No-life isurace matematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig
Overview No-life isurace from a fiacial perspective: for a premium a isurace compay commits itself to pay a sum if a evet as occured Cotract period Result elemets Te balace seet Premium Icome Losses Loss ratio Costs Policy older sigs up for a isurace Policy older pays premium. Isurace compay starts to ear premium retrospective prospective Durig te duratio of te policy, some of te premium is eared, some is ueared How muc premium is eared? How muc premium is ueared? Is te ueared premium sufficiet? Premium reserve, prospective
Wy does it work?? Cliet Cliet Result elemets Te balace seet Premium Icome Losses Loss ratio Costs Isurace compay Cliet - Cliet Ecoomic risk is trasferred from te policyolder to te isurer Due to te law of large umbers (may almost idepedet cliets), te loss of te isurace compay is muc more predictable ta tat of a idividual Terefore te premium sould be based o te expected loss tat is trasferred from te policyolder to te isurer Muc of te course is about computig tis expected loss...but first some isurace ecoomics 3
Isurace matematics is fudametal i isurace ecoomics Te result drivers of isurace ecoomics: Result elemets: Result drivers: Risk based pricig, + Isurace premium reisurace Iteratioal ecoomy for example iterest rate level, + fiacial icome risk profile for example stocks/o stocks risk reducig measures (for example istallig burglar alarm), risk selectio (cliet beaviour), cage i legislatio, weater peomeos, demograpic factors, - claims reisurace measures to icrease operatioal efficiecy, IT-systems, - operatioal costs wage developmet = result to be distributed amog te owers ad te autorities Tax politics 4
Premium icome Earig of premium adjustmets take years i o-life isurace: Maturity Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year Year patter Ja Feb Mar Apr May Ju Jul Aug Sep Oct Nov Dec Ja Feb Mar Apr May Ju Jul Aug Sep Oct Nov Dec Jauary 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % February 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % Marc 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % April 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % May 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % Jue 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % July 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % August 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % September 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % October 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % November 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % December 8 % 0,3 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,7 % 0,3 % Sum 0 % % 3 % 6 % 9 % 3 % 7 % % 8 % 35 % 4 % 50 % 58 % 65 % 7 % 78 % 83 % 88 % 9 % 94 % 97 % 99 % 00 % 00 % Assumes tat premium adjustmet is implemeted Jauary st. Assumes tat te portfolio s maturity patter is evely distributed durig te year
Loss ratio Sows ow muc of te premium icome is spet to cover losses Amouts i 000 000 NO 0 Writte gross premium 450 - ceded reisurace premium -70 Cage i reserve for ueared gross premium -0 -cage i reisurace sare of ueared premium 5 Net premium icome 095 Amouts i 000 000 NO 0 Paid claims gross -870 - Reisurace sare of paid claims gross 0 cage i gross claims reserve -00 -cage i reisurace part of gross claims reserve 00 Net claims costs -850 Gross Net Icurred losses 070 (-870-00) 850 Eared premium 340 (450-0) 095 Loss ratio 79.9% 77.6% Wat does te differece i loss ratio gross ad et tell us?
Overview Importat issues Models treated Curriculum Duratio (i lectures) Wat is drivig te result of a olife isurace compay? isurace ecoomics models Lecture otes 0,5 Poisso, Compoud Poisso How is claim frequecy modelled? ad Poisso regressio Sectio 8.-4 EB,5 How ca claims reservig be modelled? Cai ladder, Beruetter Ferguso, Cape Cod, Note by Patrick Dal How ca claim size be modelled? Gamma distributio, logormal distributio Capter 9 EB How are isurace policies priced? Geeralized Liear models, estimatio, testig ad modellig. CRM models. Capter 0 EB Credibility teory Bulma Straub Capter 0 EB Reisurace Capter 0 EB Solvecy Capter 0 EB Repetitio 7
Overview of tis sessio Te Poisso model (Sectio 8. EB) Some importat otios ad some practice too of claim frequecies (Sectio 8.3 EB) 8
Poisso Itroductio Some otios Pure premium = likeliood of claim evet (claims frequecy) * ecoomic cosequece of claim evet (claim severity) Wat is te likeliood of a claim evet? It depeds!!...o risk exposure (extet ad ature of use) object caracteristics (quality ad ature of object) subject caracteristics (beaviour of user) geograpical caracteristics (for example weater coditios ad traffic complexity) Tese depedecies are ormally adled troug regressio, were te umber of claims is te respose ad te factors above are te explaatory variables Let us start by lookig at te Poisso model 9
Te world of Poisso (Capter 8.) Number of claims Poisso Some otios I k- I k I k+ t 0 =0 t k- t k- t k t k+ t k =T Wat is rare ca be described matematically by cuttig a give time period T ito small pieces of equal legt =T/ O sort itervals te cace of more ta oe icidet is remote Assumig o more ta evet per iterval te cout for te etire period is N=I +...+I, were I j is eiter 0 or for j=,..., If p=pr(i k =) is equal for all k ad evets are idepedet, tis is a ordiary Beroulli series Pr( N )! p!( )! ( p), for 0,,..., Assume tat p is proportioal to ad set is a itesity wic applies per time uit p were 0
Te world of Poisso T T T T T p p N ) ( ) (! ) ( )!!(! ) ( )!!(! ) Pr( T e T e T N! ) ( ) Pr( I te limit N is Poisso distributed wit parameter T Some otios Poisso
Te world of Poisso Let us proceed removig te zero/oe restrictio o Ik. A more flexible specificatio is Poisso Some otios Pr( I 0) o( ), Pr( I ) o( ), Pr(Ik ) o( ) k Were o() sigifies a matematical expressio for wic o( ) 0 as 0 k It is verified i Sectio 8.6 tat o() does ot cout i te limit Cosider a portfolio wit J policies. Tere are ow J idepedet processes i parallel ad if is te itesity of policy j ad Ik j te total umber of claims i period k, te Pr( I k 0) J j ( ) j ad Pr( I k ) J i i ji ( j) No claims Claims policy i oly
Te world of Poisso 3 Bot quaities simplify we te products are calculated ad te powers of idetified J j j k j j J k o o 3 3 3 3 3 3 ) ( ) ( ) Pr( ) ( ) ( ) )( ( ) )( )( ( ) ( 0) Pr( I I It follows tat te portfolio umber of claims N is Poisso distributed wit parameter J T J T J J / )... ( were, )... ( We claim itesities vary over te portfolio, oly teir average couts Some otios Poisso
We te itesity varies over time A time varyig fuctio adles te matematics. Te biary variables I,...Ik are ow based o differet itesities k ( t) dt T T k 0 were (t),..., were k ( t k ) for k,..., We I,...Ik are added to te total cout N, tis is te same issue as if differet policies apply o a iterval of legt. I oter words, N must still be Poisso, ow wit parameter T T as 0 0 were te limit is ow itegrals are defied. Te Poisso parameter for N ca also be writte ( t) dt, Ad te itroductio of a time-varyig fuctio does t cage tigs muc. A time average takes over from a costat (t) Poisso Some otios 4
Te itesity Te Poisso distributio Claim umbers, N for policies ad N for portfolios, are Poisso distributed wit parameters T Policy level ad JT Portfolio level is a average over time ad policies. Poisso models ave useful operatioal properties. Mea, stadard deviatio ad skewess are E( N), sd( N) ad skew( ) Te sums of idepedet Poisso variables must remai Poisso, if N,...,NJ are idepedet ad Poisso wit parameters te,..., J Poisso Some otios Ν N... N J ~ Poisso... ) ( J 5
Cliet Policies ad claims Poisso Some otios Policy Isurable object (risk) Claim Isurace cover Cover elemet /claim type
Poisso Car isurace cliet Some otios Car isurace policy Policies ad claims Isurable object (risk), car Claim Tird part liability Isurace cover tird party liability Legal aid Driver ad passeger acidet Fire Isurace cover partial ull Teft from veicle Teft of veicle Rescue Accessories mouted rigidly Isurace cover ull Ow veicle damage Retal car
Some otes o te differet isurace covers o te previous slide: Tird part liability is a madatory cover dictated by Norwegia law tat covers damages o tird part veicles, propterty ad perso. Some isurace compaies provide additioal coverage, as legal aid ad driver ad passeger accidet isurace. Poisso Some otios Partial Hull covers everytig tat te tird part liability covers. I additio, partial ull covers damages o ow veicle caused by fire, glass rupture, teft ad vadalism i associatio wit teft. Partial ull also icludes rescue. Partial ull does ot cover damage o ow veicle caused by collisio or ladig i te ditc. Terefore, partial ull is a more affordable cover ta te Hull cover. Partial ull also cover salvage, ome trasport ad elp associated wit disruptios i productio, accidets or disease. Hull covers everytig tat partial ull covers. I additio, Hull covers damages o ow veicle i a collisio, overtur, ladig i a ditc or oter sudde ad uforesee damage as for example fire, glass rupture, teft or vadalism. Hull may also be exteded to cover retal car. Some otes o some importat cocepts i isurace: Wat is bous? Bous is a reward for claim-free drivig. For every claim-free year you obtai a reductio i te isurace premium i relatio to te basis premium. Tis cotiues util 75% reductio is obtaied. Wat is deductible? Te deductible is te amout te policy older is resposible for we a claim occurs. Does te deductible impact te isurace premium? Yes, by selectig a iger deductible ta te default deductible, te isurace premium may be sigificatly reduced. Te iger deductible selected, te lower te isurace premium. How is te deductible take ito accout we a claim is disbursed? Te isurace compay calculates te total claim amout caused by a damage etitled to disbursemet. Wat you get from te isurace compay is te te calculated total claim amout mius te selected deductible. 8
009 J 009 M 009 M 009 J 009 S 009 N 009 + 00 F 00 A 00 J 00 A 00 O 00 D 0 J 0 M 0 M 0 J 0 S 0 N 0 + 0 F 0 A 0 J 0 A 0 O 0 D Poisso ey ratios claim frequecy Some otios Te grap sows claim frequecy for all covers for motor isurace Notice seasoal variatios, due to cagig weater coditio trougout te years 35,00 Claim frequecy all covers motor 30,00 5,00 0,00 5,00 0,00 5,00 0,00 9
009 J 009 M 009 M 009 J 009 S 009 N 009 + 00 F 00 A 00 J 00 A 00 O 00 D 0 J 0 M 0 M 0 J 0 S 0 N 0 + 0 F 0 A 0 J 0 A 0 O 0 D Poisso ey ratios claim severity Some otios Te grap sows claim severity for all covers for motor isurace 30 000 Average cost all covers motor 5 000 0 000 5 000 0 000 5 000 0 0
009 J 009 M 009 M 009 J 009 S 009 N 009 + 00 F 00 A 00 J 00 A 00 O 00 D 0 J 0 M 0 M 0 J 0 S 0 N 0 + 0 F 0 A 0 J 0 A 0 O 0 D Poisso ey ratios pure premium Some otios Te grap sows pure premium for all covers for motor isurace 6 000 Pure premium all covers motor 5 000 4 000 3 000 000 000 0
009 J 009 M 009 M 009 J 009 S 009 N 009 + 00 F 00 A 00 J 00 A 00 O 00 D 0 J 0 M 0 M 0 J 0 S 0 N 0 + 0 F 0 A 0 J 0 A 0 O 0 D Poisso ey ratios pure premium Some otios Te grap sows loss ratio for all covers for motor isurace 40,00 Loss ratio all covers motor 0,00 00,00 80,00 60,00 40,00 0,00 0,00
009 J 009 M 009 M 009 J 009 S 009 N 009 + 00 F 00 A 00 J 00 A 00 O 00 D 0 J 0 M 0 M 0 J 0 S 0 N 0 + 0 F 0 A 0 J 0 A 0 O 0 D ey ratios claim frequecy TPL ad ull Poisso Some otios Te grap sows claim frequecy for tird part liability ad ull for motor isurace 35,00 Claim frequecy ull motor 30,00 5,00 0,00 5,00 0,00 5,00 0,00 3
009 J 009 M 009 M 009 J 009 S 009 N 009 + 00 F 00 A 00 J 00 A 00 O 00 D 0 J 0 M 0 M 0 J 0 S 0 N 0 + 0 F 0 A 0 J 0 A 0 O 0 D 009 J 009 M 009 M 009 J 009 S 009 N 009 + 00 F 00 A 00 J 00 A 00 O 00 D 0 J 0 M 0 M 0 J 0 S 0 N 0 + 0 F 0 A 0 J 0 A 0 O 0 D ey ratios claim frequecy ad claim severity Poisso Some otios Te grap sows claim severity for tird part liability ad ull for motor isurace 70 000 Average cost tird part liability motor 5 000 Average cost ull motor 60 000 50 000 0 000 40 000 5 000 30 000 0 000 0 000 0 000 5 000 0 0 4
Poisso (Capter 8.3) Some otios How varies over te portfolio ca partially be described by observables suc as age or sex of te idividual (treated i Capter 8.4) Tere are owever factors tat ave impact o te risk wic te compay ca t kow muc about Driver ability, persoal risk averseess, Tis radomeess ca be maaged by makig a stocastic variable Tis extesio may serve to capture ucertaity affectig all policy olders joitly, as well, suc as alterig weater coditios Te models are coditioal oes of te form N ~ Poisso ( T ) ad Ν ~ Poisso ( JT ) Policy level Portfolio level Let E( ) ad sd( ) ad recall tat E( N ) var( N ) T wic by double rules i Sectio 6.3 imply E ( N) E( T ) T ad var( N) E( T ) var( T ) T T Now E(N)<var(N) ad N is o loger Poisso distributed 5
Poisso Te rule of double variace Some otios Let X ad Y be arbitrary radom variables for wic ( x) E( Y x) ad var( Y x) Te we ave te importat idetities E( Y) E{ ( X )} Rule of double expectatio ad var(y) E{ ( X )} var{ ( X )} Rule of double variace Recall rule of double expectatio E( E( Y all y all x x)) yf X, Y all x ( E( Y (x, y)dxdy x)) f X y all y all x ( x) dx f X, Y all x all y yf Y X (x, y)dxdy (y x)dy f all y yf Y X ( y) dy ( x) dx E( Y) 6
wikipedia tells us ow te rule of double variace ca be proved Poisso Some otios
Poisso Te rule of double variace Some otios Var(Y) will ow be proved from te rule of double expectatio. Itroduce Y ( x) ad ote tat E( Y) E(Y) wic is simply te rule of double expectatio. Clearly ( Y ) (( Y Y) ( Y)) ( Y Passig expectatios over tis equality yields were var( Y) B B B 3 Y) ( Y) ( Y B E( Y Y), B E( Y), B3 E( Y Y)( Y), wic will be adled separately. First ote tat ( x) E{( Y ( x)) x} E{( Y ad by te rule of double expectatio applied to E{ ( x)} E{( Y Y) B. Te secod term makes use of te fact tat rule of double expectatio so tat Y) x} ( Y Y) E ( Y) by te Y)( Y). 8
Poisso Te rule of double variace Some otios B var( ) Y var{ ( x)}. Te fial term B3 makes use of te rule of double expectatio oce agai wic yields were B3 E{ c( X )} c( X ) { E( Y E{( Y Y)( Y ) x} E{( Y Y) x}( Y ) x) Y)}( Y ) { Y Y)}( Y ) 0 Ad B3=0. Te secod equality is true because te factor is fixed by X. Collectig te expressio for B, B ad B3 proves te double variace formula ( Y ) 9
Poisso Some otios Specific models for Pr( N ) Pr( N 0 are adled troug te mixig relatiosip ) g( ) d Gamma models are traditioal coices for ad g() i Pr( N )Pr( ) ad detailed below Estimates of ca be obtaied from istorical data witout specifyig g(). Let,..., be claims from policy olders ad T,...,TJ teir exposure to risk. Te itesity if idividual j is te estimated as / T. Ucertaity is uge. Oe solutio is ad j wj j were wj j Ti i wj ( j ) c j ( ) were c wj Ti j i j T i j (.5) (.6) i j j Bot estimates are ubiased. See Sectio 8.6 for details. 0.5 returs to tis. 30
Te egative biomial model Te most commoly applied model for mu is te Gamma distributio. It is te assumed tat G Gamma( ) were G ~ Gamma( ) Here is te stadard Gamma distributio wit mea oe, ad fluctuates aroud wit ucertaity cotrolled by. Specifically Sice emerges i te limit. E( ) ad sd( ) / sd( ) 0 as, te pure Poisso model wit fixed itesity Poisso Some otios Te closed form of te desity fuctio of N is give by Pr( N ( ) ) ( ) ( ) p ( p) were p T bi(, ) for =0,,... Tis is te egative biomial distributio to be deoted. Mea, stadard deviatio ad skewess are E( N) T, sd( N) T ( T / ), skew(n) / T/ T ( T Were E(N) ad sd(n) follow from (.3) we is iserted. Note tat if N,...,NJ are iid te N+...+NJ is bi (covolutio property). / ) (.9) 3
Poisso Fittig te egative biomial Some otios Momet estimatio usig (.5) ad (.6) is simplest tecically. Te estimate of is simply i (.5), ad for ivoke (.8) rigt wic yields 0 / so tat /. If, iterpret it as a ifiite or a pure Poisso model. Likeliood estimatio: te log likeliood fuctio follows by isertig j for i (.9) ad addig te logaritm for all j. Tis leads to te criterio L(, ) j j j log( log( ) ( j j ) {log( ( )) log( )} )log( T were costat factors ot depedig o ad ave bee omitted. j ) 3