1998 CASUALTY LOSS RESERVE SEMINAR Intermediate Track II - Techniques SEPTEMBER 28, 1998

RESERVE MODELS Average Hindsight Reserve Method (Slides 2 - 12) Average Incremental Paid Method (Slides 13 - 19) Bornhuetter-Ferguson Method (Slides 20 - 30) Slide 1

AVERAGE HINDSIGHT RESERVE METHOD Goal: Estimate The Average Future Settlement Value Per Claim For . Recent Accident Years, Both For Claims Already Reported . and Future Claim Reports Based On: Projected Ultimate Losses and Hindsight . Average Values For More Mature Accident Years Hindsight (Current Ultimate-Historical Payments) is the . Implied Unpaid Amount Slide 2

AVERAGE HINDSIGHT RESERVE METHOD Data Needed Cumulative Paid Loss Triangle Cumulative Closed (Paid) Claim Count Triangle Estimated Ultimate Number of Claims for All Accident Years Estimated Ultimate Losses For Several Mature Accident Years Slide 3

AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Cumulative Paid Losses ($000) Accident Months Of Development . Year 12 24 36 48 60 72 84 Ultimate 1991 $50.0 $80.0 $98.2 $107.8 $113.2 $117.2 $119.7 $119.7 1992 60.2 97.0 118.5 130.7 136.6 141.0 143.8 1993 75.5 120.1 147.0 162.4 171.0 178.7 1994 91.9 147.1 180.2 197.0 220.1 1995 115.0 184.1 226.4 * 1996 146.5 233.4 * 1997 181.1 * *To be estimated Slide 4

AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Cumulative Number Of Closed Claims Accident Months Of Development . Year 12 24 36 48 60 72 84 Ultimate* 1991 50 75 88 94 97 99 100 100 1992 55 83 97 104 107 109 110 1993 63 94 110 118 122 125 1994 70 105 123 131 140 1995 80 120 141 160 1996 93 139 185 1997 105 210 *Estimated using claim count development factors. Slide 5

Slide 6 (1) (2) (3) (4) (5) (6) (7) (8) AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Calculation Of Average Outstanding Losses At 36 Months Purpose: Project Future Settlement Dollars For 1995 Number Estimated Estimated Number of to Settle Average Estimated Paid Future Ultimate Closed Beyond Future Accident Ultimate Losses at Payments Number Claims at 36 Mos* Payment . Year Losses 36 Months = (2)-(3) of Claims 36 Months =(5)-(6) =(4)/(7) (1) (2) (3) (4) (5) (6) (7) (8) 1991 $119,700 $98,200 $21,500 100 88 12 $1,792 1992 143,800 118,500 25,300 110 97 13 1,946 1993 178,700 147,000 31,700 125 110 15 2,113 1994 220,100 180,200 39,900 140 123 17 2,347 *Includes IBNR Claims Fitted forecasted value for AY 1995 = $2,549 Slide 6

AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Estimated Ultimate Losses: Accident Year 1995 (1) Forecasted Average Future Payment [Slide 6] = $2,549 (2) Number of Future Claims to Settle [Slide 5] (Ultimate - Closed Claims) = 160 - 141 = 19 (3) Estimated Future Loss Payments [(1) x (2)] = $ 48,431 (4) Paid Losses to Date [Slide 4] = $226,400 (5) Estimated Ultimate Losses [(3) + (4)] = $274,831 Slide 7

Slide 8 (1) (2) (3) (4) (5) (6) (7) (8) AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Calculation Of Average Outstanding Losses At 24 Months Purpose: Project Future Settlement Dollars For 1996 Number Estimated Estimated Number of to Settle Average Estimated Paid Future Ultimate Closed Beyond Future Accident Ultimate Losses at Payments Number Claims at 24 Mos* Payment . Year Losses 24 Months = (2)-(3) of Claims 24 Months =(5)-(6) =(4)/(7) (1) (2) (3) (4) (5) (6) (7) (8) 1992 $143,800 $97,000 $46,800 110 83 27 $1,733 1993 178,700 120,100 58,600 125 94 31 1,890 1994 220,100 147,100 73,000 140 105 35 2,086 1995 274,831 184,100 90,731 160 120 40 2,268 *Includes IBNR Claims Fitted forecasted value for AY 1996 = $2,484 Slide 8

AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Estimated Ultimate Losses: Accident Year 1996 (1) Forecasted Average Future Payment [Slide 8] = $2,484 (2) Number of Future Claims to Settle [Slide 5] (Ultimate - Closed Claims) = 185 - 139 = 46 (3) Estimated Future Loss Payments [(1) x (2)] = $ 114,264 (4) Paid Losses to Date [Slide 4] = $233,400 (5) Estimated Ultimate Losses [(3) + (4)] = $347,664 Slide 9

Slide 10 (1) (2) (3) (4) (5) (6) (7) (8) AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Calculation Of Average Outstanding Losses At 12 Months Purpose: Project Future Settlement Dollars For 1997 Number Estimated Estimated Number of to Settle Average Estimated Paid Future Ultimate Closed Beyond Future Accident Ultimate Losses at Payments Number Claims at 12 Mos* Payment . Year Losses 12 Months = (2)-(3) of Claims 12 Months =(5)-(6) =(4)/(7) (1) (2) (3) (4) (5) (6) (7) (8) 1993 $178,700 $75,500 $103,200 125 63 62 $1,665 1994 220,100 91,900 128,200 140 70 70 1,831 1995 274,831 115,000 159,831 160 80 80 1,998 1996 347,664 146,500 201,164 185 93 92 2,187 *Includes IBNR Claims Fitted forecasted value for AY 1997 = $2,397 Slide 10

AVERAGE HINDSIGHT RESERVE METHOD XYZ Auto Insurance Company Estimated Ultimate Losses: Accident Year 1997 (1) Forecasted Average Future Payment [Slide 10] = $2,397 (2) Number of Future Claims to Settle [Slide 5] (Ultimate - Closed Claims) = 210 - 105 = 105 (3) Estimated Future Loss Payments [(1) x (2)] = $ 251,385 (4) Paid Losses to Date [Slide 4] = $181,100 (5) Estimated Ultimate Losses [(3) + (4)] = $432,785 Slide 11

AVERAGE HINDSIGHT RESERVE METHOD ADVANTAGES Relatively Unaffected by Changes in Case Reserving Practices Can Easily Adjust Trend Assumption Allows Separate Analysis of Frequency and Severity DISADVANTAGES Sensitive to Payment Pattern Shifts Averages Highly Variable When Only a Few Claims May be Insufficient if Business has Significantly . . Changed (Example: Retentions Dramatically Increase) Too “Formula” Driven Slide 12

AVERAGE INCREMENTAL PAID METHOD Incremental Paid Losses per Ultimate Claim (Actual) Accident Months Of Development . Year 0-12 12-24 24-36 36-48 48-60 60-72 72-84 1991 500 300 182 96 54 40 25 1992 547 335* 195 111 54 40 1993 604 357 215 123 69 1994 656 394 236 120 1995 719 432 264 1996 792 470 1997 862 * 335 = (97,000-60,200) / 110 Slide 13

AVERAGE INCREMENTAL PAID METHOD Trend Factors Accident Months Of Development . Year 0-12 12-24 24-36 36-48 48-60 60-72 72-84 1992/91 9.4% 11.7% 7.1% 15.6% 0.0% 0.0% 1993/92 10.4% 6.6%* 10.3% 10.8% 27.8% 1994/93 8.6% 10.4% 9.8% - 2.4% 1995/94 9.6% 9.6% 11.9% 1996/95 10.2% 8.8% 1997/96 8.8% Select 9.0% 9.0% 9.0% 9.0% 9.0% 9.0% 9.0% * 6.6% = 357 / 335 - 1 Slide 14

AVERAGE INCREMENTAL PAID METHOD Incremental Paid Losses per Ultimate Claim (On-level) Accident Months Of Development . Year 0-12 12-24 24-36 36-48 48-60 60-72 72-84 1991 914 503 280 136 70 48 27 1992 917 515* 275 144 64 44 1993 929 504 278 146 75 1994 926 510 280 131 1995 931 513 288 1996 941 512 1997 940 Select N/A 510 280 140 70 46 27 * 515 = 335 x (1.09 ^ 5) Slide 15

AVERAGE INCREMENTAL PAID METHOD Incremental Paid Losses per Ultimate Claim (Projected) Accident Months Of Development . Year 0-12 12-24 24-36 36-48 48-60 60-72 72-84 Total 1991 0 1992 27 27 1993 46 29 75 1994 70 50 32 152 1995 140 76 55 35 306 1996 280 152 83* 60 38 613 1997 510 305 166 91 65 42 1,179 * 83 = 70 x (1.09 ^ 2) Slide 16

AVERAGE INCREMENTAL PAID METHOD Projected Ultimate Losses (1) (2) (3) (4) (5) Paid Projected Projected Projected Projected Severity Future Ultimate Ultimate Ultimate Accident as of Severity Severity Claims Losses ($000) . Year 12/31/97 (Slide 16) (1) x (2) (Slide 5) (3) x (4) 1991 $1,197 $0 $1,197 100 $119.7 1992 1,282 27 1,309 110 144.0 1993 1,368 75 1,443 125 180.4 1994 1,407 152 1,559 140 218.3 1995 1,415 306 1,721 160 275.4 1996 1,262 613 1,875 185 346.9 1997 862 1,179 2,041 210 428.6 Slide 17

AVERAGE INCREMENTAL PAID METHOD Summary Step 1: Project Ultimate Claims Step 2: Calculate Historical Severities Step 3: Select Trend Factor Step 4: Calculate On-level Severities Step 5: Select On-level Severities Step 6: Project Future Severities Step 7: Multiply Total Severities by Ultimate Claims Slide 18

AVERAGE INCREMENTAL PAID METHOD ADVANTAGES Allows separate analysis of frequency and severity trends. Can be modified to account for changes affecting accident year severity (e.g. . deductibles, benefit changes). Model can accommodate different trends by accident year, calendar year, or . development age. DISADVANTAGES Very dependent upon estimate of future inflation rates. Less accurate for low frequency lines of business. Could be distorted if payout patterns change. Slide 19

BORNHEUTTER-FERGUSON METHOD Data Needed Earned Premium or Exposure By Year A priori Expected Loss Ratio or Pure Premium For Each Year An Estimate of the Percent of Dollars Unreported, Usually . Based on Loss Development Factors (LDFs) Slide 20

BORNHUETTER-FERGUSON METHOD “IBNR” RESERVES As of the evaluation date, there are four categories of future claims activity that . may not be reflected in either claim payments or case reserves. 1. Losses Not Yet Reported To The Company 2. Claims In Transit (Reported But Not Recorded) 3. Future Development On Known Open Claims 4. Reopenings on Claims Currently Closed The Bornhuetter-Ferguson method and most accident year methods estimate “broad” IBNR which includes all four categories. NOTE: The sum of (1) and (2) is termed “True” or “Pure” IBNR. Slide 21

BORNHUETTER-FERGUSON METHOD Basic Formulas Expected Losses = Loss Ratio x Earned Premium = Pure Premium x Exposure = Frequency x Severity IBNR Reserve = IBNR Factor x Expected Losses Slide 22

BORNHUETTER-FERGUSON METHOD XYZ Auto Insurance Company Accident Year . 1995 1996 1997 (1) Earned Premium $1,250 $1,600 $2,000 (2) Expected Loss Ratio 65% 70% 75% (3) Initial Expected Losses (1) x (2) $813 $1,120 $1,500 (4) Development Factor 1.350 1.650 2.000 (5) IBNR Factor [1 - [1 / (4)] 26% 39% 50% (6) IBNR Reserve (3) x (5) $211 $437 $750 (7) Reported Losses $600 $700 $1,000 (8) Estimated Ultimate Losses (6) + (7) $811 $1,137 $1,750 Slide 23

BORNHUETTER-FERGUSON METHOD IBNR Factor Derivation IBNR Factor = IBNR / Ultimate Losses = Ultimate - Incurred to Date Ultimate = 1.000 - (Incurred to Date/Ultimate) = 1.000 - [1.000/(LDF-to-Ultimate)] = 1.000 - Percent Unreported = Percent Unreported Slide 24

BORNHUETTER-FERGUSON METHOD XYZ Auto Insurance Company Accident Year . 1995 1996 1997 (1) Ultimate Losses a. Expected (Slide 23, Line 3) 813 1,120 1,500 b. Estimated (Slide 23, Line 8) 811 1,137 1,750 (2) Reported Losses at 12/31/97 a. Expected (Slide 23, Line 3 - Line 6) 602 683 750 b. Actual 600 700 1,000 (3) Loss Ratio a. Expected (Slide 23, Line 2) 65.0% 70.0% 75.0% b. Estimated (Slide 23, Line 8/Line 1) 64.9% 71.1% 87.5% Slide 25

BORNHEUTTER-FERGUSON METHOD Considerations Premium Adequacy and Expected Loss Ratios - Sources: pricing assumptions, historical data such as Schedule P, industry data Changes in Operations: - Reinsurance - Longer-Tailed Lines (LDF selection more critical) - Underlying Limits, Deductibles - Claims Made versus Occurrence - Claims Handling Changes in Mix of Business That May Impact Either Loss Ratios, and/or Development Patterns Slide 26

BORNHUETTER-FERGUSON METHOD Advantages Easy to Use Compromises Between Loss Development and Expected Loss Ratio Methods Avoids Overreaction: Doesn’t apply Development Factors to an Unusual Claim Occurrence Suitable for New or Volatile Lines of Business Can be Used with No Internal Loss History Can also be Used with Paid Data Disadvantages Depends on Expected Loss Ratio or A Priori Pure Premium Requires Development Factors Slide 27

BORNHUETTER-FERGUSON METHOD XYZ Auto Insurance Company Illustration Of Tempering Effect One Extra Large Claim Expected of $150 EXPECTED LOSS RATIO METHOD (1) Earned Premium $2,000 $2,000 (2) Expected Loss Ratio 75% 75% (3) Projected Ultimate Losses (1) x (2) $1,500 $1,500 LOSS DEVELOPMENT METHOD (4) Actual Reported to Date $750 $900 (5) Development Factor 2.00 2.00 (6) Projected Ultimate Losses (4) x (5) $1,500 $1,800 Slide 28

BORNHUETTER-FERGUSON METHOD XYZ Auto Insurance Company Illustration Of Tempering Effect One Extra Large Claim Expected of $150 BORNHUETTER-FERGUSON METHOD (1) Earned Premium $2,000 $2,000 (2) Expected Loss Ratio 75% 75% (3) Expected Ultimate Losses (1) x (2) $1,500 $1,500 (4) Actual Reported to Date $750 $900 (5) Development Factor 2.00 2.00 (6) IBNR Factor 1 - [1 / (5)] 50% 50% (7) Projected Ultimate Losses $1,500 $1,650 (4) + [(3) x (6)] Slide 29

Slide 30 (2) Loss Development $1,500 $1,800 BORNHUETTER-FERGUSON METHOD XYZ Auto Insurance Company Illustration Of “Tempering” Effect One Extra Large Claim Method Expected of $150 (1) Expected Loss Ratio Method $1,500 $1,500 (2) Loss Development $1,500 $1,800 (3) Bornhuetter-Ferguson $1,500 $1,650 Slide 30