The Mack-Method and Analysis of Variability. Erasmus Gerigk

Transcription

1 The Mac-Method and Analyss of Varablty Erasmus Gerg

2 ontents/outlne Introducton Revew of two reservng recpes: Incremental Loss-Rato Method han-ladder Method Mac s model assumptons and estmatng varablty Estmatng Rs Margns Model testng Example Further research: extensons and refnements

3 Prelmnary remars Depends on the type of avalable data There are plenty of methods around Many deal wth projectng trangulated data In ths presentaton we focus entrely han-ladder types of methods Manfold termnology n the lterature

4 Incremental Loss-Rato Method Loo at loss rato ncrements: Exposure ,72 % 2% 3% 7% 4% 6% ,34 22% 57% 9% 3% 8%? 200 5,6 6% 7% 4% 5% ,789 2% 3% 5%? ,784 3% 7% ,445 9%????? and complete the rectangle Method s old and appears under many names????????

5 Incremental Loss-Rato Method Usually done through weghted averages: Exposure ,72 % 2% 3% 7% 4% 6% ,34 22% 57% 9% 3% 8% 6.2% 200 5,6 6% 7% 4% 5% 6.0% 6.2% ,789 2% 3% 5% 7.4% 6.0% 6.2% ,784 3% 7%.6% 7.4% 6.0% 6.2% ,445 9% 3.2%.6% 7.4% 6.0% 6.2% m 2 =3.2% m 3 =.6% = n+ n+ = mˆ = mˆ = n+ n+ v = v = S v

6 Summary: han-ladder Method Loo at ndvdual development factors: = = f 3 = ,29,406,784 2, ,840 2,273 2,568 2,994 3, ,84,424, ,743 2,043 2,456 3, ,364,953 2,288 2,75 3, ,23 3,039 3,56 4,282 5,343 fˆ calculate weghted averages = and complete the rectangle n n, = = F = n, n ˆ =,, + =, fˆ =, +,

7 Graphcal nterpretaton addtve projecton:, +, v = v + m 250% 200% 50% 00% 50% 0% Loss ratos Development years

8 Graphcal nterpretaton multplcatve projecton: log,, + = log + log f 0,000, Losses n $'000, log-scale Development years

9 Practcal ssues ) an any of the methods be expected to lead to reasonable results? 2) How to decde between the two models? 3) How to select the slope n the graphs,.e. the development factor? Is there a best way? 4) Both recpes provde a pont estmate. How to deal wth the requrement of rs margns?

10 The Mac-Method Is a textboo example of a proper statstcal model wth precse model assumptons and estmators. Through model-assumptons we wll re-nvent both methods In the Mac-Method, both procedures are extended to nclude varablty estmates Recall the graphcal nterpretatons. Many formal expressons correspond to vsble phenomena

11 Addtve model assumptons The Mac-method maes three model assumptons about the payments S n a partcular underwrtng / development year: (AM ) S v E = m n; n (AM 2) The payments S are ndependent for all, (AM 3) Var S v = s v 2 n; n (AM 3) may remnd you of the Indvdual Model

12 Estmatng the model parameters The ey results are mˆ = S v s Best Lnear Unbased Estmator of m sˆ 2 n n + = = v S v mˆ 2 s Unbased Estmator of 2 ˆ s All further results follow from straghtforward algebrac manpulatons

13 Further consequences There are straghtforward estmators for the standard errors of the ncrements mˆ There are closed-formula expressons for the standard error of the ultmate loss For worng wth spreadsheets, there are neat recursons-formulas It s possble to estmate the random error and the estmaton error separately

14 han-ladder model assumptons ;,,, = + n n f E K ;,,, 2,, = + n n Var σ K The underwrtng years {,, n } are globally ndependent,.e. the sets {,, n } are ndependent for j (L ) (L 2) (L 3) Model assumptons loo a bt more dffcult, because of condtonal expectaton Exposure measure s here the loss n the precedng perod

15 han-ladder estmators The results are very smlar to the addtve case. However, the proofs are more sophstcated. Key results: f s Best Lnear Unbased Estmator of + = f,, ˆ s Unbased for 2,,, 2 ˆ ˆ = + = n f n σ 2 σ And we have nce recursons for the standard error of the ultmate loss estmates (hard to prove!)

16 Rs Margns The total standard error enables us to calculate a stand-alone rs margn, e.g. at a 75% suffcency level The standard error and thus the rs margn depends only on the orgnal trangle The underlyng sgmas are often qute volatle and should not be used mechancally

17 From s.e. to Rs Margns Be aware of the propertes of your dstrbuton Popular choce: log-normal ft Frequency Mean V=75% Mean = maxmum possble margn 25.5% 75% Percentle 90% Percentle NB: upper bound by one-sded hebyshev P 2 [ X E( X ) α s. e.( X )] ( + α ) Alas, an nequalty of such generalty cannot lead to sharp results.

18 Intermedate summary Mac method provdes two self-contaned ways of obtanng central estmates as well as varaton estmates from trangulated data It s probably the smplest method avalable for dervng varablty estmates from trangles

19 aveats The relablty of the estmates depends on how well the data s descrbed by the model assumptons Whch model, f any, shall be preferred? Not yet clear how outlers can be dealt wth wthn the model, n partcular the calculaton of the standard errors more wor to be done by the actuary: model chec and dealng wth outlers

20 Testng the model assumptons A lot of nformaton can be extracted from the approprate graphs,.e. loss ratos or log-scaled dollars Qucly checed: parallel behavour of graphs and obvous outlers More accurate: resdual analyss from regresson theory Regresson approach wors for both the addtve and the multplcatve model

21 Regresson analyss (addtve) For a fxed development perod, loo agan at (AM ) E ( S ) = v m (AM 2) Independence (AM 3) ( S ) v s 2 Var = A statstcan s someone who calls that a heterosedastc regresson wthout ntercept

22 Model chec For each development year Plot ( v ; S ). Does t loo lnear? Plot the standardsed resduals S, v mˆ v aganst the exposure. There should be no pattern!

23 More on testng: calendar-year effects alendar year-effects have many causes, e.g. Inflaton hange n clams handlng hange n legslaton... They are actng on the dagonal!

24 Test for calendar-year effects Plot the standardsed resduals S, sˆ v mˆ v 2 + n for each calendar year. If all resduals have the same sgn, ths could ndcate a calendar-year effect. Ideally, there should be no pattern at all!

25 What to do wth outlers and cy-effects? Sum only over selected parts of the trangle. All estmators can be adjusted n a straghtforward way. In theory: ( w, S ) 2 S mˆ = sˆ =, ˆ w, v m ( w v ) I v w, Smlar for the other formulas Same for han-ladder 2

26 A bref example umulatve payments per development year Exposure ,24,240,353,396,42,488,545,608, ,39,85,337,597,677,79,788,869,887, ,055,097,25 996, ,276,452,584,627,70,727, , ,338,69,808 2,69 2,98 2, , ,036,566,724 2,054 2,50 2, ,72 272,2 4,642 2,600 2,89 3, ,34 62,35 2,30 2,592 2, ,6 56 2,849 5,32 5, ,789,737 3,958 5, ,836,567 4, ,936 - umulatve loss ratos per development year Exposure ,24 02% % 5% 7% 23% 27% 32% 34% 35% 994,39 04% 7% 40% 47% 5% 57% 64% 66% 67% % 8% 87% 94% 04% 04% 9% 24% 27% 996,89 30% 69% 07% 22% 33% 37% 43% 45% 47% 997,682 5% 44% 80% 0% 07% 29% 3% 4% 998 2,020 5% 5% 78% 85% 02% 06% % 999 2,72 0% 44% 7% 96% 06% 6% ,34 7% 49% 00% 2% 25% 200 5,6 0% 55% 99% 0% ,789 30% 68% 95% ,836 20% 60% ,936 0%

27 Graphcal run-off Loss ratos Losses n $'000, log-scale 80% 60% 0,000 40% 20% 00% 80%,000 60% 40% 20% 0% Development years Development years

28 Resdual analyss Plot of the addtve resduals per calendar year Ths could be a Y-effect. In fact, more nformaton s requred.

29 Resdual analyss loser loo at the ncremental loss-ratos: Loss rato ncrements per development year % 9% 4% 2% 6% 5% 5% % 2% % 3% 23% 7% 4% 6% 7% 2% % % 32% 6% 8% 0% 0% 5% 5% 3% % 39% 39% 5% % 4% 6% 2% 2% 997 5% 39% 36% 2% 7% 2% 2% 0% 998 5% 36% 26% 8% 6% 5% 4% 999 0% 34% 26% -75% % 0% % 42% 5% 2% 3% 200 0% 45% 44% 0% % 38% 26% % 40% % Do not sum over these values

30 oncluson Proper applcaton of the Mac-Method s not mechancal Judgemental adjustments can be ncorporated nto calculaton of standard errors It mght provde useful nformaton for establshng loss reserves It s all very smple

31 Somethng to read For the han-ladder Method: Mac, T [993] Dstrbuton free calculaton of the standard error of chan ladder reserve estmates In: ASTIN Bull. 23, Mac, T [994] Measurng the varablty of chan ladder reserve estmates In: AS Forum Sprng 994, pp (and many more publcatons ) For the Incremental Loss-Rato Method: Mac, T [997, 2002] Schadenverscherungsmathemat In: Karlsruhe Verlag Verscherungswrtschaft

32 Further refnements. Weghted regresson of development factors and sgmas for smoothng and extendng the run-off 2. Munch han-ladder 3. orrelaton between trangles