Liability Valuation and Optimal Asset Allocation

Transcription

1 Liabiliy Valuaion and Opimal Asse Alloaion Joahim Inkmann Finanial Markes Group London Shool of Eonomis Houghon Sree London WC2A 2AE Unied Kingdom David Blake Pensions Insiue Cass Business Shool 106 Bunhill Row London EC1Y 8TZ Unied Kingdom Firs draf: This version: Absra: Curren approahes o asse-liabiliy managemen employ a sequene of disin proedures o value liabiliies and deermine he asse alloaion. Firs, a disoun rae ha is usually diaed by aouning sandards is used o value liabiliies. Seond, he asse alloaion is deermined by maximizing some objeive funion in he surplus of asses over liabiliies, aken as given he valuaion of liabiliies. We inrodue a model ha allows for he join valuaion of liabiliies and he deerminaion of he opimal asse alloaion using disoun raes ha appropriaely refle defaul risk. We fous on he ase of a defined benefi pension plan. JEL Classifiaion: G11, G23, G28 Keywords: Asse-liabiliy managemen, liabiliy valuaion, asse alloaion, surplus, defaul, disoun rae j.inkmann@lse.a.uk, d.blake@iy.a.uk. We graefully aknowledge ommens of pariipans in he Workshop on he RTN Nework on Finaning Reiremen in Europe in Louvain-La-Neuve, he UBS Pensions Researh Seminar a LSE and an Eonomis Seminar a Mannheim Universiy.

2 1. Inroduion This paper deals wih liabiliy valuaion and opimal asse alloaion and he key word is he onjunive and. Boh in praie and in he heoreial lieraure, liabiliy valuaion and asse alloaion are ypially reaed as ompleely separae issues, despie lip servie o he onrary. A lassi example is defined benefi pension fund liabiliies and asses. Wha we end o observe is he projeed fuure liabiliy ash flows being disouned using a se of disoun raes ha fail o refle he rue risk aahed o ha liabiliy sream. A whole range of disoun raes are used in praie: high qualiy (AA orporae bonds (as required by various aouning sandards: FAS87 (US, FRS17 (UK and IAS19 (inernaional; he weighed average reurn on a noional porfolio of sauory referene asses (as required by he UK Minimum Funding Requiremen; and he weighed average expeed reurn on he aual porfolio of asses supporing he liabiliies (as used in mos auarial valuaions; for more deails, see, e.g., Blake (2001. A he same ime, he asse alloaion is generally hosen quie independenly of he projeed liabiliy sream. Typially, a leas in Anglo-Saxon ounries, he pension plan rusees hoose (or are advised by heir invesmen onsulan o hoose a high weighing in equiies in order o benefi from he equiy risk premium and hene lower he os o he plan sponsor of providing pensions. In oher ounries, e.g., many in oninenal Europe, pension funds are enouraged o inves heavily in governmen bonds in order o help governmens finane heir naional deb. Some would argue (e.g., finanial eonomiss suh as Bodie (1995 and radial auaries suh as Exley, Meha and Smih (1997, Gold (2001 and Bader and Gold (2003 ha pension funds should be enirely invesed in bonds on he grounds ha pension funds should no ake risks wih he sponsoring ompany s shareholders funds and ha pension paymens are bond-like in naure. The Boos pension fund in he UK was suffiienly persuaded by his argumen ha beween April 2000 and July 2001 i swihed all is asses ino bonds (see Blake (2003a, There migh well be raional explanaions for why erain of he above praies emerged. For example, AA orporae bonds were hosen for disouning purposes under FAS87 beause his was he asse lass ha US insurane ompanies used when aking over he pension obligaions of insolven US ompanies. The UK aouning sandard FRS17 adoped he same disoun rae even hough AA orporae bonds are no a signifian invesmen aegory in he UK, aouning for only 7% of UK bonds ousanding in 2000, he year FRS17 was announed. Similarly, he adopion of reurns on he asses in he pension fund o disoun liabiliies was inended o minimize any asse-liabiliy mismah, bu leads o he unomforable impliaion ha a pension fund an redue he value of is liabiliies by invesing in a riskier asse lass. The underlying impression from all of his is ha here is a lak of onsiseny beween he way in whih liabiliies are valued and he way in whih he asse alloaion is deided. We will argue in his paper ha hey mus be joinly deermined, oherwise poenial inonsisen- 2

3 ies emerge. The valuaion of he liabiliies depends on a disoun rae (more preisely a disoun erm sruure ha depends, in urn, on he asse alloaion hosen. We are aware of only one auhor who expliily disusses he relaionship beween he disoun rae and asse alloaion deisions, namely Peersen (1996 in an empirial sudy of US pension plans. He argues ha a porfolio shif from low-risk o high-risk asse lasses should be aompanied by an inrease in he disoun rae on pension liabiliies. He also poins ou ha he disoun rae should inrease wih a dereasing funding raio. While he laer argumen is onfirmed by his empirial analysis, he evidene for he former urns ou o be somewha mixed. On he one hand, firms end o inrease he disoun rae wih a higher equiy alloaion relaive o ash; on he oher hand, hey end o inrease he disoun rae even more wih an inreasing bond alloaion relaive o ash, a finding ha is learly inonsisen wih his line of reasoning. VanDerhei (1990 onsiders he possibiliy of defauling on he pension promise in his derivaion of fair-value insurane premiums for US defined benefi pension plans overed by he Pension Benefi Guarany Corporaion (PBGC. While he does no aoun expliily for he asse alloaion of he pension plan, he inludes he funding raio (of asses o liabiliies as an explanaory variable in he defaul probabiliy regression and repors he aniipaed negaive impa. He also alulaes insurane premiums as he produ of he esimaed defaul probabiliy and he esimaed sale of defaul. In relaed work, Carroll and Niehaus (1998 invesigae he impa of unfunded pension liabiliies on orporae deb raings, wih higher raings usually assoiaed wih smaller defaul spreads. They find ha raings inrease wih over-funding and derease even more so wih under-funding. Over he las few years, wih mos pension funds having o deal wih under-funding problems aused by a ombinaion of equiy marke delines and a legay of sponsor onribuion holidays, pension defiis have been repeaedly ied by raing agenies as a reason for downgrading aions. 1 The ouline of he paper is as follows. Seion 2 presens a model of a pension plan s liabiliies and heir dependene on he forward erm sruure of disoun raes. The laer are deomposed ino a risk-free omponen and a spread refleing he defaul risk of failing o deliver he promised pensions and his is mos likely o be riggered by he insolveny of he sponsoring ompany. This defaul risk will be quanified in erms of he defaul probabiliy and he reovery rae in he even of defaul (boh of whih will depend on he raio of pension plan asses o liabiliies and he ne worh of he sponsor. Following Sharpe and Tin (1990, he objeive funion for he pension plan s asse alloaion problem is a mean-variane funion of he surplus raio. In onras wih previous work, we show ha maximizing his objeive funion wih respe o he asse alloaion simulaneously generaes an appropriae defaul spread. Sine he defaul spread depends on he value of he defaulable pension 1 Sandard & Poor s published a wah lis of 12 European ompanies in February 2003 beause of onerns abou he unfunded pensions. Among hese ompanies were Deushe Pos, Linde, Mihelin, Sainsbury, Rolls Roye, and Thyssen-Krupp. The subsequen downgrading of Thyssen-Krupp from BBB o junk bond grade BB+ has led o a onroversial disussion beween S&P and he German pension indusry. 3

4 laim iself, i is endogenous in he sense of Duffie and Singleon (1999 and his onsiderably ompliaes he opimizaion problem. We onsider a numerial example in Seion 3 under he assumpion ha reurns and yields are independen and idenially mulivariae normal disribued in order o shed more ligh on he relaionship beween defaul spreads and he opimal asse alloaion. The main findings are: The appropriae defaul spread dereases wih an inreasing funding raio and inreasing sponsoring ompany ne worh. The appropriae defaul spread dereases wih inreasing risk aversion (or an inreasing willingness o pay high expeed onribuions and leads o a lower equiy weighing. The spread on AA orporae bonds, he mos ommon asse lass used in he disouning of pension plan liabiliies in he US and UK, is unlikely o be an appropriae defaul spread in he valuaion of mos pension liabiliies. 2. Asse-Liabiliy Modeling We onsider he defined benefi pension plan of a sponsoring ompany whose own finanial srengh an be summarized in period 2 by an exogenously given ne worh V(. The se-up ha we desribe is a ypial asse and liabiliy managemen (ALM exerise for defined benefi pension plans, one ha is ondued on a regular basis, for example, during he preparaion of he annual finanial saemens or he riennial auarial valuaion. ALM involves wo issues: valuaion and asse alloaion. While he asses in he pension fund are valued more or less oninuously by he finanial markes, he urren value of pension liabiliies is deermined by disouning all fuure pension paymens ha he ompany has promised o is employees. The ruial ask in his alulaion is he deerminaion of he disoun rae ha appropriaely refles he risk of failing o deliver he promised pension paymens in he fuure in he absene of a perfe insurane vehile. 2.1 Asses and Liabiliies Curren ALM approahes rely on an exogenously given disoun rae F for he valuaion par of he ALM exerise. For example, F ould be he urren yield of a AA orporae bond wih long mauriy. In his ase, he auarial liabiliy of he pension plan wih N members and an arual rae based on he sixiehs sale is as follows (see Blake, 2003b, Cairns, 2003 L N (,hi = Y(,hi ( ( W 0 W h e pr(,hi hi ( 1+ F pr( T i, l l ( 1 F i= 1 i l= 1 + i 60 L (1 where he index i refers o an employee who joined he pension plan years ago and has o i work h addiional years before reahing he reiremen age in period T = + h. In he ure i i i 2 The model is wrien in disree ime, as are mos of he relevan priing (e.g. Das and Sundaram, 2000, Cohrane, 2001 and asse alloaion (e.g. Sharpe and Tin, 1990 models o whih we refer. 4

5 ren period, he employee has arued pension benefis equal o he share, e i 60, of his projeed final salary, whih equals he urren salary, Y (,h i, saled up by W ( 0 W( h i, he (age-relaed inrease in earnings over his remaining working life. F is a real disoun rae ha aouns for (prie inflaion. Noe ha he las sum in (1 is he uni prie of a real annuiy ha is bough a reiremen age. We assume wihou loss of generaliy ha eah employee lives for a maximum of L years in reiremen, whih means ha he probabiliy of surviving from o + s, pr (, s, falls o zero for all s > hi + L. From now on we assume again wihou loss of generaliy ha N = 1 o simplify noaion. The index i will be dropped aordingly. Our subsequen analysis is based on wo imporan modifiaions of (1. Firs, he assumed ime-invariane of he disoun rae is dropped. Seond, he appropriae defaul spread will be derived raher han aken as given. In order o ahieve hese modifiaions, F will be replaed by forward raes ha are deomposed ino defaul-free forward raes and defaul spreads. Le F (,s, n be he n-period forward rae a ime for he finanial ransaion in period s. Following, e.g., Das and Sundaram (2000, F,s,n is deomposed as + ( ( = ( 1+ G(,s,n ( 1 D(,s,n 1 + F,s,n + (2 where G (,s, n denoes he defaul-free forward rae and (,s,n his definiion, (1 an be expressed more generally for N = 1 as D he defaul spread. Given L (,h = Y(,h ( ( W 0 W h e 60 pr(,h ( 1+ F(,0,h h L pr( T, l l= 1 ( 1+ F(,h, l l. (3 I will be onvenien o work wih one-period forward raes, whih we denoe more ompaly as F(,m F(,m, 1. They are relaed, of ourse, o n-period forward raes by s + n 1 n ( ( + F,s,n = ( 1+ ( m= s 1 F,m. (4 =. Log pries of zero- We denoe wih small leers he logs of variables. Thus, we wrie f (,m g(,m + d(, m f (,m = log( 1+ F(,m, g (,m = log( 1+ G(,m and d (,m log( 1+ D(,m oupon defaul-free and defaulable bonds are denoed q (,m and p (,m orresponding one-period log reurns are wrien as s ( + 1,m and ( 1,m = wih r +. In erms of oneperiod log forward raes, log pries and one-period log reurns are defined as m 1 n= 0 m 1 n= 0, respeively. The q (,m = g(,n s( + 1,m = q( + 1,m 1 q(,m = g(,0 ( g( + 1,n 1 g(,n (,m = f( m 1 n= 1 m 1 n= 1, (5 ( ( ( ( ( ( ( p,n, r + 1,m = p + 1,m 1 p,m = f,0 f + 1,n 1 f,n. (6 Using hese properies he urren value of he pension liabiliies (3 an be rewrien as 5

6 L (,h = U(,h,e pr( T, l exp( p(,h + l wih (,h,e Y(,h L l= 1 ( ( W 0 e U = pr(,h. (7 W h 60 Mahing hese liabiliies in he pension fund are asses wih marke value A(. The asseliabiliy posiion of a defined benefi pension plan is usually summarized by he surplus, S(,h = A( L(, h, or by he funding raio, C (,h = A( L(,h. 2.2 Pension Plan Defaul We are ineresed in he deerminaion of he forward spread erm sruure ha is relevan for he ompuaion of (7. In line wih he lieraure on he priing of orporae bonds (see e.g. Duffie and Singleon, 1999, we derive defaul spreads for he valuaion of he pension laims ha appropriaely ake ino aoun he possibiliy of defaul. We define defaul for a orporae defined benefi pension plan as he even in whih he ombined value of he plan asses and he ne worh of he sponsoring ompany is insuffiien o over he value of he pension liabiliies. We assume ha he plan has no defauled by period. Thus we fous on he likelihood of he fuure even ( + 1,h 1 > A( V( 1 S ( 1,h 1 < V( + 1 L + + (8 We make he assumpion ha any reovery value in he even of defaul in period + 1 will be relaed o he period value of pension liabiliies. Thus, we assume, in he even of defaul, ha pension righs are frozen a heir defaul dae (i.e. disoninuane dae value, alhough asses oninue o grow. In order o refle his assumpion, he fuure ne worh of he sponsoring ompany is relaed o he urren liabiliies by V (8 hen beomes ( + 1,h 1 L(,h S ( 1 = τ( + 1 L(,h +. Defaul ondiion < τ( + 1. (9 We will refer o he variable on he lef-hand-side as he disoninuane surplus raio (DSR. I an be derived from he urren funding raio C (,h of pension plan asses and liabiliies as ( 1,h 1 L(,h S K L + = ( 1+ w(,k exp( x( + 1,k C(,h ( 1+ u v(,h, exp( r( k= 1 l= 1 l + 1,h + l (10 A (, ( where refers o onribuions 3 o he plan asses as a perenage of w,k denoes he pension fund s proporionae alloaion o asse lass k and x ( + 1,k = log( 1+ X( + 1,k is he one-period log reurn for asse lass k 1, K, K v,h,l follow from (7 as v (,h, pr( T, l ( 1+ F(,h, l pr( T,l l ( 1+ F(,h,l 1 =. The weighs ( L L l = wih l v(,h, l = 1. (11 l= 1 l= 1 3 These ould beome negaive in he ase where he plan member dies before reiremen. 6

7 Equaion (10 shows ha he growh rae of he auarial liabiliy is proporional o he oneperiod log reurn of a value-weighed porfolio of defaulable bonds wih mauriies beween h + 1 and h + L. The faor u = U( + 1,h 1,e + 1 U(,h,e 1 in he liabiliy growh formula inorporaes ime- and age-speifi wage inflaion as well as hanges in survival probabiliies and benefi arual. Sine his paper fouses on he relaionship beween disoun raes and asse alloaion, we simplify maers by assuming ha hese hanges an be summarized by some onsan and known growh faor, herefore any sub-indies are omied for u. This resriion will simplify he noaion and helps o fous on he ore problem. We are ineresed in he probabiliy of defaul as defined in (9. We also noe ha he raio of he fuure asse value and he ompany s ne worh o he urren liabiliy value is a naural measure of he reovery rae in he sense of relaing available asses o frozen pension liabiliies. Thus, he random variables ( + 1 = 1S [ ( + 1,h 1 L(,h < τ( 1] ( 1 = ( A( V( 1 L(, h ( 1 = 1 D( D( + 1 R( 1 D + R + + (13 K + + (14 (12 are of pariular ineres ( 1 [ ] in (12 is he indiaor funion ha akes a value of uniy if he argumen in parenheses is rue and zero else. Before deriving he expeaions of hese variables we have o disuss he available ondiioning informaion. We assume ha he sohasi disoun faor ha pries all raded asses in he eonomy is a funion of random variables Z + 1 M ( 1 M( Z + 1 (ha migh, for example, approximae onsumpion growh as in a onsumpion-based asse priing model: +. The sohasi disoun faor (or priing kernel is he same for all raded asses. For example, he pries of one-period defaul-free and defaulable bonds are given by he fundamenal priing equaions [ [ Z ( q(,1 E M( 1 Z ( p(,1 E M( + 1 K( 1 exp = + ] (15 exp = + (16 ] ha relae ime + 1 payoffs o ime pries by an appliaion of he priing kernel. We assume ha addiional informaion is available onerning he finanial sae of he sponsoring ompany. The ne worh of he sponsoring ompany depends boh on his informaion and on he variables desribing he sohasi disoun faor. Thus, he relevan ondiional expeaion of Z ( + 1 τ( Z, Z τ is (,1 = E[ τ( + 1 Z,Z ] τ and he orresponding ondiional expeaions of he random variables (12-(14 are π (,1 = E[ D( + 1 Z, Z ] (,1 = E R( + 1 D( + 1 [ = 1,Z, Z ] (,1 = E[ K( + 1 Z, Z ] = 1 π(,1 + π(,1 ρ(,1 ρ (18 Π (19 (17 7

8 where in (18 we also ondiion on he defaul even. 4 Equaion (17 defines he ondiional defaul probabiliy. Equaion (18 defines he ondiional expeed reovery rae in he even of defaul. Equaion (19 defines he expeed DFR as he sum of he expeed ondiional reovery raes in he evens of survival and defaul, weighed by heir respeive probabiliy. K +, τ ( + m ondi- We denoe m period ahead expeed values of D ( + m, R ( + m, ( m ional on urren informaion Z, a s π (,m, ρ (,m, Π (,m and τ (,m. 2.3 Defaul Spreads Z For he derivaion of he defaul spread erm sruure we employ a ondiional mean independene (CMI assumpion for M ( + 1 and K ( + 1 where he ondiioning informaion inludes Z and he orporae-speifi variables Z : [ M( 1 K( + 1,Z,Z ] = E M( 1 [ Z, Z E + + ]. (Assumpion CMI Using CMI, we have he following imporan resul for he raio of (16 o (19: exp p Π ( (,1 (,1 = E M = E E = E E K ( ( Π(,1 M( + 1 Z Π(,1 M( + 1 Z Π(,1,Z,Z Z = E E M,K ( + 1 ( + 1 E K( + 1 Π K Π ( + 1 (,1 [ Z,Z ] (,1 Z = E[ M( + 1 Z ] = exp q(,1 Z, Z Z Z (. (20 CMI is used o obain he firs expression in he las row. The impliaions of CMI are similar o hose from using risk-neural probabiliies insead of objeive probabiliies for priing purposes. Under he objeive probabiliy measure expression, (16 an be deomposed ino E [ M( + 1 K( + 1 Z ] = E[ M( + 1 Z ] E[ K( + 1 Z ] + ov( M( + 1,K ( + 1 Z * * * E [ M( 1 K( + 1 Z ] = E [ M( + 1 Z ] E [ K( + 1 Z ] * * measure, whih implies exp( p(,1 E [ K( 1 Z ] = E [ M( + 1 Z ] = exp( q(, 1 (,m = f(,m g(, m. This redues o + under he risk-neural probabiliy +. While his expression is similar o (20, he laer does no require a ransiion from he risk-neural o he objeive probabiliy measure, whih is a huge advanage in praial implemenaions. 5 One migh ask why Z is no par of Z. This exlusion resriion is, of ourse, ruial for CMI. In he presen onex, we an jusify he exlusion beause he defaulable pension laims we are disussing are no raded. They simply serve as a vehile for deermining he appropriae defaul spreads for he valuaion of pension liabiliies. The defaul spreads d follow immediaely from a sraighforward generalizaion of (20 o all 4 Equaion (17 an be rewrien as π(+1 = Pr[R(+1<1 Z] whih is a definiion of he (ondiional on informaion Z value-a-risk (VaR, while ρ(+1 = E[R(+1 R(+1<1,Z] in (18 defines he omplemen of he orresponding expeed shorfall (ES. 5 Das and Sundaram (2000, for example, assume ha he defaul probabiliy (17 will be larger under he risk-neural measure bu a he same ime rea he reovery rae in he even of defaul (18 as invarian agains he applied measure. Bu (18 will hange when (17 hanges beause i ondiions on defaul. 8

9 mauriies m: exp( p(,m = exp( q(,m Π(,m. Saring from m = 0 reursively using g(,m = ( q(,m + 1 q(,m and f(,m ( p(,m + 1 p(,m d d, spreads are alulaed = as (,0 f(,0 g(,0 = p(,1 + p(,0 + q(,1 q(,0 = [ p(,1 ln Π(,1 ] p(,1 = ln Π( (,1 = f(,1 g(,1 = p(,2 + p(,1 + q(,2 q(,1 = ln Π(,2 d(,0 (,2 = f(,2 g(,2 = p(,3 + p(,2 + q(,3 q(,2 = ln Π(,3 d(,0 d(,1 L d =,1 where he firs line makes use of he erminal ondiions (,0 0 ( 0 p = and q,0 =. More generally, he following wo equaions ompleely desribe he defaul spread erm sruure d (,0 log( Π(, 1 = m 1 n= 0 (,m = log( Π(,m + 1 d( d,n for m > 0 (22 (21 These spreads are endogenous (Duffie and Singleon, 1999 in he sense ha hey depend on he value of he defaulable laim iself. The spreads depend on he disoninuane surplus raio whih iself depends on he spread. Combining and rearranging equaions (21 and (22, he defaul spread erm sruure an be expressed as follows (for m > 0 m Π ( = 1,m exp d(,n (23 n= 0 where Π (, m is dereasing in m and Π (,m = 1 if and only if (,n 0 d =, for all n < m. 2.4 Opimal Asse Alloaion Having inrodued an asse-liabiliy modeling framework ha provides an inerdependeny of valuaion and asse alloaion, an opimaliy rierion for hoosing he asse alloaion w (,k, for k = 1, K,K, and he onribuion rae in (10 needs o be disussed. A he same ime, he equilibrium defaul spread (whih appropriaely refles he likelihood ha he promised pension paymens anno be delivered is deermined. The deerminaion of he opimal values of hese variables will be arried ou in wo sages. Firs, he seleed objeive funion is opimized wih respe o he opimal asse alloaion. Seond, he onribuion rae is deermined o ahieve erain funding arges for he pension fund over a given ime horizon, usually known as a onrol period. For example, may be hosen o ahieve 100% funding over a hree-year onrol period. The following analysis fouses on he firs opimizaion problem ha is solved for some given. Oher sudies solve hese problems simulaneously. For example, Haberman and Sung (1994 fous on he derivaion of an opimal onribuion sraegy ha simulaneously minimizes onribuion rae risk, unexpeed deviaions from a argeed onribuion rae, and solveny risk, unexpeed deviaions from a argeed funding level. Neverheless, a wo-sage proess is ommon in praial implemenaions of ALM. 9

10 Sharpe and Tin (1990 opimize a mean-variane funion in he raio of fuure surplus o urren asses. Similarly, we propose a mean-variane objeive funion in he DSR (10, whih is he raio of fuure surplus o urren liabiliies. A mean-variane objeive funion in he surplus wih an exogenously given defaul spread is widely used for plan asse alloaion deisions and herefore provides a naural saring poin for he presen paper 6. Alhough his objeive funion is wrien in erms of he firs and seond momens of he saled surplus disribuion, higher momens are usually par of he objeive funion in he presen onex beause hey affe he defaul spread as is lear from (12 and (13. We herefore assume ha he pension fund minimizes he ondiional variane of he DSR, given ha he ondiional mean of he DSR equals some given E [ DSR] and porfolio weighs sum o uniy min w( w ( V [ y( + 1 ] w( s.. w ( E [ y( + 1 ] = E[ DSR] and w ( 1K = 1 where (24 ( + 1 = exp( x( + 1 ( 1+ C(,h ( 1+ u v (,h exp( r( 1,h, x ( + 1 = ( x( + 1,1, K, x( + 1,K, ( = ( w(,1, K, w (,K, v (,h = ( v(,h,1, K,v(,h,L, r ( + 1,h = ( r( + 1,h + 1, K,r( + 1,h + L y + w 1 K denoes a Kx1 veor of ones and E and V are he expeaion and variane operaors ondiional on ime informaion Z, Z. A smaller E [ DSR] orresponds o a more risk averse behavior and a higher willingness on he par of he sponsor o pay onribuions. Solving (24 for he opimal asse alloaion is ompliaed by he fa ha he defaul spreads enering he surplus equaion hemselves depend on he alloaion w( as well. As a onsequene of his dependeny, any hange in he asse alloaion will immediaely affe he urren value of liabiliies, whih eners he objeive funion in he denominaor of C (, h. This is he main differene from asse alloaion problems in he radiion of Markowiz (1952 for he ase wihou liabiliies and Sharpe and Tin (1990 for he ase wih liabiliies. The dependeny beomes lear by noing ha he firs-order ondiions dj dw,k for J = w V y + 1 w wih respe o he porfolio ( ( a minimum of he objeive funion ( ( [ ( ] ( weigh w (,k involve he sum of derivaives of he form J( w(,k + J( d(,m d( d(,m dw(,k for mauriies m = 0, K,h + L 1. Deermining he signs of he wo omponens of he derivaives requires assumpions regarding he disribuion of asse reurns and forward raes. Deriving he sign of he expression d ( d(,m dw(,k = exp( d(, m 1 ρ,m dπ,m dw,k Π,m dρ,m dw,k remains a diffiul analyial ask even [( ( ( ( ( ( ( ] for given disribuional assumpions. 3. A Numerial Example In order o shed more ligh on he relaionship beween opimal asse alloaion and defaul spreads, we onsru a simple example involving an individual who lives for wo periods, 6 Alhough we use a mean-variane objeive funion, here are many ypes of objeive funions involving (10, inluding hose for muli-period alloaion deisions, ha are ompaible wih he idea of deriving simulaneously he opimal asse alloaion and he appropriae defaul spread. 10

11 one period of work and one period of reiremen, so ha h = L = 1. We also assume ha here are only wo asse lasses, so ha K = 2. Finally we impose he simplifying assumpion of a ime- and mauriy-invarian forward rae urve for he numerial example, i.e. ( ( ( f,0 = f,1 = f + 1,0. Wih his assumpion we need o alibrae jus one single defaul-free forward rae, g (, 0, and obain jus one single defaul spread, d (,0, ha an be onvenienly ompared wih he hisorial spread of he yield of a AA orporae bond wih long mauriy over g(,0. The DSR (10 hen beomes ( + 1,0 = w ( y( + 1 = w ( [ exp( x( + 1 ( 1+ C(,1 ( 1+ u exp( r( + 1,2 ] L(,1 A ( ( ( ( ( w exp x exp( f(,0 + f(,1 ( 1+ u exp( f(,0 + f(,1 f( + 1,0 U(,1,e w ( [ exp( x( + 1 ( 1+ ( C (,1 exp( 2d(,0 ( 1+ u exp( g(,0 d(,0 ] S = = 0 + (25 where he funding raio has been replaed by he erm in large round parenheses in he seond line and by C 0 (,1 exp( 2d(, 0 raio for a risk-free pension plan wih zero defaul spread, (,1 A( U(,1,e exp( 2g(,0 We assume ha he veor ( (,0, x( + 1,1, x( + 1, 2 mulivariae normal disribued as in he hird line, making use of he definiion of he funding C 0 =. g is independen and idenially x x g(,0 µ 0 σ00 ( + 1,1 ~ N µ 1, σ10 σ11 ( + 1,2 µ σ σ σ (26 whih allows us o look a unondiional momen funions and o drop he ime index. The mean and variane omponens of he objeive funion (24 simplify o E m v v [ y] m A ml [ ] A L AL 2 2 = and V y = v + v v wih (27 = mx ( 1 C( 1 exp( 2d( 0, ml = ( 1+ u my exp( d( = v X ( 1 C( 1 exp( 4d( 0, vl v Y ( 1+ u exp( 2d( 0 = v ( 1+ ( 1 u C( 1 exp( 3d( 0 A + 2 A + AL XY + =, [ ] where denoes he Kroneker produ, C( 1 = C0 ( 1E exp( 2g( 0, and using he following momens derived from he properies of he log normal disribuion = E[ exp( x + 2g( 0 ] = { m X,i} = { exp( µ i + 2µ σ ii + 2σ 00 + σ i0 } i= 1,2 i 1, = E[ exp( g( 0 ] = exp( µ σ00, v Y = V[ exp( g( 0 ] = my ( exp( σ00 1 V[ exp( x + 2g( 0 ] = { m X,im X,j ( exp( σij + 4σ00 + 2σi0 + 2σ j0 } i,j= 1, 2 = COV[ exp( x + 2g( 0,exp( g( 0 ] = { m X,im Y ( exp( σi0 + 2σ00 } i 1, 2. m X 2 = m v Y 5 2 X = 1 v XY 1 = 11

12 Mos imporanly, assumpion (26 allows us o derive explii expressions for he wo omponens of he defaul spread exp( d( 0 = 1 π( 1 + π( 1 ρ( 1 π ( [ ] ( E DSR + τ 1 1 = Φ and ρ( 1 = τ( 1 w V[ y] w, namely [ ] + τ( 1 w V[ y] w 2 v AL v A φ E DSR + ma + (28 w V[ y] w π( 1 using properies of he sandard normal disribuion (see, e.g., Gourieroux and Monfor, 1995, h. B.3.4.b wih p.d.f. φ ( and.d.f. ( Φ. We now opimize (24 wih (27 using an ieraive proedure based on he following idea. We an generae an explii soluion for he opimal asse alloaion in (24 if we ondiion on a * fixed spread di in ieraion i. Call his w di, whih is derived e.g. by Cohrane (2001, p. 85 using he erms = E y d y d E y d, = E y d V y d γ = 1 V y d as α [ i ] [ i ] [ i ] β [ i ] [ i ] 1 2 and 2 [ i ] 1 2 ( γ E[ DSR] β E[ y d ] + ( α βe[ DSR] * 1 i 12 w di = V[ y di ] 2. (29 αγ β Thus, we iniialize he ieraions by ompuing (29 for a saring value of d 1 = 0, plug w d1 * ino (28, ompue = log 1 π 1 + π 1 ρ 1 and w d in he nex ieraion and oninue d 2 ( ( ( ( wih his ieraive proedure unil boh he asse alloaion and he defaul spread onverge in he sense ha heir las updae is smaller han Of ourse, his ieraive proess will yield he same soluion as he dire opimizaion of (24 by means of numerial opimizaion mehods, bu, in he presen onex, is muh less ompliaed. 2 * We ondu wo experimens for pension plans wih an assumed funding raio (using a zero defaul spread of C 0 ( 1 = and ( C 0 =. These pariular hoies are moivaed by our desire o presen a se of inerior soluions o he opimizaion problem. I will beome lear below ha a funding raio muh below ( C 0 = will imply bond shor selling while a funding raio muh above C 0 ( 1 = will generae zero defaul spreads given our hoies for he oher parameers. In boh experimens we onsider a range of possible defaul hresholds τ ( 1 from 0.00 o Reall ha τ ( 1 denoes he ondiional expeaion of he ompany s fuure ne worh per uni of urren pension liabiliies. Thus, we fous on ompanies wih a omparably large burden of pension liabiliies. We do his beause i is preisely hese ompanies whih fae a signifian defaul risk in he sense of having a high defaul probabiliy and a low reovery rae for heir pension plans in he ase of defaul. Boh experimens are based on ime- and age-speifi wage inflaion and onribuions = We mah he momens of he asse reurns and he defaul-free yield o sample momens ompued from ime series daa of US marke indies. We use he real yield of a Treasury bond wih 30 years mauriy for u = 0.02 g ( 0 and real oal reurn indies for US Treasury bonds of all mauriies (JPM index and US equiies (MSCI index for he wo asses. The firs hoie implies ha one period in our example has a lengh ha equals he ypial average mauriy of orporae pension liabiliies. 12

13 Table 1 onains desripive saisis for all he variables we need, ogeher wih Moody s (real yield index for AA raed orporae bonds wih mauriies 20 years and above. This index is frequenly used for deermining he disoun rae for he alulaion of orporae pension liabiliies in he USA. The average yield on his index exeeds he average yield on 30- year Treasury bonds by 1.05 perenage poins. 7 We will use his spread as a benhmark agains whih we ompare he endogenously derived defaul spread of our model. The sample period is Deember 1988 (he monh he firs annual oal reurn ould be ompued from he JPM index whih sared in Deember 1987 o February 2002 (he monh he US Treasury sopped publishing yields on bonds wih 30 years mauriy. Figure 1 presens he yield and reurn daa. Table 1: Desripive Saisis of he Daa Index Mean Volailiy Correlaion Marix Treasury 3.65% 0.78% JPM 4.96% 4.76% MSCI 11.23% 15.23% Moody's 4.70% 0.62% Noes: Sample period is Deember 1988 February Treasury: annual real yields on US Treasury seuriies wih 30 years mauriy. JPM: oal annual real reurn on JP Morgan US Treasury index for all mauriies. MSCI: oal annual real reurn on MSCI USA equiy index. Moody's: annual real yields on AA orporae bonds wih mauriies 20 years and above. The means are annualized geomeri means. The volailiies are annualized sandard deviaions. Daa soure: Daasream Figure 1: Annual Toal Real Reurns and Yields Annual Perenages De De-90 De-92 De-94 De-96 De-98 De Treasury JPM MSCI Moody's Noes: The graph displays annualized means of he real yield of a US Treasury bond wih 30 years mauriy (Treasury, he real yield of he Moody s index of AA raed orporae bonds wih mauriies 20 years and above (Moody s, he real reurn of he JP Morgan US Treasury bond index for all mauriies (JPM and he real reurn of he MSCI US equiy index (MSCI over he period Deember 1988 February Ideally, we would have liked o use he yield on 20-year Treasury bonds as a mah for he Moody s index, bu he definiion of he 20-year index hanged over he sample period. We were herefore fored o use 30-year Treasury bonds as he nex bes alernaive. 13

14 The resuls of he wo experimens are depied in Figures Tables in he Appendix onain he daa underlying hese Figures. We firs disuss movemens along he urves (i.e. for a given [DSR wihin eah graph (i.e. for a given funding raio 1, hen disuss move- E ] C 0 ( mens beween urves (i.e. for a differen E [ DSR] resuls beween graphs (i.e. for differen funding raios C 0 ( 1. wihin eah graph and finally ompare he For a given pension funding raio ( C 0 ( 1 and expeed disoninuane surplus raio (e.g. E [ DSR] = 0, higher ompany ne worh (orresponding o a higher defaul hreshold has wo effes. Firs, i lowers he defaul probabiliy (sine insolveny is less likely: his is shown by he downward sloping urves in Figures 2 and 3. Seond, i inreases he reovery rae (sine we assume he pension fund an laim up o 100% of he ne worh of he ompany: his is shown by he upward sloping urves in Figures 4 and 5. The ombined effe is ha he defaul spread is lower he higher he ompany s ne worh as shown by he downward sloping urves in Figures 6 and 7. Sine he defaul spread is inreasing in he defaul probabiliy and dereasing in he reovery rae (see (19, his resul follows mehanially from Figures 2-5. As he spread falls, he expeed value of he liabiliies rises and ges loser o is promised value, i.e. he value of he liabiliies using a zero defaul spread. The opimal equiy weighing inreases wih he ompany s ne worh beause a higher alloaion o equiies (wih heir higher expeed reurns inreases he expeed surplus and his is needed o mah he higher level of liabiliies in he denominaor of he DSR, ( 1,0 L(,1 S +, implied by he lower defaul spread (see (25. This explains he upward sloping urves in Figures 8 and 9. The posiive relaionship ha we find beween ompany ne worh and pension fund equiy weighing is no, however, onsisen wih he analyses of Blak (1980 and Tepper (1981 or he empirial findings of Bodie e al. (1985 whih show ha profiable axpaying ompanies will aemp o redue heir ax liabiliies by invesing bonds. This differene in resuls is explained by he absene in our model of he disoring effe of axes. For a given pension funding raio and defaul hreshold, an inrease in he E [DSR] raises he defaul probabiliy when he defaul hreshold is low and lowers he defaul probabiliy when he defaul hreshold is high: he urves in Figures 2 and 3 inerse wih he E DSR [ ] = urve highes o he lef of he inerseion and he E [ DSR] = 0 urve highes o he righ of he inerseion. This is explained as follows. An inrease in E [ DSR] inreases boh he mean and variane of he disribuion (see equaion (24. This has he effe of flaening he densiy funion around he mean (whih iself shifs o he righ and faening he densiy funion in he ails (ompare he dashed and solid urves in he upper panel of Figure B1 in Appendix B. There mus exis a defaul hreshold for whih he area under eah urve o he lef of his hreshold (whih measures he defaul probabiliy is he same (his is given by he inerseion poin in he lower panel of Figure B1. For lower defaul hresholds, he defaul probabiliy inreases when E [DSR] inreases; he opposie holds for higher defaul hresholds. This explains why he downward sloping lines in Figures 2 and 3 inerse: hey orrespond o differen umulaive disribuion funions wih differen means and varianes. 14

15 Figure 2: Defaul Probabiliies π ( 1 for Differen E [ DSR] and ( τ ; C 0 ( 1 = Defaul Probabiliy Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 Noes: C 0 (1 is he funding raio of he pension plan assuming a zero defaul spread. E[DSR] denoes he argeed mean disoninuane surplus raio. The defaul hreshold is he raio of he sponsoring ompany s fuure ne worh o he urren value of liabiliies. Figure 3: Defaul Probabiliies π ( 1 for Differen E [ DSR] and ( 1 τ ; C 0 ( 1 = Defaul Probabiliy Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 Noes: Cf. Figure 2. 15

16 Figure 4: Reovery Raes ρ ( 1 for Differen E [ DSR] and ( 1 τ ; C 0 ( 1 = Reovery Raes Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 Noes: Cf. Figure 2. Figure 5: Reovery Raes ρ ( 1 for Differen E [ DSR] and ( 1 τ ; C 0 ( 1 = Reovery Raes Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 Noes: Cf. Figure 2. 16

17 Figure 6: Defaul Spreads d ( 0 for Differen E [ DSR] and ( 1 τ ; C 0 ( 1 = Defaul Spread Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 AA Noes: Cf. Figure 2. Figure 7: Defaul Spreads d ( 0 for Differen E [ DSR] and ( 1 τ ; C 0 ( 1 = Defaul Spread Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 AA Noes: Cf. Figure 2. 17

18 Figure 8: Equiy Weighings for Differen E [ DSR] and ( 1 τ ; C 0 ( 1 = Equiy Weighing Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 Noes: Cf. Figure 2. Figure 9: Equiy Weighings for Differen E [ DSR] and ( 1 τ ; C 0 ( 1 = Equiy Weighing Defaul Threshold E[DSR] = E[DSR] = E[DSR] = 0.00 Noes: Cf. Figure 2. 18

19 Figure 10: Effiien Fronier in DSR; C 0 ( 1 = E[DSR] Vola[DSR] Defaul Thres. = 0.02 Defaul Thres. = 0.08 Noes: Cf. Figure 2. Vola[DSR] denoes he volailiy or sandard deviaion of he DSR Figure 11: Effiien Fronier in DSR; C 0 ( 1 = E[DSR] Vola[DSR] Defaul Thres. = 0.02 Defaul Thres. = 0.08 Noes: Cf. Figure

20 For a given pension funding raio and defaul hreshold, an inrease in E [DSR] lowers he reovery rae (see Figures 4 and 5. This is beause he value of he ompany s equiy and he equiy in he pension fund mus now fall by larger amouns before he pension fund beomes insolven and so he pension fund, perversely, has a greaer shorfall o reover. Noe ha here is no rossover of he urves in his ase: from he seond equaion in (28, he size of ρ (1 is dominaed by he firs erm τ ( 1 + ma so he impa of hanges in π (1, whih only affe he seond erm, is small. For a given pension funding raio and defaul hreshold, an inrease in E [DSR] inreases he defaul spread (see Figures 6 and 7. This indiaes ha he negaive impa of he reovery E [ DSR] on he defaul spread dominaes he posiive rae (reovery raes fall wih inreasing impa of he defaul probabiliy (defaul probabiliies are lower wih inreasing E [DSR] o he lef of he inerseion poins in Figures 2 and 3 9 on he defaul spread. For a given pension funding raio and defaul hreshold, an inrease in he E[DSR ] 8 (whih, as menioned above, orresponds o a derease in risk aversion naurally resuls in a higher equiy weighing (see Figures 8 and 9. In he ase of ( of 0.05, he equiy weighing is around 72% when E[ DSR] E [ DSR] = 0. C 0 = and a defaul hreshold = and around 95% when Sharpe (1976 and Bodie e al. (1987 show ha firms faing finanial diffiulies or emporary ash flow shorages have an inenive o raise he required disoun rae by invesing in equiies o lower boh repored liabiliies and he onribuion rae o he plan. In our numerial example, a higher disoun rae always follows from a more aggressive invesmen sraegy (an inrease in E[DSR] ha resuls in an inrease in equiy weighings whaever he finanial srengh of he sponsoring ompany. The posiive relaionship beween equiy weighings and he defaul spread, aniipaed by ommon sense, holds for movemens beween urves for a given defaul hreshold (f. Figures 6 and 8, and Figures 7 and 9. By onras, movemens along a given urve whih hold E [ DSR] onsan and vary he defaul hreshold show a negaive relaionship beween he defaul spread and equiy weighings (f. he urves labeled E [ DSR] = 0 in Figures 6 and 8, and Figures 7 and 9. For a given E [DSR] and defaul hreshold, higher funding raios lead o lower defaul probabiliies so long as defaul probabiliies are smaller han 0.5. For defaul probabiliies larger han 0.5, higher funding raios inrease he defaul probabiliy (see Tables A1 and A2 in Appendix A. This is beause a probabiliy 0.5, E [ DSR] is exaly equal o he defaul hreshold regardless of he funding raio. Thus, he umulaive disribuion funions inerse a probabiliy 0.5 as is lear from (28 (and Figure B1 in Appendix B. Probabiliies ha are relaively higher o he lef of he inerseion swih o being relaively lower o he righ of he inerseion. Comparing Figures 2 and 3, he urves have a ommon fixed poin passing hrough a 8 The resul also follows immediaely from he relaionship beween VaR and ES (f. foonoe 4: If π(+1 = Pr[R(+1<1 Z] dereases wih inreasing E[DSR], ρ(+1 = E[R(+1 R(+1<1,Z] has o derease as well by definiion. 9 To he righ of he inerseion poins he effes of he defaul probabiliy and he reovery rae on he spread are reinforing and he spread unambiguously inreases wih inreasing E[DSR]. 20

21 defaul probabiliy of 0.5, bu he urves in Figure 3 are seeper han hose in Figure 2. Van- Derhei (1990 obains a signifian negaive oeffiien on he funding variable in a logi regression for he defaul probabiliy using a sample of US pension plans erminaed beween 1981 and This resul is onsisen wih our findings for defaul probabiliies below 0.5. For a given E [DSR] and defaul hreshold, a 1.5 perenage poin inrease in he funding raio implies approximaely a 1.5 perenage poin inrease in he reovery rae (he urves in Fiure 5 lie approximaely 1.5 perenage poins above Figure 4. Regardless of he defaul probabiliy being above or below 0.5, a higher funding raio unambiguously implies a lower defaul spread for a given E[DSR] and defaul hreshold (he urves in Figure 7 are lower han he urves in Figure 6 whih indiaes ha he reovery rae has a greaer impa on he defaul spread han on he defaul probabiliy. This is onsisen wih empirial resuls from Peersen (1996 who repors a highly signifian negaive (albei small in absolue value oeffiien on he funding variable in a disoun rae regression using US daa from Figures 6 and 7 also plo he average AA orporae bond yield for he sample period. Only under very pariular irumsanes i is opimal o disoun pension liabiliies using he AA orporae bond yield: Figure 6 shows hese o be C 0 ( 1 = , E [ DSR] = 0 and ( 1 τ approximaely equal in our example. The figures also show ha in general he AA orporae bond yield is likely o be an inappropriae disoun rae for valuing pension liabiliies. Finally, Figures 10 and 11 display effiien froniers in mean volailiy graphs for he opimized disoninuane surplus raio. An inrease in he funding raio implies an upward shif of he effiien fronier. An inrease in he ne worh of he sponsoring ompany does no affe he effiien fronier o any signifian exen. 4. Conlusion This paper hallenges urren praie in asse-liabiliy managemen (ALM in a fundamenal way. We have shown ha any exogenously deermined disoun rae is unlikely in general o be suiable for deermining he value of liabiliies sine i will no refle he rue risk of failing o deliver he promised fuure liabiliy paymens. The appropriae disoun rae will depend on a number of faors, he mos imporan of whih are: he asse alloaion, he funding raio and he finanial srengh of he guaranor of he liabiliies suh as he orporae sponsor of a defined benefi pension plan. In order o fous on he main relaionship beween disoun raes and asse alloaion, we inrodued some simplifiaions, whih we would like o abandon in fuure work: We would like o exend he heoreial framework by esablishing a relaionship beween he plan sponsor s ore business and he finanial srengh of he pension plan. Webb (2004 reas he defii of a orporae pension plan as orporae deb wih funding re- 21

22 quiremens and prioriy rules in he even of ompany insolveny and examines he impa of he plan sponsor s finanial posiion on he pension plan s invesmen poliy. One way of onsidering hese dependenies in our framework would be o relae he hreshold defining defaul o he plan sponsor s finanial srengh in a redued form approah. We would like o examine he asse-liabiliy modeling exerise from he poin of view of he differen sakeholders in he orporae pension plan, prinipally he sponsor and he members. Eah of hese differen viewpoins involves differen (possibly iner-emporal objeive funions, differen risk aversion parameers and hene differen opimal asse alloaions, equilibrium disoun raes and liabiliy valuaions. Anoher ineresing line of researh using our mehodology would be o analyze he role of a pension plan insurane sheme, whih already exiss in he US (Pension Benefi Guarany Corporaion, PBGC 10 and has reenly been inrodued in he UK (Pension Proeion Fund. Finally, an empirial appliaion of our model remains an imporan ask. In an empirial analysis we ould abandon he kind of disribuional assumpions we imposed in he numerial example, i.e. i.i.d. normal reurns and yields. To esimae he ondiional expeaion omponens of he defaul spread, one ould use nonparameri esimaion ehniques as proposed by Saille (forhoming for he esimaion of ondiional VaR and ES risk measures. The firs order ondiions for he opimal porfolio weighs given ondiioning informaion migh be solved along he lines of he nonparameri Kernel-M esimaion approah suggesed by Brand (1999 and Brand and Aï-Sahalia (2001. Daa on pension plan defauls and reovery raes in he even of defaul is no required beause he defaul spread is endogenous and herefore ompleely desribed by he evoluion of he sum of pension plan asses and he ne worh of he sponsoring ompany relaive o he liabiliies of he pension sheme. 10 The PBGC an ake up o 30% of he ne worh of a ompany in he even of a pension plan defaul. 22

23 Referenes Bader, L. and J. Gold (2003: Reinvening Pension Auarial Siene, The Pension Forum, Soiey of Auaries, January. Blak, F. (1980: The Tax Consequenes of Long-run Pension Poliy, Finanial Analyss Journal, 36,1-28. Blake, D. (2001: UK Pension Fund Managemen: How is Asse Alloaion Influened by he Valuaion of Liabiliies?, Merrill Lynh Commenary, 25 January. Blake, D. (2003a: Pension Shemes and Pension Funds in he Unied Kingdom, Oxford Universiy Press, Oxford. Blake, D. (2003b: UK Pension Fund Managemen Afer Myners: The Hun for Correlaion Begins, Journal of Asse Managemen, 4, Bodie, Z., J. O. Ligh, R. Mørk, and R. A. Taggar (1985: Corporae Pension Poliy: An Empirial Invesigaion, Finanial Analyss Journal, 41, Bodie, Z., J. O. Ligh, R. Mørk, and R. A. Taggar (1987: Funding and Asse Alloaion in Corporae Pension Plans: An Empirial Invesigaion, in Z. Bodie, J. Shoven, and D. Wise (eds., Issues in Pension Eonomis, Naional Bureau of Eonomi Researh and he Universiy of Chiago Press. Bodie, Z. (1995: On he Risks of Soks in he Long Run, Finanial Analyss Journal, May/June Brand, M. W. (1999: Esimaing Porfolio and Consumpion Choie: A Condiional Euler Equaion Approah, Journal of Finane, 54 (5, Brand, M. W., and Y. Aï-Sahalia (2001: Variable Seleion for Porfolio Choie, Journal of Finane, 56 (4, Cairns, A. J. G. (2003: Pension Fund Mahemais, Disussion Paper PI-0315, The Pension Insiue. Carrol, T. J., and G. Niehaus (1998: Pension Plan Funding and Corporae Deb Raing, Journal of Risk and Insurane, 65 (3, Cohrane, J. H. (2001: Asse Priing, Prineon Universiy Press. Das, S. R., and R. K. Sundaram (2000: A Disree-Time Approah o Arbirage-Free Priing of Credi Derivaives, Managemen Siene, 46 (1, Duffie, D., and K. J. Singleon (1999: Modeling Term Sruures of Defaulable Bonds, Review of Finanial Sudies, 12 (4, Exley, C.J., S. Meha, and A. D. Smih (1997: The Finanial Theory of Defined Benefi Pension Shemes, Insiue & Fauly of Auaries, 28 April. Gold, J. (2001: Aouning/Auarial Bias Enables Equiy Invesmen by Defined Benefi Pension Plans, Disussion Paper , Pensions Researh Counil. Gourieroux, C., and A. Monfor (1995: Saisis and Eonomeri Models, Volume Two, Cambridge: Cambridge Universiy Press. Haberman, S., and J.-H. Sung (1994: Dynami Approahes o Pension Funding, Insurane: Mahemais and Eonomis, 15, Markowiz, H. (1952: Porfolio Seleion, Journal of Finane, 7, Peersen, M.A. (1996: Alloaing Asses and Disouning Cash Flows, in Pensions, Savings and Capial Markes, Pension and Welfare Benefis Adminisraion, U.S. Deparmen of Labor,

24 Saille, O. (forhoming: Nonparameri Esimaion of Condiional Expeed Shorfall, Mahemaial Finane. Sharpe, W. F. (1976: Corporae Pension Funding Poliy, Journal of Finanial Eonomis, 3, Sharpe, W. F., and L. G. Tin (1990: Liabiliies A New Approah, Journal of Porfolio Managemen, 16 (2, Tepper, I (1981: Taxaion and Corporae Pension Poliy, Journal of Finane, 36, VanDerhei, J. L. (1990: An Empirial Analysis of Risk-Relaed Insurane Premiums for he PBGC, Journal of Risk and Insurane, 57, Webb, D. (2004: Sponsoring Company Finane and Invesmen and Defined Benefi Pension Sheme Defiis, UBS Pensions Researh Programme Disussion Paper No.23, Finanial Markes Group, London Shool of Eonomis. Wilson, T. (1997: Porfolio Credi Risk (I, Risk, 10 (9. 24

25 Appendix A Table A1: Simulaion Resuls for he Numerial Example Based on C 0 ( 1 = τ ( 1 E [ DSR] V [ DSR] π ( 1 ρ ( 1 ( 0 d Bond Equiy AA AA AA Noes: C 0 (1 is he funding raio of he pension plan assuming a zero defaul spread. The firs olumn onains he defaul hreshold, i.e. he sponsoring ompany s fuure ne worh per uni of he urren value of liabiliies. AA in he firs olumn refers o he benhmark resuls obained from using he exogenous hisorial defaul spread of AA bonds. The seond and hird olumns presen he mean and volailiy of he opimized disoninuane surplus raio. d(0 is he endogenous defaul spread wih omponens π(1, he defaul probabiliy, and ρ(1, he reovery rae. Bond and Equiy refer o he opimized asse alloaion. 25