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/whatexactlybayesianstatisticsitsrolepricinggeneralsardana/
https://www.linkedin.com/pulse/whatexactlybayesianstatisticsitsrolepricinggeneralsardana/
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 selfinsurers it may change. But for beginners just remember till earned premium.
Example: Suppose this is my triangle based on accident year.
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

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 BornhuetterFerguson
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

fcumu 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%

11/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 11/f represents how much is your claim going to develop in future which is dependent on Collateral data i.e. Loss ratios.
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 11/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. 11/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 when you multiply your initial ultimate claims with your remaining percentage that is yet to be developed ( i.e. 11/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.
Follow me on LinkedIn: Kamal Sardana
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 me on LinkedIn: Kamal Sardana
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