Implied Equity Duration: A New Measure of Equity Risk *

Transcription

1 Implied Equiy Duraion: A New Measure of Equiy Risk * Paricia M. Dechow The Carleon H. Griffin Deloie & Touche LLP Collegiae Professor of Accouning, Universiy of Michigan Business School Richard G. Sloan Vicor L. Bernard PricewaerhouseCoopers LLP Collegiae Professor of Accouning and Finance, Universiy of Michigan Business School Mark T. Soliman Ph.D. Candidae, Universiy of Michigan Business School This Version: May 2002 Correspondence: Richard G. Sloan Universiy of Michigan Business School 701 Tappan Sree Ann Arbor, MI sloanr@umich.edu Phone: (734) Fax: (734) Key Words: Duraion, Asse Pricing, Risk KEL classificaion: G12; G14; M41 * We are graeful for commens from workshop paricipans a UC Berkeley, Emory Universiy, Universiy of Michigan, MIT, UCLA and Universiy of Souhern California. Thanks also o Paul Michaud for programming assisance. Sloan and Dechow acknowledge financial suppor provided by he Michael A. Sakkinen Research Scholar Fund a he Universiy of Michigan Business School.

2 Absrac We derive an expression for implied equiy duraion by adaping he radiional expression for bond duraion and develop an algorihm for is empirical esimaion. We find ha he sandard empirical predicions and resuls for bond duraion hold for our measure of implied equiy duraion. Sock reurn volailiies and beas are increasing in implied equiy duraion. Moreover, esimaes of common shocks o expeced equiy reurns exraced using our measure of implied equiy duraion capure a srong common facor in sock reurns. We also show ha book-omarke raio represens a special case of our expression for implied equiy duraion ha imposes resricive assumpions on he evoluion of fuure cash flows. Consequenly, our implied equiy duraion framework provides an explanaion for he empirical properies of he book-o-marke relaed facor documened Fama and French (1993). Empirical ess confirm ha he common facor relaed o our more general measure of implied equiy duraion dominaes and subsumes he common facor relaed o book-o-marke.

3 Inroducion Techniques for analyzing he risk characerisics of fixed income securiies have evolved wihin a heoreically rigorous framework based on he discouned expecaions of he fuure cash flows of he securiies. Consrucs such as duraion and convexiy are well esablished for fixed income securiies and are embraced by academics and praciioners alike. The analysis of equiy securiies, in conras, has evolved in a relaively ad hoc manner. Following disappoinmen wih he performance of equilibrium pricing models such as he CAPM, academics and praciioners have adoped empirically moivaed procedures for he analysis of equiy risk. For example, following Fama and French (1993), a popular academic approach o modeling he risk characerisics of sock reurns is hrough a hree-facor model incorporaing a marke-relaed facor, a size-relaed facor and a book-o-marke-relaed facor. Similarly, praciioners have embraced he noion of classifying socks on he basis of marke capializaion and he exen o which hey exhibi he syle characerisics of value and growh. We bridge his gap in he analysis echniques for fixed income and equiy securiies by developing an implied equiy duraion measure ha provides boh a heoreically jusifiable and empirically powerful echnique for he analysis of equiy securiy risk. 1 We begin by developing a measure of implied equiy duraion based on Macaulay s radiional measure of bond duraion. The primary obsacle in implemening he bond duraion formula for equiies is in he esimaion of he expeced fuure cash disribuions for equiies. We develop a wo-sage procedure o faciliae his ask. Firs, using simple forecasing models based on hisorical financial daa, we esimae he expeced fuure cash flows for a finie forecas horizon. Second, we assume ha he remaining value implici in he observed sock price will be disribued as a level perpeuiy beyond our finie forecas horizon. We hen apply he sandard 1

4 duraion formula o compue our measure of implied equiy duraion. We recognize ha our esimaion procedure for implied equiy duraion represens a simple approximaion based on relaively crude forecasing assumpions. Neverheless, he resuling duraion esimaes perform well in empirical ess, and our basic framework is easily adaped o incorporae more sophisicaed forecasing models. Empirical ess demonsrae he effeciveness of our measure of implied equiy duraion in explaining he risk characerisics of equiy securiy reurns. Implied equiy duraion is srongly posiively relaed wih sock reurn volailiies and beas and has incremenal explanaory power over pas volailiies/beas in forecasing fuure volailiies/beas. Moreover, esimaes of common shocks o expeced equiy reurns exraced using our measure of implied equiy duraion capure a srong common facor in sock reurns. We also show ha book-o-marke raio represens a special case of our expression for implied equiy duraion ha imposes resricive assumpions on he evoluion of fuure cash flows. Consequenly, our implied equiy duraion framework provides a rigorous explanaion for he empirical properies of he book-omarke-relaed facor documened in Fama and French (1993). Empirical ess confirm ha he common facor relaed o our measure of implied equiy duraion dominaes and subsumes he common facor relaed o book-o-marke. The remainder of he paper is organized as follows. The nex secion discusses our measure of implied equiy duraion and our empirical predicions. Secion 2 describes our daa, Secion 3 presens our resuls and secion 4 concludes. 1. Implied Equiy Duraion: Definiion, Measuremen and Predicions 1.1. Definiions The radiional measure of duraion (D) for a bond is he Macaulay duraion formula: 2

5 D = T CF (1 r P = 1 + ) (1) where CF denoes he cash flow a ime, r denoes he yield o mauriy and P denoes he bond price. This measure of duraion is a weighed average of he imes o each of he respecive cash flows on he bond, where he weighs represen he relaive conribuions of he cash flows o he bond s value. Inuiively, duraion represens he average mauriy of he bond s promised cash flows. The primary role of duraion in he analysis of fixed income securiies is as a measure of bond price sensiiviy o changes in he yield o mauriy. Differeniaing he expression for he value of a bond wih respec o he yield o mauriy gives: P r D = P 1+ r (2) Inuiively, his resul indicaes ha he relaion beween bond prices changes and changes in bond yields is a simple funcion of duraion: 2 P P D r 1+ r (3) The expression D 1+ r is ofen referred o as he modified duraion, and i provides a simple measure of he sensiiviy of bond prices changes o yield changes. Exending he duraion concep o equiies inroduces wo key problems: 1. A bond ypically makes a finie number of cash paymens, while he sequence of paymens on equiy is poenially infinie. 3

6 2. The amoun and iming of he cash paymens on a bond are usually specified in advance and subjec o lile uncerainy, while he paymens on equiy are no specified in advance and can be subjec o grea uncerainy. To address he firs problem, we pariion he duraion formula in equaion (1) ino wo pars, a finie forecasing horizon of lengh T and an infinie erminal expression: D = T CF (1 + r) CF (1 + r) T CF (1 + r) P CF (1 + r) CF (1 + r) CF (1 r P = 1 = 1 = T + 1 = T ) + T = 1 = T + 1 (4) Since we are now dealing wih equiy, P denoes he marke capializaion of equiy (sock price muliplied by shares ousanding), CF denoes he ne cash disribuions o equiy holders and r denoes he expeced reurn on equiy. Equaion (4) expresses equiy duraion as he valueweighed sum of he duraion of he finie forecasing horizon cash flows and he duraion of he infinie erminal cash flows. Nex, we assume ha he erminal cash flow sream consiss of a level perpeuiy wih a value equal o he difference beween he observed marke capializaion implici in he sock price and he presen value of he cash flows over he finie forecas period, so ha: CF = ( P ( 1+ r) = 1 CF (1 r) = T T ) (5) Recognizing ha he duraion of a level perpeuiy beginning in T periods is T+(1+r)/r, and subsiuing (5) ino (4) simplifies our expression for equiy duraion o: D T CF CF ( P ) ( 1+ r) (1 + r) = 1 (1 r) + ( T + ) P r P = 1 + = (6) T 4

7 The assumpion ha he cash flow sream for an equiy securiy can be pariioned ino a finie forecasing period and an infinie erminal expression is sandard in he equiy valuaion lieraure. The assumpion ha he erminal cash flows are realized as a level perpeuiy is less sandard. More commonly, he erminal cash flows are assumed o grow a a consan erminal rae, such as he expeced macroeconomic growh rae. We make he level perpeuiy assumpion for racabiliy and wihou loss of generaliy. As long as he forecasing horizon is long enough o exhaus plausible opporuniies for firm-specific or indusry-specific super-normal growh, he erminal growh rae will be a cross-secional consan, and so will no be an imporan source of cross-secional variaion in implied equiy duraion. Because he erminal cash flow perpeuiy is inferred from he observed sock, we refer o he resuling measure of equiy duraion as implied equiy duraion. In oher words, our measure of equiy duraion is based on invesors consensus expecaions, as refleced in sock prices, raher han on necessarily raional forecass of fuure cash flows. The discussion above deals wih he infinie cash flow problem. The second problem in implemening equaion (6) is he forecasing of he finie period cash disribuions, CF 0 T. Our forecasing model is based on recen research indicaing ha accouning-based performance measures provide effecive informaion variables for forecasing fuure cash flows (Nissim and Penman 2001). We begin wih he accouning ideniy ha expresses ne cash disribuions o equiy in erms of earnings and book value of equiy: 3 CF E ( BV BV 1) (7) = where E represens accouning earnings a he end of period and BV represens he book value of equiy a he end of period. Re-arranging he righ-hand side of equaion (7) gives: 5

8 E ( BV BV 1) CF = BV 1 (8) BV 1 BV 1 Equaion (8) indicaes ha o forecas ne cash disribuions o equiy, one needs o firs forecas: (i) Reurn on equiy (ROE) denoed by E /BV -1 ; and (ii) Growh in equiy, denoed by (BV -BV -1 )/ BV -1. I is well esablished ha ROE follows a slowly mean revering process [Sigler 1968, Penman 1991]. Moreover, boh economic inuiion and empirical evidence sugges ha he mean o which ROE revers approximaes he cos of equiy [Nissim and Penman 2001]. We herefore model ROE as a firs-order auoregressive process wih an auocorrelaion coefficien based on he long-run average rae of mean reversion in ROE and a long-run mean equal o he cos of equiy. To forecas growh in equiy, we rely on he resuls in Nissim and Penman (2001) indicaing ha pas sales growh is a beer indicaor of fuure equiy growh han pas equiy growh. Sales growh follows a mean revering process similar o ROE, bu mean reversion in sales growh ends o be more rapid [see Nissim and Penman (2001)]. Economic inuiion suggess ha he mean o which sales growh revers should approximae he long-run macroeconomic growh rae. 4 We herefore model growh in equiy as a firs-order auoregressive process, wih an auocorrelaion coefficien equal o he long-run average rae of mean reversion in sales growh and a mean equal o he long-run GDP growh rae. Implemenaion of our esimaion procedure for implied equiy duraion requires four financial variables and four forecasing parameers as inpus. We summarize hese inpus in Table 1. The four financial variables are book value (boh curren and lagged one year), sales 6

9 (boh curren and lagged one year), earnings (curren) and marke capializaion (curren). The four forecasing parameers are he auocorrelaion coefficien for ROE, he auocorrelaion coefficien for sales growh, he cos of equiy and he long-run GDP growh rae. We conduc our analysis using annual daa and obain he required financial variables from he annual COMPUSTAT files. Using pooled daa over our sample period, we obain average esimaes of he auocorrelaion coefficiens for ROE and sales growh of 0.57 and 0.24 respecively. The long-run averages for cos of equiy and GDP growh rae are based on he long-run averages repored by Ibboson (1999) of (approximaely) 12% and 6% respecively. Noe ha we use a naïve forecas of he cos of equiy ha assumes i o be a cross-secional consan. Such a naïve assumpion is necessary however, o avoid he possibiliy ha we could induce our empirical resuls hrough sysemaic variaion in he cos of equiy capial. By assuming ha he cos of equiy is a cross-secional consan, we ensure ha our measure of implied equiy duraion is driven solely by differences in he iming of he expeced fuure cash flows. 5 Finally, we use a finie forecas horizon of en years, because mos of he mean reversion in sales growh and ROE is complee afer 10 years. We emphasize ha hese forecasing procedures are relaively crude. For example, cerain forecasing parameers have been shown o vary sysemaically as a funcion of indusry membership and oher firm characerisics. However, our immediae goal is o inroduce he concep of implied equiy duraion and demonsrae he abiliy of a relaively parsimonious empirical esimaion procedure o produce an effecive measure of implied equiy duraion. [Table 1 here] 7

10 We illusrae our implied equiy duraion esimaion procedure using wo represenaive firm-years from our sample in able 2. The firs example in panel A is for Alaska Air in 1999 and is designed o be illusraive of low duraion equiy. The second example in panel B is for Amazon.com in 1999 and is designed o be illusraive of high duraion equiy. Values for he required forecasing variables are lised a he op lef of each panel and he forecasing parameers, which are assumed o be he same across firms, are lised a he op righ of each panel. Forecass of cash flows and heir presen values are derived for he en-year forecas horizon. The growh rae is derived by revering pas sales growh o he long-run mean of 6% using he auocorrelaion coefficien of Similarly, ROE is derived by revering pas ROE o is long run mean of 12% using he auocorrelaion coefficien of Applying he forecas growh raes o lagged book value generaes he forecass of fuure book values. Applying he forecas ROEs o he lagged book value forecass generaes he earnings forecass. Cash flow forecass are hen backed ou from earnings and book value forecass using equaion (7). The duraion of he finie forecas cash flows is equal o he raio of he ime weighed presen value o he presen value of he forecas cash flows. The weigh assigned o he finie period duraion is equal o he raio of he presen value of he forecas cash flows o he marke capializaion. The duraion of he erminal cash flows is always equal o [i.e., T + (1+r)/r = /0.12=19.33]. The weigh assigned o he erminal duraion is simply one minus he weigh assigned o he finie period duraion. Implied equiy duraion is compued by aking he weighed sum of he finie and erminal period duraions. The compuaion for Alaska Air indicaes ha 64% of he value implici in he curren price is expeced o be realized during he finie forecas period. Alaska Air s forecas ROE 8

11 exceeds is forecas growh rae in every year of he finie forecas period, which resuls in posiive cash disribuions in each of hese periods. This resuls in a relaively low implied equiy duraion figure of jus 10.0 years for Alaska Air. The compuaion for Amazon.com indicaes ha he cash flows realized during he forecas period amoun o 21% of he value implici in he curren price. In Amazon s case, he negaive curren ROE and high growh rae combine o generae cash flows over he finie forecas period ha are mosly negaive and have a negaive ne presen value. Implici in his negaive cash flow is he necessiy for Amazon.com o raise addiional capial over he finie forecasing horizon. No unil he eighh year of he finie period does he ROE exceed he growh rae in book value, which is wha is required for posiive cash disribuions. As a consequence of he negaive weighing on he finie forecas period duraion, Amazon s implied equiy duraion of 23.0 years exceeds he erminal period duraion of years. Thus, duraion ends o be low for firms wih high ROE, low growh and low marke valuaions and high for firms wih low ROE, high growh and high marke valuaions. [Table 2 here] 9

12 1.2. Equiy duraion and he earnings-o-price raio and book-o-marke raios Finance praciioners and academics frequenly use earnings-o-price and book-o-marke raios as equiy syle and risk characerisics. Our measure of implied equiy duraion is closely relaed o hese valuaion raios. We demonsrae he links by considering some special cases of he implied equiy duraion formula in equaion (6). These special cases all involve he assumpion ha he ne cash disribuions over he finie forecasing period ake he form of a level annuiy, denoed A. The duraion of a level annuiy of lengh T is given by: (1 + r) T D A = (9) T r (1 + r) 1 and he presen value of a level annuiy of amoun A and lengh T is given by: PV A 1 1 T (1 + r) = A (10) r Subsiuing hese wo equaions ino equaion (6) and simplifying yields: A (1 + r) D = T + r T r P (11) This expression highlighs he fac ha implied equiy duraion is decreasing in he magniude of he ne cash disribuions paid over he finie forecas horizon. Differeniaing (11) wih respec o A gives: D A = T r P (12) 10

13 Duraion is decreasing in he magniude of he annuiy, wih he rae of decrease being larger for longer forecas horizons, lower discoun raes and lower sock valuaions. Equaion (11) is he key o undersanding he relaion beween implied equiy duraion, he earnings-o-price raio and he book-o-marke raio. Recall from equaion (8) ha he ne cash disribuions received over he finie forecas horizon can be expressed as: CF = BV 1 E BV 1 ( BV BV BV 1 1 ) If we assume ha growh in equiy is zero for all finie forecas periods (i.e., BV = BV -1 for 0 T) and perfec persisence of curren ROE over he forecas period (i.e., E BV = BV E for 0 T), hen CF = E 0 for 0 T. The amoun of he annuiy for he finie forecas horizon is now equal o earnings a he beginning of he forecas horizon, and equaion (11) becomes: (1 + r) E0 T D = T + r P r (13) Here we see ha here is a negaive relaion beween implied equiy duraion and he earnings-oprice raio. So he earnings-o-price raio will be a good proxy for equiy duraion in firms where growh in equiy is low and ROE is highly persisen. Table 2 provides esimaes of implied equiy duraion for Alaska Air and Amazon.com based on (13) labeled earnings-o-price approximaion. The equaion (13) approximaion undersaes duraion for Alaska Air. This occurs because Alaska Air has a high curren ROE and mainaining he ROE over he finie horizon resuls in he higher cash disribuions. On he oher hand, equaion (13) oversaes duraion for Amazon, his occurs because Amazon has a very negaive ROE and mainaining he ROE over he finie forecas horizon resuls in more required capial infusions. 11

14 To see he relaion beween implied equiy duraion and he book o marke raio, assume ha growh in equiy is again zero over he forecas period bu ha ROE immediaely mean revers o he cos of capial in he firs year of he forecas period (i.e., E BV 1 = r for 0 T). Equaion (8) now simplifies o CF = r BV0. The amoun of he annuiy for he finie forecas horizon is equal o book value a he beginning of he forecas horizon muliplied by he cos of capial, and implied equiy duraion becomes: (1 + r) BV0 D = T + T r P (14) In his special case, here is a simple negaive relaion beween implied equiy duraion and he book-o-marke raio. The book-o-marke raio will be a good proxy for duraion for firms where growh in equiy is low and ROE is rapidly mean revering. Table 2 provides esimaes of implied equiy duraion for Alaska Air and Amazon.com based on (14) labeled book-o-marke approximaion. The approximaion based on equaion (14) undersaes duraion for boh Alaska Air and Amazon.com. For Alaska Air, he undersaemen arises because he implici assumpion of no growh in equaion (14) resuls in higher cash disribuions in he forecas period. For Amazon, he undersaemen arises because he implici assumpions of no growh and he immediae mean reversion o a posiive ROE resul in higher cash disribuions in he finie forecas period. The close links beween our measure of implied equiy duraion and hese popular valuaion raios sugges ha hey may serve as useful proxies for equiy duraion. We explore his possibiliy more fully in our empirical ess. 12

15 1.3 Empirical Predicions The primary empirical implicaion of duraion sems from he relaion beween ex pos holding period reurns and changes in expeced reurn. Denoing holding period reurns as h and changes in expeced reurn as r, equaion (3) indicaes ha he influence of a changes in expeced reurn on he ex pos holding period is: h = P P D r 1+ r (15) Empirical verificaion of he relaion in (15) is difficul, because changes in expeced equiy reurns are no direcly observable. Neverheless, we can use (15) o generae predicions concerning he role played by duraion in ransmiing expeced reurn volailiy o holding period reurn volailiy. Firs, defining volailiy in erms of he sandard deviaion (σ), we can use (15) o deermine he impac of volailiy in expeced reurns on he volailiy of holding period reurns: σ D 1+ r ( h) σ ( r ) (16) Noe ha equaion (16) only models he role of expeced reurn shocks on volailiy. I ignores oher poenial sources of volailiy, such as volailiy aribuable o cash flow shocks. Equaion (16) indicaes ha he impac of expeced reurn volailiy on holding period reurn volailiy is greaer for long duraion socks. This leads o our firs empirical predicion: P1: The volailiy of equiy holding period reurns is increasing in equiy duraion. Our firs predicion relaes o he oal volailiy of equiy reurns. However, asse-pricing heory suggess ha non-diversifiable volailiy consiues a more relevan measure of risk. In 13

16 paricular, he capial asse pricing model indicaes ha only sysemaic risk (β) ha is relaed o movemens in he marke porfolio should be priced. Defining h m as he ex pos holding-period reurn on he marke porfolio, D m as he duraion of he marke porfolio and r m as he expeced reurn on he marke porfolio, we can use (15) o deermine he impac of common shocks o expeced reurns, ( r m ) on sysemaic risk (β(h,h m )): m ( h, h ) σ D (1 + r ) β ( h, h ) β (17) 2 m m = ( r, rm σ ( hm ) Dm (1 + r) ) The final erm in (17) represens he sensiiviy of changes in he expeced reurn on he equiy securiy o changes in he expeced reurn on he marke porfolio. There is a large body of empirical evidence documening srong common shocks o expeced equiy reurns [e.g., Campbell and Shiller (1988), Campbell and Mei (1993)]. Thus, we expec he final erm o be posiive and close o one for he ypical equiy securiy. Equaion (17) indicaes ha he impac of common expeced reurn volailiy on holding period reurn volailiy is increasing in he duraion of he equiy securiy relaive o he duraion of he marke porfolio. Equaion (17) forms he basis for our second predicion: P2: Equiy beas compued from holding-period reurns are increasing in he duraion of he equiy relaive o he duraion of he marke porfolio. Tess of our second predicion build on evidence in Campbell and Mei (1993) and Cornell (1999). Campbell and Mei use a log-linear approximaion of reurns o esimae he proporion of he variaion in bea aribuable o common variaion in cash flows versus common variaion in expeced reurns. They find ha he beas are largely aribued o common innovaions in expeced reurns. Thus, heir evidence implies ha equaion (17) should capure 14

17 an imporan deerminan of bea. Cornell anicipaes our second predicion by recognizing ha Campbell and Mei s resuls imply ha equiy duraion should be an imporan deerminan of beas. He presens preliminary ess in his respec by correlaing beas wih earnings-o-price raios, dividend-o-price raios and growh forecass. Cornell provides mixed and indirec evidence in suppor of P2. We build on Cornell s resuls by consrucing more direc ess of P2. Our second predicion ress on he assumpion ha some shocks o expeced reurns are common across securiies. However, i does no necessarily rule ou he case of idiosyncraic shocks o expeced reurns. For example, liquidiy has been proposed as an imporan deerminan of expeced reurns [e.g., Amihud amd Mendelson (1983)]. Therefore, evens having an impac on a firm s liquidiy, such as changes in exchange lising, addiion/removal from an index and he lising of derivaive securiies, may resul in idiosyncraic shocks o expeced reurns. Denoing h f and r f as he firm-specific componens of realized and expeced reurns respecively and subsiuing ino (16) yields: σ D ( h ) σ ( ) f r f 1+ r (18) from which we generae our hird predicion: P3: The sandard deviaion of he idiosyncraic componen of realized holding-period reurns is increasing in equiy duraion. Our firs hree predicions concern associaions beween equiy duraion and common measures of volailiy. Our remaining predicions concern he abiliy of equiy duraion o capure a unique common facor in sock reurns. We esimae a facor relaed o duraion using wo alernaive procedures. Our firs procedure uses a sraighforward regression approach ha 15

18 aemps o direcly esimae he common shocks o expeced reurns hrough cross-secional regressions of holding period reurns on duraion: h i = D i α + γ + (1 + r) ε i (19) The model in (19) is esimaed separaely for each calendar monh in our sample. Comparing equaion (19) o equaion (15), we see ha if duraion is esimaed wihou error and shocks o expeced reurns are common across equiies, hen α =0 and γ = r. The inuiion behind his regression is ha we can infer he common shock o expeced reurns be observing he differenial holding period reurns on socks of differing duraions. We make wo predicions wih respec o he γ esimaes: P4: The γ esimaes from equaion (19) are negaively correlaed wih he holding period reurns on he marke porfolio. P5: The γ esimaes from equaion (19) are negaively correlaed wih he holding period reurns on long duraion bonds. P4 follows direcly from he observaion ha γ measures he change in he common expeced reurn on equiies. Increases in he expeced reurn on equiies should lead o reducions in equiy prices and lower holding period reurns on equiies. Thus, we should observe a negaive correlaion beween γ and he reurns on he marke porfolio. P5 is more enuous, since i requires commonaliy in he expeced reurn shocks across socks and bonds. If shocks o he risk free rae of reurn are a significan source of shocks o he expeced reurns on boh socks and bonds, hen here should be a negaive correlaion beween γ and long duraion bond reurns. 16

19 However, if shocks o expeced reurns on equiies are largely aribuable o shocks o he equiy premium, hen we will sill find suppor for P4, bu no necessarily P5. Exising academic research has focused on hree significan common facors in sock reurns: a marke facor, a facor relaed o firm size and a facor relaed o he book-o-marke raio [Fama and French 1993]. Our procedure for consrucing a duraion-relaed facor uses he Fama and French approach of consrucing a mimicking porfolio for duraion. Tha is, we ake he difference beween he monhly reurns on socks wih high versus low duraions. This relaively crude facor esimaion procedure resuls in a loss of efficiency relaive o he regression procedure. However, using his procedure allows us o direcly compare our duraion facor o he book-o-marke facor creaed by Fama and French Recall from he previous secion ha he book-o-marke raio can be inerpreed as a crude duraion proxy. Our objecive is o assess he relaive abiliy of our measure of implied equiy duraion o capure a common facor in expeced reurns. Accordingly, we es he following wo predicions: P6: A mimicking porfolio for duraion capures srong common variaion in sock reurns. P7: A mimicking porfolio for duraion subsumes a mimicking porfolio for book-o-marke in capuring common variaion in sock reurns. 2. Daa Our sample includes all firms wih available daa from he NYSE, Amex and NASDAQ from 1963 hrough Financial saemen daa are obained from he COMPUSTAT annual apes. Earnings are measured using income before exraordinary iems (annual daa iem #18). Marke value of equiy is calculaed by muliplying price as of he fiscal year end (annual daa iem #199) wih he number of shares ousanding as of he fiscal year end (daa iem #25). Book 17

20 value of common equiy (BV) represens he par value of common sock, reasury sock, addiional paid in capial and reained earnings as of he fiscal year end (annual daa iem #60). Observaions wih negaive book value of equiy are deleed from he sample. Sales growh is calculaed as he one-year discree growh rae in annual ne sales (annual daa iem #12). Sock reurns are drawn from he Cener for Research on Securiies Prices (CRSP) daily ape. We use he CRSP value-weighed index wih dividends as our measure of he marke reurn. The excess monhly marke reurn is equal o he monhly marke reurn less he one-monh reasury bill rae. We compue hree measures of sock reurn volailiy all using weekly holding period reurns over a wo-year period. Firs we compue he sandard deviaion of oal monhly sock reurns (σ), second we esimae a marke model regression for each firm and use he bea (β), and hird we use he marke model regression residual sandard deviaion (σ f ). For each firm-year, we compue volailiy using boh hisorical and forward daa. The hisorical esimaes employ daa from he wo-year period ending a he end of he fiscal year from which we obain our financial daa. The forward esimaes use daa from he wo-year period beginning a he end of he fiscal year from which we obain our financial daa. To be included in our final sample, a firm mus have non-missing values for all he required variables from COMPUSTAT and mus have a leas some of he required reurn daa available on CRSP. This sample consiss of 126,870 firm-year observaions. Of hese observaions, daa is available o compue a leas one of he volailiy merics for 102,684 observaions. We also winsorize he one-percen ails of each of he financial raios compued using he COMPUSTAT daa o reduce he influence of exreme ouliers. 18

21 Finally, we obain daa on monhly percen long-erm governmen bond reurns from Ibboson Associaes. We consruc our excess long-bond reurn series by subracing he onemonh Treasury bill rae, measured a he beginning of he monh. 3. Resuls 3.1. Descripive Saisics Panel A of Table 3 repors univariae saisics on our implied equiy duraion variable. Implied equiy duraion has a mean of 15.1 years and a sandard deviaion of 4.1 years. The lower quarile value is 13.3 and he upper quarile value is Thus, for mos firms duraion is somewha below 19.3 years, he value of duraion in he special case where no cash disribuions are made in he finie forecas horizon. Mos firms herefore disribue jus a small proporion of he value represened by heir sock price during he 10-year finie forecas period. However, he minimum value of duraion is 16.8 years, indicaing ha here are excepions. A negaive value for duraion requires ha he presen value of he cash flows over he finie forecas horizon exceed he marke value of equiy. One explanaion for such a siuaion is ha he sock is underpriced. An alernaive explanaion is ha our forecasing model has incorrecly forecas ha pas profiabiliy will coninue ino he fuure. A he oher exreme, he maximum value of duraion is 32.0 years. For duraion o be so much greaer han 19.3 years, he negaive presen value of he finie forecas period cash flows mus be large relaive o he marke capializaion. Panel B of Table 3 repors he correlaions beween implied equiy duraion and relaed financial variables. The correlaions are generally srong and are consisenly of he expeced signs. Implied equiy duraion is srongly negaively correlaed wih book-o-marke (Pearson=- 0.67; Spearman=-0.73) and earnings-o-price (Pearson=-0.79; Spearman=-0.76). We also find 19

22 ha implied equiy duraion is posiively correlaed wih sales growh (Pearson=0.20; Spearman=0.19). Ceeris paribus, higher sales growh implies more near-erm invesmen and longer duraion. I is also noeworhy ha he correlaions beween book-o-marke and earningso-price (Pearson=0.57; Spearman=0.58) are lower han he respecive correlaions of each of hese variables wih duraion. In oher words, duraion synhesizes common variaion in booko-marke and earnings-o-price. Book-o-marke, earnings-o-price and sales growh have all been proposed as empirical proxies for unidenified common risk facors in sock reurns. The correlaions in Table 3 are consisen wih implied equiy duraion represening he underlying common facor represened by each of hese variables Volailiy Resuls [Table 3 here] The firs hree predicions oulined in Secion 1.3 concern he relaion beween implied equiy duraion and sock reurn volailiy. This secion presens he resuls of ess of hese predicions. We begin in Table 4 by providing evidence on he associaion beween implied equiy duraion and hisorical sock reurn volailiy. Table 5 hen provides evidence on he abiliy of duraion o forecas fuure sock reurn volailiy. Panel A of Table 4 presens correlaions beween our esimaes of implied equiy duraion and esimaes of he sandard deviaion of weekly sock reurns. We also repor correlaions for relaed financial variables. Consisen wih our firs predicion, P1, implied equiy duraion has a srong posiive correlaion wih sock reurn volailiy (Pearson=0.19, Spearman=0.23). Book-omarke, earnings-o-price, sales growh and marke capializaion also have significan correlaions wih sock reurn volailiy. However, in he case of book-o-marke, earnings-o- 20

23 price and sales growh, he correlaions are much weaker han hey are for implied duraion. Moreover, he sign of he correlaions for hese variables are he same as he sign of heir correlaions wih implied equiy duraion. The resuls for hese variables are herefore consisen wih hem serving as noisy proxies for duraion. For marke capializaion, however, he correlaions wih sock reurn volailiy are negaive and he Spearman correlaion, is sronger han he corresponding reurn for implied duraion. The srong negaive correlaions for marke capializaion canno be explained by a duraion proxy sory, and are probably aribuable o he greaer cash flow volailiy of smaller, less diversified firms. Panels B and C of Table 4 look a he correlaions beween implied equiy duraion and he sysemaic and firm-specific componens of volailiy respecively. Consisen wih P2, here is a srong posiive correlaion beween relaive duraion and bea (Pearson=0.12; Spearman=0.19). The correlaions for book-o-marke, earnings-o-price and sales growh are somewha weaker, and are of he same sign as heir respecive correlaions wih duraion. The resuls for hese variables are again consisen wih hem serving as noisy proxies for duraion. In conras, he sign of he correlaions on marke capializaion swiches from negaive o posiive from panel A o panel B. Small firms have higher oal volailiy, while large firms have higher sysemaic volailiy. This resul is consisen wih he higher reurn volailiy of small firms arising from higher firm-specific volailiy in heir underlying cash flows. Finally, Panel C repors he correlaions for he firm-specific componen of sock reurn volailiy (σ f ). Consisen wih P3, here is a srong posiive correlaion beween implied duraion and σ f (Pearson=0.18; Spearman=0.22). Again, he correlaions for book-o-marke, earnings-o-price and sales growh are somewha weaker. Finally, he correlaions for marke 21

24 capializaion are large and negaive, confirming he conjecure ha he higher reurn volailiy of small firms arises from higher firm-specific cash flow volailiy. [Table 4 here] Table 5 invesigaes he abiliy of implied equiy duraion o forecas fuure sock reurn volailiy. We use he same measures of sock reurn volailiy as Table 4, bu he measures are now esimaed using weekly sock reurns in he wo years following he compuaion of implied equiy duraion. Insead of reporing correlaions, we repor regressions of our volailiy merics on implied equiy duraion. This approach allows us o include lagged values of he volailiy merics as compeing explanaory variables. For our esimaes of implied equiy duraion o be useful from a forecasing perspecive, hey mus have incremenal explanaory power over lagged values of he volailiy merics. Panel A of Table 5 provides evidence of he hypohesized posiive relaion beween implied equiy duraion and fuure sock reurn volailiy. Panels B and C confirm ha he posiive relaion exends o boh he sysemaic and firm-specific componens of reurn volailiy. Finally, we find ha he implied equiy duraion sill loads wih a significan posiive coefficien when we include lagged values of he respecive volailiy merics in he regressions. Thus, implied equiy duraion is incremenally useful in forecasing fuure sock reurn volailiy and is componens. [Table 5 here] In summary, we provide hree key findings concerning he relaion beween implied equiy duraion and sock reurn volailiy. Firs, we find srong evidence of he hypohesized posiive relaion beween implied equiy duraion and sock reurn volailiy. Second, we show ha associaions of book-o-marke and earnings-o-price wih sock reurn volailiy is consisen 22

25 wih hese variables serving as noisy proxies for duraion. Finally, we show ha implied equiy duraion is incremenally useful over pas sock reurn volailiy in forecasing fuure sock reurn volailiy Common Facor Resuls In our nex se of ess, we examine wheher duraion represens a significan common facor in sock reurns. In secion 1.3, we derived he following cross-secional relaion beween monhly holding period reurns and duraion (see equaion 19): h i = D (1 + r) i α + γ + ε i The coefficien γ from hese monhly cross-secional regressions provides an esimae of he change in expeced reurn ( r) for monh. We predic ha γ, our esimae of ( r), will be negaively correlaed wih he excess monhly marke reurn. Empirical esimaion of his regression is subjec o several specificaion issues. Firs, he relaion is only approximae and no valid for large values of r (he convexiy propery described in foonoe 2). This should no creae a serious problem, since our esimaion uses monhly daa, and monhly changes in expeced reurn are unlikely o be large enough o creae serious violaions of he lineariy assumpion. Second sock reurns are also deermined by cash flow shocks. This omied variable has he poenial o bias our γ esimaes if cash flow shocks are correlaed wih expeced reurn shocks. Third, here is an errors-in-variables problem arising from our use of empirical esimaes for duraion (D) and expeced reurns (r). This problem will cause he inercep in he regression o be posiive and he slope o be biased oward zero, hus undersaing he magniude 23

26 of he esimaed changes in expeced reurns. We have no a priori reasons o expec ha any of hese specificaion issues will bias our empirical ess in favor of our predicions. Panel A of Table 6 and Figure 1A repor he disribuional properies of our esimaes of change in expeced reurn on equiies ( r). The r esimaes range from a low of 0.82% o a high of 1.51%. The low of 0.82% occurred in Ocober of 1969, a monh in which he marke rose by over 5%. The high of 1.51% occurred in June of 1970, a monh when he marke fell by over 11%. During he bes monh for he marke in our sample period (Ocober 1974), he marke rose by over 16% and r was less han 0.5%. Conversely, during he wors monh for he marke in our sample period (Ocober 1987), he marke fell by over 22% and r exceeded 0.5%. Thus, our analysis suggess ha our lower bound esimaes of r exhibi subsanial emporal variaion. Moreover, significan shocks o expeced reurns are associaed wih significan shocks o holding period reurns of he opposie sign, consisen wih he predicions of basic valuaion heory. Figure 1A indicaes ha he disribuion of r is righ-skewed (skewness=0.54) and highly lepokuric (kurosis=5.04). I is well known ha monhly marke reurns are lef skewed and lepokuric. Our resuls sugges ha hese properies in reurns can be aribued, a leas in par, o relaed properies in he disribuion of shocks o expeced reurns. [Table 6 and Figure 1 here] P4 and P5 predic a negaive correlaion beween r and boh he marke reurn and he excess long bond reurn. Panel B of Table 6 repors hese correlaions. For he marke reurn, boh he Pearson and Spearman correlaions are srongly negaive (-0.45 and 0.45 respecively) and suppor P4. Visual confirmaion of he negaive correlaion beween our esimaes of r and he marke reurn are provided in Figures 1B (monhly realizaions) and 1C (12-monh moving 24

27 averages). For he long bond reurn, however, he correlaions are negaive bu are no saisically significan. This laer resul is somewha puzzling. One explanaion for he resul is ha shocks o he risk-free componen of expeced equiy reurns are exremely small relaive o shocks o he equiy premium. However, he relaively srong correlaion beween he marke reurn and he long-bond reurn is difficul o reconcile wih his explanaion. Alernaively, shocks o he risk-free rae may be correlaed wih shocks o shor-erm cash flows ha are greaer for shor duraion equiies and hence confound he repored correlaions. We nex es predicions P6 and P7. For hese ess we form a duraion mimicking porfolio (HDMLD) by aking he difference beween he reurns on socks wih high duraion and he reurns on socks wih low duraion each monh in exacly he same manner as Fama and French (1993) use o calculae heir book-o-marke mimicking porfolio (HML). We also creae a size facor (SMB) and a book-o-marke facor (HML) using he exac procedures described in Fama and French (1993). 6 Panel B of Table 6 compares he correlaion of HDMLD wih he marke reurn and he excess bond reurn. Consisen wih P6, HDMLD has a srong posiive correlaion wih he marke reurn. This correlaion is sronger han eiher HML or SMB. However, r has he sronges correlaion wih he marke reurn. This is consisen wih r represening our mos efficien esimae of he common facor in reurns relaed o duraion. As would be expeced, r is highly negaively correlaed wih our mimicking porfolio for duraion (Pearson=-0.73 and Spearman=-0.72) and posiively correlaed wih he mimicking porfolio for book-o-marke (Pearson=0.57, Spearman=0.57). 25

28 Panel C of able 6 provides ess of P7. Model 1 indicaes ha 14 percen of he variaion in excess monhly marke reurns is explained by he SMB and HML mimicking facors. In models 2 and 3 we add duraion-relaed facors ( r and HDMLD). The resuls indicae ha our duraion facors subsume he explanaory power of HML. Boh r and HDMLD load wih a significan coefficien and he coefficien on HML falls close o zero and is no longer saisically significan. This is consisen wih P7. The R 2 is highes (24 percen) when r is used as he duraion facor (Model 2). This resul is comforing, because he r esimaes are derived from he underlying heoreical relaion beween duraion and reurns raher han an ad hoc mimicking facor. Overall, hese resuls are consisen wih book-o-marke and is associaed HML facor serving as noisy proxies for equiy duraion-relaed effecs in sock reurns. Our refined proxies for duraion lead o significan improvemens in explanaory power. 4. Conclusions In his paper, we develop an expression for implied equiy duraion and provide a simple algorihm for he is empirical esimaion. We show ha he sandard empirical predicions and resuls for bond duraion hold for our measure of equiy duraion and ha equiy duraion represens an imporan common facor in sock reurns. We documen ha sock reurn volailiy and sock beas are boh increasing in implied equiy duraion. We also show how empirical esimaes of equiy duraion can be used o impue he common shocks o he expeced equiy reurns. Our resuls sugges ha he book-o-marke raio provides a crude proxy for equiy duraion, and ha he Fama and French (1993) book-o-marke facor can be inerpreed as a noisy duraion facor. Fama and French 1995 presen loose argumens o he effec ha heir 26

29 book-o-marke facor capures a financial disress facor. We presen a igher se of argumens and empirical resuls indicaing ha a duraion-relaed facor represens a more naural explanaion. We also acknowledge ha our cash flow forecasing model is crude. Improvemens in he forecasing model should lead o improved measures of equiy duraion and more refined esimaes of expeced reurn shocks. Finally, our measure of implied equiy duraion provides a naural and defensible ranking of socks syle characerisics on he value/growh dimension ha is popular among praciioners. Currenly, index providers such as Sandard and Poor s, Dow Jones and Russell compee o provide he bes indices of value and growh socks. 7 Ye heir growh and value classificaions are based on ad hoc reasoning and daa-moivaed saisical procedures. By combining informaion abou expeced growh, expeced profiabiliy and curren sock price ino a single and rigorously developed measure, implied equiy duraion provides an aracive alernaive o he ad hoc measures of value and growh proposed by praciioners. 27

30

31 REFERENCES Amihud, Y. and H. Mendelson, Asse Pricing and he Bid-Ask Spread, Journal of Financial Economics. Vol 17, Campbell, J. and J. Mei, Where Do Bea Come From? Asse Price Dynamics and he Sources of Sysemaic Risk, The Review of Financial Sudies. Vol 6, Campbell, J. and R. Shiller, Sock Prices, Earnings and Expeced Dividends, Journal of Finance, Vol. 43, Cornell, B., Risk, Duraion, and Capial Budgeing: New Evidence on Some Old Quesions, Journal of Business. Vol 72, Fama, E.F. and K.R. French, The Cross-Secion of Expeced Sock Reurns, Journal of Finance 47, Fama, E.F. and K.R. French, Common Risk Facors in he Reurns on Socks and Bonds, Journal of Finance 33, Fama, E.F. and K.R. French, Size and Book-o-Marke Facors in Earnings and Reurns, Journal of Finance Gould, J. B., and E. H. Sorensen, A Facor in Equiy Pricing, Journal of Porfolio Managemen; New York; Fall Ibboson Associaes.Socks, Bonds, Bills and Inflaion Yearbook. Nissim, D. and S. H. Penman, Raio Analysis and Equiy Valuaion: From Research o Pracice, Review of Accouning Sudies 6, Penman, S. H., An Evaluaion of Accouning Rae-of-Reurn, Journal of Accouning, Audiing and Finance, Vol 6. Spring Sigler, G.J Capial and Raes of Reurn in Manufacuring Indusries, Princeon Universiy Press, Princeon, NJ. 29

32 TABLE 1 Summary of Financial Variables and Forecasing Parameers Used in he Esimaion of Implied Equiy Duraion Panel A: Financial Variables Financial Variable Compusa Definiion Book Value of Equiy (BV) Daa Iem 60 Earnings (E) Daa Iem 18 = Income before exraordinary iems Sales (S) Daa Iem 12 Marke Capializaion Daa Iem 199 x Daa Iem 25 Panel B. Forecasing Parameers Forecasing Parameer Value Auocorrelaion Coefficien for Reurn on Equiy 0.57 Cos of Equiy Capial 0.12 Auocorrelaion Coefficien for Growh in Sales/Book Value 0.24 Long-Run Growh Rae in Sales/Book Value 0.06 The auocorrelaion coefficiens are based on pooled auoregressions for Reurn on Equiy and Sales Growh using a sample of 139,404, observaions over Compusa years 1950 o The Cos of Equiy Capial and Long-Run Growh Raes are based on heir long-run hisorical averages. 30

33 TABLE 2 Panel A: The Compuaion of Implied Equiy Duraion for Alaska Air Group and Amazon.com for 1999 Calculaion of Implied Equiy Duraion for Alaska Air in 1999 Inpu daa ($millions, excep percenages) Forecasing Parameers Price (P 0 ) Auocorr. Coeff. for ROE 57% Lagged Book Value (B -1 ) Cos of equiy capial (r) 12% Book Value (B 0 ) Auocorr. Coeff. for Growh 24% Growh rae (S 0 -S -1 )/S % Long-Run Growh Rae 6% Earnings (E 0 ) Forecas Model Time Period () Growh Rae 9.70% 6.89% 6.21% 6.05% 6.01% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% ROE (E /B -1 ) 17.00% 14.85% 13.62% 12.93% 12.53% 12.30% 12.17% 12.10% 12.06% 12.03% 12.02% BV , , , , , , , , , E =B -1 *ROE CF =B -1 +E -BV PV(CF ) *PV(CF ) Σ(PV(CF )) Terminal PV Σ(*PV(CF )) 2, Year Duraion 4.80 Terminal Duraion Year Weigh 0.64 Terminal Weigh 0.36 Implied Equiy Duraion years Earnings-o-Price Approximaion 3.03 Years Book-o-Marke Approximaion 5.76 Years 31

34 TABLE 2 - coninued Panel B: The Compuaion of Implied Equiy Duraion for Alaska Air Group and Amazon.com for 1999 Calculaion of Implied Equiy Duraion for Amazon.com in 1999 Inpu daa ($millions, excep percenages) Forecasing Parameers Price (P 0 ) 8, Auocorr. Coeff. for ROE 57% Lagged Book Value (B -1 ) Cos of equiy capial (r) 12% Book Value (B 0 ) Auocorr. Coeff. for Growh 24% Growh rae (S 0 -S -1 )/S % Long-Run Growh Rae 6% Earnings (E 0 ) Forecas Model Time Period () Growh Rae % 45.10% 15.38% 8.25% 6.54% 6.13% 6.03% 6.01% 6.00% 6.00% 6.00% ROE (E /B -1 ) % % % % % % -6.21% 1.62% 6.08% 8.63% 10.08% BV E =B -1 *ROE (719.97) (773.84) (620.07) (384.80) (212.54) (102.54) (33.87) CF =B -1 +E -BV (893.92) (679.50) (421.59) (244.10) (134.06) (66.78) (25.38) PV(CF ) (798.14) (541.69) (300.08) (155.13) (76.07) (33.83) (11.48) *PV(CF ) (798.14) 1,083.39) (900.24) (620.52) (380.33) (203.01) (80.35) Σ(PV(CF )) -1, Terminal PV 10,806 Σ(*PV(CF )) -3, Year Duraion 2.06 Terminal Duraion Year Weigh (0.21) Terminal Weigh 1.21 Implied Equiy Duraion Earnings-o-Price Approximaion Book-o-Marke Approximaion years years years 32

35 TABLE 3 Descripive Saisics for Esimaes of Implied Equiy Duraion (Duraion) and Oher Relaed Equiy Securiy Characerisics Panel A: Univariae Saisics Obs Mean Sd. Min. Lower Median Upper Max. Dev. Quarile Quarile Duraion Book-o-Marke Earnings-o-Price Sales Growh Marke Cap Panel B: Correlaions (Pearson above he diagonal, Spearman below he diagonal) Duraion Book-o- Marke Earnings-o- Price Sales Growh Marke Cap. Duraion Book-o-Marke Earnings-o-Price Sales Growh Marke Cap See Table 2 for he calculaion of duraion for fiscal year. Book-o-Marke is calculaed as book value of equiy divided by he marke value of equiy measured a he end of fiscal-year. Earnings-o-Price is earnings divided by he marke value of equiy measured a he end of fiscal-year. Sales Growh is calculaed as (Sales Sales -1 ) /Sales -1, where is he curren fiscal year. Marke Capializaion (Marke Cap.) is he marke value of equiy measured a he end of fiscal-year. 33