Pricing of Guaranteed Products for Defined Benefit Pension Funds.

Transcription

1 Mark Saxonov New York Life Invesmen Managemens We will discuss mahemaical modeling and pricing of "Sable Value" financial producs offered o pension plan providers. hese producs arge a risk-averse populaion of invesors, who have a significan preference for a sabiliy of reurns. We offer a quaniaive mehodology ha reflecs a decision making process and applicable in he siuaion of incomplee marke where radiional arbirage free argumens are invalid Pricing of Guaraneed Producs for Defined Benefi Pension Funds. We will discuss mahemaical modeling and pricing of "Sable Value" financial producs offered o pension plan providers. hese producs arge a risk-averse populaion of invesors, who have a significan preference for a sabiliy of reurns. We offer a quaniaive mehodology ha reflecs a decision making process and applicable in he siuaion of incomplee marke where radiional arbirage free argumens are invalid As a firs approximaion hese producs can be described as deb insrumens wih some specific cashflow and conrac arrangemens. In a majoriy of cases, a conrac does no erminae a once. Each conrac is comprised of a number of paricipans who have a righ o erminae (may be for unrelaed personal reasons). For his reason, erminaion of he conrac by he invesor (pu opion) may randomly deviae from he opimal. Subsequen mahemaical formalizaion will lead o a randomized sopping-ime problem. A predicabiliy of reurns is an essenial componen of porfolio manager performance. he cos of a produc from he guaranee provider s poin of view depends on volailiy of reurn. A he same ime he sabiliy of reurn is also an invesor's objecive. Consequenly, a porfolio managemen abiliy o generae reasonable reurn subduing volailiy should be facored in he produc pricing. his is a difficul problem since marke is incomplee and inefficien, and arbirage free argumens are no applicable Forunaely, a corporae pricing rouine gives us a clue of how o approach he problem. I urns ou ha required capial may serve as an indicaor of a company's volailiy olerance and reurn on capial as a company's measure of profiabiliy. As an imporan byproduc, a soluion will necessary generae he invesmen sraegy. We will sar from a small numerical example which we hope will clarify imporan poins emphasizing necessiy of proper quaniaive porfolio managemen and exposing weaknesses of a radiional duraion managemen approach. hen we will show soluion of some sochasic opimizaion models applicable o he above. Finally, we will discuss pricing models relaed o he Guaranee Invesmen Conracs.

2 Minimum Complexiy Example Consider a simples asse-liabiliy model wih a sochasic inerference. Assume ha conrac sipulaes paymens of wo premiums - P a he incepion of he conrac, and - P 1 a momen laer. Liabiliy paymen L is paid a he ime ma and conrac herefore erminaed. Assume he following simplified financial environmen. Yield curve is fla a r = 5% Expec a random jump N(m,sigma) a second afer invesmen decision is made. From ha ime on ineres rae does no change. m =7%,sigma =% Iniial premium P =$4 million paid a ime zero. Second and he las premium P 1 =$6 million expeced a ime 1 =.1 righ afer he jump. Conrac maured a ma = wih a liabiliy paymen L = $36 million One has o make an invesmen decision of how o allocae he exising asses (firs premium) o maximize a presen value of a fuure surplus. Second premium would be invesed unil mauriy since he ineres rae does no change afer he firs jump. he following formulas are sraigh-forward resuls of he assumpions: Presen value of he ending surplus for a realized rae r PV r P e P e L e r ( ma ) + r ( ma ) ( r r )( ma ) r ma (, ) = + 1 Expeced value of he fuure surplus is m ( r m) σ σ + + ( ) m + EV() = P e + P e L 1 Variance of he fuure surplus is a bi more cumbersome σ 1 σ Var( V ( )) = P e + P e m + ( r m) + ( ) m + 1 m + r m + σ P P e ( EV( )) + 1 ( ).5 ( )

3 Old radiional Invesmen Decision 1. Ignore premium. Inves in he bulle mauring a ma Old radiional Invesmen Decision. Calculae modified duraion of he liabiliy. ma r r1 ( ma L e 1 P1 e ) ma ModDur = = 65 ma r L e P ma Inves exising asses in he bulle mauring a 65 years. 1 Now we will invesigae if indeed any of above soluions offers a reasonable sraegy. Figure 1 6 Expeced Presen Value of Fuure Surplus by Iniial Invesmen V alue d e c E xpe Mauriy of he Iniial Invesmen Above we ploed expeced presen values of a fuure surplus depending on a lengh of he iniial invesmen. Clearly, he old radiional decisions bring abou inadequae resuls. Wha follows from his observaion ha if long invesmen (>1) is no available- keep money in cash. However in his analysis we ignored randomness concenraing on he expeced values. Wha abou volailiy of he reurn?

4 Figure 4 -nd percenile of Presen Value of Surplus e n ile pe rc ṉ d Mauriy of he Iniial Invesmen Figure suggess ha a shor iniial invesmen may generae a significan loss, or pu i differenly creaes a significan value a risk. From his perspecive, a risk averse invesor would inves as long as possible however giving up an expeced reurn. A his ime assume ha company's inernal requiremen demands allocaion of a risk capial covering losses wih 98% confidence. hus, a cash (very shor) invesmen sraegy requires $7 million of capial. Consider now risk capial as an equiy invesmen (Figure 3). More risky sraegy, more capial should be allocaed. Now, i urns ou ha shor invesmen would bring beer reurn on equiy nowihsanding higher capial requiremen, han invesmen ino years mauriy. If excepionally long mauriies are no available, cash invesmen is superior again.

5 Figure Expeced Reurn on Equiy O E R d e c E xpe Mauriy of he Iniial Invesmen Finally consider a siuaion when risk capial decision is based on he availabiliy of capial and capial is allocaed regardless of he calculaed risk. In his case we solve for an invesmen sraegy ha generaes maximum reurn subjec o loss resriced by he capial. Assume ha $ million is allocaed as risk capial. In his case Figure 3 shows ha an opimal invesmen is 3 years bulle. Figure 4.1 Expeced Reurn on Equiy.1.8 O E R d e c E xpe Mauriy of he Iniial Invesmen

6 Summary Risk Capial allocaion is an indicaor of a company risk olerance. More risk averse company more risk capial required o open line of business. Reurn on Equiy requiremen is a company's desired profiabiliy. Produc is profiable if a maximum expeced reurn is greaer han he company s requiremen. Ideally, an opimal conrol problem solves for he bes sraegy, maximizing expeced reurn on equiy and assuming ha he iniial capial is allocaed according o he company requiremen. In pracice he opimizaion problem ofen oo difficul o solve. A simplified problem solved wih a hope ha i s soluion would produce a feasible resul.

7 Mahemaical heory of a Sable Value Producs Logical Srucure and Classificaions of Guaraneed Invesmen Conracs Srucure of GIC Business 1 Direc Plan Sponsor Plan Sponsor- Consulan (Advisory) 3 Plan Sponsor urned over auhoriy o Discreionary Manager Guaranee Provider 1 1% Plan Sponsor Decision making 1% Plan Sponsor + Recommendaion from advisors 3 1% Invesmen manager, who is given his auhoriy hrough an agreemen he char above classified GICs by he degree o which he invesmen manager is involved in he process of making economical decisions. Consider how i works in is simples form. We look a he case 3 as he mos general one. Case 3 (Fully Discreional Pension Plan Manager) A plan sponsor - PS corporaion, on behalf of is employees plan paricipans, has o inves $M in he sable value secor of he 41K pension plan. We assume M = where I is a se of all pension plan paricipans. PS hires an invesmen manager i I M i

8 company IM o manage he enire process. IM eners ino a conrac wih an insurance company GP (Guaranee Provider). In he resuling agreemen GP ges money in reurn for an obligaion o pay M b1 + rg upon mauriy of he conrac assuming no wihdrawals have been made. For he illusraive purposes assume ha he ineres rae 1 r is a consan and is sipulaed by he paricipaing sides a he incepion of he conrac. he plan sponsor has he righ o wihdraw money a any ime. For each dollar invesed a he incepion, he amoun available for wihdrawal a ime is equal o b1+ rg. However here are usually imporan srings aached. If he wihdrawal is iniiaed by he IM, he plan sponsor has o pay an early wihdrawal fee F. No fee is paid if he wihdrawal is a plan paricipan iniiaed even. he money could be wihdrawn (before mauriy) from he guaraneed accoun a ime < for hree differen reasons. 1. A ime, IM realizes ha an obligaion of he Guaranee Provider GP is worh less han M b1 + rg F. In his case a sound economic decision is o iniiae a erminaion of he conrac and wihdraw money.. An employee x of he PS decides ha he guaraneed rae is oo low for he curren marke siuaion and requess he IM o wihdraw his porion of he accoun. Since his is an acion iniiaed by an employee, no early wihdrawal penaly is imposed. Accordingly, he employee ges back his invesmen a a guaraneed value of Mx b1 + rg and he accoun value is reduced by he same amoun. Mx b 1 + rg. Here M x denoes he iniial invesmen by x a ime. 3. An employee x of PS decides o wihdraw money for reasons no relaed o he marke siuaion (his may include reiremen or change of employmen.) As we shall see laer, he mahemaical racabiliy of he wihdrawal is very differen for each case. Open window opion he nex common feaure, which may be added o he conrac, is an open window opion. his feaure is familiar o many homebuyers as he lock-in-rae opion for house financing. Under he open window clause he plan manager (or plan sponsor) may deposi money during a cerain period of ime (open window) wih he crediing rae esablished a he beginning of he period. Synheic GIC his produc separaes guaranee on he book value from he ownership of he asses. In he case of a regular GIC, he guaranor has an obligaion o pay he enire sum requesed for he wihdrawal by he plan sponsor. In he case of a Synheic GIC he guaranor has o subsidize a wihdrawal if he asses porfolio (no owned by he guaranor) has 1 he consan ineres rae assumpion is local and is made here only for an illusraive purpose. In fac he mehodology of assigning conracual ineres raes is essenial parin cerain ypes of conrac classificaions.

9 insufficien funds o cover he wihdrawal reques and he amoun of wihdrawal does no exceed he guaraneed value. Paricipaing (Par) and Non-Paricipaing (Non-Par) GICs he Guaraneed Invesmen Conrac is said o be paricipaing if he plan sponsor paricipaes in he profi or loss of he asses porfolio hrough he adjusmen of he guaraneed conrac rae. If he guaraneed rae does no depend on porfolio performance we say ha he conrac is non-paricipaing. Mahemaical Models For all he consideraions below we assume ha ineres rae erm-srucure is described by a sochasic differenial equaion dr = b ( r ) d + σ ( r ) dw (1) wih respec o a sandard Wiener process w, which is defined on a complee probabiliy space { Ω, P} wih filraion. lq F >, r is an insananeous ineres rae, i.e. a risk free ineres rae paid for shor erm borrowing. We also assume ha he probabiliy measure P is risk neural. Accordingly we will calculae he price of a securiy as a discouned cashflow. here are wo differen mahemaical problems, which from he clien's (plan sponsor) poin of view are associaed wih he ype of he conrac. For he non-paricipaing conracs he clien is ineresed only in he size of he fee he is charged by he guaranee provider. If he conrac is paricipaing, he clien would reques ha he guaranee provider s managemen of he porfolio would maximize plan sponsor s invesmen. herefore we will discuss below a Pricing Problem he calculaion of he fair marke value of he fee charged by he guaranee provider and a Porfolio Managemen Problem - he opimal sraegy he porfolio manager has o follow o benefi his clien he mos. 1. Paricipaing GIC wih no wihdrawal opions (Porfolio Managemen Problem) Assume ha he conrac sars a ime = when clien makes a deposi L. he guaranee provider purchases porfolio A of asses. Denoe by M a marke price of such a porfolio a momen. We assume ha L = M. Assume ha invesmen income from porfolio A is equal o r + s where r -is a spo ineres rae and s is a qualiy spread. Dynamic of he marke value of he porfolio A is described by a following equaion dm = M (( r + s) d D dr )

10 his equaion simply saes ha change in porfolio price consiss of wo componens. he firs componen is an invesmen income, which is defined by he amoun of ineres earned hrough a coupon paymen over he ime inerval d. he second componen is a change of price due o he ineres rae shif. his change is proporional o a porfolio effecive duraion D (By definiion) a he ime. he guaraneed value of liabiliy L is defined as L = L e. Here r L is a guaraneed rae calculaed radiionally as L L 1 M r = MAX( r, ln( ) + ( r + s)) L Where r L is a minimum guaraneed rae. he equaion is derived from ( )( r ) ( )( r + s) ( )( r ) ( )( r + s) L e = M e if Le < M e () L he las equaion makes a coninuous readjusmen o he guaraneed rae in such a way ha final value of asses and liabiliy porfolios would be equal each oher if he marke condiions remain unchanged unil mauriy of he conrac. he equaion () implies ha he final liabiliy value should converge o a final marke value of he asses porfolio in a case of an adequae porfolio performance. Denoe l = ln( L ); m = ln( M ) By he Io formula we have L z lds s dm = ( D dr + ( r + s) d) + 5. D ( dr ) hus we are geing a following equaion for a marke value dynamic dm = ( D b ( r) d+ ( r+ s) d) + 5. D ( σ ( r)) d D σ ( r) dw dm = ( D b ( r ) + ( r + s) + 5. D ( σ ( r )) ) d D σ ( r ) dw (3) Equaion (3) describes a fixed income porfolio dynamic. Even hough i looks quie simplisic, here are no significan aberraions from he realiy. Mos resricive (implici) assumpion ha has o be made o jusify (3) is a coninuous rebalancing of he porfolio.

11 Asses Managemen Opimizaion Problem he nex sep is a choice of crierions ha evaluae a porfolio manager and a guaranee provider performance. he mos sraighforward one is o maximize liabiliy value a he end of he horizon. Opimizaion Problem 1. 1 (Maximize Ending Liabiliy Value) r L max El ( ) u L 1 = MAX( r, ( m l) + ( r + s)) l dl = r d dm = ( u b( r) + ( r + s) + 5. u ( σ ( r)) ) d u σ ( r) dw U = { u: u [ D, D ] Here D D, are low and upper boundary allowed for he duraion D of he asses porfolio. hose funcions are deerminisic and usually are he conrac-sipulaed values Since we consider a paricipaing conrac, he final oal earning would be in large degree defined by a porfolio performance iself. herefore as an approximaion and as a reasonable compromise for he invesor would be a following problem, which is easier o solve analyically. Opimizaion Problem 1. (Expeced Ending Porfolio Value) max Em ( ) u dm = ( u b( r) + ( r + s) + 5. u ( σ ( r)) ) d u σ ( r) dw U = { u: u [ D, D ] (4) (4 ) he problem has o be reformulaed for an infinie ime horizon if an evergreen conrac is considered Wih no expiraion dae

12 Opimizaion Problem 1.3(Infinie ime horizon) b ( ) max Lim Em g b ( ) or max Lim El g u u dm = ( u b( r) + ( r + s) + 5. u ( σ ( r)) ) d u σ ( r) dw U = { u: u [ D, D ] he crieria 1.1 and 1. may no adequaely reflec he basic premises of he produc. he guaraneed invesmen conracs coninue o occupy a significan porion of he pension fund marke because of he higher reurn-o-volailiy raio, no o he reurn on he invesmen per se as would suggesed by 1.1 or 1.. Considering his, we will inroduce a CAPM 3 - ype crieria. We assume ha he marke is risk averse and demands an addiional reurn from more volaile securiies. his is o say ha he plan sponsor would have esimaed a performance of he porfolio manager by looking no only a he porfolio reurn ( m m ) bu also a he hisorically esimaed volailiy. he las concepion couldn be direcly inroduced wihin he coninuous ime framework. Assume ha he reurn is measured a imes =, 1 = Δ,... n =. A hisorical esimae of he volailiy is 1 1 n i+ i n 1 m m F m mi ( ) (5). n i= Δ HG Δ n KJ i Here in order o shoren he noaions we denoe m = m i I is naurally herefore o sae ha porfolio managemen purposes o minimize a mahemaical expecaion of (5) ogeher wih (4) or (4 ). A differen opimal sraegy is generaed by a new problem where he crierions (4) and (5) are mixed ogeher. Opimizaion Problem 1.4 n i+ i m m max{ El ( ) ( ( ) E m α + β m )} u n Δ i= dm = ( u dr + ( r + s) d) +. u ( dr) U = { u: u [ D, D ] F HG 5 I KJ 3 Capial Asse Pricing Model

13 here are a leas wo problem wih he problem 1.3. he firs is a robusness and sabiliy of he soluion.. If Δ is small he volailiy componen n i+ i m m E m F I ( ) m n i= Δ HG KJ could significanly supersede he marke value componen Em ( ). Anoher problem is ha minimizaion of hisorical volailiy does no necessary coincides wih he clien goals. I is more in sync wih a GIC salesman argumens han wih a clien ineres.. Assuming ha he clien main objecive is o maximize profi and minimize volailiy he opimal problem could be reinsaed as follows:

14 Opimizaion Problem 1.6 max{ α El ( ) + β ( El ( ) ( El ( )) } u c dm = ( u dr + ( r + s) d) +. u ( dr) U = { u: u [ D, D ] h 5 Anoher version is may be considered is Opimizaion Problem 1.6 max u R S Em ( ) ( Em ( ) ( Em ( )) U V W I looks like a more difficul problem, hough more in line wih CAPM argumens.. Non-Paricipaing GIC wih a wihdrawal opion (Pricing Problems) For non-paricipaing GICs he liabiliy is a guaraneed conac value calculaed on he base of he guaraneed conac rae. his rae is a parameer of he conrac and is assumed o be a consan unil he mauriy of he conrac. Plan sponsor of such a GIC has an opion (pu) o wihdraw money a any ime a his discreion. However, wo cases of wihdrawal are idenified and separaed. If a plan paricipan iniiaes wihdrawal, no penaly for an early wihdrawal is imposed. In his case he Guaraneed Conrac Value is reduced by he amoun wihdrawn. If he plan sponsor iniiaed a wihdrawal on behalf of he plan paricipans he Guaraneed Conrac value is reduced by he amoun wihdrawn plus early wihdrawal penaly. Noaions, Assumpions and Preliminary Informaion Consider a Guaraneed Invesmen Conrac paying coupon f wih a coninuously compounded ineres 4. Assume ha Level of a risk free ineres rae is equal o r a he ime > as described by (1). 4 his means ha coupon paymen over he infiniesimal inerval Δ is equal o Δ f M Δ

15 he conrac-holder is eniled o a guaraneed value g a ime if decided o wihdraw money. Conrac has a par value of $1 and g is a guaraneed value ha will be mandaory wihdrawn a mauriy. We assume ha if erminaion of he conrac is happened due o he plan manager decision hen here is a penaly which reduces he guaraneed value by g f. Non-arbirage ransacion frees rading in a risk-neural world. Denoe by a fixed random momen (no necessarily opimal) when he conrac holder decided o wihdraw money. Denoe V ( x, ) he price for such a conrac, calculaed as a mahemaical expecaion of a discouned fuure cashflow. We herefore have R s f V ( x, ) = ES expf xu du fs exp xu du I g( ) + I< g ( ) H I K F I z z + HG z U KJ V c h W I is naurally o assume ha he manager would chose momen o maximize V ( x, ). herefore if he has a full discreion over he process he will behave accordingly. In his case he price for he conrac should be calculaed as a soluion of he opimizaion problem Opimizaion Problem.1 (Wihdrawal is on he plan manager own discreion) dx = bds + σdw x = x s s R s f V( x, ) = max ES z expf xu du fs exp xu du I g( ) + I < g ( ) H z I K F I + HG z KJ c h U V W Opimaliy Condiions I is shown in [1] ha v(x,) saisfies he Non-Linear Parial Differenial Equaion. Equaion 1 1 V V V g V + [ σ + b + xv + f + v g ( )] + = x x [ a] = max( a, ) + f g = I g + I g () ( ) < ( ) his in urn is equivalen o he following hree condiions:

16 Equaion V g v g > Lv = g = v Lv 1 v v v where Lv = σ + b + x v x x his problem is very difficul o solve numerically. We consider a modificaion of he problem which drasically reduces he complexiy. Conracholder is eniles o reques a wihdrawal. Assume ha plan paricipan's wihdrawal decision is no based on a marke siuaion. He may wihdraw money because of a coningency relaed o a reiremen, job securiy, and oher social evens. For he simplificaion of he analysis assume ha if plan paricipan decided o wihdraw money, he will wihdraw a oal guaraneed value and he Guaraneed Conrac is erminaed. Assume ha he inensiy of wihdrawal is r( x, ). his means ha a ime probabiliy ha he plan paricipan will erminae he conrac on he inerval (,+d) is rx (, ) exp( r( x, u) du) d u For he sake of simpliciy assume ha =. We reformulae he opimizaion problem by inroducing an addiional variable y, and assuming perpeuiy in he paymens period =. I is no difficul o see ha he las assumpion does no cause a loss of generaliy. o achieve he acual mauriy a one has o choose unresricedly high inensiy funcion in he small viciniy of. he opimizaion problem.1 may be rewrien as

17 Opimizaion Problem. z b sg b g v( x,y) = maxe[ exp y fs ds+ exp y g( )] dx = bds + σdw x = x s dy = xd y = y Now consider an individual rajecory ω where he bond is scheduled o be called a ime (ω). he condiional conribuion of his rajecory o he crieria of he Opimizaion Problem. is z b sg b g exp y fs ds+ exp y g( )] Assume now ha ogeher wih managemen call, he bond may be called due o he irraional cause defined by inensiy funcion r( x, ) herefore he expeced conribuion from he individual rajecory is z z z z { g f ( )} + exp( rx (, d ) ) exp y fds s + exp y b sg b g z { }] [ r( x, ) exp( r( x, s) ds) exp y fs ds + exp y g () d = s = b sg b g I + II For he given rajecory he conrac will be erminaed eiher due o he manager's call or he plan paricipans decision o wihdraw money. If he plan manager decides ha is an opimal ime o wihdraw, wo differen evens may happen. Firs akes place when he conracholder does no wihdraw before and he firs par (I) of he expression above evaluaes he expeced conribuion from his even. In his case exp( rx (, d ) ) is he probabiliy ha he wihdrawal of he bond will be a plan manager iniiaed even. he second even akes place when he conracholder decided o wihdraw before. Accordingly he second par (II) is a conribuion from such an even. Afer some ransformaions we have

18 z b g z f I + II = exp( rx (, ) ) exp ( ) + exp (, ) ) KJ ( + (, ) d y g { y rxs sds f rx g ( )) d} F HG z I Now we may ge rid of y and reurn o previous noaions. z z f I + II = exp( ( rx (, ) + ) ) ( ) + exp ( (, ) + ) ) KJ ( + (, ) x d g { rxs s xs ds f rx g ( )) d} F HG z I he price of he enire conrac wih given random wihdrawal ime is a mahemaical expecaion of he conribuions of he individual rajecories v ( x, ) = E(I + II ) Effecive marke will price he conrac by choosing he call ime o a maximum disadvanage of a bondholder. herefore we obain price of he bond as a resul of he Opimizaion Problem.3 v( x, ) = max v ( x, ) or f v( x, )= maxe[exp( z ( r( x, ) + x ) d) g ( ) + F I {exp z z (( rx,) s + x ) ds) ( f + rx (,) g()) d}] HG s s KJ Applying Equaion 3 o his problem we obain a Differenial Equaion for vx (, ) Equaion 3 g f v v v f v(, x) + [ σ b ( x+ r) v+ f + rg+ v g ] x x [ a] = min( a, ) +

19 Pricing Wih Saionary Processes Assume ha x is a saionary process. his means ha coefficiens σ and b in he equaion 1 do no depend on ime. Assume also ha he inensiy funcion rx (, ) is a funcion only of he sae x, i.e. rx (, ) = rx ( ) We have 1 σ v + bv + ( x+ r) v+ f + rg = where x < x and vx ( ) = g; v ( x) = ; c c c I is a second order ordinary differenial equaion wih free boundary condiions. I can be shown ha his equaion has a unique bounded soluion. REFERENCES 1. N.V. Krylov Conrolled Diffusion Processes. New York,Springer-Verlag,198. J. Cox, J.Ingersoll and S.Ross A heory of he erm Srucure of Ineres Raes. Economerica 53,1985, pp

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