Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel
A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference Winnipeg, Augus 1-4, 212. * Join work wih Emmanuel Hamel, Universié Laval. 1
Absrac A renewal model for he aggregae discouned paymens and expenses assumed by he insurer is proposed for he medical malpracice insurance, where he real ineres raes could be sochasic and he dependency is examined hrough he heory of copulas. As a firs approach o his problem, we presen formulas for he firs wo raw momens and he firs join momen of his aggregae risk process. Examples are given for exponenial claims ineroccurence imes and he dependency is illusraed by an Archimedean copula, in which he auocovariance and he auocorrelaion funcions are also examined. Keywords : Aggregae discouned paymens; Copulas; Join and raw momens; Medical malpracice; Renewal process; Sochasic ineres rae. 2
Overview on Medical Malpracice Insurance Definiion : Medical malpracice is generally defined as a professional negligence by ac or omission by a healh care provider in which he reamen provided falls below he acceped sandard of pracice in he medical communiy and evenually causes injury or deah o he paien, wih mos cases involving medical error. Premiums : - Medical malpracice insurers ake several facors ino accoun when seing premiums, and hese are usually charged o individuals, groups of pracice, hospials or governmens. 3
- One of he main facor is he ype of work a healh-care provider does. Some specialies have a significanly higher rae of claims han ohers and will hus pay higher premiums, such as in neurosurgery and obserics/gynecology. - Anoher imporan facor is he region where a provider pracices. Indeed sandards and regulaions for medical malpracice vary by counry and even by jurisdicions wihin counries, which is paricularly apparen in USA. - Among he oher facors ha are usually considered by he insurer are : some degree of experience raing, adminisraive expenses, liigaion expenses, fuure invesmen income, profi margin sough, insurance business cycle, supply and demand. - The physician professionals claims experience is oo variable over shor ime periods bu presens more sabiliy for hospials. 4
Type of insurance : - Premiums will also vary depending on he ype of insurance coverage choosen for medical malpracice. - There are essenially wo primary ypes of insurance coverage for medical malpracice : claims-made and occurrence policies. - Claims-made insurance, like auo or home insurance, provides coverage for incidens ha occur while he policy is in force. However, an imporan condiion is ha he claim mus also be filed while he policy is in force for he inciden o be covered. For his ype of insurance, a ail coverage is highly recommended o cover incidens ha have no been repored o he company during he policy erm. 5
- Occurrence coverage policies differ from he claims-made coverage by he fac ha hey cover any inciden ha occurs while he policy is in force, no maer when he claim is filed. - As generally observed wihin his insurance marke, he firs ype of insurance is subsanially less expensive in he very firs years bu by he fourh or fifh year i reaches a maure level a abou 95% of he cos of an occurrence policy. - Claims-made policies are wha are normally issued by mos insurance carriers nowadays. In spie of ha, he decision beween a claims-made and an occurrence policy will obviously depend of wha is bes suied for he specific needs of he insured eniy. * In his research, only he claims-made policies will be considered. 6
Dependency : - The business line Medical malpracice is characerized by a srong degree of uncerainy under many aspecs ofen relaed. - Many empirical observaions seem o show ha here is a posiive dependency beween he delay from he recepion o he selemen of he claim, he final paymen of he claim and he amoun of expenses allocaed o he claim. - The discoun raes used o acualize he paymen of he claims and he expenses are no necessarily independen. - To represen he dependencies menionned previously, he heory of copulas seems o be mos suiable and has been largely applied in he acuarial lieraure since he las decade. 7
A renewal model, wih copula and sochasic ineres rae Moivaed mainly by he works of Léveillé & Garrido (21) and Léveillé & Adékambi (211) on discouned compound renewal sums, we presen a sochasic model for he medical malpracice insurance where he couning process is an ordinary renewal process, he discoun facors relaed o he paymens and he expenses may be sochasic and dependen, and he dependencies are evenually governed by copulas. Hence consider he following aggregae discouned paymens and expenses process Z ( ) = Z 1 ( ) + Z 2 N( ) =: D 1 ( T k + τ k ) X k + D 2 ( T k + τ k )Y k k=1 N( ) k=1 8
where { τ k, k } is a sequence of coninuous posiive independen and idenically disribued (i.i.d.) random variables, such ha τ k represens he iner-occurrence ime beween he ( k 1)-h and k-h claims. { } is a sequence of random variables such ha T k, k T k = k τ k, T =, and hen T k represens he occurrence ime of i=1 he claims received by he insurer. { τ k, k } is a sequence of coninuous posiive i.i.d. random variables, independen of he τ k, such ha τ k is he ime from T k aken by he insurer o pay he k-h claim. 9
{ X k, k } is a sequence of posiive i.i.d. random variables, independen of he T k, such ha X k represens he deflaed amoun of he claim effecively paid by he insurer. { Y k, k } is a sequence of posiive i.i.d. random variables, independen of he T k, such ha Y k represens he deflaed amoun of he expenses incurred by he insurer o fix he paymen corresponding o he k-h claim. { } is an ordinary renewal process generaed by he N ( ), iner-occurrence imes { τ k, k }, which represens he number of claims received by he insurer in [,]. 1
The random variables X k, Y k and τ k are evenually dependen and his dependency relaion is generaed by a copula C( u 1,u 2,u 3 ), where ( u 1,u 2,u 3 ) [,1] 3, which has posiive measures of dependence and concordance. D i ( ) = exp δ ( u)du, i = 1,2, is he discoun facor a = corresponding o Z i ( ) and δ i ( ) is he force of ne ineres which could be deerminisic or sochasic. Moreover, we will assume ha { δ 1 ( ), } and { δ 2 ( ), } could be dependen bu are independen of he processes { N ( ), }, { τ k, k }, { X k, k } and { Y k, k }. * Here, we make he choice of no represening he possible dependency beween he discoun facors by anoher copula in order no o weigh down our model. 11
Firs and second raw momens of Z () The following heorem gives an inegral expression for he firs momen of Z, in agreemen wih our hypoheses. Theorem 1 : Consider he discouned aggregae paymens and expenses process, such as assumed previously. Then, for sochasic forces of ineres δ 1 ( ) and δ 2 ( ), he firs momen of Z is given by : = E X τ = v E D 1 u + v E Z ( ) dm u df τ ( v) + E Y τ = v E D 2 ( u + v) dm( u) df τ ( v), where m( ) is he renewal funcion. 12
Corollary 1 : For posiive consan forces of ineres δ 1 and δ 2 Theorem 1 yields E Z = E e δ 1 τ X e δ1v dm v + E e δ 2 τ Y e δ 2v dm v. Example 1: Assume ha he deflaed amouns X k and Y k have respecively, for x >, Pareo disribuions F X ( x) = 1 α1 β 1 β 1 + x, F Y x = 1 α2 β 2 β 2 + x, where β 1 >, β 2 >, α 1 > 2 and α 2 > 2, 13
and ha he ineroccurrence imes of he claims τ k and he delays τ k have respecively, for >, exponenial disribuions F τ ( ) = 1 e λ, F τ = 1 e λ, where λ >, λ >. Furhermore assume ha he dependency relaion beween X k, Y k and τ k is generaed by he Archimedian copula = 1 1 1 ( 1 u i ) γ C u 1,u 2,u 3 where u 1 = F X 3 i=1 1 γ =:1 f 1 γ ( x), u 2 = F Y ( y), u 3 = F τ ( ) and γ 1. ( u 1,u 2,u 3,γ ), 14
Applying Corollary 1, and wih he help of a sofware such as Maple, he preceding ideniy for E Z numerically. [ ] can be calculaed Hence, if we consider he paricular case where δ 1 =.2, δ 2 =.3, α 1 = 3, α 2 = 5, β 1 = β 2 = 1, λ = 1, λ = 2 and γ = 2, hen we ge from Corollary 1 he following funcion for he firs raw momen of Z, E Z ( ) ( 24.48) 1 e.2 + ( 8.1) 1 e.3. * The nex heorem gives an inegral expression for he second momen of Z where he assumed dependencies of he model are also presen in each erm of his expression. 15
Theorem 2 : Consider he discouned aggregae paymens and expenses process, such as assumed previously. Then, for sochasic forces of ineres δ 1 ( ) and δ 2 ( ), he second momen of Z is given by: E Z 2 ( ) = E X 2 τ = w + E Y 2 τ = w E D 2 1 E D 2 2 ( v + w) dm v ( v + w) dm v +2 E XY τ = w E D 1 ( v + w)d 2 v + w +2 E X τ = w E X τ = w v E D 1 ( u + w)d 1 u + v + w df τ df τ ( w) ( w) dm v dm( u)dm v df τ df τ ( w) ( w)df τ w 16
+ 2 E Y τ = w E Y τ = w v E D 2 ( v + w)d 2 u + v + w + 2 E X τ = w E Y τ = w v E D 1 ( v + w)d 2 u + v + w + 2 E X τ = w E Y τ = w v E D 2 ( v + w)d 1 u + v + w dm( u)dm v dm( u)dm v dm( u)dm v df τ df τ df τ ( w)df τ w ( w)df τ w ( w)df τ w 17
Corollary 2 : For posiive consan forces of real ineres δ 1 and δ 2, Theorem 2 yields ( ) E Z 2 = E e 2δ 1 τ X 2 e 2δ1v dm v + E e 2δ 2 τ Y 2 e 2δ 2v dm v + 2E e ( δ 1+δ 2 ) τ XY e δ 1+δ 2 v dm v v + 2 E 2 e δ 1 τ X e δ 1( u+2v ) v dm( u)dm( v) + E 2 e δ 2 τ Y e δ 2( u+2v ) dm( u)dm v +E e δ 1 τ X E e δ 2 τ Y e δ 1+δ 2 v v e δ 1u + e δ 2u dm( u)dm v. 18
Example 2 : Consider he same disribuions, copula and parameers such as given in Example 1, hen by combining he resuls of Example 1 and Corollary 2 we ge ( ) E Z 2 22.75 1 e.4 + 2.5 1 e.6 + 11.6 1 e.5 + 2{ ( 773, 762.5) 1 2e.2 + e.4 + 36, 45.29 1 2e.3 + e.6 + ( 33, 48.66) 1 e.2 e.3 + e.5 }. 19
From Examples 1 and 2, we ge he following able for he expecaion and sandard deviaion of Z ( ), and for a premium based on he sandard deviaion principle. Table 1: E Z, σ Z, Π Z = E Z + σ Z ( ). 1 2 3 4 5 6 7 E Z.72 1.43 2.12 2.8 3.46 4.1 4.73 σ Z 32.53 64.32 95.4 125.8 155.52 184.59 213. Π Z 33.25 65.75 97.52 128.6 158.98 188.69 217.73 If ime is measured in unis of year and he paid amouns and he expenses are boh measured in unis of $1,, hen we noe ha he premium charged by he insurer is very expensive and increases subsanially wih he insurance coverage period. 2
Remark : The choice of he Archimedian copula used in he previous examples is arbirary bu presens posiive measures of concordance and dependence as our model requires i. To verify ha, hereafer we compue (numerically) hree classical measures corresponding o he copula used in our examples, precisely he rivariae Kendall s au τ 3 defined by { } 1 τ 3.354,,1 τ 3 = 1 3 8 C ( u 1,u 2,u 3 )dc u 1,u 2,u 3 [ ] 3 he rivariae Spearman s rho ρ 3 defined by, ρ 3 = 8 C( u 1,u 2,u 3 )du 1 du 2 du 3 1 ρ 3.44, [,1] 3 21
he mulivariae upper ail dependence coefficien λ U 1..h h+1..3 defined by λ U 1..h h+1..3 = lim u 1 n h i= n n i ( 1 )i ϕ 1 ( iϕ ( u) ) n h n h i ( 1 )i ϕ 1 ( iϕ ( u) ) n i=1, h = 1,2, where ϕ ( u) is he generaor of his Archimedian copula = ln 1 ( 1 u) γ ϕ u ϕ 1 which implies ha 1 λ 2,3 1,2 U.84, λ 3 U.49. ( u) = 1 1 e u γ 1, 22
Firs join momen beween Z () and Z (+h) In his secion, we presen an inegral expression for he covariance beween Z ( ) and Z ( + h ), where he erms depending on h are also highly affeced by he dependencies of he model. Theorem 3 : Consider he discouned aggregae paymens and expenses process, such as assumed previously. Then, for sochasic forces of ineres δ 1 ( ) and δ 2 ( ), he firs join momen beween Z and Z ( + h) is given by : E Z ( )Z ( + h) = E Z 2 + E X τ = w E X τ = w +h v E D 1 ( v + w)d 1 u + v + w v dm( u)dm v df τ ( w)df τ w 23
+ E Y τ = w E Y τ = w +h v E D 2 ( v + w)d 2 u + v + w v + E X τ = w E Y τ = w +h v E D 1 ( v + w)d 2 u + v + w v + E Y τ = w E X τ = w +h v E D 2 ( v + w)d 1 u + v + w v dm( u)dm v dm( u)dm v dm( u)dm v df τ df τ df τ ( w)df τ w ( w)df τ w ( w)df τ w 24
Corollary 3 : For posiive consan forces of ineres δ 1 and δ 2, Theorem 3 yields +h v E Z ( )Z ( + h) = E Z 2 ( ) + E 2 e δ 1 τ X e δ 1( u+2v ) dm( u)dm v +E e δ 1 τ X v +h v + E 2 e δ 2 τ Y e δ 2( u+2v ) dm( u)dm v +h v v E e δ 2 τ Y e ( δ 1+δ 2 )v e δ1u + e δ 2u dm( u)dm( v). v 25
Example 3 : Again, using he same assumpions and he resuls obained in Examples 1 and 2, we ge E Z ( )Z ( + h) E Z 2 ( ) + 1,498,176 e.2 1 e.2 + ( 72,899.93) e.3 1 e.3 1 e.3h e.2 1 e.2h + 33, 48.66 1 e.2h { + e.3 1 e.3h e.5 2 e.2h e.3h }, ( ) where E Z 2 is given in Example 2. 26
By using he daa of Table 1, we obain he following able for he auocovariance and auocorrelaion funcions of Z ( ), where we define = C ov Z C,h (, Z ( + h) ), ρ,h = C,h. σ Z ( ) σ Z + h Table 2 : C(,h), ρ(,h). h 1 2 3 4 5 6 7 272.7 364.5 434.4 4982.7 591. 6816.8 773.6.995.9874.9858.9848.9842.9837.9833 C 1,h ρ 1,h This able corroboraes he srong linear correlaion observed beween he values of E[ Z ( ) ] in Table 1. Obviously, his las funcion is concave and ends (approximaively) o he value 32.58 as. 27
Conclusion A renewal model for medical malpracice insurance has been proposed. This model incorporaes sochasic ineres raes and a copula o esablish he dependence beween he paymen of he claim, he expenses and he delay beween he receip and he selemen of he claim. Inegral formulas has been given for he firs wo raw momens and he firs join momen of our risk process. The auocorrelaion funcion has also been examined, as well as he incidence of our model on he premium. Several imporan challenges arise now from his model, such as he calibraion of he copulas ha will characerize adequaely he dependency relaions wihin our problem and he choice of he discoun raes ha will bes represen he yields expeced by he insurer, o menion only hose. 28
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