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Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method Calculation of the proportionality factors. The cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
5 4,456
6
5
6
The d run-off triangle Year of origin 0 1 2 3 4 5 6
0 1.362 1.384 1.381 1.396 1.378
#DIV/0!
1 1.009 1.008 1.010 1.017
#DIV/0!
0.9957
pj
Tot. Loss Tol. Increm.
Development year 2 3 4 1.004 1.002 1.005 1.005 1.002 1.004
0.9981
0.9953
98.914% 98.914%
98.914% 0.000%
1.00434 98.914% 0.000%
1.00186 99.344% 0.430%
1.00474 99.529% 0.185%
24,832 169% 24,832 169%
27,087 185% 2,255 15%
20,525 140% (6,562) -45%
20,614 141% 89 1%
14,595 100% 38 0%
DEVELOPMENT FACTORS DEVELOPMENT RATIOS DEVELOPMENT PERCENTAG 14,664 100% 98 1%
14,664 100% 0%
4 7 10
5 21
6
4
5
6
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 1,163 1,292 1,474 1,678 1,865
Development year 2 3 39 17 37 24 53 22 103
Completion of the cumulative run-off triangle Method 1 (using the proportionality factors p i,i+1) Year of origin
0
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1
Development year 2 3 1
The chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
0 1 2 3 4 5 6
3,209 3,367 3,871 4,239 4,929 5,217
4,372.00 4,659.00 5,345.00 5,917.00 6,794.00
4,411.00 4,696.00 5,398.00 6,020.00 0.00
4,428.00 4,720.00 5,420.00 6,046.15 0.00 0.00
4,435.00 4,730.00 5,430.07 0.00 0.00
4,456.00 4,752.40 5,455.78 0.00 0.00 0.00
4,456.00 4,752.40 5,455.78 0.00 0.00 0.00
1 4,372 4,659 5,345 5,917 6,794 0.00
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 5,430.07 6,020 6,046.15 6,057.38 0.00 0.00 0.00 0.00 0.00 0.00
5 4,456 4,752.40 5,455.78 6,086.06 0.00 0.00
6 4,456.00 4,752.40 5,455.78 6,086.06 0.00 0.00
1 1,163 1,292 1,474 1,678 1,865 (5,217.00)
Development year 2 3 4 39 17 7 37 24 10 53 22 10.07 103 26.15 11.24 ### -
5 21 22.40 25.71 28.68 -
Method 2 (using the proportionality factors p 1,i) Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
6 -
Estimated Reserves Year of origin Reserve Reserve 0 0 0 1 22 22 2 36 36 3 66 4 (6,794.00) 5 -5,217 -5,217 Total: The d-triangle
Loss Reserving Techniques in Non-Life (E. Van den Borre)
2
The chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
Year of origin 0 1 2 3 4 5 6
0 1.362 1.384 1.381 1.396 1.378 -
1 1.009 1.008 1.010 1.017 #DIV/0!
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 4 1.004 1.002 1.005 1.005 1.002 1.005 1.004 1.002 1.005 1.004 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!
3
5 1.000 1.000 1.000 #DIV/0! #DIV/0! #DIV/0!
6
The chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
DEVELOPMENT FACTORS DEVELOPMENT RATIOS DEVELOPMENT PERCENTAGES
141.5%
Loss Reserving Techniques in Non-Life (E. Van den Borre)
4
The chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
5
The chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
6
The chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
The (old) chain ladder method Here the proportionality factors are the mean of empirical proportionality factors. Calculation of the proportionality factors. The cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
5 4,456
6
5
6
5 21
6
6,250
7,000
6,050
f(x) = 1.4039538088x 1.3814972301x - 90.3107918789
6,500
f(x) = 1.0404611982x 1.011835442x - 147.2697736397
5,850 5,650
6,000
5,450 5,250
5,500
5,050 5,000
4,850 4,650
4,500
4,450 4,250
4,000 3,000
3,500
4,000
4,500
5,000
5,500
4,000
4,500
5,000
5,500
6,000
6,500
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 1.362 1.384 1.381 1.396 1.378
1 1.009 1.008 1.010 1.017
pj
Development year 2 3 4 1.004 1.002 1.005 1.005 1.002 1.004
1.00435
1.00185
1.00474
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 1,163 1,292 1,474 1,678 1,865
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 39 17 37 24 53 22 103
7
4 7 10
The chain ladder method (2)
Fortis Bank Insurance Corporate Actuarial Control
Completion of the cumulative run-off triangle Method 1 (using the proportionality factors p i,i+1) Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372.00 4,659.00 5,345.00 5,917.00 6,794.00 0.00
Development year 2 3 4 4,411.00 4,428.00 4,435.00 4,696.00 4,720.00 4,730.00 5,398.00 5,420.00 5,430.03 6,020.00 6,046.17 6,057.35 0.00 0.00 0.00 0.00 0.00
5 4,456.00 4,752.40 5,455.74 6,086.03 0.00 0.00
6 4,456.00 4,752.40 5,455.74 6,086.03 0.00 0.00
6 4,456.00 4,752.40 5,455.74 6,086.03 0.00 0.00
Method 2 (using the proportionality factors p 1,i) Year of origin 0 1 2 3 4 5 6
1 4,372 4,659 5,345 5,917 6,794 0.00
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 5,430.03 6,020 6,046.17 6,057.35 0.00 0.00 0.00 0.00 0.00 0.00
5 4,456 4,752.40 5,455.74 6,086.03 0.00 0.00
1 1,163 1,292 1,474 1,678 1,865 (5,217.00)
Development year 2 3 4 39 17 7 37 24 10 53 22 10.03 103 26.17 11.18 ### -
5 21 22.40 25.71 28.68 -
0 3,209 3,367 3,871 4,239 4,929 5,217
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Estimated Reserves Year of origin Reserve Reserve 0 0 0 1 22 22 2 36 36 3 66 66 4 -6,794 5 -5,217 -5,217 Total: -11,886.8
6 -
(Official Chain Ladder : )
The d-triangle
Loss Reserving Techniques in Non-Life (E. Van den Borre)
8
The chain ladder method (2)
Fortis Bank Insurance Corporate Actuarial Control
Year of origin 0 1 2 3 4 5 6
0 1.362 1.384 1.381 1.396 1.378 -
1 1.009 1.008 1.010 1.017 #DIV/0!
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 4 1.004 1.002 1.005 1.005 1.002 1.005 1.004 1.002 1.005 1.004 1.002 1.005 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!
9
5 1.000 1.000 1.000 1.000 #DIV/0! #DIV/0!
6
The chain ladder method (2)
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
10
The chain ladder method (2)
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
11
The chain ladder method (2)
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
12
The chain ladder method (2)
Fortis Bank Insurance Corporate Actuarial Control
The (old) chain ladder method Here the proportionality factors are the mean of empirical proportionality factors. Calculation of the proportionality factors. The cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
1.38093
1.01143
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
5 4,456
6
5 4,456.00 4,752.40 5,455.78 6,086.06 6,947.08 7,366.66
6 4,456.00 4,752.40 5,455.78 6,086.06 6,947.08 7,366.66
0 5 ,2 6 ,0 7
6 5 ,0 3 0 .4 1 f(x)= -7 8 5 9 8 .3 1 f(x)= 0 2 7 9 4 4 .0 1 f(x)= 3 -7 2 8 9 6 6 0 ,5 .0 1 f(x)= 2 4 5 3 8 0 ,8 5
1.00434
1.00186
1.00474
0 ,6 5 ,0 6 0 ,4 5
Completion of the cumulative run-off triangle
0 ,2 5 ,0 5
,0 5
5 ,0 0 5 ,8 4
0 5 ,6 4 0 ,5 4 0 ,5 4
0 5 ,2 4 ,0 4 ,0 8 ,0 5 ,0 4 ,0 3
Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372.00 4,659.00 5,345.00 5,917.00 6,794.00 7,204.33
Reserves
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 4 4,411.00 4,428.00 4,435.00 4,696.00 4,720.00 4,730.00 5,398.00 5,420.00 5,430.07 6,020.00 6,046.15 6,057.38 6,871.67 6,901.52 6,914.34 7,286.69 7,318.34 7,331.94
0 0.004713 0.006559 0.01085507 0.022036 4591 4672 4863 5175 5673 0 22.01795 31.89587 56.1749909 125.0083 0.0 24.6 35.6 62.7 139.6
13
cape code
Fortis Bank Insurance Corporate Actuarial Control
PREMIUM 4591 4672 4863 5175 5673 6431
4,456.00 4,752.40 5,455.78 6,086.06 6,947.08 7,366.66
LOSS RATIO
35,063.99 31405 111.65%
0 0.004713 0.006559 0.010855 0.022036 0.291809
PREMIUM 4,591 4,672 4,863 5,175 5,673 6,431
0.0 22.0 31.9 56.2 125.0 1876.6
0 25 36 63 140 2095 2358
0.291809 6431 1876.623 2095.3
Loss Reserving Techniques in Non-Life (E. Van den Borre)
14
cape code
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method Calculation of the proportionality factors. The cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
Year of origin 0 1 2 3 4 5 6
Year of origin 0 1 2 3 4 5 6
Year of origin 0 1 2 3 4 5 6
Year of origin 0 1 2 3
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
0 1.362 1.373 1.376 1.382 1.381
1 1.009 1.008 1.009 1.011
0 3,209 3,367 3,871 4,239 4,929
1 4,372 4,659 5,345 5,917 6,811
0 3,209 3,367 3,871 4,239
1 4,372 4,659 5,345 5,833
0 3,209 3,367 3,871
1 4,372 4,659 5,316
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
5 4,456
6
5
6
5
6
4
5
6
4
5
6
Development year 2 3 4 1.004 1.002 1.005 1.005 1.002 1.004
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,727 5,398 5,422 5,970
Development year 2 3 4,411 4,428 4,696 4,714 5,390
Development year 2 3 4,411 4,701
15
Boni-Mali (chain ladder)
Fortis Bank Insurance Corporate Actuarial Control
4 5 6
Year of origin
5
6
Development year 2 3 4 #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF!
5 #REF! #REF! #REF! #REF! #REF!
6 #REF! #REF! #REF! #REF! #REF!
Previous estimation (without any correction for year 0) Development year Year of origin 0 1 2 3 4 0 3,209 #REF! #REF! #REF! #REF! 1 3,367 #REF! #REF! #REF! #REF! 2 3,871 #REF! #REF! #REF! #REF! 3 4,239 #REF! #REF! #REF! #REF! 4 4,929 6,811 6,871.97 6,902.91 6,913.82
5 #REF! #REF! #REF! #REF! 6,913.82
6 #REF! #REF! #REF! #REF! 6,913.82
Year of origin 0 1 2 3 4
1 4,372 4,587
0 3,209 3,367 3,871 4,239 4,929
1 #REF! #REF! #REF! #REF! #REF!
Development year 2 3
4
0 1 2 3 4 5 6
0 3,209 3,367
Previous estimation (without correction for year 0) Development year Year of origin 0 1 2 3 4 0 3,209 #REF! #REF! #REF! #REF! 1 3,367 #REF! #REF! #REF! #REF! 2 3,871 #REF! #REF! #REF! #REF! 3 4,239 #REF! #REF! #REF! #REF! 4 4,929 6,811 6,871.97 6,902.91 6,913.82
Loss Reserving Techniques in Non-Life (E. Van den Borre)
16
0.5% 5 #REF! #REF! #REF! #REF! 6,948.39
6 #REF! #REF! #REF! #REF! 6,948.39
Boni-Mali (chain ladder)
Fortis Bank Insurance Corporate Actuarial Control
Year of ori 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 1,163 1,292 1,474 1,678 1,865
Year of ori 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929
1 1,163 1,292 1,474 1,678 1,882
Year of ori 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239
1 1,163 1,292 1,474 1,594
Year of ori 0 1 2 3
0 3,209 3,367 3,871
1 1,163 1,292 1,445
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 39 17 37 24 53 22 103
Development year 2 3 39 17 37 24 53 24 53 -
Development year 2 3 39 17 37 18 45
Development year 2 3 39 42
17
4 7 10
5 21
6
4 7 7
5
6
4
5
6
4
5
6
-
Boni-Mali (chain ladder)
Fortis Bank Insurance Corporate Actuarial Control
4 5 6
Year of ori 0 1 2 3 4 5 6
#REF! #REF! #REF!
#REF!
#REF! #REF! #REF!
#REF!
0 3,209 3,367
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 1,163 1,220
Development year 2 3
18
4
5
6
Boni-Mali (chain ladder)
Fortis Bank Insurance Corporate Actuarial Control
Difference
3 (2) 50 187 237
Difference
6 8 204 218
Difference
(5) 182 177
Difference
72
Loss Reserving Techniques in Non-Life (E. Van den Borre)
19
Boni-Mali (chain ladder)
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
20
Boni-Mali (chain ladder)
Formation Caritat, 2008
The chain ladder method, Average cost Number of claims
Year of origin 0 1 2 3 4 5 6
Year of origin 0 1 2 3 4 5 6
Development year 2 3 1,047.5 1,047.7 1,028.7 1,028.9 967.8 970.1 986.8
0 1,043.4 1,043.0 965.1 977.0 1,099.0 1,076.3
1 1,045.5 1,027.1 967.9 984.7 1,118.5
4 1,047.7 1,028.7
5 1,047.7
6
1.00316
1.00139
1.00089
0.99990
1.00000
0 1,043 1,043 965 977 1,099 1,076
1 1,045.50 1,027.10 967.90 984.70 1,118.50 1,079.70
Development year 2 3 1,047.50 1,047.70 1,028.70 1,028.90 967.80 970.10 986.80 987.68 1,120.06 1,121.05 1,081.20 1,082.16
4 1,047.70 1,028.70 970.01 987.58 1,120.94 1,082.06
5 1,047.70 1,028.70 970.01 987.58 1,120.94 1,082.06
6 1,047.70 1,028.70 970.01 987.58 1,120.94 1,082.06
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
4 4,435 4,730
5 4,456
6
1.38093
1.01143
0 3,209 3,367
1 4,372.00 4,659.00
5 4,456.00 4,752.40
6 4,456.00 4,752.40
Paid losses
Year of origin 0 1 2 3 4 5 6
Year of origin 0 1
Loss Reserving Techniques in Non-Life ([email protected])
Development year 2 3 4,411 4,428 4,696 4,720 5,398 5,420 6,020
1.00434
1.00186
1.00474
Development year 2 3 4,411.00 4,428.00 4,696.00 4,720.00
4 4,435.00 4,730.00
21
Chain ladder, average cost
Formation Caritat, 2008
2 3 4 5 6
3,871 4,239 4,929 5,217
5,345.00 5,917.00 6,794.00 7,204.33
5,398.00 6,020.00 6,871.67 7,286.69
5,420.00 6,046.15 6,901.52 7,318.34
5,430.07 6,057.38 6,914.34 7,331.94
5,455.78 6,086.06 6,947.08 7,366.66
5,455.78 6,086.06 6,947.08 7,366.66
0 4,975 5,135 5,681 6,272 7,326 7,353
1 4,629 4,949 5,631 6,198 7,087
4 4,456 4,750
5 4,456
6
0.96955
0.97646
0.99513
0.99740
1.00000
0 4,975 5,135 5,681 6,272 7,326 7,353
1 4,629.00 4,949.00 5,631.00 6,198.00 7,087.00 7,129.07
Development year 2 3 4,497.00 4,470.00 4,783.00 4,760.00 5,492.00 5,470.00 6,131.00 6,101.12 6,920.15 6,886.42 6,961.23 6,927.30
4 4,456.00 4,750.00 5,455.78 6,085.25 6,868.51 6,909.29
5 4,456.00 4,750.00 5,455.78 6,085.25 6,868.51 6,909.29
6 4,456.00 4,750.00 5,455.78 6,085.25 6,868.51 6,909.29
Incurred losses
Year of origin 0 1 2 3 4 5 6
Year of origin 0 1 2 3 4 5 6
0 1 2 3 4 5
Development year 2 3 4,497 4,470 4,783 4,760 5,492 5,470 6,131
NUMBER AV. COST (I) AV. COST (TOT. COSTTOT. COSTLATEST RES (I) 1,047.70 4.25 4.25 4456 4456 4,456 1,028.70 4.62 4.62 4750 4752 4,730 20 970.01 5.62 5.62 5456 5456 5,420 36 987.58 6.16 6.16 6085 6085 6,020 65 1,120.94 6.13 6.20 6869 6947 6,794 75 1,082.06 6.38 6.80 6908 7360 5,217 1,691 Total
Loss Reserving Techniques in Non-Life ([email protected])
22
1,886
Chain ladder, average cost
Formation Caritat, 2008
AVERAGE Paid losses
Year of ori 0 1 2 3 4 5 6
Year of ori 0 1
0 3.08 3.23 4.01 4.34 4.48 4.85
1 4.18 4.54 5.52 6.01 6.07
1.37541
1.01013
0 3.08 3.23
1 4.18 4.54
Loss Reserving Techniques in Non-Life ([email protected])
Development year 2 3 4 4.21 4.23 4.23 4.56 4.59 4.60 5.58 5.59 6.10
1.00330
1.00196
6
5 4.25 4.62
6 4.25 4.62
1.00474
Development year 2 3 4.21 4.23 4.56 4.59
23
5 4.25
4 4.23 4.60
Chain ladder, average cost
Formation Caritat, 2008
2 3 4 5 6
4.01 4.34 4.48 4.85
5.52 6.01 6.07 6.67
5.58 6.10 6.14 6.73
5.59 6.12 6.16 6.76
5.60 6.13 6.17 6.77
5.62 6.16 6.20 6.80
5.62 6.16 6.20 6.80
5 4.25
6
5 4.25 4.62 5.62 6.16 6.13 6.38
6 4.25 4.62 5.62 6.16 6.13 6.38
AVERAGE Incurred losses
Year of ori 0 1 2 3 4 5 6
Year of ori 0 1 2 3 4 5 6
0 4.77 4.92 5.89 6.42 6.67 6.83
1 4.43 4.82 5.82 6.29 6.34
0.96618
0.97530
0 4.77 4.92 5.89 6.42 6.67 6.83
1 4.43 4.82 5.82 6.29 6.34 6.60
Development year 2 3 4 4.29 4.27 4.25 4.65 4.63 4.62 5.67 5.64 6.21
0.99412
0.99751
1.00000
Development year 2 3 4.29 4.27 4.65 4.63 5.67 5.64 6.21 6.18 6.18 6.14 6.44 6.40
4 4.25 4.62 5.62 6.16 6.13 6.38
RES (P) 22 36 65 153 2,143 2,419
Loss Reserving Techniques in Non-Life ([email protected])
24
Chain ladder, average cost
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method, Variant I Estimation of the linear least square parameters The d-triangle Year of origin
0 1.362 1.384 1.381 1.396 1.378
0 1 2 3 4 5 6
Method 1 (using Excel functions) a0 column 0: column 1: column 2: column 3:
1.3714
a1
0.0027
b1
1.0069
a2
0.0001
b2
1.0042
a3
0.0005
column 4: Method 2 (using formulas) column 0: n xi yi xiyi
column 2:
column 3:
5.0 10.0 6.9 13.8 30.0 0.0044 1.3714 4.0 6.0 4.0 6.1
xi)2
14.0
a1
0.0027
b1
1.0069
n xi yi xiyi
3.0 3.0 3.0 3.0
xi)2
5.0
a2
0.0001
b2
1.0042 2.0
Loss Reserving Techniques in Non-Life (E. Van den Borre)
5
1.0016 1 (no least square regression needed)
xi)2 a0 b0 n xi yi xiyi
n
4 1.005
0.0044
b0
b3
column 1:
1 1.009 1.008 1.010 1.017
Development year 2 3 1.004 1.002 1.005 1.002 1.004
25
Chain ladder, Variant I
Fortis Bank Insurance Corporate Actuarial Control
xi yi xiyi xi)2 a3 b3 column 4:
1.0 2.0 1.0 1.0 0.0005 1.0016 1.0047 (no least square regression needed)
Completion of the d-triangle Method 1 (using Excel functions) Year of origin 0 1 2 3 4 5 6
0 1.3624 1.3837 1.3808 1.3958 1.3784 1.3934
1.0206
Development year 2 3 1.0039 1.0016 1.0051 1.0021 1.0041 1.0027 1.0046 1.0032 1.0047 1.0037 1.0048 1.0043
4 1.0047 1.0047 1.0047 1.0047 1.0047 1.0047
5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 1.0089 1.0079 1.0099 1.0174 1.0179 1.0206
Development year 2 3 1.0039 1.0016 1.0051 1.0021 1.0041 1.0027 1.0046 1.0032 1.0047 1.0037 1.0048 1.0043
4 1.0047 1.0047 1.0047 1.0047 1.0047 1.0047
5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 4,372 4,659 5,345 5,917 6,794 7,270
Development year 2 3 4,411 4,428 4,696 4,720 5,398 5,420 6,020 6,048 6,916 7,420 7,455
4 4,435 4,730 5,434 6,067 7,487
5 4,456 4,752 5,460 6,096 7,523
1 1,163 1,292 1,474 1,678 1,865 2,053
Development year 2 3 39 17 37 24 53 22 103 28 122 (6,916) 150 36
4 7 10 14 19 32
5 21 22 26 29 35
1 1.0089 1.0079 1.0099 1.0174
Method 2 (using formulas) Year of origin 0 1 2 3 4 5 6
0 1.3624 1.3837 1.3808 1.3958 1.3784 1.3934
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
26
Chain ladder, Variant I
Fortis Bank Insurance Corporate Actuarial Control
6 Estimated Reserves Year of origin Reserve 0 0 1 22 2 40 3 76 4 -6,794 5 2,306 Total: -4,350
Loss Reserving Techniques in Non-Life (E. Van den Borre)
27
Chain ladder, Variant I
Fortis Bank Insurance Corporate Actuarial Control
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
28
Chain ladder, Variant I
Fortis Bank Insurance Corporate Actuarial Control
6
6
6 4,456 4,752 5,460 6,096 7,523
6 Loss Reserving Techniques in Non-Life (E. Van den Borre)
29
Chain ladder, Variant I
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
30
Chain ladder, Variant I
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method, Variant II Calculation of the weighted average of each column (w ij =( i+j+1)2) The d-triangle Year of origin 0 1 2 3 4 5 6
0 0.930 0.964 0.991 0.988 0.967
1 0.971 0.966 0.975 0.989
Development year 2 3 4 0.994 0.997 1.000 0.995 0.998 0.996
5
6
Calculation d0 t-j j
5 0 wi0
i 0 1 2 3 4
di0 1 2 3 4 5
Calculation d1 t-j
0.9305 0.9638 0.9912 0.9882 0.9674
wi0*di0 0.930452 1.927556 2.973596 3.952806 4.836882
d0
0.974753
di1
wi1*di1 1.942968 2.899374 3.901261 4.94595
5
j
1 wi1
i 0 1 2 3
2 3 4 5
0.9715 0.9665 0.9753 0.9892 d1
0.977825
Calculation d2 t-j j
4 2 wi2
i 0 1 2
di2 3 4 5
wi2*di2 0.9940 2.981988 0.9952 3.980765 0.9960 4.979971
d2 Calculation d3 t-j j
0.995227
3 3
Loss Reserving Techniques in Non-Life (E. Van den Borre)
31
Chain ladder, Variant II
Fortis Bank Insurance Corporate Actuarial Control
wi3
i 0 1
di3 4 5
wi3*di3 0.9969 3.987472 0.9979 4.989496
d3 Calculation d4 t-j j
0.997441
2 4 wi4
i 0
di4 5
wi1*di4 1.0000
5
d4
1
Completion of the d-triangle d0 0.9748 d1
0.9778
d2
0.9952
d3
0.9974
d4
1.0000 1.0000
d5
Year of origin 0 1 2 3 4 5 6
0 0.930 0.964 0.991 0.988 0.967 0.975
1 0.971 0.966 0.975 0.989 0.978 0.978
Development year 2 3 0.994 0.997 0.995 0.998 0.996 0.997 0.997 0.997 0.997
4 1.000 1.000 1.000 1.000 1.000 1.000
5 1.000 1.000 1.000 1.000 1.000 1.000
6
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 5,406 6,020 6,643 4,973 -
5 4,456 4,730 5,406 -
6 4,456 4,730 5,406 -
The completed cumulative triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794 5,085
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5
0 3,209 3,366 3,869 4,236 4,925 5,212
1 1,163 1,292 1,474 1,678 1,865 (132)
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 39 17 37 24 53 22 103 (6,020) (151) (6,643) (113) (4,973) 32
4 7 10 (14) -
5 21 -
6 Chain ladder, Variant II
Fortis Bank Insurance Corporate Actuarial Control
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
33
Chain ladder, Variant II
Fortis Bank Insurance Corporate Actuarial Control
Estimated Reserves Year of origin Reserve 0 0 1 0 2 -14 3 -6,020 4 -6,794 5 -5,217 Total: -18,045
Loss Reserving Techniques in Non-Life (E. Van den Borre)
34
Chain ladder, Variant II
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method, Variant III The logarithm of the cumulative run-off triangle. The cumulative run-off triangle Development year Year of origin 0 1 2 3 0 3,209 4,372 4,411 4,428 1 3,367 4,659 4,696 4,720 2 3,871 5,345 5,398 5,420 3 4,239 5,917 6,020 4 4,929 6,794 5 5,217 6
4 4,435 4,730
5 4,456
6
4 8.40 8.46
5 8.40
6
ln(Cumulative triangle) Year of origin
0 8.07 8.12 8.26 8.35 8.50 8.56
0 1 2 3 4 5 6
1 8.38 8.45 8.58 8.69 8.82
Development year 2 3 8.39 8.40 8.45 8.46 8.59 8.60 8.70
Estimation of the lineair least square parameters Method 1 (using Excel functions) column 0: 0.104684923 a0 8.050189709 column 1: 0.112066733 column 2:
a1
8.360432289 0.107227495
column 3:
a2
8.374896106 0.1010739
column 4:
a3
8.383298922 0.064397587
column 5:
a4
8.397282895 Err:504
a5
Err:504
Estimation of the cost level adjustment factors 0.1047 0.1121 0.1072 0.1011 0.0644 Err:504
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1.110361 1.118588 1.113187 1.106358 1.066516 Err:504
35
Chain ladder, Variant III
Fortis Bank Insurance Corporate Actuarial Control
Cost level adjustment of the incremental run-off triangle The incremental run-off triangle Development year Year of origin 0 1 2 3 0 3,209 1,163 39 17 1 3,367 1,292 37 24 2 3,871 1,474 53 22 3 4,239 1,678 103 4 4,929 1,865 5 5,217 6 n-(i+j)
4 7 10
5 21
6
4 1.00 0.00
5 0.00
6
4 7.47 10.00
5 Err:504
6
(n = 6)
Year of origin
0 5.00 4.00 3.00 2.00 1.00 0.00
1 4.00 3.00 2.00 1.00 0.00
0 5,416.14 5,117.99 5,299.26 5,226.27 5,472.97 5,217.00
1 1,820.79 1,808.31 1,844.32 1,876.99 1,865.00
0 1 2 3 4 5 6
Development year 2 3 3.00 2.00 2.00 1.00 1.00 0.00 0.00
The x'-run-off triangle Year of origin 0 1 2 3 4 5 6
Completion of the x'-run-off-triangle column 0 j 0 n-j 6 w x'i,0 i i,0 0 1 2 3 4 5 6 x'0
1 2 3 4 5 6 7 28
5,416.14 5,117.99 5,299.26 5,226.27 5,472.97 5,217.00 0.00
Development year 2 3 53.80 20.81 45.85 26.55 59.00 22.00 103.00
wi,0 x'i,0 5416.143 10235.98 15897.79 20905.07 27364.84 31302 0 111121.8
3968.6364
Loss Reserving Techniques in Non-Life (E. Van den Borre)
36
Chain ladder, Variant III
Fortis Bank Insurance Corporate Actuarial Control
column 1 j n-j
1 5 wi,01
i 0 1 2 3 4 5 x'1
wi,0 x'i,1 1,820.79 1,808.31 1,844.32 1,876.99 1,865.00 0.00
3641.578 5424.924 7377.299 9384.949 11190 0 37018.75
1371.0648
column 2 j n-j
2 4 wi,2
i 0 1 2 3 4 x'2 column 3 j n-j
x'i,2 3 4 5 6 7 25
53.80 45.85 59.00 103.00 0.00
wi,2 x'i,2 161.3953 183.3996 294.9947 618 0 1257.79
20.81 26.55 22.00 0.00
wi,3 x'i,3 83.23397 132.763 132 0 347.997
50.311581 3 3 wi,3
i 0 1 2 3 x'3
x'i,3 4 5 6 7 22
15.818044
column 4 j n-j
4 2 wi,4
i 0 1 2 x'4
x'i,4 5 6 7 18
wi,4 x'i,4 7.47 37.32807 10.00 60 0.00 0 97.32807
5.4071151
column 5 j n-j
5 1 wi,5
i 0 1 x'5
x'i,1 2 3 4 5 6 7 27
x'i,5 6 7 13
wi,5 x'i,5 Err:504 0.00
Err:504 0 Err:504
Err:504
Loss Reserving Techniques in Non-Life (E. Van den Borre)
37
Chain ladder, Variant III
Fortis Bank Insurance Corporate Actuarial Control
column 6 j n-j
6 0 wi,6
i 0 x'6
x'i,6
wi,6 x'i,6
7 7
0.00
0 0
0
The completed x'-run-off triangle Year of origin 0 1 2 3 4 5 6
0 5,416.14 5,117.99 5,299.26 5,226.27 5,472.97 5,217.00
Development year 2 3 53.80 20.81 45.85 26.55 59.00 22.00 103.00 15.82 50.31 15.82 50.31 15.82
4 7.47 10.00 5.41 5.41 5.41 5.41
5 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
6 0.00 0.00 0.00 0.00 0.00 0.00
Development year 2 3 -3.00 -2.00 -2.00 -1.00 -1.00 0.00 0.00 1.00 1.00 2.00 2.00 3.00
4 -1.00 0.00 1.00 2.00 3.00 4.00
5 0.00 1.00 2.00 3.00 4.00 5.00
6
Development year 2 3 39.00 17.00 37.00 24.00 53.00 22.00 103.00 17.50 56.01 19.36 62.35 21.42
4 7.00 10.00 5.77 6.15 6.56 7.00
5 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
6
Development year 2 3 4 4,411.00 4,428.00 4,435.00 4,696.00 4,720.00 4,730.00 5,398.00 5,420.00 5,425.77 6,020.00 6,037.50 6,043.65 6,850.01 6,869.37 6,875.93 6,813.00 6,834.42 6,841.42
5 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
6 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
1 1,820.79 1,808.31 1,844.32 1,876.99 1,865.00 1,371.06
Reversion of the cost level adjustment i+j-n Year of origin 0 1 2 3 4 5 6
0 -5.00 -4.00 -3.00 -2.00 -1.00 0.00
1 -4.00 -3.00 -2.00 -1.00 0.00 1.00
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209.00 3,367.00 3,871.00 4,239.00 4,929.00 5,217.00
1 1,163.00 1,292.00 1,474.00 1,678.00 1,865.00 1,533.66
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209.00 3,367.00 3,871.00 4,239.00 4,929.00 5,217.00
1 4,372.00 4,659.00 5,345.00 5,917.00 6,794.00 6,750.66
Loss Reserving Techniques in Non-Life (E. Van den Borre)
38
Chain ladder, Variant III
Fortis Bank Insurance Corporate Actuarial Control
Estimated Reserves Year of origin Reserve 0 Err:504 1 Err:504 2 Err:504 3 Err:504 4 Err:504 5 Err:504 Total: Err:504
Loss Reserving Techniques in Non-Life (E. Van den Borre)
39
Chain ladder, Variant III
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method, Variant IV The logarithm of the cumulative run-off triangle. The cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
0 8.07 8.12 8.26 8.35 8.50 8.56
1 8.38 8.45 8.58 8.69 8.82
Development year 2 3 4,411 4,428 4,696 4,720 5,398 5,420 6,020
4 4,435 4,730
5 4,456
6
4 8.40 8.46
5 8.40
6
ln(Cumulative triangle) Year of origin 0 1 2 3 4 5 6
Development year 2 3 8.39 8.40 8.45 8.46 8.59 8.60 8.70
Estimation of the lineair least square parameters Method 1 (using Excel functions) a0 column 0: 0.104684923 column 1: column 2: column 3: column 4: column 5:
b0
8.050189709
a1
0.112066733
b1
8.360432289
a2
0.107227495
b2
8.374896106
a3
0.1010739
b3
8.383298922
a4
0.064397587
b4
8.397282895
a5
Err:504
b5
Err:504
Estimation of the cost level adjustment factors wj j w0 w1 w2
0 1 2
25
1.110361
16
1.118588
9
1.113187 1.106358 1.066516
w3
3
4
w4
4
1
Loss Reserving Techniques in Non-Life (E. Van den Borre)
40
Chain ladder, Variant IV
Fortis Bank Insurance Corporate Actuarial Control
w5
5 sum
0 55
0.1046849
'
Err:504 Err:504 Err:504
w0
27.75902 17.8974
0.1120667
'
Err:504
w1
0.1072275 0.1010739
' '
Err:504 Err:504
w2 w3
10.01869 4.425434
0.0643976
'
Err:504
w4
1.066516
Err:504
'
Err:504
w5
Err:504 Err:504
sum Cost level adjustment of the incremental run-off triangle The incremental run-off triangle Development year Year of origin 0 1 2 3 0 3,209 4,372 4,411 4,428 1 3,367 4,659 4,696 4,720 2 3,871 5,345 5,398 5,420 3 4,239 5,917 6,020 4 4,929 6,794 5 5,217 6 n-(i+j)
4 4,435 4,730
5 4,456
6
4 1 0
5 0
6
(n = 5)
Year of origin 0 1 2 3 4 5 6
0 5 4 3 2 1 0
1 4 3 2 1 0
0 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
1 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Development year 2 3 3 2 2 1 1 0 0
x'-run-off triangle Year of origin 0 1 2 3 4 5 6
Development year 2 3 4 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
5 6 Err:504 #VALUE! Err:504
Completion of the x'-run-off triangle
Loss Reserving Techniques in Non-Life (E. Van den Borre)
41
Chain ladder, Variant IV
Fortis Bank Insurance Corporate Actuarial Control
column 0 j n-j
0 6 wi,0
i 0 1 2 3 4 5 6 x'0 column 1 j n-j
wi,0 x'i,0 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Err:504 1 5 wi,01
i 0 1 2 3 4 5
x'i,1 2 3 4 5 6 7 27
x'1
wi,0 x'i,1 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Err:504
column 2 j n-j
2 4 wi,2
i 0 1 2 3 4
x'i,2 3 4 5 6 7 25
x'2
wi,2 x'i,2 Err:504 Err:504 Err:504 Err:504 Err:504
Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Err:504
column 3 j n-j
3 3 wi,3
i 0 1 2 3 x'3
x'i,0 1 2 3 4 5 6 7 28
x'i,3 4 5 6 7 22
wi,3 x'i,3 Err:504 Err:504 Err:504 Err:504
Err:504 Err:504 Err:504 Err:504 Err:504
Err:504
column 4
Loss Reserving Techniques in Non-Life (E. Van den Borre)
42
Chain ladder, Variant IV
Fortis Bank Insurance Corporate Actuarial Control
j n-j
4 2 wi,4
i 0 1 2
x'i,4 5 6 7 18
x'4
wi,4 x'i,4 Err:504 Err:504 Err:504
Err:504 Err:504 Err:504 Err:504
Err:504
column 5 j n-j
5 1 wi,5
i 0 1
x'i,5 6 7 13
x'5 column 6 j n-j
wi,5 x'i,5 Err:504 Err:504
Err:504 Err:504 Err:504
Err:504 6 0 wi,6
i 0 x'6
x'i,6 7 7
wi,6 x'i,6 #VALUE! #VALUE! #VALUE!
#VALUE!
The completed x'-run-off triangle Year of origin 0 1 2 3 4 5 6
0 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
1 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Development year 2 3 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
4 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
5 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
Development year 2 3 -4.00 -3.00 -3.00 -2.00 -2.00 -1.00 -1.00 0.00 0.00 1.00 1.00 2.00 2.00 3.00
4 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
5 -1.00 0.00 1.00 2.00 3.00 4.00 5.00
6 0.00 1.00 2.00 3.00 4.00 5.00 6.00
Reversion of the cost-level adjustment i+j-n Year of origin 0 1 2 3 4 5 6
0 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00
1 -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00
The completed incremental run-off triangle
Loss Reserving Techniques in Non-Life (E. Van den Borre)
43
Chain ladder, Variant IV
Fortis Bank Insurance Corporate Actuarial Control
Year of origin 0 1 2 3 4 5 6
0 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
1 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Development year 2 3 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
4 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
5 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
Development year 2 3 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
4 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
5 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
6 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 2,062.00 2,031.00 2,164.00 2,320.00 2,462.00 2,651.00 3,084.00
1 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504 Err:504
Estimated Reserves Year of origin Reserve 0 0.0 1 Err:504 2 Err:504 3 Err:504 4 Err:504 5 Err:504 6 Err:504 Total: Err:504
Loss Reserving Techniques in Non-Life (E. Van den Borre)
44
Chain ladder, Variant IV
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method Calculation of the proportionality factors. The final loss run-off triangle Year of origin 0 1 2 3 4 5 6
pj
0 4,975 5,135 5,681 6,272 7,326 7,353
1 4,629 4,949 5,631 6,198 7,087
1.0314
1.0241
0.96955
0.97646
Development year 2 3 4 4,497 4,470 4,456 4,783 4,760 4,750 5,492 5,470 6,131
1.0049
0.99513
1.0026
0.99740
5 4,456
6
1.0000
1.00000
DEVELOPMENT FAC
Completion of the cumulative run-off triangle Method 1 (using the proportionality factors p i,i+1) Year of origin 0 1 2 3 4 5 6
TOTAL
1 4,629.00 4,949.00 5,631.00 6,198.00 7,087.00 7,129.07
Development year 2 3 4 4,497.00 4,470.00 4,456.00 4,783.00 4,760.00 4,750.00 5,492.00 5,470.00 5,455.78 6,131.00 6,101.12 6,085.25 6,920.15 6,886.42 6,868.51 6,961.23 6,927.30 6,909.29
LATEST PALATEST IN 4,456.00 4,456.00 4,730.00 4,750.00 5,420.00 5,470.00 6,020.00 6,131.00 6,794.00 7,087.00 5,217 7,353 32,637 35,247
LATEST P/ ULT. PAID ULT. INC. 1 4,456.00 4,456.00 0.995789 4,752.40 4,750.00 0.990859 5,455.78 5,455.78 0.981895 6,086.06 6,085.25 0.958657 6,947.08 6,868.51 0.709506 7,366.66 6,909.29 0.925951 35,064 34,525
0 4,975 5,135 5,681 6,272 7,326 7,353
Loss Reserving Techniques in Non-Life (E. Van den Borre)
45
5 4,456.00 4,750.00 5,455.78 6,085.25 6,868.51 6,909.29
6 4,456.00 4,750.00 5,455.78 6,085.25 6,868.51 6,909.29
ULT. P/I, 1 1.0005046 1.00000129 1.00013342 1.01143965 1.06619617 1.01561652
The Munich chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
Calculation of the proportionality factors. The final loss run-off triangle Year of ori 0 1 2 3 4 5 6
DEVELOPMENT FAC pj
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
0.7241
0.9887
1.38093
1.01143
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
0.9957
1.00434
0.9981
1.00186
46
6
0.9953
1.00474
Completion of the cumulative run-off triangle Method 1 (using the proportionality factors p i,i+1) Development year Year of ori 0 1 2 3 4 0 3,209 4,372.00 4,411.00 4,428.00 4,435.00 1 3,367 4,659.00 4,696.00 4,720.00 4,730.00 2 3,871 5,345.00 5,398.00 5,420.00 5,430.07 3 4,239 5,917.00 6,020.00 6,046.15 6,057.38 4 4,929 6,794.00 6,871.67 6,901.52 6,914.34 5 5,217 7,204.33 7,286.69 7,318.34 7,331.94 6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
5 4,456
DEVELOPMENT FACTORS
5 4,456.00 4,752.40 5,455.78 6,086.06 6,947.08 7,366.66
6 4,456.00 4,752.40 5,455.78 6,086.06 6,947.08 7,366.66
The Munich chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
DEVELOPMENT FACTORS
Loss Reserving Techniques in Non-Life (E. Van den Borre)
47
The Munich chain ladder method
Fortis Bank Insurance Corporate Actuarial Control
The Vylder's Least Square Method. Calculation of the proportionality factors. The cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
5 4,456
6
5 21
6
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 1,163 1,292 1,474 1,678 1,865
The regression dataset X line 3,209 3,367 3,871 4,239 4,929 5,217 1,163 1,292 1,474 1,678 1,865 39 37 53 103 17 24 22 7 10 21
Development year 2 3 39 17 37 24 53 22 103
0 column 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 0 1 2 0 1 0
L0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 3 3 3 4 4 5
4 7 10
1 L1
1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1
2 L2
0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0
3 L3
0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0
0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0
RAPPORT DÉTAILLÉ
Loss Reserving Techniques in Non-Life (E. Van den Borre)
48
De Vylder regression (1)
Fortis Bank Insurance Corporate Actuarial Control
Statistiques de la régression Coefficient de 0.9955403258 Coefficient d
0.9911005403
Coefficient d
0.9822010806
Erreur-type
243.97432754
Observations
21
ANALYSE DE VARIANCE Degré de liberté Somme des carrés Moyenne des carrés Régression
10
66289131.8
6628913.18
Résidus
10
595234.725
59523.4725
Total
20
66884366.6
Coefficients
F
Valeur critique de F
111.366372 6.827498E-009
Erreur-type
Statistique t Limite Probabilité inférieureLimite pour supérieure seuil de confiance pour seuil = 95% de confiance = 95%
1685.825
377.525692
4.46545768
0.00120588 844.645341872
L0
-1664.825
288.100982
-5.7786162
0.00017807 -2306.7539874 -1022.89601
L1
-1605.825
288.100982
-5.5738269
0.00023613 -2247.7539874 -963.896013
L2
-1423.825
290.672087 -4.89838915
0.00062461 -2071.4827682 -776.167232
L3
-1169.825
296.305791 -3.94803286
0.002739 -1830.0354428 -509.614557
L4
-605.7
308.605826 -1.96269788
0.07808574 -1293.3166285
81.9166285
C0
3531.175
288.100982
12.2567267 2.3935E-007 2889.24601258
4173.10399
C1
1102.575
288.100982
3.82704354
0.00333494 460.646012577
1744.50399
C2
-161.75
290.672087 -0.55646898
0.59012478 -809.40776819
485.907768
C3
-100
296.305791 -0.33748918
0.74272519 -760.21044275
560.210443
C4
-42
308.605826 -0.13609594
0.89444621 -729.61662848
645.616628
Constante
ANALYSE DES RÉSIDUS Observation
RÉPARTITION DES PROBABILITÉS
Prévisions X
Résidus
Centile
X
1
3552.175
-343.175
2.38095238
7
2
3611.175
-244.175
7.14285714
10
3
3793.175
77.825
11.9047619
17
4
4047.175
191.825
16.6666667
21
5
4611.3
317.7
21.4285714
22
6
5217
0
26.1904762
24
7
1123.575
39.425
30.952381
37
8
1182.575
109.425
35.7142857
39
9
1364.575
109.425
40.4761905
53
10
1618.575
59.425
45.2380952
103
11
2182.7
-317.7
50
1163
12
-140.75
179.75
54.7619048
1292
13
-81.75
118.75
59.5238095
1474
14
100.25
-47.25
64.2857143
1678
15
354.25
-251.25
69.047619
1865
16
-79
96
73.8095238
3209
17
-20
44
78.5714286
3367
18
162
-140
83.3333333
3871
19
-21
28
88.0952381
4239
20
38
-28
92.8571429
4929
21
21 6.8212E-013
97.6190476
5217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
2527.00466
49
De Vylder regression (1)
Fortis Bank Insurance Corporate Actuarial Control
The incured projected run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,552 3,611 3,793 4,047 4,611 5,217
1 1,124 1,183 1,365 1,619 2,183 2,788
3,531
1102.575
Development year 2 3 4 (141) (79) (21) (82) (20) 38 100 162 220 354 416 474 918 980 1,038 1,524 1,586 1,644
-161.75
(100)
5 21 80 262 516 1,080 1,686
(42)
-
4 28 (28)
-
6
Estimated Reserves Year of origin Reserve 0 0 1 80 2 482 3 4 4017 5 9228 Total: Residuals Year of origin 0 1 2 3 4 5 6
0 (343) (244) 78 192 318 -
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 39 109 109 59 (318)
Development year 2 3 180 96 119 44 (47) (140) (251)
50
5
6
De Vylder regression (1)
Fortis Bank Insurance Corporate Actuarial Control
4 L4
0 C0
0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
1 C1
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 C2
0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
Loss Reserving Techniques in Non-Life (E. Van den Borre)
3 C3
0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0
4 C4
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0
51
5 C5
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
De Vylder regression (1)
Fortis Bank Insurance Corporate Actuarial Control
mite inférieure Limite poursupérieure seuil de confiance pour seuil= de 95,0% confiance = 95,0% 844.645342
2527.00466
-2306.75399 -1022.89601 -2247.75399 -963.896013 -2071.48277 -776.167232 -1830.03544 -509.614557 -1293.31663
81.9166285
2889.24601
4173.10399
460.646013
1744.50399
-809.407768
485.907768
-760.210443
560.210443
-729.616628
645.616628
Loss Reserving Techniques in Non-Life (E. Van den Borre)
52
De Vylder regression (1)
Fortis Bank Insurance Corporate Actuarial Control
(1,665) (1,606) (1,424) (1,170) (606) -
1,686
Loss Reserving Techniques in Non-Life (E. Van den Borre)
53
De Vylder regression (1)
Fortis Bank Insurance Corporate Actuarial Control
The Vylder's Least Square Method. Estimation of the parameters The triangle of the model Year of origin
0
1
Development year 2 3 p3S0 p4S0
0 p0S0 1 p0S1
p1S0
p2S0
p1S1
p2S1
p3S1
p4S1
2 p0S2 3 p0S3
p1S2
p2S2
p3S2
p4S2
p1S3
p2S3
p3S3
4 p0S4 5 p0S5
p1S4
p2S4
4
5 p5S0
6 p6S0
p5S1
p1S5
6 p0S6 The incremental run-off triangle Year of origin
0 3,209 3,367 3,871 4,239 4,929 5,217
0 1 2 3 4 5 6
Iteration 1. 2. 3. 4. 5. rescaled
p
Development year 2 3 39 17 37 24 53 22 103
1 1,163 1,292 1,474 1,678 1,865
p
p
p
4 7 10
p
p
0.50000 0.56732 0.57067 0.57078 0.57078
0.30000 0.22930 0.21817 0.21778 0.21777
0.20000 0.01033 0.00934 0.00933 0.00933
0.10000 0.00384 0.00345 0.00344 0.00344
0.05000 0.00166 0.00149 0.00149 0.00149 sum pj
0.70767
0.27000
0.01156
0.00427
0.00184
The resulting parameters pr 0.70767 Loss Reserving Techniques in Non-Life (E. Van den Borre)
Sr
5 21
6
S
S
S
S
S
S
0.04000 0.00421 0.00377 0.00376 0.00376
4984 5574 5586 5586 5586
5303 5892 5902 5903 5903
6090 6767 6780 6780 6780
6957 7451 7463 7463 7463
8894 8610 8626 8626 8626
10434 9196 9142 9140 9140
0.80657 0.00466
4505
5468
6019
6956
6435
7372
4,505.4751 54
De Vylder regression (2)
Fortis Bank Insurance Corporate Actuarial Control
pr pr
0.27000
Sr
5,468.4782 6,019.1588
0.01156
Sr
pr
0.00427
Sr
6,956.2030
pr
0.00184
Sr
6,435.3532
0.00466
Sr
7,372.1232
pr
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,188.37 3,869.86 4,259.55 4,922.67 4,554.08 5,217.00
1 1,216.46 1,476.47 1,625.15 1,878.15 1,737.52 1,990.45
Development year 2 3 52.10 19.24 63.24 23.35 69.61 25.70 80.44 29.70 74.42 27.48 85.25 31.48
4 8.30 10.07 11.09 12.81 11.85 13.58
5 21.00 25.49 28.06 32.42 30.00 34.36
6
Development year 2 3 4,456.94 4,476.18 5,409.57 5,432.92 5,954.31 5,980.02 6,881.26 6,910.97 6,366.02 6,393.50 7,292.70 7,324.18
4 4,484.48 5,442.99 5,991.10 6,923.78 6,405.36 7,337.76
5 4,505.48 5,468.48 6,019.16 6,956.20 6,435.35 7,372.12
6
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,188.37 3,869.86 4,259.55 4,922.67 4,554.08 5,217.00
1 4,404.83 5,346.33 5,884.71 6,800.82 6,291.60 7,207.45
Loss Reserving Techniques in Non-Life (E. Van den Borre)
55
De Vylder regression (2)
Fortis Bank Insurance Corporate Actuarial Control
The linear Arithmetic Separation Method In this example the seperation method starts from the non-cumulative triangle. We assume therefore that the triangle that was given always has an estimate of one accident at the end of year 0. Estimation of the parameters The triangle of the model Year of origin 0 1 2 3 4 5 6
0 r r r r r r
Development year 2 3 r r r r r r r
1 r r r r r
4 r r
5 r
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6 Calculation of r5 and 5 5 r5 Calculation of r4 and 4 4 r4 Calculation of r3 and 3 3 r3 Calculation of r2 and 2 2 r2 Calculation of r1 and 1 1 r1 Calculation of r0 and 0 0 r0
0 3,209 3,367 3,871 4,239 4,929 5,217
1 1,163 1,292 1,474 1,678 1,865
Development year 2 3 39 17 37 24 53 22 103
4 7 10
5 21
7,238.00 0.0029 6,710.47 0.0012 5,790.86 0.0032 5,240.32 0.0093 4,606.46 0.2526 4,390.78 0.7308
Loss Reserving Techniques in Non-Life (E. Van den Borre)
56
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
Estimation of the parameters corresponding to the future calender years. The values resulting The known from the values lin. regression 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12
The regression parameters a b The resulting parameters r0 r1 r2 r3 r4 r5
4,391 4,606 5,240 5,791 6,710 7,238
4,156 4,759 5,361 5,964 6,567 7,170 7,773 8,375 8,978 9,581 10,184 10,787 11,390
602.82 4,155.77 inflation 0.7308 0 0.2526 1
4,391 4,606
4.9%
0.0093 2 0.0032 3
5,240
13.8%
5,791
10.5%
0.0012 4 0.0029 5
6,710
15.9%
7,238
7.9%
6
7,773
7.4%
7
8,375
7.8%
8
8,978
7.2%
9
9,581
6.7%
10
10,184
6.3%
11
10,787
5.9%
12
11,390
5.6%
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5
Loss Reserving Techniques in Non-Life (E. Van den Borre)
0 3,209 3,367 3,871 4,239 4,929 5,217
1 1,163 1,292 1,474 1,678 1,865 1,963
57
Development year 2 3 39 17 37 24 53 22 103 25 72 27 78 29
4 7 10 9 10 11 12
5 21 23 24 26 28 30
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
58
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794 7,180
59
Development year 2 3 4,411 4,428 4,696 4,720 5,398 5,420 6,020 6,045 6,866 6,893 7,258 7,286
4 4,435 4,730 5,429 6,055 6,904 7,298
5 4,456 4,753 5,454 6,081 6,932 7,328
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
od
sume therefore that the
6
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
60
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
73.08498 25.2551 0.928756 0.31916 0.121877 0.290135
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
61
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
62
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
63
Linear Arithmetic Separation
Fortis Bank Insurance Corporate Actuarial Control
The linear Arithmetic Separation Method In this example the seperation method starts from the non-cumulative triangle. We assume therefore that the triangle that was given always has an estimate of one accident at the end of year 0.
Estimation of the parameters The triangle of the model Development year Year of origin 0 1 2 3 4 5 6
0 r r r r r r
1 r r r r r
2 r r r r
3 r r r
4 r r
5 r
6
The incremental run-off triangle Development year Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Calculation of r5 and 5 5 r5
7,238.00 0.0029
Calculation of r4 and 4 4 r4
6,710.47 0.0012
Calculation of r3 and 3 3 r3 Calculation of r2 and 2 2 r2 Calculation of r1 and 1 1 r1 Calculation of r0 and 0 0 r0
1 1,163 1,292 1,474 1,678 1,865
2 39 37 53 103
3 17 24 22
4 7 10
5 21
6
5,790.86 0.0032 5,240.32 0.0093 4,606.46 0.2526 4,390.78 0.7308
Loss Reserving Techniques in Non-Life (E. Van den Borre)
64
Exp. Arithmetic separation
Fortis Bank Insurance Corporate Actuarial Control
Estimation of the parameters corresponding to the future calender years. The known values 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12
4,391 4,606 5,240 5,791 6,710 7,238
The values resulting from the lin. regression 4266.70513 4746.21776 5279.62029 5872.96915 6533.00137 7267.21115 8083.93492 8992.446 10003.0599 11127.2513 12377.7848 13768.8592 15316.2691
The regression parameters m 1.112385 b 4,266.71 The resulting parameters r0 0.7308 0 r1 0.2526 1 r2 r3 r4 r5
inflation 4390.78 4606.46
4.9%
0.0093 2 0.0032 3
5240.32
13.8%
5790.86
10.5%
0.0012 4 0.0029 5
6710.47
15.9%
7238.00
7.9%
6
8083.93
11.7%
7
8992.45
11.2%
8
10003.06
11.2%
9
11127.25
11.2%
10
12377.78
11.2%
11
13768.86
11.2%
12
15316.27
11.2%
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 1,163 1,292 1,474 1,678 1,865 2,042
Development year 2 3 39 17 37 24 53 22 103 26 75 29 84 32
65
4 7 10 10 11 12 14
5 21 23 26 29 32 36
6
Exp. Arithmetic separation
Fortis Bank Insurance Corporate Actuarial Control
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 4,372 4,659 5,345 5,917 6,794 7,259
Development year 2 3 4,411 4,428 4,696 4,720 5,398 5,420 6,020 6,046 6,869 6,898 7,342 7,374
66
4 4,435 4,730 5,430 6,057 6,910 7,388
5 4,456 4,753 5,456 6,086 6,942 7,424
6
Exp. Arithmetic separation
Fortis Bank Insurance Corporate Actuarial Control
The Linear Geometric Separation Method In this example the seperation method starts from the non-cumulative triangle. We assume therefore that the triangle that was given always has an estimate of one accident at the end of year 0. Estimation of the parameters The triangle of the model Year of origin 0 1 2 3 4 5 6
0 r r r r r r
1 r r r r r
Development year 2 3 r r r r r r r
4 r r
5 r
6
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Calculation of r5 and 5 5 r5
129.10 0.1627
Calculation of r4 and 4 4 r4
103.68 0.0723
Calculation of r3 and 3 3 r3 Calculation of r2 and 2 2 r2 Calculation of r1 and 1 1 r1 Calculation of r0 and 0 0 r0
1 1,163 1,292 1,474 1,678 1,865
Development year 2 3 39 17 37 24 53 22 103
4 7 10
5 21
6
82.46 0.2011 77.27 0.5513 71.46 16.2619 68.05 47.1533
Loss Reserving Techniques in Non-Life (E. Van den Borre)
67
Linear Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
Estimation of the parameters corresponding to the future calender years. The known values 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12
The regression parameters a b
68.1 71.5 77.3 82.5 103.7 129.1
r3 r4 r5
59.6 71.2 82.9 94.5 106.1 117.7 129.4 141.0 152.6 164.3 175.9 187.5 199.2
11.63 59.59
The resulting parameters r0 47.1533 0 r1 16.2619 1 r2
The values resulting from the lin. regression
68.0547 71.4611
0.5513 2 0.2011 3
77.2735
0.0723 4 0.1627 5
103.6776
6
129.3777
7
141.00822
8
152.63869
9
164.26916
10
175.89964
82.4585 129.1011
11
187.53011
12
199.16058
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 39 17 37 24 53 22 103 26 71 28 78 31
1 1,163 1,292 1,474 1,678 1,865 2,104
68
4 7 10 9 10 11 12
5 21 21 23 25 27 29
6
Linear Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
69
Linear Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 4,411 4,428 4,696 4,720 5,398 5,420 6,020 6,046 6,865 6,894 7,399 7,429
1 4,372 4,659 5,345 5,917 6,794 7,321
70
4 4,435 4,730 5,429 6,056 6,905 7,441
5 4,456 4,751 5,452 6,081 6,931 7,470
6
Linear Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
The Exponential Geometric Separation Method In this example the seperation method starts from the non-cumulative triangle. We assume therefore that the triangle that was given always has an estimate of one accident at the end of year 0. Estimation of the parameters The triangle of the model Year of origin 0 1 2 3 4 5 6
0 r r r r r r
1 r r r r r
Development year 2 3 r r r r r r r
4 r r
5 r
6
The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Calculation of r5 and 5 5 r5
129.10 0.1627
Calculation of r4 and 4 4 r4
103.68 0.0723
Calculation of r3 and 3 3 r3 Calculation of r2 and 2 2 r2 Calculation of r1 and 1 1 r1 Calculation of r0 and 0 0 r0
Development year 2 3 39 17 37 24 53 22 103
1 1,163 1,292 1,474 1,678 1,865
4 7 10
5 21
6
82.46 0.2011 77.27 0.5513 71.46 16.2619 68.05 47.1533
Loss Reserving Techniques in Non-Life (E. Van den Borre)
71
Exp. Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
Estimation of the parameters corresponding to the future calender years. The known values 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 12
The regression parameters m b The resulting parameters r0 r1 r2 r3 r4 r5
68.1 71.5 77.3 82.5 103.7 129.1
The values resulting from the lin. regression 63.146 71.569 81.117 91.938 104.202 118.103 133.857 151.714 171.953 194.891 220.890 250.356 283.754
1.13 63.15
47.1533 0 16.2619 1
68.0547
0.5513 2 0.2011 3
77.2735
0.0723 4 0.1627 5
103.6776
6
133.857
7
151.714
8
171.953
9
194.891
10
220.890
11
250.356
12
283.754
71.4611 82.4585 129.1011
The completed incremental run-off triangle Year of origin 0 1 2 3 4 5
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 39 17 37 24 53 22 103 27 74 31 84 35
1 1,163 1,292 1,474 1,678 1,865 2,177
72
4 7 10 10 11 12 14
5 21 22 25 28 32 36
6
Exp. Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
6
Loss Reserving Techniques in Non-Life (E. Van den Borre)
73
Exp. Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 2 3 4,411 4,428 4,696 4,720 5,398 5,420 6,020 6,047 6,868 6,898 7,477 7,512
1 4,372 4,659 5,345 5,917 6,794 7,394
74
4 4,435 4,730 5,430 6,058 6,911 7,526
5 4,456 4,752 5,454 6,086 6,942 7,562
6
Exp. Geometric Separation
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method (Mack estimation) Calculation of the proportionality factors. The cumulative run-off triangle Year of origin
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
1.38093 1.41205
1.01143 1.02253
1.00434 1.01097
1.00186 1.00660
1.00474 1.00474
1.00000
24,832 19,615
27,087 27,087
20,525 20,525
14,568 14,568
9,165 9,165
4,456 4,456
0 1.362 1.384 1.381 1.396 1.378
1 1.009 1.008 1.010 1.017
1.100 0.026 0.000 0.943 0.032
0.028 0.057 0.012 0.211
0.001 0.003 0.000
0.000 0.000
0.5254 0.5254
0.1026 0.1026
0.0021 0.0021
0.0007 0.0007
0.0000 0.0002
Process Variance Multiplier Annual 0.38905 Reserve 0.49462
0.10259 0.10556
0.00211 0.00298
0.00066 0.00087
0.00021 0.00021
Parameter Varia Annual 0.00001
0.00000
0.00000
0.00000
0.00000
0 1 2 3 4 5 6
pj
sums
5 4,456
6
-
The d-triangle Year of origin 0 1 2 3 4 5 6
Development year 2 3 4 1.004 1.002 1.005 1.005 1.002 1.004
5
6
Error term
Loss Reserving Techniques in Non-Life (E. Van den Borre)
75
Mack coefficients
Fortis Bank Insurance Corporate Actuarial Control
Reserve
0.00001
0.00000
0.00000
0.00000
0.00000
3 0.00000 2.23E-008 6.72E-008 6.72E-008 6.72E-008 6.72E-008
4 0.00000 2.23E-008 6.72E-008 1.68E-007 1.68E-007 1.68E-007
5 0.00000 2.23E-008 6.72E-008 1.68E-007 3.79E-006 3.79E-006
6 0.00001 2.2321E-008 6.7215E-008 1.6775E-007 0.00000379 1.3738E-005
Covariance Matrix - Estimation Error Only 2 3 4456 4752 2 4456 0.4432034763 0.472684 3 4752 0.4726837526 1.518064 4 5456 0.5426441605 1.742748 5 6086 0.6053332619 1.944079 6 6947 0.6909720811 2.219115
4 5456 0.542644 1.742748 4.99323 5.570074 6.358093
5 6086 0.605333 1.944079 5.570074 140.3817 160.242
6 6947 0.69097208 2.2191154 6.35809338 160.242046 663.007524
4 2.23 0.36 0.63 1.00 0.21 0.11
5 11.85 0.08 0.13 0.21 1.00 0.53
6 25.75 0.04 0.07 0.11 0.53 1.00
Matrix of Estimation Error Factors
2 3 4 5 6
0.00000 0.00000 0.00000 0.00000 0.00001
2 0.00000 2.23210E-008 2.23210E-008 2.23210E-008 2.23210E-008 2.23210E-008
Correlation Matrix - Estimation Error Only 2 0.67 2 0.67 1.00 3 1.23 0.58 4 2.23 0.36 5 11.85 0.08 6 25.75 0.04
Loss Reserving Techniques in Non-Life (E. Van den Borre)
3 1.23 0.58 1.00 0.63 0.13 0.07
76
Mack coefficients
Fortis Bank Insurance Corporate Actuarial Control
Diagonal LDF Reserves Ultimate Proc Mult Proc Sdev 4,456 1.00000 0 4,456 4,730 1.00474 22 4,752 0.00021 0.99 5,420 1.00660 36 5,456 0.00087 2.18 6,020 1.01097 66 6,086 0.00298 4.26 6,794 1.02253 153 6,947 0.10556 27.08 5,217 1.41205 2,150 7,367 0.49462 60.36 32,637
4.42% 6.09% 6.44% 17.69% 2.81%
0.67 1.23 2.23 11.85 25.75
2,427 66.34
4456 4752.397 5455.784 6086.065 6947.084
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Param Sdev
4456 0.443203 0.472684 0.542644 0.605333 0.690972
77
4752.397 0.472684 1.518064 1.742748 1.944079 2.219115
5455.784 0.542644 1.742748 4.99323 5.570074 6.358093
2.73% 34.22162 6086.065 0.605333 1.944079 5.570074 140.3817 160.242
6947.084 0.690972 2.219115 6.358093 160.242 663.0075
Mack coefficients
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
78
Mack coefficients
Fortis Bank Insurance Corporate Actuarial Control
Param Sdev
Total Sdev 2.97% 3.44% 3.38% 7.74% 1.20%
1.19 2.50 4.81 29.56 65.63
5.33% 6.99% 7.28% 19.31% 3.05%
1.41%
75
3.08%
Loss Reserving Techniques in Non-Life (E. Van den Borre)
79
Mack coefficients
Fortis Bank Insurance Corporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
80
Mack coefficients
Fortis Bank Insurance Corporate Actuarial Control
The chain ladder method (Mack estimation) Calculation of the proportionality factors. The cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 4,372 4,659 5,345 5,917 6,794
0 0.734 0.723 0.724 0.716 0.725
1 0.991 0.992 0.990 0.983
0.724
0.991
0
1
Development year 2 3 4 4,411 4,428 4,435 4,696 4,720 4,730 5,398 5,420 6,020
5 4,456
6
5
6
5
6
The 1/d-triangle Year of origin 0 1 2 3 4 5 6 median
Development year 2 3 4 0.996 0.998 0.995 0.995 0.998 0.996
0.996
0.998
0.995
The 1/d-triangle Year of origin
+ = +
0 1 2 3 4 5 6 Diagonal
N0 1 2 3 4
+ + -
N+ 1 3 2
Loss Reserving Techniques in Non-Life (E. Van den Borre)
+ =
Development year 2 3
+ -
N 1 1 2 1 1 Total :
n 1 1 1 3
81
=
4
m 1 2 2 4 3
E(N) 0 0 0 1 1
0.5000 0.5000 1.2500 0.7500 3.0000
Var(N) 0.75 0.75 3.44 1.69 6.63
Mack coefficients (2)
Fortis Bank Insurance Corporate Actuarial Control
1 1 -
inf 0 -1.19741 -1.19741 -2.383937 -1.796115 - 2.04
1 1
1 -
1
-
1 1
1 1 -
1 -
1 -
-
sup 2.20 2.20 4.88 3.30 8.04
Loss Reserving Techniques in Non-Life (E. Van den Borre)
82
Mack coefficients (2)
Fortis Bank Insurance Coporate Actuarial Control
Christophides linear model The incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
1 1,163 1,292 1,474 1,678 1,865
Development year 2 3 39 17 37 24 53 22 103
4 7 10
5 21
6
4 1.95 2.30
5 3.04
6
Log (incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 8.07 8.12 8.26 8.35 8.50 8.56
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 7.06 7.16 7.30 7.43 7.53
Development year 2 3 3.66 2.83 3.61 3.18 3.97 3.09 4.63
Christophides (lin regr)
83
Fortis Bank Insurance Coporate Actuarial Control 0 Year of origin 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 3 3 3 4 4 5
Development year 0 1 2 3 4 5 0 1 2 3 4 0 1 2 3 0 1 2 0 1 0
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Pij
Yij
3,209 1,163 39 17 7 21 3,367 1,292 37 24 10 3,871 1,474 53 22 4,239 1,678 103 4,929 1,865 5,217
8.074 7.059 3.664 2.833 1.946 3.045 8.122 7.164 3.611 3.178 2.303 8.261 7.296 3.970 3.091 8.352 7.425 4.635 8.503 7.531 8.560
a0
1 a1
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 a2
0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
3 a3
0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0
Christophides (lin regr)
4 a4
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0
5 a5
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 b1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0
2 b2 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0
3 b3 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0
4 b4 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
5 b5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
84
Fortis Bank Insurance Coporate Actuarial Control RAPPORT DÉTAILLÉ Statistiques de la régression Coefficient de d 0.9987258137 Coefficient de d 0.997453251 Coefficient de d 0.894906502 Erreur-type 0.1752879878 Observations 21 ANALYSE DE VARIANCE Degré de libertéSomme des Moyenne carrés des carrésFValeur critique de F Régression 11 120.3402 10.94002 391.657 1E-010 Résidus 10 0.307259 0.030726 Total 21 120.6475
Constante a0 a1 a2 a3 a4 a5 b1 b2 b3 b4 b5
Coefficients Erreur-typeStatistique Limite inférieure Probabilité t Limite supérieure pour Limite seuil inférieure de pour confiance Limite seuilpour de supérieure =confiance seuil 95% depour confiance = 95% seuil de = 95,0% confiance = 95,0% 8.5006458854 0.135777 62.60718 3E-014 8.19811 8.803177 8.198115 8.80317699 -0.5535203733 0.156237 -3.54282 0.00533 -0.90164 -0.2054 -0.901638 -0.2054026 -0.393095772 0.156237 -2.51602 0.0306 -0.74121 -0.04498 -0.741214 -0.044978 -0.2817166289 0.158676 -1.77542 0.10621 -0.63527 0.071836 -0.635269 0.07183607 0.0368376361 0.163967 0.224665 0.82676 -0.3285 0.402179 -0.328503 0.40217865 0 0 65535 #NUM! 0 0 0 0 0.0590319176 0.221724 0.266241 0.79546 -0.435 0.553063 -0.434999 0.55306314 -0.967384032 0.110862 -8.72603 5E-006 -1.2144 -0.72037 -1.2144 -0.7203684 -4.2328969857 0.120809 -35.038 9E-012 -4.50208 -3.96372 -4.502076 -3.963718 -5.0570984181 0.134197 -37.6842 4E-012 -5.35611 -4.75809 -5.356108 -4.7580893 -5.9030901917 0.156237 -37.7829 4E-012 -6.25121 -5.55497 -6.251208 -5.5549724 -4.9026030744 0.206992 -23.685 4E-010 -5.36381 -4.4414 -5.363809 -4.441397
ANALYSE DES RÉSIDUS Observation
Prévisions Yij
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Résidus
Christophides (lin regr)
85
Fortis Bank Insurance Coporate Actuarial Control 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
7.9471255121 6.9797414801 3.7142285264 2.890027094 2.0440353204 3.0445224377 8.1075501134 7.1401660814 3.8746531277 3.0504516953 2.2044599217 8.2189292565 7.2515452245 3.9860322708 3.1618308384 8.5374835215 7.5700994894 4.3045865357 8.5006458854 7.5332618534 8.559677803
0.126589 0.079017 -0.050667 -0.056814 -0.098125 -4E-016 0.014227 0.023781 -0.263735 0.127602 0.098125 0.042339 0.04419 -0.01574 -0.070788 -0.185401 -0.144742 0.330142 0.002246 -0.002246 0
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
86
Fortis Bank Insurance Coporate Actuarial Control Calculation of the variance-covariance matrix The future design matrix 0 Year of origin i Development year j Pij 1 5 ln(Ŷ1,5) 2 4 ln(Ŷ2,4)
a0
1 a1
2 a2
3 a3
4 a4
5 a5
1 b2
2 b3
3 b4
4
0
1
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
2
5
ln(Ŷ2,5)
0
0
1
0
0
0
0
0
0
1
3
3
ln(Ŷ4,3)
0
0
0
1
0
0
0
1
0
0
3
4
ln(Ŷ4,4)
0
0
0
1
0
0
0
0
1
0
3
5
ln(Ŷ4,5)
0
0
0
1
0
0
0
0
0
1
4
3
ln(Ŷ4,3)
0
0
0
0
1
0
0
1
0
0
4
4
ln(Ŷ4,4)
0
0
0
0
1
0
0
0
1
0
4
5
ln(Ŷ4,5)
0
0
0
0
1
0
0
0
0
1
2
ln(Ŷ5,2)
0
0
0
0
0
1
1
0
0
0
3
ln(Ŷ5,3)
0
0
0
0
0
1
0
1
0
0
4
ln(Ŷ5,4)
0
0
0
0
0
1
0
0
1
0
5
ln(Ŷ5,5)
0
0
0
0
0
1
0
0
0
1
5 5 5 5
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
5
b5
87
Fortis Bank Insurance Coporate Actuarial Control
X=
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
Christophides (lin regr)
6 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0
7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
8 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0
9 10 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0
88
Fortis Bank Insurance Coporate Actuarial Control
X'=
1 2 3 4 5 6 7 8 9 10
1 1 0 0 0 0 0 0 0 1
1 0 1 0 0 0 0 0 1 0
1 0 1 0 0 0 0 0 0 1
1 2 3 4 5 6 7 8 9 10
1 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
3 ### ### ### ### ### ### ### ### ### ###
4
(X'X)=
2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
3 ### ### ### ### ### ### ###
4
1 2 3 4 5 6 7
1 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
(XT'X) = -1
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 0 0 1 0 0 0 1 0 0
1 0 0 1 0 0 0 0 1 0
1 0 0 1 0 0 0 0 0 1
1 0 0 0 1 0 0 1 0 0
1 0 0 0 1 0 0 0 1 0
1 0 0 0 1 0 0 0 0 1
1 0 0 0 0 1 1 0 0 0
### ### ### ### ### ### ### ### ### ###
5 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
8 ### ### ### ### ### ### ### ### ### ###
9 ### ### ### ### ### ### ### ### ### ###
10 ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ###
5 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
8 ### ### ### ### ### ### ###
9 ### ### ### ### ### ### ###
10 ### ### ### ### ### ### ###
Christophides (lin regr)
1 0 0 0 0 1 0 1 0 0
1 0 0 0 0 1 0 0 1 0
1 0 0 0 0 1 0 0 0 1
1
1
89
Fortis Bank Insurance Coporate Actuarial Control 8 9 10
Loss Reserving Techniques in Non-Life (E. Van den Borre)
#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
### ### ###
### #VALUE! ### #VALUE! ### #VALUE!
#VALUE! #VALUE! #VALUE!
Christophides (lin regr)
#VALUE! ### ### ### #VALUE! ### ### ### #VALUE! ### ### ###
90
Fortis Bank Insurance Coporate Actuarial Control X (X'X)-1 = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Loss Reserving Techniques in Non-Life (E. Van den Borre)
1 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
3 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
4 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
5 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
Christophides (lin regr)
7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
8 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
9 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
10 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
91
Fortis Bank Insurance Coporate Actuarial Control
X (X'X) X'= -1
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
3 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
4 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
5 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
8 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
9 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
10 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
11 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
12 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
13 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
14 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
15 ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
0.0307258787 (result of the regression analysis)
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
92
Fortis Bank Insurance Coporate Actuarial Control
2 X (XT X)-1 X'=
Loss Reserving Techniques in Non-Life (E. Van den Borre)
#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
Christophides (lin regr)
#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
8.50064589
0 -1
1 0
2 0
3 0
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
4 5 0 0.1
1 -1
2 -4
3 -5
4 -6
93
Fortis Bank Insurance Coporate Actuarial Control i
j 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5
5 4 5 3 4 5 2 3 4 5 1 2 3 4 5
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Yi,j 3.20 2.32 3.32 3.48 2.63 3.63 4.27 3.44 2.60 3.60 7.59 4.33 3.50 2.66 3.66
var(Yi,j) #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
E(xi,j) var(xi,j) ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
(xi,j) #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
Christophides (lin regr)
Constante a0 a1 a2 a3 a4 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0 8.50064589 0 0 0 0 0
a5 b1 b2 b3 b4 0 0 0 0 0 0 0 0 0 -6 0 0 0 0 0 0 0 0 -5 0 0 0 0 0 -6 0 0 0 0 0 0 0 -4 0 0 0 0 0 -5 0 0 0 0 0 -6 0 0 0 0 0 0.1 -1 0 0 0 0.1 0 -4 0 0 0.1 0 0 -5 0 0.1 0 0 0 -6 0.1 0 0 0 0
94
Fortis Bank Insurance Coporate Actuarial Control The completed incremental run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Development year 1 2 3 1,163 39 17 1,292 37 24 1,474 53 22 1,678 103 ### 1,865 #VALUE! ### #VALUE! #VALUE! ###
4 7 10 ### ### ### ###
5 21 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
6
4 4,435 4,730 ### ### ### ###
5 4,456 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
6
The completed cumulative run-off triangle Year of origin 0 1 2 3 4 5 6
0 3,209 3,367 3,871 4,239 4,929 5,217
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Development year 1 2 3 4,372 4,411 4,428 4,659 4,696 4,720 5,345 5,398 5,420 5,917 6,020 ### 6,794 #VALUE! ### #VALUE! #VALUE! ###
Christophides (lin regr)
95
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
96
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
97
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
98
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
99
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
100
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
101
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
102
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
103
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
104
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
105
Fortis Bank Insurance Coporate Actuarial Control
5 -5
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
106
Fortis Bank Insurance Coporate Actuarial Control b5 -5 0 -5 0 0 -5 0 0 0 -5 0 0 0 0 -5
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
107
Fortis Bank Insurance Coporate Actuarial Control
Loss Reserving Techniques in Non-Life (E. Van den Borre)
Christophides (lin regr)
108
