The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies


 April Lambert
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1 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, Ulm, Germany phone: , fax: Andreas Richer Assisan Professor Illinois Sae Universiy, Kingsridge C. B, Normal, IL, USA Jochen Ruß Managing Direcor, Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081Ulm, Germany This version: 5/30/2005 * aending and presening auhor.
2 2 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling, Andreas Richer, Jochen Ruß This Version: 5/30/2005 Absrac Tradiional life insurance policies in many markes are sold wih minimum ineres rae guaranees. In producs which are predominan e.g. in he German marke, here is a socalled cliquesyle (or yearbyyear) guaranee, where he guaraneed reurn mus be credied o he policyholder s accoun each year. Usually, life insurers ry o provide his guaraneed rae of ineres plus some sable surplus on he policyholder s accoun every year by applying he socalled average ineres principle: Building up reserves in years of good reurns on asses and using hese reserves o keep surplus sable in years of low reurns. In he curren low ineres environmen, insurance companies in many counries are forced o provide high guaraneed raes of ineres o accouns o which a big porion of pas years surplus has already been credied. This developmen illusraes he addiional risk ha a cliquesyle guaranee incurs, compared wih a poinopoin guaranee, by limiing he insurance company s flexibiliy. So far, only very lile lieraure exiss ha deals wih hese guaranees. The primary focus of mos exising lieraure in his area is on he fair (i.e. riskneural) valuaion of life insurance conracs. Since mos insurers do no apply riskneural (or riskminimizing) hedging sraegies, an analysis of he resuling risks seems very imporan. Therefore, he presen paper will concenrae on he risk a conrac imposes on he insurer, measured by shorfall probabiliies under he socalled realworld probabiliy measure P. We develop a raher general model and analyze he impac ineres rae guaranees have on he risk exposure of he insurance company and how defaul risks depend on characerisics of he conrac, on he insurer s reserve siuaion and asse allocaion, on managemen decisions, as well as on regulaory parameers. In paricular, he ineracion of he parameers is analyzed yielding resuls ha should be of ineres for insurers as well as regulaors. Keywords: wih profi life insurance, ineres rae guaranees, shorfall probabiliy, Mone Carlo simulaion,
3 3 1. Inroducion Tradiional (i.e. no variable) life insurance policies in many markes are sold wih minimum ineres rae guaranees. Quie common are guaranees on a poinopoin basis: A mauriy of a conrac, he policyholder is guaraneed he amoun equivalen o he resul of a process which credis a cerain minimum ineres rae o he insured s accoun in every single year. However, an insurer s invesmen underperformance (relaive o he guaraneed rae) for some of he years of a conrac s lifeime would be olerable so long as he minimum amoun is me in he end. This provides he insurer wih he possibiliy o compensae bad invesmen resuls by posiive resuls in oher years. On he oher hand, of course, his ype of guaranee poses risk on he policyholder as i leaves he insurer wih considerable flexibiliy in crediing ineres o specific accouns. In so far as here is discreion wih respec o he accouns o which ineres is credied or wih respec o wheher reurns are passed on o he insureds a all, his ype of guaranee ses incenive o meeing shorerm obligaions while a he same ime neglecing young conracs. This incenive problem can be reduced by means of incorporaing a differen ype of ineres rae guaranee: In he case of a socalled cliquesyle guaranee, as, e.g., required by he German regulaory framework, he guaraneed reurn mus be credied o he policyholder s accoun each year. Obviously, he resuling reducion in risk for he policy owner comes a he cos incurred by he reducion of he insurer s flexibiliy in is invesmen decisions. Usually, as long as his is permied by he marke, life insurers ry o provide he guaraneed rae of ineres plus some surplus on he policyholders accoun every year. Insurers apply a sraegy which is ofen referred o as he average ineres principle (see,
4 4 e.g., Grosen and Jorgensen, 2000): Companies aemp o hold he surplus credied o he policyholders accoun as consan as possible, in order o signal sabiliy and low risk compared o oher personal invesmen opions an insured would have. This is achieved hrough building up of reserves (mosly asse valuaion reserves) in years of good reurns on asses and using hese reserves o keep surplus sable in years of low (or even negaive) reurns on asses. A reasonable model of he disribuion mechanism in wih profi life insurance conracs should include his averaging mechanism. Superficially, he longerm use of he sraegy described above suggess ha he minimum ineres rae guaranee is obsolee. For a significan period in he pas, i seemed as if he minimum guaraneed ineres raes required by regulaors were so low ha insurance companies would exceed hese values anyway wihou a all perceiving he minimum requiremen as a resricion. Consequenly, i appears ha unil raher recenly life insurers have no charged a premium for an ineres rae guaranee (see Grosen and Jorgensen, 2002, p. 64). The process of averaging reurns over ime worked raher well since marke ineres raes were, over a long ime span, significanly higher han he guaraneed raes. In recen years, however, low marke ineres raes and plunging sock markes have caused rouble for insurance companies. In he changed environmen, hey now have o provide comparably high guaraneed reurns o accouns o which already a subsanial amoun of he surplus of pas years has been credied. Under hese circumsances, minimum ineres rae guaranees have suddenly become a hrea o insurers solvency. These developmens illusrae he relevance of analyses of he impac of ineres rae guaranees on hese conracs and heir ineracion wih oher parameers. A key raionale
5 5 for he regulaion of insurance markes is o reduce or limi insurers risk of insolvency. Minimum ineres requiremens, however, obviously generae a resricion which may increase insolvency risk. Paricular emphasis herefore needs o be pu on he inerdependence beween ineres rae guaranees and he likelihood of defaul. A number of papers have recenly addressed ineres rae guaranees, such as Briys and de Varenne (1997), Grosen and Jorgensen (2000), Jensen e al. (2001), Milersen and Persson (2001), Hansen and Milersen (2002), Grosen and Jorgensen (2002), Bacinello (2003), and Tanskanen and Lukkarinen (2003). For a poinopoin guaranee framework, Briys and de Varenne (1997) compue closedform soluions for marke values of liabiliies and equiies. In heir model he policy owner receives a guaraneed ineres and is also credied a bonus, deermined as a cerain fracion of ne financial gains (when posiive). They provide an equilibrium condiion, which reflecs he inerdependencies beween hese wo parameers, assuming fair valuaion of he conrac in a riskneural evaluaion framework. The paper also addresses 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
6 6 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 of only limied explanaory value. Milersen and Persson (2001) 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 (2001) derive numerical resuls on he influence of various parameers on he conrac value 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). Conrasing he mechanism discussed in Milersen and Persson (2001), 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).
7 7 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 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 policyholders disadvanage. While some of he abovemenioned papers incorporae he risk of a conrac s or he insurer s defaul, he primary focus is on he fair (i.e. riskneural) valuaion of he life insurance conrac. Since mos insurers do no or can no apply opimal hedging sraegies in heir asse allocaion, an analysis of he resuling risks seems very imporan. Therefore, his work will concenrae on he risk a conrac imposes on he insurer, measured by means of shorfall probabiliies under he socalled realworld probabiliy measure P. We are ineresed in he impac ineres rae guaranees have on he exposure of he insurance company and how defaul risks depend on characerisics of he conrac, on he insurer s reserve siuaion and asse allocaion, on managemen decisions, as well as on regulaory parameers. We will assume cliquesyle guaranees hroughou his paper. Cerain oher feaures of our model are moivaed by he German regulaory framework, bu model specificaions could easily be changed o reflec oher counries siuaions. The paper is organized as follows. In secion 2, we inroduce our model. We use a simplified illusraion of he insurer s financial siuaion. Before we describe he surplus disribuion mechanism, we presen our model for he asses and he insurance conrac.
8 8 Secion 2 concludes wih inroducing shorfall probabiliies as he relevan risk measure for his work. In secion 3, we presen he resuls of our analysis. We examine he influence of he abovemenioned parameers on shorfall probabiliies and analyze heir ineracion. The resuls should be of ineres for insurers as well as for regulaors. Secion 4 gives a shor summary of he mos imporan resuls and an oulook on furher research opions. 2. The model framework This Secion inroduces our model. We keep i as simple as possible o be able o focus on he basic effecs. Firs, we consider he reserve siuaion of he insurance company s balance shee. Then, we inroduce our model for he financial marke and refer o some specific aspecs of German regulaion. Aferwards, he insurance conrac considered and he corresponding liabiliies are defined. Our analysis of he ineracion of asses and liabiliies akes ino accoun he abiliy of insurance companies o build up and dissolve hidden reserves over ime. We assume ha insurance companies can buy and sell asses in order o reduce hidden reserves wihou any resricions. However, he decision wheher an increase in he marke value of asses increases he book value or he hidden reserves is subjec o some resricions. Finally, we define shorfall probabiliies as he relevan risk measure for he following analysis. 2.1 The insurer s iniial siuaion We use a simplified illusraion of he insurer s financial siuaion given in figure 1.
9 9 Asses Liabiliies A L R A A Figure 1 Model of he insurer s financial siuaion By A, we denoe he marke value of he insurer s asses a ime. The liabiliy side comprises wo enries: L is he ime book value of he policyholders accoun or, in oher words, he policy reserve which also coincides wih he book value of he asses. The second accoun is he reserve accoun R which is given by R = A L. Alhough i migh consis of oher componens as well, e.g. firm s capial, we refer o R as he asse valuaion reserves or hidden reserves. Paymens o equiy holders are paid ou and herefore leave he company. This is refleced by subracing he corresponding amoun from A. To simplify noaion, we assume ha such paymens occur only once a year, a imes = 1,2, K, T, where T denoes he considered ime horizon. 2.2 The financial model The insurer s asses are invesed in a porfolio conaining socks and bonds. We hink of boh as risky asses wih known expeced rae of reurn, volailiy and correlaion. We assume a finie ime horizon T and a complee, fricionless and coninuous marke. Ignoring paymens o equiy holders for a momen, we le A follow a geomeric Brownian moion da A = μ ( ) d + σ ( ) dw, (1)
10 10 where W denoes a Wiener process on some probabiliy space (Ω,Σ,P) wih a filraion F, o which W is adaped. Boh, μ and σ are deerminisic bu can be ime dependen. In our numerical analysis in Secion 3, we assume μ = 8% and σ = 20% for he sock porion of he porfolio as well as μ = 5% and σ = 3.5% for he bond porion of he porfolio. Furhermore, we assume sock and bond reurns o be slighly negaively correlaed wih a correlaion coefficien of ρ = Thus, drif and volailiy of he porfolio can be calculaed for any given asse allocaion. For a given A 0 > 0, he soluion of (1) is given by 2 σ ( s) μ ( s) ds+ σ ( s) dws A = A0e (2) and, hence, we have 2 σ ( s) μ ( s) ds A = A 1 e σ ( s) dw s (3). If here are any dividend paymens D o equiy holders a ime, we le A  denoe he value of he asses before hese paymens leave he company and A + he value of he asses afer hese paymens lef he company. Thus, we ge (for = 1,2, K, T ) 2 σ ( s) μ ( s) ds A = A 1 e σ ( s) dw s and A + = A D (4), which can be used handily in Mone Carlo algorihms. 3 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%.
11 11 The porion of socks conained in A is denoed by s. We do no consider any ransacion fees for buying and selling asses. In our numerical analysis, we assume s = s o be consan. 2.3 The insurance conrac For he sake of simpliciy, we look a a very simple life insurance conrac, a singlepremium ermfix insurance and ignore any charges. The premium P is paid a = 0. A benefi is paid a ime T, no maer if he insured is sill alive or no. Thus, here are no moraliy effecs o be considered. The benefi paid a ime T depends on he developmen of he insurer s liabiliies and is given by L P T. 4 L Regulaory and legal requiremen In wha follows, we include imporan feaures of he curren German regulaory and legal framework in our model. Neverheless, specific aspecs of oher counries could be considered analogously. Currenly, German life insurance companies guaranee policyholders a minimum rae of ineres of g = 2.75%. 5 This guaranee is given as a cliquesyle guaranee for he whole erm of he policy and may no be reduced even if regulaors change he guaraneed 4 5 We ignore moraliy effecs as well as charges for any moraliy benefi ha exceeds he policy value a ime of deah. Alhough his migh seem inappropriae in he analysis of a life insurance company s asses and liabiliies, i is jusified under he following wo assumpions: 1) The risk premiums are calculaed properly such ha he insurer incurs no significan profi or loss upon deah. 2) On average, new business roughly compensaes for moraliy and surrenders. Under hese seady sae assumpions, he erm fix conrac and he corresponding asses are an approximaion for he liabiliies and asses of he whole insurance company. More precisely, here is a maximum rae of reurn, policy reserves may be calculaed wih. Since his rae is used for almos all producs and since surrender values have o be close o policy reserves, his implies ha insured have a year by year guaranee of his ineres rae on heir accoun value.
12 12 ineres rae for new business. Thus, all policies ha have been sold when guaraneed raes were higher are sill eniled o he guaraneed rae ha prevailed when he conracs were sold: 3.25% or even 4% p.a. Furhermore, he law requires ha a leas δ = 90% of he earnings on book values have o be credied o he policyholders accouns. This socalled minimum paricipaion rae was inroduced o make sure ha policyholders are no pu a a disadvanage compared o he shareholders. 2.5 Developmen of he financial siuaion over ime As menioned above, we consider a yearbyyear guaranee on he liabiliies. Given g he liabiliies a ime 1, he guaraneed liabiliies a ime are given by L 1 L L = L (1 ) (5). g 1 + g Since earnings on book value are subjec o accouning rules, hey are no necessarily equal o he earnings on marke value A A + 1. For insance, by using he lower of cos or marke principle, a company in Germany can hide a rise in a sock price in order o increase asse valuaion reserves. This can, however, only be done wih pars of he earnings in marke value since he decision wheher an increase in he marke value of asses should increase he book value or he hidden reserves is subjec o some resricions. These resricions are differen for every asse class and herefore raher complex. In our model, we simplify by assuming ha a leas a porion y of he increase in marke value has
13 13 o be idenified as earnings in book values in he balance shee. 6 This means ha a leas he + amoun y( ) δ has o be credied o he policy reserve. The parameer y herefore A A 1 represens he degree of resricion in asse valuaion given by he regulaor. Furhermore, he insurer can reduce reserves (i.e. increase he book value of asses) wihou any resricions by selling asses whose marke value exceeds he book value. The decision, which surplus (i.e. ineres exceeding he guaraneed rae) is given o he insured has o be made by he insurance company s managemen every year. Our general model allows for any managemen decision rule a ime ha is F measurable, i.e. ha depends only on informaion available a ime. Therefore, i would be possible o analyze he effec of differen surplus disribuion mechanisms on he financial siuaion of he insurance company. In he numerical analysis, however, we will focus on one disribuion mehodology ha seems o prevail in Germany: In he pas, German insurance companies used o credi a raher consan rae of ineres o he policy reserves over years. When ineres raes came down and sayed low and sock markes plunged, hey used he hidden reserves ha had been accumulaed in earlier years o keep he surplus sable. Only when he reserves reached a raher low level, hey sared reducing he surplus. Therefore, we apply he following decision rule: A arge rae of ineres z > g is credied o he policy reserves, as long as he socalled reserve quoa R x = says wihin a given range [ b] L a;. Only if he reserve quoa becomes oo low (oo high) will he surplus be reduced (increased). 6 Noe ha for y=0, he insurance company is oally free in deermining he earnings on book values.
14 14 If he arge rae of ineres z is given o he insured a ime (i.e. a surplus of ( z g) L 1 is credied o he insured s accoun), he liabiliies L are given by g ( 1+ z) L 1 = L + ( z g) L 1 L (6). = As menioned above, our model also allows for dividends ha are paid o he owners of he insurance company. Again, he general model allows for any F measurable dividend paymen. For he numerical examples, we assume ha he dividend amouns o a porion α of any surplus credied o he policy reserves. Thus, if he arge rae z is given o he insured, we ge ( z g) L 1 A α (7) + = A and ( 1+ z) L 1 R (8). + = A The condiion for he reserve quoa a x b, i.e. a R ( 1 + z) L 1 b is fulfilled if and only if (( 1 + a)( 1 + z) + ( z g) ) L 1 A (( 1 + b)( 1 + z) + α( z g) ) L 1 α (9). In his case, exacly he arge rae of ineres z is credied o he insurance conracs. For he oher cases, we use he following decision rules: 7 7 Oher F measurable decision rules ha may apply for cerain companies can also be implemened in he model.
15 15 If crediing he arge rae z leads o a reserve quoa below a and crediing he guaraneed rae g leads o a reserve quoa above a, hen he company credis exacly ha rae of ineres o he policy holders ha leads o x = a. Hence, we have L a + α [ L ] ( + g) L + A (1 + g) ( a) = (10) and A + α 1 + a + α [ A (1 + g) ( 1 + a) L ] = A 1 (11). If even crediing he guaraneed rae of ineres leads o a reserve quoa level below a, i.e., ( 1+ a)( 1+ g) L 1 A (12), < hen he guaraneed rae of ineres is provided o he policyholders and he equiy holders do no receive anyhing, i.e., ( 1+ g) L 1 L and + A = A (13). = If crediing he arge rae of ineres z leads o a reserve quoa above he upper limi b, he company credis exacly ha rae of ineres o he policyholders ha mees he upper reserve quoa boundary x = b, i.e., L b + α [ L ] ( + g) L + A (1 + g) ( b) = (14) and
16 16 A + α 1 + b + α [ A (1 + g) ( 1 + b) L ] = A 1 (15). Finally, we wan o check wheher hese rules comply wih he minimum paricipaion + rae, i.e. wheher a leas he amoun y( ) δ is credied o he policy reserves. A A 1 Whenever our decision rules lead o a violaion of his rule, i.e. + [ y( A A ) gl ] L L < gl + δ (16), we increase he surplus such ha he minimum paricipaion rae is me by leing + ( + g) L + [ y( A A ) gl ] L δ (17), = and A + + [ δ y( A A ) gl ] 8 = A α 1 1 (16). 2.6 Shorfall We consider he life insurance company o defaul if a any balance shee dae =1,2,,T, he marke value of he asses is below he book value of he liabiliies, i.e., if R < 0. 9 We le he sopping ime τ be he firs balance shee dae, where a defaul happens or τ = T+1 if 0 1, K,T. R { } 8 Noe ha his can only happen, if he insurance company does no have enough freedom o hide asse price gains in hidden reserves, i.e. if he amoun δ y( A ) + A 1 ha has o be shown as an increase in book value as described earlier in his secion, leads o a higher book value han desired by he insurer. 9 Recen change in German legislaion allowed for socalled negaive hidden reserves, i.e. book values above marke values under cerain circumsances. In his model, we neglec he resuling effecs.
17 The shorfall probabiliy p P( T ) F 17 = τ is defined as he probabiliy ha a shorfall occurs a some balance shee dae afer he curren ime, given he informaion available a ime. I depends on parameers describing he regulaory framework, i.e. he guaraneed rae of ineres g, he minimum paricipaion rae δ, he resricion in asse valuaion y, parameers describing he insurance company s financial siuaion and managemen decisions, i.e. he curren reserve siuaion x, he porion of socks in he asse porfolio s, 10 he porion of surplus ha is paid ou o equiy holders α, he arge rae of ineres z, he arge range for he reserve quoa [ a;b], capial marke parameers, i.e. drif μ and volailiy σ of he asse porfolio, and he considered ime horizon, i.e. he remaining erm o mauriy of he produc T. 3. Analysis In wha follows, we will analyze he effec of he differen parameers on he shorfall probabiliy. I can easily be shown analyically ha whenever a parameer is changed ha leads o an increase in liabiliies and does no influence (or even decrease) he developmen 10 This could easily be replaced by some asse allocaion sraegy if we allow a changing asse allocaion.
18 18 of he asses, he shorfall probabiliy increases, and vice versa. Therefore, p is (ceeris paribus) increasing in g, z, δ, and α and decreasing in x, a, and b. In wha follows, we will perform more deailed analyses focusing on he ineracion of several parameers. Since in hese cases no analyical soluions exis, we use Mone Carlo simulaion mehods performing 10,000 simulaions per analyzed combinaion of parameers in order o calculae he shorfall probabiliy. For all our calculaions we fix = 10 unless saed oherwise. T, [ ; b] = [ 5%;30% ] a, δ = 90%, and α = 3%, 3.1 The influence of he iniial reserve siuaion In a firs sep, we calculae he shorfall probabiliy p 0 (x) as a funcion of he insurer s iniial reserve quoa for differen values of he guaraneed rae of ineres g { 2.75%, 4% } 11 and differen asse allocaions ( s { 10%, 30% } ). We assume he arge rae of ineres z o equal 5% and consider differen values for he resricion in asse valuaion y. For values of y close o 100%, i urns ou ha for companies wih low iniial reserves shorfall probabiliies are close o 100%. This is no very surprising because in his case here is almos no chance for he insurance company o increase is reserve accoun R over ime. In years of high asse reurns, almos all earnings are eiher given o policyholders or shareholders leaving lile poenial o build up reserves. On he oher hand, in years of low asse reurns, reserves are reduced o provide he guaraneed rae of ineres o he 11 This corresponds o he rae applicable in Germany o new business saring January 1, 2004 (2.75%) and o conracs aken ou beween January 1995 and June 2000 (4%).
19 19 policyholders accouns. This shows ha high values of y pose a high risk on an insurance company, i.e. highly resricive accouning rules in connecion wih high minimum paricipaion raes would be very dangerous for insurance companies offering cliquesyle guaranees. This issue will be analyzed in more deail in Secion 3.6. Figure 2 shows he shorfall probabiliy as a funcion of he iniial reserve siuaion x for a arge rae of ineres z=5% and differen values of he guaraneed rae of ineres (g = 2.75% and g = 4%) and differen sock raios (s = 10% and s = 30%) assuming ha a leas y = 50% of he earnings on marke value have o be idenified as earnings on book value. g = 2.75% g = 4% 100% 100% 90% 90% 80% 80% shorfall probabiliy 70% 60% 50% 40% 30% 20% shorfall probabiliy 70% 60% 50% 40% 30% 20% 10% 10% 0% 0% 5% 10% 15% 20% 25% 30% 0% 0% 5% 10% 15% 20% 25% 30% iniial reserve quoa iniial reserve quoa s=10% s=30% s=10% s=30% Figure 2 Shorfall probabiliy as a funcion of he insurer's iniial reserve quoa for y=50% Of course, he shorfall probabiliy is decreasing wih increasing iniial reserves. One can see from boh picures ha companies wih low reserves have o have low sock raios in order o keep shorfall probabiliies low. I is no surprising ha, all oher hings equal, conracs wih a guaraneed rae of ineres of 4% pose a higher risk o he insurance company han hose wih a guaraneed
20 20 rae of 2.75%, since here is less freedom in crediing profis and building up reserves. I is noiceable, hough, ha he absolue values of he shorfall probabiliies are raher high in boh cases. Furher analysis shows ha changing y from 50% o 0% hardly changes he resuls a all. This suggess ha i seems sufficien o have some (as opposed o complee) freedom in deermining earnings on book values, e.g. he possibiliy o hide 50% of he earnings on marke value. Since y = 50% seems o be a reasonable value ha is consisen wih accouning rules ha prevail, e.g., in Germany we keep his parameer fixed for he remainder of his analysis unless saed oherwise. 3.2 The ineracion of reserve siuaion and asse allocaion We now assume ha he insurance company has some olerable level of shorfall probabiliy, here p0 ( x, s) = 10%. We analyze he ineracion of iniial reserve siuaion and asse allocaion by calculaing which combinaions of x and s lead o his given shorfall probabiliy. This answers he quesion of which reserve quoa is necessary o back a given asse allocaion, or (equivalenly) which asse allocaion is admissible for a given reserve siuaion. Figure 3 displays he ineracion beween he insurer s iniial reserve quoa and he sock raio s for differen values of he guaraneed rae of ineres (g = 2.75% and g = 4%) and differen values of he arge disribuion (z = 5% and z = 6%).
21 21 g=2,75% g=4% 30% 30% 25% 25% iniial reserve quoa 20% 15% 10% iniial reserve quoa 20% 15% 10% 5% 5% 0% 0% 5% 10% 15% 20% 25% 30% 35% 0% 0% 5% 10% 15% 20% 25% 30% 35% sock raio sock raio z=5% z=6% z=5% z=6% Figure 3 Insurer s iniial reserve quoa as a funcion of he sock raio s For a olerable shorfall probabiliy of 10% and conracs wih guaranee g = 2.75% and arge rae of ineres z = 5%, we can see from he lef par of Figure 3 ha for a sock raio of s = 10% iniial reserves of x = 7% are sufficien. If he insurer wans o inves 30% of he asses in socks, 25% iniial reserves are required. Since socks and bonds are boh risky asses ha are no perfecly correlaed, i is no he porfolio wih 0% socks ha minimizes risk. In our model, any asse allocaion wih a sock raio below he minimum risk porfolio is dominaed as i represens greaer risk while providing lower reurns han he higher sock raios. As a rule of humb, figure 3 shows ha for any increase of 1% in sock raio, abou 1% more iniial reserves are necessary. Companies wih lower iniial reserves should have a lower porion of socks in heir asse porfolio. Clearly, conracs wih a higher guaraneed rae of ineres are of higher risk for he insurance company han conracs wih a low guaranee. In paricular, for conracs wih a
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