Reserving of Insurance Products using Bayesian Statistics:
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.
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 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.
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.
Follow us on LinkedIn : Actuary Sense
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
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