹              5,500.00 ₹               12,500.00 2018 ₹           33,750.00 ₹            17,250.00 ₹               51,000.00 2019 ₹           44,500.00 ₹            62,975.00 ₹            1,07,475.00

IBNER = (Ultimate ACPC – ACPC) * Reported Claims
Pure IBNR = Ultimate ACPC * Unreported Claims
IBNR = IBNER + Pure IBNR

This is the most straightforward method to estimate the IBNR components. There have been various researches and papers written to estimate this using other statistical techniques as well. Interested people can look at the papers available on CAS/IFoA websites or any other Actuarial Forums.

Note: On behalf of Actuary Sense, i would like to thank Ujjaval Agarwal for writing this article in a very easy language which is easily interpreted by our readers. Mr. Ujjaval is an Actuarial Consultant with RSA and prior to that associated with PwC