Stochastic Claims Reserving under Consideration of Various Different Sources of Information


 Chrystal Nichols
 1 years ago
 Views:
Transcription
1 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 Happ geb. am n Tübngen Hamburg, Jul 2014
2 Vorstzender: Prof. Dr. Bernhard Arnold (Unverstät Hamburg) Erstgutachter (Supervsor): Prof. Dr. Mchael Merz (Unverstät Hamburg) Zwetgutachter (CoSupervsor): Prof. Dr. Maro V. Wüthrch (ETH Zürch) Datum der Dsputaton:
3 Acnowledgements Durng my dploma studes n mathematcs and busness admnstraton at the Eberhard Karls Unverstät Tübngen Prof. Dr. Mchael Merz started teachng at unversty as an assstant professor at the department of busness admnstraton. Ths gave me the chance to attend hs lecture seres on selected topcs n actuaral scence. Vstng these lectures I got a fundamental nsght n practcal problems n quanttatve rs management and nsurance and how they can be approached by mathematcal statstcal concepts. Ths awaened my deep nterest n actuaral scence. At ths pont I would le to express my deepest grattude to my supervsor Mchael Merz for gvng me the chance to pursue a PhD at the faculty of busness admnstraton n Hamburg. Not only dd he support me n scentfc ssues but also n personal matters. I am deeply grateful to my cosupervsor Prof. Dr. Maro V. Wüthrch from ETH Zürch for hs great support and nvaluable advce n our jont contrbutons. He permanently supported me wth hs vast nowledge and experence n actuaral scence. Moreover, I would le to than my coauthor René Dahms for hs valuable collaboraton. My thans also go to the whole team, namely T. Gummersbach, J. Heberle, NhaNgh Huynh, A. Johannssen, A. RuzMerno and A. Thomas at the char of mathematcs and statstcs n busness admnstraton at Unverstät Hamburg under the admnstraton and supervson of Mchael Merz for ts support. I would le to than Marco Bretg for hs valuable dscussons. Fnally, I than my wfe Svetlana for her confdence and support n all those years. Sebastan Happ
4
5 Contents 1 Introducton 1 2 Reservng Problem Insurance Contracts and Process of Clams Settlement Data Bass n a Nonlfe Insurance Company Classcal Vew Extended Vew Predcton Problem Inflaton Predcton Precson Mean Squared Error of Predcton Clams Development Result Classcal DstrbutonFree Clams Reservng Methods General Notaton Chan Ladder Method Bayes Chan Ladder Method Complementary Loss Rato Method Bornhuetter Ferguson Method Munch Chan Ladder Method (Bayesan) Lnear Stochastc Reservng Methods Lnear Stochastc Reservng Methods Classcal Clams Reservng Methods as LSRMs Parameter Estmaton for LSRMs Predcton of Future Clam Informaton Bayesan Lnear Stochastc Reservng Methods Classcal Bayesan Clams Reservng Methods as Bayesan LSRMs Predcton of Future Clam Informaton Credblty for Lnear Stochastc Reservng Methods I
6 II Contents Mean Squared Error of Predcton Specal Case: Mean Squared Error of Predcton for the Bayes CL Method Clams Development Result Specal Case: Clams Development Result for the Bayes CL Method Example Bayesan LSRM Conclusons PadIncurred Chan Reservng Method Notaton and Model Assumptons Oneyear Clams Development Result Expected Ultmate Clam at Tme J Mean Squared Error of Predcton of the Clams Development Result Sngle Accdent Years Aggregated Accdent Years Example PIC Reservng Method Conclusons PadIncurred Chan Reservng Method wth Dependence Modelng Notaton and Model Assumptons Ultmate Clam Predcton for Known Parameters Θ Estmaton of Parameter Θ Predcton Uncertanty Example PIC Reservng Method wth Dependence Modelng Conclusons Solvency Regulatory Requrements on Reserves MaretValue Margn Solvency Captal Requrements Fnal Regulatory Reserves Smplfcatons for Regulatory Solvency Requrements Example for Regulatory Reserves Conclusons and Outloo 125 Data Sets 133
7 Lst of Fgures 2.1 Generc tme lne of the clams settlement process Classcal vew (extended vew): Generc runoff trapezod of the mth LoB (clam nformaton) for m {1,..., M} and ncremental clams payments (clam nformaton) of accdent year and development year wth + = I Data set D I observable at tme I and data set D I+1 observable at tme I Reserves R I based on D I at tme I, updated reserves R I+1 based on D I+1 at tme I + 1 and the resultng clams development result CDR M,I σfelds (sets of observatons): B,  all clam nformaton n accdent year up to development year, D  all clam nformaton up to development year, D n  all clam nformaton up to accountng year n and D n  the unon of all nformaton n D and D n σfelds (sets of observatons): D  all clam nformaton up to development year, D n  all clam nformaton up to accountng year n and D n  the unon of all nformaton n D and D n Development factors for BUs 1 3 n the classcal LSRM and credblty development factor F 0 I,Cred {0,..., 10} for BU Cumulatve clams payments P,j and ncurred losses I,j observed at tme t = J both leadng to the ultmate loss P,J = I,J Updated cumulatve clams payments P,j and ncurred losses I,j observed at tme t = J Emprcal densty for the oneyear CDR (blue lne) from smulatons and ftted Gaussan densty wth mean 0 and standard devaton (dotted red lne) QQplot for lower quantles q (0, 0.1) to compare the left tal of the emprcal densty for the oneyear CDR wth the left tal of the ftted Gaussan densty wth mean 0 and standard devaton III
8 IV Lst of Fgures 6.1 Correlaton estmators ˆρ l for ρ l for l {0, 1, 2, 3} as a functon of the number of observatons used for the estmaton Reserves consst of BEL, MVM (together satsfyng accountng condton) and SCR (satsfyng the nsurance contract condton) The calbrated lognormal dstrbuton wth µ = and σ = used as an approxmaton for the dstrbuton of the quantty S21 M + BEL21 and correspondng expected value, VaR and ES for the securty level α = Bestestmate valuaton of labltes BEL 20, maretvalue margn MVM 20 (together satsfyng accountng condton) and solvency captal requrements SCR 20 (satsfyng the nsurance contract condton) leadng to the overall reserves
9 Lst of Tables 2.1 Classcal rs characterstcs: Reserves and CDR and the correspondng frst two moments Reserves and predcton uncertanty Indvdual LoB and overall CDR uncertanty Ultmate clam predcton and predcton uncertanty for the oneyear CDR calculated by the ECLR method for clams payments and ncurred losses (cf. Dahms [16] and Dahms et al. [18]) and by the PIC method, respectvely Ratos msep 1/2 CDR /msep1/2 Ultmate calculated by the ECLR method for clams payments and ncurred losses (cf. Dahms et al. [18]) and calculated by the PIC method, respectvely Lefthand sde: development trangle wth cumulatve clams payments P,j ; rghthand sde: development trangle wth ncurred losses I,j ; both leadng to the same ultmate clam P,J = I,J Uncorrelated case and three explct choces for correlatons Clams reserves n the classcal PIC model and PIC model wth dependence Predcton uncertanty msep 1/2 for the classcal PIC model and the PIC model wth dependence Predcted ncremental clam nformaton for LoB 1, 2 and Expected pattern of BEL for calendar years n = 20,..., Cumulatve clams payments Incurred losses Busness unt Busness unt Busness unt Cumulatve clams payments P,j, + j 21, from a motor thrd party lablty Incurred losses I,j, + j 21, from a motor thrd party lablty V
10
11 1 Introducton Recent developments n (fnancal) marets have shown that unexpected negatve events may have a tremendous mpact on a wde range of fnancal nsttutons such as bans, funds, nvestment and nsurance companes. Often such events are followed by serous problems rangng from economc depresson wth hgh unemployment rates, a decrease n common wealth and bad medcal mantenance to socal rots. Governmental authortes and regulatory nsttutons have been establshed to adopt and develop regulatory framewors for the fnancal ndustry, n order to reduce negatve mpact of such events and to avod collateral damage on other parts of the economy n the future. In Germany the Federal Fnancal Supervsory Authorty (BaFn) supervses bans, fnancal servces provder, nsurance companes as well as securtes tradng. Moreover, n response to the fnancal crss the European Unon (EU) created the European System of Fnancal Supervson, whch conssts of three European Supervsory Authortes: 1. European Banng Authorty (EBA) for the European banng sector 2. European Insurance and Occupatonal Pensons Authorty (EIOPA) for the nsurance sector 3. European Securtes and Marets Authorty (ESMA) for securtes tradng For the banng sector the correspondng regulatory framewor called Basel II was developed by the Basel Commttee on Banng Supervson and s currently replaced by ts successor Basel III. Insurance companes n Europe are controlled by the regulatory framewor called Solvency II. In Swtzerland the regulaton of all fnancal nsttutons ncludng nsurance companes s provded by the Swss Fnancal Marets Authorty (FINMA) wth the correspondng regulaton framewors Basel II and Basel III for the banng sector and the Swss Solvency Test (SST) for the nsurance ndustry. For an nsurance company there are two ways a regulatory framewor can be looed at. a) From the perspectve of nvestors and the management: The functon and the exstence of the company must be mantaned n the md/long term run to generate earnngs for the nvestors and the management. Moreover, these earnngs should be maxmzed 1
12 2 1 Introducton (proft maxmzaton). b) From the perspectve of regulatory authortes: Fnancal lqudty of the nsurance company must be provded even n tmes of extreme fnancal dstress and phases of an extraordnary accumulaton of clam compensaton payments. The ablty of the nsurer to pay losses has to be mantaned n almost all realstc scenaros to prevent losses for the polcyholders and to elmnate wderangng negatve effects on the whole economy. Smlar to Basel II, the Solvency II regulatory framewor s subdvded nto three man pllars to ncorporate the man deas of the regulatory authortes pont of vew: Pllar I: Mnmum Standard and Implementaton Maret consstent valuaton of assets and labltes Internal models, bestestmate reserves, techncal provsons, solvency captal requrements, target captal and own funds Pllar II: Supervsor Revew and Control Group supervson Supervsory revew process Governance Pllar III: Dsclosure Supervsory transparency Accountablty Reportng and dsclosure For detals on the techncal standards, further gudelnes and nformaton, see the Webste of EIOPA 1. For the basc structure of the SST we refer to the Webste of FINMA 2. In ths thess we focus on the the frst pllar. Moreover, one has to dstngush between lfe and nonlfe nsurance busness, snce the contract specfcatons, rs drvers and payoff patterns and hence the methodologc means of approachng and modelng lfe and nonlfe contract labltes dffer substantally. For an llustraton of ths fact we refer to the examples gven n Chapter 7 n Wüthrch Merz [62]. An ntroducton on stochastc models n lfe nsurance can be found n Gerber [26] and Koller [35]. It s crucal to eep n mnd that from now on throughout the thess we wll strctly deal wth nonlfe nsurance busness. The frst pllar n nonlfe nsurance has been subject to many quanttatve scentfc studes, see Wüthrch Merz [63] and [62] for an overvew, snce t s drectly assocated wth the 1 https://eopa.europa.eu/actvtes/nsurance/solvency 2
13 3 problem of the management and quantfcaton of (random) future cash flows. These cash flows typcally arse from assets and clams payments, see Wüthrch Merz [62]. The correspondng feld of study to analyze (random) rs outcomes and assocated loss lablty cash flows n nsurance wth mathematcal and statstcal methods s called actuaral scence. Actuaral scence comprses the followng aspects: 1. Evaluaton of (random) outstandng loss lablty cash flows and settngup of suffcent reserves to meet these labltes 2. Evaluaton of assets and ts assocated rs 3. Level of premums n polces 4. Rensurance 5. Asset lablty management (ALM) comprsng all prevous aspects All stated aspects have an mpact on the process of future cash flows and are therefore crucal for management purposes n nsurance companes. That means that actuaral scence s drectly assocated wth the central problem n nsurance companes of predctng future cash flows. Therefore, the crucal tas and man goal of actuares s the predcton of (random) future cash flows. Among the fve aspects stated above we focus n ths thess on the frst aspect,.e. the feld of predctng future outstandng loss labltes. In actuaral scence ths feld s called clams reservng. Clams reservng belongs to the man tass of a nonlfe actuary, snce clams reserves are the bggest poston on a balance sheet of a nonlfe nsurance company and must therefore be predcted very precsely. Therefore, n ths doctoral thess we wll focus on the tas of predctng future loss labltes and calculatng the correspondng reserves needed to cover these outstandng loss labltes n nonlfe nsurance companes. For ths predcton problem there are often varous sources of nformaton avalable. Most classcal clams reservng methods are very lmted w.r.t. the sources of nformaton they can ncorporate. We present n ths thess two powerful models whch can cope wth several sources of nformaton n a mathematcally consstent way. The frst model generalzes most wdely used dstrbutonfree clams reservng methods. Ths provdes a new perspectve and new possbltes for dstrbutonfree clams modelng and s subject to Part II of ths thess. The second method s an mportant representatve of the class of dstrbutonal clams reservng methods whch can cope wth two dfferent data sources often avalable n nsurance practce. Ths s subject to Part III. The thess s closed up by Part IV dscussng some central aspects of clams reservng under new solvency requrements le Solvency II or SST.
14 4 1 Introducton Outlne Ths thess s dvded nto four parts: Part I: In the frst part (Chapter 2) the classcal clams reservng problem s ntroduced. We consder the assocated general predcton problem and pont out whch data sources have been used n classcal as well as n stateoftheart clams reservng methods for the predcton of future loss labltes. Moreover we show how the ncorporated predcton uncertanty s classcally quantfed n long term as well as n short term rs consderatons. Part II: In the second part (Chapters 3 and 4) we brefly present wdely used classcal clams reservng methods. Followng Dahms [17] and Dahms Happ [15] all these methods are then merged n a general stateoftheart dstrbutonfree clams reservng framewor n Chapter 4. Ths model framewor comprses almost all dstrbutonfree clams reservng methods. Moreover, t allows for the ncorporaton of varous sources of nformaton for the predcton process and hence provdes a new perspectve and possbltes of dstrbutonfree clams reservng. Part III: In contrary to Part II ths part s subject to dstrbutonal clams reservng. In the model class of dstrbutonal clams reservng methods we consder n Chapters 5 and 6 an mportant representatve, the padncurred chan (PIC) reservng method presented n Merz Wüthrch [46]. Followng Happ et al. [30] and Happ Wüthrch [31] we consder for ths method the quantfcaton of the oneyear reservng rs and generalze the classcal PIC method so that dependence structures n the data can be approprately captured. Moreover, the whole predctve dstrbuton of the clams development result s derved va MonteCarlo (MC) methods. Part IV: In ths part (Chapter 7) we pont out central regulatory requrements ncluded n recent solvency framewors le SST or Solvency II. These solvency requrements are not coherent wth most classcal clams reservng methods. We pont out smplfcaton methods proposed n the SST and show how they mae most clams reservng methods accessble for these solvency requrements. We close up ths part by presentng an example where reserves are calculated regardng the SST reservng requrements.
15 2 Reservng Problem 2.1 Insurance Contracts and Process of Clams Settlement An nsurance contract s an agreement of two partes: For a fxed payment (nsurance premum) the nsurer (nsurance company) oblges to pay a fnancal compensaton to the nsured (polcyholder) n the case of an occurrence of some well defned (random) future event n a well specfed tme perod. In the case of such an event at a certan date (occurrence date) durng the nsured perod, the nsured person reports the clam to the nsurance company at the socalled reportng (notfcaton) date. The tme between the occurrence and the reportng date s called reportng delay. After the reportng of the clam the nsurance company verfes whether all nsurance contract specfcatons are fulflled so that the nsurer has to provde coverage of the clam. If ths s the case, the nsurance company starts payments for the fnancal compensaton of the clam n accordance to the contract specfcatons. Ths clams settlement process typcally conssts of one or more payments to the polcyholder. It ends wth the closure date where no further clams payments are expected and the clam s (presumably) completely settled and closed. The tme lne of typcal nonlfe nsurance clams from occurrence to the fnal settlement s llustrated n Fgure 2.1. Tme delays from occurrence to notfcaton and from settlement process to the premum nsured perod occurrence notfcaton closure clams payments Fgure 2.1: Generc tme lne of the clams settlement process tme closure date are typcal for nonlfe nsurance clams and can be caused by dfferent reasons: Delays when ncurred clam events are not mmedately reported to the nsurance company Fnal clam amounts are determned over a long perod of tme (up to several decades) Jurdcal nspecton of a clam. The lablty of the nsurance company to pay for the clam 5
16 6 2 Reservng Problem s to be determned Court decsons leadng to payment adjustments, reverse transactons of already pad compensatons or addtonal clams payments These tme delays often lead to a very slow clams settlement process wth clams payments far n the future (up to several decades). Ths shows that the very nature of nsurance busness (.e. underwrtng rss through nsurance contracts) often causes a very slow settlement process and the predcton of ths process becomes a central pont of nterest. For a more detaled dscusson on that topc, see Wüthrch Merz [63]. General Remar: In nonlfe nsurance busness many clam characterstcs (occurrence date, frequency of clams, severty of a clam, clam settlement pattern, clams payments, etc.) are subject to randomness and can not be predcted wthout uncertanty. Hence, probablty theory and statstcs provde sutable mathematcal tools for dealng wth those clam characterstcs. Thereby, t s assumed that the very nature and the behavor of these clam characterstcs do not change too fast over tme. Ths assumpton s requred to utlze past observatons for predctng purposes and to reveal systematc propertes (behavor) of the quanttes under consderaton. For ths reason we model all quanttes of nterest n a stochastc framewor as random varables, whch are defned on a common probablty space (Ω, D, P). 2.2 Data Bass n a Nonlfe Insurance Company In general, nsurance companes group polces (nsurance contracts) wth smlar rs characterstcs or comparable contract specfcatons nto suffcently homogeneous nsurance portfolos. Ths s often done by Lnes of Busness (LoB), but can be subdvded further nto smaller unts. Typcal LoBs are: Motor thrd party, product lablty, prvate and commercal property, commercal lablty, health nsurance, etc. An nsurance company has to put provsons asde, n order to cover future loss labltes arsng from these grouped nsurance portfolos. For ths reason an accurate predcton of future loss labltes and the assocated cash flows n the clam settlement process s of central nterest. Ths predcton can be based on varous sources of nformaton Classcal Vew In the classcal vew the predcton of future loss labltes s often based on the nformaton of the past observed development of the settlement process tself. Classcal clams reservng
17 2.2 Data Bass n a Nonlfe Insurance Company 7 lterature often assumes that an nsurance company has, after groupng of ndvdual contracts, M 1 nearly homogeneous portfolos. All clams, whch occur n year, are called clams n accdent year {0,..., I}, where I s the current year. The number of years between accdent year and the year of the actual clams payment s called development year {0,..., J}, wth J beng the total number of development years. It s usually assumed that I J and that all clams are completely settled n development year J,.e. there are no clams payments beyond development year J. For models consderng clams payments beyond development year J by means of socalled tal factors, see Mac [40] and Merz Wüthrch [42]. We denote all payments for accdent year and development year n the mth portfolo (m {1,..., M}) by S, m and say that all clams payments Sm, wth + = n and n {0,..., I + J} belong to accountng year n. Ths notaton s called ncremental clams representaton n the actuaral lterature, because we consder clams payments S, m n accdent year and development year of the mth portfolo. In the actuaral lterature (cf. Wüthrch Merz [63]) the cumulatve clams payments representaton of the clam settlement process s also used. In ths representaton one consders cumulated amounts n accdent year up to development year defned by C m, := S,j, m (2.1) j=0 where all clams payments whch belong to accdent year up to development year n the mth portfolo are aggregated. At tme n {0,..., I + J} all clams payments S, m wth + n and 1 m M are observed and generate the σfeld D n := σ { S, m + n, 0 I, 0 J, 1 m M } = σ { C, m + n, 0 I, 0 J, 1 m M }. Moreover, we denote the resultng fltraton by D := (D n ) 0 n I+J leadng to the probablty space wth fltraton (Ω, D, D, P). The two representatons (ncremental or cumulatve representaton) are commonly used n the clams reservng lterature, and t manly depends on the model choce whether the ncremental or the cumulatve representaton s used. The settlement process of the mth portfolo n the ncremental as well as n the cumulatve clams payments representaton s llustrated n clams development (runoff) trapezods where accdent years {0,..., I} and development years {0,..., J} are gven by the rows and the columns, respectvely. Ths means the ncremental clams payments n accdent year and development year of the mth portfolo are postoned n the th row and the th column n the mth development trapezod, see Fgure 2.2. (2.2) We wll see n Chapter 4 that the ncremental clams payments representaton s an approprate choce for almost all dstrbutonfree clams reservng methods. Moreover, the ncremental representaton s advantageous f one s nterested n the valuaton of outstandng loss labl
18 8 2 Reservng Problem development years 1 m J 0 1 accdent years... D I S m, I... to be predcted Fgure 2.2: Classcal vew (extended vew): Generc runoff trapezod of the mth LoB (clam nformaton) for m {1,..., M} and ncremental clams payments (clam nformaton) of accdent year and development year wth + = I tes va valuaton portfolos, see Wüthrch Merz [62]. However, we swtch to the cumulated clams payments representaton, f helpful (Chapters 5 and 6) Extended Vew Besde the clam settlement process data there are often other sources of nformaton avalable for the predcton of loss labltes: Settlement processes of other correlated portfolos Data of collectves whch may nfluence the settlement process under consderaton Incurred losses: Clams payments plus ndvdual case dependent loss reserves Pror ultmate clam estmates: Ths nformaton may nclude prcng arguments Insured volume Number and sze of contracts etc. Recent publcatons n actuaral scence consder new models whch allow for ncludng some of these sources of nformaton n a mathematcally consstent way, see for example Dahms [17] and Merz Wüthrch [46]. In these models S, m and Cm, do not necessarly only correspond
19 2.2 Data Bass n a Nonlfe Insurance Company 9 anymore to ncremental clams payments and cumulatve clams payments (.e. nformaton from the clam settlement process). They may also represent some other sources of nformaton stated above, for example ncurred losses data, see the PIC reservng method n Merz Wüthrch [46], or pror ultmate clam estmates, see the Bornhuetter Ferguson (BF) method n Mac [39]. Therefore, t s necessary to extend the denotaton of S, m of the classcal vew, snce we focus n the actuaral contrbutons of ths thess on such new model classes, see Chapters 4 6. Throughout the thess S, m denotes the mth (m {1,..., M}) clam nformaton of accdent year {0,..., I} and development year {0,..., J} and not necessarly only the clams payments as t s convenent n classcal clams reservng methods. These clam nformaton may besde the clams payments process contan ncurred losses, see Merz Wüthrch [46] and Dahms [16], receved premum and the average loss rato, see Bühlmann [11], pror ultmate clam estmates, see Mac [39] and Arbenz Salzmann [6], clam volume nformaton, see Dahms [17], or other addtonal sources of nformaton. By a slght abuse of notaton we wll call also m {1,..., M} the mth clam nformaton by dentfyng the ndex m wth ts assocated clam nformaton S, m. In the extended vew some clam nformaton S, m do not generate any loss lablty cash flows n the future and thus do not have to be predcted. Therefore, we defne M := { m M S m, generates loss lablty cash flows }. (2.3) By defnton M s the set of clam nformaton whch generate cash flows, see (2.3), and s therefore of central nterest for clams reservng and rs management. Remars 2.1 (Set M) In most classcal clams reservng methods, each clam nformaton m M s gven by the clams payments of an nsurance portfolo of a specfc LoB, see Chapter 3 for examples. However, ths s not always the case. In Example 1 n Dahms [17] there s a clam nformaton m M of subrogaton payments. Ths shows that M may besde the clams payments of dfferent LoBs also contan other clam nformaton whch also generate cash flows. That means that the clam nformaton n M are not explctly restrcted to clams payments of dfferent nsurance portfolos. However, for a smpler nterpretaton of the set M one may thn of each clam nformaton m M as clams payments arsng from an nsurance portfolo of a certan LoB. As a consequence of the defnton of M, the set of all clam nformaton {1,..., M} s dvded nto dsjont subsets M {1,..., M} and M c = {1,..., M}\M. The clam nformaton m M have already been dscussed above. The set M c of clam nformaton s not of central nterest for rs management and clams reservng, because t does not generate any loss lablty cash flows. However, clam nformaton out of M c are utlzed n many models for the predcton of clam nformaton m M under consderaton,.e. they contan nformaton, whch are requred for
20 10 2 Reservng Problem the predcton of clam nformaton m M. To name only a few of them, an ultmate clam estmate (as a clam nformaton m M c ) s ncorporated for the predcton of clams payments (as a clam nformaton m M) n the BF method, see Secton 3.5, ncurred losses are used for the predcton of clams payments n the extended complementary loss rato (ECLR) method, see Dahms [16], and n the PIC reservng method, see Chapters 5 and 6, or volume measures are ncluded for clams payments predctons n the addtve loss reservng (ALR) method n Merz Wüthrch [44]. In analogy to the classcal vew, clam nformaton m {1,..., M} n the extended vew are also llustrated n development (runoff) trapezods, see Fgure 2.2. Notatonal Conventon: Unless otherwse ndcated we wor n ths thess wthn the extended vew,.e. we assume that a set of M 1 clam nformaton (sources of nformaton) s avalable today,.e at tme I. In ths extended vew all clam nformaton m M generate loss lablty cash flows and hence have to be predcted, whereas clam nformaton m M c are used only for the predcton of clam nformaton m M. 2.3 Predcton Problem As mentoned n the prevous secton nsurance companes often have varous sources of nformaton (clam nformaton) for the predcton of future loss labltes cash flows S, m wth m M. We wor n the extended vew,.e. we assume that M 1 clam nformaton m {1,..., M} (as mentoned above we dentfy m by ts correspondng clam nformaton S, m ) are avalable today (at tme I). The set of clam nformaton generatng cash flows s denoted by M {1,..., M}. A reservng actuary has to predct today (at tme I) and at all future tmes up to the fnal runoff,.e. at tmes n {I,..., I + J 1}, the outstandng loss lablty cash flows. These are gven for clam nformaton m M and accdent year {I J + 1,..., I} at tme n {I,..., I + J 1} by (an empty sum s defned by zero) R m n := J j=n +1 S m,j. By summaton of (2.4a) over all clam nformaton of nterest,.e. aggregated outstandng loss labltes of accdent year gven by R n := m M R m n = m M J j=n +1 S m,j (2.4a) m M, we obtan the (2.4b)
benefit 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 informationAnalysis 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationIntrayear Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error
Intrayear Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor
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 informationModelling The Claims Development Result For Solvency Purposes
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
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 informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationHOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA*
HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* Luísa Farnha** 1. INTRODUCTION The rapd growth n Portuguese households ndebtedness n the past few years ncreased the concerns that debt
More informationReturn decomposing of absoluteperformance multiasset class portfolios. Working Paper  Nummer: 16
Return decomposng of absoluteperformance multasset class portfolos Workng Paper  Nummer: 16 2007 by Dr. Stefan J. Illmer und Wolfgang Marty; n: Fnancal Markets and Portfolo Management; March 2007; Volume
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 informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More information1.1 The University may award Higher Doctorate degrees as specified from timetotime in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
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 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 informationTo manage leave, meeting institutional requirements and treating individual staff members fairly and consistently.
Corporate Polces & Procedures Human Resources  Document CPP216 Leave Management Frst Produced: Current Verson: Past Revsons: Revew Cycle: Apples From: 09/09/09 26/10/12 09/09/09 3 years Immedately Authorsaton:
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
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 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 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 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 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 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 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 informationHollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )
February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs
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 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 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 informationReporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide
Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB
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 Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6  The Time Value of Money. The Time Value of Money
Ch. 6  The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21 Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important
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 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 informationLIFETIME INCOME OPTIONS
LIFETIME INCOME OPTIONS May 2011 by: Marca S. Wagner, Esq. The Wagner Law Group A Professonal Corporaton 99 Summer Street, 13 th Floor Boston, MA 02110 Tel: (617) 3575200 Fax: (617) 3575250 www.ersalawyers.com
More informationDI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice
DI Fund Suffcency Evaluaton Methodologcal Recommendatons and DIA Russa Practce Andre G. Melnkov Deputy General Drector DIA Russa THE DEPOSIT INSURANCE CONFERENCE IN THE MENA REGION AMMANJORDAN, 18 20
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 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 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 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 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 informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
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 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 informationAmeriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration
CED0105200808 Amerprse Fnancal Servces, Inc. 70400 Amerprse Fnancal Center Mnneapols, MN 55474 Incomng Account Transfer/Exchange/ Drect Rollover (Qualfed Plans Only) for Amerprse certfcates, Columba mutual
More informationNordea G10 Alpha Carry Index
Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and
More informationTime Value of Money Module
Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the
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 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 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 informationAbstract # 0150399 Working Capital Exposure: A Methodology to Control Economic Performance in Production Environment Projects
Abstract # 0150399 Workng Captal Exposure: A Methodology to Control Economc Performance n Producton Envronment Projects Dego F. Manotas. School of Industral Engneerng and Statstcs, Unversdad del Valle.
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 information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationTesting Adverse Selection Using Frank Copula Approach in Iran Insurance Markets
Journal of mathematcs and computer Scence 5 (05) 5458 Testng Adverse Selecton Usng Frank Copula Approach n Iran Insurance Markets Had Safar Katesar,, Behrouz Fath Vajargah Departmet of Statstcs, Shahd
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 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 informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationMeasuring portfolio loss using approximation methods
Scence Journal of Appled Mathematcs and Statstcs 014; (): 45 Publshed onlne Aprl 0, 014 (http://www.scencepublshnggroup.com/j/sjams) do: 10.11648/j.sjams.01400.11 Measurng portfolo loss usng approxmaton
More informationThe Use of Analytics for Claim Fraud Detection Roosevelt C. Mosley, Jr., FCAS, MAAA Nick Kucera Pinnacle Actuarial Resources Inc.
Paper 18372014 The Use of Analytcs for Clam Fraud Detecton Roosevelt C. Mosley, Jr., FCAS, MAAA Nck Kucera Pnnacle Actuaral Resources Inc., Bloomngton, IL ABSTRACT As t has been wdely reported n the nsurance
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 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 informationA Master Time Value of Money Formula. Floyd Vest
A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes.
More informationCovariatebased pricing of automobile insurance
Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 José Antono Ordaz (Span), María del Carmen Melgar (Span) Covaratebased prcng of automoble nsurance Abstract Ths
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 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 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 informationIMPACT ANALYSIS OF A CELLULAR PHONE
4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng
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 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 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 informationConstruction Rules for Morningstar Canada Target Dividend Index SM
Constructon Rules for Mornngstar Canada Target Dvdend Index SM Mornngstar Methodology Paper October 2014 Verson 1.2 2014 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationOperational risk quantification for loss frequency using fuzzy simulation
COMPUTER MODELLIG & EW TECHOLOGIES 04 8(C) 57758 Lu Shuxa, M Haje Operatonal rs quantfcaton for loss frequency usng fuzzy smulaton Shuxa Lu,*, Haje M School of Busness Admnstraton, Hebe ormal Unversty
More informationSmall pots lump sum payment instruction
For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested
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 informationPerformance attribution for multilayered investment decisions
Performance attrbuton for multlayered nvestment decsons 880 Thrd Avenue 7th Floor Ne Yor, NY 10022 212.866.9200 t 212.866.9201 f qsnvestors.com Inna Oounova Head of Strategc Asset Allocaton Portfolo Management
More informationLatent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006
Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
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 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 informationMultiple discount and forward curves
Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of
More information