Calculating a Loss Ratio for
Commercial Umbrella
Todd Cheema, FCAS, MAAA
CARe, Philadelphia, PA June 6-7, 2011
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Todd Cheema, FCAS, MAAA
Overview
This session will present an alternate method of arriving at a loss ratio for
pricing for commercial umbrella.
Outline

Discuss the weaknesses of the traditional method, which applies average
trend and development factors to aggregate losses by accident year or
policy year.

Discuss an alternative method that applies line of business specific trend
and development factors to individual losses, while also addressing the
issue of free cover and shock losses.

Discuss the differences between primary and umbrella layers form
frequency and severity trends.
Todd Cheema, FCAS, MAAA
3
Traditional Method

Losses are aggregated by PY or AY

Aggregate losses are trended by an excess trend

Aggregate trended losses are developed to ultimate

LR by PY or AY = selected ultimate loss divided by on-level premium

Select a LR for the prospective Treaty Year
Todd Cheema, FCAS, MAAA
4
Weaknesses of the Traditional method

LR could be understated due to free cover - (Skewed dist will have
mode<mean)

LR could be overstated due to shock losses

The underlying loss can be AL, Prem/Ops, or Prod, therefore applying
average trends and average development factors can lead to misleading
results
Todd Cheema, FCAS, MAAA
5
Example of Loss Ratio Calculation for
Commercial Umbrella Using the
Traditional method
Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's)
(1)
Accident
Ultimate
Year
Earn Prem
2003
87,000
2004
92,000
2005
95,000
2006
96,000
2007
93,000
2008
89,000
2009
83,000
2010
79,000
Total 03-10
714,000
Total 03-08
552,000
Notes :
(2)
On-Level
Factor
1.250
1.080
1.020
0.980
0.970
0.970
0.980
0.990
(7)=(4)*(5)+[1-(6)]*(3)*(9)
Todd Cheema, FCAS, MAAA
(3)=(1)*(2)
(4)
Loss &
Trended
ALAE
On-level EP @ 3/31/10
108,750
36,156
99,360
10,487
96,900
73,883
94,080
34,493
90,210
7,924
86,330
9,449
81,340
3,465
78,210
0
735,180
175,857
575,630
172,393
(5)
Frequency
& Severity
Trend
1.436
1.393
1.358
1.314
1.257
1.197
1.135
1.092
(6)
%
Reported
90.4%
82.0%
69.8%
56.1%
40.4%
25.0%
2.6%
0.3%
(7)
(8)=(7)/(6)
Trended
Trended
Ult Loss & Ult Loss &
ALAE
ALAE Ratio
58,709
54.0%
26,274
26.4%
119,376
123.2%
72,257
76.8%
44,998
49.9%
53,519
62.0%
55,613
68.4%
50,853
65.0%
481,599
65.5%
375,132
65.2%
(9) a-priori LR (yrs 03-08) = 65.2%
Selected
6
65.2%
Comparison of Trended Loss from LOB
Specific and Average Trend Factors
Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's)
Accident
Year
2003
2004
2005
2006
2007
2008
2009
2010
Total 03-10
Total 03-08
Notes :
(1)
(2)
(3)=(1)*(2)
Comm Umb with U/L Auto Liab
Loss
AL Freq &
Tr Loss
& ALAE
Sev Trend
& ALAE
24,637
1.221
30,088
3,386
1.358
4,598
36,093
1.148
41,435
29,479
1.124
33,148
6,032
1.180
7,118
1,364
1.218
1,661
3,465
1.092
3,784
0
0
104,457
121,832
100,992
118,048
(4)
(5)
(6)=(4)*(5)
(7)=(3)+(6)
(8)
Comm Umb with U/L Gen Liab
Trended Loss & ALAE
Loss
GL Freq &
Tr Loss
Individual
Aggregate
& ALAE
Sev Trend
& ALAE
Loss Method Loss Method
11,518
1.874
21,586
51,675
51,921
7,101
1.971
13,994
18,592
14,614
37,790
1.259
47,579
89,013
100,311
5,014
1.597
8,008
41,156
45,318
1,892
1.672
3,164
10,282
9,959
8,085
1.149
9,290
10,951
11,315
0
0
3,784
3,932
0
0
0
0
71,401 Calculation
103,622
225,454
237,370
71,401 on Slide 10
103,622
221,670
233,438
(2) & (5) = (Sum of Severity Trend Loss & ALAE, capped at pol limits / Sum of Loss & ALAE) * Freq Trend
* Adj for Clms that Trend into the Layer
Todd Cheema, FCAS, MAAA
7
Comparison of Ultimate Loss from LOB
Specific and Average Development
Factors
Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's)
Accident
Year
2003
2004
2005
2006
2007
2008
2009
2010
Total 03-10
Total 03-08
(1)
(2)
(3)=(1)/(2)
Comm Umb with U/L Auto Liab
Loss
Ult Loss
& ALAE
% Rept
& ALAE
24,637
98.9%
24,919
3,386
97.8%
3,464
36,093
95.6%
37,773
29,479
91.3%
32,287
6,032
75.9%
7,952
1,364
51.2%
2,664
3,465
5.0%
68,757
0
2.0%
0
104,457
177,816
100,992
109,059
Todd Cheema, FCAS, MAAA
(4)
(5)
(6)=(4)/(5)
(7)=(3)+(6)
(8)
Comm Umb with U/L Gen Liab
Ultimate Loss & ALAE
Loss
Ult Loss
Individual
Aggregate
& ALAE
% Rept
& ALAE
Loss Method Loss Method
11,518
85.5%
13,478
38,397
39,983
7,101
72.7%
9,764
13,228
12,789
37,790
54.7%
69,110
106,883
105,815
5,014
35.4%
14,182
46,469
61,504
1,892
19.6%
9,665
17,617
19,601
8,085
9.6%
84,101
86,765
37,764
0
1.1%
0
68,757
135,855
0
0.4%
0
0
0
71,401
200,299
378,116
413,310
71,401
200,299
309,359
277,456
8
Using Individual Umbrella Losses

Trend each loss by a ground-up trend by the line of business from
underlying loss, and cap at policy limits.

Develop losses with underlying GL losses separately from losses with
underlying AL losses.

Cap losses at various limits, and use exposure relativities to arrive at an
uncapped loss. (An unpublished paper by Dr R. Bender documents this
methodology).

Reconcile individual losses with the losses from the triangle.

Trending of ALAE only claims .

Account for losses that trend into the layer.

Recognize signal reserves.
Todd Cheema, FCAS, MAAA
9
Using Individual Umbrella Losses –
Example of Trend
Trend for AY 2005 for Commercial Umbrella Claims with U/L GL
(1)
AY 2005
Loss
Loss #1
Loss #2
Loss #3
Loss #4
Loss #5
Loss #6
Total
AY
2005
(2)
(3)
(4)
Reported
ALAE
86,250
11,500
230,000
11,500
87,400
92,000
518,650
Policy
Limit
10,000,000
25,000,000
25,000,000
25,000,000
10,000,000
5,000,000
Reported
Attach
Loss
Point
1,000,000 5,761,500
1,000,000 1,366,200
1,000,000 25,000,000
773,950
1,000,000
1,000,000 2,300,000
1,000,000 2,070,000
37,271,650
(9)=(3)+(4)
(12)
Adjust for
Frequency Claims that
Severity
Trend In
Trend
Trend
2.71%
0.97%
21.4%
Reported
L&ALAE
37,790,300
(10)=(8)
(11)
(8)
(7)
(6)
(5)
Ground-up
Sev Trend
Sev Tr
Sev Tr
Cumulative
Sev Trend Rept Loss Rept ALAE in Layer
53.2%
125,458
45.5% 8,835,159
78.5%
16,728
45.5% 2,441,833
0.4%
334,554
45.5% 25,000,000
103.3%
16,728
45.5% 1,580,356
64.5%
127,131
45.5% 3,800,122
66.5%
133,822
45.5% 3,465,568
21.4%
754,419
45,123,038
(14)=(9)*[1+(12)]
Trended
Reported
Total
L&ALAE
Trend Adj
25.90% 47,578,903
(13)
Notes : (5) = 6.2% GL Severity trend per year from 7/1/05 to 9/1/11
(8)=[(6)+(7)]/[(3)+(4)]-1
(7)=(4)*[1+(5)]
Todd Cheema, FCAS, MAAA
(6) = Min{[(2)+(3)]*[1+(5)]-(2),(1)}
(13)=[1+(10)]*[1+(11)]*[1+(12)]-1
10
Using Individual Umbrella Losses –
Discussion of Trend

Frequency trend - Does the frequency trend observed in the primary layer
apply to the umbrella layer?

Severity trend – Losses are a combination of the economic loss and a
punitive damage loss. As the mix between economic loss and punitive
damages shifts as the attachment point increases, how should the
severity trend be adjusted for umbrella layers?

Adjustment for claims that trend into the umbrella layer – What is the
best way to quantify this adjustment?
Todd Cheema, FCAS, MAAA
11
Partial Loss Ratio Using Individual Losses
with Underlying Auto Liability
Commercial Umbrella with U/L AL - Treaty Eff Date = 9/1/2010 (amounts in '000's)
(1)
Accident
Ultimate
Year
Earn Prem
2003
87,000
2004
92,000
2005
95,000
2006
96,000
2007
93,000
2008
89,000
2009
83,000
2010
79,000
Total 03-10
714,000
Total 03-08
552,000
Notes :

(2)
On-Level
Factor
1.250
1.080
1.020
0.980
0.970
0.970
0.980
0.990
(7)=(4)*(5)+[1-(6)]*(3)*(9)
(3)=(1)*(2)
(4)
Loss &
Trended
ALAE
On-level EP @ 3/31/10
108,750
24,637
99,360
3,386
96,900
36,093
94,080
29,479
90,210
6,032
86,330
1,364
81,340
3,465
78,210
0
735,180
104,457
575,630
100,992
(5)
Frequency
& Severity
Trend
1.221
1.358
1.148
1.124
1.180
1.218
1.092
(6)
%
Reported
98.9%
97.8%
95.6%
91.3%
75.9%
51.2%
5.0%
2.0%
(9) a-priori LR (yrs 08-08) = 23.8%
(7)
(8)=(7)/(6)
Trended
Trended
Ult Loss & Ult Loss &
ALAE
ALAE Ratio
30,381
27.9%
5,129
5.2%
42,460
43.8%
35,095
37.3%
12,301
13.6%
11,688
13.5%
22,167
27.3%
18,242
23.3%
177,464
24.1%
137,055
23.8%
Selected
This example uses only the umbrella losses with underlying Comm Auto
Liability losses and 100% of the EP, therefore a partial LR is calculated
Todd Cheema, FCAS, MAAA
12
23.8%
Partial Loss Ratio Using Individual Losses
with Underlying General Liability
Commercial Umbrella with U/L GL - Treaty Eff Date = 9/1/2010 (amounts in '000's)
(1)
Accident
Ultimate
Year
Earn Prem
2003
87,000
2004
92,000
2005
95,000
2006
96,000
2007
93,000
2008
89,000
2009
83,000
2010
79,000
Total 03-10
714,000
Total 03-08
552,000
Notes :

(2)
On-Level
Factor
1.250
1.080
1.020
0.980
0.970
0.970
0.980
0.990
(7)=(4)*(5)+[1-(6)]*(3)*(9)
(3)=(1)*(2)
(4)
Loss &
Trended
ALAE
On-level EP @ 3/31/10
108,750
11,518
99,360
7,101
96,900
37,790
94,080
5,014
90,210
1,892
86,330
8,085
81,340
0
78,210
0
735,180
71,401
575,630
71,401
(5)
Frequency
& Severity
Trend
1.874
1.971
1.259
1.597
1.672
1.149
(6)
%
Reported
85.5%
72.7%
54.7%
35.4%
19.6%
9.6%
1.1%
0.4%
(9) a-priori LR (yrs 03-08) = 37.4%
(7)
(8)=(7)/(6)
Trended
Trended
Ult Loss & Ult Loss &
ALAE
ALAE Ratio
27,501
25.3%
24,128
24.3%
64,003
66.1%
30,754
32.7%
30,298
33.6%
38,474
44.6%
30,091
37.0%
29,123
37.2%
274,371
37.3%
215,157
37.4%
Selected
This example uses only the umbrella losses with underlying General
Liability losses and 100% of the EP, therefore a partial LR is calculated
Todd Cheema, FCAS, MAAA
13
37.4%
Capping at Various Limits
Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's)
Exposure Rating for Commercial Auto Liability
Loss Ratio
Layer
Relativity
Layer
5,000,000
xs 0
65.6%
20,000,000
xs 5,000,000
10,000,000
xs 0
81.8%
15,000,000
xs 10,000,000
15,000,000
xs 0
90.3%
10,000,000
xs 15,000,000
Todd Cheema, FCAS, MAAA
Loss Ratio
Relativity
34.4%
18.2%
9.7%
14
Total
100.0%
100.0%
100.0%
Partial Loss Ratio Using Capping of
Individual Losses with U/L Auto Liability
Commercial Umbrella with U/L AL - Treaty Eff Date = 9/1/2010 (amounts in '000's)
Individual Loss & ALAE are Capped at $10M
(1)
Accident
Ultimate
Year
Earn Prem
2003
87,000
2004
92,000
2005
95,000
2006
96,000
2007
93,000
2008
89,000
2009
83,000
2010
79,000
Total 03-10
714,000
Total 03-08
552,000
Notes :

(2)
(3)=(1)*(2)
On-Level
Trended
Factor
On-level EP
1.250
108,750
1.080
99,360
1.020
96,900
0.980
94,080
0.970
90,210
0.970
86,330
0.980
81,340
0.990
78,210
735,180
575,630
(7)=(4)*(5)+[1-(6)]*(3)*(9)
(9) a-priori LR (AY 03-08) = 21.6%
(4)
(5)
Cap = $10M Frequency
Loss&ALAE & Severity
@ 3/31/10
Trend
24,053
1.037
3,386
1.358
36,093
1.140
27,618
1.000
6,032
1.180
1,364
1.218
3,465
1.092
0
102,011
98,546
(6)
%
Reported
98.9%
97.8%
95.6%
91.3%
75.9%
51.2%
5.0%
2.0%
(7)
(8)=(7)/(6)
Trended
Trended
Ult Loss & Ult Loss &
ALAE
ALAE Ratio
25,212
23.2%
5,080
5.1%
42,066
43.4%
29,379
31.2%
11,822
13.1%
10,761
12.5%
20,468
25.2%
16,555
21.2%
161,343
21.9%
124,320
21.6%
(10) Selected w/L&ALAE Cap at $10M
(11)
% of Loss > $10M Cap
(12)=(10)*{1-[1/(11]}Selected Unlim LR
This example uses only the umbrella losses with underlying Comm Auto
Liability losses and 100% of the EP, therefore a partial LR is calculated
Todd Cheema, FCAS, MAAA
15
21.6%
18.2%
26.4%
Loss Ratio Using Individual Losses –
Summary of Capping at Various Limits
Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's)
Cap
5,000,000
10,000,000
15,000,000
Uncapped
Selected
Notes :
(1)
(2)
(3)=(1)*{1-[1/(2)]}
(4)
(5)
(6)=(4)*{1-[1/(5)]}
(7)
Commercial Umbrella with U/L AL
Commercial Umbrella with U/L GL
Total
Capped Loss
% of Loss Uncapped Loss Capped Loss % of Loss Uncapped Loss Uncapped Loss
& ALAE Ratio XS of Cap & ALAE Ratio & ALAE Ratio XS of Cap
& ALAE Ratio
& ALAE Ratio
17.3%
34.4%
26.3%
27.6%
35.5%
42.8%
69.1%
21.6%
18.2%
26.4%
33.9%
21.3%
43.1%
69.5%
23.3%
9.7%
25.8%
37.0%
13.0%
42.6%
68.3%
23.8%
0.0%
23.8%
37.4%
0.0%
37.4%
61.2%
26.3%
42.8%
69.1%
(2),(5) : % of loss XS of Cap is calculated
from exposure curves
Method Using Losses Aggreagted by AY
Adjustment for Free Cover

Look at actual vs. expected claim counts to help select where to cap

Adjust development factors to account for capping at different limits

Loss and ALAE behave differently at various capping limits
Todd Cheema, FCAS, MAAA
16
65.2%
3.9%
Thank you
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Todd Cheema, FCAS, MAAA
18