NBER WORKING PAPER SERIES DISCOUNTING PENSION LIABILITIES: FUNDING VERSUS VALUE. Jeffrey R. Brown George G. Pennacchi

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1 NBER WORKING PAPER SERIES DISCOUNTING PENSION LIABILITIES: FUNDING VERSUS VALUE Jeffrey R. Brown George G. Pennacchi Working Paper hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachuses Avenue Cambridge, MA June 2015 We hank Joshua Rauh and paricipans in he 2015 NBER conference on Reiremen and Healh Benefis in he Public Secor for helpful feedback and suggesions. Disclosure: Brown is a rusee for TIAA, a financial services company ha provides services o numerous public secor reiremen plans. The views expressed herein are hose of he auhors and do no necessarily reflec he views of he Naional Bureau of Economic Research. NBER working papers are circulaed for discussion and commen purposes. They have no been peerreviewed or been subjec o he review by he NBER Board of Direcors ha accompanies official NBER publicaions by Jeffrey R. Brown and George G. Pennacchi. All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi, including noice, is given o he source.

2 Discouning Pension Liabiliies: Funding versus Value Jeffrey R. Brown and George G. Pennacchi NBER Working Paper No June 2015 JEL No. G20,H55,J26 ABSTRACT We argue ha he appropriae discoun rae for pension liabiliies depends on he objecive. In paricular, if he objecive is o measure pension under- or over- funding, a defaul-free discoun rae should always be used, even if he liabiliies are hemselves no defaul-free. If, insead, he objecive is o deermine he marke value of pension benefis, hen i is appropriae ha discoun raes incorporae defaul risk. We also discuss he choice of a defaul-free discoun rae. Finally, we show how cos-of-living adjusmens (COLAs) ha are common in public pensions can be accouned for and valued in his framework. Jeffrey R. Brown Deparmen of Finance Universiy of Illinois a Urbana-Champaign 515 Eas Gregory Drive Champaign, IL and NBER brownjr@illinois.edu George G. Pennacchi College of Business Universiy of Illinois 4041 BIF, Box 25, MC E. Gregory Drive Champaign, IL gpennacc@illinois.edu

3 Discouning Pension Liabiliies: Funding versus Value Jeffrey R. Brown and George G. Pennacchi 1. Inroducion There is a long-sanding debae among financial economiss, pension acuaries and oher ineresed paries abou he appropriae rae o use for discouning fuure pension liabiliies. The Governmen Accouning Sandards Board (GASB) has long used rules ha relae he discoun rae o he expeced reurn on pension plan asses, a posiion ha has been defended by numerous acuaries and plan sponsors. 3 In conras, mos financial economiss argue ha liabiliies should be discouned using a rae ha reflecs he risk of he liabiliies (Novy-Marx and Rauh, 2011; Brown and Wilcox 2009; Lucas and Zeldes 2009; Andonov, Bauer, and Cremers, 2014). In he special case where he public pension benefis are considered free of defaul risk, such as in he presence of srong consiuional proecions, financial economiss have argued ha he appropriae rae o use is a defaul-free one (e.g., Brown and Wilcox 2009). In his paper, we make a simple bu imporan concepual poin. We argue ha here is an imporan difference beween he appropriae measure of a plan s funding saus and he appropriae measure of is marke value. Specifically, we argue ha he correc discoun rae for deermining a plan s funding saus is he defaul-free rae: in a deparure from prior sudies in financial economics, we argue ha his is rue regardless of wheher he liabiliies are defaulfree or no. In conras, a defaul-risky discoun rae is he appropriae rae o use for measuring he marke value of he liabiliies. 3 In 2012, GASB issued new sandards for discouning public pension liabiliies. Whereas he previous sandards discouned liabiliies wih he expeced reurn on plan asses, he new sandard uses a blended rae. Roughly speaking, he new sandard permis plans o use he expeced reurn assumpion for discouning so long as hey are adequaely funded using ha discoun rae. For underfunded plans, he sandard calls for discouning he unfunded porion using a defaul free rae. As wih he prior sandard, he new GASB sandard has no grounding in financial heory. As such, i is no clear wha quesion, if any, he resuling measure of funding acually answers.

4 2 Measuring underfunding and placing a marke value on liabiliies are clearly wo disinc conceps, and are useful for answering differen quesions. For example, he use of a defaul-free discoun rae is informaive o paricipans who wan o know how much money he plan would need o be assured ha he plan will be able o pay promised benefis. This would also be a relevan measure if he plan wished o offload is liabiliies o an insurance company ha inends o make good on he fuure benefi paymens. For example, a number of defined benefi (DB) plans, including General Moors, have recenly ransferred heir currenly-accrued pension liabiliies o an insurance company (Prudenial, in he GM case) ha will hen provide he reiremen annuiies o he sponsor s employees. The insurance company aking over he plan would raionally require he sponsor o pay he difference beween he accrued plan liabiliies discouned a he defaul-free rae and he plan s asses (plus any adminisraive coss). If he insurance company received his paymen along wih he plan s curren asses, i would permi he insurance company o inves in exacly hose defaul-free securiies ha guaranees is abiliy o mee fuure obligaions o he sponsor s employees. Thus, for he purposes of deermining how much i would cos he sponsor o sell off is pension liabiliies, discouning he liabiliies a he defaul-free rae is needed o deermine he paymen o he acquirer of he liabiliies. However, here are also cases for which he marke value of he liabiliy is imporan. Curren or poenial plan paricipans migh wan o know he marke value of pension liabiliies (raher han he promised value) when hey are making decisions abou he value of pension benefis being offered by wo differen employers. Curren employees may also wan o know he marke value of pension promises when hey are deciding when o reire: if heir employer is likely o defaul on pension promises, hey may wan o work longer o increase heir savings from curren salary. The marke value of a DB plan s promises would also be informaive o

5 3 individuals faced wih a choice beween a DB and a DC plan, a choice ha a number of saes provide o heir public employees. Prior research analyzing he financial condiion of public pension funds ofen fails o disinguish beween accouning for pension liabiliies o compue a pension fund s level of (under-) funding and deermining he marke value of a pension fund s liabiliies. We demonsrae ha hese are disinc exercises, hough previous work frequenly reas hem as being he same. Specifically, i has been common in academic sudies o jusify he use of a pension discoun rae based on he likelihood ha a sae or local governmen will defaul on is pension obligaions: a higher probabiliy of defaul would warran a higher discoun rae. We apply he sandard Meron (1974) srucural credi risk model o he conex of a defined-benefi pension fund. We show ha discouning he liabiliy wih a defaul-free rae has he aracive propery ha he size of he oal liabiliy (funded plus unfunded) is no affeced by he plan s funding saus. We furher show ha i has he desirable propery ha he measure of fundedness declines o zero as plan asses fall o zero. For comparison, we show ha he marke value of he liabiliy can have odd properies as a sysem of funding measuremen: specifically, he size of he oal (funded plus unfunded) liabiliy can vary wih he degree of funding, and ha funding levels asympoically approach 100 percen as asses approach zero. Our argumen, herefore, is ha he appropriae measure of funding uses defaul-risk-free raes. This naurally leads o he pracical quesion of how o measure defaul-free discoun raes. There are a leas hree reasons ha he mos commonly-used proxies he yields on deb securiies issued by he federal governmen are occasionally criicized for being biased measures of he economy s defaul-free raes. Firs, governmen bonds are no ruly defaul-free. Alhough he risk of a U.S. governmen defaul is no high, i is also no zero, as is evidenced by

6 4 he very exisence of a marke for credi defaul swaps on governmen bonds. Second, here are sae income ax advanages o U.S. governmen bonds, suggesing ha some invesors in high marginal ax rae saes migh be willing o accep a lower yield compared o oher securiies wih he same defaul risk. Third, governmen bond yields migh be lowered by heir high liquidiy relaive o oher fixed income asses. Our paper discusses hese facors and heir quaniaive imporance as a way of bounding he measure of a defaul-free rae. In he end, we find ha he effecs are roughly offseing and ha Treasury yields are a reasonable approximaion o a defaul-free rae. Having esablished ha he appropriae way o discoun pension liabiliies for purposes of measuring funding saus is o use (a proxy for) he defaul-free rae, we hen urn o he quesion of how o incorporae cos-of-living adjusmens (COLAs) ha are common in public secor plans. Were COLAs pure CPI-indexaion, and were here a deep marke for CPI-indexed defaulfree bonds available in he economy, one could simply use he erm srucure of defaul-free real yields on hese indexed bonds, raher han he erm srucure of defaul-free nominal yields. However, as we discuss below, mos public pensions in he U.S. do no use sraigh CPI indexaion for heir COLAs. Raher, COLAs are conracually subjec o upper and lower bounds (caps and floors) on inflaion. Thus, we discuss how one can use marke prices of inflaion derivaives o compue he defaul-free values of hese promised, sae coningen COLAs. This paper proceeds as follows: In secion 2, we provide a brief discussion of issues ha arise when choosing an appropriae discoun rae for public pensions and he debae ha surrounds his issue. In secion 3, we apply he Meron (1974) model o a discussion of Defined Benefi (DB) pension liabiliies and use i o illusrae he disincion beween measuring funded

7 5 saus and valuing liabiliies. In secion 4, we discuss how one should hink abou he choice of a defaul-free rae o use for discouning. We acknowledge hree common criicisms of he use of he yield curve for U.S. Treasuries, presen relevan evidence, and ulimaely conclude ha Treasury yields are a close enough measure of defaul-free raes ha hey represen a reasonable proxy. In secion 5, we urn o he issue of how o accoun for he presence of COLAs ha are common in public pensions. We show ha COLAs ofen have he same promised paymens as paricular combinaions of inflaion derivaives, so ha he price of hese derivaives represens he marke cos of replicaing he COLA paymens. We provide examples of how his approach can be used o value several COLAs ha are common in he public secor. Secion 6 concludes. 2. Background on Public Pension Funding Measures Unlike privae pensions in he U.S., which are required o mee funding sandards imposed by he federal governmen, sae and local plans are no subjec o exernally imposed funding rules. Despie he lack of compulsion, mos saes have adoped policies for funding heir public employee pensions. The funding policies are ypically based on sandards issued by he Governmen Accouning Sandards Board (GASB), an independen organizaion responsible for esablishing accouning and financial reporing sandards for sae and local governmens. GASB has no enforcemen auhoriy and is pronouncemens only have any legally binding naure if he sae or local governmen chooses o volunarily adop he sandards as law or regulaion. Because saes have no exernal legal requiremen o fund pensions, he level of funding is a policy decision. Alhough full funding is one naural benchmark, i is worh noing ha economic heory does no dicae ha full-funding is opimal. If a populace wishes o engage in

8 6 some form of inergeneraional redisribuion, hen ransferring resources across cohors can be achieved hrough under or over funding pensions, jus as i can be done hrough adjusing he level of oher deb obligaions or, even more generally, hrough wheher resources are spen on consumpion versus invesmen. Indeed, he larges public pension in he U.S. he Social Securiy sysem has operaed in a manner closer o a rue pay-as-you-go sysem han as a funded sysem. Alhough saes are less able o susain pure unfunded sysems han is he U.S., owing o he fac ha i is much easier o move capial and labor ou of a fiscally disressed sae han ou of he counry, hey noneheless have some policy flexibiliy in choosing how much o fund. A reason i is imporan o acknowledge ha a funding raio is a policy choice raher han a requiremen is ha i allows one o separae he measuremen of he level of funding from he choice of he level of funding. One can accep ha he appropriae way o measure funding is o use a defaul-free rae wihou i necessarily following ha he opimal funding level for a sae is o always be 100 percen funded according o his measure. This paper is focused on he issue of how o measure he level of funding, wihou aking a view on wha he opimal level of funding should be. In conras o GASB sandards ha rely upon expeced asse reurns, numerous financial economiss have argued ha he appropriae rae is one adjused for he risk of he liabiliies being discouned. Several auhors have hen made he argumen ha public pensions are close o defaul-free. For example, Brown and Wilcox (2009) use several hisorical case sudies o argue ha public pension benefis have lile defaul risk. The discoun rae assumpions maer for he measuremen of funding saus. For example, Munnell, Aubrey and Cafarelli (2014) esimaed ha he acuarial value of asses for

9 7 he 150 sae and local plans in heir daabase was $2.9 rillion a he end of Using he prior GASB sandards ha allow for discouning based on he expeced reurn of plan asses, hey calculaed he presen value of liabiliies a $4.1 rillion. This 72 percen funded raio, however, clearly oversaes he financial healh of hese plans because of he use of a high discoun rae. Governmen bond yields were quie low in 2013: on December 21, 2013, he yield on a 30-year Treasury was only 3.96 percen, and he yield on shorer-erm bonds was even lower. Assuming a fla four percen erm srucure, Munnell, Aubry and Cafarelli calculae an aggregae liabiliy of $6.8 rillion, for an average funding raio of approximaely 43 percen. In a deparure from mos exising papers in financial economics (including one coauhored by one of he curren auhors), i is our conenion ha he appropriae discoun rae o use for measuring funding shorfalls is he defaul-free rae, even if he liabiliies are no defaulfree. We discuss his more in he nex secion wih he help of a simple model. Before urning o his discussion, we noe ha we are focusing on discouning he Accumulaed Benefi Obligaion (ABO) raher han he Projeced Benefi Obligaion (PBO). The general disincion is ha he ABO represens he liabiliy ha he plan has accrued o dae and is measured a curren earnings levels. The PBO, in conras, accouns for he fac ha many of oday s paricipans have earned a benefi ha will no be payable unil some fuure dae and will, a ha ime, be based on a higher earnings level. We focus on he ABO for he reasons oulined in Brown and Wilcox (2009) and Bulow (1982), including ha he ABO measure has he effec of reaing salary and pension accruals similarly. In oher words, employers are no required o repor a higher salary oday jus because employees are likely o receive a higher salary in he fuure. Similarly, we believe i is appropriae o accoun for he effec of fuure 4 hp://crr.bc.edu/wp-conen/uploads/2014/06/slp_39.pdf

10 8 salary increases on pension liabiliies only afer hose salary increases have become effecive. Because we are focused only on he ABO, we do no need o worry abou adjusing discoun raes o reflec he risk of liabiliies ha migh arise from uncerain earnings growh. Lucas and Zeldes (2006, 2009) discuss he implicaions of PBO hedging for boh discoun rae selecion and porfolio allocaion. 3. A Simple Model for Measuring Pension Funding To illusrae he disincion beween measuring liabiliies and placing a marke valuaion on hem, we apply he sandard Meron (1974) srucural credi risk model o he conex of a defined-benefi pension fund. To make our poin ransparen, we firs consider a highly simplified case in which a pension plan promises a single lump-sum benefi paymen a one fuure dae. The model makes he following four simplifying assumpions: 1) The pension fund invess in risky asses ha have a curren dae marke value of A and a rae of reurn volailiy (annual sandard deviaion) of. 2) The pension fund promises a single reiremen benefi equal o X payable a dae +. 3) The coninuously-compounded, defaul-free ineres rae (yield) is consan, equal o y. 4) The sponsor of he pension fund is expeced o conribue o he fund s asses in amouns ca per uni ime unil dae +. 5 If a dae + he pension fund has sufficien asse value, pension paricipans receive heir promised paymen of X. Insead, if A + is worh less han X, he pension fund defauls and he paricipans receive only A +. Le us denoe he dae marke value of he pension fund s liabiliies as L. L is he curren fair marke value ha invesors would pay for he possibiliy of receiving he defaul- 5 There may be more realisic ways of specifying fuure conribuions. Our assumpion ha conribuions are a fixed proporion of curren asses migh describe he phenomenon ha when pension asses experience losses during a marke downurn, he sponsor is likely o be financially weak and reduce conribuions.

11 9 risky cash flow of min X, A a he fuure dae +. Similar o Meron (1974), he marke value of his defaul-risky pension liabiliy equals y 1 2 where d1 A Xe c 2 c y L Xe N d2 A e N d1 (1) ln / / and d2 d1. N() is he sandard normal cumulaive disribuion funcion. Noe from equaion (1) ha L is a funcion of five quaniies: 1) he curren marke value of asses, A ; 2) he rae a which new asses are expeced o be conribued o he fund, c; 3) he volailiy of asses, ; 4) he ime unil he pension s benefi is due o be paid, ; and 5) he promised pension benefi discouned a he defaul-free ineres rae, y Xe. Equaion (1) provides several immediae insighs. One is ha he marke value of liabiliies, L, can never be worh more han he promised paymen discouned a he defaul-free rae, y Xe. Furher, as he value of he pension fund s curren asses grows, L converges o y Xe : lim L A y Xe (2) The quaniy y Xe plays a naural role in he concep of a fully-funded pension plan for y a leas wo reasons. Firs, if he curren value of pension asses equals A = Xe, hen here exiss an invesmen sraegy ha can guaranee o paricipans heir full paymen of X, wihou any need for he sponsor o make fuure asse conribuions o he fund (c=0). This sraegy consiss of invesing all of he pension asses in he defaul-free securiy a rae y. Indeed, if his plan were o ransfer is currenly-accrued liabiliies o an insurance company, so ha i would no be making any new conribuions afer he ransfer, i would need o op up is asses o equal y A = Xe in order for he insurance company o implemen his riskless invesmen sraegy.

12 10 Second, we argue ha he raio of he defaul-risky marke value of liabiliies o is defaul free value is a good measure of he relaive risk faced by a pension paricipan. This y y c quaniy L / Xe N d A / Xe e N d y, and A / Xe y defaul-free rae, A / Xe 2 1 is a funcion of only four facors: c,,. Thus, he pension funding raio for which liabiliies are discouned a he, emerges as he naural facor for measuring pension risk. Moreover, i is clear ha his definiion of a funding raio delivers he sensible resul ha as he value of pension asses declines, he funding raio goes o zero: A 0 y lim A / Xe 0 (3) If, insead, pension liabiliies were discouned a a risk-adjused rae, say Y, ha refleced Y heir rue defaul risk, hen Xe L. However, using his liabiliy measure o accoun for he pension plan s funding raio, A / L, is problemaic. I is also sraighforward o show ha such a value of liabiliies leads o he illogical resul ha he pension plan becomes fully funded as he value of asses shrink: lim A / L 1 (4) A 0 A simple numerical example illusraes he imporance of his disincion. Consider a plan ha has a single nominal liabiliy of X = $1,000 ha is payable in en years. If he curren coninuously-compounded yield on a 10-year defaul-free bond is y = 3.0 percen, hen invesing Xe -y =$1,000 e = $ in his bond oday, wihou any need for he sponsor o make fuure conribuions (c = 0), would guaranee ha he plan could make good on is liabiliy. Now suppose he plan was underfunded, having asses of only A = $500 on hand oday. Using he defaul-free rae, he plan would be X e -y - A = $ underfunded, or a funding

13 11 raio of A /( X e -y ) = 67.5 percen. Conrary o how GASB discouns, his level of underfunding is he same regardless of how he $500 is invesed. We argue ha his 67.5 percen is he appropriae measure of a plan s funding saus because i ells paricipans and oher sakeholders how much money he plan would need oday o pay full promised benefis in he fuure. This measure would also be helpful o deermine he fuure conribuion rae, c, ha would be expeced o close his funding gap over some fuure horizon. 6 Nex, consider he consequence of incorporaing defaul risk when discouning pension liabiliies. Suppose ha he rae of reurn on he plan s asses has an annual sandard deviaion of = 20 percen. If he sponsor is expeced o make addiional conribuions o he fund a he rae of c = 2 percen per year, he curren marke value of liabiliies calculaed from equaion (1) is L = $ This marke value is equivalen o discouning he plan s promised liabiliies by Y = 7.41 percen. Tha is, L = $ = X e -Y =1000e If one hen discouned he plan s benefi promises by a rae ha reflecs he rue likelihood of defaul and used he resul o measure funding, hen he plan would be overfunded by $500 $ = and is funding raio would be A /( X e -Y ) = 104.9%. While discouning benefi promises by a risk-adjused rae leads o a disored measure of funding, informaion on he L =$477 curren marke value of liabiliies would be quie valuable, for example, o an employee deciding beween he DB plan and, say, a $600 immediae conribuion o a DC plan. If he DB plan was fully funded and defaul-free, he individual migh 6 The conribuion rae a which he risk-neural expeced asse value equals liabiliies by he ime ha benefis are promised o be paid is given by he value of c such ha (A /( X e -y ))e c = 1, or c = ln(x e -y /A )/. For our example, his is c = 3.93 percen.

14 12 raionally choose he $741 DB over a $600 DC. 7 Bu if he DB plan is underfunded, he raional choice migh be he $600 DC raher han he $477 marke value of he DB. Thus, he marke value of he DB liabiliy conains useful informaion. Bu i is no a paricularly robus measure of funding levels, i.e., how much money he plan sponsor would need o se aside oday o pay promised benefis. For example, if one is ineresed in measuring funding saus, i makes lile sense o effecively reward he plan sponsor for underfunding heir plan. As indicaed in he model above, if a plan sponsor does no fund a all, and if his drives he probabiliy of he plan sponsor making good on is promises oward zero, hen he marke value of he liabiliy can disappear compleely. Tha is useful informaion o someone rying o value heir fuure benefi, bu i serves lile purpose as a measure of he funding shorfall. In pracice, Novy-Marx and Rauh (2011) consider wo candidae raes as being poenially appropriae for discouning sae pension liabiliies. One is a ax-adjused general obligaion municipal bond rae. They argue ha such a rae is appropriae if a sae is jus as likely o defaul on is pension obligaions as i is o defaul on is general obligaion bonds. Their second candidae discoun rae is based on he logic of Brown and Wilcox (2009) who documen ha a majoriy of saes pension obligaions are proeced by sae consiuional guaranees. I is argued ha hese saes have a lower probabiliy of defauling on heir pension obligaions han of defauling on heir municipal bonds. Therefore, hey argue ha he appropriae discoun rae for pension liabiliies should be less han a municipal bond rae and closer o a nearly defaul-free zero-coupon U.S. Treasury rae. 8 The main poin of our paper is 7 For purposes of his illusraion, we are ignoring oher differences beween DB and DC which migh influence his choice, such as access o annuiized income in reiremen. 8 Similar reasoning on accouning for pension liabiliies is found in Andonov, Bauer, and Cremers (2014): public pension funds should use lower discoun raes han privae pension funds, because public plan benefis are virually

15 13 ha his discussion is relevan o compuing he marke value of public pension liabiliies, bu ha only he defaul-free rae is relevan for measuring he degree of underfunding. We argue ha here is no need o jusify he choice of he liabiliy discoun rae based on he municipaliy s likelihood of defaul because finance heory suppors he discouning of promised pension obligaions by a defaul-free rae, wheher he pension fund sponsor is he U.S. governmen, he Sae of Illinois, or he Ciy of Deroi. 9 As we discuss in he following secions, he insighs obained from he simple model in equaion (1) carry over o more realisic models where benefis are promised a many fuure daes. For he purpose of measuring a plan s funding saus, each fuure promised benefi paymen should be discouned using he yield on he defaul-free bond ha maures a he benefi s paymen dae. Furhermore, when promised benefis are sae coningen, as hey are in he case of COLAs subjec o limis on fuure inflaion, hese promises should be valued as if hey were defaul-free. If here exiss defaul-free securiies ha can replicae he plan s saeconingen benefi promises, hen he cos of hese securiies should be used o value liabiliies when calculaing funding saus. In oher words, while valuaion for he purpose of measuring funding saus should accoun for he sae-coningen risk of he plan s conraced benefis, i should no accoun for benefis defaul risk which would consiue a violaion or renegoiaion of he benefi conrac. 4. How o Measure Defaul-Free Raes To accoun for a pension plan s funding level, we argued in he previous secion ha liabiliies should be valued as if fuure promised paymens were defaul free, unless explici free of risk as accrued benefis are usually backed by consiuional guaranees; in conras, members of privae plans sill risk losing par of heir pensions if he firm eners bankrupcy. 9 One excepion is he case where pension benefi promises conain explici provisions for wrie-downs if pension asse values are insufficien.

16 14 conracual provisions allow for changes in hese paymens. Our raionale is ha full funding should imply ha he plan has sufficien asse value o implemen an invesmen sraegy ha guaranees paymen of he plan s promised reiremen benefis. If he plan s promised benefis can be replicaed by a paricular porfolio of securiies, hen he minimum cos of his porfolio equals he plan s fully-funded asse value. In urn, he difference beween his porfolio s cos and he plan s curren asse value equals he plan s underfunding. Equivalenly, should he sponsor erminae he plan and ransfer is accrued liabiliies o an insurance company, he company s cos of implemening an invesmen sraegy ha guaranees paymen of he benefis is he value of he replicaing porfolio. As we discuss below, here exis several nominal and inflaion-relaed securiies ha can replicae many promised plan benefis. 10 In his secion, we consider purely nominal liabiliies and hen cover inflaion-relaed cos-of-living adjusmens (COLAs) in he following secion. The simple model in he previous secion assumed a single fuure promised paymen, X, and a single defaul-free ineres rae, y. The qualiaive insighs of ha model exend o paymens promised a muliple fuure daes. Suppose he curren dae is 0 and a plan promises paymens each year for T fuure years, = 1,, T, wih corresponding amouns X, = 1,, T. Then he curren dae 0 defaul-free value of he plan s liabiliies is (5) T 0, X y 1 e where y0, denoes he dae 0 coninuously-compounded yield o mauriy on a defaul-free zerocoupon bond ha maures a dae. Consequenly, if he plan had asse value sufficien o 10 If here do no exis securiies ha perfecly replicae some ypes of plan benefis, he asse value required for full funding can be modified o equal he cos of purchasing he securiy porfolio ha bes hedges promised fuure benefi paymens. Presumably, such a markeable securiy porfolio ha mos closely maches he plan s promised paymens will hedge he lion s share of he liabiliies sysemaic risks so ha unhedged risks should be mosly diversifiable. A large insurance company may be willing o guaranee he plan s liabiliies for he cos of his hedge porfolio because he plan s unhedged risks can be diversified away if he insurance company has oher business lines wih uncorrelaed risks.

17 15 purchase T defaul-free bonds wih mauriy values X, = 1,, T, i could guaranee paymen of he benefis. Thus, expression (5) represens he plan s fully-funded asse value. Equivalenly, a porfolio of defaul-free bonds whose value and duraion equals ha of (5) is sufficien o pay fuure benefis and immunizes he plan from ineres rae risk. Lewis and Pennacchi (1999) exend he model of he previous secion o find he marke value a DB plan s liabiliies or, equivalenly, he fair cos insuring is liabiliies from defaul. Their model permis benefi paymens and conribuions from a plan sponsor a muliple fuure daes and assumes defaul occurs if he plan s sponsor (e.g., a corporaion or a municipaliy) declares bankrupcy a he same ime ha he pension plan is underfunded. Consisen wih (5), he model assumes ha here exiss a duraion-maching bond porfolio ha replicaes he plan s promised benefis. The erm srucure of defaul-free yields changes randomly according o he Vasicek (1977) no-arbirage model. The insigh of he previous secion carries over o his exended model in ha he criical measure of he plan s funding saus is he raio of he marke value of he plan s asses o he value of he duraion-mached, defaul-free bond porfolio ha replicaes he plan s promised fuure benefis. 11 The issue ha we now address is how o deermine he appropriae se of defaul-free yields. A naural saring poin is yields on Treasury securiies. The erm srucure of zerocoupon Treasury yields can be obained from Treasury STRIPS or he zero-coupon yields implici in he prices of Treasury coupon-paying noes and bonds. 12 However, several objecions migh be made for using unadjused Treasury yields o discoun pension promises. We briefly 11 Similarly, a sae variable ha deermines he sponsor s likelihood of bankrupcy is he raio of he sponsor s oal non-pension plan asses o he defaul-free value of he sponsor s non-pension plan liabiliies. A similar model bu wihou random ineres raes is developed in Pennacchi and Lewis (1994). 12 STRIPS (Separae Trading of Regisered Ineres and Principal Securiies) are individual coupons or principal paymens of Treasury noes and bonds ha rade separaely and represen a rue zero-coupon Treasury securiy. Zero-coupon yields implici in Treasury noe and bond prices can be calculaed using a boosrapping mehod.

18 16 discuss hese concerns, and use he exising lieraure o place bounds on adjusmens ha migh be made when using a Treasury yield curve o proxy defaul-free raes. 4.1 U.S. Governmen Deb is no Defaul-Risk-Free One criicism of using Treasury securiies as proxies for defaul-free asses is ha hey, like any sovereign deb, are no ruly free from defaul risk. Evidence ha a leas some invesors believe U.S. Treasuries are defaul risky comes from he credi defaul swap (CDS) marke. 13 CDS are insurance conracs agains defaul. If an invesor purchases a defaul-risky bond and also buys CDS proecion agains he bond s defaul, his combined invesmen approximaes a defaul-free bond whose annual yield equals he defaul-risky bond s promised yield minus he annualized CDS spread. Figure 1 shows U.S. Treasury CDS spreads repored by Bloomberg over he period from December 2007 o December See he Appendix for deails, including Bloomberg codes, on he daa used in his paper. The average CDS spreads over his sample period were 22.9, 34.6, and 46.0 basis poins for conrac mauriies of one, five, and en years, respecively. CDS spreads increased sharply following he Lehman Brohers bankrupcy, peaking in February of There was also an especially large spike for spreads of he one-year conrac in July 2011 jus before he resoluion of a deb ceiling crisis on July 31, CDS spreads rose o a smaller exen prior o he culminaion of a second deb ceiling crisis ha was resolved on Ocober 17, In summary, based on evidence from CDS spreads, yields on Treasuries migh conain roughly a 20 o 50 basis poin defaul risk premium, depending on mauriy. 13 I is ineresing ha CDS on U.S. Treasuries are quoed in euros, raher han U.S. dollars, suggesing ha foreign invesors migh be mos concerned abou defaul risk. Since CDS spreads are expressed as annual premiums per amoun of noional principal, hey are largely unaffeced by he currency denominaion. 14 The U.S. Treasury was scheduled o exhaus is borrowing auhoriy on Augus 2, On Augus 5, 2011, Sandard & Poor s downgraded he U.S. governmen s long-erm credi raing from AAA o AA+. 15 A parial governmen shudown began on Ocober 1, 2013.

19 U.S. Governmen Deb Provides Sae-Tax Advanages Saes ha ax invesmen income ypically exemp ineres from federal governmen bonds from he ax base. According o sandard ax clienele heory, high-marginal-ax-rae invesors would be willing o accep a lower pre-ax yield on U.S. governmen deb han hey are willing o accep on an oherwise idenical, bu axable, asse. This would drive he yield on governmen bonds down and herefore require ha one adjus he yields by 1/(1- ), where is he relevan sae ax rae on ineres income. However, he ax clienele heory does no appear o be empirically relevan. Using daa from he Federal Reserve Bank of S. Louis as of April 1, 2014, ou of he $12,877 million of U.S. federal deb ousanding (a figure ha excludes governmen holdings of is own deb, such as in he Social Securiy rus funds), jus over $6 billion (47 percen of oal) was held by foreign invesors, including foreign governmens for which ax consideraions do no apply. Anoher 2.7 billion (21 percen) was held by Federal Reserve Banks, who also do no face a ax wedge beween governmen and corporae bonds. This leaves less han one-hird of all U.S. governmen deb held by privae U.S. invesors. These privae invesors, in urn, are dominaed by pension funds, life insurance companies, and oher insiuional invesors. As noed by Cochrane (2015), he marke for Treasury deb is heavily segmened, wih few axable invesors holding any deb. Non-axable invesors do no experience any ax advanage o holding Treasuries relaive o oher asses, and hus here is lile reason o hink ha U.S. Treasury yields are significanly affeced by he sae ax exempion. Thus, o a firs approximaion, we believe i is appropriae o make no adjusmen o Treasury yields o accoun for heir preferenial sae ax saus. 4.3 Liquidiy Premia for U.S. Governmen Deb

20 18 Anoher objecion o using unadjused Treasury yields as defaul-free raes relaes o Treasury securiies high liquidiy which ha makes hem especially aracive for invesors ha may have needs o rade. Treasuries can be quickly convered o cash or easily used as collaeral for borrowing via repurchase agreemens, feaures which raise heir prices and lower heir yields relaive o a less-liquid defaul-free securiy. However, if a pension fund ends o buy and hold bonds unil hey maure, i may be able o form a replicaing porfolio wih defaul-free securiies ha are less liquid han U.S. Treasuries. Doing so would allow i o purchase defaul free securiies a lower prices and higher yields. Longsaff (2004) provides evidence of such a possibiliy. He examines yields on bonds issued by a U.S. governmen agency, Refcorp. 16 Unlike oher governmen agencies such as Fannie Mae or Freddie Mac, Refcorp bonds are fully collaeralized by U.S. Treasury securiies and, hence, have he same credi risk as Treasuries. Over his April 1991 o March 2001 sample period, he average difference in Refcorp bond yields relaive o equivalen mauriy Treasury yields was 13.1 basis poins a he 10-year mauriy and 16.3 basis poins a he 30-year mauriy. However, hese yield spreads ended o be higher during periods of financial marke sress or flighs o liquidiy. Oher research finds ha, even among differen Treasury securiies, liquidiy premia affec yields. Recenly aucioned on-he-run Treasury noes and bonds end o have yields ha are 5 o 10 basis poins lower han similar mauriy bu seasoned off-he-run Treasury noes and bonds. 17 Thus, if a pension plan does no benefi from he greaer liquidiy of on-he-run Treasuries, i would be jusified o discoun nominal paymens using yields of off-herun securiies or equivalen credi risk agency securiies, such as Refcorp bonds. 16 Refcorp is he funding vehicle for he Resoluion Trus Company ha was esablished by he 1989 Financial Insiuions Reform, Recovery, and Enforcemen Ac (FIRREA). 17 Evidence of an off-he-run versus on-he-run yield spread includes Krishnamurhy (2002) and Graveline and McBrady (2011).

21 Ne Effec: Treasuries are a Reasonable Proxy Saring wih Treasury yields, he above evidence suggess ha i may be appropriae o subrac basis poins o remove he effec of a small defaul-risk premium and hen add basis poins o remove he average effec of he liquidiy premium. While Figure 1 seems o indicae ha Treasuries credi risk spread rises during periods of financial disress, research also finds ha heir relaive liquidiy is highes during such periods, so ime variaion in hese wo effecs migh parially offse each oher. We furher argued ha he marginal invesor in Treasury securiies is unlikely o be axable, given he large share of governmen bonds held by foreign governmens and ax-exemp insiuional invesors. The ne effec of hese adjusmens is close o a wash: if anyhing, he previous evidence migh sugges ha a rue defaul-free rae in he U.S. economy is up o 35 basis poins below he yields on governmen bonds. Addiional evidence on wheher Treasury yields are a reasonable proxy for defaul free raes migh be o compare he effecive yields on synheic defaul-free bonds creaed from CDSinsured, high credi qualiy corporae bonds. We carried ou a very rough, exploraory analysis by obaining yields on all exising AAA-raed, non-callable corporae bonds. 18 Currenly, he only wo corporaions wih ousanding AAA-raed, noncallable bonds are Johnson & Johnson and Microsof. To see how yields on heir corporae bonds compare o similar Treasuries, we mached each corporae bond ha currenly has a leas four years unil mauriy o a Treasury noe or bond wih he mos similar mauriy dae and coupon rae. Daa on yields for each corporae bond and maching Treasury bond were obained from Bloomberg. Summary saisics from his exercise are repored in Table We limied he analysis o only AAA bonds because here migh be more noise and idiosyncrasies in using lowergrade bonds. We also examine only non-callable bonds because callable bonds have random mauriies and heir yields are elevaed by a call premium. All ousanding U.S. Treasury securiies are non-callable.

22 20 The sample period over which yield daa was available for each corporae and maching Treasury bond is given in he firs column of Table 1. Columns 2, 3, and 4 give each corporae bond s mauriy dae, coupon rae, and average yield over he sample period, respecively. The same daa iems for each bond s maching Treasury are given in columns 5, 6, and 7. In he las column 8 is he difference in he average yields of he corporae bond and is maching Treasury, which we refer o as he average spread. Table 1 liss he bonds by corporaion and mauriy dae. As is indicaed in column 8, he corporae Treasury spread ends o rise wih he bonds ime unil mauriy. 19 Averaged over all corporae bonds, he average spread is abou 60 basis poins for each corporae issuer. However, if we limi he comparison o only bonds wih a curren ime unil mauriy of 10 years or less, hen he average spread is 29.8 basis poins for he (six) Johnson & Johnson bonds and 48.7 basis poins for he (four) Microsof bonds. The reason ha we focus on bonds of 10 years or less is ha daa on CDS spreads for hese corporae bonds were available only for CDS conrac mauriies of 5 and 10 years. 20 The las four rows in Table 1 compare hese corporaions spreads for 5-year and 10-year CDS conracs o hose of he U.S. Treasury. Johnson & Johnson and Treasury CDS spreads are over he sample period beginning in December of However, since Microsof issued is firs corporae bond in 2009, is CDS is compared o ha of Treasuries for a shorer sample period. As can be seen from he able, he differences are small, beween 0 and 10 basis poins. By subracing his difference in relaively higher corporae CDS spreads from heir relaively higher yields, we can roughly esimae ha he ne yield from creaing a synheic 19 Spreads ha rise wih he ime unil mauriy would be expeced if he long-run defaul risk of he federal governmen is less han ha of hese currenly AAA-raed corporaions. 20 While we only have daa on CDS spread for mauriies up o 10 years, like he corporae bond Treasury yield spread we migh expec ha he difference beween corporae and Treasury CDS spreads would also grow wih mauriies beyond 10 years.

23 21 defaul-free 10-year Johnson & Johnson bond is abou 30 5 = 25 basis poins greaer han ha for a defaul-free synheic Treasury. Similarly, he ne yield from creaing a synheic, defaulfree 10-year Microsof bond is abou = 39 basis poins greaer han ha for a defaul-free Treasury. Therefore, while our previous evidence suggesed ha using CDS o make Treasuries defaul-free would subrac up o 35 basis poins from raw Treasury yields, his evidence from less-liquid corporae bonds suggess ha he raw Treasury yield is approximaely he correc reurn from creaing a synheic defaul-free bond from AAA corporaes. 21 Thus, a solid argumen could be made ha i is appropriae o use U.S. Treasuries as he base case for measuring he degree of pension underfunding. 5. Accouning for Cos-of-Living Adjusmens The discussion above indicaes ha using he U.S. Treasury yield curve o discoun nominal pension liabiliies, and comparing his o he marke value of plan asses, is an appropriae way o measure he degree of underfunding of a plan. However, he exercise assumes ha pension liabiliies are fixed nominal liabiliies. In fac, many public pension plans have cosof-living adjusmens (COLAs), an issue ha can complicae he discouning process. Figure 2 shows he number and percenage of he 127 plans in NASRA s Public Fund Survey wih various ypes of COLAS. 36 plans use a pos-reiremen benefi adjusmen ha is no a rue COLA a all, bu raher is a pre-deermined rae of auomaic increase in nominal benefis. Illinois is one such example: curren reirees receive an annual 3 percen auomaic 21 Anoher way of seeing his is ha, for bonds of 10 years or less, he average yield on Johnson & Johnson and Microsof bonds over heir mached Treasuries are 30 and 49 basis poins, respecively. Since Table 1 shows he 10- year CDS spreads for Johnson & Johnson and Microsof are 51 and 60 basis poins, respecively, hese CDS insured bonds would have a ne yield of = -21 basis poins and = -11 basis poins below raw Treasury yields. So a raw Treasury yield migh be only slighly higher han he ne yield on a synheic defaul-free corporae bond.

24 22 annual increase regardless of he rae of inflaion. In such cases, one can apply he nominal Treasury yield curve direcly o he growing nominal sream of benefis. 45 plans use a COLA ha is based on he CPI. If hese plans were fully indexed, i.e., if benefis were simply real raher han nominal, hen one could use a real yield curve, raher han a nominal yield curve, o discoun benefis. However, very few of he plans are purely inflaionindexed. 38 of he 45 plans have some sor of ceiling on he inflaion adjusmen, ranging from a low of 1.5% in he Kenucky ERS o a high of 6% in he Connecicu SERS plan. The mos common ceiling is 3%, wih 15 plans capped a his level. Five plans repor floors, alhough we suspec ha more plans han his have an implici floor of zero. I would be misleading o use a real erm srucure o discoun a sream of benefis ha is subjec o floors and ceilings on he inflaion adjusmen. Unlike prior research ha values COLAs by making ad hoc assumpions regarding fuure inflaion, we show paricular ypes of limied pension COLAs can be replicaed and valued using he marke prices of recenly available inflaion derivaives, such as inflaion swaps, inflaion caps, and inflaion floors. Few, if any, assumpions regarding he process for inflaion need o be made when aking his approach. In addiion, because marke daa is available a high frequency, immediae updaes on he cos of COLAs can be recognized. This approach is useful for a leas wo purposes. Firs, i allows us o conver he COLA provisions ino a fixed nominal price ha can hen be discouned back using he nominal defaulfree erm srucure. Second, i provides a ool for evaluaing he cos savings associaed wih reforms ha reduce COLAs. This laer poin is highly relevan o he curren policy environmen: Munnell, Aubry, and Cafarelli (2014) repor ha beween 2010 and 2013, 30 plans across 17 saes eiher reduced, suspended or eliminaed COLAs for exising employees. Of

25 23 hese, 13 also reduced COLAs for curren reires. Alhough hese changes were legally challenged in a leas 12 of he saes, he cus were upheld in mos of he saes. 5.a Fixed Rae COLAs Valuing auomaic fixed-rae COLAs, such as a compounded or simple 3% annual increase is sraighforward and can be done using he erm srucure of defaul-free nominal bond yields. To illusrae his calculaion, we assume for simpliciy ha each fuure year s annuiy paymen is made in a lump sum a he end of each year, raher han paid monhly hroughou he year. If he COLA is compounded a he rae r, hen for each $1 of base year s annuiy, he cashflow paid years in he fuure is (1+r) 1. If y 0, is he iniial, dae 0 annually-compounded yield o mauriy on a defaul-free zero-coupon bond mauring a dae, hen he presen value of his fuure year s promised COLA paymen is simply [(1+r) 1]/(1+ y 0, ). Over a T-year reiremen horizon, he presen value of all fuure years COLA paymens are T 1 r 1 (6) 1 1 y Expression (6) is he value of he COLA per $1 of base annuiy value. Expressing i as a proporion of he presen value of he oal nominal annuiy received over T years, i.e., he presen value of a $1 per year benefi wihou a COLA, leads o: 0, T 1 r 1 T 1 / 1 1 y 1 0, 1 y0, (7) Expression (7) is hen he defaul-free value of a fixed rae, compounded COLA over a T-year period as a proporion of he value of he nominal annuiy over ha same period. The corresponding expression for a non-compounded (simple) COLA a he fixed rae r is

26 24 T T r 1 / 1 1 y 1 0, 1 y0, (8) We calculae (7) and (8) using Treasury securiy daa compiled by Gürkaynak, Sack, and Wrigh (2007). They provide daily esimaes of zero-coupon Treasury yields from daily prices of Treasury coupon noes and bonds. Because heir esimaes use daa on off-he-run Treasury securiies, liquidiy premia ha are highes for on-he-run Treasury securiies are miigaed. 22 Hence, hese yields reflec wha a pension fund would earn on more illiquid, seasoned Treasury securiies. The single mos common fixed rae ha pension funds choose for auomaic COLAs is 3 percen, boh when his rae is compounded and when i is no. Therefore, we compare he value of 3 percen compounded and simple (non-compounded) COLAs for reiremen horizons of 10, 20, and 30 years. In each case he COLA values are expressed as a percenage of he value of he same-horizon reiremen annuiy ha does no conain he COLA. We make hese calculaions for each day saring from he beginning of Ocober 2004 unil February These days values are averaged over each monh and graphed in Figure 3. As would be expeced, Figure 3 indicaes ha he difference beween compounded and simple COLAs grows as he reiremen horizon lenghens. The average compounded versus simple COLAs per nominal benefi a he 10, 20, and 30 year horizons are 17.0% versus 15.6%, 32.5 versus 27.1%, and 48.2 versus 36.8%, respecively. Relaive COLA values are also higher during periods when he yield curve was generally lowes, such as during he deph of he financial crisis in lae 2008 and early 2009, as well as he mos recen period of Federal Reserve quaniaive easing. 22 Their daily esimaes of he zero-coupon bond yield curve derive from prices of Treasury coupon securiies ha exclude he wo mos recenly issued securiies wih mauriies of wo, hree, four, five, seven, 10, 20, and 30 years.

27 25 5.b Fully Inflaion-Indexed COLAs Inflaion-indexed COLAs provide fuure cashflows ha are random in erms of nominal paymens bu fixed in erms of real purchasing power. Previous research such as Novy-Marx and Rauh (2011) esimaes COLAs ied o he Consumer Price Index (CPI) by making an assumpion regarding fuure inflaion and discouning he resuling expeced nominal cashflows by nominal yields. Alernaively, one migh discoun a CPI-indexed annuiy paymen using he erm srucure of real yields, raher han nominal yields. Daily zero-coupon real yield curves esimaed from U.S. Treasury Inflaion Proeced Securiy (TIPS) coupon prices are available for mauriies from 2 o 20 years and could be used. 23 However, TIPS are less liquid han nominal U.S. Treasuries, and here is evidence ha heir yields became unrealisically high (and prices unreasonably low) during sress periods such as he financial crisis. 24 An alernaive o using TIPS o value CPI-indexed COLAs is o use an inflaion derivaive securiy known as a zero-coupon inflaion swap. Zero-coupon inflaion swaps are he mos common and liquid ype of inflaion derivaive. Also, raders in inflaion swap conracs are subjec o regulaory collaeral requiremens ha are inended o minimize counerpary defaul risk. 25 A zero-coupon inflaion swap is really a forward conrac in which here is a single fuure exchange beween wo paries where one pary pays a fixed paymen and he oher pays an inflaion-linked paymen. For a swap negoiaed a dae 0 and mauring a he end of year, he ne paymen from he poin of view of he fixed-rae payer is (1+k 0, ) - CPI /CPI 0, where k 0, is he inflaion swap rae negoiaed a dae 0 on his -year inflaion swap and CPI is he CPI a 23 This daa is available a hp:// 24 For evidence ha TIPS were underpriced relaive o nominal Treasuries and inflaion swaps, see Haubrich, Pennacchi, and Richken (2012) and Fleckensein, Longsaff, and Lusig (2014). 25 Under curren inernaional regulaions, mos sandard swaps, such as ineres rae swaps, are required o be processed hrough cenral clearinghouses ha se margin requiremens. However, inflaion swaps, as well as crosscurrency swaps and swapions, are exemp from cenral clearing requiremens. These uncleared, over-he-couner derivaives are subjec o differen collaeral requiremens ha are designed, in principle, o be greaer han hose for cenrally-cleared swaps. See Basel Commiee on Banking Supervision (2015).