estimating tail parameters
TRANSCRIPT
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Parameters for a Loss DistributionEstimating the Tail
CAS Spring MeetingSeattle, Washington
May 17, 2016
Alejandro Ortega, FCAS, CFA
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The Problem
• Estimate the Distribution of Losses for 2016
• We can use this to measure economic capital or regulatory capital (Solvency II)
• Test Different Reinsurance Plans• Could be a book of Business – Auto,
Marine, Property, or an entire company
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Data
• Analysis Date – 13 Nov 2015• Premium 2010 – 2014• Loss Data 2010 – 2014 • Data as of Sep 30, 2015• Why don’t we use 2015 data
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Summary of Loss Data
YearNumber of
Claims Amount Paid2010 330 3,057,507 2011 312 3,177,057 2012 256 2,849,844 2013 272 3,571,991 2014 367 4,680,122
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Summary of Data Assumptions
• Size of the book is unchanged• If it has, we would calculate
Frequency of Claims, instead of number
• Loss Trend is zero – 0%• If it isn’t, we need to adjust the
historical data• If we have a sufficiently large book, or
market data we can use a specific loss trend
• Otherwise, a general inflation estimate can be used
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Summary of DataAmount Paid
1,000
2,000
3,000
4,000
5,000
2010 2011 2012 2013 2014
Amount Paid
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Summary of DataNumber of Claims
0
100
200
300
400Number of Claims
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YearNumber
of Claims Amount Paid Severity2010 330 3,057,507 9,265
2011 312
3,177,057 10,183
2012 256
2,849,844 11,132
2013 272
3,571,991 13,132
2014 367
4,680,122 12,752
• Severity appears to be increasing• Speak to Underwriting, Product
and Claims
Summary of Data
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Summary of DataNumber of Claims
(Quarterly)
Mean 77 Median 77 Min 49 Max 114 Std Dev 15
• There does not appear to be a trend• We assumed exposures is constant over this period
0
20
40
60
80
100
120
2010Q1 2011Q1 2012Q1 2013Q1 2014Q1
Number of Claims
Claims Mean
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Summary of DataNumber of Claims
(Quarterly)
Mean 77 Median 77 Min 49 Max 114 Std Dev 15
• Rolling Averages are helpful for looking for trends• Use annual periods to minimize seasonality effects
0
20
40
60
80
100
120
2010Q1 2011Q1 2012Q1 2013Q1 2014Q1
Number of Claims
Claims Mean Rolling 8
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Steps
• Estimate Frequency Distribution• Generally, estimate frequency
and apply to projected exposures
• Estimate Severity Distribution• First the Meat (belly) of the
distribution• Then the Tail
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Frequency Distributions
• Negative Binomial• Poisson• Overdispersed Poison• Binomial
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Frequency Parameters
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Selected Parameters370.68
Actual DataMean 76.9Standard Deviation
15.4
Negative Binomial
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Frequency Parameters
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Selected Parameters370.68
EstimatedMean 77.3Standard Deviation
15.5
Actual DataMean 76.9Standard Deviation
15.4
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Frequency Distribution
Simulation 1
0
40
80
120
2010Q1 2011Q1 2012Q1 2013Q1 2014Q1Actual Claims Simulated Negative Binomial
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Frequency Distribution
Simulation 2
0
40
80
120
2010Q1 2011Q1 2012Q1 2013Q1 2014Q1Actual Claims Simulated Negative Binomial
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Frequency Distribution
Simulation 3
0
40
80
120
2010Q1 2011Q1 2012Q1 2013Q1 2014Q1Actual Claims Simulated Negative Binomial
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Frequency Distribution
Simulation 4
0
40
80
120
2010Q1 2011Q1 2012Q1 2013Q1 2014Q1Actual Claims Simulated Negative Binomial
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Steps
• Estimate Frequency Distribution• Generally, estimate frequency
and apply to projected exposures
• Estimate Severity Distribution• First the Meat (belly) of the
distribution• Then the Tail
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Severity Distributions
• Weibull• Gamma• Normal• LogNormal• Exponential• Pareto
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Severity - Parameters
• Parameter Estimation• Maximum Likelihood
Estimators• Graph Distribution of Losses
with Estimated Distribution• Graphical representation
confirms if the modeled distribution is a good fit
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Steps
• Estimate Frequency Distribution• Generally, estimate frequency
and apply to projected exposures
• Estimate Severity Distribution• First the Meat (belly) of the
distribution• Then the Tail
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Excess Mean
For all Losses greater than the average amount by which they exceed
• For any
When this increases with respect to you have a fat tailed distribution
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Excess Mean - Normal
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93
-
250
500
- 500 1,000 1,500 2,000u
Excess Mean Normal
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Excess Mean - Exponential
25
200
-
250
500
- 500 1,000 1,500 2,000u
Excess Mean Exponential
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Excess Mean - Weibull
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454
-
500
1,000
1,500
2,000
- 500 1,000 1,500 2,000u
Excess Mean Weibull
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Excess Mean - LogNormal
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622
-
500
1,000
1,500
2,000
- 500 1,000 1,500 2,000u
Excess Mean Lognormal
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Excess Mean - Pareto
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Linear
832
-
500
1,000
1,500
2,000
0 500 1000 1500 2000u
Excess Mean Pareto
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Fat Tailed
Embrechts:Any Distribution with a Fat TailIn the Limit, has a Pareto distribution
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The girth (size, fatness) of the tail is controlled by the parameter (Xi)
Variance does Not ExistMean Does not Exist
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Fat Tailed
Embrechts:Any Distribution with a Fat TailIn the Limit, has a Pareto distribution
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InsuranceFinance (eg. stocks)
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Pareto Distribution
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Data - Severity
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Claim Quarter AmountAmount
with Trend1 2010Q1 14,101 14,101 2 2010Q1 1,824 1,824 3 2010Q1 688 688
… … ... …1534 2014Q4 25 25 1535 2014Q4 32,574 32,574 1536 2014Q4 15,380 15,380 1537 2014Q4 1,016 1,016
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Data - Severity
Largest Claims
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Claim Quarter Amount1305 2014Q2 126,434
415 2011Q1 135,387 1392 2014Q3 149,925 1055 2013Q3 153,900 1423 2014Q3 166,335
225 2010Q3 181,881 1381 2014Q3 254,864 1310 2014Q2 510,060
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Severity
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0%
20%
40%
60%
80%
100%
- 200,000 400,000 600,000
Empirical
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Survival Function – Log Log Scale
35
𝑆 (𝑥 )=1−𝐹 (𝑥)
0.05%0.10%0.20%0.39%0.78%1.56%3.13%6.25%
12.50%25.00%50.00%
100.00% 1 8 128 2,048 32,768 524,288
Empirical Log Log Scale
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Survival Function – Log Log Scale
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Look for the Turn
The Tail of the Pareto Distribution is a Line on a log log graph
0.05%
0.10%
0.20%
0.39%
0.78%
1.56%
3.13%
6.25%
12.50%
10,000 80,000
Empirical Log Log Scale
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Survival Function – Log Log Scale
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𝑆 (𝑥 )=1−𝐹 (𝑥)
0.05%
0.10%
0.20%
0.39%
0.78%
1.56%
3.13%
6.25%
40,000 320,000
Empirical Log Log Scale
The Tail of the Pareto Distribution is a Line on a log log graph
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Survival Function – Log Log Scale
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𝑆 (𝑥 )=1−𝐹 (𝑥)
0.05%
0.10%
0.20%
0.39%
0.78%
1.56%
3.13%
6.25%
40,000 320,000
Empirical Log Log Scale
The Tail of the Pareto Distribution is a Line on a log log graph
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Survival Function – Log Log Scale
39
Don’t pick a line based on just the end points0.05%
0.10%
0.20%
0.39%
0.78%
1.56%
3.13%
6.25%
40,000 320,000
Empirical Log Log Scale
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Survival Function – Log Log Scale
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Don’t pick a line based on just the end points0.05%
0.10%
0.20%
0.39%
0.78%
1.56%
3.13%
6.25%
40,000 320,000
Empirical Log Log Scale
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Select Parameters - Severity
You can try different ’s to see if the results are similar (or different) should be deep enough in the tail, that the graph is a lineWant enough points past , so we have confidence in
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Select , determine
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Claim Quarter Amount Empirical 1348 2014Q3 49,302 95.12%
592 2011Q4 49,479 95.19%
105 2010Q2 49,885 95.25%
50,000 95.26%
686 2012Q1 50,862 95.32%
682 2012Q1 51,085 95.38%
639 2011Q4 51,160 95.45%
𝐹 (𝑥 )=𝑟𝑎𝑛𝑘𝑖𝑛𝑔𝒏+𝟏 Number of Claims
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First Parameter Selection
Select first, then The selected is too high
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0.05%0.10%0.20%0.39%0.78%1.56%3.13%6.25%
40,000 320,000
Empirical F(x)
Pareto S(x)
1.00
= 2,000
Empirical Log Log Scale
0.01%0.02%0.04%0.08%0.16%0.31%0.63%1.25%2.50%5.00%
40,000 320,000
1.00
= 1,500
Empirical Log Log Scale
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Second Parameter Selection
Better. The slope is steeper, but it’s still too shallow
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0.05%0.10%0.20%0.39%0.78%1.56%3.13%6.25%
40,000 320,000
Empirical F(x)
Pareto S(x)
1.00
= 2,000
Empirical Log Log Scale
0.01%0.02%0.04%0.08%0.16%0.31%0.63%1.25%2.50%5.00%
40,000 320,000
0.50
= 6,000
Empirical Log Log Scale
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Third Parameter Selection
Now, it drops too quickly
45
0.05%0.10%0.20%0.39%0.78%1.56%3.13%6.25%
40,000 320,000
Empirical F(x)
Pareto S(x)
1.00
= 2,000
Empirical Log Log Scale
0.01%0.02%0.04%0.08%0.16%0.31%0.63%1.25%2.50%5.00%
40,000 320,000
0.20
= 15,000
Empirical Log Log Scale
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Fourth Selection of Parameters
This is much betterLet’s look a little lower and higher
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0.05%0.10%0.20%0.39%0.78%1.56%3.13%6.25%
40,000 320,000
Empirical F(x)
Pareto S(x)
1.00
= 2,000
Empirical Log Log Scale
0.01%0.02%0.04%0.08%0.16%0.31%0.63%1.25%2.50%5.00%
40,000 320,000
0.35
= 9,000
Empirical Log Log Scale
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Fifth Selection of Parameters
This is also quite good47
0.05%0.10%0.20%0.39%0.78%1.56%3.13%6.25%
40,000 320,000
Empirical F(x)
Pareto S(x)
1.00
= 2,000
Empirical Log Log Scale
0.01%0.02%0.04%0.08%0.16%0.31%0.63%1.25%2.50%5.00%
40,000 320,000
0.30
= 10,500
Empirical Log Log Scale
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Sixth Selection of Parameters
Excellent Fit from 40k to about 120k, not as good for the last 7 points or so
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0.01%0.02%0.04%0.08%0.16%0.31%0.63%1.25%2.50%5.00%
40,000 320,000
0.40
= 8,200
Empirical Log Log Scale
0.05%0.10%0.20%0.39%0.78%1.56%3.13%6.25%
40,000 320,000
Empirical F(x)
Pareto S(x)
1.00
= 2,000
Empirical Log Log Scale
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0.01%0.02%0.04%0.08%0.16%0.31%0.63%1.25%2.50%5.00%
40,000 320,000
0.35
= 9,000
Empirical Log Log Scale
Final Selection of Parameters
300k was the size of the expected largest claim
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0.01%
0.02%
0.05%
0.10%
0.20%
0.39%
0.78%
1.56%
3.13%
6.25% 40,000 320,000
S(x) emprico - logarithmo
S(x) empiricoS(x) Pareto
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Largest Expected Loss
Largest Loss 99.93%ile
0.30 10,500 277,0000.35 9,000 304,0000.40 8,200 358,000
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Resumen Severidad
We estimate a distribution for the bely• U
For the Tail, we use Pareto with the following parameters:
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Steps
• Estimate Frequency Distribution• Generally, estimate frequency
and apply to projected exposures
• Estimate Severity Distribution• First the Meat (belly) of the
distribution• Then the Tail
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Assumptions
Book is Homogenous• Homes, Apartments
Useful Exposure Base• Cars, Homes, Revenue
Make Loss Trend Adjustments• Market Inflation, or more detailed, if available
Model Risk• Exposures, Inflation
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Thank You