Dynamc Asse Lably Managemen: A Prof Tesng Model for Swss Penson Funds Bersch Ludmla Ebelng Sven Rechln Andreas ECOFIN Invesmen Consulng AG Sgnausrasse 6 P.O. Box 832 urch Swzerland Tel: +4 43 499 3939 Fax: +4 43 499 394 E-mal: ludmla.bersch@ecofn.ch sven.ebelng@ecofn.ch andreas.rechln@ecofn.ch Absrac Prof Tesng an esablshed concep n he lfe nsurance ndusry s developed and appled o self-governed self-nsured DC penson schemes n Swzerland. The dynamcs of he penson scheme s populaon are modelled by Markov chans. Ths allows for accuraely capurng he mpacs of dfferen personnel polces such as reducon or ncrease of saff promoon of early reremen ec. Prof profles cash flows fundng raos and oher key characerscs are calculaed for DC schemes akng no accoun he Swss regulaory framework. I s shown ha he close o realy modellng of he penson scheme s dynamcs whn he Prof Tesng Model provdes an approprae and relable bass for asse-lably managemen and helps o deermne he opmal nvesmen sraegy. Keywords Rsk Conrol Prof Tesng Asse Lably Managemen Dynamc Penson Fund Modellng Page / 3
Inroducon. General Informaon abou Swss Penson Funds The prncple of occupaonal provsons for reremen deah and dsably n Swzerland has been pu hrough s paces by he me he so-called Three Pllar Sysem was added o he Swss Consuon n 972. The Swss Federal Old Age and Survvng Dependans Insurance AHV n conuncure wh he dsably nsurance IV forms he Frs Pllar. Togeher wh he Second Pllar mandaory occupaonal provsons n force snce 985 s supposed o provde adequae coverage for he assured and hs famly. Coverage s consdered adequae when benefs provded n accordance wh he Swss Federal Law on Occupaonal Reremens Survvors and Dsably Penson Plans BVG ogeher wh AHV allow mananng he accusomed sandard of lvng. Ths condon s consdered sasfed when he benefs provded by he wo pllars oal around 6% of he former salary assumng all requred conrbuons have been pad. Accordng o he BVG every employer has o provde all hs employees wh a guaraneed mnmum penson scheme n a fully funded plan. The maory of penson schemes provde more generous benefs han requred under BVG ha manly serves as a benchmark. However BVG regulaons and mnmum benefs mus be ncluded n he company plan and are guaraneed. The schemes generally provde he followng benefs: Reremen pensons he benef formula s ofen 6% o 7% of fnal salary afer 4 years servce. Alhough reremen pensons are no legally or formally ndexed penson schemes usually gran ncreases on an ad hoc bass; Pre- and pos-reremen spouses pensons of 6% and orphans pensons of % o 2% based on he member s proeced/acual penson. A deah benef of one or wo mes salary s somemes pad where here are no survvors pensons; Long-erm dsably emporary and lfeme dsably pensons equal o servce proeced reremen pensons; The vesed amoun or vesed benef for leavng employees. For defned conrbuon DC schemes he vesed amoun wll be he oal of he employee s and employer s conrbuons plus neres creded unl he day of leavng. For defned benef DB schemes wll be he value of he accrued benefs based on he rao of compleed pensonable servce o proeced pensonable servce. The possbly o pledge or draw money from he company s penson scheme o nves n propery used by an employee and hs or her famly. The amoun whch can be pledged or whdrawn depends on he vesed benef. Top-ha penson plans and specal arrangemens exs for senor execuves very ofen n form of prof sharng sock opon plans equy lnked savng accouns and benefs for a salary hgher han he specfed value. Larger plans are self-nsured or parally covered hrough nsurance of rsk benefs. Law does no resrc acuaral mehods. Swss penson and rsk premums are arff raed bu profsharng bonuses are pad. Penson mones are requred by law o be deposed n a foundaon kep compleely separaely from he company and s balance shee. Page 2 / 3
In hs paper we focus on self-governed or self-nsured penson schemes se up as foundaons.e. legal enes n her own rgh governed by a board of rusees. The am of hs paper s o creae a sochasc Markov based model for Swss penson schemes ha capures he complexy of penson schemes and hus goes beyond usng smply Mone Carlo smulaons. Ths allows nvesgang he nfluence of dfferen economcal facors on he fundng rao and conrbuon rae..2 Prof Tess n Lfe Insurance and for Penson Schemes For more han hry years prof esng has been recognzed as a maor ool avalable o acuares nvolved n produc developmen. Many papers have been wren on he use of prof ess n he calculaon of gross premums. The algebra for developng prof profles s well documened n papers such as Smar [977] Darbellay/Veraguh [998] or Koller [2]. A prof es uses proecon mahemacs o esablsh he prospecve prof profle of a polcy on a gven se of assumpons. The resulng prof profle s dscouned a a rsk rae o gve he presen value of fuure profs PVFP. A maor srengh of prof esng over convenonal gross premum formulae s s ably o ncorporae he sauory reservng bass ndependen of he assumed rae of reurn on one he hand and o use an emprcal bass such as even probables on he oher hand. The prof profle derved from a prof es s he sream of profs whch flow from he polcy over s lfeme. Prof or surplus s defned as he excess of ncome over expendures durng any perod where expendures nclude he necessary ncrease n valuaon reserves. The shape of he prof profle vares grealy from produc o produc and wll depend upon he assumpons ncorporaed no he prof es. In a lfe nsurance company pars of he prof are dsrbued o shareholders and/or polcyholders n a penson scheme however could be only dsrbued o s members as a bonus or used for conrbuon holdays. There are wo mporan aspecs of rsk for penson funds: he probably of havng a fundng rao below one shorfall rsk and he absolue amoun of beng underfunded. To esmae he behavour of hese mporan characerscs we use he dea of Prof Tesng and adop he sandpon of he penson fund exper. Thus we calculae he expeced sauory prof or loss for each year of he proecon horzon n he same manner as he or she does for he acuaral balance of a penson scheme..3 Invesmen Obecves for Penson Schemes and Prof Tesng The BVG formulaes resrcons and obecves for Swss penson schemes. Frs of all shall be guaraneed ha he penson oblgaons could be me permanenly. Ths obecve s subdvded no a shor-erm a md-erm and a long-erm componen: In every year of a plannng-perod he fund needs o have enough lqud asses o mee he penson paymens already due Vesed benefs due need o be pad Assured benefs need o be covered by he asses suffcen fundng rao. The penson scheme has o acheve he lowes possble and sable level of conrbuons. All hese obecves have o be reaed for he acual srucure and he expeced evoluon of he populaon of he penson scheme. Therefore he nvesmen prncple should be o maxmze he rae of asse reurn bu consderng he fnancal rsk capacy of he penson scheme and he regulaory sauory resrcons. The fnancal rsk capacy of he penson scheme s Page 3 / 3
deermned by lqudy requremens and developmen and commmen of capal s nduced by penson oblgaons lables and free pars of he rsk reserves. Prof esng s a very powerful vehcle o make sure he nvesmen sraegy sus such obecves. 2 Dynamc Managemen of Swss Penson Funds 2. The Markov Model The Markov Model s an exremely useful nsrumen o model lfe nsurances and annues snce provdes a level of generaly suable for our purpose. In lfe nsurance applcaons of he Markov model one has o deermne acuaral values of paymens ha are conngen upon he say n ceran saes or upon ransons beween he saes. Transons beween saes are governed by he adoped ranson probables. They sasfy he so-called Markov propery whch mples ha he fuure course of he sochasc process does no depend on earler vsed saes. The evoluon of he populaon of a penson fund s modelled wh an nhomogeneous dscree me Markov chan snce ranson probables largely depend on he age of he nsured. To hs respec our paper uses he dea of Ruhdorfer [988]. To follow he dynamc evoluon of a penson fund s necessary o calculae he value of ncomes and expendures of he fund. Furhermore s necessary o compue he presen value of all fnancal flows ha are nvolved. For he purpose of dscounng and accumulaon we wll use he usual compounded neres rae model. There are fve groups of parcpans n a self-governed penson scheme n Swzerland: employees sae A dsably pensonss sae I early and elderly rered pensonss saes EP and P wdow s and wdower s pensons sae W orphan s pensons and pensons for chldren of rerees and dsabled persons. We do no model he laer ype of pensons. They are pad as a lump sum equal o he presen value of such a penson. There are wo absorbng saes n he model: he saes Deah and Resgnaon. They are grouped ogeher on he sae dagram and used as a reserve o hre new employees accordng o he employer s personnel polcy nroduced hrough dsmssal or resgnaon raes and he new generaon densy. Fgure shows he ranson dagram for he Markov model of a Swss penson scheme each arrow denoes a possble ranson from one sae o anoher. The dashed arrow from he sae Deah o he sae W represens new wdows and wdowers hred n he Sae W n case of deah of penson scheme members such as employees dsabled persons or rerees. There are no benefs n case of deah of wdows and wdowers. Page 4 / 3
Sae Rered Pensoner early & elderly Recrumen new employees Employees Sae Resgnaon wh vesng benefs Sae Dsably Pensoner Sae Deah Wdower Fgure. Markov model sae dagram for a penson scheme. Arrows reurnng o he same sae denoe he possbly o say n he same sae e.g. o be an employee and no o ge dsabled or rered for a perod of me. Recrumen s shown as a separae sae o sress he fac ha only new employees can ener he penson scheme from ousde. The saes I dsabled EP and P rered can be reached only hrough he sae A employees. We do no model specal cases such as a merger beween wo frms or an negraon of a porfolo wh dsabled pensoners and rerees no he penson scheme. On he oher hand rerees and dsabled can no cancel her polces and leave alve he penson scheme bu only va he sae Deah. As we have already menoned earler new wdows and wdowers ener he scheme from ousde bu her age and gender depend on he dead nsured person and her number can no be vared by personnel polcy. The probably o be remarred for a wdower s zero so she or he can no leave he penson scheme alve. A more dealed descrpon of sae spaces for Markov models relevan o lfe polces can be found n Wolhus [994]. Benef calculaons wh Markov chans n lfe nsurance are descrbed n Koller [2]. 2.2 Lably Scenaros In general here are hree possble scenaros for he number of employees n sae A n he penson scheme: Closed scheme no new employees: he number of persons n he sae A decreases dependng on ranson probables especally dsmssal and early reremen raes. Open balanced scheme: he number of persons n sae A s consan growh rae % or grows wh a small annual growh rae of say % - 2%. Reducon of he number of employees due o a ceran personnel polcy for example rered employees are no or only parally subsued by new employees. The mpac of each scenaro on cash flow profles wll be analyzed separaely. We wll no work wh a combnaon of dfferen scenaros. Transon probables from sae A o he saes I and P are gven n he acuaral bases. Dsmssal raes and early reremen probables can vary o a grea exend accordng o he personnel polcy of he employer and are srongly age dependen. Page 5 / 3
The number of new employees enerng he penson scheme a he end of each year s calculaed wh respec o a chosen scenaro. The new generaon densy deermnes he age and gender of each new employee. Ths densy s also defned as a par of he personnel polcy. As a rule n Swzerland each new employee ransfers hs or her vesng benef from he former employer o he new penson scheme. We nvesgae a ypcal defned conrbuon DC scheme. Is benefs are somewha hgher han he BVG requres. Conrbuons are age dependen and here s a guaraneed mnmum neres rae of 3.25% per annum. Penson paymens and salares are ndexed o nflaon. There are no conrbuon holdays or bonus paymens. Frs we focus on cash flows. They are an mporan characersc of a penson scheme. They conss of nflows conrbuons accordng o he benef plan and vesng benefs from former employers brough by new employees mnus ouflows. Ouflows are dfferen lump sum benefs and pensons due o be pad durng a year and also vesng benefs for resgned or dsmssed employees. The whole value of ouflows depends on he age srucure of he penson scheme dsmssal or resgnaon raes and he number of rerees spouses and dsabled persons. We wll calculae cash flow profles for he open balanced penson scheme wh a growh rae of % usng four dfferen new generaon denses and he same resgnaon raes. The early reremen raes used are sascally observed and represenave daa for Swss penson schemes. The resuls for cash flows are shown n Fgure 2 and he appled denses n Fgure 3. Ths example shows he sensvy of he Markov model for dfferen ypes of personnel polcy. Cash Flow Profles for Dfferen New Generaon Denses CHF mln - -5-2 -25-3 -35-4 -45-5 2 3 4 5 6 7 8 9 Year Fgure 2: Cash flow profles for dfferen new generaon denses. CF Densy CF Densy 2 CF Densy 3 CF Densy 4 Page 6 / 3
New Generaon Denses 5% % Denses o 3 2% 9% 6% 3% 8% 6% 4% 2% Densy 4 % 8 23 28 33 38 43 48 53 58 % Age of Jonng Employee Densy Densy 2 Densy 3 Densy 4 Fgure 3. New generaon denses used n Fgure 2. The denses and 2 are ypcal for penson schemes for cvl servans e.g. he densy 3 s more general: he probably o ener he penson scheme s very small for employees older han 4. The densy 4 s for he age of 3 and zero oherwse. One concludes ha he possble profle of he new generaon densy should be seleced carefully snce s mpac s consderable. 2.3 Profles of Cash Flows and Sauory Reserves of Penson Funds Now we urn o he mpac of dfferen dsmssal personnel polces on he cash flows and sauory reserves profles. New employees wll be hred accordng o densy s profles for males and females are shown n Fgure 4 ogeher wh he appled dsmssal raes. Page 7 / 3
6% 4% 2% % -2% -4% -6% -8% -% -2% New Generaon Denses and Dsmssal Raes 8 23 28 33 38 43 48 53 58 Female: Enrance rae Female: Dsmssal rae Age Male: Enrance rae Male: Dsmssal rae Fgure 4: New generaon denses usng densy for males and females and dsmssal raes. The macroeconomc parameers ha where used are as follows: he annual nflaon rae s % he annual wage growh rae s % and he oal reurn from nvesmens s 3% p.a.. There are four scenaros for he evoluon of he penson scheme populaon:. The penson scheme s open and he growh rae for he number of acve members s %. 2. The penson scheme s closed.e. no new employees a all. 3. The penson scheme s open bu rered and early rered employees are no subsued wh new employees so he number of acve members s decreasng durng he proecon perod. 4. The penson scheme s open and he number of acve members s ncreasng lnearly a 2% per year. Table summarzes he values for dfferen nsured groups n he penson scheme. The hrd scenaro s no shown n Table as s values are exacly beween hose of scenaro and 2 bu Fgures 5 and 6 show all values. Page 8 / 3
Growh Rae 2% Growh Rae % Closed Scheme T T5 T T5 T T5 T Acves 4467 494 536 4467 4467 3256 234 Dsabled 99 236 25 234 24 227 2 Rerees 69 692 92 69 885 687 837 Spouses 56 59 68 59 677 589 67 Toal 6835 7433 894 6982 7269 576 522 Toal Reserves CHF mln 34 44 723 37 64 269 345 Table. Evoluon of penson scheme populaon durng proecon horzon. Cash Flow Profles for Dfferen Personnel Polces Mo. CHF -5 - -5-2 -25-3 -35-4 -45-5 2 3 4 5 6 7 8 9 T Proecon sep Open: growh rae % Closed scheme Open wh reducon Open: growh rae 2% Fgure 5: Cash flow profles for dfferen scenaros. Page 9 / 3
Sauory Reserves for Dfferen Personnel Polces CHF mln '8 '7 '6 '5 '4 '3 '2 2 3 4 5 6 7 8 9 T proecon sep Open: growh rae % Closed scheme Open wh reducon Open: growh rae 2% Fgure 6: Evoluon of sauory reserves for dfferen scenaros. 2.4 Prof Tesng n ALM An mporan applcaon of prof esng s he smulaneous dynamc Asse and Lably Managemen ALM. Penson schemes wh prof sharng or ad hoc conrbuon holdays can be modelled only by ALM wh prof esng o ensure ha n each macroeconomc scenaro here are enough acuaral provsons and reserves o guaranee sauory benefs n each year durng he proecon horzon. The formulaon of he obecves and her ransformaon n an nvesmen sraegy are subec o several resrcons e.g. a frame for he fundng rao solvably drecves resrcons for ceran asse classes. The realsaon of nvesmen conceps depends on he nvesmen opporunes offered by he fnancal markes. The penson oblgaons resrc he obecves and he sraegy o he exen ha hey deermne he developmen and he commmen of capal he conrbuon sream and he flow of penson paymens. 3 Fnal Resuls and Conclusons The presened Markov based prof esng model allows o manage a wde varey of Swss penson schemes. If he ranson probables and parameers for he lably par of he model e.g. dynamcs of new enrans salares ec. are esmaed correcly he smulaed resuls of he lqudy requremens and lables wll be sasfyng. Page / 3
4 References. Smar I.C. [977] Prcng and Profably n a Lfe Offce Journal of The Insue of Acuares 4 Par 2. 2. Darbellay Paul-Anone and Chrsne Veraguh [998] Méhodes d évaluaon d une compagne d assurance ve e de ses produs Melungen der Schwezerschen Akuarverengung Hef. 3. Koller Mchael [2] Sochassche Modelle n der Lebensverscherung Sprnger Verlag. 4. Ruhdorfer Mahas [988] Smulaonen n der bereblchen Alersversorgung: En Markov-Modell und Abschäzungen für de Anzahl der Smulaonen Dsseraon Unversä der Bundeswehr München Fakulä für Elekroechnk Insu für Mahemak und Daenverarbeung. 5. Wolhus H. [994] Lfe Insurance Mahemacs The Markovan Model Educaon Seres Volume 2 CAIRE Brussels. 6. Janssen J. [994] Sem-Markov Modelsaon for he Fnancal Managemen of Penson Funds 4 h AFIR Inernaonal Colloquum p. 369 387. Page / 3
Page 2 / 3 5 APPENDI: Sae Space and Formulae for he Markov Chan for Penson Scheme Populaon The sequence of random varables Τ wh Τ Ν defned on he same probably space wh he fne dscree sae space : 2 h s called Markov chan of he frs order wh he sae space and he nal dsrbuon p f and only f { } { } { } T T P P p P + + + + + K K A. The ranson probably from sae a me o sae a me + vares n me and he Markov chan s no homogeneous any more as s he case n mos lfe nsurance applcaons: { } T P + : A.2 h Π : h s he number of all possble saes n he Markov chan. The h x h marx of ranson probables s denoed by Π s a sochasc marx means ha A.3 ~ ~ ~ ~ 22 2 x x x x T T T S S S B ît A ît A îs A.4 where A denoes he Sae A Acve B s he Sae I dsabled ogeher wh he Saes EP and P rerees. Dsabled and rerees are combned ogeher n hs sub-marx o show he sochasc marx n s compac form. S sands for he Sae wh dsmssed or resgned employees and T sands for he Sae Deah.
~ S s he new generaon densy a me as a vecor wh values for he whole age nerval of he Sae A as depced n Fgures 3 and 4 as a replacemen for resgned employees ~ T s lke ~ S bu as a replacemen for dead employees durng he year. xs and x T are he quoes o calculae for each year he values are n [. Π s a sub-marx for all saes n A o be an acve and o say as an acve Π 2 s a sub-marx for ransons beween A and B o be an acve and o ge dsabled or o ge rered f he age s a leas 58-6 wh he age of 65 all acves become rered Π 22 descrbes ransons whn B o be dsabled a frs o de as a dsabled or/and hen o ge rered a he age of 65 for persons older han 65: o say as reree or o de. Havng he nal dsrbuon p smply he age dsrbuon n each sae n he penson scheme we can calculae he developmen of he Markov chan n each pon of me : p + p Π T A.5 The sae space of our Markov chan s shown on he Fgure. The Saes I EP and P are grouped o B n he above descrpon. Acve members dsabled members or rerees can no vs he Sae W afer her deah bu a new spouse could be hred from ousde n he pensons scheme wh a dfferen gender and age dependen on ha of he defunc. On he oher hand spouses can no vs he Saes A I EP or P herefore we do no model he suaon when a marred couple s employed a he same frm and he wfe for example connues o work afer he deah of her husband and s an acve and a wdow a he same me n he penson scheme. Page 3 / 3