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Stochastic Claims Reserving Methods
in Insurance
Corrections August 20, 2009
Mario V. W¨
uthrich

Michael Merz

Wiley Finance
ISBN 978-0-470-72346-3

2
Example 3.55 (MSEP in the enhanced CL model), pages 83-88.
(k)
There was a small error in the calculation of the σ
bj in the spreadsheet. This has an influence
on Tables 3.16, 3.17, 3.19 and 3.20. The corrected numerical values are given below.
cj (1)
F
cj (2)
F
cj (3)
F
σ
cj (1)
σ
cj (2)
σ
cj (3)

0

1

2

3

4

5

6

7

8

9

1.44152

1.02784

1.01123

1.00572

1.00477

1.00249

1.00082

1.00200

1.00095

1.00009

1.44152

1.02784

1.01123

1.00572

1.00477

1.00249

1.00082

1.00200

1.00095

1.00009

1.44152
15.64668
15.64695
15.64695

1.02784
8.68378
8.68378
8.68378

1.01123
3.87261
3.87261
3.87261

1.00572
2.18487
2.18487
2.18487

1.00477
2.13834
2.13834
2.13834

1.00249
2.08537
2.08537
2.08537

1.00082
0.82833
0.82833
0.82833

1.00200
2.47425
2.47425
2.47425

1.00095
1.07543
1.07543
1.07543

1.00009
0.12801
0.12801
0.12801

Table 3.16: Estimated parameters in the enhanced CL Model 3.24 for Portfolio A.

cj (1)
F
cj (2)
F
cj (3)
F
σ
cj (1)
σ
cj (2)
σ
cj (3)

0

1

2

3

4

5

6

7

8

9

1.43998

1.03310

1.01168

1.00632

1.00463

1.00415

1.00102

1.00026

1.00088

0.99996

1.43998

1.03310

1.01168

1.00632

1.00463

1.00415

1.00102

1.00026

1.00088

0.99996

1.43998
11.03187
11.03355
11.03355

1.03310
10.86383
10.86383
10.86383

1.01168
3.52681
3.52681
3.52681

1.00632
2.24192
2.24192
2.24192

1.00463
2.35329
2.35329
2.35329

1.00415
2.63728
2.63728
2.63728

1.00102
1.10593
1.10593
1.10593

1.00026
0.30254
0.30254
0.30254

1.00088
0.69056
0.69056
0.69056

0.99996
0.05593
0.05593
0.05593

Table 3.19: Estimated parameters in the enhanced CL Model 3.24 for Portfolio B.

August 20, 2009 (M. W¨
uthrich, ETH Z¨
urich & M. Merz, Uni T¨
ubingen)

20
231
898
1’044
1’731
2’747
4’489
6’804
14’024
90’796
122’784

7
8
9
10
11
12
13
14
15
16
Total

64
543
1’582
1’574
1’958
2’169
2’565
3’170
5’668
10’239
14’030

322.0%
235.2%
176.1%
150.7%
113.1%
79.0%
57.1%
46.6%
40.4%
11.3%
11.4%



(CL,2) 1/2
d
msep
[ Ci,J |DI C
i,J
59
510
1’468
1’470
1’838
2’055
2’428
3’031
5’448
9’884
12’435

300.4%
220.8%
163.5%
140.9%
106.2%
74.8%
54.1%
44.5%
38.8%
10.9%
10.1%

`
´
d Ci,J |DI 1/2
Var
59
510
1’468
1’470
1’836
2’051
2’418
3’012
5’402
8’779
11’547

300.4%
220.8%
163.4%
140.8%
106.1%
74.6%
53.9%
44.3%
38.5%
9.7%
9.4%

process error1/2
1
12
45
52
87
137
224
340
701
4’540
4’615

5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
3.8%

param. pred. error1/2
23
187
589
560
674
693
826
928
1’564
2’673
6’497

115.8%
80.9%
65.5%
53.7%
38.9%
25.2%
18.4%
13.6%
11.2%
2.9%
5.3%

˛
”1/2

(CL,2) ˛
d C
d
Var
˛ DI
i,J

Table 3.17: Reserves and conditional MSEP in the enhanced CL Model 3.24 for Portfolio A.

CL reserves

i

3

August 20, 2009 (M. W¨
uthrich, ETH Z¨
urich & M. Merz, Uni T¨
ubingen)

-4
91
166
320
961
1’445
2’650
3’749
8’224
45’877
63’479

7
8
9
10
11
12
13
14
15
16
Total

19
242
319
558
1’233
1’454
1’839
2’151
4’740
6’284
9’211

d
msep
[ Ci,J |DI (C
i,J
-485.2%
265.7%
192.0%
174.6%
128.4%
100.7%
69.4%
57.4%
57.6%
13.7%
14.5%

(CL,2) 1/2
)

18
228
294
520
1’160
1’382
1’739
2’059
4’559
6’068
8’278

-454.1%
249.8%
176.9%
162.7%
120.7%
95.7%
65.6%
54.9%
55.4%
13.2%
13.0%

`
´
d Ci,J |DI 1/2
Var
18
228
293
519
1’159
1’380
1’734
2’051
4’544
5’626
7’948

-453.9%
249.7%
176.3%
162.5%
120.6%
95.6%
65.4%
54.7%
55.3%
12.3%
12.5%

process error1/2
0
6
23
29
49
74
130
182
373
2’273
2’316

-12.0%
6.2%
13.7%
9.1%
5.1%
5.1%
4.9%
4.9%
4.5%
5.0%
3.6%

param. pred. error1/2

7
82
124
202
419
452
599
625
1’295
1’633
4’038

-171.1%
90.5%
74.7%
63.3%
43.7%
31.3%
22.6%
16.7%
15.7%
3.6%
6.4%


«1/2
(CL,2)
d C
d
Var
|DI
i,J

Table 3.20: Reserves and conditional MSEP in the enhanced CL Model 3.24 for Portfolio B.

CL reserves

i

4

August 20, 2009 (M. W¨
uthrich, ETH Z¨
urich & M. Merz, Uni T¨
ubingen)

5
Example 4.24, revisited, page 113.
The unconditional MSEP was calculated with the standard deviation parameter σi instead of
the variance parameter σi2 . The corrected numbers are
i



1/2

msepC

i,J |Ci,I−i

0
1
2
3
4
5
6
7
8
9
Total

Go
d
C
i,J



1/2

msepC

i,J

16’391
21’602
23’714
37’561
51’584
68’339
82’516
129’667
309’586
359’869



Go
d
C
i,J



1/2 ` c ∗ ´
Ri (ci )
i

msepR

17’526
22’279
25’875
42’139
58’825
81’644
105’397
162’982
363’331
427’850

17’527
22’282
25’879
42’153
58’862
81’745
105’626
163’852
372’199
435’814

Table 4.8: MSEP under the assumptions of Lemma 4.21 and Model 4.14.

Example 8.36 (MSEP in the multivariate ALR model), pages 303-306.
In Table 8.11 on page 303 the correct values are given by the following table (the corrected
values are given in blue color):
i

subportfolio A
reserves

subportfolio B
reserves

1
2
3
4
5
6
7
8
9
10
11
12
13
Total

2’348
5’923
9’608
13’717
26’386
40’906
80’946
143’915
283’823
594’362
1’077’515
1’806’833
2’225’221
6’311’503

-142
-747
1’193
893
3’154
3’243
10’087
21’058
55’625
111’151
235’757
568’114
1’038’295
2’047’680

portfolio
reserves
(k = 1)
2’206
5’176
10’801
14’610
29’541
44’149
91’032
164’973
339’448
705’513
1’313’272
2’374’947
3’263’516
8’359’183

portfolio
reserves
(k = 2)
2’206
5’196
10’815
14’677
29’723
44’749
91’808
165’709
340’160
706’398
1’313’647
2’376’160
3’264’815
8’366’062

portfolio
reserves
(k = 3)
2’206
5’196
10’815
14’677
29’723
44’753
91’813
165’715
340’166
706’405
1’313’653
2’376’170
3’264’826
8’366’119

portfolio
reserves
overall calc.
2’262
5’442
10’356
13’821
28’266
41’604
84’451
153’693
328’700
659’509
1’246’294
2’325’704
3’223’750
8’123’852

In Table 8.12 on page 304 the correct values are given by the following table (the corrected
values are given in blue color):
i

1
2
3
4
5
6
7
8
9
10
11
12
13
Total

subportfolio A
`
´
d C (1) |DI 1/2
Var
i,J
133
471
1’640
5’381
12’669
14’763
17’819
23’840
30’227
43’067
51’294
64’413
80’204
131’444

5,7%
7,9%
17,1%
39,2%
48,0%
36,1%
22,0%
16,6%
10,6%
7,2%
4,8%
3,6%
3,6%
2,1%

subportfolio B
`
´
d C (2) |DI 1/2
Var
i,J

portfolio
´
`
d Ci,J |D N 1/2
Var
I

portfolio
`
´
d Ci,J |D N 1/2
Var
I

portfolio
`
´
d Ci,J |D N 1/2
Var
I

portfolio

444
1’134
2’418
2’552
4’743
5’043
6’682
7’989
14’366
21’419
28’466
40’112
51’955
77’162

(k = 1)
483
21,9%
1’289
24,9%
2’783
25,8%
6’420
43,9%
14’781
50,0%
17’227
39,0%
20’537
22,6%
27’112
16,4%
36’978
10,9%
53’848
7,6%
67’390
5,1%
91’552
3,9%
107’567
3,3%
174’596
2,1%

(k = 2)
483
21,9%
1’289
24,8%
2’783
25,7%
6’421
43,7%
14’782
49,7%
17’233
38,5%
20’544
22,4%
27’118
16,4%
36’985
10,9%
53’854
7,6%
67’404
5,1%
91’569
3,9%
107’580
3,3%
174’624
2,1%

(k = 3)
483
21,9%
1’289
24,8%
2’783
25,7%
6’421
43,7%
14’782
49,7%
17’234
38,5%
20’544
22,4%
27’118
16,4%
36’985
10,9%
53’854
7,6%
67’404
5,1%
91’569
3,9%
107’580
3,3%
174’624
2,1%

calculation
512
22,6%
1’275
23,4%
2’851
27,5%
6’196
44,8%
14’656
51,8%
17’020
40,9%
20’133
23,8%
26’640
17,3%
37’860
11,5%
53’978
8,2%
69’957
5,6%
94’860
4,1%
110’223
3,4%
179’043
2,2%

-313,1%
-151,8%
202,7%
285,9%
150,3%
155,5%
66,3%
37,9%
25,8%
19,3%
12,1%
7,1%
5,0%
3,8%

overall

In Table 8.13 on page 305 the correct values are given by the following table (the corrected
values are given in blue color):
August 20, 2009 (M. W¨
uthrich, ETH Z¨
urich & M. Merz, Uni T¨
ubingen)

6

i

subportfolio A

«1/2
AD ˛˛
[
(1)
d C
˛ DI
Var
˛
i,J

1
2
3
4
5
6
7
8
9
10
11
12
13
Total

149
375
1’074
2’916
6’710
7’859
10’490
12’953
16’473
24’583
30’469
38’904
42’287
172’174

6,3%
6,3%
11,2%
21,3%
25,4%
19,2%
13,0%
9,0%
5,8%
4,1%
2,8%
2,2%
1,9%
2,7%

subportfolio B

«1/2
AD ˛˛
[
(2)
d C
˛ DI
Var
˛
i,J
507
985
1’538
1’547
2’615
2’750
3’584
4’000
6’934
9’520
13’116
20’318
23’687
74’052

-357,2%
-131,9%
128,9%
173,3%
82,9%
84,8%
35,5%
19,0%
12,5%
8,6%
5,6%
3,6%
2,3%
3,6%

portfolio
˛


AD ˛ N 1/2
d C
[
Var
˛ DI
i,J

portfolio
˛


AD ˛ N 1/2
d C
[
Var
˛ DI
i,J

portfolio
˛


AD ˛ N 1/2
d C
[
Var
˛ DI
i,J

portfolio

(k = 1)

(k = 2)

(k = 3)

calculation
576
25,5%
1’086
19,9%
1’898
18,3%
3’383
24,5%
7’640
27,0%
8’807
21,2%
11’283
13,4%
13’734
8,9%
19’446
5,9%
27’814
4,2%
36’798
3,0%
51’665
2,2%
54’980
1,7%
203’909
2,5%

549
1’103
1’809
3’515
7’810
9’087
11’887
14’510
19’523
28’861
36’975
50’834
54’274
207’119

24,9%
21,3%
16,7%
24,1%
26,4%
20,6%
13,1%
8,8%
5,8%
4,1%
2,8%
2,1%
1,7%
2,5%

549
1’103
1’809
3’515
7’810
9’090
11’890
14’513
19’527
28’865
36’982
50’843
54’282
207’157

24,9%
21,2%
16,7%
23,9%
26,3%
20,3%
13,0%
8,8%
5,7%
4,1%
2,8%
2,1%
1,7%
2,5%

549
1’103
1’809
3’515
7’810
9’090
11’890
14’513
19’527
28’865
36’982
50’843
54’282
207’157

24,9%
21,2%
16,7%
23,9%
26,3%
20,3%
13,0%
8,8%
5,7%
4,1%
2,8%
2,1%
1,7%
2,5%

In Table 8.14 on page 306 the correct values are given by the following table (the corrected
values are given in blue color):
i

subportfolio A
1/2

msep
[ (1) ˛
˛D
C
i,J

1
2
3
4
5
6
7
8
9
10
11
12
13
Total

200
602
1’961
6’120
14’337
16’724
20’677
27’131
34’424
49’589
59’660
75’250
90’670
216’613

I

8,5%
10,2%
20,4%
44,6%
54,3%
40,9%
25,5%
18,9%
12,1%
8,3%
5,5%
4,2%
4,1%
3,4%

subportfolio B
1/2

msep
[ (2) ˛
˛D
C
i,J

674
1’502
2’866
2’984
5’416
5’744
7’583
8’935
15’952
23’440
31’342
44’965
57’100
106’947

I

-475,0%
-201,1%
240,3%
334,3%
171,7%
177,1%
75,2%
42,4%
28,7%
21,1%
13,3%
7,9%
5,5%
5,2%

portfolio
1/2 ˛
˛D N
C

msep
[

i,J

(k =
731
1’696
3’319
7’319
16’717
19’477
23’729
30’751
41’815
61’094
76’868
104’718
120’484
270’891

I

1)
33,1%
32,8%
30,7%
50,1%
56,6%
44,1%
26,1%
18,6%
12,3%
8,7%
5,9%
4,4%
3,7%
3,2%

portfolio
1/2 ˛
˛D N
C

portfolio
1/2 ˛
˛D N
C

msep
[

i,J

(k =
731
1’697
3’319
7’320
16’718
19’484
23’737
30’757
41’823
61’102
76’883
104’737
120’499
270’938

msep
[

i,J

I

2)
33,1%
32,7%
30,7%
49,9%
56,2%
43,5%
25,9%
18,6%
12,3%
8,6%
5,9%
4,4%
3,7%
3,2%

(k =
731
1’697
3’319
7’320
16’718
19’484
23’737
30’757
41’823
61’102
76’883
104’738
120’499
270’939

August 20, 2009 (M. W¨
uthrich, ETH Z¨
urich & M. Merz, Uni T¨
ubingen)

portfolio
overall

I

3)
33,1%
32,7%
30,7%
49,9%
56,2%
43,5%
25,9%
18,6%
12,3%
8,6%
5,9%
4,4%
3,7%
3,2%

calculation
770
34,1%
1’675
30,8%
3’425
33,1%
7’059
51,1%
16’528
58,5%
19’163
46,1%
23’079
27,3%
29’972
19,5%
42’562
12,9%
60’723
9,2%
79’045
6,3%
108’017
3,8%
123’174
3,8%
271’358
3,3%

overall

7
Minor Corrections.
c2 should be replaced by σb ; and we choose σ
• Page 174, Table 5.1: σ
c9 = σ
c8 .
j
j
• Page 218, Formula (6.54): Replace N by N (for Gaussian distribution).
• Page 239, Line -12: Add (VaR) after Value-at-Risk.
• Page 279, Remarks 8.20, 3rd bullet point: The last sentence should read as follows:
d (εi,J , εi,J )(k)
In practical applications, it is thus important to verify whether the estimates Cov
are invertible or not and to modify those estimates (e.g. by extrapolation like in the example below) which are not invertible.
(n,m)

(n,m)

• Page 301: There are missing hats in the formulas for ϕ
b12
and ϕ
b11
formulas are given by
)
(
(n,m) 2
(n,m) ϕ
b
(n,m)
11
b10 , (n,m)
ϕ
b12
= min ϕ
ϕ
b10
)
(
(n,m) 2
(n,m) ϕ
b
(n,m)
, 10
b9
ϕ
b11
= min ϕ
(n,m)
ϕ
b

. The correct

9

• Page 318, 1st displayed formula: The prime ’ in the last term of the first formula on page
318 has to be removed. This means change Tj−1 (m, n)0 to Tj−1 (m, n).

Acknowledgment. We are very grateful to Zhang Lianzeng, Stephen J. Mildenhall and James
Quinn who pointed out these errors.

August 20, 2009 (M. W¨
uthrich, ETH Z¨
urich & M. Merz, Uni T¨
ubingen)

8

August 20, 2009 (M. W¨
uthrich, ETH Z¨
urich & M. Merz, Uni T¨
ubingen)

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error

in

the

chain

ladder

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August 20, 2009 (M. W¨
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ubingen)

18

Bibliography

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August 20, 2009 (M. W¨
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ubingen)


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