INCOME PROTECTION IN CASE OF DISABILITY - FRENCH CONTEXT AND TARIFF METHODOLOGY ABSTRACT KEYWORDS

Transcription

1 INCOME PROTECTION IN CASE OF DISABILITY - FRENCH CONTEXT AND TARIFF METHODOLOGY BY RICHARD LAMBERT rchard.lambert@genre.com ABSTRACT Ths artcle descrbes the three llars of French ncome rotecton and rovdes the reader wth a straghtforward method for clams eerence ratng of Short-Term Dsablty ) wthn the scoe of self-emloyed ncome rotecton n France Thrd Pllar). Insurance roducts wth short and medum deferred erods currently sold n the French market encourage stochastc modelng of based on a smle state model actve or dsabled lve) snce hgh ncdence and reactvaton rates consderably reduce the mact on remum of other causes generally nvolved n a mult state model occurrence of death, Long-Term Dsablty or retrement). Although the rad recovery develoment of actually calls for a contnuous-tme actuaral model, the resent aer rooses an alternatve n dscrete-tme for a sreadsheet calculaton of remums and mathematcal reserve. KEYWORDS Sécurté Socale Françase, Conventon Collectve, Salared Emloyees, Self-emloyed, Income Protecton, Long-Term Dsablty LTD), Short-Term Dsablty ), Accdent, Sckness, Hostalzaton, Deferred Perod, GLM for Incdence and Reactvaton Rates, Negatve Bnomal, Log-Logstc, Model Desgn, Data Formattng, Oeratve Mode, Occuatonal Classes, Rsk Sngle and Level Premum, Mathematcal Reserve.

2 . INTRODUCTION Techncal notes of ndvdual Short-Term and Long-Term Dsablty tarff sold n the French market usually show a ga between actuaral theory and ractce, but also between underwrtng gudelnes and clams eerence. The major role layed by the French Socal Securty system n the ncome rotecton of salared emloyees and self-emloyed largely contrbutes to ths ga snce the frst and second llars of the French Socal Securty have enforced art of the dsablty defntons used n ndvdual nsurance sold n the thrd llar. The resent aer addresses the queston of Short-Term Dsablty tarff ) through the followng sectons: - Secton 2 dscusses the three llars of French ncome rotecton; - Secton 3 rooses a stochastc model of the French ncdence and reactvaton rates; - Secton 4 descrbes the necessary stes for analysng and modelng clams eerence; - Secton 5 resents a remum and reserve model n dscrete-tme; - A Concluson s gven n Secton THREE PILLARS OF FRENCH INCOME PROTECTION Secton 2 of ths artcle gves a broad outlne of the three llars of French ncome rotecton. 2.. Frst Pllar "Régme Général" of the French Socal Securty System The French Socal Securty System created n 945) rovdes a comulsory dsablty cover for salared emloyees and for the self-emloyed that s nterdeendent through generatons and rofessons. The urose of ths cover s to comensate for loss of ncome resultng from a reducton n caacty to work and the nablty to engage n ganful emloyment due to medcal causes. It s fnanced two-thrds by the emloyer and one-thrd by the emloyee, f salared, or totally by the self-emloyed. In the followng dscusson, accdents at work and dseases acqured whle erformng the normal dutes of one s rofesson are ecluded Short-Term Dsablty Incaacté de Traval Enttlement to a dsablty beneft s evaluated on an "own occuaton" bass and requres aroval from the famly doctor. Socal Securty's medcal advsors conduct random queres to confrm the dsablty status of the clamant. The beneft s ad after a deferred erod of three days for salared emloyees and after seven days for the self-emloyed reduced to three days n case of hostalzaton). For salared emloyees, the monthly beneft s equal to 50% of the average last three months gross salary lmted to 50% of the "PMSS" Plafond Mensuel de la Sécurté Socale; n 2006, PMSS = 2,589/month). The beneft s ad on a daly bass and lmted to 360 days over one or several ncaactes wthn each erod of three consecutve years. For the self-emloyed the same rule ales to the yearly beneft by relacng "last three months" by "last three years" and "PMSS" by "PASS" =2 PMSS) n the above-mentoned defnton, and wth a guaranteed mnmum beneft snce there s no mnmum salary. For salared emloyees, an addtonal rotecton ad by the emloyer "Mensualsaton" law, 978) comletes the above-mentoned beneft to a mamum of 90% of gross salary from the st to the 90th day of dsablty, whch s reduced to 66% from the 9st to the 80th day of dsablty. For both the self-emloyed and salared emloyees, the beneft s ndeed every three months to the comany's wages for salared emloyees, and to PASS for the self-emloyed Long-Term Dsablty Invaldté Permanente Enttlement to a beneft s made by agreement between the Socal Securty's medcal advsor and the clamant's famly doctor after medcal "consoldaton" of dsablty at a ermanent level or after three consecutve years n Short-Term Dsablty. The artes should agree on the followng levels by order: Functonal dsablty 0 Fn ) Pure hysologc and mental scale "Dammum emergens"); and 2

3 Occuatonal dsablty 0 Oc ) Professonal scale based on an "any sutable occuaton" dsablty defnton and takng nto account the nature of the functonal dsablty, but also the educatonal level of the clamant Artcles L34.-3 and R34.2 of Socal Securty Code). 3 Fnally, the resultant dsablty level Rs = Fn 2 Oc determnes the dsablty category and thus the yearly beneft equal to ercent of the average last 0 years gross salary but lmted to ercent of the PASS lus 5% er deendent chld wth a mamum of 0% more: st Category Partal and Permanent Dsablty: 33 % Rs < 66% and = 30% 2 nd Category Total and Permanent Dsablty: Rs 66% and = 50% 3 rd Category Total and Irreversble Loss of Autonomy: Rs 66% lus the clamant requres assstance of a thrd erson n order to erform the actvtes of daly lvng. = 70%. The Long-Term Dsablty LTD) beneft, ad on a monthly bass, s PASS ndeed and ends at age 60. It can be revewed f the dsablty level Rs moves to another category reductons are rare). After the age of 60, the enson system takes over. In addton: The Socal Securty System requres that the clamant must have comleted some mnmum amount of tme at work. Income taes and lmted socal contrbutons are alcable to the beneft. The frst llar ncludes other rotectons, such as death, health, enson, accdent at work, and dseases acqured whle erformng the normal dutes of one s rofesson. These are regulated by other defntons Second Pllar Grou Income Protecton on To of the Frst Pllar for Salared Emloyees The second llar conssts of a Collectve Agreement "Conventon Collectve") between an emloyer and ts salared emloyees. It s negotated by comany or lne of ndustry and regulated by the Labor Code see also "Evn" Law 989). In 2005, 28 Agreements were n force n France. Ths comulsory cover, wthout medcal underwrtng, comletes the frst llar to between 70% and 90% of gross salary wth a lmt of four or eght PASS. Dsablty defntons are generally n lne wth those of the frst llar. Products of the market often roose: a Short-Term Dsablty ) beneft generally relacng the "Mensualsaton" law,.e., from the 90th day of dsablty or later; and a LTD beneft equal to Mn 3 Rs / 2;00% ) for Rs 33% no beneft for Rs < 33% ). The beneft s ad untl age 60 or 65. The tarffether standardzed for small comanes or talored by age, gender and occuaton dstrbutons for larger oness eressed n a flat rate for all wages and artally fnanced by the emloyer between 50% and 70%). Eclusons are mnmal, mostly for sucde and self-nflcted njury, automoble accdents caused by alcohol or drug use, and njures ncurred durng rots, motor racng, gamblng, or rofessonal cometton. Wth bllon n grou dsablty benefts ad n 2005, three grous of layers are cometng. Provdent Insttutons governed by the French Socal Securty Code 45.3%; Insurance Comanes governed by the French Insurance Code 33.3% ncludng rensurance of Mutual Socetes and Provdent nsttutons); and Mutual Socetes governed by the French Mutualty Code 2.3%. 3

4 Mutual Socetes and Provdent Insttutons roft from ther hstorcal oston n the second llar sometmes drectly desgnated n the Collectve Agreement) when nsurance comanes roose nnovatve roducts and effcent olcy admnstraton. In addton: The remum s ta-deductble from emloyee's ncome / emloyer's result, but the beneft s tble. The cover s not transferable f the emloyee leaves the comany. In order to comlete the frst llar, most benefts of the second llar generally nclude other rotectons such as death, health, enson cover, and remum waver Thrd Pllar: Indvdual Income Protecton for Salared Emloyees and the Self-Emloyed For salared emloyees, the thrd llar comletes the second one or relaces t f the Collectve Agreement fals to roose a grou lan. Characterstcs are: beneft defnton and deferred erod often n lne wth second llar roducts; age at entry and occuatonal dfferentated level remum wth medcal underwrtng; lum sum nstead of annuty sometmes roosed for LTD; and lan not fnanced by the emloyer. For the self-emloyed, ndvdual ncome rotecton wth medcal underwrtng acts as a second llar snce there s no Collectve Agreement. It s often sold n addton to death cover see also "Madeln Law" 994). Characterstcs are: short deferred erod dfferentated by cause, e.g., for sckness, accdent, and hostalzaton, the deferral erod would be 30/0/3 days; cover and beneft untl age 65, sometmes wth a reducton from age 60 to 65; age at entry and occuatonal dfferentated tarff wth three or four classes mostly corresondng to retal tradng, the craft ndustry and academc self-emloyed sometmes wth hyscans and nurses as subgrous); and standard nsurance eclusons, such as for re-estng condtons; n some cases, eclusons for back dsease or hostalzaton for a mental dsorder. Wth bllon of gross remum for the ndvdual dsablty market n 2005, nsurance comanes are the major layers n ths segment, whch s more roftable than the second llar. The llustraton below show the three llars of French ncome rotecton for Salared Emloyees and the Self-Emloyed. Salared emloyees % Gross Salary % Gross Salary 00% LTD 00% 90% 90% Self-emloyed LTD 80% 3 rd Pllar - Facultatve /LTD nsurance 80% 70% 70% 2 nd Pllar - Collectve Agreement 60% "Mensualsaton" Grou /LTD nsurance 60% 50% 50% 40% 40% st LTD st Pllar Pllar 30% Socal Securty st and 2 nd Categores 30% Socal Securty 3 rd Pllar - Facultatve /LTD nsurance st Pllar Socal Securty LTD st Pllar st and 2 nd Categores Socal Securty t=0 day t<=3 years Dsablty t=0 day t<=3 years Dsablty Acc. or Sck. "Consoldaton" Duraton Acc. or Sck. "Consoldaton" Duraton 4

5 3. A MODEL OF CLAIMS EXPERIENCE The thrd secton of ths artcle s devoted to a statstcal methodology that can be used to model French ncdence and reactvaton rates when analyzng clams eerence. The ntent s to rovde the bass of an oeratve mode Secton 4) arorate for a tarff. 3.. Assumtons In ths secton t s assumed that dsablty net sngle remum can be eressed as the eected value of a tme-contnuous stochastc rocess. For a healthy actve) nsured of age at olcy ssue: a a = µ du where: 65 u+ λ a u tu u + u v tu + u v dt 0 u v s the annual fnancal dscount factor; u s the robablty er unt of tme that an actve nsured of age olcy ssue) wll reman actve at age + u ; a µ s the robablty er unt of tme that an actve nsured of age +u olcy ssue) wll become dsabled at age + u ncludng deferred erod); and + s the robablty er unt of tme that a dsabled nsured of age u + u wll reman dsabled at age + u + t. The ntegral n brackets s the actuaral value resent eected value) at age + u of a contnuous annuty ayable to a dsabled nsured untl recovery, LTD or death, wth a mamum of λ years. a Thus, the sngle remum estmaton amounts to model µ and +u aroach usng Generalsed Lnear Models GLMs) s gven. + u. In the followng secton, an 3.2. Stochastc Background GLMs consst of a wde range of dstrbutons n whch the relatonsh lnk functon) between the random effect eected value of the observatons) and the systematc comonent elanatory covarates) rovdes great advantages comared to smle dstrbuton fttng GLMs have more degrees of freedom) or to a classcal lnear regresson GLMs are not restrcted to usng only the Normal dstrbuton). Formally, the dstrbuton of the deendent varables Y ) n reresentng the observaton to be redcted n each tarff cell ) eonental famly defned by the densty functon: where a φ), b θ ) and y,φ ) f y, φ ) =, ndeendent, dentcally dstrbuted d) and =,..., n, belongs to the two arameters b θ ) φ ) y θ θ = e c y, φ ) a, c are functons of the canoncal arameter θ and the dserson arameter φ. In addton, eonental famly allows eressng the varance as a functon of the eected value: where: X s the Model Desgn see below) t β β β,..., β ) 0, k = E Y ) = b θ ) = g β ) and Var Y ) = b θ ) a φ) µ X = s the k + - vector of the model arameters to be estmated; and 5

6 g s a functon lnkng the eected value = E Y ) µ to the lnear redctor X β. The urose of modelng s to fnd β and φ, whch mamze the Lkelhood functon: L Y d y, y2,..., yn, θ, φ ) = f y, θ, φ ) The Reweghted Least Square algorthm aled to the Log-Lkelhood ermts dentfcaton of the best estmates of β. Ths algorthm must be teratvely combned wth the mamum lkelhood equaton wth resect to any addtonal arameter reresentng φ n the followng α or σ ). In some cases, before modelng the rocess must offset the eosure e.g., ncdence rate model) and/or oerate a transformaton of the deendent varable e.g., reactvaton rate model). Let s see an eamle of each model Incdence Rate: a µ +u If the number of clams Y ) n = observed n the tarff cell,..., n) = s suosed to follow a Negatve Bnomal dstrbuton wth log-lnk functon, then the Log-Lkelhood s: wth C ndeendent from β Reactvaton Rate: Ln + u If the reactvaton tme n days of clams have a hazard rate t ) n = n X β { L y, X, β, α )} = y ln{ α e } [ X β y + α ) ln{ + α e } + C ] n m T ) + = = observed n the tarff cell,..., n ) h followng a Log-Logstc dstrbuton wth logt-lnk functon: = s suosed to Then the Log-Lkelhood s: X β σ e σ h t ) = t X β σ σ σ + t e where: t ) Ln n m { L t, X, β, σ )} = w f t ) + w ) S t ) + = f s the " reactvaton" robablty densty on date t. It holds for the n "not censored" observatons: t ) t t ) = h t ) h u) f du ; S s the robablty densty of remanng dsabled on date t. It holds for the m "rght censored" observatons. Ths functon s also known as the " cumulatve dstrbuton": S ITT t ) F t ) = ; 6

7 w s the uncensored ndcator of the observaton, whch takes the value 0 f the observaton s "rght censored" and f t s not. An observaton s called "not censored" f termnates wthn the observaton wndow and "rght censored" f carres on after the end of the observaton wndow Model Desgn and lnk functon The model desgn s eressed through a rectangular matr X = [ X ] n called Desgn Matr, where each = selected covarate added to the model and each of ts ossble transformatons olynomal, log, crossed and nested effects, etc.) s a column of the Desgn Matr makng the lnear regresson form X β an 2 ntegral art of the model. For eamle: Age + Age) + Gender Und. Year) + { Occu.Class j} Two nvertble forms of the Desgn Matr are ossble: Over-Parameterzed or Sgma-Restrcted. Then the lnk functon g nsures the mlementaton of the regresson n the model establshng a drect relaton between the elanatory covarates and the eected value of the deendent varable Model Testng Several methods allow checkng the goodness of ft rovded by the model as well as the adequacy of the covarates and of the resdual dstrbuton = observed redcted): Plots of redcted versus observed values straght lne y= = good ft) or redcted versus resduals symmetrcal sread = good ft); Plots of the normal robablty dstrbuton versus scaled devance resduals dstrbuton straght lne y= = good ft); Plots of Leverages, whch dentfy tarff cells dstant from the center of the observatons no Leverage outlers = good ft). Lkelhood-Rato tests and 3: Contrbuton to the model of each covarate ) taken ndvdually and 3) by removng t from the full model all covarates); Model robustness Tests f the model holds = robust) on subsamles 4 j= 7

8 4. OPERATIVE MODE FOR MODELING CLAIMS EXPERIENCE The fourth secton of ths artcle revews the necessary stes for modelng clams eerence from data etracton to the fnal model testng of µ and 4.. Scoe and Product Feature a +u +. u The oeratve mode resented n ths secton stemmed from the followng roduct desgn: Thrd llar of the French ncome rotecton for self-emloyed Indvdual busness; Comlementary to Death and LTD st and 2 nd category of French Socal Securty; 0, 8, 5, 30, 90 days deferred erod n case of Accdent; 8, 5, 30, 90 days deferred erod n case of Sckness; Reducton n Accdent and Sckness deferred erods n case of hostalzaton; Cover lmted to age 65 and beneft lmted to,095 days and to age 65. Eclusons as usual of Thrd Pllar olces see secton 2.2); Standard medcal underwrtng wth three occuatonal classes and substandard rsks; Level remum by age at olcy nceton Data formattng Data formattng consstes of drawng u two searate tables accordng to the analyss requred Table for analyss of ncdence rates Ths table s the result of a breakdown of the hstory of each olcy n rsk status wth the related clams eerence. It contans the followng nformaton: Start and end dates as well as eosure by fracton of year of the rsk status annversary date of brth or of olcy nceton, date of addendum); All varables remanng unchanged wthn each erod defned by a rsk status age, gender, deferred erod, nsured daly benefts, occuatonal class etc); Any clam that may have affected the rsk status Table for analyss of reactvaton rates Ths table of clams ncludes censorng ndcators wthn the bounds of the observaton wndow and contans the followng nformaton: Dates of occurrence recognzed by the medcal advsor of the nsurance comany for enttlement to a beneft) and frst day of beneft ayment after the deferred erod); All varables remanng unchanged wthn the rsk status affected by the clam see related crtera of the table for ncdence rates); Medcal cause Accdent or Sckness) trggerng the clam and also any degrees of functonal as well as rofessonal dsablty; Date of reactvaton f any wthn the observaton wndow and ts censored ndcator. The tables should be desgned searately by causes trggerng cover Sckness and Accdent, both derved n wth or wthout hostalzaton where Sckness s defned as the comlementary to accdent over all causes. 8

9 4.3. Sequence for Model Buldng The followng stes are requred to fnd out the best model of µ and resectvely a +u + u ). The sequence should aly to Sckness and Accdent searately and be derved n wth or wthout hostalzaton snce most olces of the French market roose a deferred erod secfcally for hostalzaton. Dstrbuton fttng on aggregated data. The arametrc dstrbuton most arorate for the nature of the data s selected. Ths ste ncludes an analyss of the relaton between eected value and functon of varance Quas-lkelhood). The Kalan Meyer roduct-lmt method may hel to select the best dstrbuton for Analyss of varance for smle effect to the model ANOVA. +. u a Each effect s tested alone n order to select the best subset of covarates dscrmnatng µ and +u resectvely ). Co models and Co tests may hel to select the best subset of covarates for + u + u. A frst selecton of outlers s made. Analyss of varance for multle effects to the model MANOVA Cross effects are tested. They often act as a correctve term of the stand alone effects n the model. Ths ste may be an alternatve to covarates falng the ANOVA tests. A second selecton of outlers s made. a Parametrc model of µ and +u t u + u GLM Ths ste s dscussed n Secton 3. In addton t should nclude an analyss of the Devance functon. a Notes: The deferred erod has a double functonalty. In addton to dscrmnate µ and +u covarate acts as a rorty retenton) Eamles of Model Desgn Negatve Bnomal wth Log-Lnk for ncdence rates by Accdent wth hostalzaton + u Model Desgn Gender Age Reached Accdent Deferred Perod Accdent Occuatonal Class: Coeffcent Estmate Std Err. Wald) LR) LR3) Intercet % Gender = "F" % 0.0% 0.0% Gender = "M " Age reached % 0.00% 0.00% 2 Age reached ) % 3.7%.24% Acc. Def. Per % 0.0% 0.02% Occ. Class = % 0.07% 0.03% Occ. Class = % 0.02% 0.03% Occ. Class = Occ. Class = % 0.92% 0.4% α % -- --, ths 9

10 Log-Logstc wth Logt-Lnk for reactvaton rate by Sckness wthout hostalzaton Model Desgn Age at tme of Clam Sckness Occuatonal Class Sckness Deferred Perod: Coeffcent Estmate Std Err. Wald) LR) LR3) Intercet % Age Clamed % 0.00% 0.00% Occ. Class = % 0.00% 0.00% Occ. Class = % 0.52% 0.34% Occ. Class = %.03% 0.96% Occ. Class = Sck. Def. Per % 0.0% 0.02% σ % Occuatonal Classes Lmts of the classes There are two ossble ways of defnng the lmts of occuatonal classes. Strategc orentaton Usual classfcaton The lmts of the classes are re-defned by eerence of underwrters n accordance wth the strategc orentaton of the roduct n the market. In addton to a msclassfcaton, ths method often leads to classes wth heterogeneous sze a large class for "referred rsks" and a small class 4 for substandard rsks. Premum weghted orentaton Occuatonal classes are defned n accordance wth the remum volume reresented by each class of the ortfolo. The classfcaton s made by scorng the ncdence rate of each occuaton. The cumulatve remum volume by order of occuaton n the rankng defnes the lmts of each classe. Ths method should aly to mature and stable ortfolos only Dual Classfcaton Most roducts sold n the French market oerate wth a sngle occuatonal classfcaton, whch s convenent for roducng a aer tarff. Nevertheless, n most of the cases t aears very clearly that occuatonal classes, when desgned by underwrters, are often nfluenced by a rsk erceton of each occuaton that s orented towards ncdence and/or accdental comonents. Unconscously the reactvaton and sckness comonents are layng a second role even when the classfcaton s desgned by eermented underwrters. The evdence based eerence s often dfferent, and a dual occuatonal classfcaton aears to be necessary n order to avod a bas. The Model Desgns of ncdence and reactvaton rates mentoned n secton 4.4 are bult on 22- occuatonal classes: accdent, sckness) ncdence, reactvaton). Ths means 3 4 =8 nstead of 3 classes, whch s too comlcated to sell. In ractce a fnal rankng and groung of the 22-occuatonal classfcaton must be oerated. The rankng/groung must alternate wth the remum calculaton n a recursve rocess. 0

11 5. ACTUARIAL METHODOLOGY FOR BUILDING A TARIFF The ffth secton of ths artcle resents a remum model n dscrete-tme. The remum s drawn u usng the models for ncdence and reactvaton rates as descrbed n Secton Defnton of a Smlfed Actuaral Model The rad recovery develoment of actually calls for a contnuous-tme actuaral model of remum hazard rate on a daly bass), that s dffcult to acheve wth a sreadsheet calculaton. For ths reason, t has been settled on the hyothess of a dscrete weekly tme ), whch governs the followng actuaral model: = 52 + s, age n weeks of the nsured, wth s 52 and = ω ; b, mamum duraton beneft n weeks as er olcy condtons b 56 ; = + ) 52 + ) 52 d = d weekly techncal fnancal dscount rate wth, yearly rate); weekly beneft ndeaton rate from the ayment of the frst weekly beneft after the deferred erod wth d, yearly rate); q = q ) 52 yearly rate and age n years); robablty of death wthn the week for an actve nsured aged wth = )52 robablty of becomng dsabled wthn the week for an actve nsured aged wth, yearly rate and age n years); q = q ) 52 robablty of death wthn the year for a dsabled nsured aged wth yearly rate and age n years); l l q r + = +, weekly change n ortfolo of actve nsured oulaton where ) ) r = b k q ) l k+ ) q k+ ) k + ) H k+ ) + j, j ) k = b k= 2 l k 2 q 2 reresents returns to work and therefore ayment of remums) wth k k k H k+ j, j ) j= j= H y, j q, q, j b ), the weekly robablty of reactvaton for all causes ncludng death) at the end of the j th week of for a clamant aged y, rovded that he was dsabled at the end of the th week. The oulaton of weekly reactvaton s defned as the dfference of dsabled ortfolo, of dsabled nsured whose,. It has been suosed observed between the begnnng and the end of the erod [ ) and havng survved to the erod [ ) dsablty occurred before age r j

12 that dsablty occurred wthn the erod [, ) return to actve oulaton Commutaton Factors 52 D = l, ω + N = D + y= ω nsured Note: = D ); N 52 y+ y= y et ä D or clamant stll dsabled at week b dd not N =, resent value commutaton factors for actve k ) d D = + b k + d l q 2 and N = D H k + ) + j, j ), resent + k= + j= value commutaton factors for dsabled nsured; k ) b k N + d ä = = H k + ) + j j ),, resent value of a weekly beneft of EUR, ad n D k= + j= advance for u to b weeks of ; = N + a ä, resent value of a weekly beneft of EUR, ad n arrears for u to b D weeks of Premum and Mathematcal Reserve The remum for an nsured aged.e. = 52 + s weeks), coverng n case of a weekly allowance of EUR ad after deferred erod for u to b weeks of dsablty and u to the age of ω years, are as follows: Rsk Premum: ä + a RP = 2 and RP = 52 + ) + 26 RP D y y= 52 + D52 + y + Def. Per. 365 Sngle Premum: 52 ω RP D + 26 SP y y= = D y and SP = SP Def. Per Level Premum: SP L P = D N N52 ω + ) and SP Def. Per LP = D ω 52 + D52 y+ y= 2

13 Fnally, the mathematcal reserve s eressed as follows: t V = SP + t LP ä + ω D52 y+ y= + t D t + t ) = SP D, ω 52 7 D52 y+ y= 52 + Def. Per. ω + t ) t ) 6. CONCLUSION Ths artcle adresses most of the centers of cometences nvolved n the French ncome rotecton n general and n Short-Term Dsablty of self-emloyed n artcular: Medcal and fnancal underwrtng, eclusons, olcy condtons, based eerence evdence for modelng ncdence and reactvaton rates, actuaral methodology of remum and reserve calculaton as well as legal and commercal envronments. These key factors cannot be overlooked n a commercal success. In ractce, for modelng ncdence and reactvaton rates, the roducton and clams data have to be formatted n order to rovde daly rsk eosure, clam censorng ndcators and tme deendent covarates. Then, after selecton of covarates dscrmnatng the tarff ncdence and/or reactvaton rates), a dstrbuton and a lnk functon arorate for the nature of the observatons must be chosen. Statstcal ackages rovde algorthms for smle and multle analyses of varance as well as for model fttng and testng. And fnally, before mlementaton n the remum model, results must be adjusted by the eerence of underwrters and clam managers snce some effects cannot be redcted e.g., socal contet, ant-dscrmnaton and tton laws, and sometmes eclusons). Further estmatons show that the level remum calculaton, when evaluated on a smle state model n dscrete-tme weekly bass wth Ecel) as resented n ths artcle devates from less than one ercent from the mult state model n contnuous-tme Hazard rate wth Male). The smlfed model has consderable advantage because t s more fleble to eerence data rovded by olcy admnstraton systems of comanes oeratng n the market. Furthermore the smlfed model has been ostvely tested n the French commercal envronment. Eamles of further actuaral methods are gven by Haberman and Ptacco Actuaral Models for Dsablty Insurance). BIBLIOGRAPHY Code des assurances, Code de la Mutualté, Code de la Sécurté Socale, Code du Traval. htt:// and La rotecton socale dans l'entrerse. Argus de l'assurance, Gude de l'assurance de Groue. BCAC, La révoyance en entrerse. Caher Pratque de l'argus de l'assurance, 200. GLM. McCullagh & J.A. Nelder, 2 nd Edton, 989. GLM and etenson. Hardn and Hlbe, 200. Analyss of Survval Data. Co and Oakes, 998 Survval analyss: Technques for censored and truncated data. Klen and Moeschberger, Actuaral Models for Dsablty Insurance. Haberman and Ptacco,