A Family of Chain-Ladder Factor Models for Selected Link Ratios

Transcription

1 A Family of Chain-Ladder Factor Models for Selected Lin Ratios by Manolis Bardis, Ali Maidi, and Daniel Murphy AbSTRACT The models of Mac (993 and Murphy (994 are expanded to a continuously indexed family of chain-ladder models by broadening the variance structure of the error term. It is shown that, subect to certain restrictions, an actuary s selected report-toreport factor can be considered the best linear unbiased estimate for some member of this family. The approach given in Murphy (994 yields a mean square error estimate of the unpaid claim liability that is consistent with the actuary s selections. KEYwORdS Chain ladder, Mac, Murphy, variance, mean square error, reserve ris, regression VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 43

2 Variance Advancing the Science of Ris. Introduction The chain-ladder variance formulas first proposed by Dr. Thomas Mac (993 are based upon all-year volume-weighted average report-to-report factors ( lin ratios or factors and an assumed variance structure that is proportional to the development period s initial loss. Under the regression approach of Daniel Murphy (994 it was shown that the proportional variance structure assumption is sufficient for the weighted average lin ratio to be considered the best linear unbiased estimate (BLUE of such a chain-ladder model. In practice, however, the actuary selects factors. Factor selection is an important component of actuarial analysis that utilizes actuarial udgment in its consideration of those and other averages as well as additional information gleaned from benchmar lin ratios, industry trends, discussions with company management, etc. Although much research has been dedicated to framing the chain-ladder method within a statistical structure, 3 little ground is devoted to the treatment of the uncertainty of the unpaid claim estimates when the selected factors differ from some prescribed formula. The few treatments on the subect tend to adopt a bifurcated approach, that is, one which supplements the expected value estimates from one model with variability estimates from a different model. A Bayesian perspective can be exploited to combine point and uncertainty estimates derived from bifurcated models. For example, Verrall (007 assumes the actuary selects volume-weighted average lin ratios from An alternative variance assumption for which the simple average lin ratio is the BLUE solution was also provided. For a mandate on the requirement to exercise udgement in selecting lin ratios, see, for example, Friedland (009. For a survey of how a group of actuaries selected factors under test conditions see Blumsohn and Laufer ( For stochastic research related to the chain-ladder method, see Bardis, Maidi, and Murphy (008, Buchwalder et al. (006, Mac (993, 994, Mac, Quarg, and Braun (006, Mac (999, Murphy (994, Venter (006, Wright (990 and Barnett and Zenwirth (000 in the references. Other prominent research includes Christofides (997, Panning (006, Rehman and Klugman (009 (regression; England and Verrall (00 (bootstrapping; Verrall (004, 007 (Bayesian. the most recent five years but derives variation estimates that reflect information from all years, not ust the most recent five. Verrall s approach holds promise as actuaries become more comfortable with the Bayesian perspective, which can be useful for combining statistics and udgment but which requires prior distributions and sophisticated statistical software. An approach with which actuaries do appear comfortable is based on scaling. Panning (006 argues that loss reserve uncertainty under his method is scalable. By that he means that his method s coefficient of variation (CV is applicable to reserves that have been estimated in different ways (Panning 006. Scaling is an actuarial technique utilized in a wide variety of applications. In stochastic analysis the authors are aware that it is common practice to apply a CV based on the Mac method to a chain-ladder point estimate that is based on selected factors other than the all-year volume-weighted average. The authors are concerned that bifurcated point and variability estimates may underestimate the volatility of the underlying claims process. This paper taes a more direct approach. We show how, under certain restrictions on the selected lin ratio, a chain ladder model can be formulated such that the actuary s selection can be considered a consistent unbiased estimate of the model. Our chain ladder models are similar to those of Mac and Murphy, but allow for a broader set of weights by expanding the domain of the exponent of the beginning value of loss to the entire real line. Using classical regression analysis, variability estimates fall out of the same model. This overcomes the scaling disconnect alluded to above. We also believe our approach is more accessible to practicing actuaries than Verrall s Bayesian approach. Although a drawbac of our approach is that our mean square error formulas are more complicated than those of Mac and Murphy, this should not be unexpected for models that allow for a continuum of selected factors rather than ust the standard averages. Despite the higher degree of difficulty, our formulas can be calculated in a spreadsheet. To the authors nowledge, this is the first paper to posit models that reflect the chain-ladder method 44 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

3 A Family of Chain-Ladder Factor Models for Selected Lin Ratios in practice, i.e., when selected factors are other than the volume-weighted or simple averages. The authors believe that by associating the actuary s choice with a model, the selected lin ratio can better be bactested against the observable data, which can add more insight into the reserving exercise. We caution, however, that it is not necessarily possible to identify a chain-ladder model in our framewor that is consistent with every potential selected factor. Restrictions are defined in the paper. Of course, the results of our chain ladder model are subect to model error. As with all stochastic models, the actuary must assess the applicability of the indications relative to his or her understanding of the model s assumptions, familiarity with the triangle and other data, and the udgment underlying the factor selections. The remainder of this paper is organized as follows. In Section we present a family of models that generalizes those in Mac (993; 999 and Murphy (994 and is consistent with the practical implementation of the chain ladder method, because it allows for conformance with a broad set of udgmentally selected factors. In Section 3 we give formulas for the expected value and mean square error of chain ladder proections from selected factors. In Section 4 we demonstrate the concepts and calculations in a wored-through, spreadsheet-based example. Section 5 is a summary. Appendix A includes proofs of our results. Appendix B compares our model s recursive formulas with those of Mac (999.. A chain-ladder model for udgmentally selected lin ratios Adopting notation commonly found in the literature, we denote the observed triangle of positive cumulative losses 4 by D = {C i, i I, I}. A model equivalent to the chain-ladder method is for i I and I. Under these assumptions it is well nown (first shown by Aiten 935 that the best linear unbiased estimate (BLUE of the lin ratio f from age to age + given triangle D, denoted, is a weighted average of the observed fˆ lin ratios: where the weights I ˆ : ˆ ( α f = f ( α : = w F i, i, i= w α i, = are functions of the a and F i, C α i, I α C, = C, = C i+ i, ( 3 ( 4 are the observed lin ratios based on the triangle. Model ( describes a family of models indexed by a continuous parameter a R. This family contains the models given in Mac (993; 999 and Murphy (994 as special cases, where those authors propose that the a indices assume the values 0,, and, at most. 5 Murphy (994 demonstrated that for the member indexed by a = the weighted average lin ratio is the best linear unbiased estimate consistent with the model s parameter f; for the a = member, the simple average lin ratio is a consistent estimator; for a = 0, a consistent lin ratio is the slope of a simple regression line through the origin. Model ( allows the domain of possible values for a to encompass the entire real line rather than ust the values 0,, and. As a result, a continuum of selected factors has the potential to be consistent with Model (. Put another way, Model ( allows for an actuary s selected lin ratio that is α (,,,, C = fc + C σε i+ i i i independent random variablesε i have mean 0 and variance 4 Losses can refer cumulative paid or case incurred amounts, cumulative counts, or any triangular array of data subect to the given assumptions., 5 See also Barnett and Zehnwirth (000. Murphy considers 0,, and. Barnett and Zehnwirth consider and, denoting the exponent by delta (d. In his original paper (993 Mac only considered a =. Mac (999 reframed his model in terms of lin ratios rather than cumulative loss and extended a to also include 0 and ; given the new model formulation, the simple average corresponds to a = 0 in Mac (999. VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 45

4 Variance Advancing the Science of Ris different from the simple or volume-weighted average to be, nevertheless, a linear unbiased estimate of a statistical model consistent with the chain-ladder method. We refer to Model ( as the chain-ladder factor model (CLFM. With that as bacground, there remain the following questions: When a selected lin ratio is not one of the usual averages, how does one find a member of the CLFM family for which it could be considered consistent? How does one calculate the value and the ris of a point estimate under the CLFM framewor, and what additional assumptions are needed? To help answer these questions we introduce the lin ratio function, a concept fundamental to CLFM theory and results... The lin ratio function Definition: Given observations of loss at the beginning and end of development period, the lin ratio function LR (a is a mapping on the real line given by I = ( α LR α : w F, α R ( 5 i, i, i= (a where w i, and F i, are defined in (3 and (4 above. 6 The lin ratio function calculates weighted averages of the observed lin ratios, where the weights depend on the exponent of loss at the beginning of the period. We begin our investigation of the lin ratio function by considering its asymptotic properties as a ±. Lemma : Asymptotic properties of the lin ratio function Consider for a given triangle D and development period the set of all possible values of linear estimates ( as a function of a real valued parameter a R. Let aymin and aymax denote the accident years 6 We may sometimes omit the subscript when the context of development period is understood. with the smallest and largest values of loss, respectively, as of the beginning of development period : aymin = min{ Ci, } and i aymax = max{ C i I i},. i Then lim a LR (a = F and lim LR (a = aymin, a - F aymax,. In the case of ties for accident years having the smallest or largest beginning value C i,, lim a LR (a = mean{f i, i aymin } and lim a - LR (a = mean{f i, i aymax }. The proof can be found in the appendix. Lemma says that the best linear unbiased estimate of a lin ratio for a given development period approaches the lin ratio experienced by the accident year with the smallest/largest value of loss at the beginning of the development period as index a approaches + /-. To illustrate, suppose losses as of the beginning and end of development period for five accident years are as shown in Table. The largest and smallest values of loss as of the beginning of the period are highlighted in yellow. The lin ratio function corresponding to these losses is graphed in Figure. As predicted by Lemma, the graph is asymptotic to the line y =.500, the lin ratio corresponding to accident year 3, and to the line y =.0, the lin ratio corresponding to accident year 5. The blue line corresponds to the volume-weighted average lin ratio (a = ; the red line to the simple average (a =. The lin ratio function need not be monotonic. Indeed, change the ending value of accident year 5 to Table. development period losses C i, = = F i, i = i = i = i = i = volume weighted avg..65 simple avg CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

5 A Family of Chain-Ladder Factor Models for Selected Lin Ratios Figure. Lin ratio function Lin Ratio Function Table data LR(α Year 5 would still have the smallest beginning value so its lin ratio, now.45, would still be the asymptote. The new non-monotonic lin ratio function, graphed in Figure, has a minimum somewhere in the vicinity of a = 6. From Figures and it should be clear that not all possible lin ratios (abscissa are achievable from a given triangle. In fact, the maximum or minimum empirical lin ratio may not even be achievable (the.983 lin ratio for accident year 4 is literally off the chart in Figure. Mathematically stated, the image of the lin ratio function is not the entire real line. In other words, many lin ratio selections would be inconsistent for any member of the CLFM family Figure. Lin ratio function: Accident year 5 as of years Lin Ratio Function Accident year 5 as of years = 500 LR(α VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 47

6 Variance Advancing the Science of Ris We adopt the usual chain ladder convention of developing the current diagonal. For accident year i with current diagonal value C i, and a selected lin ratio fˆ, the expected value at the end of the first future development period is Ĉ i, + = Ĉ i, fˆ. This estimate is clearly unbiased if fˆ is unbiased because C i, is a scalar. The expected value at the end of the next development period is Ĉ i, + = Ĉ i, + fˆ +. Expected value estimates for subsequent development periods are iterated in a similar fashion. The estimate Ĉ i, + will be unbiased if we assume that the product of the two estimates fˆ and fˆ + equals the product of the two underlying parameters f and f +. Note that this assumption is implicit in chain ladder calculations where, say, a higher than average lin ratio on the current diagonal has no bearrelative to a given triangle D. 7 This brings us to our next definition, that of a reasonable lin ratio. Definition: A lin ratio lr is reasonable with respect to a given triangle D if there exists a member of the a-indexed CLFM family for which lr can be calculated as in (5. We denote the set of all reasonable development period lin ratios by LR (D: LR D = lr lr = LR forsome, ( α α R. given triangle D Noting that large values of a may lead to impractically large factors C a/ in the error term of (, we recommend limiting a to a prudently bounded interval; we selected [-8, 8] udgmentally. A selected lin ratio may be associated with more than one value of a (e.g., in Figure the blue, volumeweighted line crosses the graph at more than one point. That is to say, there may be more than one member of the CLFM family whose best linear unbiased estimate is the selected factor. We suggest the following procedure for selecting the selection-consistent alpha value. Definition: The selection-consistent alpha of a reasonable lin ratio lr is the smallest positive solution a [-8, 8] of the equation lr = LR (a, or, if no positive solution exists, the smallest solution in absolute value. Mathematically this is expressed as αˆ ( > = ( min α 0 lr LR α, max max α 0 lr LR α. = By convention, if the selected lin ratio lr is the volume-weighted average we set â = ; for the simple average we set â =. Given a selected lin ratio lr, the selectionconsistent member of the CLFM family can be determined by finding positive and negative solutions a of the equation I I α α lr = lr C C C i, i, i, + i= i= ( 6 7 It is hoped that the actuary would rely on information beyond the triangle to ustify such a selection. and selecting the smallest positive value if one exists or the negative value closest to the origin. According to traditional actuarial thining, the variability of proected loss increases as the beginning value of loss increases, i.e., the value of a in the exponent of C i in model ( should be positive. A negative value of a would say that the variability of proected losses is inversely proportional to the beginning value, a seemingly counterintuitive result. However, we have found contexts in which such a counterintuitive result is not unreasonable. For example, given a boo of first party business with low policy limits, case reserves for obvious limits losses would tend to be more certain than reserves on smaller claims. For that situation it would not be unreasonable to find the variability of losses at the end of a calendar period to be inversely proportional to the beginning value of loss. We only suggest that actuaries stay open to the story that data have to tell. 3. CLFM chain-ladder proection formulas CLFM formulas are recursive because that allows for maximum flexibility in selecting different family members from one period to the next. 3.. Expected value formulas 48 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

7 A Family of Chain-Ladder Factor Models for Selected Lin Ratios ing on the factors selected to develop that year going forward. 8 The expected value of the sum of all accident years combined at development age is the sum of the estimates of the individual accident years at the same age. 3.. Standard error formulas The first step in woring with loss variation over a given development period is estimating the scale parameters s, which can easily be found using weighted least squares available in virtually all popular statistical pacages. Equivalently, for each development period the data can be transformed into ordinary least squares (OLS form by dividing the beginning and ending values of loss by the beginning value raised to the power a/. As transformed, model ( is α α C C = fc C + σε. ( 7 i, + i, i, i, i, The formula for calculating an estimate ŝ of s can be found in any good statistical text. In the example we illustrate this approach using the LINEST function in Excel. The next step is to estimate the variability of the selected factors fˆ. The estimate of the conditional variance of those factors, which we denote by D, 9 is by definition the quantity ( f f f D D : = E( ( ˆ E ( ˆ. As with the estimates of s, these estimates are also standard outputs of regression software. 0 8 Mac (993 proved that weighted average loss development factors are uncorrelated. His proof is an unconditional result, however, that does not necessarily hold conditionally for a specific triangle. Indeed, it is possible to simulate triangles that have correlated development factors, yet where all assumptions in ( are satisfied. 9 We use the delta operator D to denote parameter ris and the gamma operator G for process ris. 0 We also use Excel s LINEST function for this estimate. Alternatively one could use the formula (Mac 999, p. 363 ˆ σ ( f = n Σ where α i= ci, weights w i, Standard error formulas for an individual accident year Consider an individual accident year i and its estimate Ĉ i, at age. The mean square error of the estimate Ĉ i, is the sum of parameter ris and process ris: mse ˆ E ˆ ( C C C D i, = ( i, i, (( ii, ii, D = E Cˆ C E Cˆ (( E C D D ii, ii, = ( D + E C E C D ii, ( ii, : = ( C C ii+ Γ, ( ii, Parameter ris (denoted D and process ris (denoted G, notation borrowed from the literature, can be calculated recursively according to the formulas shown next Parameter ris: Variance of the estimate of the mean future value of loss For the first period after the current diagonal (s =, ( C C f ( 8 i, + = i, ( because C is a constant. For s =, 3,... i, = + C f fˆ C µ i, + s i, + s + s + s i, + s + ( f+ s ( + 9 C ( i, s where µ i s : E C i s D. Formula (9 is consistent with the formula in Mac (999 for a =,, +, + except for the third term, which Mac excludes. = Derived in Appendix A. See Appendix B for more information. VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 49

8 Variance Advancing the Science of Ris 3... Process ris: Variance of the deviation of future value of loss from its mean For the first period after the current diagonal For subsequent periods Γ = α σ Γ C C i+ i ( C = ( C D + ˆ. ( 0,, Ψ α α s E i, + s i, + s + s ( i, + s E ( i, + s Γ C, C σ + f Γ C. + s + s i, + s As noted in the proof in Appendix A, the process ris calculation, drawing upon the Law of Total Variation, involves the expectation E(C a which is not the same as E(C a. Since E(C is a readily available quantity, Y is our helper function which, when multiplied by E(C a, yields E(C a. For example, since E(X = E (X + Var(X, E(X /E (X = + cv (X, so Y(, = +. Clearly Y(, = ; and Y(0, = as well. For higher raw moments, the ratio of E(C a to E(C a depends on the distribution; for the normal distributions it is a polynomial in. We adopt that simplification for our purposes. Therefore, for nonnegative integer values n of alpha we define Y as = Ψακ, n n ( n... n κ.! = 0 even ( For a > 0 but not an integer, we define Y(a, to be the linear interpolation between Y([a], and Y([a] +, where [x] denotes the greatest integer function. For negative values of a we recommend approximating Y using simulation Standard error formulas for all accident years combined Recursive variance formulas for all accident years combined become slightly more complicated because at each new age an additional accident year is included. 3 See Bardis, Maidi, and Murphy (008 for more details. I For ages =, 3,..., let X = Σ C be the i= I + i, sum of the future losses for accident years that have I not yet matured to age. Let M : = Σ i = I + µ denote i, the expected value of X and let Xˆ = Σ I C ˆ i= I + i, be its chain ladder estimate Parameter ris: Variance of the estimate of the mean future value of total loss For =, only the most recent accident year is included in the total, so the parameter ris of the total is equal to the parameter ris of the most recent year: D (X = D ( f? C. For = 3, 4,..., I, = ( + + X M C f f X I +, + ( f ( X Process ris: Variance of X Model ( assumes all accident years are independent. Therefore the process variance of the sum of the future values as of a given age is the sum of the process variances: Γ 4. An example I ( X Γ C 3 = ( i,. i= I + We consider the triangle of RAA data analyzed in Mac (993, Barnett and Zehnwirth (000 and elsewhere in the literature and illustrate spreadsheet calculations of process ris and parameter ris within the CLFM framewor. We selected simple and volumeweighted average lin ratios for a few ages, and udgmental selections for other periods to demonstrate the concepts. Losses, lin ratios, simple and volumeweighted averages and the selections are shown in Table. The mean and standard error estimates based on this triangle D, the selected factors, and the CLFM formulas are summarized in Table 3. We will illustrate the CLFM calculations for a few representative entries. 4.. Expected value calculations Table 4 shows the proected chain-ladder values based on the latest diagonal and the selected factors. 50 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

9 A Family of Chain-Ladder Factor Models for Selected Lin Ratios Table. RAA data Losses AY/Age ,0 8,69 0,907,805 3,539 6,8 8,009 8,608 8,66 8, ,85 5,396 0,666 3,78 5,599 5,496 6,69 6, ,40 8,99 3,873 6,4 8,735,4,863 3, ,655,555 5,766,66 3,45 6,083 7,067 5,09 9,565 5,836,69 5,955 6,80 6,53 6,445,70,935 5, ,00 0,946,34 8,35 6,947 3, 9 3,33 5,395 0,063 Lin Ratios AY/Dev. Period to to 3 3 to 4 4 to 5 5 to 6 6 to 7 7 to 8 8 to 9 9 to Simple average Volume-weighted average Selected Table 3. CLFM calculations for representative entries AY/Age Estimated Ultimate Current Diagonal Estimated Unpaid Total Ris CV 8,834 8,834 6,858 6, % 3 4,09 3, % 4 8,78 7,067, % 5 9,006 6,80,86, % 6 9,583 5,85 3,73, % 7 7,874,34 5,560,80 39.% 8 4,66 3,,54 5, % 9 6,0 5,395 0,85 6, % 0 50,866,063 48,803 8, % All 46,387 60,987 85,400 8, % VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 5

10 Variance Advancing the Science of Ris Table 4. Proected loss by accident year and age AY\ Age =Ultimate 6, ,888 4,09 4 8,04 8,59 8,78 5 7,78 8,33 8,74 9, ,675 8,46 9,06 9,404 9, ,469 6,33 6,809 7,398 7,7 7, ,78 9,643,90,8 3,60 4,045 4,66 9 8,759,68 3, 4,63 5,45 5,778 6,06 6,0 0 6,99 7,485 35,043 4,76 45,9 47,836 49,5 50,40 50,866 All (X 6,99 36,44 6,99 88,40 6,5 48,405 8,64 08,77 7,553 For example, for accident year 0 the proected value in the first future diagonal is the product of the diagonal value and the - selected factor (,063? 8.06 = 6,99. For the next diagonal the proected value is the product of the age proection and the -3 selected factor (6,99?.64 = 7,485. The values in the bottom row ( All are the sums of the values in their respective columns. 4.. Variability calculations 4... Selection-consistent alphas The simple average was selected for development period - and the volume-weighted average for periods -3 and 6-7. Accordingly, the respective selectionconsistent alphas are and by convention. For the remaining selections the selection-consistent alphas are the solutions of Equation (6, which we solved in Excel with a Newton-Raphson technique. 4 The values of a shown in Table 5 thus identify selectionconsistent members of the CLFM family s We chose the OLS approach to illustrate how to carry out the CLFM calculations in Excel. For exam- ple, for period 3-4, a =.58 (Table 5, the data for the transformed model (7 are given in Table 6, and the LINEST estimate for s is The 9 to 0 development period has only one observation, insufficient for regression; we used Mac s suggested heuristic [0, p. 363] s n - = min (s 4 n - /s n - 3, min (s n - 3, s n -. Table 7 summarizes the s estimates for all development periods D (f For the standard error of the selected lin ratio, denoted in our paper as D ( f, either refer to the output of the software employed LINEST 5 in our case or use the formula [(Mac 999 p. 363; see σˆ footnote 0] ( f = which we did for the n α Σ c i= i, problematic 9-0 development period. Table 8 summarizes these estimates Parameter ris (D for proected loss Parameter ris is estimated recursively in an analogous fashion to the expected value. Table 9 displays the parameter ris estimates by accident year as of each future evaluation and for all accident years combined. 4 For development years 9-to-0, where we do not have sufficient data to perform a regression, we selected a selection-consistent alpha equal to the one calculated for the 8-to-9 development years. 5 LINEST labels the estimate of s as se y and the standard error of the slope parameter as se. 5 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

11 A Family of Chain-Ladder Factor Models for Selected Lin Ratios Table 5. Selection-Consistent alpha to to 3 3 to 4 4 to 5 5 to 6 6 to 7 7 to 8 8 to 9 9 to Table 6. Transformed data for OLS regression AY/Age D (C for an individual accident year To illustrate how we calculate these parameter ris estimates for an individual accident year, let s wor with accident year 0. For the first period after the current diagonal (i = 0 and = we use Formula (8, the actual loss in Table, and the lin ratio uncertainty estimate from Table 8: ( 0 = 0 = C C f, ,, = 7, 04, 303 For the next development period we use Formula (9, the estimated proected loss µ 0, from Table 4, the selected lin ratio in Table, Table 8 and the result of the previous calculation: = + + C µ f f C f 0, 3 0, 0, = + 6, , 04, , 04, 303 = 96, 434, 086. ( C 0, Estimates for the remaining ages are iterated in a similar fashion Parameter ris: D (X for all accident years combined For all accident years combined, the parameter ris for age is identical with the parameter ris for accident year 0 alone: D (X = 7,04,303. For age 3, we use Formula (: = + 3 ( 9, ( X M C f fˆ X f X + + =, +, ,, , 04, 303 = 00, 34, 585. The value for M = E(X comes from Table 4, the actual diagonal value C 9, from Table 3 and the value of D (X from the previous recursion step. Estimates for the remaining ages are iterated in a similar fashion Process ris (G for proected loss Table 0 summarizes the process ris estimates by accident year and for all accident years combined. The process ris estimates for all accident years combined is the sum of the process ris estimates for the individual accident years. The process ris estimates for individual accident years are calculated recursively. We illustrate with accident year 0. Table 7. s estimates to to 3 3 to 4 4 to 5 5 to 6 6 to 7 7 to 8 8 to 9 9 to , Table 8. D (f to to 3 3 to 4 4 to 5 5 to 6 6 to 7 7 to 8 8 to 9 9 to VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 53

12 Variance Advancing the Science of Ris Table 9. Parameter ris estimates D (C i, AY\ Age =Ultimate 0 3 5,438 7,76 4 7,980 97,45 0, ,384 39,54 588, ,33 6 3, ,65 54, , ,83 7 0,7 387,36 553,40 599,78 690, ,388 8,537,83,33,5 3,357,455 3,89,96 4,80,95 4,460,856 4,543, ,048,564,07,44,93 3,007,04 3,375,358 3,6,38 3,80,39 3,880, ,04,303 96,434,086 37,84,68 453,68,9 566,69,078 66,580,03 660,54, ,5, ,94,670 All 7,04,303 00,34, ,6, ,70,855 68,63,67 68,5,87 73,569, ,890,48 78,0,374 Table 0. Process ris estimates G (C i, AY\ Age =Ultimate , 56, ,040 47,84 436,009 5,068,664,3,30,6,884,65,68 6,88,86,708,7,96,36 3,99,55 3,58, ,566,556,85 3,435,687 3,700,44 3,974,44 4,048,36 8 9,974,886 4,867,845 0,88,56 3,495,904 5,5,93 6,397,37 6,886,46 9 5,980,499 6,050,554,85,400 9,880,893 33,037,943 35,408,793 36,84,6 37,506, ,8,730,77,,088,839,654,69 3,95,360,699 4,886,849,06 5,307,76,777 5,686,53,97 5,896,87,944 6,006,08,70 All 648,8,730,733,0,587,865,680,069 3,963,78,50 4,94,933,76 5,370,93,9 5,755,054,995 5,969,54,73 6,080,07, CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

13 A Family of Chain-Ladder Factor Models for Selected Lin Ratios For the first period after the current diagonal (i = 0 and =, we use Formula (0, the actual loss in Table, and the scale parameter estimate from Table 7: = = = Γ α C C 0 0 σ ˆ, , 8, 730.,, For the next development period ( = 3 we use Formula (: Γ ( C = E( C D Γ Ψ α, E α 0, 3 0, + ˆ f Γ ( C0,. 000 = 6, 99, , 8, 730 =, 77,, 088. ( C0, ( C0, σˆ because Y(a, when a =. For the process ris at age = 4 where a 3 =.58 we linearly interpolate between Y(, = and Y(, = + where κ =, 77,, 088 7, 485 =. 5 and get Y(.58,.5 = + (.58 - (.5 =.36. So Γ ( C = E( C D Γ Ψ α, E α3 0, 4 03, 3 + f Γ 3 ( C0, = 7, , 77,, 088 =, 839, 654, 69. ( C0, 3 ( C0, 3 σˆ 3 Estimates for the remaining ages are iterated in a similar fashion Comparison of the CLFM vs. the Mac method The question of how the CLFM and Mac results compare often arises. 6 As we understand the popular practice of the method of Mac (993, the Mac method CV assuming weighted average lin ratios and all years in the triangle would be applied to the point estimate based on a different set of factors. The Mac method CV from the RAA data is 5.6%. 7 This is about half the CLFM CV in Table 3. Thus, the CLFM ris estimate would be about twice the value of the ris estimate from the Mac method as we understand its common implementation in practice. 5. Summary This paper presents a family of models that is consistent with the implementation of the chain-ladder method as used in practice. Our approach is different from the methods of Mac (993, 999 and Murphy (994 because, whereas their models assume that the selected chain-ladder lin ratio is a volume-weighted or simple average, our model accepts an actuary s udgmentally selected factor as a fundamental input. By enlarging the domain of the exponent of the chain ladder method s explanatory variable (the value of loss at the beginning of the development period in its influence on modeling loss development variability, our approach allows for many more selected lin ratios than ust the usual averages to be considered best linear unbiased estimates within a chain-ladderconsistent stochastic model. As a result, point estimates and ris estimates of unpaid claim liabilities can be calculated simultaneously. This avoids the need to scale chain ladder point estimates based on one model (selected factors with CVs based on a different model (e.g., volume-weighted or simple averages or with CVs based on a different methodology entirely (e.g., bootstrapping. Our approach can be implemented in a spreadsheet, thus avoiding the need for more sophisticated statistical software. The theory of our approach and illustrated in the example suggests that scaling a chain-ladder point estimate with a Mac method CV based on the allyear volume-weighted average will understate the 6 Most recently by a reviewer of the paper. 7 This cv can be produced by the formula in Mac (993 or by the approach herein, where unity a is selected for all development periods. VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 55

14 Variance Advancing the Science of Ris standard error of the proections; the greater the difference between the actuary s selections and the volumeweighted averages, the greater the understatement. It goes without saying that to model loss development within the CLFM family does not eliminate model ris, an inescapable side effect of any statistical model by definition. The authors also caution that it is not necessarily possible to identify a CLFM family member that is consistent with every potential lin ratio selection. Refer to the constraints outlined in the paper. Various reviewers have suggested that the alpha index that identifies a member of a CLFM family can be considered a parameter rather than an index and therefore some component of the model ris might possibly be quantified by an estimate of that parameter s estimation ris. The authors had indeed investigated that wor stream within a maximum lielihood context. Although the mathematics was interesting, that research thread was abandoned because there was no guarantee that the lielihood maximizing value of alpha would index the CLFM member consistent with the actuary s selection. Others may find this wor stream more fruitful, but our primary goal was to identify selection-consistent models that cater to the needs of practitioners who select development factors based on udgment on a daily basis. For diagnostics regarding the selections relative to potential trends in the triangle, we refer the reader to our first paper (Bardis, Maidi, and Murphy 008. The authors also wish to point out the CLFM framewor assumes that the only available data that might shed light on lin ratio uncertainty is the triangle alone. When exogenous data help determine factor selection, unpaid claim estimate uncertainty will undoubtedly be improved by incorporating additional sources of pertinent quantifiable information within a broader model that is not limited to the triangle alone. We anticipate much research in that area in the future. The authors want to than Tom Ghezzi and the many reviewers for their helpful comments and suggestions. References Aiten, A. C., On Least Squares and Linear Combinations of Observations, Proceedings of the Royal Society of Edinburgh 55, 935, pp Bardis, E. T., A. Maidi, and D. Murphy, Manually Adustable Lin Ratio Model for Reserving, Casualty Actuarial Society E-Forum, Fall 008. Barnett, G., and B. Zehnwirth, Best Estimates for Reserving, Proceedings of the Casualty Actuarial Society 87, Part, 000, pp Blumsohn, G., and M. Laufer, Unstable Loss Development Factors, Casualty Actuarial Society E-Forum, Spring 009. Buchwalder, M., H. Bühlmann, M. Merz, and M. V. Wüthrich, The Mean Square Error of Prediction in the Chain Ladder Reserving Method (Mac and Murphy Revisited, ASTIN Bulletin 36, 006, pp Christofides, S., Regression Models Based on Log-Incremental Payments in Claims Reserving Manual vol., Edinburgh: Faculty and Institute of Actuaries, 997. England, P. D., and R. J. Verrall, Stochastic Claims Reserving in General Insurance, British Actuarial Journal 8, 00, pp Friedland, J., Estimating Unpaid Claims Using Basic Techniques, Arlington, VA: Casualty Actuarial Society, 009. Mac, T., Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates, ASTIN Bulletin 3, 993, pp Mac, T., Measuring the Variability of Chain Ladder Reserve Estimates, Casualty Actuarial Society Forum, Spring ( 994, pp Mac, T., G. Quarg, and C. Braun, The Mean Square Error of Prediction in the Chain Ladder Reserving Method A Comment, ASTIN Bulletin 36, 006, pp Mac, T., The Standard Error of Chain Ladder Reserve Estimates: Recursive Calculation and Inclusion of a Tail Factor, ASTIN Bulletin 9, 999, pp Murphy, D., Unbiased Loss Development Factors, Proceedings of the Casualty Actuarial Society 8, 994, pp. 54. Panning, W., Measuring Loss Reserve Uncertainty, Casualty Actuarial Society Forum, Fall 006. Rehman, Z., and S. Klugman, Quantifying Uncertainty in Reserve Estimates, Casualty Actuarial Society E-Forum, Spring 009. Venter, G., Discussion of the Mean Square Error of Prediction in the Chain Ladder Reserving Method, ASTIN Bulletin 36, 006, pp Verrall, R. J., A Bayesian Generalized Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving, North American Actuarial Journal 8:3, 004, pp Verrall, R. J., Obtaining Predictive Distributions for Reserves which Incorporate Expert Opinion, Variance, 007, pp Wright, T. S., A Stochastic Method for Claims Reserving in General Insurance, Journal of the Institute of Actuaries 7, 990, pp CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

15 A Family of Chain-Ladder Factor Models for Selected Lin Ratios Appendix A Proof of Lemma (Lin Ratio Function. We first note that for arbitrary a we have I α w i i= =. ( A, Without loss of generality we can assume C aymin, < C i, for i I -. It is now sufficient to prove that w a aymin, as a. This can be proven by re writing the weight as I I aymin, aymin,, aymin, C, = = α α α w = C C = C α ( C C. aymin,, Obviously (C aymin /C, < for all aymin. Thus all terms converge to 0 except for = aymin, so that S I- = C? (C, aymin /C, a C aymin, as a. That proves w a aymin as a and subsequently w a 0 i, as a for all i aymin based on (A. The proposition is then obvious: lim a LR (a = F. aymin,. The proof that lim a - LR (a = F aymax, is similar to. The generalization to the case where the accident years having the minimum/maximum beginning values of loss are not unique is obvious, as the limits of the corresponding weights are as well. Proof of the Parameter Ris Formulas single accident year For the first period after the current diagonal, Ĉ i, + = fˆ C i,, so D (C i, + = C i, D ( f because C i, is a constant. For s > periods after the current diagonal, Ĉ i, + s = fˆ Ĉ, so based on the law of + s - i, + s - total variance : ( = = ( ˆ ˆ C EVar C C i, + s i, + s i, + s + Var E Cˆ Cˆ i, + s i, + s ˆ E( C Var fˆ i, + s + s + Var( Cˆ E( fˆ i, + s + s = Var( fˆ + E Cˆ s ( i, + s + Var( Cˆ f i, + s + s = Var( fˆ + s Var( ˆ C + E + ( Cˆ i, s i, + s + f + s ( i+ s Var C ˆ, i, + s + s + s ( i, + s = + µ f f C + ( f C + s ( i+ s,. Proof of the Process Ris Formulas single accident year For the first period after the current diagonal, a G(C i, + = C i, s. For s > periods after the current diagonal, process ris can be calculated recursively according to the formula Γ = + C f Γ C C + σ. E α + s D i, + s + s i, + s i, + s s VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 57

16 Variance Advancing the Science of Ris Proof: For the first period after its current age (s = the process ris for C i, + is a direct result of assumption (: = α σ Γ C C i, + i, because C a is a nown constant. i, For s > we again rely on the law of total variance : ( i, + s = ( ( i, + s + ( ( i, + s Γ C EVar C D Var EC D α+ s = E( C D Var E f C D i, + s + ( + s i, + s α+ s = + E C D σ f Γ C i, + s + s + s i, + s As explained in the text, in practice we favor α α s + s approx imating E( C D i, + s with ( E( C D i, + s + Ψ, where factor Y is a function of a and the coefficient of variation. For estimates of G we replace all unnown quantities by their best estimates: f by fˆ, s by ŝ, etc. Again we note here that ŝ and fˆ both depend on â. However, we drop the functional notation ŝ (â and fˆ (â for convenience of presentation. Proof of the Parameter Ris Formulas all accident years combined For = 3, 4,..., Xˆ = fˆ -? (Xˆ - + C I - +, -, where I - + is the only accident year that has matured as of age -. By employing the law of total variance mentioned above, we have: = ( ( ˆ ˆ + ( ( ˆ ˆ X EVar X X Var EX X = EVar( fˆ Xˆ + C Xˆ ( I +, + Var E( fˆ ( Xˆ + C + Xˆ I, = E ( Xˆ + C fˆ Xˆ I +, Var( ( ( ˆ ˆ + Xˆ + C Var E f X I +, = + = (( ˆ f E X C I +, ( f Xˆ C ( I +, + Var + { Var( Xˆ + E ( Xˆ + C } f I +, + f Var ( Xˆ + I +, ( X = M + C f f + ( f X ( because C I - +, - is a constant. Proof of the Process Ris Formulas all accident years combined The formula for process ris is straightforward since all accident years are assumed to be independent and the process variance of the sum of the losses for all accident years is the sum of the process variance of each accident year. 58 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE

17 A Family of Chain-Ladder Factor Models for Selected Lin Ratios Appendix B The Mac (999 model is based on the assump- tions that E(F C = f and Var ( FC = σ where, α C for simplicity, we omit his accident year index i and assume that all weights are equal to. Mac (999 calculates standard error recursively as follows: = ( + + s.e. ˆ ˆ C C s.e. F s.e. fˆ fˆ s.e. ˆ. + Case : Volume-weighted average lin ratios ( C In the Mac framewor the volume-weighted average case is achieved for a =. Thus s.e. s.e. ˆ ˆ ˆ σ C C s.e. fˆ + C ˆ = + = + Cˆ Cˆ Cˆ s.e. σ + ˆ + f s.e. ( Cˆ + f ˆ f ˆ s.e. C ˆ. ( I Within the CLFM framewor the volume-weighted average case is also achieved for a =. The CLFM formula for mean square error (mse in Mac s notation (s.e. is = + s.e. Ĉ C Γ C (from(9 and (, ˆ + ˆ respectively C f f ( C = + ( f ( C + E( Cˆ σ + Γ ( C fˆ { } s.e. ( Cˆ = Cˆ + = E( C ˆ C f σ + ˆ ( + fˆ C C [ ( + Γ ( ]+ ( f ( C = σ + + E C ˆ C ˆ f f ˆ + f σ C + ( f ( C s.e. ( Cˆ + Cˆ ( f + fˆ s.e. ( Cˆ. ( II The last cross-variance term in (II, i.e., D ( f D (C, is not included in the Mac s volume-weighted average formula (I. This is a well-nown result. 8 Case : Simple average lin ratios In the Mac framewor the simple average case is achieved for a = 0. Thus = ( + + s.e. ˆ ˆ C C s.e. fˆ fˆ σ s.e. + ( Cˆ ˆ ˆ C C Cˆ s.e. ( fˆ = σ + s.e. ( + fˆ s.e. Cˆ. ( III + Within the CLFM framewor the simple average case is achieved for a =. Again using Mac s notation, the CLFM mse formula is s.e. ˆ ( C C C + = ( + + Γ ( + = Cˆ f ˆ { ( + f C f C ( + ( ( } { } + E( C ˆ Γ C σ + ( ˆf 8 See, for example, Buchwalder et al. (006. VOLUME 6/ISSUE CASUALTY ACTUARIAL SOCIETY 59

18 Variance Advancing the Science of Ris ˆ C Cˆ = E( f σ + ( fˆ [ C Γ C f C ( ]+ ( ( + + = ( C C + ( E Γ + ˆ σ Cˆ ˆ f Cˆ s.e. f C ( f + + s.e. ˆ ˆ ˆ ˆ ( C C C f f s.e. Cˆ + = σ + ( + ( + ( f ( C + Γ ( C σ. ( IV So the difference between the CLFM and Mac formulas for mean square error in the simple average case is comprised of the last two cross-variance terms in (IV, i.e., D (f D (C + G (C s. As far as the authors can tell, this comparison is a new result. In both cases, mse estimates based on Mac s formulas will be smaller than those based on CLFM formulas by a magnitude equal to the additional terms. For most relatively stable triangles, the crossvariance terms will have relatively little impact. But when there is considerable volatility in the empirical loss ratios, the magnitude of the cross-variance terms can be significant. (This was demonstrated for the volume-weighted case in the example. In the straight average case the other term G (C s not included in Mac s formula can overshadow the cross-variance term, as it does with the Example data (analysis omitted above. When the udgmentally selected lin ratio is not one of these two cases, the differences between the CLFM and Mac mse estimators will depend on the proximity of the selection to the straight average and volume-weighted average cases. 60 CASUALTY ACTUARIAL SOCIETY VOLUME 6/ISSUE