Modelling The Claims Development Result For Solvency Purposes


 Bryce Lambert
 1 years ago
 Views:
Transcription
1 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes Mchael Merz Maro V. Wüthrch Abstract We assume that the clams lablty process satses the dstrbutonree chanladder model assumptons. For clams reservng at tme we predct the total ultmate clam wth the normaton avalable at tme and smlarly at tme we predct the same total ultmate clam wth the (updated) normaton avalable at tme. The clams development result at tme or accountng year ( ] s then dened to be the derence between these two successve predctons or the total ultmate clam. n [6 0] we have analyzed ths clams development result and we have quanted ts predcton uncertanty. Here we smply mody and llustrate the results obtaned n [6 0]. We emphasze that these results have drect consequences or solvency consderatons and were (under the new rsadusted solvency regulaton) already mplemented n ndustry. Keywords. tochastc lams Reservng hanladder Method lams evelopment Result Loss Experence ncurred Losses Pror Accdent Years olvency Mean quare Error o Predcton.. NTROUTON We consder the problem o quantyng the uncertanty assocated wth the development o clams reserves or pror accdent years n general nsurance. We assume that we are at tme and we predct the total ultmate clam at tme (wth the avalable normaton up to tme ) and one perod later at tme we predct the same total ultmate clam wth the updated normaton avalable at tme. The derence between these two successve predctons s the socalled clams development result or accountng year ( ]. The realzaton o ths clams development result has a drect mpact on the prot & loss (P&L) statement and on the nancal strength o the nsurance company. Thereore t also needs to be studed or solvency purposes. Here we analyze the predcton o the clams development result and the possble luctuatons around ths predcton (predcton uncertanty). Bascally we answer two questons that are o practcal relevance: asualty Actuaral ocety EForum Fall
2 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes (a) n general one predcts the clams development result or accountng year ( ] n the budget statement at tme by 0. We analyze the uncertanty n ths predcton. Ths s a prospectve vew: how ar can the realzaton o the clams development result devate rom 0? Remar: we dscuss below why the clams development result s predcted by 0. (b) n the P&L statement at tme one then obtans an observaton or the clams development result. We analyze whether ths observaton s wthn a reasonable range around 0 or whether t s an outler. Ths s a retrospectve vew. Moreover we dscuss the possble categorzaton o ths uncertanty. o let us start wth the descrpton o the budget statement and o the P&L statement or an example we reer to Table. The budget values at an. year are predcted values or the next accountng year ( ] o ths accountng year ( ].. The P&L statement are then the observed values at the end Postons a) and b) correspond to the premum ncome and ts assocated clams (generated by the premum lablty). Poston d) corresponds to expenses such as acquston expenses head oce expenses etc. Poston e) corresponds to the nancal returns generated on the balance sheet/assets. All these postons are typcally wellunderstood. They are predcted at an. year (budget values) and one has ther observatons at ec. 3 year n the P&L statement whch descrbes the nancal closng o the nsurance company or accountng year ( ]. asualty Actuaral ocety EForum Fall
3 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes budget values at an. year P&L statement at ec. 3 year a) premums earned b) clams ncurred current accdent year c) loss experence pror accdent years d) underwrtng and other expenses e) nvestment ncome ncome beore taxes Table : ncome statement n $ 000 However poston c) loss experence pror accdent years s oten much less understood. t corresponds to the derence between the clams reserves at tme t and at tme t adusted or the clam payments durng accountng year ( ] or clams wth accdent years pror to accountng year. n the sequel we wll denote ths poston by the clams development result (R). We analyze ths poston wthn the ramewor o the dstrbutonree chanladder (L) method. Ths s descrbed below. hortterm vs. longterm vew n the classcal clams reservng lterature one usually studes the total uncertanty n the clams development untl the total ultmate clam s nally settled. For the dstrbutonree L method ths has rst been done by Mac [7]. The study o the total uncertanty o the ull runo s a longterm vew. Ths classcal vew n clams reservng s very mportant or solvng solvency questons and almost all stochastc clams reservng methods whch have been proposed up to now concentrate on ths long term vew (see WüthrchMerz [9]). However n the present wor we concentrate on a second mportant vew the shortterm vew. The shortterm vew s mportant or a varety o reasons: asualty Actuaral ocety EForum Fall
4 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes the shortterm behavour s not adequate the company may smply not get to the longterm because t wll be declared nsolvent beore t gets to the long term. A shortterm vew s relevant or management decsons as actons need to be taen on a regular bass. Note that most actons n an nsurance company are usually done on a yearly bass. These are or example nancal closngs prcng o nsurance products premum adustments etc. Relected through the annual nancal statements and reports the shortterm perormance o the company s o nterest and mportance to regulators clents nvestors ratng agences stocmarets etc. ts consstency wll ultmately have an mpact on the nancal strength and the reputaton o the company n the nsurance maret. Hence our goal s to study the development o the clams reserves on a yearly bass where we assume that the clams development process satses the assumptons o the dstrbutonree chanladder model. Our man results Results and 3.5 below gve an mproved verson o the results obtaned n [6 0]. e FelceMorcon [4] have mplemented smlar deas reerrng to the random varable representng the YearEnd Oblgatons o the nsurer nstead o the R. They obtaned smlar ormulas or the predcton error and vered the numercal results wth the help o the bootstrap method. They have notced that ther results or aggregated accdent years always le below the analytcal ones obtaned n [6]. The reason or ths s that there s one redundant term n (4.5) o [6]. Ths s now corrected see ormula (A.4) below. Let us menton that the deas presented n [6 0] were already successully mplemented n practce. Predcton error estmates o YearEnd Oblgatons n the overdspersed Posson model have been derved by VAP [5] n a eld study on a large sample o talan MTPL companes. A eld study n lne wth [6 0] has been publshed by AAMAME []. Moreover we would also le to menton that durng the wrtng o ths paper we have learned that smultaneously smlar deas have been developed by Böhm Glaab []. asualty Actuaral ocety EForum Fall
5 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes. METHOOLOGY. Notaton We denote cumulatve payments or accdent year { 0 K } { 0 K } untl development year by. Ths means that the ultmate clam or accdent year s gven by. For smplcty we assume that (note that all our results can be generalzed to the case > ). Then the outstandng loss labltes or accdent year { 0 K } at tme t are gven by and at tme t they are gven by R (.) R. (.) Let denote the clams data avalable at tme { ; and } (.3) t and { and } { } ; ; (.4) denote the clams data avalable one perod later at tme t. That s we go one step ahead n tme rom to we obtan new observatons { } ; on the new dagonal o the clams development trangle (c. Fgure ). More ormally ths means that we get an enlargement o the eld generated by the observatons generated by the observatons.e. ( ) ( ) to the eld. (.5). strbutonree chanladder method asualty Actuaral ocety EForum Fall
6 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes We study the clams development process and the R wthn the ramewor o the wellnown dstrbutonree L method. That s we assume that the cumulatve payments satsy the assumptons o the dstrbutonree L model. The dstrbutonree L model has been ntroduced by Mac [7] and has been used by many other actuares. t s probably the most popular clams reservng method because t s smple and t delvers n general very accurate results. accdent development year accdent development year year 0 K K year 0 K K 0 0 M M M M Fgure : Loss development trangle at tme t and t Model Assumptons. umulatve payments n derent accdent years { 0 K } are ndependent. ( ) 0 are Marov processes and there exst constants > 0 > 0 such that or all and 0 we have [ ] E (.6) asualty Actuaral ocety EForum Fall
7 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes ( ) Var. (.7) Remars. n the orgnal wor o Mac [7] there were weaer assumptons or the denton o the dstrbutonree L model namely the Marov process assumpton was replaced by an assumpton only on the rst two moments (see also WüthrchMerz [9]). The dervaton o an estmate or the estmaton error n [0] was done n a tmeseres ramewor. Ths mposes stronger model assumptons. Note also that n (.7) we requre that the cumulatve clams are postve n order to get a meanngul varance assumpton. Model Assumptons. mply (usng the tower property o condtonal expectatons) [ ] E and [ ] E. (.8) Ths means that or nown L actors we are able to calculate the condtonally expected ultmate clam gven the normaton and respectvely. O course n general the L actors ramewor o the L method ths s done as ollows: are not nown and need to be estmated. Wthn the. At tme t gven normaton the L actors are estmated by 0 where. (.9) 0. At tme t gven normaton the L actors are estmated by asualty Actuaral ocety EForum Fall
8 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes where 0 0. (.0) Ths means the L estmates at tme use the ncrease n normaton about the clams development process n the new observed accountng year ( ] based on the addtonal observaton. and are thereore Mac [7] proved that these are unbased estmators or and moreover that m and m l ( m or ) are uncorrelated random varables or l (see Theorem n Mac [7] and Lemma.5 n [9]). Ths mples that gven s an unbased estmator or [ ] L (.) E wth and gven L (.) s an unbased estmator or [ ] E wth. Remars.3 The realzatons o the estmators realzatons o 0 K are nown at tme t but the 0 K are unnown snce the observatons K durng the accountng year ( ] are unnown at tme. asualty Actuaral ocety EForum Fall
9 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes When ndces o accdent and development years are such that there are no actor products n (.) or (.) an empty product s replaced by. For example and. The estmators are based on the L estmators at tme and thereore use the ncrease n normaton gven by the new observatons n the accountng year rom to..3 ondtonal mean square error o predcton Assume that we are at tme that s we have normaton and our goal s to predct the random varable. Then gven n (.) s a measurable predctor or. At tme we measure the predcton uncertanty wth the socalled condtonal mean square error o predcton (MEP) whch s dened by E (.3) That s we measure the predcton uncertanty n the L ( P[ ]) dstance. Because s measurable ths can easly be decoupled nto process varance and estmaton error: ( ) ( [ ] Var E ). (.4) Ths means that s used as predctor or the random varable and as estmator or the expected value [ ] [ ] E at tme. O course the condtonal expectaton E s nown at tme (.e. the L actors are nown) t s used as predctor asualty Actuaral ocety EForum Fall
10 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes or and the estmaton error term vanshes. For more normaton on condtonal and uncondtonal MEP s we reer to hapter 3 n [9]:.4 lams development result (R) We gnore any prudental margn and assume that clams reserves are set equal to the expected outstandng loss labltes condtonal on the avalable normaton at tme and respectvely. That s n our understandng best estmate clams reserves correspond to condtonal expectatons whch mples a selnancng property (see orollary.6 n [8]). For nown L actors thereore used as predctor or expectaton [ ] the condtonal expectaton [ ] E s used as predctor or result (true R) or accountng year ( ] E s nown and s at tme. mlarly at tme the condtonal s dened as ollows.. Then the true clams development enton.4 (True clams development result) The true R or accdent year { K } n accountng year ( ] s gven by ( ) E [ R ] ( X E [ R ] ) E [ ] E [ ] R (.5) where by X denotes the ncremental payments. Furthermore the true aggregate s gven R ( ). (.6) Usng the martngale property we see that asualty Actuaral ocety EForum Fall
11 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes [ R ( ) ] 0 E. (.7) Ths means that or nown L actors equal to zero. Thereore or nown L actors the expected true R (vewed rom tme ) s we reer to ( ) R as the true R. Ths also ustes the act that n the budget values o the ncome statement poston c) loss experence pror accdent years s predcted by $0 (see poston c) n Table ). The predcton uncertanty o ths predcton 0 can then easly be calculated namely R ( 0) Var R ( ) ( ) E [ ]. (.8) For a proo we reer to ormula (5.5) n [0] (apply recursvely the model assumptons) and the aggregaton o accdent years can easly be done because accdent years are ndependent accordng to Model Assumptons.. Unortunately the L actors are n general not nown and thereore the true R s not observable. Replacng the unnown actors by ther estmators.e. replacng the expected ultmate clams E [ ] and E [ ] wth ther estmates and respectvely the true R or accdent year ( ) n accountng year ( ] predcted/estmated n the L method by: s enton.5 (Observable clams development result) The observable R or accdent year { K } n accountng year ( ] ( ) R X R R s gven by (.9) asualty Actuaral ocety EForum Fall
12 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes where R and aggregate R s gven by R are dened below by (.) and (.) respectvely. Furthermore the observable ( ) R. (.0) Note that under the Model Assumptons. gven R s an unbased estmator or [ R ] s an unbased estmator or [ R ] ( ) (.) E and gven R ( ) (.) E. Remars.6 We pont out the (nonobservable) true clams development result ( ) approxmated by an observable clams development result R ( ) R s. n the next secton we quanty the qualty o ths approxmaton (retrospectve vew). Moreover the observable clams development result R ( ) s the poston that occurs n the P&L statement at ec. 3 year. Ths poston s n the budget statement predcted by 0. n the next secton we also measure the qualty o ths predcton whch determnes the solvency requrements (prospectve vew). We emphasze that such a solvency consderaton s only a oneyear vew. The remanng runo can or example be treated wth a costocaptal loadng that s asualty Actuaral ocety EForum Fall
13 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes based on the oneyear observable clams development result (ths has or example been done n the wss olvency Test). 3. MEP OF THE LAM EVELOPMENT REULT Our goal s to quanty the ollowng two quanttes: R ( ) E R ( ) 0 (3.) R 0 R ( ) ( ( E R )) R. (3.) The rst condtonal MEP gves the prospectve solvency pont o vew. t quantes the predcton uncertanty n the budget value 0 or the observable clams development result at the end o the accountng perod. n the solvency margn we need to hold rs captal or possble negatve devatons o ( ) R rom 0. The second condtonal MEP gves a retrospectve pont o vew. t analyzes the dstance between the true R and the observable R. t may or example answer the queston whether the true R could also be postve ( we would now the true L actors) when we have an observable R gven by $ (see Table ). Hence the retrospectve vew separates pure randomness (process varance) rom parameter estmaton uncertantes. n order to quanty the condtonal MEP s we need an estmator or the varance parameters. An unbased estmate or s gven by (see Lemma 3.5 n [9]) 0. (3.3) asualty Actuaral ocety EForum Fall
14 Modellng The lams evelopment Result For olvency Purposes 3. ngle accdent year n ths secton we gve estmators or the two condtonal MEP s dened n (3.)(3.). For ther dervaton we reer to the appendx. We dene Δ (3.4) Φ (3.5) Ψ (3.6) and Φ Ψ Φ Γ. (3.7) We are now ready to gve estmators or all the error terms. Frst o all the varance o the true R gven n (.8) s estmated by R ar V Ψ. (3.8) The estmator or the condtonal MEP s are then gven by: Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety EForum Fall
15 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes Result 3. (ondtonal ME estmator or a sngle accdent year) We estmate the condtonal MEP s gven n (3.)(3.) by R ( 0) ( ) ( ) Γ Δ (3.9) R ( ) ( Φ Δ ) R ( ). (3.0) Ths mmedately mples that we have ( 0) R ( ) R Var R ( ) ( R ) ( ) R R. (3.) Note that ths s ntutvely clear snce the true and the observable R should move nto the same drecton accordng to the observatons n accountng year ( ]. However the rst lne n (3.) s slghtly msleadng. Note that we have derved estmators whch gve an equalty on the rst lne o (3.). However ths equalty holds true only or our estmators where we neglect uncertantes n hgher order terms. Note as already mentoned or typcal real data examples hgher order terms are o neglgble order whch means that we get an approxmate equalty on the rst lne o (3.) (see also dervaton n (A.)). Ths s smlar to the ndngs presented n hapter 3 o [9]. 3. Aggregaton over pror accdent years When aggregatng over pror accdent years one has to tae nto account the correlatons between derent accdent years snce the same observatons are used to estmate the L actors and are then appled to derent accdent years (see also ecton 3..4 n [9]). Based on the denton o the condtonal MEP or the true aggregate R around the aggregated observable R the ollowng estmator s obtaned: asualty Actuaral ocety EForum Fall
16 Modellng The lams evelopment Result For olvency Purposes Result 3. (ondtonal MEP or aggregated accdent years part ) For aggregated accdent years we obtan the ollowng estmator R R sep m (3.) > > Λ Φ R R 0 wth Λ. (3.3) For the condtonal MEP o the aggregated observable R around 0 we need an addtonal denton. Φ Φ Ξ. (3.4) Result 3.3 (ondtonal MEP or aggregated accdent years part ) For aggregated accdent years we obtan the ollowng estmator 0 R sep m (3.5) > > Λ Ξ R 0 0. Note that (3.5) can be rewrtten as ollows: Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety EForum Fall
17 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes ( 0) m sep (3.6) R ( ) R ( ) R ( ) Var ( R ( ) ) > > 0 ( ) R R ( ). ( Ξ Φ ) Hence we obtan the same decouplng or aggregated accdent years as or sngle accdent years. Remars 3.4 (omparson to the classcal Mac [7] ormula) n Results we have obtaned a natural splt nto process varance and estmaton error. However ths splt has no longer ths clear dstncton as t appears. The reason s that the process varance also nluences the volatlty o and hence s part o the estmaton error. n other approaches one may obtan other splts e.g. n the credblty chan ladder method (see Bühlmann et al. [3]) one obtans a derent splt. Thereore we mody Results whch leads to a ormula that gves nterpretatons n terms o the classcal Mac [7] ormula see also (4.)(4.3) below. Result 3.5 asualty Actuaral ocety EForum Fall
18 Modellng The lams evelopment Result For olvency Purposes For sngle accdent years we obtan rom Result 3. R 0 Δ Γ (3.7). / / / For aggregated accdent years we obtan rom Result 3.3 R R ) ( (0) 0 (3.8). / / 0 > > We compare ths now to the classcal Mac [7] ormula. For sngle accdent years the condtonal MEP o the predctor or the ultmate clam s gven n Theorem 3 n Mac [7] (see also Estmator 3. n [9]). We see rom (3.7) that the condtonal MEP o the R consders only the rst term o the process varance o the classcal Mac [7] ormula and or the estmaton error the next dagonal s ully consdered ) ( but all remanng runo cells ) ( are scaled by /. For aggregated accdent years the condtonal MEP o the predctor or the ultmate clam s gven on page 0 n Mac [7] (see also Estmator 3.6 n [9]). We see rom (3.8) that the condtonal MEP o the R or aggregated accdent years consders the estmaton error or the next accountng year ) ( and all other accountng years ) ( are scaled by /. Hence we have obtaned a derent splt that allows or easy nterpretatons n terms o the Mac [7] ormula. However note that these nterpretatons only hold true or lnear approxmatons (A.) otherwse the pcture s more nvolved. Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety EForum Fall
19 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes 4. NUMERAL EXAMPLE AN ONLUON For our numercal example we use the dataset gven n Table. The table contans cumulatve payments or accdent years { 0 K8} at tme 8 and at tme 9. Hence ths allows or an explctly calculaton o the observable clams development result Table : Runo trangle (cumulatve payments n $ 000) or tme 8 and 9 asualty Actuaral ocety EForum Fall
20 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes Table summarzes the L estmates and o the agetoage actors as well as the varance estmates or 0 K 7. nce we do not have enough data to estmate Usng the estmates clams labltes 7 (recall and R at tme ) we use the extrapolaton gven n Mac [7]: 4 { }. (4.) 7 mn we calculate the clams reserves R or the outstandng t and X R or X R at tme t respectvely. Ths then gves realzatons o the observable R or sngle accdent years and or aggregated accdent years (see Table 3). Observe that we have a negatve observable aggregate R at tme o about $ (whch corresponds to poston c) n the P&L statement n Table ). R X R R ( ) Total Table 3: Realzaton o the observable R at tme t n $ 000 The queston whch we now have s whether the true aggregate R could also be postve we had nown the true L actors at tme t (retrospectve vew). We thereore asualty Actuaral ocety EForum Fall
21 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes perorm the varance and MEP analyss usng the results o ecton 3. Table 4 provdes the estmates or sngle and aggregated accdent years. On the other hand we would le to now how ths observaton o $ corresponds to the predcton uncertanty n the budget values where we have predcted that the R s $ 0 (see poston c) n Table ). Ths s the prospectve (solvency) vew. We observe that the estmated standard devaton o the true aggregate R s equal to $ 65 4 whch means that t s not unlely to have the true aggregate R n the range o about $ ± Moreover we see that the square root o the estmate or the MEP between true and observable R s o sze $ (see Table 4) ths means that t s lely that the true R has the same sgn as the observable R whch s $ Thereore also the nowledge o the true L actors would probably have led to a negatve clams development experence. Moreover note that the predcton 0 n the budget values has a predcton uncertanty relatve to the observable R o $ whch means that t s not unlely to have an observable R o $ n other words the solvency captal/rs margn or the R should drectly be related to ths value o $ R V ar ( R) m sep R R cov Total Table 4: Volatltes o the estmates n $ 000 wth: Mac asualty Actuaral ocety EForum Fall
22 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes R estmated reserves at tme t c. (.) V ar estmated std. dev. o the true R c. (3.8) ( R) estmated R (3.0) and (3.) between true and observable R c. m predcton std. dev. o 0 compared to R ( ) sep R 0 Mac c. (3.9) and (3.5) o the ultmate clam c. Mac [7] and (4.3) Note that we only consder the oneyear uncertanty o the clams reserves runo. Ths s exactly the short term vew/pcture that should loo ne to get to the long term. n order to treat the ull runo one can then add or example a costocaptal margn to the remanng runo whch ensures that the uture solvency captal can be nanced. We emphasze that t s mportant to add a margn whch ensures the smooth runo o the whole labltes ater the next accountng year. Fnally these results are compared to the classcal Mac ormula [7] or the estmate o the condtonal MEP o the ultmate clam by n the dstrbutonree L model. The Mac ormula [7] gves the total uncertanty o the ull runo (long term vew) whch estmates Mac ( ) E (4.) and Mac E (4.3) see also Estmator 3.6 n [9]. Notce that the normaton n the next accountng year (dagonal ) contrbutes substantally to the total uncertanty o the total ultmate clam over pror accdent years. That s the uncertanty n the next accountng year s $ and asualty Actuaral ocety EForum Fall
23 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes the total uncertanty s $ Note that we have chosen a shorttaled lne o busness so t s clear that a lot o uncertanty s already contaned n the next accountng year. Generally speang the porton o uncertanty whch s already contaned n the next accountng year s larger or shorttaled busness than or longtaled busness snce n longtaled busness the adverse movements n the clams reserves emerge slowly over many years. one chooses longtaled lnes o busness then the oneyear rs s about /3 o the ull runo rs. Ths observaton s nlne wth a European eld study n derent companes see AAMAME []. APPENX A. PROOF AN ERVATON Assume that a are postve constants wth >> a then we have ( ) a a (A.) where the rghthand sde s a lower bound or the lethand sde. Usng the above ormula we wll approxmate all product terms rom our prevous wor [0] by summatons. ervaton o Result 3.. We rst gve the dervaton o Result 3. or a sngle accdent year. Note that the term Δ s gven n ormula (3.0) o [0]. Henceorth there remans to derve the terms Φ and Γ. For the term Φ we obtan rom ormula (3.9) n [0] ( ) ( ) asualty Actuaral ocety EForum Fall
24 Modellng The lams evelopment Result For olvency Purposes (A.) Φ where the approxmatons are accurate because >> or typcal clams reservng data. For the term Γ we obtan rom (3.6) n [0] (A.3) Γ Φ Ψ. Henceorth Result 3. s obtaned rom (3.8) (3.4) and (3.5) n [0]. ervatons o Results 3. and 3.3. We now turn to Result 3.. All that remans to derve are the correlaton terms. We start wth the dervaton o Λ (ths ders rom the calculaton n [6]). From (4.4) (4.5) n [6] we see that or < the cross covarance term o the estmaton error [ ] [ ] R E R E s estmated by resampled values gven whch mples Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety EForum Fall
25 Modellng The lams evelopment Result For olvency Purposes E (A.4) E E E E. Note that the last two lnes der rom (4.5) n [6]. Ths last expresson s now equal to (see also ecton 4.. n [6]). Next we use (A.) so we see that the last lne can be approxmated by Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety EForum Fall
26 Modellng The lams evelopment Result For olvency Purposes. Next we note that hence ths last term s equal to. Hence pluggng n the estmators or and at tme yelds the clam. Hence there remans to calculate the second term n Result 3.. From (3.3) n [0] we agan obtan the clam usng that >> or typcal clams reservng data. o there remans to derve Result 3.3. The proo s completely analogous the term contanng Λ was obtaned above. The term Ξ s obtaned rom (3.7) n [0] analogous to (A.3). Ths completes the dervatons. 5. REFERENE [] AAMAME (007). AAMAME study on nonle long tal labltes. Reserve rs and rs margn assessment under olvency. October [] Böhm H. Glaab H. (006). Modellerung des KalenderahrRsos m addtven und multplatven chadenreserverungsmodell. Tal presented at the German ATN olloquum. [3] Bühlmann H. e Felce M. Gsler A. Morcon F. Wüthrch M.V. (008). Recursve credblty ormula or chan ladder actors and the clams development result. Preprnt ETH Zurch. [4] e Felce M. Morcon F. (006). Process error and estmaton error o yearend reserve estmaton n the dstrbuton ree chanladder model. Ale Worng Paper Rome November 006. Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety EForum Fall
Analysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
More informationCostofCapital Margin for a General Insurance Liability Runoff
CostofCaptal Margn for a General Insurance Lablty Runoff Robert Salzmann and Maro V Wüthrch Abstract Under new solvency regulatons, general nsurance companes need to calculate a rsk margn to cover possble
More informationPrediction of Disability Frequencies in Life Insurance
Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng Fran Weber Maro V. Wüthrch October 28, 2011 Abstract For the predcton of dsablty frequences, not only the observed, but also the ncurred but
More informationPrediction of Disability Frequencies in Life Insurance
1 Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng 1, Fran Weber 1, Maro V. Wüthrch 2 Abstract: For the predcton of dsablty frequences, not only the observed, but also the ncurred but not yet
More informationIstituto Italiano degli Attuari Riunione di Seminario Attuariale. A Collective Risk Model for Claims Reserve Distribution
Isttuto Italano degl Attuar Runone d Semnaro Attuarale Unverstà Cattolca del Sacro Cuore Mlano, 12 Maggo 2011 A Collectve Rsk Model for Clams Reserve Dstrbuton no Savell Full Professor of Rsk Theory Catholc
More informationStochastic Claims Reserving under Consideration of Various Different Sources of Information
Stochastc Clams Reservng under Consderaton of Varous Dfferent Sources of Informaton Dssertaton Zur Erlangung der Würde des Dotors der Wrtschaftswssenschaften der Unverstät Hamburg vorgelegt von Sebastan
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationStress test for measuring insurance risks in nonlife insurance
PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n nonlfe nsurance Summary Ths memo descrbes stress testng of nsurance
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationUnderwriting Risk. Glenn Meyers. Insurance Services Office, Inc.
Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether
More informationTrafficlight a stress test for life insurance provisions
MEMORANDUM Date 006097 Authors Bengt von Bahr, Göran Ronge Traffclght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationTrafficlight extended with stress test for insurance and expense risks in life insurance
PROMEMORIA Datum 0 July 007 FI Dnr 07117130 Fnansnspetonen Författare Bengt von Bahr, Göran Ronge Traffclght extended wth stress test for nsurance and expense rss n lfe nsurance Summary Ths memorandum
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationThe CoxRossRubinstein Option Pricing Model
Fnance 400 A. Penat  G. Pennacc Te CoxRossRubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te noarbtrage
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationAnalysis of the provisions for claims outstanding for nonlife insurance based on the runoff triangles
OFFE OF THE NSURANE AND PENSON FUNDS SUPERVSORY OMMSSON Analyss of the provsons for clams outstandng for nonlfe nsurance based on the runoff trangles Ths Report has been prepared n the nformaton Systems
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationA Model of Private Equity Fund Compensation
A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationENTERPRISE RISK MANAGEMENT IN INSURANCE GROUPS: MEASURING RISK CONCENTRATION AND DEFAULT RISK
ETERPRISE RISK MAAGEMET I ISURACE GROUPS: MEASURIG RISK COCETRATIO AD DEFAULT RISK ADIE GATZERT HATO SCHMEISER STEFA SCHUCKMA WORKIG PAPERS O RISK MAAGEMET AD ISURACE O. 35 EDITED BY HATO SCHMEISER CHAIR
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More informationScale Dependence of Overconfidence in Stock Market Volatility Forecasts
Scale Dependence of Overconfdence n Stoc Maret Volatlty Forecasts Marus Glaser, Thomas Langer, Jens Reynders, Martn Weber* June 7, 007 Abstract In ths study, we analyze whether volatlty forecasts (judgmental
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationEstimation of Dispersion Parameters in GLMs with and without Random Effects
Mathematcal Statstcs Stockholm Unversty Estmaton of Dsperson Parameters n GLMs wth and wthout Random Effects Meng Ruoyan Examensarbete 2004:5 Postal address: Mathematcal Statstcs Dept. of Mathematcs Stockholm
More informationRisk Model of LongTerm Production Scheduling in Open Pit Gold Mining
Rsk Model of LongTerm Producton Schedulng n Open Pt Gold Mnng R Halatchev 1 and P Lever 2 ABSTRACT Open pt gold mnng s an mportant sector of the Australan mnng ndustry. It uses large amounts of nvestments,
More informationSTAMP DUTY ON SHARES AND ITS EFFECT ON SHARE PRICES
STAMP UTY ON SHARES AN ITS EFFECT ON SHARE PRICES Steve Bond Mke Hawkns Alexander Klemm THE INSTITUTE FOR FISCAL STUIES WP04/11 STAMP UTY ON SHARES AN ITS EFFECT ON SHARE PRICES Steve Bond (IFS and Unversty
More informationExperiment 5 Elastic and Inelastic Collisions
PHY191 Experment 5: Elastc and Inelastc Collsons 8/1/014 Page 1 Experment 5 Elastc and Inelastc Collsons Readng: Bauer&Westall: Chapter 7 (and 8, or center o mass deas) as needed 1. Goals 1. Study momentum
More informationMultiplePeriod Attribution: Residuals and Compounding
MultplePerod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens
More informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 200502 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 148537801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More informationStatistical Methods to Develop Rating Models
Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and
More informationModelling the Claims Development Result for Solvency Purposes
Modelling the Claims Development Result for Solvency Purposes Michael Merz, Mario V. Wüthrich Version: June 10, 008 Abstract We assume that the claims liability process satisfies the distributionfree
More informationGeneralized Linear Models for Traffic Annuity Claims, with Application to Claims Reserving
Mathematcal Statstcs Stockholm Unversty Generalzed Lnear Models for Traffc Annuty Clams, wth Applcaton to Clams Reservng Patrca Mera Benner Examensarbete 2010:2 Postal address: Mathematcal Statstcs Dept.
More informationDscreteTme Approxmatons of the HolmstromMlgrom BrownanMoton Model of Intertemporal Incentve Provson 1 Martn Hellwg Unversty of Mannhem Klaus M. Schmdt Unversty of Munch and CEPR Ths verson: May 5, 1998
More informationECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management
ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C Whte Emerson Process Management Abstract Energy prces have exhbted sgnfcant volatlty n recent years. For example, natural gas prces
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationRisk Measurement and Management of Operational Risk in Insurance Companies from an Enterprise Perspective
FRIEDRICHALEXANDER UNIVERSITÄT ERLANGENNÜRNBERG RECHTS UND WIRTSCHAFTS WISSENSCHAFTLICHE FAKULTÄT Rsk Measurement and Management of Operatonal Rsk n Insurance Companes from an Enterprse Perspectve
More informationTHE USE OF RISK ADJUSTED CAPITAL TO SUPPORT BUSINESS DECISIONMAKING
THE USE OF RISK ADJUSTED CAPITAL TO SUPPORT BUSINESS DECISIONMAKING By Gary Patrk Stefan Bernegger Marcel Beat Rüegg Swss Rensurance Company Casualty Actuaral Socety and Casualty Actuares n Rensurance
More informationVasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio
Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of
More informationAn Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services
An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsnyng Wu b a Professor (Management Scence), Natonal Chao
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More informationThe impact of hard discount control mechanism on the discount volatility of UK closedend funds
Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closedend funds Abstract The mpact
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationThe Current Employment Statistics (CES) survey,
Busness Brths and Deaths Impact of busness brths and deaths n the payroll survey The CES probabltybased sample redesgn accounts for most busness brth employment through the mputaton of busness deaths,
More informationSorting Online Reviews by Usefulness Based on the VIKOR Method
Assocaton or Inormaton Systems AIS Electronc Lbrary (AISeL) Eleventh Wuhan Internatonal Conerence on e Busness Wuhan Internatonal Conerence on ebusness 5262012 Sortng Onlne Revews by Useulness Based
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationEstimating Total Claim Size in the Auto Insurance Industry: a Comparison between Tweedie and ZeroAdjusted Inverse Gaussian Distribution
Estmatng otal Clam Sze n the Auto Insurance Industry: a Comparson between weede and ZeroAdjusted Inverse Gaussan Dstrbuton Autora: Adrana Bruscato Bortoluzzo, Italo De Paula Franca, Marco Antono Leonel
More informationThe Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15
The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationMethods for Calculating Life Insurance Rates
World Appled Scences Journal 5 (4): 653663, 03 ISSN 88495 IDOSI Pulcatons, 03 DOI: 0.589/dos.wasj.03.5.04.338 Methods for Calculatng Lfe Insurance Rates Madna Movsarovna Magomadova Chechen State Unversty,
More informationPragmatic Insurance Option Pricing
Paper to be presented at the XXXVth ASTIN Colloquum, Bergen, 6 9th June 004 Pragmatc Insurance Opton Prcng by Jon Holtan If P&C Insurance Company Ltd Oslo, Norway Emal: jon.holtan@f.no Telephone: +47960065
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationCourse outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy
Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT  We show that the well known euvalence between the "fundamental theorem of
More informationFinancial Mathemetics
Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationLoss analysis of a life insurance company applying discretetime riskminimizing hedging strategies
Insurance: Mathematcs and Economcs 42 2008 1035 1049 www.elsever.com/locate/me Loss analyss of a lfe nsurance company applyng dscretetme rskmnmzng hedgng strateges An Chen Netspar, he Netherlands Department
More informationA Critical Note on MCEV Calculations Used in the Life Insurance Industry
A Crtcal Note on MCEV Calculatons Used n the Lfe Insurance Industry Faban Suarez 1 and Steven Vanduffel 2 Abstract. Snce the begnnng of the development of the socalled embedded value methodology, actuares
More informationOLA HÖSSJER, BENGT ERIKSSON, KAJSA JÄRNMALM AND ESBJÖRN OHLSSON ABSTRACT
ASSESSING INDIVIDUAL UNEXPLAINED VARIATION IN NONLIFE INSURANCE BY OLA HÖSSJER, BENGT ERIKSSON, KAJSA JÄRNMALM AND ESBJÖRN OHLSSON ABSTRACT We consder varaton of observed clam frequences n nonlfe nsurance,
More informationSimon Acomb NAG Financial Mathematics Day
1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Comovement between the ponts Postve
More informationTransition Matrix Models of Consumer Credit Ratings
Transton Matrx Models of Consumer Credt Ratngs Abstract Although the corporate credt rsk lterature has many studes modellng the change n the credt rsk of corporate bonds over tme, there s far less analyss
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationEstimating Total Claim Size in the Auto Insurance Industry: a Comparison between Tweedie and ZeroAdjusted Inverse Gaussian Distribution
Avalable onlne at http:// BAR, Curtba, v. 8, n. 1, art. 3, pp. 3747, Jan./Mar. 2011 Estmatng Total Clam Sze n the Auto Insurance Industry: a Comparson between Tweede and ZeroAdjusted Inverse Gaussan
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationStochastic epidemic models revisited: Analysis of some continuous performance measures
Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,
More information