Calculating a Loss Ratio for Commercial Umbrella · Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's) Comm Umb with U/L Auto Liab Comm Umb with U/L Gen Liab Ultimate

Calculating a Loss Ratio for Commercial Umbrella Todd Cheema, FCAS, MAAA

CARe, Philadelphia, PA June 6-7, 2011

Todd Cheema, FCAS, MAAA

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This session will present an alternate method of arriving at a loss ratio for pricing for commercial umbrella.

Outline

n  Discuss the weaknesses of the traditional method, which applies average trend and development factors to aggregate losses by accident year or policy year.

n  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.

n  Discuss the differences between primary and umbrella layers form frequency and severity trends.

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Overview


n  Losses are aggregated by PY or AY

n  Aggregate losses are trended by an excess trend

n  Aggregate trended losses are developed to ultimate

n  LR by PY or AY = selected ultimate loss divided by on-level premium

n  Select a LR for the prospective Treaty Year

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Traditional Method


n  LR could be understated due to free cover - (Skewed dist will have mode<mean)

n  LR could be overstated due to shock losses

n  The underlying loss can be AL, Prem/Ops, or Prod, therefore applying average trends and average development factors can lead to misleading results

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Weaknesses of the Traditional method

Todd Cheema, FCAS, MAAA 6

Example of Loss Ratio Calculation for Commercial Umbrella Using the Traditional method

(1) (2) (3)=(1)*(2) (4) (5) (6) (7) (8)=(7)/(6)Loss & Frequency Trended Trended

Accident Ultimate On-Level Trended ALAE & Severity % Ult Loss & Ult Loss &Year Earn Prem Factor On-level EP @ 3/31/10 Trend Reported ALAE ALAE Ratio2003 87,000 1.250 108,750 36,156 1.436 90.4% 58,709 54.0%2004 92,000 1.080 99,360 10,487 1.393 82.0% 26,274 26.4%2005 95,000 1.020 96,900 73,883 1.358 69.8% 119,376 123.2%2006 96,000 0.980 94,080 34,493 1.314 56.1% 72,257 76.8%2007 93,000 0.970 90,210 7,924 1.257 40.4% 44,998 49.9%2008 89,000 0.970 86,330 9,449 1.197 25.0% 53,519 62.0%2009 83,000 0.980 81,340 3,465 1.135 2.6% 55,613 68.4%2010 79,000 0.990 78,210 0 1.092 0.3% 50,853 65.0%

Total 03-10 714,000 735,180 175,857 481,599 65.5%Total 03-08 552,000 575,630 172,393 375,132 65.2%

Notes : (7)=(4)*(5)+[1-(6)]*(3)*(9) (9) a-priori LR (yrs 03-08) = 65.2% Selected 65.2%

Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's)


Comparison of Trended Loss from LOB Specific and Average Trend Factors

(1) (2) (3)=(1)*(2) (4) (5) (6)=(4)*(5) (7)=(3)+(6) (8)

Accident Loss AL Freq & Tr Loss Loss GL Freq & Tr Loss Individual AggregateYear & ALAE Sev Trend & ALAE & ALAE Sev Trend & ALAE Loss Method Loss Method2003 24,637 1.221 30,088 11,518 1.874 21,586 51,675 51,9212004 3,386 1.358 4,598 7,101 1.971 13,994 18,592 14,6142005 36,093 1.148 41,435 37,790 1.259 47,579 89,013 100,3112006 29,479 1.124 33,148 5,014 1.597 8,008 41,156 45,3182007 6,032 1.180 7,118 1,892 1.672 3,164 10,282 9,9592008 1,364 1.218 1,661 8,085 1.149 9,290 10,951 11,3152009 3,465 1.092 3,784 0 0 3,784 3,9322010 0 0 0 0 0 0

Total 03-10 104,457 121,832 71,401 Calculation 103,622 225,454 237,370Total 03-08 100,992 118,048 71,401 on Slide 10 103,622 221,670 233,438

Notes :


Comm Umb with U/L Auto Liab Comm Umb with U/L Gen Liab Trended Loss & ALAE

(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


Comparison of Ultimate Loss from LOB Specific and Average Development Factors

(1) (2) (3)=(1)/(2) (4) (5) (6)=(4)/(5) (7)=(3)+(6) (8)

Accident Loss Ult Loss Loss Ult Loss Individual AggregateYear & ALAE % Rept & ALAE & ALAE % Rept & ALAE Loss Method Loss Method2003 24,637 98.9% 24,919 11,518 85.5% 13,478 38,397 39,9832004 3,386 97.8% 3,464 7,101 72.7% 9,764 13,228 12,7892005 36,093 95.6% 37,773 37,790 54.7% 69,110 106,883 105,8152006 29,479 91.3% 32,287 5,014 35.4% 14,182 46,469 61,5042007 6,032 75.9% 7,952 1,892 19.6% 9,665 17,617 19,6012008 1,364 51.2% 2,664 8,085 9.6% 84,101 86,765 37,7642009 3,465 5.0% 68,757 0 1.1% 0 68,757 135,8552010 0 2.0% 0 0 0.4% 0 0 0

Total 03-10 104,457 177,816 71,401 200,299 378,116 413,310Total 03-08 100,992 109,059 71,401 200,299 309,359 277,456


Comm Umb with U/L Auto Liab Comm Umb with U/L Gen Liab Ultimate Loss & ALAE


n  Trend each loss by a ground-up trend by the line of business from underlying loss, and cap at policy limits.

n  Develop losses with underlying GL losses separately from losses with underlying AL losses.

n  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).

n  Reconcile individual losses with the losses from the triangle.

n  Trending of ALAE only claims .

n  Account for losses that trend into the layer.

n  Recognize signal reserves.

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Using Individual Umbrella Losses


Using Individual Umbrella Losses – Example of Trend

(1) (2) (3) (4) (5) (6) (7) (8)Ground-up

AY 2005 Policy Attach Reported Reported Cumulative Sev Tr Sev Tr Sev TrendLoss Limit Point Loss ALAE Sev Trend Rept Loss Rept ALAE in Layer

Loss #1 10,000,000 1,000,000 5,761,500 86,250 45.5% 8,835,159 125,458 53.2%Loss #2 25,000,000 1,000,000 1,366,200 11,500 45.5% 2,441,833 16,728 78.5%Loss #3 25,000,000 1,000,000 25,000,000 230,000 45.5% 25,000,000 334,554 0.4%Loss #4 25,000,000 1,000,000 773,950 11,500 45.5% 1,580,356 16,728 103.3%Loss #5 10,000,000 1,000,000 2,300,000 87,400 45.5% 3,800,122 127,131 64.5%Loss #6 5,000,000 1,000,000 2,070,000 92,000 45.5% 3,465,568 133,822 66.5%

Total 37,271,650 518,650 45,123,038 754,419 21.4%

(9)=(3)+(4) (10)=(8) (11) (12) (13) (14)=(9)*[1+(12)]Adjust for Trended

Reported Severity Frequency Claims that Total ReportedAY L&ALAE Trend Trend Trend In Trend Adj L&ALAE

2005 37,790,300 21.4% 0.97% 2.71% 25.90% 47,578,903

Notes : (5) = 6.2% GL Severity trend per year from 7/1/05 to 9/1/11 (6) = Min{[(2)+(3)]*[1+(5)]-(2),(1)}(7)=(4)*[1+(5)] (8)=[(6)+(7)]/[(3)+(4)]-1 (13)=[1+(10)]*[1+(11)]*[1+(12)]-1

Trend for AY 2005 for Commercial Umbrella Claims with U/L GL


n  Frequency trend - Does the frequency trend observed in the primary layer apply to the umbrella layer?

n  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?

n  Adjustment for claims that trend into the umbrella layer – What is the best way to quantify this adjustment?

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Using Individual Umbrella Losses – Discussion of Trend


Partial Loss Ratio Using Individual Losses with Underlying Auto Liability


Accident Ultimate On-Level Trended ALAE & Severity % Ult Loss & Ult Loss &Year Earn Prem Factor On-level EP @ 3/31/10 Trend Reported ALAE ALAE Ratio2003 87,000 1.250 108,750 24,637 1.221 98.9% 30,381 27.9%2004 92,000 1.080 99,360 3,386 1.358 97.8% 5,129 5.2%2005 95,000 1.020 96,900 36,093 1.148 95.6% 42,460 43.8%2006 96,000 0.980 94,080 29,479 1.124 91.3% 35,095 37.3%2007 93,000 0.970 90,210 6,032 1.180 75.9% 12,301 13.6%2008 89,000 0.970 86,330 1,364 1.218 51.2% 11,688 13.5%2009 83,000 0.980 81,340 3,465 1.092 5.0% 22,167 27.3%2010 79,000 0.990 78,210 0 2.0% 18,242 23.3%

Total 03-10 714,000 735,180 104,457 177,464 24.1%Total 03-08 552,000 575,630 100,992 137,055 23.8%


Commercial Umbrella with U/L AL - Treaty Eff Date = 9/1/2010 (amounts in '000's)

n  This example uses only the umbrella losses with underlying Comm Auto Liability losses and 100% of the EP, therefore a partial LR is calculated


Partial Loss Ratio Using Individual Losses with Underlying General Liability


Accident Ultimate On-Level Trended ALAE & Severity % Ult Loss & Ult Loss &Year Earn Prem Factor On-level EP @ 3/31/10 Trend Reported ALAE ALAE Ratio2003 87,000 1.250 108,750 11,518 1.874 85.5% 27,501 25.3%2004 92,000 1.080 99,360 7,101 1.971 72.7% 24,128 24.3%2005 95,000 1.020 96,900 37,790 1.259 54.7% 64,003 66.1%2006 96,000 0.980 94,080 5,014 1.597 35.4% 30,754 32.7%2007 93,000 0.970 90,210 1,892 1.672 19.6% 30,298 33.6%2008 89,000 0.970 86,330 8,085 1.149 9.6% 38,474 44.6%2009 83,000 0.980 81,340 0 1.1% 30,091 37.0%2010 79,000 0.990 78,210 0 0.4% 29,123 37.2%

Total 03-10 714,000 735,180 71,401 274,371 37.3%Total 03-08 552,000 575,630 71,401 215,157 37.4%


Commercial Umbrella with U/L GL - Treaty Eff Date = 9/1/2010 (amounts in '000's)

n  This example uses only the umbrella losses with underlying General Liability losses and 100% of the EP, therefore a partial LR is calculated


Capping at Various Limits

Exposure Rating for Commercial Auto LiabilityLoss Ratio Loss RatioRelativity Relativity Total

5,000,000 xs 0 65.6% 20,000,000 xs 5,000,000 34.4% 100.0%10,000,000 xs 0 81.8% 15,000,000 xs 10,000,000 18.2% 100.0%15,000,000 xs 0 90.3% 10,000,000 xs 15,000,000 9.7% 100.0%


Layer Layer


Partial Loss Ratio Using Capping of Individual Losses with U/L Auto Liability

(1) (2) (3)=(1)*(2) (4) (5) (6) (7) (8)=(7)/(6)Cap = $10M Frequency Trended Trended

Accident Ultimate On-Level Trended Loss&ALAE & Severity % Ult Loss & Ult Loss &Year Earn Prem Factor On-level EP @ 3/31/10 Trend Reported ALAE ALAE Ratio2003 87,000 1.250 108,750 24,053 1.037 98.9% 25,212 23.2%2004 92,000 1.080 99,360 3,386 1.358 97.8% 5,080 5.1%2005 95,000 1.020 96,900 36,093 1.140 95.6% 42,066 43.4%2006 96,000 0.980 94,080 27,618 1.000 91.3% 29,379 31.2%2007 93,000 0.970 90,210 6,032 1.180 75.9% 11,822 13.1%2008 89,000 0.970 86,330 1,364 1.218 51.2% 10,761 12.5%2009 83,000 0.980 81,340 3,465 1.092 5.0% 20,468 25.2%2010 79,000 0.990 78,210 0 2.0% 16,555 21.2%

Total 03-10 714,000 735,180 102,011 161,343 21.9%Total 03-08 552,000 575,630 98,546 124,320 21.6%

Notes : (7)=(4)*(5)+[1-(6)]*(3)*(9) (10) Selected w/L&ALAE Cap at $10M 21.6%(9) a-priori LR (AY 03-08) = 21.6% (11) % of Loss > $10M Cap 18.2%

(12)=(10)*{1-[1/(11]}Selected Unlim LR 26.4%

Commercial Umbrella with U/L AL - Treaty Eff Date = 9/1/2010 (amounts in '000's)Individual Loss & ALAE are Capped at $10M

n  This example uses only the umbrella losses with underlying Comm Auto Liability losses and 100% of the EP, therefore a partial LR is calculated


Loss Ratio Using Individual Losses – Summary of Capping at Various Limits

(1) (2) (3)=(1)*{1-[1/(2)]} (4) (5) (6)=(4)*{1-[1/(5)]} (7)Total

Capped Loss % of Loss Uncapped Loss Capped Loss % of Loss Uncapped Loss Uncapped LossCap & ALAE Ratio XS of Cap & ALAE Ratio & ALAE Ratio XS of Cap & ALAE Ratio & ALAE Ratio5,000,000 17.3% 34.4% 26.3% 27.6% 35.5% 42.8% 69.1%10,000,000 21.6% 18.2% 26.4% 33.9% 21.3% 43.1% 69.5%15,000,000 23.3% 9.7% 25.8% 37.0% 13.0% 42.6% 68.3%Uncapped 23.8% 0.0% 23.8% 37.4% 0.0% 37.4% 61.2%Selected 26.3% 42.8% 69.1%

Notes : (2),(5) : % of loss XS of Cap is calculated Method Using Losses Aggreagted by AY 65.2%from exposure curves Adjustment for Free Cover 3.9%


Commercial Umbrella with U/L AL Commercial Umbrella with U/L GL

n  Look at actual vs. expected claim counts to help select where to cap

n  Adjust development factors to account for capping at different limits

n  Loss and ALAE behave differently at various capping limits

Thank you


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Documents

Calculating a Loss Ratio for Commercial Umbrella · Commercial Umbrella - Treaty Eff Date = 9/1/2010 (amounts in '000's) Comm Umb with U/L Auto Liab Comm Umb with U/L Gen Liab Ultimate