Chapter 2 The Basics of Pricing with GLMs

Transcription

1 Chapter 2 The Bascs of Prcng wth GLMs As descrbed n the prevous secton, the goal of a tarff analyss s to determne how one or more key ratos Y vary wth a number of ratng factors Ths s remnscent of analyzng how the dependent varable Y vares wth the covarates (explanatory varables) x n a multple lnear regresson Lnear regresson, or the slghtly larger general lnear model, s not fully sutable for non-lfe nsurance prcng, though, snce: () t assumes normally dstrbuted random errors, whle the number of nsurance clams follows a dscrete probablty dstrbuton on the non-negatve ntegers, and clam costs are non-negatve and often skewed to the rght; () n lnear models, the mean s a lnear functon of the covarates, whle multplcatve models are usually more reasonable for prcng, cf Sect 13 Generalzed lnear models (GLMs) s a rch class of statstcal methods, whch generalzes the ordnary lnear models n two drectons, each of whch takes care of one of the above mentoned problems: Probablty dstrbuton Instead of assumng the normal dstrbuton, GLMs work wth a general class of dstrbutons, whch contans a number of dscrete and contnuous dstrbutons as specal cases, n partcular the normal, Posson and gamma dstrbutons Model for the mean In lnear models the mean s a lnear functon of the covarates x In GLMs some monotone transformaton of the mean s a lnear functon of the x s, wth the lnear and multplcatve models as specal cases These two generalzaton steps are dscussed n Sects 21 and 22, respectvely GLM theory s qute recent the basc deas were ntroduced by Nelder and Wedderburn [NW72] Already n the frst 1983 edton of the standard reference by Mc- Cullagh and Nelder there s an example usng motor nsurance data; n the second edton [MN89] ths example can be found n Sects 841 and 1283 But t was not untl the second half of the 90 s that the use of GLMs really started spreadng, partly n response to the extended needs for tarff analyss due to the deregulaton of the nsurance markets n many countres Ths process was facltated by the publcaton of some nfluental papers by Brtsh actuares, such as [BW92, Re94, HR96]; see also [MBL00], wrtten for the US Casualty Actuaral Socety a few years later E Ohlsson, B Johansson, Non-Lfe Insurance Prcng wth Generalzed Lnear Models, EAA Lecture Notes, DOI / _2, Sprnger-Verlag Berln Hedelberg

2 16 2 The Bascs of Prcng wth GLMs Some advantages of usng GLMs over earler methods for rate makng are: GLMs consttute a general statstcal theory, whch has well establshed technques for estmatng standard errors, constructng confdence ntervals, testng, model selecton and other statstcal features GLMs are used n many areas of statstcs, so that we can draw on developments both wthn and wthout of actuaral scence There s standard software for fttng GLMs that can easly be used for a tarff analyss, such as the SAS, GLIM, R or GenStat software packages In spte of the possblty to use standard software, many nsurance companes use specalzed commercal software for rate makng that s provded by major consultng frms 21 Exponental Dsperson Models Here we descrbe the exponental dsperson models (EDMs) of GLMs, whch generalze the normal dstrbuton used n the lnear models In our dscusson of the multplcatve model n (13) and (14) weusedaway of organzng the data wth the observatons, y 1, 2,, K, havng one ndex per ratng factor Ths s sutable for dsplayng the data n a table, especally n the twodmensonal case y j, and wll therefore be called tabular form In our general presentaton of GLMs, we rather assume that the data are organzed on lst form, wth the n observatons organzed as a column vector y = (y 1,y 2,,y n ) Besdes the key rato y, each row of the lst contans the exposure weght w of the tarff cell, as well as the values of the ratng factors The transton from tabular form to lst form amounts to decdng on an order to dsplay the tarff cells; a smple example s gven n Table 21 As further llustraton, consder agan the moped nsurance example, for whch Table 12 contans an mplct lst form; n Table 22 we repeat the part of that table that gves the lst form for analyss of clam frequency Lst form corresponds to the way we organze the data n a data base, such as a SAS table Tabular form, on the other hand, s useful for demonstratve purposes; hence, the reader should try to get used to both forms By the assumptons n Sect 12, the varables Y 1,,Y n are ndependent, as requred n general GLM theory The probablty dstrbuton of an EDM s gven Table 21 Transton from tabular form to lst form n a 2 by 2 case Marred Male Female Yes y 11 y 12 No y 21 y 22 Marred Gender Observaton 1 Yes M y 1 2 Yes F y 2 3 No M y 3 4 No F y 4

3 21 Exponental Dsperson Models 17 Table 22 Moped tarff on lst form (clam frequency per mlle) Tarff Covarates Duraton Clam cell Class Age Zone (exposure) frequency x 1 x 2 x 3 w y by the followng frequency functon, specalzng to a probablty densty functon n the contnuous case and a probablty mass functon n the dscrete case, { } y θ b(θ ) f Y (y ; θ,φ)= exp + c(y,φ,w ) (21) φ/w Here θ s a parameter that s allowed to depend on, whle the dsperson parameter φ>0sthesameforall The so called cumulant functon b(θ ) s assumed twce contnuously dfferentable, wth nvertble second dervatve For every choce of such a functon, we get a famly of probablty dstrbutons, eg the normal, Posson and gamma dstrbutons, see Example 21 and (23) and (26) below Gven the choce of b( ), the dstrbuton s completely specfed by the parameters θ and φ The functon c(,, ), whch does not depend on θ, s of lttle nterest n GLM theory Of course, the expresson above s only vald for the y that are possble outcomes of Y the support; for other values of y we tactly assume f Y (y ) = 0 Examples of support we wll encounter s (0, ), (, ) and the non-negatve ntegers Further techncal restrctons are that φ>0, w 0 and that the parameter space must be open, e, θ takes values n an open set, such as 0 <θ < 1 (whle a closed set such as 0 θ 1 s not allowed)

4 18 2 The Bascs of Prcng wth GLMs An overvew of the theory of the so defned exponental dsperson models can be found n Jörgensen [Jö97] We wll presuppose no knowledge of the theory for EDMs, and our nterest n them s restrcted only to the role they play n GLMs Remark 21 If φ was regarded as fxed, (21) would defne a so called one-parameter exponental famly, see, eg, Lndgren [L93, p 188] If, on the other hand, φ s unknown then we usually do not have a two-parameter exponental famly, but we do have an EDM Example 21 (The normal dstrbuton) Here we show that the normal dstrbuton used n (weghted) lnear models s a member of the EDM class; note, though, that the normal dstrbuton s seldom used n the applcatons we have n mnd Nevertheless, assume for the moment that we have a normally dstrbuted key rato Y The expectaton of observaton s denoted μ, e, μ = E(Y ) Lemma 11 shows that the varance must be w -weghted; the σ 2 of that lemma s assumed to be the same for all n lnear models We conclude that Y N(μ,σ 2 /w ), where w s the exposure Then the frequency functon s { y μ μ 2 f Y (y ) = exp /2 } σ 2 + c(y,σ 2,w ), (22) /w were we have separated out the part of the densty not dependng on μ, c(y,σ 2,w ) = 1 ( w y 2 ) 2 σ 2 + log(2πσ 2 /w ) Ths s an EDM wth θ = μ, φ = σ 2 and b(θ ) = θ 2 /2 Hence, the normal dstrbuton used n (weghted) lnear models s an EDM; the unweghted case s, of course, obtaned by lettng w Probablty Dstrbuton of the Clam Frequency Let N(t) be the number of clams for an ndvdual polcy durng the tme nterval [0,t], wth N(0) = 0 The stochastc process {N(t); t 0} s called the clams process Beard, Pentkänen and Pesonen [BPP84, Appendx 4] show that under assumptons that are close to our Assumptons 12 13, plus an assumpton that clams do not cluster, the clams process s a Posson process Ths motvates us to assume a Posson dstrbuton for the number of clams of an ndvdual polcy durng any gven perod of tme By the ndependence of polces, Assumpton 11, we get a Posson dstrbuton also at the aggregate level of all polces n a tarff cell So let X be the number of clams n a tarff cell wth duraton w and let μ denote the expectaton when w = 1 Then by Lemma 11 we have E(X ) = w μ, and so X follows a Posson dstrbuton wth frequency functon f X (x ; μ ) = e w μ (w μ ) x, x = 0, 1, 2, x!

5 21 Exponental Dsperson Models 19 We are more nterested n the dstrbuton of the clam frequency Y = X /w ;nthe lterature, ths case s often (vaguely) referred to as Posson, too, but snce t s rather a transformaton of that dstrbuton we gve t a specal name, the relatve Posson dstrbuton The frequency functon s, for y such that w y s a non-negatve nteger, f Y (y ; μ ) = P(Y = y ) = P(X = w y ) = e w μ (w μ ) w y (w y )! = exp{w [y log(μ ) μ ]+c(y,w )}, (23) where c(y,w ) = w y log(w ) log(w y!) Ths s an EDM, as can be seen by reparameterzng t through θ = log(μ ), f Y (y ; θ ) = exp{w (y θ e θ ) + c(y,w )} Ths s of the form gven n (21), wth φ = 1 and the cumulant functon b(θ ) = e θ The parameter space s μ > 0, e, the open set <θ < Remark 22 Is the Posson dstrbuton realstc? In practce, the homogenety wthn cells s hard to acheve The expected clam frequency μ of the Posson process may vary wth tme, but ths s not necessarly a problem snce the number of clams durng a year wll stll be Posson dstrbuted A more serous problem s that there s often consderable varaton left between polces wthn cells Ths can be modeled by lettng the rsk parameter μ tself be the realzaton of a random varable Ths leads to a so called mxed Posson dstrbuton, wth larger varance than standard Posson, see, eg, [KPW04, Sect 463]; such models often ft nsurance data better than the standard Posson We wll return to ths problem n Sects 34 and 35 It s a reasonable requrement for a probablstc model of a clam frequency to be reproductve, n the followng sense Suppose that we work under the assumpton that the clam frequency n each cell has a relatve Posson dstrbuton If a tarff analyss shows that two cells have smlar expectaton, we mght decde to merge them nto just one cell Then t would be very strange f we got another probablty dstrbuton n the new cell Luckly, ths problem wll not arse on the contrary, the relatve Posson dstrbuton s reproduced on the aggregated level, as we shall now show Let Y 1 and Y 2 be the clam frequency n two cells wth exposures w 1 and w 2, respectvely, and let both follow a relatve Posson dstrbuton wth parameter μif we merge these two cells, the clam frequency n the new cell wll be the weghted average Y = w 1Y 1 + w 2 Y 2 w 1 + w 2 Snce w 1 Y 1 + w 2 Y 2 s the sum of two ndependent Posson dstrbuted varables, t s tself Posson dstrbuted Hence Y follows a relatve Posson dstrbuton wth

6 20 2 The Bascs of Prcng wth GLMs exposure w 1 + w 2 and the parameter s, by elementary rules for the expectaton of a lnear expresson, μ As ndcated above, a parametrc dstrbuton that s closed under ths type of averagng wll be called reproductve, a concept coned by Jörgensen [Jö97] In fact, ths s a natural requrement for any key rato; fortunately, we wll see below n Theorem 22 that all EDMs are reproductve 212 A Model for Clam Severty We now turn to clam severty, and agan we shall buld a model for each tarff cell, but for the sake of smplcty, let us temporarly drop the ndex The exposure for clam severty, the number of clams, s then wrtten w Recall that n ths analyss, we condton on the number of clams so that the exposure weght s non-random, as t should be The dea s that we frst analyze clam frequency wth the number of clams as the outcome of a random varable; once ths s done, we condton on the number of clams n analyzng clam severty Here, the total clam cost n the cell s X and the clam severty Y = X/w In the prevous secton, we presented a plausble motvaton for usng the Posson dstrbuton, under the assumptons on ndependence and homogenety However, n the clam severty case t s not at all obvous whch dstrbuton we should assume for X The dstrbuton should be postve and skewed to the rght, so the normal dstrbuton s not sutable, but there are several other canddates that fulfll the requrements However, the gamma dstrbuton has become more or less a de facto standard n GLM analyss of clam severty, see, eg, [MBL00, p 10] or [BW92, Sect 3] As we wll show n Sect 233, the gamma assumpton mples that the standard devaton s proportonal to μ, e, we have a constant coeffcent of varaton; ths seems qute plausble for clam severty In Sect 35 we wll dscuss the possblty of constructng estmators wthout assumng a specfc dstrbuton, startng from assumptons for the mean and varance structure only For the tme beng, we assume that the cost of an ndvdual clam s gamma dstrbuted; ths s the case w = 1 One of several equvalent parameterzatons s that wth a so called ndex parameter α>0, a scale parameter β>0, and the frequency functon f(x)= βα Γ(α) xα 1 e βx ; x>0 (24) We denote ths dstrbuton G(α, β) for short It s well known that the expectaton s α/β and the varance α/β 2, see Exercse 24 Furthermore, sums of ndependent gamma dstrbutons wth the same scale parameter β are gamma dstrbuted wth the same scale and an ndex parameter whch s the sum of the ndvdual α, see Exercse 211 SofX s the sum of w ndependent gamma dstrbuted random varables, we conclude that X G(wα, β) The frequency functon for Y = X/w s

7 21 Exponental Dsperson Models 21 then f Y (y) = wf X (wy) = (wβ)wα Γ(wα) ywα 1 e wβy ; y>0, and so Y G(wα, wβ) wth expectaton α/β Before transformng ths dstrbuton to EDM form, t s nstructve to re-parameterze t through μ = α/β and φ = 1/α In Exercse 25 we ask the reader to verfy that the new parameter space s gven by μ>0 and φ>0 The frequency functon s 1 f Y (y) = f Y (y; μ, φ) = Γ(w/φ) ( w μφ { y/μ log(μ) = exp + c(y,φ,w) φ/w ) w/φ y (w/φ) 1 e wy/(μφ) } ; y>0, (25) where c(y,φ,w) = log(wy/φ)w/φ log(y) log Γ(w/φ)WehaveE(Y) = wα/(wβ) = μ and Var(Y ) = wα/(wβ) 2 = φμ 2 /w, whch s consstent wth Lemma 11 To show that the gamma dstrbuton s an EDM we fnally change the frst parameter n (25)toθ = 1/μ; the new parameter takes values n the open set θ < 0 Returnng to the notaton wth ndex, the frequency functon of the clam severty Y s { } y θ + log( θ ) f Y (y ; θ,φ)= exp + c(y,φ,w ) (26) φ/w We conclude that the gamma dstrbuton s an EDM wth b(θ ) = log( θ ) and hence we can use t n a GLM Remark 23 By now, the reader mght feel that n the rght parameterzaton, any dstrbuton s an EDM, but that s not the case; the log-normal dstrbuton, eg, can not be rearranged nto an EDM 213 Cumulant-Generatng Functon, Expectaton and Varance The cumulant-generatng functon s the logarthm of the moment-generatng functon; t s useful for computng the expectaton and varance of Y, for fndng the dstrbuton of sums of ndependent random varables, and more For smplcty, we once more refran from wrtng out the subndex for the tme beng The momentgeneratng functon of an EDM s defned as M(t) = E(e ty ), f ths expectaton s fnte at least for real t n a neghborhood of zero For contnuous EDMs we fnd, by usng (21), E(e ty ) = e ty f Y (y; θ,φ)dy = { } y(θ + tφ/w) b(θ) exp + c(y,φ,w) dy φ/w

8 22 2 The Bascs of Prcng wth GLMs { } b(θ + tφ/w) b(θ) = exp φ/w { y(θ + tφ/w) b(θ + tφ/w) exp φ/w } + c(y,φ,w) dy (27) Recall the assumpton that the parameter space of an EDM must be open It follows, at least for t n a neghborhood of 0, e, for t <δfor some δ>0, that θ + tφ/w s n the parameter space Thereby, the last ntegral equals one and the factor precedng t s the moment-generatng functon, whch thus exsts for t <δ In the dscrete case, we get the same expresson by changng the ntegrals n (27) to sums By takng logarthms, we conclude that the cumulant-generatng functon (CGF), denoted Ψ(t), exsts for any EDM and s gven by b(θ + tφ/w) b(θ) Ψ(t)=, (28) φ/w at least for t n some neghborhood of 0 Ths s the reason for the name cumulant functon that we have already used for b(θ) As the name suggests, the CGF can be used to derve the so called cumulants; ths s done by dfferentatng and settng t = 0 The frst cumulant s the expected value; the second cumulant s the varance; the reader who s not famlar wth ths result s nvted to derve t from well-known results for moment-generatng functons n Exercse 28 We use the above property to derve the expected value of an EDM as follows, recallng that we have assumed that b( ) s twce dfferentable, Ψ (t) = b (θ + tφ/w); E(Y) = Ψ (0) = b (θ) The second cumulant, the varance, s gven by Ψ (t) = b (θ + tφ/w)φ/w; Var(Y ) = Ψ (0) = b (θ)φ/w As a check, let us see what ths yelds n the case of a normal dstrbuton: here b(θ) = θ 2 /2, b (θ) = θ and b (θ) = 1, whence E(Y) = θ = μ and Var(Y ) = φ/w = σ 2 /w as t should be In general, t s more convenent to vew the varance as a functon of the mean μ We have just seen that μ = E(Y) = b (θ), and snce ths s assumed to be an nvertble functon, we may nsert the nverse relatonshp θ = b 1 (μ) nto b (θ) to get the so called varance functon v(μ) = b (b 1 (μ)) Now we can express Var(Y ) as the product of the varance functon v(μ) and a scalng and weghtng factor φ/w Here are some examples Example 22 (Varance functons) For the relatve Posson dstrbuton n Sect 211 we have b(θ ) = exp(θ ), by whch μ = b (θ ) = exp(θ ) Furthermore b (θ ) = exp(θ ) = μ, so that v(μ ) = μ and Var(Y ) = μ /w, snce φ = 1 In the case w = 1 ths s just the well-known result that the Posson dstrbuton has varance equal to the mean

9 21 Exponental Dsperson Models 23 Table 23 Example of varance functons Dstrbuton Normal Posson Gamma Bnomal v(μ) 1 μ μ 2 μ(1 μ) The gamma dstrbuton has b(θ ) = log( θ ) and b (θ ) = 1/θ so that μ = 1/θ, as we already knew Furthermore, b (θ ) = 1/θ 2 = μ 2 so that Var(Y ) = φμ 2 /w The varance functons we have seen so far are collected n Table 23, together wth the bnomal dstrbuton that s treated n Exercses 26 and 29 We summarze the results n the followng lemma, returnng to our usual notaton wth ndex for the observaton number Lemma 21 Suppose that Y follows an EDM, wth frequency functon gven n (21) Then the cumulant generatng functon exsts and s gven by Ψ(t)= b(θ + tφ/w ) b(θ ), φ/w and μ = E(Y ) = b (θ ); Var(Y ) = φv(μ )/w, where the varance functon v(μ ) s b ( ) expressed as a functon of μ, e, v(μ ) = b (b 1 (μ )) Remark 24 Recall that n lnear regresson, we have a constant varance Var(Y ) =φ, plus possbly a weght w whch we dsregard for a moment The most general assumpton would be to allow Var(Y ) = φ, but ths would make the model heavly over-parameterzed In GLMs we are somewhere n-between these extremes, snce the varance functon v(μ ) allows the varance to vary over the cells, but wthout ntroducng any new parameters The varance functon s mportant n GLM model buldng, a fact that s emphaszed by the followng theorem Theorem 21 Wthn the EDM class, a famly of probablty dstrbutons s unquely characterzed by ts varance functon The practcal mplcaton of ths theorem s that f you have decded to use a GLM, and hence an EDM, you only have to determne the varance functon; then you know the precse probablty dstrbuton wthn the EDM class Ths s an nterestng result: only havng to model the mean and varance s much smpler than havng to specfy an entre dstrbuton

10 24 2 The Bascs of Prcng wth GLMs The core n the proof of ths theorem s to notce that snce v( ) s a functon of the dervatves of b( ), the latter can be determned from v( ) by solvng a par of dfferental equatons but b( ) s all we need to specfy the EDM dstrbuton n (21) The proof can be found n Jörgensen [Jö87, Theorem 1] We saw n Sect 211 that the relatve Posson dstrbuton had the appealng property of beng reproductve The next theorem shows that ths holds for all dstrbutons wthn the EDM class Theorem 22 (EDMs are reproductve) Suppose we have two ndependent random varables Y 1 and Y 2 from the same EDM famly, e, wth the same b( ), that have the same mean μ and dsperson parameter φ, but possbly dfferent weghts w 1 and w 2 Then ther w-weghted average Y = (w 1 Y 1 + w 2 Y 2 )/(w 1 + w 2 ) belongs to the same EDM dstrbuton, but wth weght w = w 1 + w 2 The proof s left as Exercse 212, usng some basc propertes of CGFs derved n Exercse 210 From an appled pont of vew the mportance of ths theorem s that f we merge two tarff cells, wth good reason to assume they have the same mean, we wll stay wthn the same famly of dstrbutons From ths pont of vew, EDMs behave the way we want a probablty dstrbuton n prcng to behave One may show that the log-normal and the (generalzed) Pareto dstrbuton do not have ths property; our concluson s that, useful as they may be n other actuaral applcatons, they are not well suted for use n a tarff analyss 214 Tweede Models In non-lfe actuaral applcatons, t s often desrable to work wth probablty dstrbutons that are closed wth respect to scale transformatons, or scale nvarant Let c be a postve constant, c>0, and Y a random varable from a certan famly of dstrbutons; we say that ths famly s scale nvarant f cy follows a dstrbuton n the same famly Ths property s desrable f Y s measured n a monetary unt: f we convert the data from one currency to another, we want to stay wthn the same famly of dstrbutons the result of a tarff analyss should not depend on the currency used Smlarly, the nference should not depend on whether we measure the clam frequency n per cent or per mlle We conclude that scale nvarance s desrable for all the key ratos n Table 13, except for the proporton of large clams, for whch scale s not relevant It can be shown that the only EDMs that are scale nvarant are the so called Tweede models, whch are defned as havng varance functon v(μ) = μ p (29) for some p The proof can be found n Jörgensen [Jö97, Chap 4], upon whch much of the present secton s based, wthout further reference

11 21 Exponental Dsperson Models 25 Table 24 Overvew of Tweede models Type Name Key rato p<0 Contnuous p = 0 Contnuous Normal 0 <p<1 Non-exstng p = 1 Dscrete Posson Clam frequency 1 <p<2 Mxed, non-negatve Compound Posson Pure premum p = 2 Contnuous, postve Gamma Clam severty 2 <p<3 Contnuous, postve Clam severty p = 3 Contnuous, postve Inverse normal Clam severty p>3 Contnuous, postve Clam severty Wth the notable excepton of the relatve bnomal n Exercse 26, forwhch scale nvarance s not requred snce y s a proporton, the EDMs n ths book are all Tweede models In Table 24 we gve a lst of all Tweede models and the key ratos for whch they mght be useful The cases p = 0, 1, 2 have already been dscussed Tweede models wth p 2 are often suggested as dstrbutons for the clam severty, especally p = 2, but also the nverse normal dstrbuton (p = 3) s sometmes mentoned n the lterature The class of models wth 1 <p<2 s nterestng: these so called compound Posson dstrbutons arse as the dstrbuton of a sum of a Posson dstrbuted number of clams whch follow a gamma dstrbuton; hence, they are proper for modelng the pure premum, wthout dong a separate analyss of clam frequency and severty Note that the compound Posson s mxed (nether purely dscrete nor purely contnuous), havng postve probablty at zero plus a contnuous dstrbuton on the postve real numbers Jörgensen and Souza [JS94] analyze the pure premum for a prvate motor nsurance portfolo n Brazl, and get the value p = 137 from an algorthm they desgned for maxmum lkelhood estmaton of p A bt surprsng s that for 0 <p<1 no EDM exsts Negatve values of p, fnally, are allowed and gve contnuous dstrbutons on the whole real axs, but to the best of our knowledge no applcaton n nsurance has been proposed From now on, we only dscuss the case p 1, whch covers our applcatons, and we start by presentng the correspondng cumulant functon b(θ) e θ, for p = 1; b(θ) = log( θ), for p = 2; (210) p 2 1 [ (p 1)θ](p 2)/(p 1), for 1 <p<2 and p>2 The parameter space M θ s { <θ<, for p = 1; M θ = <θ<0, for p>1 (211)

12 26 2 The Bascs of Prcng wth GLMs The dervatve b (θ) s gven by { b e (θ) = θ, for p = 1; [ (p 1)θ] 1/(p 1), for p>1, (212) wth nverse { log(μ), p = 1; h(μ) = p 1 1 μ (p 1), p>1 (213) These results are taken from Jörgensen [Jö97]; some are also verfed n Exercse The Lnk Functon We have seen how to generalze the normal error dstrbuton to the EDM class, and now turn to the other generalzaton of ordnary lnear models, concernng the lnear structure of the mean We start by dscussng a smple example, n whch we only have two ratng factors, one wth two classes and one wth three classes On tabular form, we let μ j denote the expectaton of the key rato n cell (, j), where the frst factor s n class and the second s n class j Lnear models assume an addtve model structure for the mean: μ j = γ 0 + γ 1 + γ 2j (214) We recognze ths model from the analyss of varance (ANOVA), and recall that t s over-parameterzed, unless we add some restrctons In an ANOVA the usual restrcton s that margnal sums should be zero, but here we chose the restrcton to force the parameters of some base cell to be zero Say that (1, 1) s the base cell; then we let γ 11 = γ 21 = 0, so that μ 11 = γ 0, and the other parameters measure the mean departure from ths cell Next we rewrte the model on lst form by sortng the cells n the order (1, 1); (1, 2); (1, 3); (2, 1); (2, 2); (2, 3) and renamng the parameters, β 1 = γ0, β 2 = γ12, β 3 = γ22, β 4 = γ23 Wth these parameters the expected values n the cells are as lsted n Table 25 Next, we ntroduce so called dummy varables through the relaton { 1, f βj s ncluded n μ x j =, 0, else

13 22 The Lnk Functon 27 Table 25 Parameterzaton of a two-way addtve model on lst form Tarff cell μ β β 1 +β β 1 +β β 1 +β β 1 +β 2 +β β 1 +β 2 +β 4 Table 26 Dummy varables n a two-way addtve model Tarff cell x 1 x 2 x 3 x The values of the dummy varables n ths example are gven n Table 26 Note the smlarty to Table 25 Wth these varables, the lnear model for the mean can be rewrtten μ = 4 x j β j = 1, 2,,6, (215) j=1 or on matrx form μ = Xβ, where X s called the desgn matrx, ormodel matrx, and μ 1 x 11 x 12 x 13 x 14 μ 2 x 21 x 22 x 23 x 24 β 1 μ = μ 3 μ 4, X = x 31 x 32 x 33 x 34 x 41 x 42 x 43 x 44, β = β 2 β 3 (216) μ 5 x 51 x 52 x 53 x 54 β 4 μ 6 x 61 x 62 x 63 x 64 So far the addtve model for the mean has been used; next we turn to the multplcatve model μ j = γ 0 γ 1 γ 2j that was ntroduced n Sect 13 By takng logarthms we get log(μ j ) = log(γ 0 ) + log(γ 1 ) + log(γ 2j ), (217) and agan we must select a base cell, say (1, 1), where γ 11 = γ 21 = 1 Let us now do a transton to lst form smlar to the one we just performed for the addtve model,

14 28 2 The Bascs of Prcng wth GLMs by frst lettng β 1 = log γ0, β 2 = log γ12, β 3 = log γ22, β 4 = log γ23 By the ad of the dummy varables n Table 26, wehave log(μ ) = 4 x j β j ; = 1, 2,,6 (218) j=1 Ths s the same lnear structure as n (215), the only dfference beng that the lefthand sde s log(μ ) nstead of just μ In general GLMs ths s further generalzed to allow the left-hand sde to be any monotone functon g( ) of μ Leavng the smple two-way example behnd, the general tarff analyss problem s to nvestgate how the response Y s nfluenced by r covarates x 1,x 2,,x r Introduce r η = x j β j ; = 1, 2,,n, (219) j=1 where x j as before s the value of the covarate x j for observaton In the ordnary lnear model, μ η ; n a GLM ths s generalzed to an arbtrary relaton g(μ ) = η, wth the restrcton that g( ) must be a monotone, dfferentable functon Ths fundamental object n a GLM s called the lnk functon, snce t lnks the mean to the lnear structure through g(μ ) = η = r x j β j (220) We have seen that multplcatve models correspond to a logarthmc lnk functon, a log lnk, j=1 g(μ ) = log(μ ), whle the lnear model uses the dentty lnk μ = η, e, g(μ ) = μ Note that the lnk functon s not allowed to depend on For the analyss of proportons, t s common to use a logt lnk, ( ) μ η = g(μ ) = log (221) 1 μ Ths lnk guarantees that the mean wll stay between zero and one, as requred n a model where μ s a proporton, such as the relatve bnomal model for the key

15 22 The Lnk Functon 29 rato proporton of large clams The correspondng GLM analyss goes under the name logstc regresson In a GLM, the lnk functon s part of the model specfcaton In non-lfe nsurance prcng, the log lnk s by far the most common one, snce a multplcatve model s often reasonable For a dscusson of the merts of the multplcatve model, see Sect 13 Interactons may cause departure from multplcatvty An example s motor nsurance, where young men are more accdent-prone than young women, whle n mdlfe there s lttle gender dfference n drvng behavor In Sect 362 we wll show how ths problem can be handled wthn the general multplcatve framework For further dscusson on multplcatve models, see [BW92, Sects 21 and 311] Wth the excepton of the analyss of proportons, we wll assume a multplcatve model throughout ths text, wth possble nteractons handled by combnng varables as n the age and gender example We end ths secton by summarzng the GLM generalzaton of the ordnary lnear model and the partcular case that s most used n our applcatons Weghted lnear regresson models: Y follows a normal dstrbuton wth Var(Y ) = σ 2 /w ; The mean follows the addtve model μ = j x j β j Generalzed Lnear Models (GLMs): Y follows an EDM wth Var(Y ) = φv(μ )/w ; The mean satsfes g(μ ) = j x j β j, where g s a monotone functon Multplcatve Tweede model, subclass of GLMs: Y follows a Tweede EDM wth Var(Y ) = φμ p /w, p 1; The mean follows the multplcatve model log(μ ) = j x j β j 221 Canoncal Lnk* In the GLM lterature one encounters the concept of a canoncal lnk; ths concept s not so mportant n our applcatons, but for the sake of completeness we shall gve a bref orentaton on ths subject Frst we note that we have been workng wth several dfferent parameterzatons of a GLM; these parameters are unque functons of each other as llustrated n the followng fgure: θ b ( ) μ g( ) η (222)

16 30 2 The Bascs of Prcng wth GLMs Here b( ), and hence b ( ), are determned by the structure of the random component, unquely determned by the choce of varance functon The lnk functon g( ), on the other hand, s part of the modelng of the mean and n some applcatons there are several reasonable choces The specal choce g( ) = b 1 ( ) s the canoncal lnk From(222) we fnd that the canoncal lnk makes θ = η It turns out that usng the canoncal lnk smplfes some computatons, but the name s somewhat msleadng snce t may not be the natural choce n a partcular applcaton On the other hand, some of the most common models actually use canoncal lnks Example 23 (Some canoncal lnks) The normal dstrbuton has μ = b (θ) = θ and the dentty lnk g(μ) = μ that s used n the lnear model s the canoncal one In the Posson case we have μ = b (θ) = e θ, by whch the log lnk g(μ) = log μ s canoncal So for ths mportant EDM, the multplcatve model s canoncal In the case of the gamma dstrbuton b (θ) = 1/θ and so the canoncal lnk s the nverse lnk g(μ) = 1/μ A problem wth ths lnk s that, as opposed to the log lnk, t may cause the mean to take on negatve values McCullagh and Nelder [MN89, Sects 841 and 1283], suggests usng ths lnk for clam severty, but as far as we know t s not used n practce 23 Parameter Estmaton So far, we have defned GLMs and studed some of ther propertes It s now tme for the most mportant step: the estmaton of the regresson parameters n (220), from whch we wll get the relatvtes the basc buldng blocks of the tarff Before dervng a general result, we wll gve an ntroducton to the subject by treatng an mportant specal case 231 The Multplcatve Posson Model We return to the smple case n Sect 131, wth just two ratng factors For easy reference, we repeat the multplcatve model on tabular form from (13) μ j = γ 0 γ 1 γ 2j (223) In GLM terms, we say that we use a log lnk In addton to ths, we assume that the clam frequency Y j has a relatve Posson dstrbuton, wth frequency functon gven on lst form n (23) We rewrte ths on tabular form and note that the cells are ndependent due to Assumpton 11, mplyng that the log-lkelhood of the whole sample s the sum of the log-lkelhoods n the ndvdual cells l = w j {y j [log(γ 0 ) + log(γ 1 ) + log(γ 2j )] γ 0 γ 1 γ 2j }+c, (224) j

17 23 Parameter Estmaton 31 where c does not depend on the γ parameters By n turn dfferentatng l wth respect to each γ, we get a system of equatons for the maxmum lkelhood estmates (MLEs), the so called ML equatons The result s exactly the MMT equatons n (16) (18); ths s no surprse f we recall that Jung [Ju68] derved these equatons as the MLEs of a Posson model, cf Remark 11 In general, under a GLM model for the clam frequency wth a relatve Posson dstrbuton and log lnk (multplcatve model), the ML equatons are equal to the equatons of the method of margnal totals (MMT), and hence the resultng estmates are the same, see Exercse General Result We now turn to the general case of fndng the MLEs of the β-parameters n a GLM The estmates are based on a sample of n observatons The ndvdual observatons follow the EDM dstrbuton gven on lst form n (21) and by ndependence, the log-lkelhood as a functon of θ s l(θ; φ,y) = 1 w (y θ b(θ )) + φ c(y,φ,w ) (225) It s clear that the dsperson parameter φ does not affect the maxmzaton of l wth respect to θ a smlar observaton should be famlar from the lnear regresson model, where φ s denoted σ 2 We can therefore gnore φ here, but we wll return to the queston of how to estmate t later, n Sect 311 The lkelhood as a functon of β, rather than θ, could be found by the nverse of the relaton μ = b (θ ), combned wth the lnk g(μ ) = η = j x j β j The dervatve of l wth respect to β j s, by the chan rule, l = β j = 1 φ l θ θ β j = 1 φ ( w y w b (θ ) ) θ β j ( w y w b (θ ) ) θ μ μ η η β j (226) By the mentoned relaton μ = b (θ ) we have μ / θ = b (θ ) The dervatve of the nverse relaton s smply the nverse of the dervatve, and so θ / μ = 1/v(μ ), snce by defnton v(μ ) = b (θ ) Furthermore, μ / η =[ η / μ ] 1 = 1/g (μ ) Fnally, from η = j x j β j we get η / β j = x j Puttng all ths together gves the result, l β j = 1 φ w y μ v(μ )g (μ ) x j, (227)

18 32 2 The Bascs of Prcng wth GLMs the so called score functon By settng all these r partal dervatves equal to zero and multplyng by φ, we get the ML equatons w y μ v(μ )g (μ ) x j = 0, j = 1, 2,,r (228) It mght look as f the soluton s smply μ = y, but then we forget that μ = μ (β) also has to satsfy the relaton gven by the regresson on the x s, e, μ = g 1 (η ) = g 1 ( j x j β j ) (229) It s only the so called saturated model, where there are as many parameters as there are observatons, that allows the soluton μ = y It s nterestng to note that the only property of the probablty dstrbuton that affects the ML equatons (228) s the mean and the varance, through the lnk functon g and the varance functon v; for further dscusson on ths ssue, see Sect 35 Example 24 (Tweede models) In a tarff analyss, we typcally use the Tweede models of Sect 214, where v(μ) = μ p, n connecton wth a multplcatve model for the mean, e, g(μ ) = log(μ ), mplyng that g (μ ) = 1/μ Then the general ML equatons n (228) smplfy to y μ w μ p 1 x j = 0 w μ p 1 y x j = w μ p 1 μ x j, (230) where the μ s are connected to the β s through { μ = exp x j β j } (231) Compared to the Posson case p = 1, models wth p>1 wll downwegh both sdes of the rght-most equaton n (230) byμ p 1, gvng less weght to cells wth large expectaton In the end, we are not nterested n the β s, but rather the relatvtes γ These are found by the relaton γ j = exp{β j } Introduce the dagonal matrx W wth the parameter-dependent weghts w = j w v(μ )g (μ ), on the dagonal and zeroes off the dagonal, W = dag( w ; = 1,,n) Then (228) may be wrtten on matrx form X Wy = X Wμ, (232)

19 23 Parameter Estmaton 33 where X s the desgn matrx, cf the example n (216) In the case of the normal dstrbuton wth dentty lnk, t s readly seen that w = w, and f we furthermore let all weghts w = 1, then (232) s reduced to the well-known normal equatons of the lnear model X y = X Xβ, as we would expect snce the lnear model s a specal case of the GLM Here we have used the fact that μ = Xβ when we use the dentty lnk Except for a few specal cases, the ML equatons must be solved numercally The nterested reader s referred to Sect 323 for an ntroducton to numercal methods for solvng the ML equatons and to Sect 324 for the queston whether ML equatons really gve a (unque) maxmum of the lkelhood 233 Multplcatve Gamma Model for Clam Severty In Sect 231 we dscussed estmaton for clam frequency; our next mportant specal case s clam severty Wth the data on lst form, n cell we have w clams and the clam severty Y, whch s assumed to be relatve gamma dstrbuted wth densty gven by (25) Let us have a closer look at the relaton between the mean μ and the varance n ths case By Lemma 21 and Table 23 we have E(Y ) = μ and Var(Y ) = φμ 2 /w Hence, Var(Y ) E(Y ) 2 = φ w, (233) whch means that the coeffcent of varaton (CV) s constant over cells wth the same exposure w Ths s, of course, a consequence of havng a constant CV n the underlyng model for ndvdual clams Such a constant CV, e, a standard devaton proportonal to the mean, s plausble and n any case much more realstc than havng a constant standard devaton: say that we have a tarff cell wth mean 20 and standard devaton 4, then n another cell wth the same exposure but wth mean 200 we would rather expect a standard devaton of 40 than of 4 From (230) wth p = 2 we get the ML equatons, whch n ths case are w y μ x j = w x j (234) It s nstructve to wrte these equatons on tabular form n the smple case wth just two ratng factors, usng the multplcatve model n (223): w j y j = w j ; γ j 0 γ 1 γ 2j j w j y j = w j γ j 0 γ 1 γ 2j j = 2,,k 1 ; (235) w j y j = w j γ 0 γ 1 γ 2j j = 2,,k 2

20 34 2 The Bascs of Prcng wth GLMs Ths system of equatons provdes a qute natural algorthm for estmatng relatvtes, gvng a margnal balance n the sum of the relatve devatons of the observatons from ther means: the average relatve devaton equals one for each ratng factor level or j Mack [Ma91] refers to work by Eeghen, Njssen and Ruygt from 1982 where these equatons are used for the pure premum, under the name the drect method They found that the estmates of ths method were very close to those gven by the MMT Ths s consstent wth our general experence that the choce of p n the Tweede models s not that mportant for estmatng relatvtes 234 Modelng the Pure Premum In the end, t s the model for the pure premum that gves the tarff One mght consder usng a Tweede model wth 1 <p<2 to analyze the pure premum drectly, as demonstrated by Jörgensen and Souza [JS94] However, the standard GLM tarff analyss s to do separate analyses for clam frequency and clam severty, and then relatvtes for the pure premum are found by multplyng the results The reason for ths separaton nto two GLMs s: () Clam frequency s usually much more stable than clam severty and often much of the power of ratng factors s related to clam frequency: these factors can then be estmated wth greater accuracy; () A separate analyss gves more nsght nto how a ratng factor affects the pure premum See also [BW92, MBL00] Example 25 (Moped nsurance contd) We return once more to Example 11, wth theaggregateddatagvenntable12 InExample13 we appledthe MMTdrectly to the pure premum We now carry out a separate analyss for clam frequency and severty, obtanng the relatvtes for the pure premum by multplyng the factors from these two analyses The results are lsted n Table 27 We observe that the two varables vehcle class and vehcle age affect clam frequency and severty n the same drecton, whch means that newer and stronger vehcles are not only more expensve to replace when stolen, but are also stolen more often The geographc zone has a large mpact on the clam frequency, but once a moped s stolen the cost of replacng t s not necessarly larger n one zone than another, wth a possble excepton for the largest ctes n zone 1 Note that these nterestng conclusons could not have been drawn, had we analyzed pure premum only On the other hand, the resultng estmates for pure premum are qute close to those obtaned by the MMT method, see Table 14 For zone 5 and 7, t s rather obvous that no concluson can be drawn due to the very small number of clams, and hence very uncertan estmates of clam severty But how can we know f the 23 observatons n zone 6 or the 132 n zone 3 are enough to draw sgnfcant conclusons? Ths s one of the themes n the next chapter, where we go further nto GLM theory and practce

21 24 Case Study: Motorcycle Insurance 35 Table 27 Moped nsurance: relatvtes from a multplcatve Posson GLM for clam frequency and a gamma GLM for clam severty Ratng Class Duraton No Relatvtes, Relatvtes, Relatvtes, factor clams frequency severty pure premum Vehcle class Vehcle age Zone Case Study: Motorcycle Insurance Under the headlne case study we wll present larger exercses usng authentc nsurance data; for solvng the case studes, a computer equpped wth SAS or other sutable software s requred The data for ths case study comes from the former Swedsh nsurance company Wasa, and concerns partal casco nsurance, for motorcycles ths tme It contans aggregated data on all nsurance polces and clams durng ; the reason for usng ths rather old data set s confdentalty more recent data for ongong busness can not be dsclosed The data set mccasetxt, avalable at wwwmathsuse/glmbook, contans the followng varables (wth Swedsh acronyms): AGARALD: The owners age, between 0 and 99 KON: The owners gender, M (male) or K (female) ZON: Geographc zone numbered from 1 to 7, n a standard classfcaton of all Swedsh parshes The zones are the same as n the moped Example 11 MCKLASS: MC class, a classfcaton by the so called EV rato, defned as (Engne power n kw 100) / (Vehcle weght n kg + 75), rounded to the nearest lower nteger The 75 kg represent the average drver weght The EV ratos are dvded nto seven classes as seen n Table 28 FORDALD: Vehcle age, between 0 and 99 BONUSKL: Bonus class, takng values from 1 to 7 A new drver starts wth bonus class 1; for each clam-free year the bonus class s ncreased by 1 After the frst clam the bonus s decreased by 2; the drver can not return to class 7 wth less than 6 consecutve clam free years

22 36 2 The Bascs of Prcng wth GLMs Table 28 Motorcycle nsurance: ratng factors and relatvtes n current tarff Ratng factor Class Class descrpton Relatvty Geographc zone 1 Central and sem-central parts of 7678 Sweden s three largest ctes 2 Suburbs plus mddle-szed ctes Lesser towns, except those n 5 or Small towns and countrysde, except Northern towns Northern countrysde Gotland (Sweden s largest sland) 1402 MC class 1 EV rato EV rato EV rato EV rato EV rato EV rato EV rato Vehcle age years years years 1000 Bonus class DURATION: the number of polcy years ANTSKAD: the number of clams SKADKOST: the clam cost The current tarff, the actual tarff from 1995, s based on just four ratng factors, descrbed n Table 28, where ther current relatvtes are also gven For each ratng factor, we chose the class wth the hghest duraton as base class For the nterested reader, we menton that the annual premum n the base cell (4, 3, 3, 3) s 183 SEK, whle the hghest premum s SEK (!) Problem 1: Aggregate the data to the cells of the current tarff Compute the emprcal clam frequency and severty at ths level Problem 2: Determne how the duraton and number of clams s dstrbuted for each of the ratng factor classes, as an ndcaton of the accuracy of the statstcal analyss Problem 3: Determne the relatvtes for clam frequency and severty separately, by usng GLMs; use the result to get relatvtes for the pure premum

23 Exercses 37 Problem 4: Compare the results n 3 to the current tarff Is there a need to change the tarff? Whch new conclusons can be drawn from the separate analyss n 3? Can we trust these estmates? Wth your estmates, what s the rato between the hghest pure premum and the lowest? Exercses 21 (Secton 21) Work out the detals n the dervaton of (22) 22 (Secton 21) Suppose we have three ratng factors, wth two, three and fve levels respectvely In how many ways could the tarff be wrtten on lst form? Each orderng of the cells s counted as one way of wrtng the tarff 23 (Secton 21) Show drectly, wthout usng the results n Sect 213, that the normal dstrbuton of Example 21 s reproductve; what s the value of Var(Y )? 24 (Secton 21) Verfy that the expectaton and varance of the gamma dstrbuton n (24) areα/β and α/β 2, respectvely 25 (Secton 21) Consder the reparameterzaton of the gamma dstrbuton just before (25) Show that when φ and μ run through the frst quadrant, α and β run through ther parameter space, so that the same famly of dstrbutons s covered by the new parameterzaton 26 (Secton 21) An actuary studes the probablty of customer renewal that the customer chooses to stay wth the nsurance company for one more year and how t vares between dfferent groups Let w be the number of customers n group and X the number of renewals among these, so that by polcy ndependence X Bn(w,p ) Furthermore, let p be the probablty under study, estmated by the key rato Y = X /w Is the dstrbuton of Y an EDM? If the answer s yes, determne the canoncal parameter θ, an expresson for φ, plus the functons b( ) and c( ) (In any case, we mght call ths the relatve bnomal dstrbuton) 27 (Secton 21) The Generalzed Pareto Dstrbuton (GPD) s commonly used for large clams, especally n the area of rensurance The frequency functon can be wrtten, see eg Klugman et al [KPW04, Appendx A231], f(y)= γα γ (α + y) γ +1 y>0, where α>0 and γ>0 Is ths dstrbuton an EDM? If so, determne the canoncal parameter θ and n addton φ and the functon b( )

24 38 2 The Bascs of Prcng wth GLMs 28 (Secton 21) Use well-known results for moment-generatng functons, see eg Gut [Gu95, Theorem 33], to show that f the cumulant-generatng functon Ψ(t) exsts, then E(Y) = Ψ (0) and Var(Y ) = Ψ (0) 29 (Secton 21) Derve the varance functon of the relatve bnomal dstrbuton, defned n Exercse (Secton 21) LetX and Y be two ndependent random varables and let c be a constant Show that the CGF has the followng propertes (a) Ψ cx (t) = Ψ X (ct), (b) Ψ X+Y (t) = Ψ X (t) + Ψ Y (t) 211 (Secton 21) Prove that the sum of ndependent gamma dstrbutons wth the same scale parameter β are agan gamma dstrbuted and determne the parameters Hnt: use the moment generatng functon 212 (Secton 21) Prove Theorem 22 Hnt: Use the result n Exercse (Secton 21) (a) Derve b (θ) n (212) from the b(θ) gven by (210) Then derve b (θ) for p 1 (b) Show that h(μ) n (213)sthenversetob (θ) n (212) (c) Show that an EDM wth the cumulant functon b(θ) taken from (210) has varance functon v(μ) = μ p, so that t s a Tweede model 214 (Secton 22) Use a multplcatve model n the moped example, Example 12, and wrte down the desgn matrx X for the lst form that s mplct n Table 12 For the sake of smplcty, use just three zones, nstead of seven 215 (Secton 22*) Derve the canoncal lnk for the relatve bnomal dstrbuton, defned n Exercse 26 and treated further n Exercse (Secton 23) Derve(224) and dfferentate l wth respect to each parameter to show that the ML equatons are gven by (16) (18) 217 (Secton 23) Derve the system of equatons n (235) drectly, n the specal case wth just two ratng factors, by maxmzng the log-lkelhood Hnt: use(25) and (223)

25