Trending Entry Ratio Tables

Transcription

1 Tending Enty Ratio Tables Dan Coo, NCCI CARe Semina on Reinsuance June, 2006

2 Desciption of the Issue Outline A Thought Expeiment Theoetical Setup in Geneal Two Geneal Fomulas Two Real wold Case Studies Diffeential Seveity Diffeential Fequency Compaison of Tends Implications fo Updating ELFs Questions 2

3 Desciption of the Issue Enty atios automatically adjust fo monetay inflation ove time, but this equies the shape of the seveity cuve to emain unchanged Medical and indemnity loss components may gow at diffeent ates, and so the shape of seveity distibution may change We have seen a decline in case fequency in which small claims saw a geate decline, and so the shape of the seveity distibution changed What ae easonable inteim adjustments, shot of completely ebuilding enty atio tables using new data Pesumption is pape will appea in CAS Foum which pesents this discussion in geate detail me at Dan_Coo@nccicom with any follow-up questions 3

4 A Thought Expeiment Conside 100 WC claims: 99 at $1 each 1 at $10M (Mean is about $100K) Fist, suppose diffeential seveity tend selectively impacts lage claims and esults in: 99 at $1 each stay the same 1 at $10M doubles to $20M (Tended Mean is about $200K) 4

5 A Thought Expeiment Diffeential Seveity Tend 1 New Excess atio is above Old Excess Ratio as a function of Loss Attachment Amount Excess Ratio Old New $1 $10M $20M Loss Attachment Amount 5

6 A Thought Expeiment Diffeential Seveity Tend 1 New Excess Ratio emains above Old Excess Ratio as a function of Nomalized Loss = Enty Ratio Excess Ratio Old New $1 New Mean $1 Old Mean $10M Old Mean 100 $20M New Mean Enty Ratio 6

7 A Thought Expeiment Conside the same 100 WC claims: 99 at $1 each 1 at $10M (Mean is about $100K) Now suppose diffeential fequency tend selectively emoves about half the small claims and esults in only 50 claims: 49 at $1 each 1 at $10M (Tended Mean is about $200K) 7

8 A Thought Expeiment Diffeential Fequency Tend 1 New Excess Ratio is above Old Excess Ratio as a function of Loss Attachment Amount Excess Ratio Old New $1 $10M Loss Attachment Amount 8

9 A Thought Expeiment Diffeential Fequency Tend 1 New Excess Ratio stats above but then falls below Old Excess Ratio as a function of Nomalized Loss = Enty Ratio Excess Ratio New Old $1 New Mean $1 Old Mean $10M New Mean 50 $10M Old Mean 100 Enty Ratio 9

10 A Thought Expeiment Compae Enty Ratio Table Lookup with two Scenaios, Each with a Tended Mean of $200K Loss Amount Enty Ratio (Old Line on Gaph) Enty Ratio Table Lookup Excess Ratio (New Line on Gaph) Diffeential Seveity Excess Ratio (New Line on Gaph) Diffeential Fequency Excess Ratio $ $ $ $ $100K $500K $1M $5M $10M $20M

11 A Thought Expeiment Both tend scenaios doubled the mean cost pe case But the impact on the excess atio was diffeent Suggests something genuinely diffeent is happening Suggests common intuition, that shift to a moe sevee loss distibution makes enty atio tables fo ELFs go up, is not always coect 11

12 The Geneal Theoetical Setup Let be a andom vaiable of individual claim loss amounts F = distibution function f = PDF µ = aveage cost pe case, µ = = enty atio Speadsheet fiendly enty atio table is AB whee: 0 < µ < A ( ) = F ( µ ) = µ f ( µ x) dx B ( ) = µ Excess Ratio 0 ( ) = 1 B ( ) ( 1 A ( ) ) 0 xf ( µ x) dx 12

13 13 Two Geneal Fomulas Let be a andom vaiable of tended individual claim loss amounts: ( ) ( ) ) ( ) ( lim ) ( lim ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (,,,,, f f f A A F F d d µ µ µ µ µ µ ρ µ µ δ δ ρ δ = = = = = then Excess Ratio Excess Ratio Excess Ratio Define

14 Case Study: Diffeential Seveity Obsevations: Medical is unlimited, indemnity is limited Medical dominates the tail Medical seveity has gown faste than indemnity Issue: Quantifying the impact of the change in the seveity cuve on the excess atio 14

15 Diffeential Seveity Model Calculation Steps Let P ind ()=pecent of indemnity losses at enty atio Let 1 < T ind = constant indemnity tend facto and 1 < T med = constant medical tend facto Oveall tend facto depends on enty atio h sev () = P ind ()x T ind + (1- P ind ())x T med Because P ind () deceases with, tending peseves layes of enty atio, so we continue to take A () as values of the nomalized cumulative distibution function, getting a distibution function F defined fom: = h sev () 15

16 Diffeential Seveity Model Moe Calculation Steps Calculate an additive adjustment facto = δ, ( ) = Excess Ratio ( ) Excess Ratio ( ) Do this fo each injuy type to get adjustment facto by injuy type Combine to get oveall adjustment facto using fact that combined excess atio is loss-weighted sum of excess atios by injuy type Fo einsuance, a efinement may be to allow medical inflation to vay by size of (medical) loss, T med = T med () 16

17 In Theoy, What holds fo Diffeential Seveity Models? Assume given a seveity tend facto h sev ( x) 1, dh dx sev > 0, = h sev ( ), h sev (x) such that and that f ( x) > 0 fo all x > 0, then µ > µ and δ, ( ) > 0 fo all > 0 17

18 Diffeential Seveity Model Compaison with Diect Calculation Calculation Compaison: Diffeential Tend Impact on Excess Loss Ratio diect simplified $1K $5K $10K $25K $50K $100K $250K $500K $750K $1M $25M $5M $10M $25M $50M $100M Attachment Annual medical tend facto=1095 Reflects a 7 ea Window Annual indemnity tend facto=1075 Pio to catastophic loss exclusion 18

19 Diffeential Seveity-Findings Afte 10 yeas the elative impact due to diffeential seveity tend is less than 1% at attachment $500K, but can gow to 30% at attachment $50M Testing eveals vey little diffeence in the woking layes (above $25K limitation) between applying medical and indemnity tend factos by claim and the diffeential seveity model Diffeential seveity tend is a tuning adjustment (compaed with oveall tend) suited to be elegated to a stand alone additive adjustment Thee is a ationale fo making an ongoing diffeential seveity tend adjustment a pat of the ELF update pocess, athe than awaiting ELF tables fom new seveity cuves 19

20 Diffeential Fequency Obsevations: Claim fequency has consistently declined since 1990 No clea consensus on the causes and futue of the patten Decline is not unifom by claim size smalle claims have seen geate decline Issue: Quantifying the impact of the change in the seveity cuve on the excess atio 20

21 Diffeential Fequency Model Calculation Steps Let h feq (x) = mean pecentage eduction in claim occuence ate at loss amount x Assume 0 h feq (x) 1 and h feq (x) is non-deceasing and piecewise continuous Tended PDF f ( x) = h feq E[ h ( x) f feq ( x) ( )] This gives the distibution function F fo tended losses as: F ( y) = y 0 f ( x) dx 21

22 Diffeential Fequency Model Moe Calculation Steps Calculate an additive adjustment facto δ, ( ) = Excess Ratio ( ) Excess Ratio ( ) Do this fo each injuy type to get adjustment facto by injuy type Combine to get oveall adjustment facto using fact that combined excess atio is loss-weighted sum of excess atios by injuy type Fo einsuance, model has built in capability to teat lagest claims diffeently 22

23 In Theoy, What holds fo Diffeential Fequency Models? Maximum Loss Case Assume given a fequency change facto 0 h feq dh ( x) 1, dx feq and that thee is some M 0, 0 < with E[ h f feq h feq ( )] < 1, (x) such that : ( x) = ( x) = 0 fo all x > M f hfeq ( x) f ( x), E[ h ( )], feq then µ > µ and thee is some b > 0 such that δ, ( b) < 0 and δ, ( ) 0 fo all > b 23

24 0 h f In Theoy, What holds fo Diffeential Fequency Models? No Maximum Loss Case Assume given a fequency change facto with and feq dh ( x) 1, dx ( x) > 0 fo all feq 0, 0 < x > 0 and h E[ h feq feq h ( )] < 1, (x) = η is constant ( x) feq (x) such that : f ( x) = µ x 1 F µ futhe that the limit λ = lim exists, x 1 F h feq ( x) f ( x), E[ h ( )] feq fo x lage then µ > µ and thee is some constant b > 0 such that ηλ > 1 implies that δ ηλ < 1 implies that δ,, ( ) > 0 fo all > b, and ( ) < 0 fo all > b 24

25 Diffeential Fequency Model Histoical Example Claim Fequency by Injuy Type and by Size of Loss Five-ea Change Between Policies Expiing in 1999 and 2003 Pecent Change in Lost-Time Claim Fequency pe $1M Wage-Adjusted Payoll % -22% -15% Fatal Pemanent Total Pemanent Patial -21% -24% Smallest 1/3 Middle 1/3 Lagest 1/3 Tempoay Total Medical-Only Annual Loss Size Adjustments: 42% Indemnity, 74% Medical Fequency = Lost-time claims / payoll; Payoll adjusted fo inflation Claim count detemined at fist epot NCCI states, excluding Texas Claims gouped accoding to thei 1999 claim size within injuy type 25

26 Diffeential Fequency Model Absolute Impact by Attachment Absolute Impact of Fequency Decline Excess Ratio $0 $2,500,000 $5,000,000 $7,500,000 $10,000,000 Attachm ent Reflects the impact on the Excess Ratio beyond the natual incease due to the highe ACCs Assumes countywide loss weights and mean cost pe case Reflects the fequency decline of the expeience eview 26

27 Diffeential Fequency Model Relative Impact by Attachment 000% Relative Impact of Fequency Decline Excess Ratio -500% -1000% -1500% $0 $2,500,000 $5,000,000 $7,500,000 $10,000,000 Attachment Reflects the impact on the Excess Ratio beyond the natual incease due to the highe ACCs Assumes countywide loss weights and mean cost pe case Reflects the fequency decline of the expeience eview 27

28 Diffeential Fequency Model Update Claim Fequency by Size of Loss Diffeential Patten Appeas to be Abating ea by ea Changes Ove 5 eas, Policies Expiing fom 2000 to 2004 Pecent Change in Lost-Time Claim Fequency % -7% -5% -4% Size of Loss $0 - $2K $2K - $10K $10K - $50K $50K - ove Policy Expiation ea Annual Loss Size Adjustments: 38% Indemnity, 74% Medical Fequency = Lost-time claims / payoll; Payoll adjusted fo inflation Claim count detemined at fist epot NCCI states, excluding Texas Claims gouped accoding to Policy Expiation ea 2000 claim size 28

29 Diffeential Fequency-Findings The impact of Diffeential Fequency on excess atios can be numeically upwad o downwad ove the ange of attachment points; it is small in absolute tems, but can be significant in elative tems as the attachment point inceases The patten of the fequency decline (small claims decline moe) has been consistent ove the ecent past, but the mechanism to poduce that patten and how long it may pesist emains unclea It is difficult to assess a tuning point in a tend adjustment when both the undelying mechanism is unclea and the diection of the adjustment is ambiguous 29

30 Compaison of Tends Diffeential Seveity-Indemnity vs Medical inflation Clea mechanism: medical inflation highe than wage inflation Expect patten to continue ove the nea tem Diection of the change to oveall excess atio is intuitive and unambiguous upwad Diffeential Fequency- Geate fequency decline fo small claims Expets diffe on the mechanism diving fequency decline Expets diffe on whethe the decline will continue ove the nea tem Latest expeience eview still shows a fequency decline but the diffeential patten seems to be abating Diection of the change to oveall excess atio is ambiguous 30

31 Implications fo Updating ELFs Matching one moment (mean) is geneally a good appoach, but can become poblematic when seveity distibution is subject to non-unifom changes Pape discusses matching a pecentile (eg median) as well as using enty atios to match the mean Cuently woking on a genealized AB enty atio table fomat that can match the coefficient of vaiation of a loss seveity distibution (in effect, matches fist two moments) 31

32 Moe Implications fo Updating ELFs A tend in the mean cost pe case that is much diffeent than oveall inflation suggests such a non-unifom change A change in the coefficient of vaiation o in the atio of the median to the mean also suggest a non-unifom change Open issue: When seveity distibution is subject to non-unifom changes, may want to conside adjusting the enty atio of the loss limit only fo coe inflation (meaning?) and not by the full change in the mean cost pe case 32

33 Questions? 33