ࡱ>  LNGHIJK{QQS bjbj hh`Ci""gggTrSiiifJ>JJ )T+T+T+T+T+T+TnVY+TiKAfJKK+Tggi7(TQQQKgi)TLV"QK)TQQ Q\qMQTT0TQYOYQQYR\J>K,Q.K$RKJJJ+T+TQ:JJJTKKKKYJJJJJJJJJ" -: Some important properties of chain ladder models Section I. Mack (chain ladder) is volume-weighted average ratios When we compute a ratio, y/x, what are we looking at? Data are incurred losses from Mack (1994). 012345678915,0128,26910,90711,80513,53916,18118,00918,60818,66218,83421064,2855,39610,66613,78215,59915,49616,16916,70433,4108,99213,87316,14118,73522,21422,86323,46645,65511,55515,76621,26623,42526,08327,06751,0929,56515,83622,16925,95526,18061,5136,44511,70212,93515,85275574,02010,94612,31481,3516,94713,11293,1335,395102,063 Consider the ratio of the cumulative value in AY1, DY1 to the previous value in AY1: 0123(15,0128,26910,90711,805(21064,2855,39610,666(33,4108,99213,87316,141((((((( 8269/5012 = 1.65 Lets call the number on the numerator y and the number on the denominator x. Lets look at a graph of DY1 vs DY0. What is the value 1.65?  If you plot x and y as a point on a plot, the ratio y/x is the slope of the line through the origin that passes through that point. When we think of there being a typical ratio (which we may want to use for prediction), we are also talking about a typical slope through the origin on that (x,y) plot. If our measure of typical is some kind of average (such as a weighted average, a geometric mean, an average of the most recent values, or whatever), we are talking about both an average ratio and at the same time, an average slope. Table of ratios 0-11-22-33-44-55-66-77-88-911.64981.31901.08231.14691.19511.11301.03331.00291.0092240.42451.25931.97661.29211.13180.99341.04341.033132.63701.54281.16351.16071.18571.02921.026442.04331.36441.34891.10151.11351.037758.75921.65561.39991.17081.008764.25971.81571.10541.225577.21722.72291.125085.14211.887491.7220 In statistical terms, were using sample ratios to estimate the underlying ratio, since the observed ratios are noisy. We are also saying that, given the previous cumulative, we expect that on average the next cumulative is a multiple of the previous one. That is: E(y/x |x) = r a" E(y | x ) = rx.  E() stands for  expected (underlying average) value of whatever is in parentheses, and  | means  given . So E(y|x) means  the expected value of y, given the value of x . The left side is a ratio, r, the right side is a line through the origin with slope r. They both describe the same relationship between one cumulative or incurred and the next. So if theres an underlying "ratio", its also an underlying slope. Heres one such average line, an ordinary regression line through the origin:  The slope of this line is around 2.217, which is an estimate of the underlying ratio. Residuals The residuals from this fit are the differences between the points and the line. Residual = data fit of method ; Residual trend = data trend method trend We use residuals to assess ways in which the model assumptions dont apply. Lets calculate the residual for the point with y = 8992 and x = 3410, and for the point with y = 9565, x = 1092 (these are from AYs 3 and 5 respectively). The observed value for DY1 for the point circled in red below is 8992. The fitted value (the height of the line at the x-value 3410) is 3410 ( 2.217 = 7560.  The residual, 1432 is 8992 3410 ( 2.217; the observed value in DY 1 minus the prediction from the ratio times the previous value (residual = data fit). The observation with the largest residual is circled in blue. Its observed (y) value is 9565, while the x value is 1092. Consequently, its residual is 9565 1092 ( 2.217, which gives 7144.  Above are residuals from the fitted line (ratio). Notice the downward trend! Something is clearly amiss; the line through the origin doesnt describe the relationship well. In fact, the residuals are getting smaller as the previous cumulative gets larger. Look at the fitted line again, and see how the points on the left are above it and the points on the right are mostly below it. The relationship is not a line through the origin:  Above is a line of best fit in green. Clearly a line that doesnt go through the origin is a better description of the relationship here. Normally residuals are divided by their (individual) standard deviation, so that they share a common scale the result is standardized residuals. Secondly, its important to see whether the residuals are related to other likely predictors of the observations (in which case we will see non-random trends in the residuals plotted against those predictors). One obvious thing to do is to look at residuals against the three directions (accident year, development year and calendar year), as below. The fourth plot, residuals vs fitted values, is a standard regression diagnostic. Notice that it has exactly the same appearance as the above plot only the scale labels are different!  Residual display for a ratio model for DY1 on DY0 (first pair of years). The plot against accident and calendar years are the same because we only have a single pair of years. Theres also not a lot of information in the residuals for a single pair of years patterns have to be quite strong for us to be able to pick anything up. Heres the plot of y vs x for the second pair of years (DY2 vs DY1), followed by the corresponding residuals.   Residual display for a ratio model for DY2 on DY1 (second pair of years). In the model for DY2, we can see an increasing trend against calendar (and accident) year, and a decreasing trend against fitted values again, the relationship between DY2 (y) and DY1 (x this time) is not through the origin.  Plot of y vs x for second pair of years (DY3 vs DY2).  Residual display for a ratio model for DY3 on DY2 (third pair of years).  Plot of y vs x for third pair of years.  Residual display for a ratio model for DY4 on DY3 (fourth pair of years). By now its getting hard to see much of anything going on, there are only 7 and 6 points respectively in the most recent two sets of residual plots above. We can combine the residuals together into a display against each direction. The plots against accident and calendar years will no longer be redundant, and we will be able to pick up trends in those directions. Note that the plot against development years will not show lack of fit in general, since the fitted line will go through a weighted average value of y at each development, but it will allow us to see whats going on with the spread around the line. The fitted value will more clearly show (by having an overall trend) whether theres a tendency to need an intercept.  We can see a strong overall downward trend against fitted values. This strongly suggests a need for an intercept term! The standard chain ladder ratio (Mack) The standard chain ladder ratio (Mack ratio) is a kind of weighted average of the ratios, where the weight is the previous cumulative its sometimes called a  volume weighted average . An ordinary average of a set of y values (y1, y2, & yn) is (y1 + y2 + & + yn)/n. We can write that in short form as: (Si yi )/n. A weighted average has a weight for each observation, so that an observation with more weight  affects the average more than one with less weight. It  pulls the average toward it. A weighted average looks like this: [S wi yi ] / [Swi]. A weighted average ratio is written like this: r = [S wi (yi /xi)] / [Swi]. The chain ladder/Mack model has wi = xi . That is, r = [S xi (yi /xi)] / [Sxi] = [S yi xi /xi] / [Sxi] = Syi / Sxi . In other words, if we add up the two columns and take the ratio to get the chain ladder ratio, 8269 + 4285 + 8992 + & 5012 + 106 + 3410 + & its the same as weighting the individual ratios by the first column and taking the average: 5012 1.6498 + 106 40.4245 + 3410 2.6370 + 5012 + 106 + 3410 + There are three entirely equivalent ways of looking at the same thing as a ratio of sums, and a weighted average ratio, and as a weighted average slope (weighted regression line through the origin). Because the weights applied to the ratios are the previous value (incurred or cumulative), this kind of weighted average is often called a volume-weighted average ratio.  Mack (chain ladder) is volume-weighted average ratios  Individual ratios and the standard chain ladder (Mack) ratio (red) for DY1 vs DY0. The arithmetic average of the ratios is in gray. Below are the residuals for the Mack model applied to all pairs of years.  Again, we see a strong downward trend. Note that if a line through the origin is inadequate because the actual relationship needs an intercept, then no other line through the origin will fit. That is, no ratio, no matter how you choose it, will adequately describe the development. Section II. Chain ladder does not distinguish between accident and calendar years It is a property of the chain ladder that, when applied to an incremental triangle. the incremental forecasts identical to those for the equivalent procedure where accident years and development years are interchanged. Equivalently, you get the same incremental forecasts whether cumulation, calculation of ratios and projection runs to the right, across development years or down, across accident years. This applies to any model whose forecasts reproduce those of the standard chain ladder, so Macks model and the two-way cross-classification quasi-Poisson GLM both have this property in respect of the mean forecasts. Consider 5he following (toy) triangle, which represents incremental paid losses. 201030 It turns out that the value to be forecast (in gray) = 30 ( 10 / 20 = 15 Usual calculation (working across to the right) Equivalent cumulatives: 203030 Ratio: 30/20 = 1.5 Cumulative forecast = 30 ( 1.5 = 45 Incremental forecast = 45 30 = 15. Working down rather than across Cumulating down: 201050 Ratio running down: 50/20 = 2.5 Cumulative forecast = 10 ( 2.5 = 25 Incremental forecast = 25 10 = 15. In the first case, writing everything in terms of the incrementals, the ratio is (20+10)/20 = 1 + 10/20 . The cumulative forecast is 30 ( (1 + 10/20) = 30 + 30 ( 10/20 . The incremental forecast is 30 + 30 ( 10/20 30 = 30 ( 10 / 20 . In the second case, again writing in terms of incrementals, the ratio is (20+30)/20 = 1 + 30/20 . The cumulative forecast is 10 ( (1 + 30/20) = 10 + 10 ( 30/20 . The incremental forecast is 10 + 10 ( 30/20 10 = 10 ( 30 / 20 . In the general case, the incremental forecast (calculated in either direction) if b denotes the sum of incrementals in the same development year, and c denotes the sum of incrementals in the same accident year, and a denotes the sum of all values that are both above and to the left (in earlier accident and development years), then the incremental predicted (forecast) value is (b(c)/a .                Note that if the forecast value is further into the future than the next diagonal (calendar year), that (unknown) future incremental values required for the formula are replaced with their own forecasts. An algebraic proof that the incremental forecasts are of the form (b(c)/a is given in Barnett, Zehnwirth and Dubossarsky (2005). This formula is symmetric (the formula is the same if we interchange accident and development periods i.e. transpose the incremental array, since b(c/a = c(b/a. Consequently, it is always the case that the chain ladder works the same if we treat the accident years as if they were the development years and vice-versa. This is a worrisome property, because we know that the accident and development year directions are different. It appears to make no sense to cumulate downward and take ratios running down, but in fact the usual across version makes just as much sense. Some of the consequences are discussed in detail in Barnett et al (2005). Example The data in the following example are from the triangle ABC in Barnett and Zehnwirth (2000). Note that we do not use the exposures in this example. Firstly, lets see what the residuals from a Mack model tell us about the suitability of a ratio model. We clearly see below strong changes in trend in the calendar year direction; neither Mack nor the quasi-Poisson GLM (#link to other page#) version of the chain ladder can deal with this. This is not an unusual circumstance.  Lets now examine standard chain ladder forecasts for this data. These forecasts are the same for Mack and the quasi-Poisson GLM. Incremental chain ladder forecasts for the ABC data (not exposure adjusted): 01234567891019771536381884121345348745660348424043123821252166221444012200197817853622641215889410468671448479903557624818226621800014455197921017225916818838812307483380560863849633768274002059016918198021144825348218337013104078994602324556838000261022075817056198121981026630419465012009887582627505100034286269972146917640198220565425274617750612952296786824004492533853266562119817418198319771625540819464814232810560065149456973443527114215621771719842397843292422648021904001161938294958182438433452227453225581985326304471744375400234612159737114035799866027347459377423101119864207785904004258052873011956111396459795073809581184621837976198749620064932748237932547322160015819911096483616658405235943021 Incremental chain ladder forecasts for the transpose of the ABC data: 01234567891019771536381785362101722114482198102056541977162397843263044207784962001978188412226412259168253482266304252746255408329242471744590400649327197913453415889418838818337019465017750619464826480237540042580548237919808745610468612307413104012009812952214232819040023461228730132547319816034871448833807899487582967861056001161931597371956112216001982424044799056086602326275082400651498294911403513964515819919833123835576384964556851000449254569758182799869795011096419842125224818337683800034286338533443543843602737380983616198516622226622740026102269972665627114345224745958118658401986144401800020590207582146921198215622745337742462185235919871220014455169181705617640174181771722558310113797643021 It should be noted, however, that the Mack standard deviations of forecasts (or outstandings or ultimates) and their coefficients of variation do not have the transpose property; because of the way the conditioning is set up; when transposed, the conditioning in the variance is not symmetric. The symmetry in the forecasts (specifically, that the transpose of the forecast and the forecast of the transpose of the triangle are equal) is not a desirable property. It is not an indication of robustness forecasts from ratio models are highly sensitive to particular observations, and completely insensitive to other observations. It is not an indication of suitability of ratio models in general or of the chain ladder in particular, nor of either the Mack or quasi-Poisson GLM versions of it. If we run a structurally similar symmetric model from PTF (a two-way cross-classification model with log-link), of course the model wont be suitable either:  - however, a more suitable model can be obtained by allowing for some CY trend changes (though it would still be overparameterized and suffer from some of the other deficiences mentioned in Barnett et al, 2005) This model does have symmetry in forecast standard errors and CVs. For example, the following table is the same for both the original and transposed array. CalendarMeanStandardCVYearReserveDev.Reserve19881,619,40787,0100.0519891,145,58364,12:=BQrs2 3 5 p q s    9 : < Z [ ] w x z ˾˞&hh5CJOJQJ\^JaJh1h6hh6hx#"OJQJ\^Jhx#"CJOJQJ^JaJhx#"5OJQJ\^Jhx#"hDph"Eih"Eih"Ei5 hJjw5 h5 h@5 h-5h0<h-5CJaJ112st   Fft $$Ifa$gdYM $IfgdYMgdx#"gd"Ei   $ + 2 3 5 9 ? 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" # $ % ) * + , - 9 : ? @ a b c d ŹŹwpŽkkkk h16 h\h6j3hbhbUjh\UmHnHujhbUmHnHu h\h\h\ hJjwhJjw hJjw6hJjwh"Eih1hYMh6hhCJOJQJ^JaJ#hh5CJOJQJ^JaJ# jOh5CJOJQJ^JaJ( kd1$$IflֈZ I:'''''0&6244 la]p2 # $ % , - _ ` NO_`dhlptx| $$Ifa$gd`) $Ifgd`)gdzXu   S ] _ NO_`abefijmnqruvyz}~9:<ijl{ʿʿʿʿʿʿʿʿʿخخخخخخخخخhDhDph6hzXuOJQJ\^J h1hzXuCJOJQJ^JaJhzXu5OJQJ^Jh4IhzXu5OJQJ^JhzXu5OJQJ\^JhzXuh1h*h*6h* h1h*<Ff* h->*hJjwhJjw>*h-hJjwhOJQJh h6 hJjw6H* h6H*,b.l.m.n......>/?/@/A/O///0Z0[0\0d0h00001)131w11111111D2E2F2H2"3,3G3N333333۽פӓ| jhhvkh>*h*hh[4h-h-5 h-5 h@5h@h}6 h@h@ h@6h@ h`zh}jwKh`zh`zUh-hh`zhXQih.jEh\h.Ujh.UmHnHu0n....>/@/A/[0\00011G2H2!3"3t3w3z3 $$Ifa$gd8z3{3~33xx $$Ifa$gd8{kd$$Ifl0r oo t0644 laoyt8333333444vv $$Ifa$gd8{kd$$Ifl0r oo t0644 laoyt83333(404M4N4~4444444444+5555555 6 6666666666P7Q777777z8{8|8}8~888888888ɾjhvkUmHnHu h%2Kh%2K jh%2K h%2K6h%2K hvk6hvkhvk6hvk jhdzhdzjh`)UmHnHu jh`)h`)hh`)h6 h6hh674444xx $$Ifa$gd8{kd?$$Ifl0r oo t0644 laoyt84 4!444X4}4~44444vv $$Ifa$gd8{kdץ$$Ifl0r oo t0644 laoyt8 4444xx $$Ifa$gd8{kdo$$Ifl0r oo t0644 laoyt844445*5+55566b6{666}}}}gddz{kd$$Ifl0r oo t0644 laoyt8668888888888888888899 : :::P;Q;<<gd%2Kgdvk8888888888888888888888888899999999 : :::::::::::::O;P;;;<<<<<<<5=7=>>οӸ꿴jh1h1Uh1h- hLIhLIhLIhLI6 jhLI hLI6hLI jh%2K h%2K6 h%2Kh%2Kh%2KjhvkUmHnHuhvk><<<6=7===>>>??T?V?X?Z?\?^?`?b?d?f?h?j?m?n?Ff $$Ifa$$IfgdU$gd1gds%t>>?2?5?:?R?S?T?m?n?s?????@@ @D@P@Q@V@@@@@@@@@A4A5A:A]AAAAAAAAABB"B0B7BjBkBpBwB~BBBBܼܼܼܼܼܼܼܼܼܼܼ/h-h-5B*CJOJQJ\^JaJph&h-h-5CJOJQJ\^JaJ/h-h-5B*CJOJQJ\^JaJphh-CJaJ h-5CJOJQJ\^JaJhU$hs%th1 h1h18n?s?z???????????????????????@@ @Ff9 Ff $$Ifa$ @@@@&@,@2@8@>@D@J@P@Q@V@]@d@k@r@x@~@@@@@@@@@Ff!Ff $$Ifa$@@@@@@@@@@@@@@@A AAAA"A(A.A4A5A:AAAHAFf Ff $$Ifa$HAOAVA]AcAiAoAuA{AAAAAAAAAAAAAAAAAAAAFf'Ff}# $$Ifa$AAAAB BBBBB"B)B0B7B>BEBLBRBXB^BdBjBkBpBwB~BBBFf0Ffe, $$Ifa$BBBBBBBBBBCCCC C C CCCCCCCC C'C.CFfg9$IfFf15 $$Ifa$BBBBCCCC CmCnCsCCCCCDDDDPDeDfDkDDDDDDDDE!EFEGELEdEEEEEEEEEEFF$F*F0FfFgFhFzFFّh1hLI/hU$hU$5B*CJOJQJ\^JaJph&hU$hU$5CJOJQJ\^JaJ/hU$hU$5B*CJOJQJ\^JaJphhU$CJaJ hU$5CJOJQJ\^JaJhU$hs%t hs%t67.C5CWXqr\]澶樠jh)\Uh)\hihi6UhiCJaJ hi5CJOJQJ\^JaJ hi6 hihijZlhihiUhi h906h1 hs%t6h90hs%t;KKKKKKqkkkk$Ifkdۆ$$Ifl\0 '0'R';'644 lalp(KKKKKKqhhhh $$Ifa$kd$$Ifl\0 '0'R';'644 lalp(KKKK qhhhh $$Ifa$kd$$Ifl\0 '0'R';'644 lalp(0740.061990787,89044,2770.061991549,71531,1610.061992390,44822,7140.061993279,53416,8700.061994203,82013,1950.061995145,47610,7660.07199692,4448,5940.09199744,2216,2520.14Total5,258,538240,5390.05The same would be true for the quasi-Poisson GLM version of the chain ladder. As already mentioned, symmetry is not of itself a desirable property, but if you want to impose that symmetry it would be interesting to ask why we should expect it to hold for the mean but not the standard deviation. An article on the Mack method and bootstrapping is available  HYPERLINK "http://www.insureware.com/Library/Technical/technical.php" \l "Bootstrap" here. References Barnett, G. and B. Zehnwirth (2000) Best Estimates for Reserves, PCAS No 87, p245-303. Barnett G., B. Zehnwirth and E. Dubossarsky (2005) When Can Accident Years Be Regarded As Development Years?, PCAS No 92, p249-256. Mack, Th. (1994), "Which stochastic model is underlying the chain ladder method?" Insurance Mathematics and Economics, Vol 15 No. 2/3, 1994, pp. 133-138. 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