EDWARD W. (JED) FREES 1, GLENN MEYERS 2 AND A. DAVID CUMMINGS 2 ABSTRACT

Transcription

1 DEPENDENT MULTI-PERIL RATEMAKING MODELS BY EDWARD W. (JED) FREES, GLENN MEYERS 2 AND A. DAVID CUMMINGS 2 ABSTRACT This paper onsiders insurane laims that are available by ause of loss, or peril. Using this multi-peril information, we investigate multivariate frequeny and severity models, emphasizing alternative dependeny strutures. Although dependeny models may be used for many ris management strategies, we fous on ratemaing. Motivation for this researh omes from homeowners insurane and so, for the frequeny portion, we onsider binary response models. Speifially, we examine several multivariate binary regression models that have appeared in the biomedial literature, fousing on a dependene ratio model. For multivariate severity, we use Gaussian opulas to represent dependenies among gamma regressions. We alibrate ompeting models based on a representative sample of over 400,000 reords and validate them using a held-out sample of over 350,000 reords. We find that methods that allow for ross-dependenies among perils provide important eonomi value in priing. KEYWORDS Copulas, multivariate binary regression, insurane priing.. INTRODUCTION This paper explores the use of modern statistial preditive models that an be used for priing and ratemaing in personal lines insurane. Speifially, we fous on homeowners insurane although, as desribed below, the range of appliations is broader. Homeowners represents a large segment of the personal property and asualty (general) insurane business; for example, in the US, homeowners aounted for 2% of all property and asualty insurane premiums and 25% of personal lines insurane, for a total of over $53 billions of US dollars (I.I.I. Insurane Fat Boo 2007). University of Wisonsin and Insurane Servies Offie. 2 Insurane Servies Offie. Astin Bulletin 40(2), doi: 0.243/AST by Astin Bulletin. All rights reserved.

2 700 E.W. FREES, G. MEYERS AND A.D. CUMMINGS In the traditional atuarial literature (e.g., Bowers et al. 997, Chapter 2), ratemaing is based on the individual ris model. This model deomposes a short-term ris, suh as a homeowners laim, into frequeny and amount (nown as severity ) omponents. Speifially, let r i be a binary variable indiating whether or not the ith poliyholder has an insurane laim and y i desribe the amount of the laim. Then, the laim is modeled as (laim reorded) i = r i y i. This is the basis of the individual ris model. In homeowners, typially insurers have available many harateristis of the home upon whih rates are based. For notation, let x i be a omplete set of explanatory variables that are available to the analyst. The approah adopted here is often nown as a frequeny-severity model, where models are speified for both the frequeny and severity omponents. For example, for the frequeny omponent we might fit a logit regression model with r i as the dependent variable and x i as the set of explanatory variables. Denote the orresponding set of regression oeffiients as b. For the severity omponent, we ondition on the ourrene of a laim (r i = ), and might use a gamma regression model with y i as the dependent variable and x 2i as the set of explanatory variables. Denote the orresponding set of regression oeffiients as b 2. In this frequeny-severity, also nown as a two-part, model, one need not have the same set of explanatory variables influening the frequeny and amount of response. However, it is not unommon to find overlap in the sets of explanatory variables, where variables are members of both x and x 2. Typially, one assumes that b and b 2 are not related so that the oint lielihood of the data an be separated into two omponents and run separately, as desribed above. Many atuaries interested in priing homeowners insurane are now deomposing the set of dependent variables (r i, y i ) by peril, or ause of loss (e.g., Modlin, 2005). Homeowners is typially sold as an all-ris poliy, whih overs all auses of loss exept those speifially exluded. By deomposing losses into homogenous ategories of ris, atuaries see to get a better understanding of the determinants of eah omponent, resulting in a better overall preditor of losses. Table illustrates this deomposition for a data set that we will desribe further in Setion 2. This table displays summary statistis for nine perils from a sample of 404,664 reords. This table shows that WaterNonWeather is the most frequently ourring peril whereas Liability is the least frequent. (Water- NonWeather is water damage from auses other than weather, e.g., the bursting of a water pipe in a house.) When a laim ours, Hail is the most severe peril (aording to the median severity) whereas the Other ategory is the least severe. In Table, we note that neither the frequeny nor the number sum to the totals due to ointly ourring perils within a poliy. That is, for eah poliy, we reord the laims amount for eah of the nine perils.

3 DEPENDENT MULTI-PERIL RATEMAKING MODELS 70 TABLE HOMEOWNERS SUMMARY STATISTICS Peril Frequeny (in perent) Number of Claims Median Claim Amount Fire 0.30,254 4,52 Lightning , Wind.226 4,960,35 Hail 0.49,985 4,484 WaterWeather ,42,48 WaterNonWeather.332 5,39 2,67 Liability ,000 Other 0.464, Theft-Vandalism ,287,9 Total 5.889* 23,834*,66 In a multi-peril model, one deomposes the ris into one of types ( = 9 in Table ). To set notation, define r i, to be a binary variable indiating whether or not the ith poliy-year has an insurane laim due to the th type, =,,. Similarly, y i, desribes the amount of the laim due to the th type. To relate the multi- to the single-peril variables, we have the following relationships r = - ( - r ) # f # (- r ) = max( r, f, r ) () i i, i, i, i, and i i i,. = (laim reorded) i = y = / r, # y (2) Current atuarial pratie involves modeling eah peril in isolation of the others. Thus, for example, from the full set of explanatory variables x, the analyst selets a set of variables x, to predit the frequeny and another a set x 2, to predit the severity for eah peril, =,,. This is intuitively appealing beause some preditors do well in prediting ertain perils but not others. For example, dwelling in an urban area may be an exellent preditor for the theft peril but provides little useful information for the hail peril. To implement this modeling strategy, it is straightforward in priniple to use a logisti regression for eah frequeny and gamma regression for eah severity. Although easy to interpret, this proedure uses the same dataset to alibrate several models. From a modeling point of view, this amounts to assuming that perils are independent of one another and that sets of parameters from eah peril are unrelated to one another. Although maing sets of parameters unrelated to another (sometimes alled funtionally independent) is plausible, it

4 702 E.W. FREES, G. MEYERS AND A.D. CUMMINGS seems unliely that perils are independent. Event lassifiation an be ambiguous (e.g., fires triggered by lightning) and unobserved latent harateristis of poliyholders (autious homeowners who are sensitive to potential losses due to theft-vandalism and liability) may indue dependenies among perils. Our preliminary empirial examinations in Setion 2 will also suggest that perils may be related to one another. To investigate potential dependene relationships, we retain the basi hierarhial approah of the frequeny-severity model and treat eah omponent as a multivariate response. Speifially, we first analyze the frequeny omponent and then model the severity omponent onditional on the frequeny. The multivariate multi-peril model is:. Use a multivariate binary regression model with r i = ( r i,,, r i, ) as the dependent variable. 2. Conditional on the frequeny r i, for the severity we speify a multivariate regression with y i = ( y i,,, y i, ) as the dependent variable. We will apply this modeling strategy to homeowners insurane, where a laim type may be due to fire, liability, and so forth. One ould also use this strategy to model homeowners and automobile poliies ointly. As another example, in healthare, expenditures are often broen down by diagnosti related groups. In the atuarial and insurane literatures, there has been a reent surge of interest in modeling short-term overages at the individual poliyholder level, e.g., miro-level data. Examples inlude wors by Angers, Desardins, Dionne and Guertin (2006), Bouher and Denuit (2006), Frees, Peng and Valdez (2009), Loo, Fung and Zhu (2007) and Mahmoudvan and Hassani (2009). However, these papers deal with automobile insurane insurane, not homeowners. Moreover, beause they do not split by ause of loss as we do in this paper, for the frequeny model they examine ount models suh as Poisson regression in lieu of the binary logisti regression models in this paper. Setion 3 will introdue our multivariate severity model. It is based on gamma regressions for the marginal peril distributions with a Gaussian opula to quantify the assoiation among severities. In this ontext, a strength of the opula framewor is that marginals are preserved this is important when we have relatively few oint severities upon whih to base our measures of dependene. Setion 4 will introdue multivariate binary regression models, fousing on the dependene ratio approah introdued by Eholm et al (995). Appendix Setion B desribes alternative models, inluding log-linear, quadrati exponential and alternating logisti regressions. As we disuss estimation of the models in Setions 3 and 4, we will be able to provide assessments using in-sample measures. In-sample measures inlude hypothesis testing statistis suh as t-statistis and lielihood ratio tests to udge the statistial signifiane of parameters and goodness of fit statistis suh as AIC to udge the overall model fit. Setion 5 will introdue our validation of models based on a held-out sample that is unrelated to our estimation

5 DEPENDENT MULTI-PERIL RATEMAKING MODELS 703 sample. As disussed in this setion and Appendix Setion D, insurane laims out-of-sample model validation an be diffiult due to the mixture of zeros and positive outomes, as well as positive outomes that are sewed and fattailed. Conluding remars are provided in Setion DATA To alibrate our models, we drew two random samples from a homeowners database maintained by the Insurane Servies Offie. This database ontains over 4.2 million poliyholder years. It is based on the poliies issued by several maor insurane ompanies in the United States, thought to be representative of most geographi areas in the US. These poliies were almost all for one year and so we will use a onstant exposure (one) for our models. Our in-sample, or training, dataset onsists of a representative sample of 404,664 reords taen from this database. The original database had an oversampling of laims; we adusted our sampling proedures so that the in-sample dataset ould be treated as a random sample from the population of poliies. The summary measures in this setion are based on this training sample. In Setion 5, we will test our alibrated models on a seond held-out, or validation subsample that was also randomly seleted from this database. Table in Setion summarized the main tendenies of frequeny and severity. Prior to introduing formal mathematial models that aount for dependene, we first examine the homeowners data to establish, or at least suggest, the presene of dependene among perils. 2.. Severities Dependene among ontinuous variables is more well-nown than disrete variables, so we begin with the severity portion. Table 2 presents orrelations among perils. Beause laims distributions are typially right-sewed, entries in the tables are Spearman orrelations. Reall that a Spearman orrelation is a regular (Pearson) orrelation among the rans, not the atual laims. In this way, they do not depend on the sale and so, for example, would be unhanged if we alulated the orrelation of logarithmi laims. Table 2 shows many positive orrelations but also some large negative ones. For example, the orrelation between Lightning and Liability is! The explanation for this is that orrelations are based on observed pairs of severity perils there are relatively few observations to base these orrelations upon. Table showed that there were only 23,834 laims in our data base. Moreover, it turns out that 96.% of these were from poliies with only a single laim. Thus, the number of poliies with two or more laims is small, maing the estimation of severity orrelations impreise.

6 704 E.W. FREES, G. MEYERS AND A.D. CUMMINGS TABLE 2 SPEARMAN CORRELATIONS AMONG SEVERITY PERILS Fire Lightning Wind Hail Water Weather Water Non Weather Liability Other Theft Vand Fire Lightning Wind Hail WaterWeath WaterNWeath Liability Other TheftVand Frequenies Table 3 gives the number of oint laims among perils. For example, we see that there were only 3 reords that had a Lightning and a Liability laim within the year. The (ran) orrelation between Lightning and Liability noted above,, is not meaningful as it is based on only 3 observations. TABLE 3 JOINT CLAIM COUNTS AMONG PERILS Fire Lightning Wind Hail Water Weather Water Non Weather Liability Other Theft Vand Fire Lightning Wind Hail WaterWeather WaterNWeath Liability Other TheftVand Subtotal Totals Note: Totals refer to all laims from a peril, not ust those ourring ointly with another peril. To measure assoiation in our binary frequeny data, orrelations an be misleading summary statistis. Instead, we will examine dependene ratios of the form

7 DEPENDENT MULTI-PERIL RATEMAKING MODELS 705 t = 2 Pr( r =, r2 = ) Pr( r = ) Pr( r = ) 2 (3) the ratio of the oint probability to the produt of the marginal probabilities. In the ase of independene, we would expet the dependene ratio t 2 to be. Values of t 2 > indiate positive dependene and values of t 2 < indiate negative dependene. Table 4 provides empirial dependene ratios for eah pair of perils. To understand the alulation of this table, onsider the relationship between Fire and WaterWeather. To begin, the marginal empirial probability of a fire laim is Pr( Z 254 rfire = ) = = , where from Table 3 the number of fire laims is,254. Similarly, the probability of a water due to weather laim is Pr Z 342 ( r WaterWeather = ) = = Further, the oint probability of a poliy having laims due to both Fire and WaterWeather is Pr( Z 23 r Fire =, r WaterWeather = ) = = Putting these together, the estimated dependene ratio is t = = Fire, WaterWeather # TABLE 4 DEPENDENCE RATIOS AMONG PERILS Fire Lightning Wind Hail Water Weather Water Non Weather Liability Other Theft Vand Fire Lightning Wind Hail WaterWeath WaterNWeath Liability Other TheftVand

8 706 E.W. FREES, G. MEYERS AND A.D. CUMMINGS Table 4 summarizes these alulations for all 36 pairs of perils. Here we see muh dependene among perils, typially positive dependene but some negative dependene as well. The data (and intuition) suggests dependenies among perils. As we have seen, the severity orrelations are based on only a few observations these assoiations turn out to be hard to estimate. In ontrast, for frequeny the dependene ratios are based on 404,664 observations. Although the oint probabilities are small, we an get preise estimates Dependenies among Marginal Frequeny Models The data presented (so far) do not aount for explanatory variables that ould indue orrelations. For these effets, we used many explanatory variables for eah peril (from 8 for the Other peril to 9 for the Water Weather peril) for over 00 preditors. These are a variety of geographi-based plus several standard industry variables that aount for: weather and elevation, viinity, ommerial and geographi features, experiene and trend, and rating variables. The web site Homeowners.html provides more information on these explanatory variables. Beause the fous of this paper is on the assoiation aspets, in this paper we summarize the results only for the dependeny parameters. For assessing frequeny dependenies, reall that r denotes the binary variable that indiates a laim ( y = ). In our sample, we have r i, i =,, n = 404,664, and =,, = 9. Let q i be the orresponding probability of a laim. The number of laims that is oint between the th and th perils is n / r # r Assuming independene among perils, this has mean and variane i = i i. and n / / Ee r # r o = q # q n i i i = i = / / n i i Var e r # r o = q q -`q q. i i i = i = n 2 i i i i To assess dependenies among the laim frequenies, we employ the t-statisti t = n / r # r - / n i= i i i= / n i = q q i q - `q q i i i i # q 2 i. (4)

9 DEPENDENT MULTI-PERIL RATEMAKING MODELS 707 TABLE 5 TEST STATISTICS FROM LOGISTIC REGRESSION FITS Fire Lightning Wind Hail Water Weather Water Non Weather Liability Other Theft Vand Fire Lightning Wind Hail WaterWeath WaterNWeath Liability Other TheftVand The t-statisti in equation (4) would be a standard two-sample t-statisti exept that we allow the probability of a laim to vary by poliy i. To estimate these probabilities, we fit a logisti regression model for eah peril, where the explanatory variables are peril-speifi. Eah model was fit in isolation of the others, thus impliitly using the null hypothesis of independene among perils. Table 5 summarizes the test statistis for assessing independene among the frequenies. Not surprisingly, the strongest relationship was between water damage due to weather and water damage from auses other than weather. The largest dependene ratio in Table 4, between fire and the Other ategory, was the seond largest t-statisti this indiates strong dependene even after ovariates are introdued. Interestingly, the only signifiant negative relationship was between hail and the Other ategory. For the degrees of freedom of the t-statisti, we have followed the usual rule of the number of observations minus the number of parameters. Beause our sample size is large ( n = 404,664) relative to the number of parameters, the referene distribution is essentially normal. We anowledge that the asymptoti distribution may be slightly mis-speified beause the probabilities q are only nown up to the estimated regression parameters. Thus, for some appliations, one may wish to determine the referene distribution via alternative means suh as bootstrapping. However, given our large sample size and beause we are using the statisti only for diagnosti purposes, we reommend this proedure to unover dependenies among the frequenies. 3. MULTIVARIATE SEVERITY MODEL To aommodate dependenies among laim severities, we use a parametri opula. A opula allows us to use different gamma regression models for eah type, thus permitting a diret omparison with the independene model.

10 708 E.W. FREES, G. MEYERS AND A.D. CUMMINGS One ould allow the distributions to vary by peril within the opula framewor. We use only gamma regression marginals in part to onform with industry pratie. See Frees and Wang (2005) for a longitudinal appliation that employs opulas to relate dependenies among gamma regression models. 3.. Marginal Distributions Suppose that there are potential laims for the ith poliy, y i = ( y i,, y i ). The oint distribution funtion is denoted by ^ h ^,, h, Fi ai, f, ai = Pr yi # ai f yi # ai with marginal distribution funtions F i (a i ) = F i = Pr( y i a i ). The marginals follow a gamma distribution with parameters that vary by peril and ovariates that depend on the poliy. Speifially, let q i = x i b 2, be a systemati omponent where x i is a vetor of nown explanatory variables and b 2, is a vetor of unnown parameters. Assuming a logarithmi lin funtion, the distribution F i is speified to be a gamma distribution, with mean m i = exp (q i ) and sale parameter that also varies by peril, sale. Below, we use f (., q i, sale ) to denote this density Modeling the Dependene The oint distribution funtion of laim severities an be expressed as a funtion of the marginal distributions through a opula. Suppressing the {i } subsripts, let F be the distribution funtion assoiated with the th type, y. We may write the oint distribution of laims y = ( y,, y ) as F ^a, f, a h Pr^y,, y a h = f # = COP ^F ( a ), f, F ( a ) h. = Pr^F ( y ) # F ( a ), f,f ( y ) # F ( a ) h Here, COP ( ) is the opula lining the marginals to the oint distribution. See, for example, Frees and Valdez (998) for an introdution to opulas. Let f be the density funtion assoiated with the th type. The multivariate density is f^a, f, a h = op ^F ( a ), f, F ( a ) h% f ( a ). (5) = Here, op (.) is the density funtion orresponding to the opula distribution funtion COP(.).

11 DEPENDENT MULTI-PERIL RATEMAKING MODELS 709 To illustrate, we will wor extensively with the normal (also nown as the Gaussian) opula. We may write the normal opula as - - op ( u N, f, u ) = fn_ F ( u), f, F ( u ) i % (6) = f - ( F ( u )). Here, F and f are the distribution and density funtions of the standard normal distribution, respetively. The multivariate normal density is f ( z) = ( 2p) N / 2 det S expb - z S 2 - zl. The matrix S is a orrelation matrix, with ones on the diagonal Estimation Results The opula allows for a fully parametri speifiation of the probability model. We exploit this speifiation by using maximum lielihood estimation. We assume independene among poliies and use the opula to model dependenies among perils. Using equation (5), when there are laims from all perils, the log-lielihood of the ith poliy is l i = / ln f _ yi, q, salei + ln op ( Fi,, Fi ). (7) = i N f Gaussian opulas are preserved under the marginals, so having only a subset of perils does not present a diffiulty in evaluating the lielihood expression. A broader set of diffiulties in the evaluation of the lielihood is summarized in Appendix A. To estimate the opula model, we use the explanatory variables that were developed under the independene model. As with the frequeny model, there were many explanatory variables for eah peril and so we summarize the results only for the dependeny parameters. We examined three models of assoiation by varying the speifiation of the orrelation matrix S. In the most omplex speifiation, we allowed S to be unstrutured (subet to being symmetri and invertible), resulting in ( 2 9 ) = 36 assoiation parameters to be estimated, one for eah pair of perils. In the least omplex speifiation, we speified all assoiation parameters to be equal, so that S has a struture nown as a uniform orrelation or ompound symmetry model. As an intermediate hoie, we grouped the perils into five lasses. As desribed below, this grouping resulted in twelve assoiation parameters. Using a single assoiation parameter, the maximum lielihood estimator turned out to be with a orresponding t-statisti equal to This indiates statistially signifiant positive assoiation among laims.

12 70 E.W. FREES, G. MEYERS AND A.D. CUMMINGS TABLE 6 COPULA PARAMETER ESTIMATES Assoiation Estimate t-statisti Between Groups and Between Groups and Between Groups and Between Groups and Between Groups 2 and Between Groups 2 and Between Groups 2 and Between Groups 3 and Between Groups 3 and Between Groups 4 and Within Group Within Group Note: The five groups are: () Fire, Lightning, (2) Wind, Hail, WaterWeather, WaterNonWeather, (3) Liability, (4) Other and (5) Theft. Results for the twelve parameter model appear in Table 6. Here, we see positive assoiation between most groups and within the two larger groups the exeption is between groups 3 and 5. However, none of the assoiation parameters are strongly statistially signifiant, despite estimation using 23,384 laims severities. Further, when we estimated the opula model with 36 parameters, only one of the 36 of the parameter estimates turned out to be strongly statistially signifiant (with a p-value < 0.0 this was between Fire and Water Nonweather ); thus, these parameter estimates are not reported here. We oneture that the reason is that there were relatively few poliies having oint laims that would ontribute to the opula portion of the lielihood (see Table 3). We also examined alternative opula (e.g., t-opula) and marginal regression speifiations (heavier tail than gamma). Beause we had relatively few oint laims, the out-of-sample analysis showed that this line of researh was not fruitful for our data. 4. DEPENDENCE RATIO MULTIVARIATE FREQUENCY MODEL Fortunately, in the statistis literature, there are many good approahes to modeling multivariate binary frequenies. To eep this paper ontained, Setion 4.3 and Appendix Setion B provide brief overviews of these methods with appropriate referenes for readers who wish to explore this topi further. This paper features the dependene ratio approah, introdued by Eholm, Smith and MDonald (995). This is a lielihood approah, where the lielihood

13 DEPENDENT MULTI-PERIL RATEMAKING MODELS 7 is written in terms of means and dependene ratios. See also Eholm, MDonald and Smith (2000). Suppressing the {i}subsripts, reall the mean parameters p = E r = Pr ( r = ) and similarly define higher order moments p = E r r = Pr ( r = r = ), p i = E r i r r,, p 2 = E r r 2 r. Now, if the responses are independent, then p 2 = p p 2 and so forth. To assess this, as in equation (3) we may define the dependene ratio p2 Pr( r = r2 = ) t2 = pp =. Pr( r = ) Pr( r = ) 2 2 Interpret t 2 00 to be the perentage that r and r 2 are both one (laims) under the dependene model ompared to the independene model. Similarly, define higher order dependene ratios as p 2 pp, t p p g t = i = p p p, f, t2g = ppgp. i i 2 The approah is to use regression ovariates to estimate the means p and simpler speifiations, typially onstants, to estimate the dependene ratios. 4.. Basi Lielihood We will use maximum lielihood estimation. Writing the lielihood in terms of the means p is straightforward yet tedious. To see some of the diffiulties, onsider the ase with only = 3 perils. Then, we have Pr( r=, r2=, r3= ) = E rrr 2 3= p23 Pr( r =, r2 =, r3 = 0) = E rr2 ( - r3) = p2 -p23 Pr( r =, r2 = 0, r3 = ) = p3 -p23 Pr( r =, r2 = 0, r3 = 0) = p -p2 - p3 + p23 h Pr( r = 0, r = 0, r = 0) = - ( p + p + p ) + ( p + p + p )-p So this gives eah possible lielihood outome in terms of marginal means p and dependeny parameters t and t 23. For = 9 perils, the pattern is similar. Most poliies result in zero laims for all perils: Pr( r = 0, r = 0, f, r = 0) 2 = - / p + / p - / pi + f + (-) p2g = < i< < (8)

14 72 E.W. FREES, G. MEYERS AND A.D. CUMMINGS For a poliy with a laim in the first peril and no other laims, we have Pr( r =, r = 0, f, r = 0) 2 / / = p - p + p - f + (- ) p2g. = 2 < < - Other poliies with singleton laims an be alulated via symmetry. For a laim in the first two perils and no other laims, we have Pr( r =, r =, r = 0, f, r = 0) / 2 / = 3 2 < < = p - p + p - f + (-) p. 2 2 Other poliies with two laims an be alulated via symmetry. For our data set, poliies with three and more laims represent an extremely small fration of the data. Thus, Setion 4.2 presents estimates t for eah pair of perils (, ) but assume that higher order ratios, suh as t l and t lm, are equal to one. Appendix Setion C provides further alulation details. Calulation of the lielihood estimates uses the independene model to provide initial values. For some data sets, there ould be a small issue with onstraints on the optimization. For example, beause r 2, we have p 2 = E r r 2 E r = p. Thus, p min(, ), and min, 2 # p p2 t2 # p p m. 2 Beause of the small size of our p s, this has not been an issue for the homeowners appliation Estimation Results Our fous is on estimation of assoiation between pairs of perils. For onsisteny with the severity setion, we onsider three models of assoiation, a single parameter model, a model with 36 parameters, one for eah pair of perils, and an intermediate version formed by taing groups of perils. Also for onsisteny with the severity portion, we estimate models inluding regression ovariates but do not report on this portion of the results. For the single assoiation parameter, the maximum lielihood estimator turned out to be.3325 with a standard error of , 8.6 standard errors above. This is learly statistially signifiant and indiates positive dependene. Results for a 2 parameter intermediate model are summarized in Table 7. Here, we see that all of the parameter estimates indiated positive dependene and most are statistially signifiant. To lin this to the summary statistis

15 DEPENDENT MULTI-PERIL RATEMAKING MODELS 73 TABLE 7 DEPENDENCE RATIO PARAMETER ESTIMATES Assoiation Estimate t-statisti Between Groups and Between Groups and Between Groups and Between Groups and Between Groups 2 and Between Groups 2 and Between Groups 2 and Between Groups 3 and Between Groups 3 and Between Groups 4 and Within Group Within Group Notes: The five groups are: () Fire, Lightning, (2) Wind, Hail, WaterWeather, WaterNonWeather, (3) Liability, (4) Other and (5) Theft. The t-statisti provides the number of standard errors that the estimate differs from, the null value under the independene hypothesis. presented in Table 4, Table 8 gives the parameter estimates in a matrix format. This allows one to ompare the maximum lielihood results from our dependene ratio model that inludes ovariate information to the empirial dependene ratios that are alulated without ovariate information. We also alulated the 36 parameter dependeny ratio model where eah pair of perils had a unique parameter. These maximum lielihood estimates also turned out to be lose to the empirial ones presented in Table 4 and so are not presented here. TABLE 8 MATRIX OF DEPENDENCE RATIO PARAMETER ESTIMATES Fire Lightning Wind Hail Water Weather Water Non Weather Liability Other Theft Fire Lightning Wind Hail WaterWeath WaterNWeath Liability Other Theft

16 74 E.W. FREES, G. MEYERS AND A.D. CUMMINGS 4.3. Alternative Multivariate Models There are many approahes for modeling multivariate variables with none being uniformly superior to the others. This setion douments our findings in exploring alternative models for multivariate laims. One widely used multivariate frequeny model that we did not empirially estimate is the multivariate probit. To fit this model using lielihood methods, one needs to ompute a multivariate normal distribution funtion for eah poliyholder. For our ase, this would involve a 9-dimensional multivariate normal distribution funtion evaluation for eah of the 404,664 reords. Beause of these omputational issues, multivariate probits were not further explored. We did explore a method for multivariate frequeny that is widely used in the biomedial ommunity nown as alternating logisti regressions. Alternating logisti regressions, also nown by the aronym ALR, uses generalized estimating equations (GEE) methods to estimate binary dependenies. This method is attrative in that it is available in the ommerial statistial software SAS. However, for the number of subets onsidered here it is oded ineffiiently and this anned proedure is not a reasonable alternative. More importantly, as a GEE method, alternating logisti regression does not probabilistially model dependenies but rather allows for them in the moment struture to enhane the effiieny of estimators. For data sets of the size we are onsidering, effiieny is less of an issue. Moreover, the la of a probabilisti model maes it more diffiult to speify an optimal preditor, a ey goal of ratemaing. We do not report further results for ALR, although Appendix Setion B3 ontains a brief overview with referenes for readers wishing to learn more about the alternating logisti regression approah. 5. OUT-OF-SAMPLE VALIDATION In preditive modeling, one validates a model by examining performane on an independent held-out sample of data (e.g., Hastie, Tibshirani and Friedman, 200). Standard performane measures inlude an assessment of bias, root mean square error and related measures. In insurane laims, these standard measures are not the most informative due to the high proportions of zeros (orresponding to no laim) and the sewed, fat tailed distribution of the positive values. Appendix Setion D disusses this point and develops a summary measure that we will use in this paper. We all this measure a Gini index, after a standard measure of inome inequality. As desribed in Appendix Setion D, this measure summarizes the typial profitability that an insurer will enoy for a held-out sample when taing on a new soring method relative to an existing priing struture. The larger the index, the more effetive is the soring mehanism. We do not have atual pries harged by the insurane ompanies. Moreover, beause these are interompany data, even if we did it is not lear that these pries would be

17 DEPENDENT MULTI-PERIL RATEMAKING MODELS 75 TABLE 9 COMPARISON OF ALTERNATIVE APPROACHES TO INDEPENDENCE MODEL Frequeny Model Severity Model Gini Index Independene Independene Independene Copula with 36 parameters.477 Independene Copula with parameter Dep Ratio with parameter Independene Dep Ratio with 2 parameters Independene.673 Dep Ratio with 36 parameters Independene omparable beause different ompanies use different priing strategies. Instead, as our base prie we use the independene model predited values omputed assuming no dependene among perils. Appendix Setion D provides additional details. Table 9 summarizes the out-of-sample performane of the several approahes to dependene modeling onsidered in this paper. Here, we see that introduing assoiation among severities using the opula framewor provides no additional preditive ability. This is not surprising as desribed in Setion 3, beause of the large number of perils (9), we have relatively few poliies with oint severities, meaning that it is diffiult to assess their assoiation. For other appliations, it would ertainly mae sense to investigate other parametri speifiations or hoies of opulas. Beause of this la of preditive ability, Table 9 reports only two models using opulas. Table 9 shows that introduing dependene ratios to model multivariate binary responses provides additional preditive power. All three speifiations performed well on an out-of-sample basis. Of these three, the one parameter speifiation is preferred based on the priniple of parsimony and out-of-sample performane. 6. SUMMARY AND CONCLUDING REMARKS This wor provides evidene regarding the effets of dependene on multi-peril multivariate frequeny and severity models. Our foundation essentially assumes independene among perils. We have examined several alternative models, using in-sample and out-of-sample measures to validate the model seletion. For multivariate severity, opula modeling with gamma regression marginals provided little benefit, either on an in-sample or out-of-sample basis. In ontrast, reent wor in Frees and Valdez (2008) shows that opula models of multivariate severity (in auto) an be an effetive modeling tool they onsidered three types of automobile laims. We oneture that the large dimension (number of perils is nine) of our severity response vetor ontributes to our inability to apture severity dependenies.

18 76 E.W. FREES, G. MEYERS AND A.D. CUMMINGS For multivariate frequeny, we surveyed a number of models that ould be used. For our data, the dependene ratio approah was most effetive. We used a speial ase of this framewor that features an assoiation struture similar to a orrelation matrix. The dependene ratio model displayed statistially signifiant assoiation parameters and had desirable out-of-sample performane. ACKNOWLEDGEMENT We than olleagues at the Innovative Analytis unit of the Insurane Servies Offie and an anonymous review for omments that help to improve the paper. REFERENCES ANGERS, J.-F., DESJARDINS, D., DIONNE, G. and GUERTIN, F. (2006) Vehile and fleet random effets in a model of insurane rating for fleets of vehiles. Astin Bulletin 36(), BOUCHER, J.-P. and DENUIT, M. (2006) Fixed versus random effets in Poisson regression models for laim ounts. A ase study with motor insurane. Astin Bulletin 36(), BOUCHER, J.-P. and DENUIT, M. (2008) Credibility premiums for the zero-inflated Poisson model and new hunger for bonus interpretation. Insurane: Mathematis and Eonomis 42(2), BOWERS, N.L., GERBER, H.U., HICKMAN, J.C., JONES, D.A. and NESBITT, C.J. (997) Atuarial Mathematis. Soiety of Atuaries, Shaumburg, IL. CAREY, V., ZEGER, S.L. and DIGGLE, P. (993) Modelling multivariate binary data with alternating logisti regressions. Biometria 80(3), DIGGLE, P.J., HEAGERTY, P., LIANG, K.-Y. and ZEGER, S.L. (2002) Analysis of Longitudinal Data, Seond Edition. Oxford University Press. EKHOLM, A., SMITH, P.W.F. and MCDONALD, J.W. (995) Marginal regression analysis of a multivariate binary response. Biometria 82(4), EKHOLM, A., MCDONALD, J.W. and SMITH, P.W.F. (2000) Assoiation models for a multivariate binary response. Biometris 56, FREES, E.W. and WANG, P. (2005) Credibility using opulas. North Amerian Atuarial Journal 9(2), FREES, E.W. and VALDEZ, E. (998) Understanding relationships using opulas. North Amerian Atuarial Journal 2(), -25. FREES, E.W., SHI, P. and VALDEZ, E.A. (2009) Atuarial appliations of a hierarhial insurane laims model. Astin Bulletin 39(), HASTIE, T., TIBSHIRANI, R. and FRIEDMAN, J. (200) The Elements of Statistial Learning: Data Mining, Inferene and Predition. Springer, New Yor. LIANG, K.-Y. and ZEGER, S.L. (992) Multivariate regression analyses for ategorial data. Journal of the Royal Statistial Soiety B 54(), LO, C.H., FUNG, W.K. and ZHU, Z.Y. (2007) Strutural parameter estimation using generalized estimating equations for regression redibility models. Astin Bulletin 37(2), MAHMOUDVAND, R. and HASSANI, H. (2009) Generalized bonus-malus systems with a frequeny and a severity omponent on an individual basis in automobile insurane. Astin Bulletin 39(), MODLIN, C. (2005) Homeowners modeling. Presentation at the 2005 Casualty Atuarial Soiety Seminar on Preditive Modeling, available at handouts/modlin.pdf. ZHAO, L.P. and PRENTICE, R.L. (990) Correlated binary regression using a quadrati exponential model. Biometria 77,

19 DEPENDENT MULTI-PERIL RATEMAKING MODELS 77 APPENDICES A. MULTIVARIATE SEVERITY LIKELIHOODS The opula allows for a fully parametri speifiation of the probability model. We exploit this speifiation by using maximum lielihood estimation. We assume independene among poliies and use the opula to model dependenies among perils. The diffiulty is in the pratial evaluation. There are between 0 and 20 regression ovariates for eah peril for over 00 regression parameters. There are an additional 9 sale parameters plus 36 orrelation parameters. The large number of parameters an be handled under the independene model. Under independene, we have that op (.) / and so the opula portion of the loglielihood is zero. Beause the first part of the log-lielihood is additive and with the assumption of no ommon parameters, the log-lielihood an be deomposed into = 9 separate problems. To get a better handle on the omputational diffiulties under dependene, let us examine the seond expression on the right-hand side of equation (7) in the normal opula ase. Using equation (6), we have - - F ( F ), - lnop _, f, F i = ln f `F f, F ( F ) - ln f`f ( F ) N i i n i i = - =- ln( 2p) - ln det S - ni S ni - / ln f( n i ) / = i where n i = F (F i ) and n i = (n i,, n i ). As desribed before, it is rare to enounter a poliy with laims from all perils. When there is a laim from a single peril, say the th, the ontribution to the log-lielihood is l i = ln f ( y i, q i, sale ). This is beause the opula redues to a uniform distribution over [0, ] that has logarithmi density ln op( ) = 0. When there is a laim from two perils, say the first and seond, the ontribution to the log-lielihood is l i = ln f ( y i, q i, sale ) + ln f ( y i2, q i2, sale 2 ) + ln op N (F i, F i2 ), where ln opn_ Fi, Fi 2 i =-ln( 2p) - ln det e 2 s ni n s 2-2 ^ i ni2h e o e o / s n 2 s2 o i2 ( n ). - ln f i = Note that this lielihood expression involves only oeffiients from the first two perils.

20 78 E.W. FREES, G. MEYERS AND A.D. CUMMINGS Thus, we investigate the following algorithm:. Determine initial estimates of regression and sale oeffiients assuming independene. Call these estimates b W and sale\, for =,,. 2. Assume that the regression and sale parameters are fixed. Maximize the lielihood over orrelation parameters. Call these estimates S W. 3. Update the parameter estimates for the th peril, =,,. Assume that the orrelation parameters (S W ) are fixed. Assume that the regression and sale parameters from other perils ( b W and sale \, for =,,,! ) are fixed. Find the regression and sale parameters to maximize the lielihood. 4. Return to Step 2, until onvergene. We provide the following remars. Step provides estimates under the independene model. Step 2 involves only data from poliies with two or more laims. For our data, this is only 3.9% of laims. For Step 3, eah maximization step involves only laims from that peril. B. MULTIVARIATE BINARY REGRESSION MODELS This appendix reviews ey features of multivariate binary regression models. We rely on Diggle et al. (2002), as well as Eholm et al. (995) and Liang and Zeger (992). B.. Log-Linear Model The vetor of dependent variables is r = ( r,, r ) ; there are d = 2 possible responses. One way of organizing the d possible outomes is through the suffiient statisti s() r = `r,, r, r r,, r r,, r r r - f 2 f f 2g onsisting of all singletons, possible produt pairs, and so on up to a produt of all variables. For example, if = 3, then the d = 8 outomes onsist of s( r) = `r, r, r, rr, rr, rr, rrr With a parameter vetor q = (q,, q, q 2,, q,,, q 2 ), we may write the lielihood as Pr() r = ( q) exp^s() r qh (9) = ( q) expe / q r + / qi rir + / qi rir r + f+ q2 g r r2gro. = i< i< <

21 DEPENDENT MULTI-PERIL RATEMAKING MODELS 79 Here, (q) is a saling term, so that probabilities sum to one. Equation (9) provides the basis for the log-linear model, a widely use representation for multivariate binary data. Although widely used, the log-linear model does not readily handle regression explanatory variables. Instead, the parameters are interpreted in terms of onditional odds and odds ratios. To see this, tae = 3 and use r 3 = 0. Then, Pr( r, r,0) = ( q) exp^q r + q r + q r r h From this, the odds for r, onditional on r 2 and r 3 = 0, are Pr( r = r2,0) Pr( r = 0 r,0) 2 = Pr( r =, r2,0) Pr( r = 0, r,0) 2 = exp^q + q r h. (0) 2 2 We may interpret q to be the logarithmi odds for r, onditional on all the other responses equal to zero (inluding r 2 ). We an interpret q 2 to be the assoiation between r and r 2, onditional on the values of the other dependent variables. To introdue regression explanatory variables x, one an always mae eah parameter a funtion of x. One limitation is that the parameter interpretation depends on the other responses. Another limitation is that it is diffiult to relate the parameters to the independene model, our baseline. Speial Case Quadrati Exponential Model Assuming that oeffiients assoiated with more than produt pairs are zero, Zhao and Prentie (990) introdued the quadrati exponential model qr qi rr i = i< / / () Pr() r = ( q) exp e + o. An advantage of this model is that oeffiients an be interpreted in terms of onditional log odds ratios. As with equation (0), from equation (), one an he that Pr( r = r, rl = 0, l!, ) ln * 4 = q Pr + q ( r = 0 r, r = 0, l!, ) l r. We may interpret q to be the logarithmi odds for r, onditional on all the other responses equal to zero (inluding r ). We an interpret q to be the assoiation between r and r, onditional on the values of the other dependent variables.

22 720 E.W. FREES, G. MEYERS AND A.D. CUMMINGS To introdue regression explanatory variables x, one an always mae eah parameter a funtion of x. One limitation is that the parameter interpretation depends on the number of responses. This is partiularly important in longitudinal data, where the number depends on eah subet. This is less important in our set-up where the number of perils is fixed. As mentioned earlier, perhaps the main limitation is that it is diffiult to relate the parameters to the independene model, our baseline. B.2. Bahadur s Representation For marginal regression models, we use the means p for parameters. There are several ways to speify the remaining 2 parameters. This subsetion briefly desribes Bahadur s representation. Here, seond order moments are given in terms of orrelations. Speifially, define the mean of eah response to be p = E r and a standardized version as r ) r - p = p ( - p ). Parameters are given as r = Corr (r, r ) = E r * r *, r i = E r i * r * r * and so on up to r 2 = E r * r 2 * r *. With this notation, Bahadur s representation is % r Pr() r = p (-p ) = ( - r ) / / + r r ) r ) i + ri r ) i r ) r ) + + r2 r ) r ) g 2 r ) i< i< < # g g m. (2) The strength of Bahadur s representation is that one an see how the oint probability depends on the orrelations and more omplex interations among dependent variables. The limitation is that orrelations are onstrained by marginal means in the binary model. B.3. Alternating Logisti Regressions This setion disusses an alternating logisti regression, or ALR, due to Carey et al. (993). The ALR uses generalized estimating equations (GEE) to estimate mean and assoiation parameters. The ALR algorithm is desribed that uses means and odds ratios only, and is silent on the role of the remaining parameters. To desribe the ALR proedure, we begin with the same mean parameters that one would use for a univariate binary regression model. Define p i = Pr (r i = ) to be the mean that is modeled through a systemati omponent as logit (p i ) =

23 DEPENDENT MULTI-PERIL RATEMAKING MODELS 72 x i b. Through this notation, we allow the explanatory variables (x) and regression oeffiients (b) to depend on peril. For assoiation, instead of orrelations we use the odds ratio. Speifially, define the odds ratio between r i and r i to be Pr( ri =, ri = )/ Pr( ri = 0, ri = ) i = (3) Pr( r =, r = 0)/ Pr( r = 0, r = 0) i i i i This is one under independene. As with the means, we estimate parameters that will summarize this dependene and use the relation ln i = z i A. Here, z i is a set of explanatory variables that ould be used to model the assoiation. The ALR algorithm is based on two stages, one for the regression oeffiients (b s) and one for the assoiation parameters (a s). Speifially, the first stage is the usual GEE estimation proedure with fixed assoiation parameters. To ompute the variane, reall the well-nown relation that if means and orrelations are nown, then one an ompute varianes and ovarianes. Similarly, if means and odds ratios are nown, then one an ompute varianes and ovarianes of binary variables. To see this, for the varianes, we have Var r i = p i p i 2, so the mean parameter determines the variane. For ovarianes, we have Cov (r i, r i ) = E r i r i p i p i = Pr(r i =, r i = ) p i p i. Thus, only the oint probability Pr(r i =, r i = ) needs to be determined to ompute the ovarianes. Using the relations, p i = Pr(r i =, r i = ) + Pr(r i =, r i = 0), similarly for p i and equation (3), we an solve for Pr(r i =, r i = ) and hene ompute the ovarianes. To summarize the varianes and ovarianes, define V i to be the variane-ovariane matrix of r i. With this notation, the stage one algorithm is to solve the estimating equation for b n 2p - 0 = / i Vi r p. b i i 2 ^ - h i = Stage two of the algorithm is for estimation of the assoiation parameters, fixing the regression oeffiients. To this end, we fous on the onditional mean h i = E(r i r i = y), y = 0, and define the residual R i = r i h i. The vetor of residuals R i has dimension ( ). The alulation of the onditional mean uses the expression pi - Pr( ri =, ri = ) logit ( hi) = y ln i + ln (4) -pi - p i + Pr( ri =, ri = ) that an easily be derived from straightforward alulations. In the algorithm, the seond term on the right-hand side of equation (4) is taen to be an offset term (although it involves A in the oint probability). Following