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 Ludwig-Maximilians 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 clique-syle 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 poin-o-poin 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 so-called clique-syle (or year-by-year) 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 closed-form soluions for marke values of liabiliies and equiies in a poin-o-poin 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 jus-menioned 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 clique-syle 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 so-called 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 risk-neural 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. risk-neural) 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 so-called real-world probabiliy measure P. Assuming clique-syle 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: clique-syle guaranees and poino-poin 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 clique-syle 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 Box-Muller 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 40-year 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 long-erm shorfall probabiliies. By reducing he
19 19 guaraneed rae of ineres from g = 4% o g = 2.75%, companies are able o reduce he 40-year shorfall probabiliy by abou wo hirds, independen of oher parameers. The analyses provided earlier showed ha he 10-year 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 10-year shorfall probabiliy beween company 1 and company 3 is less han 3%.This difference is increased o 15% for he 40-year 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 10-year 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 10-year 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 clique-syle 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 so-called 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 couner-producive 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 long-erm 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): Risk-Neural 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 pah-dependen paricipaing life insurance conracs. Insurance: Mahemaics and Economics, Vol. 33, No. 3, pp
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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
Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: firstname.lastname@example.org
GMWB For Life An Analysis of Lifelong Wihdrawal Guaranees Daniela Holz Ulm Universiy, Germany email@example.com Alexander Kling *) Insiu für Finanz- und Akuarwissenschafen Helmholzsr. 22, 8981 Ulm, Germany
Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
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
Inroducion Chaper 14: Dynamic D-S 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 cuing-edge
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
. 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
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
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
Middle-Eas Journal of Scienific Research 19 (8): 1080-1084, 2014 ISSN 1990-9233 IDOSI Publicaions, 2014 DOI: 10.5829/idosi.mejsr.2014.19.8.11868 Cannibalizaion and Produc Life Cycle Managemen Ali Farrukh
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
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
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.
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
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
Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying
Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weak-form of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien
Faculy of Social Sciences School of Business Corporae Finance Examinaion December 03 English Dae: Monday 09 December, 03 Time: 4 hours/ 9:00-3:00 Toal number of pages including he cover page: 5 Toal number
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec Risk-Taking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
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
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
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
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
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
No. 22/3 Money-Back Guaranees in Individual Pension Accouns: Evidence from he German Pension Reform Raimond Maurer / Chrisian Schlag Cener for Financial Sudies an der Johann Wolfgang Goehe-Universiä Taunusanlage
Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: firstname.lastname@example.org Homepage: www.hha.dk/~jel/ Firs
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
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
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
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.
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
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series
Real-Time 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
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
Agnes Joseph, Dirk de Jong and Anoon Pelsser Policy Improvemen via Inverse ALM Discussion Paper 06/2010-085 Policy Improvemen via Inverse ALM AGNES JOSEPH 1 Universiy of Amserdam, Synrus Achmea Asse Managemen
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, email@example.com Why principal componens are needed Objecives undersand he evidence of more han one
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
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
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
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,
Chaper 7. esponse of Firs-Order 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
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