The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling Universiy of Ulm, Germany phone: +49 731 5031183, fax: +49 731 5031239 alkli@mahemaik.uni-ulm.de Andreas Richer Assisan Professor Illinois Sae Universiy, Normal, IL, USA phone: +1 309 438 3316 ariche@ilsu.edu Jochen Ruß Managing Direcor Insiu für Finanz- und Akuarwissenschafen, Ulm, Germany phone: +49 731 5031233, fax: +49 731 5031239 j.russ@ifa-ulm.de very firs draf
2 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling, Andreas Richer, Jochen Russ Absrac Tradiional life insurance policies in many markes are sold wih minimum ineres rae guaranees. In he case of a so-called clique syle guaranee, 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 so-called 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. The primary focus of mos exising lieraure in his area is on he fair (i.e. risk-neural) valuaion of life insurance conracs. Since mos insurers do no apply risk-neural (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 so-called real-world 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.
3 1. Inroducion Tradiional life insurance policies in many markes are sold wih minimum ineres rae guaranees. Quie common are guaranees on a poin-o-poin basis: A mauriy of such 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 shor-erm 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 so-called clique syle 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 policyholder s accoun every year. Insurers apply a sraegy which is ofen referred o as he average ineres principle (see, e.g. [Gr/Jo 00]): Companies aemp o hold he surplus credied o he policyholder s accoun as consan as possible, 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 long-erm 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
4 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 [Gr/Jo 02], 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 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 [Br/Va 97], [Gr/Jo 00], [Je/Jo/Gr 01], [Mi/Pe 01], [Ha/Mi 02], [Gr/Jo 02], [Ba 03], and [Ta/Lu 03]. For a poin-o-poin guaranee framework, [Br/Va 97] compue closed-form 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 risk-neural 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 jus-menioned approach, [Gr/Jo 00] consider clique syle 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. [Gr/Jo 00] 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.
5 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 of only limied explanaory value. [Mi/Pe 01] 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 [Gr/Jo 00] 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, [Mi/Pe 01] derive numerical resuls on he influence of various parameers on he conrac value. 2 [Gr/Jo 02] discuss a model based upon he framework used by [Br/Va 97]. 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 above-menioned papers incorporae he risk of a conrac s or he insurer s defaul, he primary focus is on he fair (i.e. risk-neural) valuaion of he life insurance conrac. Since mos insurers do no apply risk-neural (or risk-minimizing) hedging 1 2 [Je/Jo/Gr 01] exend he findings of [Gr/Jo 00]. As one exension, among ohers, hey inroduce moraliy risk. Anoher paper ha incorporaes moraliy risk as well as he surrender opion is [Ba 03]. Conrasing he mechanism discussed in [Mi/Pe 01], 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 [Ha/Mi 02].
6 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 so-called real-world 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 clique syle guaranees hroughou his paper. Cerain oher feaures of our model are also 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 model he surplus disribuion mechanism, we model he asses and he insurance conrac. 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 above menioned 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 he model as simple as possible o be able o focus on he basic effecs. Firs, we model 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.
7 2.1 The insurer s iniial siuaion We use a simplified illusraion of he insurer s financial siuaion given in figure 1. 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 policyholder s 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. 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 ( ) + σ ( ) dw, (1) 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
8 coefficien of ρ = -0.1. 3 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 and, hence, we have 2 σ ( s) µ ( s) ds+ σ ( s) dws 0 2 0 A = A0e 2 σ ( s) µ ( s) ds+ 1 2 1 A = A 1 e σ ( s) dws. - If here are any 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+ + 1 2 1 A = A 1 e σ ( s) dws and A + = A D, which can be used handily in Mone Carlo algorihms. 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 erm-fix 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 L he insurer s liabiliies and is given by P T. 4 L 0 3 4 We used daa of a German sock index (DAX) and a German bond index (REXP) of he pas fifeen years 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%. Alhough i migh seem srange o ignore moraliy effecs as well as charges for any moraliy benefi ha exceeds he policy value a ime of deah, in he analysis of a life insurance company s asses and liabiliies, his makes sense under he following wo assumpions: 1) On average, new business roughly compensaes for
9 2.4 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. German life insurance companies are no allowed o hold a porion of socks greaer han 35% in heir asse porfolio. Therefore, we allow [ 0;0.35] s. Under German legislaion here has o be a year-by-year clique-syle guaranee on he liabiliies. Currenly, German life insurance companies guaranee he policyholders a minimum rae of ineres g = 2.75%. 5 This guaranee has o be given for he whole erm of he policy, even if he guaraneed ineres rae will be changed by he regulaors 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 exceeding he guaraneed growh of he liabiliies have o be credied o he policyholders accouns. This so-called minimum paricipaion rae was inroduced o make sure ha policyholders are no pu a a disadvanage agains shareholders. 2.5 Developmen of he financial siuaion over ime As menioned above, we consider a year-by-year guaranee on he liabiliies. Given he liabiliies L 1 a ime -1, he guaraneed liabiliies L = L (1 ). g 1 + g g L a ime are given by 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 moraliy and surrenders. 2) The risk premiums are calculaed properly such ha he insurer incurs no significan profi or loss upon deah. 5 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.
10 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 raher complex. In he model, we simplify by assuming 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 This means + ha a leas he amoun y( ) δ has o be credied o he policy reserve. The parameer A A 1 y herefore 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 reserve quoa R z > g is credied o he policy reserves, as long as he so-called x = says wihin a given range [ b] L low (oo high) will he surplus be reduced (increased). a;. Only if he reserve quoa becomes oo If he arge rae of ineres z is given o he insured a ime (i.e. a surplus of z g) L is credied o he insured s accoun), he liabiliies L are given by ( 1 g ( 1 + z) L 1 = L + (1 + g) L 1 ( z g) L 1 L. = 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 6 Noe ha for y=0, he insurance company is oally free in deermining he earnings on book values.
11 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 is given o he insured, we ge and A α ( z g) L 1 + = A ( 1+ z) L 1 R. + = 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 or A [(( + a)( 1 + z) + α ( z g) ) L ;(( 1 + b)( + z) + α( z g) ) L ] 1 1 1 1 (( 1 a)( 1 + z) + ( z g) ) L 1 A (( 1 + b)( 1 + z) + α( z g) ) L 1 + α. 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 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 + 1 1 + a + α [ L ] ( + g) L + A (1 + g) ( a) = 1 1 1 + 1 A α 1 + a + α [ A (1 + g) ( 1 + a) L ] = 1. and i.e., If even crediing he guaraneed rae of ineres leads o a reserve quoa level below a, ( 1 + a)( 1 + g) L 1 x, < a A < 7 Again, oher decision rules ha may apply for cerain companies can also be implemened in he model.
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. + = 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 A + 1 1 + b + α [ L ] ( + g) L + A (1 + g) ( b) = 1 1 1 + 1 A α 1 + b + α [ A (1 + g) ( 1 + b) L ] = 1. and Finally, we wan o check wheher hese rules comply wih he minimum paricipaion + rae, i.e. wheher a leas he amoun y( ) A A 1 Whenever our decision rules leads o a violaion of his rule, i.e. + [ y( A A ) gl ] L L < gl + δ, 1 1 1 1 δ is credied o he policy reserves. we increase he surplus such ha he minimum paricipaion rae is me by leing + ( + g) L + [ y( A A ) gl ] L δ and = 1 1 1 1 + [ y( A A ) gl ] A α. 8 + = A 1 1 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 + L + R A R < 0 or = < 1 L L. 9 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 according o Secion 0, leads o a higher book value han desired by he insurer. 9 Recen change in legislaion allowed for so-called negaive hidden reserves, i.e. book values above marke values under cerain circumsances. In his model, we neglec he resuling effecs.
13 We le he sopping ime τ be he firs balance shee dae, where a defaul happens or τ = T+1 if 0. R { 1, K,T} The shorfall probabiliy p = p x, s, g, δ, y, µ, σ, α, z, a, b, ( T )) = P( τ T F ) ( 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 he insurer s curren reserve siuaion x, he porion of socks in he asse porfolio s, 10 he guaraneed rae of ineres g, he minimum paricipaion rae δ, he resricion in asse valuaion y, he porion of he surplus ha is paid ou o equiy holders α, he arge rae of ineres z, he arge range for he reserve quoa [ a; b], and he considered ime horizon, i.e. he remaining erm o mauriy of he produc T-. Noe ha some of hese parameers describe he insurance company s financial siuaion (x, s), ohers describe he regulaory framework (g, δ, y), or he financial markes (µ, σ), and anoher se of parameers describe he behaviour of he insurance company s managemen by modelling managemen decisions (α, z, a, b). 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 of he asses, he shorfall probabiliy increases. The same is rue, if he liabiliies are no affeced and he developmen of he asses is decreased. Therefore, p is (ceeris paribus) increasing in g, z, δ, and α and decreasing in x, a, and b. Since hese resuls are, however, raher rivial, we will focus on more complex analyses considering 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 saed oherwise. T, [ a ; b] = [ 5%;30% ], δ = 90%, and α = 3%, unless 10 This could easily be replaced by some asse allocaion sraegy if we allow a changing asse allocaion.
14 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% } and differen asse allocaions ( s { 10%, 30% } ). We assume he arge rae of ineres z o equal 6% 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 policy holders accouns. This shows ha his choice of y is raher unrealisic and ha highly resricive accouning rules in connecion wih high minimum paricipaion raes would be very dangerous for insurance companies offering clique syle 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 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. 11 11 This corresponds o he rae applicable saring January 1, 2004 (2.75%) and applicable for conracs aken ou beween January 1995 and June 2000 (4%).
15 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 siuaion 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 rae of 2.75%, since here is less freedom in crediing profis and building up reserves. I is noiceable ha he absolue values of he shorfall probabiliies are raher high, implying ha arge raes of 6% are very risky in he curren marke framework. Furher analysis shows ha changing y from 50% o 0% hardly changes he resuls a all. This suggess ha i is no imporan for insurance companies o have he possibiliy o hide all heir asse reurns. I raher 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, 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 a olerable level of shorfall probabiliy, here p ( x, s) 10%. We analyze he ineracion of iniial reserve siuaion and 0 = asse allocaion by calculaing which combinaions of x and s lead o he given olerable level
16 of 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%). 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 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. A firs view i migh be surprising ha he minimum reserve quoa does no occur a a sock raio of 0%. The reason for his is however simple: Since socks and bonds are boh risky asses ha are no perfecly correlaed, i is no he porfolio wih 0% socks ha is he one wih he leas risk. In our model, any asse allocaion wih a sock raio below he minimum risk porfolio would no be favourable because i leads o a higher risk and a he same ime lower expeced reurn. 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
17 guaraneed rae of ineres of 4%, insurance companies wih a low reserve quoa (<11%) are no able o achieve a shorfall probabiliy of 10% a all. In boh picures, he disance beween he wo lines for differen arge raes of ineres z hardly depends on s and amouns o approximaely 5% of iniial reserves. Therefore, if an insurer wans o keep he shorfall probabiliy sable, he reserve quoa should be increased by 5% if he arge rae of ineres is increased by 1%. 3.3 The ineracion of surplus disribuion and asse allocaion The arge rae of ineres z and he asse allocaion (characerized by he sock raio s) are boh subjec o managemen decisions. Therefore, life insurance companies should be highly ineresed in he effec hese parameers have on he shorfall probabiliies. Again, we fix some olerable level for he shorfall probabiliy and analyze, which combinaions of z and s lead o his level. Since shorfall probabiliies are highly sensiive o changes in he reserve siuaion, companies wih 5% reserves are in a differen risk caegory han companies wih 20% reserves. While for companies wih high reserves we can deermine combinaions of s and z ha lead o a shorfall probabiliy of 5%, his is impossible for companies wih low reserves since all admissible combinaions of parameers yield higher shorfall probabiliies. Therefore, we se he olerable level of shorfall probabiliy for he company wih low reserves o p ( z, s) 25% and le p ( z, s) 5% for he company wih high reserves. These differen 0 = 0 = levels of shorfall probabiliies have o be kep in mind when looking a he absolue values of he resul shown above. Figure 4 shows he resuls for differen values of he iniial reserve quoa (x = 5% and x = 20%) and differen values of he guaraneed rae of ineres (g = 2.75% and g = 4%).
18 7,0% x=5% shorfall probabiliy = 25% 7,0% x=20% shorfall probabiliy = 5% 6,5% 6,5% arge rae of ineres 6,0% 5,5% 5,0% arge rae of ineres 6,0% 5,5% 5,0% 4,5% 4,5% 4,0% 0% 5% 10% 15% 20% 25% 30% 35% 4,0% 0% 5% 10% 15% 20% 25% 30% 35% sock raio sock raio g=2,75% g=4% g=2,75% g=4% Figure 4 Targe rae of ineres as a funcion of he sock raio s For companies wih high reserves, he guaraneed rae of ineres is of raher minor influence. An increase of he guaraneed rae of ineres from 2.75% o 4% can be compensaed by reducing he arge rae of ineres by 1% leaving he asse allocaion unchanged. Alernaively i can be compensaed by reducing he sock raio by abou 3-5% and leaving he arge rae of ineres unchanged. For insurers wih low iniial reserves, here is a significan difference beween conracs wih differen guaranees. While (in spie of he very large level of shorfall probabiliy) for a guaraneed rae of reurn of 4% only very conservaive combinaions of z and s are admissible, a guaranee rae of 2.75% allows he company o provide 5% arge disribuion while holding 20% socks (of course a he same high level of shorfall probabiliy). Furhermore, for companies wih low reserves a sligh increase in he sock raio requires a raher big decrease in he arge disribuion o keep he shorfall probabiliy sable. For example, if such a company increases heir sock raio from 15% o 20%, i should lower he arge rae of ineres for conracs wih a guaraneed rae of 2.75% by 200 basis poins. A company wih higher reserves only needs o reduce he arge rae by 50 basis poins. Thus, for companies wih low reserves, shorfall probabiliies are very sensiive wih respec o changes in he guaraneed rae of ineres and in he sock. Therefore, from a risk managemen poin of view, life insurance companies should lower heir sock raios if
19 reserves end o go down. In he pas, hese measures were no aken by mos German life insurers leading o he curren problems. 12 3.4 The ineracion of guaranees and asse allocaion Under German regulaion, companies are no allowed o hold more han 35% socks in heir asse porfolio. Regulaors impose he exac same barrier for all life insurers, considering neiher he guaraneed rae of ineres in he companies insurance liabiliies nor he reserve siuaion. The following analysis will show ha a reasonable regulaion should consider hese issues. Figure 5 shows combinaions of he guaraneed rae of ineres and he sock raio ha yield a given shorfall probabiliy. Again, we consider differen iniial reserve quoas (x = 5% and x = 20%) and differen values of he arge disribuion (z = 5% and z = 6%). 4,00% x=5% shorfall probabiliy = 25% 4,00% x=20% shorfall probabiliy = 5% guaraneed rae of ineres 3,75% 3,50% 3,25% 3,00% guaraneed rae of ineres 3,75% 3,50% 3,25% 3,00% 2,75% 0% 5% 10% 15% 20% 25% 30% 35% 2,75% 0% 5% 10% 15% 20% 25% 30% 35% sock raio sock raio z=5% z=6% z=5% z=6% Figure 5 Guaraneed rae of ineres as a funcion of he sock raio s Again, companies wih low reserves and hose wih high reserves are in a differen risk caegory. Thus, we allow for a shorfall probabiliy of 25% for companies wih low reserves and a shorfall probabiliy of 5% for companies wih high reserves. Assuming ha he sock raio exceeds he sock raio of he minimum risk porfolio, obviously higher guaranees require lower sock raios for a given shorfall probabiliy. 12 Conrary, insurers increased heir sock raio hoping o compensae for low ineres raes.
20 From he lef par of Figure 5, we can see ha an increase in he arge rae of ineres z from 5% o 6% can be compensaed by a reducion of he sock raio of abou 3%-5%. The wo curves on he lef side are raher close o each oher. This means ha changing he arge rae of ineres in companies wih low reserves has a raher minor effec. The wo curves on he righ are raher far away from each oher suggesing ha for companies wih high reserves he effec of he arge rae of ineres is very high. Combining he analyses of Secions 3.3 and 3.4, we find he following remarkable resuls: Alhough, of course, he absolue level of he shorfall probabiliies is enirely differen for companies wih low reserves and high reserves, he shorfall probabiliy of a company wih low reserves is very sensiive o changes in g and s and less sensiive o changes in z. The shorfall probabiliy of a company wih high reserves is very sensiive o changes in z and s bu less sensiive o changes in g. Therefore saic regulaions, ha impose cerain rules and limis for all companies wihou considering he individual financial srengh of a given insurer, appear o make lile sense. From a risk managemen poin of view resricions on s and g only make any sense if heir ineracion and he reserve siuaion of he company are aken ino accoun, as well. Approaches like he required Resilience Tes in he UK 13 or he recenly inroduced (bu no ye very sophisicaed) sress es in Germany seem more reasonable. 3.5 The ineracion of guaranees and surplus disribuion In he pas, when ineres raes were quie high in comparison o he guaraneed raes of ineres, guaranees seemed no o be an issue a all. The oal ineres (guaranee rae + surplus) credied o he policy reserves was held consan by German insurance companies, no only consan over ime bu also consan over all generaions of conracs, irrespecive of he corresponding guaraneed rae of reurn. Thus, if an insurer decided o credi 6% o he insured, conracs wih a guaraneed rae of 4% would have received 2% surplus while conracs wih 2.75% guaranee would have received 3.25% surplus. In 2004, for he firs ime a few insurance companies in Germany sared o provide a lower oal ineres o he policy reserves of conracs wih a high guaraneed rae of ineres and higher oal ineres o he policy reserves of conracs wih a low guaraneed rae of ineres. The reason hey gave was ha conracs wih a lower guaraneed rae will ge less in bad years and should herefore be 13 Cf. e.g., [Be 00]
21 compensaed by receiving more in good years. We now examine if, from a risk managemen poin of view, such a differeniaion in surplus disribuion beween conracs wih differen guaranees is appropriae. Figure 6 shows combinaions of he arge rae of ineres z and he guaraneed rae of ineres g for differen iniial reserve quoas (x = 5% and x = 20%). For companies wih low reserves we allow he shorfall probabiliy o be 25%, for companies wih high reserves, we allow 5% shorfall probabiliy. For boh, we assumed an asse allocaion conaining 10% of socks. 7,0% x=5% shorfall probabiliy = 25% 7,0% x=20% shorfall probabiliy = 5% 6,5% 6,5% arge rae of ineres 6,0% 5,5% 5,0% arge rae of ineres 6,0% 5,5% 5,0% 4,5% 4,5% 4,0% 4,0% 2,75% 3,00% 3,25% 3,50% 3,75% 4,00% 2,75% 3,00% 3,25% 3,50% 3,75% 4,00% guaraneed rae of ineres guaraneedrae of ineres s=10% s=10% Figure 6 Targe rae of ineres as a funcion of he guaraneed rae of ineres Since higher guaranees pose a higher risk on he insurer, from a risk managemen poin of view, i is appropriae for companies o compensae for his by providing a lower arge disribuion o he conracs wih a higher guaranee. I can be seen from he lef figure ha for companies wih low reserves, reducing he arge rae by 1 percen poin for each percenage poin of a higher guaranee keeps he shorfall probabiliy roughly consan. The char on he righ side of figure 6 is less seep, which shows ha for companies wih high reserves a lower differeniaion in surplus disribuion is appropriae. This shows ha providing a consan arge disribuion o conracs wih differen guaranees as done in he pas is accepable as long as reserves are high. Once reserves go down, he life insurance
22 companies have o reac and change heir surplus disribuion mechanism if hey wan o have conracs of equal risk in heir liabiliy porfolio. 14 3.6 The ineracion of regulaory and legal requiremens So far, our analysis focused on parameers ha are subjec o managemen decisions assuming legal requiremens o be consan parameers. In he las secion of his chaper we will examine he ineracion of he differen regulaory and legal requiremens. We fix any parameers ha are subjec o managemen decision, i.e. we se z = 6% and s = 10%, and he iniial reserves x=20% for he remainder of his secion. In our model, low values of he resricion in asse valuaion y correspond o a high level of freedom wih respec o he applicaion of accouning rules. Furhermore, low values of he minimum paricipaion rae δ correspond o a high level of freedom wih respec o he surplus disribuion. Figure 7a shows he minimum paricipaion rae δ as a funcion of he guaraneed rae of ineres g for a given shorfall probabiliy of 5%. We show his for differen values of y (y = 70% and y = 90%). Figure 7b shows he minimum paricipaion rae δ as a funcion of he resricion in asse valuaion y for differen values of he guaraneed rae of ineres (g = 2.75% and g = 4%) and a shorfall probabiliy of 7.5%. 100% x=20% shorfall probabiliy = 5% 100% x=20% shorfall probabiliy = 7.5% minimum paricipaion rae 80% 60% 40% 20% minimum paricipaion rae 90% 80% 70% 60% 50% 40% 0% 30% 2,75% 3,00% 3,25% 3,50% 3,75% 4,00% 50% 60% 70% 80% 90% 100% guaraneed rae of ineres resricion in asse valuaion y=70% y=90% g=2.75% g=4% Figure 7a Minimum paricipaion rae as a funcion of he guaraneed rae of ineres (lef) Figure 7b Minimum paricipaion rae as a funcion of he resricion in asse valuaion (righ) 14 See [Kl/Ru 04] for a more deailed analysis of he quesion of surplus differeniaion.
23 From Figure 7a we can see ha for low values of δ here is almos no difference beween he differen curves for y = 70% and y = 90%. This, if he companies have a high level of freedom in he surplus disribuion, hey obviously do no need addiional freedom in accouning rules. Furher resuls show ha for companies wih low reserves he guaraneed rae of ineres has a very high influence on he shorfall probabiliy. For hem sligh changes in g require raher major changes in δ. For companies wih high reserves (in he case y = 90%), an increase in he guaraneed rae of ineres from 2.75% o 3.25% can be compensaed by reducing he minimum paricipaion rae from 77% o 50%. Therefore, regulaors should allow for lower values of δ for conracs wih a higher guaranee. This again implies ha regulaors should allow for he oal ineres (guaranee + surplus) o be he lower, he higher he guaranee. We also performed simulaions for y 50%. In his case, however, he minimum paricipaion rae has almos no influence on he shorfall probabiliies implying ha a high degree of freedom in accouning compensaes for sric minimum paricipaion raes. Figure 7b also implies ha for all guaraneed raes of ineres, an increase in y mus be followed by less resricive regulaions on surplus disribuion. If accouning rules are o be changed bringing book values closer o marke values, he minimum surplus disribuion parameer δ has o be reduced or even abolished. Oherwise, he currenly predominan form of life insurance will impose a significan risk on life insurance companies. 15 4. Summary and Oulook Our analysis shows ha shorfall probabiliies of life insurance companies are affeced by heir financial siuaion, by regulaion, marke developmen and managemen decision. The ineracion of hese parameers is raher complex and he superimposing effecs are no always easy o undersand. Neverheless, since in recen years low ineres raes and plunging sock markes posed new risks on insurance companies, i has become more and more imporan o examine and undersand hese effecs. 15 This should be considered by regulaors upon inroducing inernaional accouning sandards.
24 We have developed a very general model ha provides a framework o analyze he ineracion of hese parameers and heir influence on he financial siuaion of an insurer. In he presen paper we have presened only a small par of he possible analyses ha can be performed wihin his model. We have, however focussed on analyzing wha we believe o be he mos imporan effecs. We found ha an insurer s shorfall probabiliies are increasing wih an increasing guaraneed rae of ineres and wih an increasing arge rae of ineres. For companies wih a low reserve siuaion he influence of he guaranee rae is higher. For companies wih high reserves, i is he arge rae ha plays he major role. From a risk managemen perspecive, insurance companies should provide a lower arge rae of ineres o conracs wih a high guaranee and a higher arge rae of ineres o conracs wih a low guaranee. Of course, shorfall probabiliies are also increasing wih a decreasing iniial reserve quoa. Concerning he asse allocaion, i is no he porfolio wih 0% socks ha conains he leas risk. The shorfall probabiliy (as a funcion of he sock raio s) achieves a minimum a a sock raio beween 9% and 12% depending on he oher parameers. Regulaors need o consider he financial siuaion of he individual insurer. Furhermore, in paricular when clique syle guaranees are included, insurers can only manage such guaranees if hey are given a cerain degree of freedom in disribuing he asses. A marke value oriened accouning in connecion wih high minimum paricipaion raes would pose an unmanageable risk on insurers ha have issued such conracs. The raher high shorfall probabiliies hroughou our analyses are raher alarming. However, i can be assumed ha he absolue values of he shorfall probabiliies are lower in all insurance markes where insurers have he possibiliy o provide erminal bonuses as well as ongoing bonuses. Therefore, inroducing a erminal bonus in our model will be he nex sep. Fuure research should also analyze he influence of a pah dependen asse allocaion In his case, he insurer s asse allocaion is given by some sraegy, meaning ha he sock porion s is given as a funcion ( x, g,, y, µ, σ, α, z, a, b, ( T )) s δ aking values in [ ;1] To furher undersand he influence of he capial markes on shorfall probabiliies i would be ineresing o perform sensiiviy analyses wih respec o drif and volailiy of 0.
25 socks and bonds. A nex sep could be he inclusion of a differen asse model wih more han wo asse classes. Furhermore, i could be ineresing o find a more deailed model for he liabiliy side of an insurer s balance shee by modelling a porfolio of differen kinds of insurance conracs including moraliy effecs. References [Ba 03] [Be 00] [Br/Va 97] 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. 461-487. Benz, S. (2000): Applied Asse Liabiliy Managemen Resilience Tesing in he Unied Kingdom. IFA-Verlag, Ulm, Germany. 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. 637-694. [Je/Jo/Gr 01] 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. 57-84. [Gr/Jo 00] [Gr/Jo 02] [Ha/Mi 02] 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. 37-57. 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. 63-91. Hansen, M. and Milersen, K. R. (2002): Minimum Rae of Reurn Guaranees: The Danish Case. Scandinavian Acuarial Journal, Vol. 2002, No. 4, pp. 280 318.
26 [Ha/Pl 81] [Kl/Ru 04] [Mi/Pe 01] [Ta/Lu 03] Harrison, J. M. and Pliska, S. R. (1981): Maringales and Sochasic Inegrals in he Theory of Coninuous Trading. Sochasic Processes and heir Applicaions 11, pp. 215 260. Kling, A. and Russ, J. (2004): Differenzierung der Überschüsse Berug an reuen Kunden oder finanzmahemaische Nowendigkei? Versicherungswirschaf, 4/2004, pp. 254-256. Milersen, K. R. and Persson, S.-A. (2001): Guaraneed Invesmen Conracs: Disribued and Undisribued Excess Reurn. Working Paper, Odense Universiy. Tanskanen A. J., Lukkarinen J. (2003): Fair valuaion of pah-dependen paricipaing life insurance conracs. Insurance: Mahemaics and Economics, Vol. 33, No. 3, pp. 595-609.