The New NCCI Hazard Groups
Greg Engl, PhD, FCAS, MAAA
National Council on Compensation Insurance
CAS Reinsurance Seminar June, 2006
Agenda
•
History of previous work
•
Impact of remapping
Current Hazard Groups
870
86
318
428
38
Number of
Classes
45.58%
67,150,463,296
II
100.00%
147,325,629,827
Total
2.46%
3,623,213,832
IV
51.10%
75,288,994,325
III
0.86%
1,262,958,374
I
% of Total
Premium
Standard
Premium
HG
Assigning Classes to HGs
•
Prior NCCI Method
•
California Approach
Prior NCCI Method
•
Hazardousness
For each state, the following seven quantities
were measured by class and expressed as ratios
to the corresponding statewide value:
¾
Claim Frequency
¾
Indemnity Pure Premium
¾
Indemnity Severity
¾
Medical Pure Premium
¾
Medical Severity
¾
Total Pure Premium
¾
Serious Severity (including Medical)
California Methodology
•
Group classes with similar loss
distributions together
Crossover
Ex ce ss Loss Fa ctors
Loss
Limit
O
P
25,000
0.643
0.660
600,000
0.098
0.082
Hazard Group
Crossove
r
California ELF Curves
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Per Accident Limit ($M)
L
o
ss E
li
m
in
at
io
n
R
a
ti
o
J
K
L
M
N
O
P
Q
R
1
2
3
4
5
6
7
8
9
10
Crossover
California ELF Curves
0.060
0.065
0.070
0.075
0.080
0.085
0.090
0.095
0.100
$1,200,000
$1,400,000
$1,600,000
$1,800,000
$2,000,000
$2,200,000
L
o
s
s
E
lim
in
a
ti
o
n
R
a
ti
o
J
M
R
HG Remapping Rationale
•
What are HGs used for?
Excess Loss Pure Premium Factors
(Applicable to New and Renewal Policies)
Per Accident
Hazard Groups
Limitation
I II III IV
$25,000
0.701 0.727 0.813 0.865
$30,000
0.675 0.703 0.794 0.850*
$35,000
0.651 0.682 0.777 0.837
$40,000
0.630 0.662 0.761 0.824*
$50,000
0.593 0.628 0.732 0.802*
$75,000
0.521 0.561 0.674 0.754
$100,000
0.466 0.509 0.628 0.714*
$125,000
0.422 0.466 0.589 0.680
$150,000
0.386 0.430 0.555 0.650
$175,000
0.355 0.399 0.525 0.623
$200,000
0.328 0.373 0.498 0.598
$250,000
0.285 0.329 0.452 0.555
$300,000
0.251 0.294 0.413 0.517
$500,000
0.167 0.205 0.308 0.406
$1,000,000
0.088 0.117 0.188 0.262
$2,000,000
0.045 0.064 0.108 0.155
$5,000,000
0.020 0.032 0.053 0.075
* Also applicable to Underground Coal Mine classifications.
Kentucky ELPPFs
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
1
2
3
4
5
Loss Limit
(millions)
EL
PPF
HG I
HG II
Kentucky ELPPFs
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
1
2
3
4
5
Loss Limit
(millions)
EL
PPF
HG I
HG II
Kentucky ELPPFs
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
200
400
600
800
1000
Loss Limit
(thousands)
EL
PPF
HG I
HG II
Kentucky ELPPFs
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
1
2
3
4
5
Loss Limit
(millions)
EL
PPF
HG I
HG II
Kentucky ELPPFs
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
1000
2000
3000
4000
5000
Loss Limit
(thousands)
EL
PPF
HG I
HG II
HG Remapping Approach
•
Makes sense to sort classes by
ELF vectors
•
Class ELF vectors approximated by
HG ELF vectors
•
ELF curves characterize loss
distribution
Excess Ratio Calculations
∑
=
(
)
)
(
A
w
i
R
i
A
i
R
µ
i
R
i
=
excess
ratio
function
for
injury
typ
e
∑
=
i
i
i
L
L
w
losses
type
injury
i
L
i
=
loss
type
injury
mean
i
i
=
µ
HG Remapping
Basic Data
•
For each class code, , we have a
vector of ELFs:
•
Credibility weight with current HG
ELF vector
c
)
,
,
,
(
25
c
K
30
c
K
5
c
M
c
x
x
x
x
=
K
Current Hazard Groups
0.000
0.050
0.100
0.150
0.200
0.250
0.200
0.300
0.400
0.500
0.600
0.700
Excess Ratio at $100K
E
x
cess R
a
ti
o
at
$1M
Current Hazard Groups
0.000
0.050
0.100
0.150
0.200
0.250
0.200
0.300
0.400
0.500
0.600
0.700
Excess Ratio at $100K
Excess Ratio at $1M
New 4 Hazard Groups
Preliminary Mapping
0.000
0.050
0.100
0.150
0.200
0.250
0.200
0.300
0.400
0.500
0.600
0.700
Excess Ratio at $100K
Excess Ratio at $1M
0.000
0.050
0.100
0.150
0.200
0.250
0.200
0.300
0.400
0.500
0.600
0.700
Excess Ratio at $100K
Excess Ratio at $1M
New 4 Hazard Groups
4 Hazard Group Comparison
Number of Classes per Hazard Group
Current Mapping
38
428
318
86
New Mapping*
296
205
281
88
HG1
HG2
HG3
HG4
* Preliminary
4 Hazard Group Comparison
Percent of Premium Per Hazard Group
Current Mapping
45.58%
51.10%
2.46%
0.86%
New Mapping*
26.70%
31.39%
37.13%
4.78%
HG1
HG2
HG3
HG4
* Preliminary
Hierarchical Collapsing of New Mapping
A
B
C
D
E
F
G
New Hazard Groups
Preliminary Mapping
Percent of Premium per HG
9.33%
17.37%
21.25%
10.14%
18.44%
18.69%
4.78%
HG A
HG B
HG C
HG D
HG E
HG F
HG G
Number of Classes per HG
55
241
160
45
224
57
88
New Hazard Groups
Preliminary Mapping
0.000
0.050
0.100
0.150
0.200
0.250
0.200
0.300
0.400
0.500
0.600
0.700
Excess Ratio at $100K
Excess Ratio at $1M
New Hazard Groups
Preliminary Mapping
0 .0 0 0
0 .0 5 0
0 .1 0 0
0 .1 5 0
0 .2 0 0
0 .2 5 0
0 .2 0 0
0 .3 0 0
0 .4 0 0
0 .5 0 0
0 .6 0 0
0 .7 0 0
E x c e s s R a ti o a t $ 1 0 0 K
Excess Ratio at $1M
0%
10%
20%
30%
40%
50%
60%
70%
No Movement
Up 1 HG
Down 1 HG
Down 2 HGs
Movement
Percent of Premium
Percent of Premium Moved
Current Mapping to New 4 Hazard Groups
(
Based on Preliminary New Mapping)
* Number above bar represents the number of classes in each category.
552
15
3
300
Movement of Classes
Based on Preliminary New Mapping
Hazard Group
I
II
III
IV
Total
Number of Classes
38
428
318
86
870
% Premium
0.9%
45.6%
51.1%
2.5%
100%
Hazard Group
38
255
3
0
296
0.9%
25.4%
0.5%
0.0%
26.7%
0
164
41
0
205
0.0%
19.6%
11.8%
0.0%
31.4%
0
9
268
4
281
0.0%
0.6%
36.3%
0.2%
37.1%
0
0
6
82
88
0.0%
0.0%
2.6%
2.2%
4.8%
Current Mapping
New Mapping
1
2
3
4
Number of Hazard Groups
Calinski and Harabasz
2000
2400
2800
3200
3600
4
5
6
7
8
9
Number of Hazard Groups
CH Statistic
Number of HGs
4
CH Statistic
2317
5
3310
6
3213
7
3442
8
3297
9
3102
Number of Hazard Groups
Cubic Clustering Criterion
85
95
105
115
125
4
5
6
7
8
9
Number of Hazard Groups
CCC St
a
ti
s
ti
c
Number of HGs
4
CCC Statistic
89
5
110
6
108
7
112
8
111
9
125
Number of Hazard Groups
Calinski and Harabasz
Number of HGs
All Classes
50% Credibility Classes
Full Credibility Classes
4
2317
793
433
5
3310
759
393
6
3213
705
450
7
3442
1025
638
8
3297
958
620
9
3102
915
584
Cubic Clustering Criterion
Number of HGs
All Classes
50% Credibility Classes
Full Credibility Classes
4
89
51
37
5
110
50
34
6
108
48
36
7
112
59
42
8
111
57
41
9
125
56
40
Three Key Ideas
•
Map based on ELFs
•
Compute ELFs by class
HG Remapping
Objective
Break
C
=
set of all class codes,
into Hazard Groups:
i
i
HG
C
=
U
HG Remapping
Basic Data
For each class code, , we have a
vector of ELFs:
c
)
,
,
,
(
25
c
K
30
c
K
5
c
M
c
x
x
x
x
=
K
Using Hazard Groups
•
(HG mean)
•
approx by for
•
Want as close as possible to
∑
∈
=
i
HG
c
c
i
i
x
HG
x
1
c
x
x
i
c
∈
HG
i
c
x
x
i
HG Remapping Method
k-means
Splits classes into HGs to minimize
∑ ∑
=
∈
−
k
i
c
HG
i
x
c
x
i
1
2
Optimal
HGs
•
% of total variance explained
•
Analogous to an R-squared
R-squared
∑
∑ ∑
−
−
−
=
=
∈
c
c
k
i
c
HG
i
c
x
x
x
x
R
i
2
1
2
2
1
Optimal HGs
•
Want well separated,
homogeneous HGs
•
Minimize within variance
Optimal HGs
•
Between variance vs. within variance
•
Have one variance for each variable
(ELFs at different attachment points)
•
Need to consider variance-covariance
matrices
Optimal HGs
Dispersion matrix of whole data
set is given by
(
x
x
) (
x
x
)
T
T
c
c
c
−
−
=
∑
Dispersion Matrix
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
=
55
54
53
52
51
45
44
43
42
41
35
34
33
32
31
25
24
23
22
21
15
14
13
12
11
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
)
1
(
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
n
T
Optimal HGs
Dispersion matrix of HG
i
is given
by
(
) (
T
c
i
)
HG
c
i
c
i
x
x
x
x
W
i
−
−
=
∑
∈
Optimal HGs
•
If we let
•
Then
(
x
x
) (
x
x
)
HG
B
i
=
i
i
−
T
i
−
(
) (
T
c
)
i
i
HG
c
c
x
x
x
B
W
x
i
+
=
−
−
∑
∈
Optimal HGs
•
Pooled within group dispersion matrix
•
Weighted between group dispersion
matrix
∑
=
=
k
i
i
W
W
1
∑
=
=
k
i
i
B
B
1
Optimal HGs
•
Between variance vs. within variance
•
T = B + W
Credibility
•
Compute class ELFs
•
Assign a credibility to each
class
History
•
Hazard Groups were last remapped
in 1993
•
Prior to that, Hazard Groups were
remapped in 1981
•
Same credibility method used both
times
Pseudo-Bühlmann
where
n
= number of claims
⎟
⎠
⎞
⎜
⎝
⎛
×
+
=
min
1
.
5
,
1
k
n
n
z
n
k
=
Pseudo-Bühlmann
•
A class with the average number of
claims gets 75% credibility.
•
Classes with twice the average
number of claims get full credibility.
Pseudo-Bühlmann Credibility
0%
20%
40%
60%
80%
100%
0
87
174
261
348
435
522
609
696
783
870
Number of Class Codes (Ordered by Size)
Pseudo-Bühlmann
Credibility
100.0%
870
Total
71.8%
101
≥
6651
Z = 100%
3.2%
18
4988-6650
90%
≤
Z < 100%
4.0%
29
3801-4987
80%
≤
Z < 90%
4.3%
35
2910-3800
70%
≤
Z < 80%
4.8%
46
2217-2909
60%
≤
Z < 70%
2.5%
34
1663-2216
50%
≤
Z < 60%
2.5%
46
1210-1662
40%
≤
Z < 50%
2.7%
56
832-1209
30%
≤
Z < 40%
1.6%
61
512-831
20%
≤
Z < 30%
1.3%
89
238-511
10%
≤
Z < 20%
1.2%
355
0-237
0%
≤
Z < 10%
% Premium
Number of
Classes
Claims per
Year
Credibility Size
Range
Number of Classes by Claim Count
833
50
20 14 3 5 6 1 0 2 0 0 0 1 0 0 1 0 0 0 1
0
100
200
300
400
500
600
700
800
0
20
40
60
80 10
0
12
0
14
0
16
0
18
0
20
0
22
0
24
0
26
0
28
0
30
0
32
0
34
0
36
0
38
0
40
0
42
0
Number of Claims (thousands)
N
u
m
b
er
of
C
lasses i
n
I
n
te
rv
al
Number of Classes by Claim Count
0
50
100
150
200
250
300
350
400
450
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Number of Claims (thousands)
N
u
m
ber
of
C
lasses i
n
I
n
te
rv
al
Number of Classes by Claim Count
0
50
100
150
200
0
50 10
0
15
0
20
0
25
0
30
0
35
0
40
0
45
0
50
0
55
0
60
0
65
0
70
0
75
0
80
0
85
0
90
0
95
0
10
00
Number of Claims
N
u
m
b
er
of
C
lasses i
n
I
n
te
rv
al
