The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees


 Bertram Jackson
 2 years ago
 Views:
Transcription
1 1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, Ulm, Germany phone: , fax: Andreas Richer Professor, Chair in Risk & Insurance LudwigMaximilians Universiy Munich, Germany phone: , fax: Jochen Ruß Managing Direcor Insiu für Finanz und Akuarwissenschafen, Ulm, Germany phone: , fax: Absrac This paper analyzes he numerical impac of differen surplus disribuion mechanisms on he risk exposure of a life insurance company selling wih profi life insurance policies wih a cliquesyle ineres rae guaranee. Three represenaive companies are considered, each using a differen ype of surplus disribuion: A mechanism, where he guaraneed ineres rae also applies o surplus ha has been credied in he pas, a slighly less resricive ype in which a guaraneed rae of ineres of applies o pas surplus, and a hird mechanism ha allows for he company o use former surplus in order o compensae for underperformance in bad years. Alhough a ouse all conracs offer he same guaraneed benefi a mauriy, a disribuion mechanism of he hird ype yields preferable resuls wih respec o he considered risk measure. In paricular, hroughou he analysis, our represenaive company 3 faces ceeris paribus a significanly lower shorfall risk han he oher wo companies. Offering srong guaranees pus companies a a significan compeiive disadvanage relaive o insurers providing only he hird ype of surplus disribuion mechanism.
2 2 1. Inroducion Many wih profi life insurance policies conain an ineres rae guaranee. Ofen, his guaranee is given on a poinopoin basis, i.e. he guaranee is only relevan a mauriy of he conrac. Oher producs (which are predominan, e.g., in he German marke), however, offer a socalled cliquesyle (or yearbyyear) guaranee. This means ha he policy holders have an accoun o which every year a cerain rae of reurn has o be credied. Typically, life insurance companies ry o provide he guaraneed rae of ineres plus some surplus on he policy holders accouns. There are differen mechanisms defining how he annual surplus can be disribued o he insured. These mechanisms vary from counry o counry and someimes from insurance company o insurance company. They can be divided in hree differen caegories and combinaions hereof: a) Surplus may be credied o he policy reserves. In his case, i is guaraneed ha his surplus will earn he guaraneed rae of ineres in fuure years. b) Surplus may be credied o a surplus accoun ha is owned by he policy holder and may herefore no be reduced anymore. Thus, here is a guaraneed ineres rae of on money ha is in his surplus accoun. c) Surplus may be credied o a erminal bonus accoun. Money ha has been credied o his accoun will only be disribued o he insured a mauriy of heir conracs bu no (or only parially) if hey cancel he conrac. Furhermore, money may be aken from his accoun in order o pay ineres rae guaranees (on he policy reserves) if in some year he reurn on asses is no sufficien o pay for hese guaranees. I is obvious ha c.p., insurance companies using differen surplus disribuion mechanisms may have a significanly differen risk profile. In he pas, his may have been of minor imporance since here was a comforable margin beween marke ineres raes and he guaraneed raes ha were ypically offered wihin life insurance policies. Recenly, however, hese margins have been significanly reduced, in paricular for conracs ha have been sold years ago wih raher high guaraneed raes. This developmen illusraes ha analyzing and
3 3 managing an insurance company s financial risks should no only be resriced o managemen of he asses bu also be concerned wih reducing risks ha resul from he produc design. A number of papers have recenly addressed ineres rae guaranees, in paricular Briys and de Varenne (1997), Grosen and Jorgensen (2000), Jensen e al. (2001), Hansen and Milersen (2002), Grosen and Jorgensen (2002), Bacinello (2003), Milersen and Persson (2003), Tanskanen and Lukkarinen (2003), Bauer e al. (2006), and Kling, Richer and Russ (2006). Briys and de Varenne (1997) compue closedform soluions for marke values of liabiliies and equiies in a poinopoin guaranee framework. In heir model he policy holder receives a guaraneed ineres and is also credied some bonus, deermined as a cerain fracion of ne financial gains (when posiive). The paper also looks a he impac of ineres rae guaranees on he company s risk exposure by analyzing ineres rae elasiciy and duraion of insurance liabiliies. Conrasing he jusmenioned approach, Grosen and Jorgensen (2000) consider cliquesyle guaranees and inroduce a model ha akes ino accoun an insurer s use of he average ineres principle. In addiion o a policy reserve (he cusomer s accoun) hey inroduce a bonus reserve, a buffer ha can be used o smoohen fuure bonus disribuions. They analyze a mechanism ha credis bonus o he cusomer s reserve based upon he curren raio of bonus reserve over policy reserve. A bonus is paid only if his raio exceeds a given hreshold. Thus, he acual disribuion of surplus indirecly reflecs curren invesmen resuls bu primarily focuses on he company s abiliy o level ou insufficien resuls in he fuure. The auhors decompose he conrac ino a risk free bond, a bonus and a surrender opion. They compue conrac values by means of Mone Carlo simulaion, and also calculae conrac defaul probabiliies for differen parameer combinaions. 1 However, hey calculae defaul probabiliies under he risk neural probabiliy measure Q. Therefore, he numerical resuls are only of limied explanaory value. Grosen and Jorgensen (2002) discuss a model based upon he framework used by Briys and de Varenne (1997). They incorporae a regulaory consrain for he insurer s asses according o which he company is closed down and liquidaed if he marke value of asses drops below a 1 Jensen e al. (2001) exend he findings of Grosen and Jorgensen (2000). As one exension, among ohers, hey inroduce moraliy risk. Anoher paper ha incorporaes moraliy risk as well as he surrender opion is Bacinello (2003).
4 4 hreshold a any poin in ime during he life of he policy. Their resuls sugges ha he inroducion of he regulaory consrain significanly reduces he value of he shareholders defaul pu opion and hereby an insurer s incenive o change is asses risk characerisics o he policy holders disadvanage. Milersen and Persson (2003) also use a cliquesyle framework and allow for a porion of excess ineres o be credied no direcly o he cusomer s accoun bu o a bonus accoun. In heir model, he ineres ha exceeds he guaraneed rae is if posiive divided ino hree porions ha are credied o he insured s accoun, he insurer s accoun, and o a bonus accoun. In case of invesmen reurns below he guaraneed rae, funds are moved from he bonus accoun ino he policy owner s accoun. Thus, he bonus accoun is available for smoohing reurns over ime. Unlike in he Grosen and Jorgensen (2000) model, however, he buffer consiss of funds ha have already been designaed o he paricular cusomer: Any posiive balance on he bonus accoun is credied o he policy owner when he conrac expires. This is used o model socalled erminal bonuses. In his seing, Milersen and Persson (2003) derive numerical resuls on he influence of various parameers on he conrac value. 2 Bauer e al. (2006) invesigae he valuaion of paricipaing conracs under he German regulaory framework. They presen a framework, in which he differen kinds of guaranees or opions incorporaed in paricipaing conracs wih ineres rae guaranees can be analyzed separaely. The pracical implemenaion of wo differen numerical approaches o price he embedded opions is discussed. The auhors find ha life insurers currenly offer ineres rae guaranees below heir riskneural value. Furhermore, he financial srengh of an insurance company considerably affecs he value of a conrac. While he primary focus in he lieraure is on he fair (i.e. riskneural) valuaion of he life insurance conrac, Kling, Richer and Russ (2006) concenrae on he risk a conrac imposes on he insurer, measured by means of shorfall probabiliies under he socalled realworld probabiliy measure P. Assuming cliquesyle guaranees, hey sudy he impac ineres rae guaranees have on he insurer s shorfall probabiliy and how defaul risks depend on charac 2 Conrasing he mechanism discussed in Milersen and Persson (2003), life insurance conracs ofen employ a disribuion policy ha does no accumulae undisribued surplus on an individual basis, bu for a greaer pool of cusomers. A model ha allows for his echnique can be found in Hansen and Milersen (2002).
5 5 erisics of he conrac, on he insurer s reserve siuaion and asse allocaion, on managemen decisions, as well as on regulaory parameers. The presen paper analyzes he numerical impac of differen surplus disribuion mechanisms on he risk exposure of a life insurance company selling wih profi life insurance policies wih an ineres rae guaranee. We employ he model framework inroduced in Kling, Richer and Russ (2006), bu exend he model such ha he differen surplus mechanisms described above and any combinaions can be compared. The model also allows for he comparison of he wo major ypes of ineres rae guaranees: cliquesyle guaranees and poinopoin guaranees, as described above. The focus of our numerical analysis, however, is on he differen surplus disribuion mechanisms. The paper is organized as follows. In Secion 2, we inroduce our model for he insurance company, in paricular a simple asse model and a represenaion of he insurer s liabiliies in a seady sae. Furhermore, we give a deailed descripion of he surplus disribuion mechanisms described above and inroduce shorfall probabiliies as he relevan risk measure for all our analyses. Secion 3 conains a variey of resuls analyzing he differen risk levels of insurers using differen surplus mechanisms as well as he impac of several parameers on hese risk levels. Secion 4 concludes and provides some oulook on possible fuure research. 2. The model framework 2.1 The insurer s financial siuaion In our model, we use a simplified illusraion of he insurer s balance shee. We expand he model from Kling, Richer and Russ (2006) so ha differen surplus disribuion mechanisms can be included: Asses Liabiliies E A L S BB R policy holders accouns reserves
6 6 A A Figure 1: Model of he insurer s financial siuaion Here, A, denoes he marke value of he insurer s asses a ime. The liabiliy side comprises five enries: E is he ime book value of he company s equiy which is assumed o be consan over ime. L is he ime book value of he policy reserves. The insurer guaranees he policy holder an annual ineres rae g on his accoun. Noe ha any surplus ha is credied o his accoun will also have o earn a leas he guaraneed rae in he fuure. S is he ime book value of he policy holders individual surplus accouns. Consisen wih German legislaion, we assume ha he guaraneed rae need no be credied on his accoun bu once surplus is disribued o his accoun i may no be reduced a any ime in he fuure, i.e., he guaraneed rae of ineres on his accoun is p.a. B is he ime book value of he bonus accoun for erminal bonuses. I is owned by he policy holders bu no on an individual basis. Pars of his accoun are paid ou o mauring conracs as a erminal bonus. I may however also be used o provide guaranees for oher accouns in he fuure. R is he reserve accoun which is given by R = A L S B E. I consiss mainly of asse valuaion reserves. Even hough here is an inernaional rend owards fair value accouning, book value accouning will sill be imporan in some counries for several reasons. In Germany, e.g., cerain minimum surplus disribuion rules imposed by he legislaor/regulaor will coninue o be based on book value earnings according o he German Commercial Code. Thus, in order o realisically model hese minimum requiremens, book values of he asses and liabiliies will be relevan even afer he inroducion of inernaional accouning sandards.
7 7 Our model allows for dividend paymens. Whenever dividends D are paid ou o equiy holders, A is reduced by he corresponding amoun. To simplify noaion, we assume ha such paymens occur annually, a imes = 1,2, K,T, where T denoes some finie ime horizon. 2.2 The asse model Similar o he approach in Kling, Richer and Russ (2006), we use a very simple model for he asses: We assume a complee, fricionless and coninuous marke. Beween dividend paymens, we le A follow a geomeric Brownian moion da A = μd σdw, (1) where W denoes a Wiener process on some probabiliy space (Ω,F,P) wih a filraion F, o which W is adaped. Boh, μ and σ are deerminisic and consan over ime. 3 Including dividend paymens D, we ge for = 1,2, K, T 2 2 σ σ μ du σdwu μ σ dw u = 1 = 1 A A e A e 2 and A = A D, where and denoe he asse value a ime jus before and immediaely afer he dividend A A paymen. Analogously, and, and, and, and and denoe he L L B B corresponding values immediaely before and immediaely afer he dividend paymen. The numerical analysis in Secion 3 assumes A o consis of socks and bonds wih s denoing he sock porion of he (coninuously rebalancing) porfolio. 2.3 Insurance Benefis and Guaranees on he Insurance Liabiliies For he sake of simpliciy, our model considers an insurance company in a seady sae : We assume ha conracs corresponding o some consan fracion ξ of he insurer s liabiliies erminae each year due o mauriy, surrender or deah. The company pays ou he corresponding S S R R 3 The model also allows for ime dependen choices of μ and σ.
8 values of he policy holders accouns, i.e. ξ ( L S B ) We assume ha he sum of premiums P 1 colleced a he beginning of year (resuling from new business as well as regular (annual) premium from old conracs) also equals ( L S B ) P ξ. 5 = Since he premiums colleced are added o he policy reserves, he value of he policy holders accouns before guaranee provision and surplus disribuion are given by L ξ, S 1 ξ ) S and B 1 ξ ) B. = ( 1 ) L 1 P 1 = ( 1 = ( 1 The values L, S and B hen depend on amoun and ype of surplus disribuion. By definiion of he differen accouns (see Secion 2.1), we ge he following lower bounds: ( g) L L 1, S S and 0. B The amoun of disribuion o he differen policy holders accouns and o equiy holders each year depends on he earnings on book value as well as decisions made by he company s managemen. Following German legislaion, we assume ha here is a minimum paricipaion rae requiring ha a leas a cerain porion δ of he earnings on book value has o be credied o he policy holders accouns. Earnings on book value are subjec o accouning rules giving insurance companies cerain freedom o creae and dissolve asse valuaion reserves. Following he approach inroduced in Kling, Richer and Russ (2006), we assume ha a leas a porion y of he increase in marke value has o be idenified as earnings in book values in he balance shee. 6 The parameer y herefore represens he degree of resricion in asse valuaion immanen in he relevan 4 We assume ha here are neiher gains nor losses due o moraliy and hus ignore deah benefis ha migh exceed he value of he policy holder s accoun. This means ha he cos of insurance, i.e. he par of he premium ha is charged for he deah benefi, is calculaed wih bes esimae moraliy raes and exacly covers any deah benefis ha exceed he policy holders accouns. 5 Since we ignore deah benefis ha exceed he policy holders accouns, of course P does no include he cos of insurance. 6 This means ha he sum of he increase in book value [( R ) ( A 1 R 1 )] exceed y ( ). A A 1 A and he dividend paymens D has o
9 9 accouning rules. Furhermore, he insurer can reduce reserves in order o increase he book value of asses wihou any resricions by selling asses whose marke value exceeds he book value. 2.4 Surplus disribuion and dividend paymens This secion deals wih he amoun of surplus ha is credied o he policy holders in any given year, whils he nex secion inroduces differen surplus disribuion mechanisms. Surplus ha is disribued o he policy holders accouns and dividends ha are paid o he shareholders are deermined by he insurance company s managemen every year. Our general model allows for any managemen decision rule a ime ha is F measurable. In he numerical analysis, however, we will focus on one decision rule ha seems o prevail in Germany: In he pas, insurance companies used o keep surplus disribuion o policy holders and dividends o shareholders raher consan over years, building and dissolving reserves in order o smoohen reurns. Only when he reserves reached a raher low level, hey sared reducing surplus. Therefore, we apply a decision rule ha considers his: As long as reserves are in a comforable range, some consan arge policy reurn rae is credied o policy holders accouns. If crediing his arge rae would lead o an uncomforably low reserve level, surplus is reduced. If reducing surplus is no sufficien, firs reserves are furher dissolved and hen he bonus accoun is reduced. On he oher hand, if crediing he arge policy reurn rae would yield o a very high level of reserves, surplus is increased above he arge policy reurn rae. The echnical deails of his simple idea are explained in he remainder of his secion: As long as he reserve quoa x R says wihin a given range [ b] = L S B E a;, a arge policy reurn rae z > g is credied o he sum of he policy holders accouns. Furhermore, equiy holders receive a arge dividend rae α of company's equiy. Thus, we ge ( z)( L S B ) PH L S B = 1, and A = A αe 1 i.e. he surplus Su provided o policy holders and he dividend paymens are given by ( L S B ) gl Su and D α E. (2) PH = z = 1 Noe ha a his poin we do no specify, how he surplus is disribued o he accouns L, S and B. This depends on he paricular disribuion mechanisms, ha will be inroduced in he nex secion.
10 10 As long as he reserve quoa remains in [ a; b], his policy is followed. If however crediing z and α o policy holders and shareholders, respecively, would lead o a reserve quoa below a or above b, surplus and dividends from (2) are reduced or increased by muliplying boh wih a consan facor If no such facor c 0 ha leads o a reserve quoa x = a or x = b. exiss, his means ha even paying no surplus and no dividends would lead o a reserve quoa below a. In his case, only he guaraneed rae of ineres is provided o he policy reserves while he surplus and he bonus accoun remain unchanged and no dividends are paid, i.e., ( g) L = c 0 L 1, S = S, B = B and A = A. This is only possible if i resuls in a reserve quoa beween 0 and a. Oherwise, he bonus accoun is reduced by he amoun needed o keep reserves a 0, i.e. we le ( ) S = S L 1, = g L, A = A, and B = A L S E and hus R = 0. Of course, his is only possible if ( ) A 1 g L S E, since oherwise, his would resul in B 0. < If he bonus accoun is no sufficien o provide he guaraneed rae of ineres, i.e. A < ( 1 ) 1 = ( ) g L if g L S E hen L 1, S = S, A = A, and = 0 which leads o negaive reserves. Our model allows for negaive reserves as long as here is enough equiy o back he liabiliies. We speak of a shorfall if here is no enough equiy, see below. Finally, we have o check in each of he cases above, wheher hese rules comply wih he resricion in asse valuaion and he minimum paricipaion rae. The resricion in asse valuaion (see foonoe 6), is violaed if X = y( A A ) ( A R ) ( A R ) D ) 0 BV : > B. In his case, we disribue he exceeding book value X BV increasing he surplus provided o he policy holders byδ X BV and he dividends by ( 1 δ ) X BV. ( ) If δ ( ) ( ) ( ) ( ) ( ) A R A 1 R 1 D L L 1 S S 1 B B 1 > 0, he surplus provided o he policy holders is increased by his amoun in order o fulfill he minimum paricipaion rule. This is achieved by reducing he dividends given o he shareholders accordingly.
11 Surplus disribuion mechanisms Once he amoun of surplus has been deermined according o he managemen decision rule given in Secion 2.4, he surplus disribuion mechanism has o be specified. We will analyze he impac of differen surplus disribuion mechanisms by considering hree differen model companies. We assume ha all companies sar ou wih he same balance shee. In paricular, we assume ha he values of E 0, L 0, S 0 and B0 are he same for each company which means ha in he pas he companies provided surplus in he same manner and only apply he differen mechanisms described below for fuure surplus. The mechanisms chosen for he fuure are: Company 1: All surplus as deermined in Secion 2.4 above is credied o he policy L reserves. In his case, he guaraneed rae of reurn also applies o pas surplus. Company 1 herefore promises cliquesyle guaranees. Noe ha his is he ype of surplus disribuion ha leads o he highes fuure liabiliies and o he leas amoun of flexibiliy for he company. accoun Company 2: All surplus as deermined in Secion 2.4 above is credied o he surplus S. The policy reserves L are increased only by he guaraneed rae of ineres. This ype of surplus disribuion provides more flexibiliy for he insurer, since for he accoun S, he guaraneed ineres is only. B Company 3: Surplus as deermined in Secion 2.4 above is credied o he bonus accoun. The policy reserves L are increased only by he guaraneed rae of ineres. This ype of surplus disribuion provides he highes degree of flexibiliy for he company, since he bonus accoun will only be disribued o he individual policy holders a mauriy of heir conracs. In he meanime, money may be aken from his accoun o pay for ineres rae guaranees if in some year he reurn on asses is no sufficien. Noe ha alhough hese hree mechanisms lead o a differen degree of flexibiliy and hus a differen risk for he insurer, he guaraneed mauriy value ha is shown o he policy holder a ouse, is he same in all cases. The differences beween he differen surplus mechanisms are illusraed by he following example of a wo year conrac wih a guaraneed rae of ineres of 5% p.a.: We assume ha he company has neiher equiy nor posiive reserves, he accouns S and B are 0 a = 0, he value of he asses A and he value of he liabiliies L are boh 100. In he firs
12 12 year, asses increase by 15. The insurance company credis he guaranee o he policy reserves, a surplus of 5 o he policy holders and hidden reserves are increased by 5. In he second year, asses remain unchanged. The final payoff for he policy holder and he insurance company s final solvency siuaion is herefore given by: Company 1: In he firs year, policy reserves are increased o 110. Thus, in he second year, a guaraneed increase of 5.5 has o be credied o he policy reserves. The final guaranee for he policy holder is The company is unable o pay is liabiliies a = 2 since he asse value is only 115. Company 2: In he firs year, policy reserves are increased by 5 and he surplus accoun S is increased by 5. Thus, in he second year, a guaraneed increase of 5.25 has o be credied o he policy reserves. The final guaranee for he policy holder consiss of he guaraneed policy reserves and he value of he surplus accoun and is hus given by The value of he company s asses a = 2 (115) is also below he value of he liabiliies (115.25), however by a slighly smaller margin. Company 3: In he firs year, policy reserves are increased by 5 and he bonus accoun B is increased by 5. In he second year, a guaraneed increase of 5.25 has o be credied o he policy reserves. The final guaranee for he policy holder consiss of he guaraneed policy reserves only and is hus given by This can be provided by reducing he bonus accoun B 2 B. The res of he bonus accoun is paid ou as erminal bonus. Thus, he value of he asses maches he payou o he insured. 2.6 Shorfall We considered a fixed ime horizon of T years. If a any balance shee dae =1,2,,T, he marke value of he asses is lower han he book value of he policy holders accouns, i.e. if A < L S B, his consiues a shorfall. We le he sopping ime τ be he firs balance shee dae wih a shorfall or τ = T1 if no shorfall occurs. Our numerical analyses in he nex secion will use he shorfall probabiliy P( τ T ) as a risk measure. In our model, here are many parameers ha have an influence on his shorfall probabiliy, in paricular parameers describing he regulaory framework (he guaraneed rae of ineres g, he minimum paricipaion rae δ, he resricion in asse valuaion
13 13 y), parameers describing he insurance company s financial siuaion and managemen decisions (he iniial reserve siuaion x:=x 0, he porion of socks in he asse porfolio s, arge dividend rae α, arge policy reurn rae z, arge range for he reserve quoa [ a;b] ), capial marke parameers, (drif μ and volailiy σ of he asse porfolio), he considered ime horizon T, he percenage ξ of he liabiliies mauring every year, and he surplus disribuion mechanism (model company 1, 2 or 3). 3. Analysis In wha follows, we will sudy he model companies inroduced above in order o analyze he effec of differen surplus disribuion mechanisms on an insurer s shorfall probabiliy. As menioned in Secion 2.2, we assume A o be a well diversified porfolio consising of socks and bonds wih s denoing he sock porion of he (coninuously rebalanced) porfolio. We assume he porfolio o follow he process (1). Furhermore, assuming an expeced reurn of 8% and a volailiy of 2 for he sock porion of he porfolio, an expeced reurn of 5% and a volailiy of 3.5% for he bond porion of he porfolio, as well as a slighly negaive correlaion (ρ = 0.1) beween sock and bond reurns, 7 he parameers of he process (1) are uniquely deermined for any given sock porion s. Since no analyical soluions for he shorfall probabiliy exis, we use Mone Carlo simulaion mehods. We generaed he normally disribued random sample required o projec he Geomeric Brownian Moion using a BoxMuller ransformaion, cf. e.g. Fishman (1996). The required uniformly disribued random sample was creaed by he random number generaor URN03 described in Karian and Dudewicz (1991). For each combinaion of parameers, 100,000 simulaions of A were performed. In each sample pah, he developmen of he insurer s balance shee over ime was calculaed, where he developmen of he accouns L, S and B was derived using he surplus mechanisms and surplus amouns described above. The Mone Carlo esimae for he shorfall probabiliy is he relaive porion of sample pahs in which a shorfall occurs. 7 As in Kling, Richer and Russ (2006), we used daa of a German sock index (DAX) and a German bond index (REXP) of he years 1988 o 2003 o ge esimaes for drif, volailiy and correlaion of socks and bonds. Since hisorical bond reurns seem o be oo high compared o curren low ineres raes, we reduced he drif for he bond porion o 5%.
14 14 If no saed oherwise we keep he following parameers fixed in his secion: We assume ha he resricion in asse valuaion is y = 5 and use a minimum paricipaion rae of δ = 9 as required by German regulaion. A = 0, we assume he balance shee o consis of 2% equiy, 91.5% policy reserves and 6.5% bonus accoun which represens a ypical balance shee of a German life insurance company. Furhermore we assume ha he insurer aims o provide a arge policy reurn rae of z = 5% o he policy holders accouns and a arge dividend rae of α = 1 as long as he reserve quoa says wihin a range of [a; b] = [5%; 3]. We assume ha conracs corresponding o ξ = 1 of he liabiliies leave he company every year and se he ime horizon of our analysis a T = 10 years. As a saring poin, we look a shorfall probabiliies as a funcion of he iniial reserve quoa and compare resuls for wo differen values of he guaraneed ineres rae, g = 2.75% and g = 4%. 8 Addiionally, we consider wo differen asse allocaions by assuming a sock raio in he porfolio of s = 1 and s = 3, respecively. The resuls are displayed in Figure 2. Ineresingly, resuls indicae ha for a given se of parameers companies 1 and 2 behave almos idenically. In oher words, he quesion of wheher he guaraneed rae of reurn or jus a guaraneed rae of is promised on pas surplus does no make a major difference under hese condiions. 9 For company 3, however, oucomes differ significanly. Generally, all oher hings equal, company 3 faces a much lower risk of shorfall, as i has he greaes flexibiliy in using former surplus as emergency funds o provide ineres guaranees in bad years. The guaraneed rae of ineres and he sock raio have a considerable impac on he likelihood of shorfall, in paricular when iniial reserves are low. The differen diagrams in Figure 2 show, ha for low reserve quoa levels, increasing he guaraneed rae from 2.75% o 4% causes an increase in he shorfall probabiliy of abou 15% and increasing he sock raio from 8 9 The curren maximum guaraneed rae for new business in he German marke is 2.75%. There are sill many older conracs in force ha have been sold wih higher raes up o 4%. This also remains rue for mos of he ses of resuls described in he following. However, he difference becomes larger if he ime horizon is increased.
15 15 1 o 3 causes an increase in he shorfall probabiliy of more han 2. Boh effecs diminish for higher iniial reserve quoas. Obviously, he shorfall probabiliy decreases as iniial reserves increase. However, he marginal effec of he iniial reserve quoa is greaer for a higher ineres guaranee. 16% g=2.75%, s=1 16% g=4%, s=1 14% 14% shorfall probabiliy 12% 1 8% 6% 4% shorfall probabiliy 12% 1 8% 6% 4% 2% 2% 5% 1 15% 2 25% 3 5% 1 15% 2 25% 3 iniial reserve quoa iniial reserve quoa 5 g=2.75%, s=3 5 g=4%, s=3 45% 45% 4 4 shorfall probabiliy 35% 3 25% 2 15% 1 shorfall probabiliy 35% 3 25% 2 15% 1 5% 5% 5% 1 15% 2 25% 3 5% 1 15% 2 25% 3 iniial reserve quoa iniial reserve quoa Figure 2: Shorfall probabiliy as a funcion of he iniial reserves for differen values of he guaraneed rae of ineres and for differen asse allocaions Figure 3 shows he shorfall probabiliy as a funcion of he guaraneed rae of ineres for differen values of he iniial reserve quoa x and he sock porion s. Our calculaions again
16 16 confirm he sraighforward proposiion ha crediing an ineres rae guaranee inflics a much higher risk on a company wih a poor iniial reserve level. For insance, given a reserve level of 2 and a sock porion of 3, company 1 can offer a guaraneed rae of 2.75% a a shorfall probabiliy of roughly 8%, whereas, all oher hings equal, a reducion in he reserve quoa o 5% would bring up he shorfall risk o abou 2. A he lower reserve quoa, company 3, however, would be able o provide he same guaraneed ineres rae wih a shorfall probabiliy of only 1. Again, ceeris paribus company 3 is characerized by a considerably lower shorfall risk. Addiionally, we find ha no only he shorfall probabiliy, bu also he marginal impac of increasing he guaraneed ineres rae is generally greaer where reserves are low. I should also be noed ha for low levels of he guaraneed ineres rae, he probabiliy of shorfall ends o zero in he case of a 1 sock porion. As he guaranee is decreased, he bond porion of he insurer s asse porfolio becomes more and more likely o be sufficien o generae he minimum ineres, while a he same ime i limis he shorfall exposure because of he lower volailiy. Whereas wih a greaer porion of socks he shorfall risk remains significanly posiive for companies 1 and 2, insurer 3 would sill be able o basically avoid shorfall risk for low guaraneed raes a leas in he case wih higher iniial reserves. Furhermore, i mus be highlighed ha given a maximum olerable shorfall probabiliy, company 3 would a he same reserve level always be able o offer a higher guaraneed rae of ineres, i.e. creae a higher guaraneed mauriy value a incepion of he conrac. For insance, consider he parameers x = 2 and s = 3. In his siuaion, a a shorfall probabiliy of 5% company 1 would only be able o promise a guaraneed rae of 2%, while company 3 could offer 4%. This, of course, pus companies 1 and 2 a a significan compeiive disadvanage, as for a given asse allocaion company 3 can offer conracs wih a higher iniial guaranee reurn wihou exposing iself o greaer risk.
17 17 x=5%, s=1 x=2, s= % 18% 16% 16% shorfall probabiliy 14% 12% 1 8% 6% shorfall probabiliy 14% 12% 1 8% 6% 4% 4% 2% 2% guaraneed rae of ineres guaraneed rae of ineres x=5%, s=3 x=2%, s= % 35% shorfall probabiliy 3 25% 2 15% 1 shorfall probabiliy 3 25% 2 15% 1 5% 5% guaraneed rae of ineres guaraneed rae of ineres Figure 3: Shorfall probabiliy as a funcion of he guaraneed rae of ineres for differen values of he iniial reserve quoa and for differen asse allocaions Considering he shorfall probabiliy as a funcion of he sock raio (for g = 2.75%), i can be noed ha he shape of his funcion is no srongly affeced by a change in he iniial reserve siuaion (from x = 5% o x = 2), cf. figure 4. However, a significan difference can be observed for low sock raios. The funcion seems o be convex in his area and concave elsewhere for all considered parameer ses. For reasons of diversificaion, of course he probabiliy of shorfall decreases in he sock raio up o a cerain poin, (roughly s =1). This means ha very low sock raios (in paricular: s = ) are sricly dominaed by greaer levels of
18 18 s which allow a higher expeced reurn a he same risk of shorfall. This effec is paricularly pronounced in a siuaion wih low iniial reserves. Given a maximum olerable shorfall probabiliy, figure 4 shows ha company 3 would a he same reserve level and for he same guaraneed rae of ineres be able o ener a riskier asse allocaion, i.e. creae a higher expeced reurn for policy holders. For an iniial reserve quoa of x = 2, a a shorfall probabiliy of 5% company 1 would only be able o allow for a sock raio of 27% while company 3 could afford 38% in socks. This, of course, pus companies 1 and 2 a a significan compeiive disadvanage, as for a given ineres guaranee company 3 can offer conracs wih a higher expeced reurn wihou exposing iself o greaer risk. 9 x=5% 9 x=2 8 8 shorfall probabiliy shorfall probabiliy sock raio company 1 sock company raio 2 company 3 Figure 4: Shorfall probabiliy as a funcion of he sock raio s for differen values of he iniial reserve quoa Figure 5 now shows how resuls reac o variaions of he ime horizon T. Again, our resuls indicae ha company 3 is considerably more sable han companies 1 and 2. For insance, even in he scenario wih low reserves (x=5%), a guaraneed rae of ineres of 4% could be provided by company 3 a a shorfall probabiliy of abou 15% wihin 40 years, whereas companies 1 and 2 offering he same guaranee would face he same risk of shorfall wihin a period of only abou 17 years. In order o reduce he 40year shorfall probabiliy o 15%, companies 1 and 2 would need o have an iniial reserve quoa of 2. I also can be seen from figure 5 ha he guaraneed rae of ineres makes a significan difference if companies are concerned abou longerm shorfall probabiliies. By reducing he
19 19 guaraneed rae of ineres from g = 4% o g = 2.75%, companies are able o reduce he 40year shorfall probabiliy by abou wo hirds, independen of oher parameers. The analyses provided earlier showed ha he 10year shorfall probabiliy of companies 1 and 2 are almos idenical while he risk of company 3 differs srongly. However, he longer he considered ime horizon T, he greaer he difference beween he hree companies: For iniial reserves of x = 5% and a guaraneed rae of g = 4%, he difference in he 10year shorfall probabiliy beween company 1 and company 3 is less han 3%.This difference is increased o 15% for he 40year shorfall probabiliy.
20 20 3 g=2.75%, x=5% 3 g=4%, x=5% 25% 25% shorfall probabiliy 2 15% 1 shorfall probabiliy 2 15% 1 5% 5% erm T erm T 2 g=2.75%, x=2 2 g=4%, x=2 18% 18% 16% 16% shorfall probabiliy 14% 12% 1 8% 6% 4% shorfall probabiliy 14% 12% 1 8% 6% 4% 2% 2% erm T erm T Figure 5: Shorfall probabiliy as a funcion of he ime horizon T for differen values of he iniial reserve quoa and he guaraneed rae. So far, we assumed he same iniial balance shee siuaion for he hree companies considered. This is equivalen o assuming ha in he pas, he companies behaved exacly he same and will change surplus disribuion in he fuure. We will now analyze wheher our resuls change afer he respecive differen surplus disribuion mechanisms have been applied for several years by he differen companies. To analyze his effec, in wha follows, we focus on socalled forward 10year shorfall probabiliies, defined as coningen probabiliies of shorfall in [; 10], given ha no shorfall occurred in he firs years. One general observaion ha can be
21 21 made from he following analyses is ha he differences beween he hree companies [; 10] forward probabiliies end o be increasing in. This can be explained by he fac ha afer years of applying differen surplus mechanisms, he insurers year iniial balance shees differ. Figure 6 depics hese forward shorfall probabiliies as a funcion of for differen levels of he guaraneed rae and for differen levels of he iniial reserve quoa. Obviously, an increase in he guaraneed rae always increases he forward shorfall risk, all oher hings equal. However, i does no seem o have a major impac on he shape of he funcion. Ineresing observaions can be made by comparing resuls for he wo differen levels of iniial reserves, x = 5% and x = 2. As shown above, a = 0, he shorfall risk is significanly lower for larger values of x. Bu whils in he low iniial reserve siuaion he forward shorfall probabiliies decrease in, hey increase when iniial reserves are higher. Alhough his may seem surprising, i is quie inuiive, if we keep in mind he design of he managemen rules deermining he amoun of surplus o be disribued each year: In he lower iniial reserves case he insurer will credi lower surplus more frequenly as reserves end oward he lower hreshold. Thus, ceeris paribus, i is more likely ha he company builds up addiional reserves, evenually decreasing is forward shorfall risk. On he oher hand, a company wih higher iniial reserves has a endency of giving high surplus and hus reducing is reserves. Thus, for large values of, i is likely ha he reserves of he wo companies will converge, given ha no shorfall occurred before. This suggess ha here is an equilibrium reserve level (and a corresponding defaul risk). If his level is chosen as he iniial reserve level, he graph should be enirely fla. So, basically, our surplus disribuion model implies ha companies ha have buil up high reserves in he pas have a endency o give par of hese reserves away o fuure cliens whereas companies wih low reserves have a endency o increase reserves in he fuure. While his may seem o be a compeiive disadvanage for a company wih greaer iniial reserves, we have o keep in mind ha his company would be more successful in providing he arge ineres rae, hus signaling greaer sabiliy and ulimaely he beer produc. On he oher hand, policy holders should selec a company wih higher reserves, since his company will provide a higher expeced surplus.
22 22 12% g=2.75%, x=5% 12% g=4%, x=5% 1 1 shorfall probabiliy 8% 6% 4% shorfall probabiliy 8% 6% 4% 2% 2% % g=2.75%, x=2 12% g=4%, x=2 1 1 shorfall probabiliy 8% 6% 4% shorfall probabiliy 8% 6% 4% 2% 2% τ τ Figure 6: Forward 10year shorfall probabiliy as a funcion of he saring ime for differen values of he iniial reserve quoa and he guaraneed rae. 4. Conclusions This paper analyzes he impac of differen surplus disribuion mechanisms on he risk exposure of a life insurance company selling wih profi life insurance policies wih a cliquesyle ineres rae guaranee. We consider hree differen ypes of disribuion mechanism: One, where he guaraneed ineres rae also applies o surplus ha has been credied in he pas, a slighly less resricive mechanism in which a guaraneed rae of ineres of applies o pas surplus, and a
23 23 hird mechanism ha allows for he company o use former surplus in order o compensae for underperformance in bad years. The resuls srongly sugges ha a disribuion mechanism similar o he one inroduced for he hird ype of company is significanly differen from he oher wo disribuion mechanisms. Throughou he analysis, our represenaive company 3 faces ceeris paribus much lower shorfall risk han he oher wo companies. Thus, a company of our ype 3 can afford a much greaer porion of socks in is asse porfolio while mainaining he same shorfall risk, compared o ype 1 and 2 companies. This means ha, while subjec o he same amoun of risk, company 3 would be able o inves in a porfolio promising greaer expeced reurns and sill offer he same guaraneed mauriy benefi. On he oher hand, as is sraighforward bu ineresing o noe, company 3 would also be able o, all oher hings equal, provide higher ineres rae guaranees han companies 1 and 2 while mainaining he same shorfall risk. These resuls should be of paricular ineres for regulaors in he European Union (EU) since on he one hand he surplus mechanism of our ype 3 is severely resriced by regulaors in some European counries, and on he oher hand companies from oher EU counries are allowed o sell producs wih his surplus mechanism across borders (and hus ino counries wih more resricive regulaion) under he socalled Freedom o provide Services Ac. This creaes a disorion of compeiion since companies required o offer srong guaranees may be pu a a significan compeiive disadvanage. This is in paricular rue for long erm conracs. Thus, our analysis suggess ha under cerain condiions severe regulaion does no only yield subopimal resuls, bu also seems o be counerproducive wih respec o goals ha would usually be considered a fundamenal purpose of regulaion. A major raionale for insurance regulaion ypically is seen in proecing he cusomers by keeping insurers solven. This is paricularly imporan in he area of life insurance where conracual relaionships are longerm and he insured become major crediors whose sakes jusify srong regulaory inervenion. Naurally, a srong case can also be made for proecing life insurance cusomers by inroducing minimum ineres rae guaranees and regulaing surplus disribuion. One mus be aware, however, ha inference wih he way surplus is disribued, decreases an insurer s flexibiliy and, all oher hings equal, increases shorfall risk.
24 24 Overall, shorfall probabiliies hroughou our analyses are alarmingly high. Parially, his may be aribuable o he fac ha insurance companies can employ risk managemen measures no considered in his work, such as adjusing he company s asse allocaion when reserves reach a criical level. Alhough modeling differen managemen decision rules and analyzing heir effec on he shorfall risk is possible wihin our model framework, his is beyond he scope of he presen paper. This ype of exension, hough, migh be an ineresing opic for fuure research poenially leading o furher valuable insigh. References Bacinello A.R. (2003): Fair Valuaion of a Guaraneed Life Insurance Paricipaing Conrac Embedding a Surrender Opion. The Journal of Risk & Insurance, Vol. 70, No. 3, pp Bauer, D., Kiesel, R., Kling, A., and Russ, J. (2006): RiskNeural Valuaion of Paricipaing Life Insurance Conracs. Insurance: Mahemaics and Economics, Vol. 39, No. 2, pp Briys, E. and de Varenne, F. (1997): On he Risk of Insurance Liabiliies: Debunking Some Common Pifalls. Journal of Risk and Insurance, Vol. 64, No. 4, pp Jensen, B., Jorgensen, P. L. and Grosen, A. (2001): A Finie Difference Approach o he Valuaion of Pah Dependen Life Insurance Liabiliies. Geneva Papers on Risk and Insurance Theory, Vol. 26, No. 1, pp Fishman, G. S. (1996): Mone Carlo; Conceps, Algorihms and Applicaions. Springer, New York. Grosen, A. and Jorgensen, P. L. (2000): Fair Valuaion of Life Insurance Liabiliies: The Impac of Ineres Rae Guaranees, Surrender Opions, and Bonus Policies. Insurance: Mahemaics and Economics, Vol. 26, No. 1, pp Grosen, A. and Jorgensen, P. L. (2002): Life Insurance Liabiliies a Marke Value: An Analysis of Insolvency Risk, Bonus Policy, and Regulaory Inervenion Rules in a Barrier Opion Framework. Journal of Risk and Insurance, Vol. 69, No. 1, pp
25 25 Hansen, M. and Milersen, K. R. (2002): Minimum Rae of Reurn Guaranees: The Danish Case. Scandinavian Acuarial Journal, Vol. 2002, No. 4, pp Karian, Z. A. and Dudewicz, E. J. (1991): Modern Saisical, Sysems, and GPSS Simulaion. Compuer Science Press, W. H. Freeman and Company, New York. Kling, A., Richer A., and Russ, J. (2006): The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies. Forhcoming Insurance: Mahemaics and Economics. Milersen, K. R. and Persson, S.A. (2003): Guaraneed Invesmen Conracs: Disribued and Undisribued Excess Reurn. Scandinavian Acuarial Journal 2003(4), p Tanskanen A. J., Lukkarinen J. (2003): Fair valuaion of pahdependen paricipaing life insurance conracs. Insurance: Mahemaics and Economics, Vol. 33, No. 3, pp
The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.
The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationAnalyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective
Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics FriedrichAlexanderUniversiy
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationOn the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes
On he Managemen of Life Insurance Company Risk by raegic Choice of Produc Mix, Invesmen raegy and urplus Appropriaion chemes Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen Working Paper Deparmen
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO
Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,
More informationFair Valuation and Risk Assessment of Dynamic Hybrid Products in Life Insurance: A Portfolio Consideration
Fair Valuaion and Risk ssessmen of Dynamic Hybrid Producs in ife Insurance: Porfolio Consideraion lexander Bohner, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen FriedrichlexanderUniversiy
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationA Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *
A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone:
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationINVESTMENT GUARANTEES IN UNITLINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE
INVESMEN UARANEES IN UNILINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN
More informationRISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction
RISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION AN CHEN AND PETER HIEBER Absrac. In a ypical paricipaing life insurance conrac, he insurance company is eniled o a
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationTHE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS
THE IPACT OF THE ECONDARY ARKET ON LIFE INURER URRENDER PROFIT Nadine Gazer, Gudrun Hoermann, Hao chmeiser Insiue of Insurance Economics, Universiy of. Gallen (wizerland), Email: nadine.gazer@unisg.ch,
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationEquities: Positions and Portfolio Returns
Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi
More informationDynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint
Dynamic Hybrid Producs in Life Insurance: Assessing he Policyholders Viewpoin Alexander Bohner, Paricia Born, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen FriedrichAlexanderUniversiy
More informationCLASSIFICATION OF REINSURANCE IN LIFE INSURANCE
CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More informationPRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES
PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationFifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance
Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of besesimae provisions... 3 2.1
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include socalled implici or embedded opions.
More informationMarkit Excess Return Credit Indices Guide for price based indices
Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semiannual
More informationII.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
More informationCVA calculation for CDS on super senior ABS CDO
MPRA Munich Personal RePEc Archive CVA calculaion for CDS on super senior AS CDO Hui Li Augus 28 Online a hp://mpra.ub.unimuenchen.de/17945/ MPRA Paper No. 17945, posed 19. Ocober 29 13:33 UC CVA calculaion
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationLEASING VERSUSBUYING
LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationGMWB For Life An Analysis of Lifelong Withdrawal Guarantees
GMWB For Life An Analysis of Lifelong Wihdrawal Guaranees Daniela Holz Ulm Universiy, Germany daniela.holz@gmx.de Alexander Kling *) Insiu für Finanz und Akuarwissenschafen Helmholzsr. 22, 8981 Ulm, Germany
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationNewton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationCannibalization and Product Life Cycle Management
MiddleEas Journal of Scienific Research 19 (8): 10801084, 2014 ISSN 19909233 IDOSI Publicaions, 2014 DOI: 10.5829/idosi.mejsr.2014.19.8.11868 Cannibalizaion and Produc Life Cycle Managemen Ali Farrukh
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationSome Quantitative Aspects of Life Annuities in Czech Republic
Some Quaniaive Aspecs of Life Annuiies in Czech Republic Tomas Cipra The conribuion deals wih some quaniaive aspecs of life annuiies when applied in he Czech Republic. In paricular, he generaion Life Tables
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationCan Individual Investors Use Technical Trading Rules to Beat the Asian Markets?
Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weakform of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien
More informationOne dictionary: Native language  English/English  native language or English  English
Faculy of Social Sciences School of Business Corporae Finance Examinaion December 03 English Dae: Monday 09 December, 03 Time: 4 hours/ 9:003:00 Toal number of pages including he cover page: 5 Toal number
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationSupplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect RiskTaking?
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec RiskTaking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
More informationImpact of scripless trading on business practices of Subbrokers.
Impac of scripless rading on business pracices of Subbrokers. For furher deails, please conac: Mr. T. Koshy Vice Presiden Naional Securiies Deposiory Ld. Tradeworld, 5 h Floor, Kamala Mills Compound,
More informationChapter 9 Bond Prices and Yield
Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value
More informationPREMIUM INDEXING IN LIFELONG HEALTH INSURANCE
Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationAnalysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer
Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of
More informationMoneyBack Guarantees in Individual Pension Accounts: Evidence from the German Pension Reform
No. 22/3 MoneyBack Guaranees in Individual Pension Accouns: Evidence from he German Pension Reform Raimond Maurer / Chrisian Schlag Cener for Financial Sudies an der Johann Wolfgang GoeheUniversiä Taunusanlage
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationTax Externalities of Equity Mutual Funds
Tax Exernaliies of Equiy Muual Funds Joel M. Dickson The Vanguard Group, Inc. John B. Shoven Sanford Universiy and NBER Clemens Sialm Sanford Universiy December 1999 Absrac: Invesors holding muual funds
More informationDouble Entry System of Accounting
CHAPTER 2 Double Enry Sysem of Accouning Sysem of Accouning \ The following are he main sysem of accouning for recording he business ransacions: (a) Cash Sysem of Accouning. (b) Mercanile or Accrual Sysem
More informationPrice elasticity of demand for crude oil: estimates for 23 countries
Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationTask is a schedulable entity, i.e., a thread
RealTime Scheduling Sysem Model Task is a schedulable eniy, i.e., a hread Time consrains of periodic ask T:  s: saring poin  e: processing ime of T  d: deadline of T  p: period of T Periodic ask T
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationAgnes Joseph, Dirk de Jong and Antoon Pelsser. Policy Improvement via Inverse ALM. Discussion Paper 06/2010085
Agnes Joseph, Dirk de Jong and Anoon Pelsser Policy Improvemen via Inverse ALM Discussion Paper 06/2010085 Policy Improvemen via Inverse ALM AGNES JOSEPH 1 Universiy of Amserdam, Synrus Achmea Asse Managemen
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationSegmentation, Probability of Default and Basel II Capital Measures. for Credit Card Portfolios
Segmenaion, Probabiliy of Defaul and Basel II Capial Measures for Credi Card Porfolios Draf: Aug 3, 2007 *Work compleed while a Federal Reserve Bank of Philadelphia Dennis Ash Federal Reserve Bank of Philadelphia
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationEstimating TimeVarying Equity Risk Premium The Japanese Stock Market 19802012
Norhfield Asia Research Seminar Hong Kong, November 19, 2013 Esimaing TimeVarying Equiy Risk Premium The Japanese Sock Marke 19802012 Ibboson Associaes Japan Presiden Kasunari Yamaguchi, PhD/CFA/CMA
More informationRelationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**
Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationMarkov Models and Hidden Markov Models (HMMs)
Markov Models and Hidden Markov Models (HMMs (Following slides are modified from Prof. Claire Cardie s slides and Prof. Raymond Mooney s slides. Some of he graphs are aken from he exbook. Markov Model
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More information