The Equity Premium: Consistent with GDP Growth and Portfolio Insurance

Transcription

1 The Equiy Premium: Consisen wih GDP Growh and Porfolio Insurance Chrisophe Faugère Universiy a Albany Julian Van Erlach Nexxus Wealh Technologies Inc. Forhcoming, The Financial Review. Minor changes and correcions may occur before publicaion. Please monior for he publicaion schedule. Please consul he published version for exac wording and paginaion before finalizing any verbaim quoes. We find ha he long-erm equiy premium is consisen wih boh GDP growh and porfolio insurance. We use a supply-side growh model and demonsrae ha he arihmeic average sock marke reurn and he reurns on corporae asses and deb depend on GDP per capia growh. The implied equiy premium maches he U.S. hisorical average over Separaely, we find ha he equiy premium racks he value of a pu opion on he S&P 500. Our heory predics a smaller equiy premium in he fuure, assuming he recen regime shifs in dividend policies, ineres raes, and ax raes are permanen. Keywords: Equiy Premium, GDP Growh, T-Bills, Downside Risk, and Porfolio Insurance. JEL Classificaion: G10, G12 Corresponding auhor: Chrisophe Faugère, Deparmen of Finance, School of Business, Universiy a Albany SUNY, Albany, NY Phone: c.faugere@albany.edu. We hank Ken Beauchemin, Hany Shawky, and David Smih for helpful commens. We also hank an anonymous referee for very consrucive and deailed commens ha led o a subsanial improvemen of his aricle. All remaining errors are our own.

2 1. Inroducion The equiy premium, ypically defined as he difference beween he S&P500 reurn and a risk-free reurn, is a criical inpu in porfolio allocaion decisions as well as in capial budgeing decisions. I is also a he hear of he ongoing policy debae abou wheher a porion of he social securiy rus fund should be invesed in he equiy marke or no. Consequenly, i is crucial for finance professionals o be able o accuraely gauge he size of he premium and o undersand he facors ha may change is value. In heir seminal paper, Presco and Mehra (1985) show ha he sandard economic growh model is unable o explain he hisorical difference beween he average sock reurn on a broad index versus risk-free bonds in he Unied Saes. This has become known as he equiy premium puzzle. Alhough he fac ha equiy reurns are higher han shorerm bond yields makes inuiive sense, he puzzle is ha sandard risk measures canno explain he size of he hisorical difference in reurns. Many aemps a modifying he sandard model fall shor of fully explaining he size of he premium (Campbell, 2003; Kercholakoa, 1996; Mehra, 2003). Alhough a few successful models have been proposed, he curren asse pricing lieraure does no agree on a soluion o he equiy premium puzzle as well as oher asse pricing anomalies. 1 Empirical approaches o he sudy of he equiy premium, alhough no ariculaing a full explanaion of he puzzle, do offer alernaive pahways in he ques for a soluion o 1 In wo recen aricles, McGraan and Presco (2000, 2001) claim o finally pu he puzzle o res. In heir 2000 aricle, hey show ha he high raio of marke value of equiy o gross naional produc (GNP) a he beginning of he year 2000 is raionally based when compuing he value of corporae angible and inangible capial asses. In he subsequen 2001 aricle, hey show ha during he poswar period, he large rise in equiy values is as prediced by he heory, once he hisorical change in he axaion of dividends and he increase in holdings in ax-deferred accouns are accouned for. However, McGraan and Presco (2001) focuses on explaining recen equiy reurns and does no explain he hisorical difference beween sock and risk-free reurns. 1

3 his imporan puzzle. Asness (2000) draws a link beween he size of he expeced premium and differences in he volailiy of socks versus bonds for horizon periods of 20 years. However, his analysis does no accoun for he size of he average premium over he full hisorical record. 2 Ibboson and Chen (2003) use a building-block approach o esimae he long-erm sock marke reurn, and relae some of he marke reurns componens o gross domesic produc (GDP) growh. However, hey do no esablish a firm heoreical basis for relaing he equiy premium direcly eiher o GDP growh or o risk. In his aricle, we show heoreically and empirically ha in he long-run, he equiy premium is direcly relaed o GDP growh and is consisen wih downside risk avoidance. Thus, we esablish ha he hisorical average value of he equiy premium is fully accouned for by risk and growh-based explanaions. These resuls in urn enable us o accuraely gauge he size of he hisorical premium and o predic changes in he premium given exrapolaed recen rends. 3 Firs, we use a supply-side growh model and develop he macroeconomic equivalen of he sandard susainable growh formula found in corporae finance exbooks o deermine he long-run average reurn on socks. The average sock reurn depends on GDP per capia growh and he earnings reenion raio as well as a premium reflecing sysemaic risk, which is linked o de-rended earnings 4 and marke-o-book volailiies and he response of dividend and share repurchase policies o hese facors. 2 Asness (2000) uses he dividend and earnings yield as proxies for expeced reurns. Alhough he approximaions may well be valid, here is evidence ha bonds may be more risky han socks for horizons of 20 years or longer when looking a ex-pos reurn volailiy, as shown by Siegel (2002). 3 I is imporan o emphasize ha we do no claim o resolve he equiy premium puzzle here, because he puzzle is really abou he failure of he sandard asse pricing model. However, i is also a fac ha no oher class of models has been so far pu forh o show he consisency of macroeconomic growh raes, risk, and oher behavioral variables wih he size of he premium, which is wha we aemp o do here. 4 Earnings are scaled (or de-rended) by book value of equiy. In oher words, we examine ROE volailiy. 2

4 We hen esablish a relaionship beween GDP growh and he required reurns on corporae asses and deb. Based on an empirical analysis a he aggregae corporae level, we obain a value for he reurn on deb idenical o he hisorical average of he 3-monh T-bill over he period Alhough his resul is surprising, i is o some exen, an arifac of Flow of Funds daa ha exhibis a survivor bias, in he sense ha i excludes defauled or bankrup companies over ime. This resul implies ha in he long-run, survivor-biased corporae deb will be risk-free as he economy grows a a consan rae, where he risk-free rae is deermined by he 3-monh T-Bill, absen any erm srucure effecs. Our firs key conclusion is ha, in he long-run, he size of he premium (as expressed by he difference beween he average sock reurn and he 3-monh T-bill) is a funcion of GDP growh and oher financial parameers such as marginal income ax raes. We empirically mach he hisorical value of he S&P500 arihmeic average over , and he hisorical equiy premium value of 8.1%. Our second key resul is ha he equiy premium is also consisen wih a compensaion for risk, when viewed in he conex of a shor-erm porfolio insurance moive. We use an opion-based approach and show ha he equiy premium is closely approximaed by a pu opion premium on a $1 real invesmen in he marke index when a long-erm invesor wishes o insure agains year-o-year marke volailiy, by using he average yearly S&P500 volailiy over Recen research argues ha he acual equiy premium is much smaller han indicaed by hisorical long-erm averages (Siegel, 1999; Jagannahan, McGraan and Scherbina, 2000; Fama and French, 2002; and De Sanis, 2004). In conras, our resul is ha he ob- 3

5 served level of he long-run equiy premium a 8.1% is fully consisen wih he observed seady-sae GDP growh and consisen wih risk explanaions as well, and hus i may consiue a legiimae inpu in capial budgeing and invesmen analyses. Noneheless, if one believes ha he 1990 s changes in dividend yields, income axes, and ineres raes represen permanen regime shifs, our porfolio insurance model does lead o a lower premium value consisen wih he lieraure, when using parameer values from ha period. 2. Long-erm sock marke reurn and GDP per-capia growh Our firs approach is o ackle he equiy premium puzzle by using a supply-side growh model. Because he equiy premium is ypically measured by appealing o arihmeic hisorical average reurns, which are sample esimaes for uncondiional expeced reurns, i is reasonable o focus on uncondiional expecaions (Fama and French, 2002; Ibboson and Chen, 2003). We appeal o he naural properies of he economy s long-run seadysae growh pah o derive our key relaionship beween he equiy premium and GDP growh. In he long run, Kaldor s (1961) sylized facs show ha he U.S. economy has been characerized by a consan nominal (and real) GDP growh rae ha equals he growh rae of capial, wih a sable facor income disribuion (labor vs. capial). In a sochasic framework, his amouns o saying ha hese year-o-year growh raes are saionary sochasic processes. We esablish a link beween macroeconomic and finance variables by posiing ha in he long run, he uncondiional expeced growh of he economy s corporae capial sock 4

6 mus equals he uncondiional expeced growh in book value of a broad sock index. 5 The change in he index s book value of equiy is driven by clean surplus accouning. We also posulae ha he financial environmen saisfies Miller and Modigliani s (1958) proposiion and ha he opimal capial srucure is achieved for he whole economy. Le us develop our noaions. Le of a broad equiy index, K denoe he capial sock, B denoe he book value V is he marke value of he index, and e represens oal earn- e+ 1 ings for he index; all a he beginning of period. Le he variable R + 1= denoe he B ex-pos ROE a he end of period. A porion b + 1 of earnings is paid ou as dividends. Le he ex-pos rae of ne sock issues a he end of period be denoed by g s, + 1. Ex-pos, he growh in oal book value of equiy is given by he following surplus equaion: B g V 1 (1 b ) R B (1 g ) B + 1 s,+ 1 = s,+ 1 (1) Equaion (1) saes ha he growh of book value for he equiy index (S&P500) comes from wo sources: he firs erm on he righ-hand side represens inernal reinvesmen. The second erm is he incremen from issuing ne new equiy a a price equal o he curren marke price adjused for he increase in supply of shares (price divided by he facor (1+g s,+ 1) ). 6 As menioned, our key assumpion is ha he expeced growh of equiy 5 I is worh noing ha our approach differs from he consumpion-based capial asse pricing model (CCAPM). In our model, agens opimizing behavior is in he background, and key behavioral parameers are considered exogenous. In addiion, he ypical soluion of a CCAPM is a pricing kernel ha involves agens condiional expecaions. 6 Using he curren price adjused ex-pos is more accurae han using nex period s price because our adjusmen precisely correcs for he price pressure due o share growh, wihou adding he noise of new marke informaion incorporaed in nex period s price. This paricular adjusmen works when sock demand elasiciies are close o one. Shleifer (1986) provides evidence of uniary sock demand elasiciy for he S&P500. 5

7 book value of a broad equiy index (S&P500) equals he expeced growh in oal capial asses a he economy level: K 1 B E = E K B (2) where E () denoes he uncondiional expecaion operaor. We assume uncondiional expecaions saisfy he following: 7 V i) The expeced marke o book raio E = 1. B ii) The expeced ROE equals he long-run average reurn on socks μ. iii) The expeced long-run growh rae of populaion n and ne new shares g s are boh consan. iv) The expeced growh rae of capial equals he long run average growh rae of GDP denoed by g. v) The expeced long-run payou raio is he long run average raio b. Using equaion (2) and condiion iv) we rivially ge: B = 1 + g B + 1 E (3) Taking uncondiional expecaions of boh sides of equaion (1), hen subsiuing equaion (1) ino (3), by using he res of he condiions above, hen equaion (3) becomes: 7 These condiions are minimal for he sock marke o be in a saionary equilibrium. Condiion i) comes from he residual accouning lieraure (e.g.; Ohlson, 1995) since companies ha grow a he same rae as he economy have zero residual earnings growh and hus marke value mus converge o book value on average. Condiion ii) in fac follows from condiion i). Philips (1999) documens ha condiion ii) seems o hold since he average S&P500 ROE has been very close o he average sock reurn over he las hiry years. Condiions iii), iv) and v) sae ha hese variables follow saionary sochasic processes. 6

8 g s, + 1 gs, + 1 V 1+ g = 1 + (1 b) μ + E COV( b+ 1, R+ 1) + COV(, ) (1 g s, + 1 ) + (1+ g s, + 1) B (3 ) Srenghening assumpion iii) we posi ha in he saionary equilibrium ne new share growh g s equaes populaion growh n. 8 Le g=g y n denoe he GDP per capia growh rae. Rearranging (3 ), and using he approximaion g s, + 1 E n, we ge: (1+ g s, + 1) μ = s, + 1 g y + COV ( b+ 1, R+ 1) COV (, ) (1+ g s, + 1) B 1 b g V (4) Equaion (4) is he cenral resul of his secion. I shows ha he long-run nominal sock reurn is a direc funcion of he GDP per capia growh rae. This reurn also depends on he reenion raio ( 1 b) and he difference beween wo covariances: 1) he covariance beween dividend payou and he index ROE and 2) he covariance beween marke o book raio and he normalized growh rae of shares nex period. 9 Thus, he long-run nominal sock reurn equals he long-run expeced growh rae of GDP per cap- 8 In he long run, aggregae sock wealh canno grow faser han GDP, o rule ou permanen bubbles. The same mus be rue on a per capia basis, given ha he disribuion of wealh is sable in he seady sae. On one hand, he supply of ne new shares has o mach a leas he growh of new invesors. On he oher hand, a growh of shares in excess of populaion growh would depress earnings-per-share and hus dividends and capial gains. This would depress sock prices permanenly. Corporaions would hen have an incenive o repurchase shares. Using he Federal Flow of Funds, over he period , he growh in oal sock marke value was 8.40%, whereas i was 7.20% for he S&P500 over he same period. Since he S&P500 was a relaively consan fracion of he overall marke value (abou 60%), and he index is on a per-share basis, i is eviden ha he difference of 1.2% represens ne share growh, abou equal o longerm populaion growh. Loderer, Cooney, and Van Dunen (1991) show ha sock prices are reduced on average by 1% around announcemens of secondary equiy offerings over a period covering This effec is consisen again wih he fac ha in he long-run ne new share growh happens a he rae of average populaion growh. 9 These wo expressions play a similar role as ha played by he covariance beween consumpion growh and equiy reurns in he sandard consumpion-based capial asse pricing, which reflecs boh he riskiness of growh as well as he ineremporal opimal consumpion choices in ligh of expeced reurns (e.g.; Hansen and Singleon, 1983). I is also worh noing ha in hese models he covariance expression reas consumpion growh essenially as an exogenous variable. 7

9 ia plus a risk premium erm (he difference of covariances), divided by he percenage of new earnings reained. Because he reenion raio ( 1 b) and seady-sae growh rae g y are deermined in he background by opimal consumpion-invesmen decisions, equaion (4) simply esablishes ha he long-run equiy reurn is a funcion of hese choices. I is imporan o noe ha expression (4) is he macroeconomic equivalen o he long-erm susainable growh formula found in sandard corporae finance exbooks (for example, Brealey, Myers and Marcus, 1999). 10 The difference wih he sandard formula is ha ours applies o he corporae secor as a whole. Furhermore, he susainable growh rae is deermined by he long-run GDP per capia growh rae and wha essenially consiues an added sysemaic risk premium. When he firs covariance COV ( b + 1, R + 1) is large, his means ha companies pay ou a greaer fracion of earnings when heir ROE is high (procyclical), which exacerbaes he volailiy of cash flows and hus price volailiy. On he oher hand, when he second covariance COV g s, + 1 (, ) (1+ g ) s, + 1 V B is large, greaer sock issuance is associaed wih periods of high marke-o-book raios (procyclical). In ha case, greaer sock issuance will bring he marke-o-book raio back down, and vice versa in periods of low valuaion. Thus, price volailiy is dampened. Our model essenially predics ha sock markes wih counercyclical aggregae dividend payous and procyclical new shares growh overall will experience a reducion in 10 Our resul agrees wih Arno and Bernsein (2002), who find ha he sock marke reurn is direcly relaed o per capia GDP growh. 8

10 he impac of marke risk and herefore a lower long-erm sock marke reurn. For example, during periods of large downside earnings volailiy and low valuaion, firms can reduce heir invesors exposure o sysemaic risk in wo ways. One mehod is o smooh ou he dividend sream by emporarily paying higher dividends. Anoher possibiliy is o boos earnings per share by issuing fewer shares or even by buying back exising shares. Any such oucome is he resul of opimizing behavior, which depends on he aggregaed preference parameers of he paricular economy. 11 The arihmeic average yearly populaion growh rae is n = 1.19% 12 and is assumed equal o he growh rae of shares g S. Using yearly daa, he arihmeic average nominal growh rae of GDP per capia over he period is g y g n = 6.65% 1.19% = 5.46%, and he average S&P500 dividend payou is 55.5%. Our esimae for he covariance beween dividend payou and ROE is 0.51%. Srikingly, his value is idenical o he value of he sample covariance beween he marke o book raio and he subsequen period (normalized) shares growh rae, over This means ha over he pe- 11 The resul in expression (4) implies ha, even hough dividend payou policy is consan, he sock marke reurn will be differen for wo economies wih widely differen business cycles and growh risks. The second covariance erm will raise he sock marke reurn for he higher risk economy, when ha covariance is negaive. Tha is, when he growh of shares is principally driven by seady financing needs of he corporaions making up he index. In ha insance, a low price-o-book environmen necessiaes higher growh of shares o mee a given financing need, alhough he need may be smaller during a conracion han during an expansion phase. 12 This rae corresponds o he growh rae of he oal U.S. populaion (source: Bureau of he Census) over he period. 13 We consruc a measure of book value for he S&P500 by using daa from Rober Shiller s Web sie We assume ha he book value of he index in 1871 was equal o is marke value. Year-o-year, we add reained earnings o he previous year s book value. Our compuaion of he ROE is hen done from year-end 1926 unil The marke-book raio for he S&P500 is assumed similar o ha of he aggregae corporae secor. The daa is from he Federal Flow of Funds over he period The missing marke-o-book values for are reconsruced by back-rending corporae marke and book values, using regressions of he log of hese variables on linear ime rends over Again, because he S&P500 was a relaively consan fracion of he overall marke value (abou 60%), an index for he number of sock shares in he S&P500 is consruced by dividing he marke value of corporae equiy by he value of S&P500 index. 9

11 riod, boh dividend payous and ne new share issuance have been counercyclical in he Unied Saes, hereby creaing coupling effecs ha offse he risk premium. Hence, we ge: 5.46% μ = = 12.27% (1 55.5%) This final value is nearly idenical o he arihmeic average nominal sock reurn value of 12.2% esimaed for example by Siegel (2002) for he period Hence, we have derived an exac analyical relaionship linking he average real sock reurn and long-erm GDP per capia growh. Examining equaion (5), we observe ha he smaller he reenion raio is, he greaer he sock reurn is for a given GDP per capia growh rae The reurn on corporae asses and reurn on deb Whereas he firs piece of he equiy premium puzzle was he reurn on he sock marke, he second piece is he reurn on deb. In his secion, we focus on deriving he average reurn on corporae deb for he economy. As an inermediae sep, we examine he economy s required reurn on corporae asses (RRCA) from he invesors sandpoin. The required reurn on corporae asses is he discoun rae ha makes he presen value of 14 If all earnings are reinvesed (b = 0), so ha no dividends or share repurchases occur and growh is financed inernally, he bes reurn ha invesors could expec o earn is GDP per capia growh. This suggess ha leverage may enhance sock reurns by allowing firms o mainain asse and earnings growh and sill boos sock reurns hrough dividends and repurchases. 10

12 expeced fuure afer-ax cash flows accruing o crediors and equiy holders equal o he marke value of he corporae asse base in he economy. 15 Again, we predic ha in he long run, he expeced growh of he capial sock a he economy level equals he growh in book value for he S&P500. Furhermore, he corporae deb-equiy raio also mus be consan in he seady sae given an environmen a la Miller and Modigliani (1958). Fuure cash flows are given by afer personal income ax T, expeced corporae dividends D, and ne ineres paymens IP on ousanding corporae deb. 16 Assuming marke efficiency, he marke value of all corporae asses mus equal he presen value of all expeced fuure cash flows ne of axes ha accrue o crediors and shareholders a he economy level. Formally, his is expressed as: MV Corporae Asses 0 = ( ) (1 T ) D + IP E (5) j= = 1 Π j= 1 + j ( 1 RRCA ) where E () again sands for he uncondiional expecaion operaor. We assume ha in every period j, he corresponding discoun rae RRCA j is consan and equal o he long-erm average RRCA given by: RRCA = μ (1 L) + rd L (6) The long-erm nominal sock reurn is again denoed by μ. The variable r D sands for he nominal reurn on corporae deb/bonds, and L sands for he fracion of he invesor s porfolio invesed in deb or alernaively corporae leverage. We hypohesize ha in he 15 Fama and French (1999) compue an IRR based on operaing and invesmen cash flows. Our approach is grounded in he seady-sae analysis of he economy. In ha conex, he fracion of dividend plus ne ineres paymens over he marke value of corporae asses should be consan, whereas is componens may no be consan. In fac, his fracion of oal paymens has been relaively consan over he period examined, whereas he relaive size of ineres paymens compared wih dividends has increased. 16 Cash flows are in nominal erms. This curren model does no incorporae capial gains axes because dividend cash flows and ineres are assumed paid forever. The ax rae, T, is an average marginal ax rae ha blends dividend income and ineres income ax raes. 11

13 long run, he sum of expeced dividends plus ineres paymens o invesors is a consan fracion λ of GDP, as hey canno ogeher grow faser han GDP. Again, le g sand for he nominal long-erm average GDP growh rae. Given an average marginal ax rae of T, he above-menioned expression (5) becomes: MV Corporae Asses 0 = λ(1 T) GDP (1 + g) 0 = = 1 ( 1+ RRCA) λ(1 T ) GDP ( ) 0 RRCA g (7) Equaion (7) is he seady-sae version of equaion (5). The resul in equaion (7) indeed is he sandard growing perpeuiy formula applied o he enire asse base of he corporae secor. Thus, he RRCA is also given by: GDP0 RRCA = g+ λ(1 T) (8) MV Corp Asses Finally, combining expressions (6) and (8), we ge an expression for he deb reurn as: GDP 0 + λ(1 ) μ (1 L) MV Corp Asses 0 D = (9) r g T We use Flow of Funds daa and Naional Income and Produc Accouns daa from he Bureau of Economic Analysis, for he nonfinancial corporae secor over he period The variables used are ne ineres paymens, dividend paymens, oal deb, marke value of equiy and book value of equiy. Average marginal ax raes on dividend and ineres income are obained from Esrella and Fuhrer for he period and from he Naional Bureau of Economic Research TAXSIM model for he period 1980 L 0 12

14 Because our marginal income ax daa is limied o , we exrapolae he 1999 axes raes for he years 2000 and To deermine leverage, we use book value of equiy. This approach is suppored by he evidence of sabiliy of book leverage over he period documened by Fama and French (1999). The average value for he book leverage raio over our period is 38.05%; he average value for he raio of he ne oal paymens o invesors over oal GDP0 marke value of asses (before axes) λ MV Corp Asses 0 equals 3.67%. The average blended marginal (dividend plus ineres) income ax rae is 35%. Table 1 below summarizes he main parameers for he model. Table 1: Esimaed parameer values for reurn on corporae asses and deb formulas g g y b T L λ GDP0 MV Corp Asses 0 μ 6.65% 5.46% 55.5% 35% 38.05% 3.27% 112% 12.27% From hese parameers and equaion (9), we obain a value for he real RRCA of 5.89%, based on a long-erm GDP nominal growh rae of 6.65% and inflaion rae of 3.14%. This resul is nearly idenical o Fama and French s (1999) esimae of 5.95% over he period Finally, using formula (10) our esimae for he nominal re- 17 Even hough here is a difference in mehods beween he wo approaches, we do no believe ha hese differences affec our conclusions in any significan way. 13

15 urn on deb is 3.74%, which is close o he real T-bill arihmeic hisorical average of 3.93% for This resul is surprising. I suggess ha he overall corporae deb may be considered risk-free in he long run! However, we mus recognize ha he Flow of Funds corporae deb daa has a survivor bias. The corporae deb daa do no reroacively correc for defauled deb, and hus he Flow of Funds repors ineres paymens ha ypically begin o shrink earlier han he repored principal does, which biases compued raes downward Growh and he long-run equiy premium Finally, we derive he difference beween he reurn on corporae equiy versus corporae bonds. Le us define Δ COV = s, + 1 ( + 1, + 1) (, ) (1+ g s, + 1) B COV b R COV g V. Based on expressions (4) and (9), he difference in reurns is given by he following formula: 1 gb n+δcov GDP0 μ r D = λ(1 T ) L ( 1 b) MV Corp Asses 0 (10) The difference depends posiively on he rae of growh of GDP, on he blended marginal income ax rae T and on sysemaic risk, and negaively on leverage and he ne growh of shares. The fac ha he difference beween socks and corporae bonds seems 18 Fama and French (1999) discuss he use of simple versus compounded reurns as discoun raes. They argue ha under cerain condiions he expeced one period simple reurn is he appropriae discoun rae. Oherwise, a more appropriae mehod is o use a weighed average of simple and compound reurns. 19 This leads o an undersaemen of he acual long-erm reurn on deb ha should probably include a defaul premium. Thus, he growh of acual issued volume of deb should exceed GDP growh by an amoun equal o he average corporae defaul rae. Noe also ha any inflaion risk and ineres rae risk are already embedded in he T-Bill rae. Furhermore, because we are examining arihmeic average reurns, capuring 1-year invesmen horizons, he inflaion and ineres rae risks may bear a smaller effec han for deb yields represening muliyear horizons. I urns ou ha using marke-value based average hisorical leverage insead of book-value leverage raio lowers our esimae of he reurn on deb even more because he value of he leverage raio is 34.23% over

16 relaed o GDP growh is sensible if we noe ha a bond is a claim o a fixed income sream, whereas a sock is a claim o boh growing dividends and earnings sreams. In secion 3, we show ha he reurn on corporae deb r D nearly equals he risk-free rae in he long run. This resul implies ha formula (10) characerizes he equiy premium as well. 20 Over he period of , and afer combining our prior esimaes for he long-erm reurn on deb and he reurn on socks, we obain a value for he premium of equiy over he 3-monh T-Bill equal o 8.3%, which closely maches he hisorical esimae of 8.1% over he examined period. 5. Porfolio insurance and he equiy premium The lieraure on he equiy premium has so far shown scany evidence of he link beween he premium and risk. Asness (2000) aemps o empirically reconcile he premium wih measures of he difference in risk beween bonds and socks and achieves some relaive success for rolling marke periods of 20 years, bu falls shor of explaining he premium over he full hisorical record. In his secion, we show ha opion pricing can help us derive a measure for he equiy premium ha is direcly relaed o observed hisorical sock reurn volailiy. In fac, we show ha he long-run equiy premium is closely relaed o an invesor s objecive of avering downside risk. Consider an invesor adoping he following long-erm sraegy: every year, inves $1 in a sock index and buy a pu opion on he index (wih a real $1 srike), sell he sock a he end of he year. This is an insance of seeking porfolio insurance by using a proec- 20 Again, if i weren for he survivor bias in he Flow of Funds corporae deb daa, we should expec ha corporae deb reurns would incorporae a defaul premium. 15

17 ive pu (Meron, Scholes, and Gladsein, 1982). The yearly maximum loss is limied o he loss of he pu premium. In he long run, he expeced sock reurn mus be he arihmeic average of he index reurn. We posulae ha yearly marke volailiy is expeced o equal he annual volailiy given by a long-erm horizon esimae. In ha case, he pu opion price remains consan. We also posi ha he index pays a known dividend. Formally, le μ be he arihmeic average real sock reurn, le r be he arihmeic average real risk-free rae, q is he real afer-ax dividend yield, and c and p are he respecive opion prices for he European call and pu. Our goal is o find he value of his pu opion. Assume ha he curren index price is $1 and he srike price is $1 (real), and using compounded raes of reurns, he pu-call pariy formula for a dividend paying sock (Hull, 2003) leads o express he pu price as: = + (11) r q p c e e Following Rubinsein (1984), we define he real risk-neural expeced fuure value of he pu-call relaionship as follows: 21 Ε ( p) =Ε ( c) + 1 e μ q (12) Given ha he expeced capial gains rae is greaer han zero, he expeced value for he pu opion mus be zero, because i is expeced o be ou-of-he-money. Therefore, we ge: Ε () c = e μ q 1 (13) 21 We are applying Rubinsein s (1984) approach o real (deflaed) values of expeced call and pu prices. Rubinsein also defines hese expecaions for a horizon h as a fracion of one year and for volailiy esimaes ha may differ for individuals compared wih he overall marke. Here we posi ha h is arbirarily close o one and ha individual esimaes are arbirarily close o he marke volailiy. 16

18 Thus, he real expeced fuure value of he call opion equals he real expeced capial gains rae. 22 We posulae ha he presen value of he expeced fuure call value discouned a he risk-free rae is a good approximaion for he curren call value, ha is: 23 r c e Ε() c (14) Using equaions (13) and (14), and plugging he resuling call price ino (11), we obain: q e p r e μ 1 (15) Hence, he equiy premium can be expressed as: μ r Ln(1 + e q p) (16) Thus, he risk premium approximaely equals he value of a European pu opion on a sock index compounded a he dividend yield rae (using he average annual sandard deviaion of prices). Applying Black and Scholes (1973) approach, we can evaluae he price of such a pu opion. I is well known ha he resuling value is independen of he expeced sock reurn and invesor preferences. 24 Assume ha he srike price and he curren sock price are boh equal o $1 and ha he opion s mauriy is 1 year. The sandard formula for an opion on a dividend paying sock index (wih axes) is given by Scholes (1976). 25 According o our sraegy, an invesor sells her sock a he end of each year. 22 I is ineresing o noe ha his same resul can be derived using Black and Scholes (1973) call opion approach o corporae equiy. In ha case, we would have o assume ha socks are iniially purchased using zero-coupon deb and ha he minimum real required reurn on deb is zero. 23 Using parameer values inroduced laer, we find ha he Black Scholes curren call price equals 6.56%, whereas our esimae is 6.50%. 24 This approach is subjec o he sandard criicism of he normaliy assumpion of he sock marke reurn disribuion (e.g., Fama 1965). 25 Scholes (1976) prices a European call opion. A pu opion can be priced using he pu-call pariy formula. 17

19 We posi ha he or she is axed a he marginal ordinary income ax rae and ha his ax rae equals he dividend income marginal ax rae T. Hence, he formula is: wih r q p = e N( d2) e N( d1) 2 r q+ σ /2 d1 = σ d2 = d1 σ q= (1 T) dividend yield (17) Thus, using he relaionship beween he pu opion and he equiy premium expressed in (16) and he pu pricing formula (17), we obain an independen esimae of he premium. We apply formula (17) wih he following parameers: he dividend ax rae corresponds o he average marginal rae over (wih 1999 values used for he years 2000 and 2001). The value for he ax rae is 40%. 26 The sandard deviaion of sock index reurns is a hisorical esimae using coninuously compounded annual real oal S&P500 reurns over We adjus oal reurns according o Hull (2003), by removing he effec of dividends on sock volailiy o accoun for he risky porion of sock prices. The value of he sandard deviaion σ is esimaed a 18.87%. Our esimae of he inflaion-adjused preax average S&P500 dividend yield is 4.20% over he same period. The value for he real risk-free rae corresponds o he T-Bill arihmeic average real rae of 0.76% over ha period as well. We hen arrive a a value for he pu opion premium p of 8.29%, and a value for he risk premium of 8.16%, which is nearly idenical o our prior esimae for he risk premium. Noe ha he insurance sraegy presened above does no fully guaranee a risk-free reurn on a yearly basis. To achieve ha goal, he invesor could for example sell a call 26 The esimaes for are from Esrella and Fuhrer (1983) and for from he NBER TAXSIM model. 18

20 opion in addiion o owning a proecive pu. However, over a long-erm invesmen horizon, socks are on average as riskless as bonds, in he sense ha hey deliver an average reurn ha is more and more cerain wih longer horizons, based on real long-erm GDP per capia growh. The difference beween riskless securiies and socks is ha riskless securiies are a perfec hedge agains shor-erm marke downside risk, whereas socks obviously are no. Our resul shows ha he long-run equiy premium reflecs a porfolio insurance moive since here would oherwise be an opporuniy for a riskless long-erm arbirage, in he sense ha pu-call pariy would be violaed on average. Our resul demonsraes ha he risk-free rae and he long-erm sock reurn are joinly defined in relaion o he premium on a pu opion. Our analysis of he reurn o corporae asses above, when combined wih he resul of his secion, leads us o offer a new view of he deb versus sock rade-off, in which bondholders and sockholders muually benefi from co-financing corporae asses a he macroeconomic level. Sockholders will benefi from leveraging asses since hey can boos equiy reurns via receiving higher dividends all he while mainaining consan asse and earnings growh raes. In his new view, i is useful o consider ha an invesor following our porfolio insurance sraegy is a deb holder. Deb holders by choosing o hold deb insead of equiy sele o earn a lower long-erm reurn (han equiy holders), because hey choose o fully insure agains he loss of heir principal in he shor run. Thus, in equilibrium, agens will be indifferen beween holding deb or equiy, because insuring one s principal is cosly. Deb holders are in effec paying a porfolio insurance premium o sockholders, and hus end-up wih a reurn equal o he risk-free rae on average. 19

21 This argumen has an ineresing implicaion regarding he radiional view of compensaion for risk. The sandard CAPM uses a boom-up approach o he equiy premium: he equiy premium is added o he risk-free rae o compensae sockholders for he exra risk borne by hem. Our logic of porfolio insurance presens a op-down view of he premium: he premium is subraced from he long-erm equiy reurn o obain he risk-free rae, because insurance is a cos. 27 This logic obviously does no violae he spiri of CAPM. 6. A shrinking equiy premium? Jagannahan, McGraan and Scherbina (2000) and Fama and French (2002) argue ha ex-pos reurns are a disored view of expeced reurns and ha a lower equiy premium should be used compared wih he hisorical average. Table 2: Esimaes of he equiy premium using he porfolio insurance model Beg. Year Div. Yield Div. Tax Rae SD Mk. Reurn Real T-Bill Es. Premium Period % 36.48% 12.61% 1.38% 5.38% Ending % 29.08% 11.68% 3.26% 4.08% December % 27.84% 10.51% 2.25% 3.87% % 29.06% 10.90% 2.89% 3.49% Period % 35.23% 12.69% 1.17% 5.44% Ending % 29.11% 12.31% 2.63% 4.49% December % 28.13% 12.03% 1.62% 4.65% % 29.18% 13.25% 1.67% 4.91% 27 In he limi case of 100% deb-financing, corporae deb holders would require he same reurn as equiy holders (or T-Bill reurn plus risk premium). However, o avoid shor-erm principal losses, debholders would have o use our porfolio insurance sraegy and sacrifice he premium. The recen rising corporae rend of using leverage o pay dividends may cause an increase in he equiy premium, since he premium is an increasing funcion of he dividend yield, for a given level of he risk-free rae and independenly of leverage risk. 20

22 Table 2 shows ha for various ime horizons, our porfolio insurance model predics an overall decline in he equiy premium before he marke crash of 2000 wih a minimum premium value around 3.5%, and an increase back o abou 5% for periods ending in These esimaes are in line wih he above-cied lieraure. The observed decline in he premium compared o he average of 8.1%, seems o originae from declining rends in dividend yields, marginal dividend income ax raes and sock marke volailiy along wih a rise in he average real T-Bill reurns since he 1980s. Because our model works as a predicor of long-erm premia, he resuls shown in Table 2 are only applicable if we assume ha hese recen rends in dividend yields, marginal ax raes and ineres raes represen permanen regime shifs. This laer view may coincide wih he Federal Reserve (Fed) recen moneary policy rend, since he Fed appears commied o keep inflaion raes and discoun raes low (alhough he slide of T- bill raes in he 1 o 2% range is also responsible for larger premia in periods ending in 2004). Regarding ax policy, marginal dividend and capial gains ax raes have reached heir lowes levels in hisory, and hiking raes back up is a virual poliical impossibiliy. On he oher hand, dividend policy seems o have shifed owards lower payous over he 1990s (Fama and French, 2001). However, invesors can easily decide o reverse he rend if hey perceive ha capial gains are exposed o greaer downside risk Parameer values are annualized monhly averages, excep for dividend axes (yearly averages) and excep for he periods saring in 1970, where he real T-Bill is averaged yearly. S&P500 sandard deviaions are annualized and based on monhly coninuously compounded real reurns. 29 Anoher poenial issue is ha share repurchases may couner he effec of lower dividend yields. Grullon and Michaely (2002) repor ha since he mid-1980s, large esablished U.S. firms did no increase dividends as much as hey could have, bu raher chose o buy back shares. Thus, in all likelihood, lower expeced dividend yields have been associaed wih greaer expeced capial gains. In ha insance, our porfolio insurance model makes he provision in equaion (13) ha he expeced fuure value of he S&P500 call opion fully reflecs hese capial gains expecaions and hus ha he equiy premium embodies hese rend expecaions as well. 21

23 One reason why our premium esimaes seem o rise slighly (conrolling for he slide in T-bill raes) for horizons ending in 2004 compared wih 2000 is ha marke volailiy did rise afer he marke crash of This effec is accenuaed for shor horizon periods (15 years or less). To smooh ou he effec of he 2000 marke crash, an inermediae range of abou 20 years may be more appropriae. Table 2 hen shows ha for horizons saring in 1982 he premium can be esimaed in he 4 o 4.5% range. 7. Conclusion We show ha he long-run equiy premium is heoreically and empirically consisen wih GDP growh and a porfolio insurance moive. We derive he long-run ex-ane equiy reurn and long-run corporae deb and asse reurns using a supply-side growh model. We arrive a a macroeconomic generalizaion of he sandard susainable growh formula used in corporae finance exbooks o deermine he long-run average reurn on socks. The long-erm average sock reurn depends on GDP per capia growh, he earnings reenion rae, and a premium linked o de-rended earnings volailiy and marke-o-book volailiy and he response of dividend and share repurchase policies o hese facors. Our model accuraely replicaes he arihmeic average hisorical reurns for he S&P500. Our firs conclusion is ha he equiy premium defined as he difference beween he S&P500 sock reurn and 3-monh T-Bill is consisen wih observed GDP growh and oher financial parameers such as marginal income ax raes. Our resul also hinges on he fac ha he Flow of Funds daa on corporae deb and S&P daa on he equiy side have an inheren survivor bias. Ineresingly, our analysis enails ha he corporae deb of surviving firms exhibis a long-erm average reurn ha 22

24 is essenially risk-free. In acualiy, because invesors ypically inves in bond porfolios ha experience failure raes, a defaul premium should be added o our esimae. Our second key resul is ha he equiy premium is consisen wih a shor-erm porfolio insurance moive. We show ha he equiy premium is closely approximaed by a pu opion premium on a real $1 invesmen in he marke index when a long-erm invesor wishes o insure agains year-o-year marke volailiy, by using he average yearly S&P500 volailiy over This resul leads us o a new view of corporae asse financing where invesors, pursuing a shor-erm insurance moive, are indifferen beween long-run equiy reurns and he comparaively lower rae on shor-erm Treasury bonds. Deb holders looking o insure heir principal essenially forego long-run equiy reurns by paying an insurance premium equal o he equiy premium. Siegel (1999), Jagannahan, McGraan and Scherbina (2000), Fama and French (2002), and De Sanis (2004) all claim ha ex-pos reurns are a disored view of expeced reurns and ha a lower equiy premium is jusified compared wih he hisorical average. Our resuls sugges ha using an 8.1% premium in valuaion formulas and capial budgeing problems may be appropriae, since he observed level of he long-run equiy premium is fully consisen wih he observed seady-sae GDP growh and consisen wih risk explanaions as well. However, if one believes ha he recen 1990 s rends in dividend yields, ineres raes, axes and inflaion represen permanen regime shifs, our model can be parameerized o yield a 3.5% equiy premium in line, for example, wih Fama and French s (2002) esimae. 23

25 Fuure research will examine he deerminans of he equiy premium s counercyclical behavior in he shor o medium-erm. In ha respec, using European opions on he S&P500 seems o be a promising avenue o characerize he equiy premium. 24

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