The Price of Gold: A Global Required Yield Theory

Transcription

1 The Price of Gold: A Global Required Yield Theory Chrisophe Faugère Assisan Professor of Finance School of Business, Universiy a Albany Albany, NY 2222 (58) and Julian Van Erlach CEO, Nexxus Wealh Technologies Inc. 7-5 Hun Ridge Dr. Clifon Park, NY 2065 (58) Forhcoming in The Journal of Invesing, Spring May, 2004 Corresponding auhor: Dr. Chrisophe Faugère, Assisan Professor, Deparmen of Finance and Research Associae a he Cener for Insiuional Invesmen Managemen, School of Business, Universiy a Albany, Albany, NY Phone: c.faugere@albany.edu.

2 The Price of Gold: A Global Required Yield Theory Absrac We consruc a gold valuaion heory based on viewing gold as a global real sore of wealh. We show ha he real price of gold varies inversely o he sock marke P/E and hus is a direc funcion of a global yield required o achieve a consan real afer-ax reurn equal o long-erm global real GDP per-capia growh. We inroduce a new exchange rae pariy rule based on he equalizaion of inverse sock marke P/Es (required yields) across naions. Foreign exchange affecs he price of gold o he exen ha required yields and Purchasing Power Pariy equalizaions do no ake place across naions in he shor run. A quarerly valuaion model is consruced using concurren economic daa ha is wihin 2% mean percenage racking error from real U.S. gold prices from Several major world evens have had a large bu fleeing impac on gold prices. Keywords: Gold Price, Sock Marke, Required yield, Forward Earnings yield, Foreign Exchange, P/E, Price-Earnings Raio.

3 Assessing he fair value of gold largely remains a mysery in Finance. While in some insances he exising lieraure has found empirical relaionships beween gold prices and macroeconomic variables such as inflaion and exchange raes, lile evidence has been offered for connecions beween gold and oher asse classes. To dae, here is no comprehensive heory of gold valuaion showing how inflaion, exchange raes and oher asse classes may ogeher affec gold pricing; or how gold and oher asse classes may be affeced by common underlying facors. In his paper, we offer a gold asse pricing heory ha reas gold as a sore of wealh. We demonsrae a heoreical and empirical link beween gold price, inflaion, and foreign exchange raes and he general valuaion of he sock marke. Our approach is based on a generalizaion of Required Yield Theory (Faugere-Van Erlach [2003]). Required Yield Theory explains he valuaion of financial asses via invesors general requiremen o earn a minimum expeced aferax real reurn equal o long-erm GDP/capia growh. We hold ha since gold fulfills he unique funcion of a global sore of value, is yield mus vary inversely o he yield required by any financial asse class, hus providing a hedge in he case where such asses are losing value. Our heory explains abou 88% of acual $USD gold prices and 92% of acual gold reurns on a quarerly basis, including he peak prices of gold, over he period. The exan lieraure has well documened empirical relaionships beween gold price and global macroeconomic variables such as inflaion and currency exchange raes. For example, Sjaasad and Scacciavillani [996] show ha afer excluding he sharp rise in gold prices in he early 980 s, abou half of he variance in $USD gold prices during he period appears o be accouned for by flucuaion in real exchange raes. Ghosh, Levin e al. [2002] find ha gold is mosly an inflaion hedge in he long run. They furher aemp o jusify shor-erm gold 3

4 price volailiy by appealing for example o changes in he real ineres rae and $USD vs. res of he world exchange raes flucuaions. On he oher hand, he empirical record weighs heavily on he side ha gold pricing apparenly is relaed neiher o GDP growh nor o oher asse classes. Lawrence [2003] concludes ha here is no saisically significan correlaion beween real reurns on gold and changes in macroeconomic variables such as GDP, inflaion and ineres raes, and ha he reurn on gold is less correlaed wih reurns on equiy and bond indices han are he reurns of oher commodiies. Sanding in conras o he above findings, Coyne [976] focuses primarily on gold as a hedging insrumen and finds ha for periods in which he gold marke was free o flucuae, gold ended o move in a direcion opposie o he price of oher financial asses. Sherman [983] makes a noed heoreical aemp a demysifying he pricing of gold. He uses a linear regression model o esimae elasiciies of demand for gold. The key explanaory facors are exchange raes and unanicipaed inflaion proxies. While several useful relaionships are sudied, hese relaionships are assumed a-priori and no heoreically derived. Barsky and Summers [988] focus on he Gold Sandard period and develop a general gold valuaion model ha views gold as a non-moneary durable good providing a sream of consumpion services over ime, like Jewelry or objecs of ar. They heoreically show a relaionship beween he inverse of he log of gold price and he real ineres rae, which seems o hold empirically over he period In heir model, gold is a non-depreciable asse earning a yield equal o a governmen bond yield. However, by rooing heir model in he Gold Sandard era, and exending heir approach o he curren era, hey are no addressing he naure of Gold as a sore of value, ha is, a hedging insrumen agains inflaion and he collapse of he value of oher asse classes. In his paper, on he oher hand, we underake he analysis of gold along his exac line. 4

5 GOLD MARKET HIGHLIGHTS AND GOLD S EMPIRICAL LINK TO STOCKS The oal aboveground value of gold in he world is currenly around $.9 rillion a $380/Troy oz. ($380/Toz. x 32,50.7 Toz./Meric on x 55,000 Mons) compared wih he approximaely $5 rillion value of he US sock marke and $22.4 rillion for US non-financial deb ousanding. Gold mining is a $3 billion per year indusry wih gold prices a $380/Toz. Given ha he price volailiy of gold is around 0% per year, i is easy o see why producion companies heavily engage in hedging heir fuure producion. Ibboson, Siegel and Love [985] esimaed ha gold bullion represened 5% of oal invesable world wealh. Today, oal world gold bullion represens 5.% of he combined US sock and bond capializaion of $37.4 rillion; and a hus a much smaller proporion of oal world wealh han he Ibboson e al. sudy. The rae of growh of gold exracion has essenially mached world populaion growh over he pas 30 years. IMF daa show ha he more developed naions populaion grew a compounded.45% from , while oal world populaion grew a compounded.89%. The global accumulaed sock of gold grew from an esimaed 98,000 ons in 974 o 45,000 ons by mid-200; implying a.46% growh rae. Thus, he world sock of gold per-capia has remained relaively sable over his period. A preliminary empirical invesigaion of he price of gold reveals a non-rivial connecion beween real gold prices and he US sock marke. Figure below shows ha gold s real price varies inversely o he S&P 500 P/E, and hus wih he earnings-o-price raio. Figure shows he high correlaion beween indexed USD real gold prices, inverse S&P 500 forward P/E raio and 0-year T-Bond, over he period

6 FIGURE abou here The heory we develop below predics and explains his high level of correlaion based on viewing gold as a global sore of value. REQUIRED YIELD THEORY APPLIED TO GOLD VALUATION Throughou he hisory of civilizaion, gold has been he single mos imporan global sore of value. To his day, i fulfills his unique funcion. For he purpose of exending Required Yield Theory o gold pricing, we posulae he following: ) The global real price of gold essenially is a real P/E raio for gold, where earnings represen purchasing power or a global price index. 2) The global real price of gold mus vary inversely o all oher main financial asse classes real P/E o preserve he real value of any invesor s capial agains adverse movemens in he values of financial asse classes. 2 3) Law of One Price: exchange rae flucuaions mus impac local currency-denominaed gold prices o eliminae poenial inernaional gold arbirages. 4) Mining supply mus be sable in relaion o supply movemens in he aboveground sock and he worldwide sock of gold per capia should no increase in he long run. Condiion ) recognizes ha even hough gold does no produce acual earnings, is primary purpose is o provide a sream of services by mainaining real purchasing power over ime. The same uni of gold can serve o purchase a represenaive baske of economic goods repeaedly. We define he forward P/E for gold as he price of gold divided by expeced nex period s GDP deflaor. I is easy o check ha he real price of gold is he same as he real forward gold P/E raio. 3 Condiion 2) insures ha gold behaves as a sore of value, ha is: capial flows o gold are dicaed by changes in he minimum expeced reurn achievable by oher asse classes. I is imporan o emphasize ha gold per-se does no require he same yield as oher asses, as i 6

7 sands ouside of he convenional realm of invesmen goals, and acs mosly as a global hedging ool agains financial downurns, and inflaion. Hence, our heory posulaes ha movemens in he global real price occur because of he precauionary demand for gold, which largely depends on changes in he inverse real P/E (or required yield) of oher asses classes combined. A consequence of his posulae is ha a decline in he value of he sock marke index does no necessarily enail fligh o gold when, for example, expeced sock earnings are also falling o mainain a consan real P/E raio. On he oher hand, fligh o gold will happen when sock marke prices are dropping faser han expeced earnings due o acceleraion of inflaion for example. In addiion, since gold is a global homogenous durable commodiy is price mus be equalized across counries afer currency conversion, which is saed in condiion 3). Finally, condiion 4) saes ha he supply of gold mus be sable so ha invesors precauionary moive is fulfilled wihou major price movemens driven by supply shocks. Indeed his condiion seems o be characerisic of he precious meal mining indusry. Laer on, we provide a formal argumen ha shows ha his mus be he case under our heory. A Model for he Global Real Price of Gold We assume ha he main alernae invesmen asse class is a sock marke index. Since gold is a global hedging ool, Condiion 2) above saes ha he global real price of gold mus vary inversely wih he global sock marke forward P/E. Our gold valuaion heory is consruced by making use of his posulae in addiion o connecing a global sock marke earnings yield o a global nominal reurn. In order o esablish his connecion, we inroduce he concep of required yield from Faugere-Van Erlach [2003]. In ha paper, he auhors heoreically show ha a any 7

8 poin in ime, a counry s afer-ax sock marke forward earnings yield can be viewed as a minimum expeced reurn. Furhermore, his forward yield mus equal a required yield given by he sum of he counry s GDP/capia long-erm growh rae and is curren expeced inflaion rae. 4 Faugere-Van Erlach [2003] erm his approach Required Yield Theory (RYT). Since his resul is applicable o any counry, i mus also apply o a GDP-weighed average of any given se of counries earnings yields. 5 In oher words, when applied o a broad represenaive se of counries o approximae he global economy, a global sock index afer-ax forward earnings yield mus equal a global required yield ha is given by: RY π () w+ = g w + w+ Where g w sands for he global GDP/capia long-erm growh rae and π w+ is he expeced global inflaion rae, each variable being deermined as respecive GDP weighed averages of a represenaive se of counries long-erm growh raes and curren expeced inflaion raes. FIGURE 2 abou here We use a value of.5% for g w based on an esimae of he average long-erm growh rae for a group of OECD counries by Priche (997). Figure 2 above visibly shows a relaionship beween hese wo consrucs for a group of OECD counries when earnings yield raios (inverse P/Es) are weighed by heir relaive GDP weighs. 6 Furhermore, since Condiion 2) saes ha he real global price of gold mus be inversely relaed o he sock index real P/E, i hen mus co-vary direcly wih he global real required yield, as implied by Required Yield Theory (RYT). Le he global gold price be denoed by Pw 8

9 and le P ~ w represen gold s real global price, ha is he price of gold divided by he curren global price index. This implies ha on an afer-ax basis we mus have: 7 ~ P w RYw = C ( + π + w+ g w + π = C ) ( + π w+ w+ ) (2) So ha gold s real price equals a linear funcion of he afer-ax required yield, up o a muliplicaive consan C. In Appendix B, we show ha he consan C can be inerpreed as he real value of a perpeuiy ha would pay an ounce of gold every year forever, when inflaion expecaions and long-erm produciviy remain unchanged. According o expression (2) above, he real price of gold will always increase when global inflaion acceleraes and/or if he longerm global economic produciviy is raised. In oher words, he real price of gold is consan as long as inflaion and produciviy remain consan. In ha case, he nominal price of gold rises wih he general level of prices. This firs resul heoreically grounds he relaionship of he real price of gold wih inflaion, long-erm GDP growh, and he sock marke P/E (via he required yield). Figure 3 below, shows he relaionship beween he global real price of gold and he global required yield. FIGURE 3 abou here Figure 3 uses he same normalizaion based on he firs quarer of 998 as done before in Figure. Noe ha beween 97 and 979, following he Gold Sandard era, our model fails o accoun for he behavior of he global real gold price. In fac, up unil 973, he gold price was sill raded a he official quoaion and laer up unil 978, here were srong selling aciviies by cenral banks. On he oher hand, from 979 up unil he hird quarer of 2002 our heory maches 9

10 he acual real global price of gold very closely, even during he spike of 980. The mean percenage racking absolue error is 2% covering he period January 980 o April Local Gold Prices and Foreign Exchange Effecs In order o exend he above resul o local currency denominaed gold prices, we mus ake ino accoun he relaive imporance of each counry in he global economy as well as he effec of currency exchange raes. To simplify he analysis we divide he global economy ino wo blocs: he home counry and res of he world defined as he bloc of counries X. Le us denoe by θ he relaive nominal GDP weigh of he domesic economy in he global economy, and le us denoe by s~ he real exchange rae ( real uni of foreign baske equals s~ real uni of domesic baske). As shown in Appendix A, we can express he real domesic price of gold as a funcion of he global required yield as follows: C RY ~ w+ P = ( + π w+ ) θ + ( θ ) ~ s (3) Expression (3) enails ha he real domesic price of gold is an increasing funcion of global inflaion. I is an increasing funcion of he GDP weigh of he home counry as long as he inflaion rae abroad is relaively high and he foreign baske of currencies is weak. The lower he GDP weigh of he home counry is, he more dependen he real domesic gold price is on he foreign exchange rae and inflaion rae. In general, relaionship (3) shows ha as he domesic currency depreciaes, he real domesic gold price appreciaes ceeris-paribus. Figure 4 below shows he resul of our formula for he U.S. Figure 4 summarizes he quarerly relaionship beween he nominal gold prices vs. he fair value of gold using equaion 0

11 (3). Our mean percenage racking error is 2%. We use a value of $8929 for he consan C, which is obained as he inercep of he regression of he raio of acual real price over he righ hand side of equaion (3) on a ime rend, on a quarerly basis over The regression has an adjused R-square of 5%. 9 FIGURE 4 abou here Figure 4- below illusraes formula (3) when we assume ha global expeced inflaion mach U.S. expecaions. The mean percenage absolue error hen drops o 9.8%. We observe he following: firs i is crucial o noe ha he acual price of gold depars from our model, mosly for reasons ha have o do wih exraordinary world evens impacing gold prices ouside of he key facors we formulaed in expression (3). The Iran hosage crisis began on November 4, 979 and was he mos covered even by all media. This was shorly followed by he Hun silver crisis. Boh evens end o increase he perceived value of gold. During he year 980, gold peaked a $666/oz., only o converge o he fair value nex quarer. The average price during 980 was $584/oz. and was 6% above he values prediced by RYT. In 983 gold rose 22% above he RYT based value, perhaps moivaed by fears during he Mexican Deb Crisis; hen converged o RYT valued gold and sayed close o he acual price from 983:QIII 986:QII. Then hrough 988:QI gold was abou % above RYT value wih only minor changes in he required yield iself. FIGURE 4. abou here

12 From hen unil 999:QII gold and RYT value remained in exremely close in accord wih he foreign exchange raes remaining unchanged from poin o poin while he required yield declined; clearly driving gold valuaion. From 999:QIII 200:QIII, gold was below RYT value by a 5% gap. Since he required yield remained sable, while he exchange rae rose agains he dollars, his accouns for he low gold vs. RYT value relaionship. The foreign exchange rae has since declined and he gold/ryt value relaionship is again aligned while he required yield has remained sable, leading o an average gap of abou 3% unil 2002:QIII. In all cases, gold has converged o he RYT prediced value. Domesic Real Gold Price, Required Yield and a New Exchange Rae Pariy Rule Due o local currencies being legal enders, invesors seek o hedge agains adverse marke movemens in he value of domesic asses using domesic currency o buy gold. In his secion, we show ha under a se of reasonable circumsances, he value of gold in erms of domesic currency will only be affeced by domesic-currency denominaed asse values. Specifically, we show ha he domesic price of gold will solely be deermined by he domesic required yield when Purchasing Power Pariy (PPP) is saisfied in conjuncion wih a new pariy condiion. In fac, we inroduce here a new exchange rae deerminaion rule ha is based on he posulae of pariy of required yields across counries. Our saring poin is o define he global required yield in relaion o each bloc s required yield. We define his relaionship as follows: RY ( + π ) ( + π ) θ (4) w+ w+ W+ = RY + + ( θ ) RYX+ ( + π + ) ( + π X+ ) 2

13 The global required yield is a weighed average of each bloc s required yield, where he weighs are modified nominal GDP weighs in he sense ha hey depend on expeced price deflaors raher han curren deflaors. 0 Since he concep of required yield is ied o he concep of forward earnings yield, i is imporan o noe ha we are also able o pu forh a new exchange rae pariy condiion ha depends on comparing sock marke P/E raios across counries. Le us denoe by s he domesic spo exchange rae ( uni foreign baske equals domesic), he expeced following period s e s + spo exchange rae, and ~ e s+ he real expeced spo rae. Assuming ha here is no risk premium associaed wih invesing in foreign asses, we have: s RY RY (5) X+ = e + s+ Expression (5) is a new pariy rule based on required yields. This rule is derived by considering ha sock indexes are homogenous commodiies and consequenly ha index values and forward earnings mus equae across naions, afer currency exchange. While Required Yield pariy and PPP (as i applies o general price indexes) are direcionally consisen, he magniude of exchange rae flucuaions implied by his new rule is far greaer han ha of PPP especially over he shor-erm. These wo rules will have similar effecs only if inflaion is sable and low in boh he domesic counry and res of he world. Using expressions (A), (3), (4) and (5), afer several manipulaions we ge: ~ s θ + ( θ ) e ~ C RY ~ + s+ P = ( + π + ) θ + ( θ ) ~ s (6) Relaionship (6) gives a comprehensive accoun of he key facors ha affec domesic real gold prices: domesic GDP/capia growh, domesic inflaion, exchange raes and relaive GDP 3

14 weigh. Thus, he domesic price is now a direc increasing funcion of he domesic required yield and domesic inflaion. The real domesic price rises when he domesic currency depreciaes, or is expeced o appreciae in he fuure. I increases as well when he home counry s GDP weigh increases, as long as boh is weigh is less han 50% and he domesic currency is expeced o appreciae srongly. The second raio on he RHS of expression (6) may in general be close o one. This is rue, for example, when he home counry GDP weigh is large. This is also rue when PPP is close o being saisfied. In he case where PPP is fully saisfied a any poin in ime, hen he second raio on he RHS does equal one, and expression (6) simply becomes: ~ P C RY + = ( + π ) + (7) This las equaion (7) embodies he fac ha when financial markes are well inegraed, hen he real domesic price of gold responds solely o domesic required yield and relaed asse value movemens. Ineresingly, in ha case he real price of gold is independen of he weigh of he domesic economy in he global economy. I is imporan o keep in mind ha his propery is rue when i is assumed ha exchange raes fully inegrae gold and sock marke valuaion differenials. Alhough no as realisic, he oher case when his resul is rue as well is a diamerically opposie siuaion where he domesic economy would be isolaed or a one-world economy. In ha case, he real domesic price of gold and he domesic required yield are independen from he foreign price of gold and required yield, and hus he resul in (7) follows direcly from equaions (2), (4) and (A). 4

15 The Local Gold Reurn In his secion, we focus on modeling gold s relaive price change, or reurn. Firs, noe ha he gold price in equaion (3) depends on he value for he consan C. A poenial hurdle is ha he value of C iself is no deermined by an ouside relaionship. In fac, as previously menioned, we use equaion (3) and he acual gold price hisory o deermine is empirical value, which can lead o some minor uncerainy in our esimae of he absolue price of gold. The advanage of deermining gold s reurn over ime is ha we avoid he problem of esimaing his consan. Le us denoe by π u + 2 he uncondiional expeced inflaion rae for he domesic counry, u based on he informaion se available a ime, and θ + is he GDP weigh based on ha same inflaion expecaion. 2 A relaionship abou he behavior of he expeced real gold reurn in he domesic counry is obained by aking he raio of wo consecuive periods using equaion (3), he following way: ~ E( P ~ ) + P = RY RY u w+ 2 w+ ( + π ( + π w+ u w+ 2 θ + ( θ ) ) ~ s u u ) θ + + ( θ+ ) ~ s e + (8) Figure 5 below illusraes his resul using U.S. daa. Figure 5 summarizes he quarerly relaionship beween he change in he real reurn on gold as prediced by RYT from equaion (8) and acual gold real reurns. We use he spo and ex-pos fuure spo raes for he $USD agains he baske of foreign currencies. 3 Once he required yield and foreign exchange effecs are aken ino accoun, he RYT valuaion model exhibis an 8% mean percenage racking error from acual real gold reurns over he period. 4 FIGURE 5 abou here 5

16 The cos Approach o Pricing Gold We now urn o explain he absolue price of gold bullion via he average producion cos. We assume ha he gold mining indusry as a whole has achieved is long-run maximum efficien capaciy, and since he indusry has large barriers o enry, he real profi margin is no shrinking o zero, bu raher converges o is long-run achievable profi given by he long-run global GDP/capia growh rae of.5%. The reason why he long-run GDP/capia growh rae deermines he indusry s profi margin is ha he long run average indusry margin mus be equal o he long-erm average corporae profi margin 5 oherwise he quaniy supplied would change o bring his relaionship ino line. If he margin were greaer, mining aciviy would pick up, and vice versa. Le ATCw sands for he global average oal cos per ounce, and τ w is he corporae ax rae in he global economy. In Appendix C, we show ha he following relaionship mus hold for he real world price of gold: P w ATCw = RYw ( τ + w+ ) (9) The pricing formula (9) saes ha he price is relaed o he average cos and required yield via a cos-plus-margin relaionship. Formula (9) is compared o acual nominal gold price for he U.S. Figure 6 below illusraes he comparison. The cos-based esimae of he gold price is ploed agains he acual gold price in Figure 6. Our mean percenage racking absolue error is 6%. Clearly, he cos-based model does no accoun for changes in required yield from he invesor s perspecive and does no generae he same volailiy as our he firs model did based on formulas (2) and (3). 6

17 FIGURE 6 abou here Precauionary Moives and Gibson s Paradox Barsky and Summers [988] provide a direc compeing heory o he model developed in his aricle. Conrary o heir resul, our model does no resolve Gibson s paradox. Gibson s paradox is he puzzling observaion during he Gold Sandard era ha ineres raes were co-varying wih he general level of prices bu no he inflaion rae as sandard economic heory would predic (Fisher [930]). Barsky and Summers argue ha he paradox ook place because in effec, he Gold Sandard links he real gold price inversely o he general price index. They heoreically prove ha he gold price is iself inversely relaed o he real reurn. Thus he price index and real reurn are posiively relaed. The reason our framework is unable o ackle he Gibson paradox phenomenon is ha during he Gold Sandard, ouside of periods of severe inflaion, he fixed converibiliy of dollars ino gold made i unnecessary for invesors o hold gold as a precauionary moive. On he oher hand, i mus be remembered ha high inflaion raes a he beginning of he 970s were responsible for he large depleion of he sock of US governmen gold reserves and he evenual collapse of he Gold Sandard. In Barsky and Summers he consumpion moive is sufficien o explain he paradox. However, even hough heir model seems o explain he movemen of gold prices over he period jus as we do, his does no refue our approach and conclusions. Recall ha our heory saes ha he real price of gold moves in conjuncion wih he expeced inflaion rae and he real reurn as measured by long-erm produciviy. Over he period considered, nominal raes on bonds were relaively fla, hus flucuaions in expeced inflaion were inversely affecing real reurns. Consequenly, Barsky and Summers do documen an inverse relaionship beween real 7

18 reurn and real price of gold, which is he same as a posiive relaionship beween he real price of gold and inflaion in our case. RYT saes ha raional invesors seek a real posiive consan reurn. Even hough his did no ake place on he bond marke a he ime mosly due o lags in expeced inflaion, RYT makes provision for his requiremen by saing ha invesors would hen urn o he equiy marke o fulfill ha condiion. On he oher hand, one of Barsky and Summers key conclusions is ha he real price of gold does no change when he real ineres rae remains consan. In our case, we predic ha even small movemens in inflaion raes will affec he real price of gold even when he real rae is unaffeced. Ulimaely, Barsky and Summers [988] model breaks down afer 995, as he record shows ha he real reurn on he 30-year reasury was relaively consan over and fell hereafer while he price of gold seadily dropped from , which we explain by a slowly declining inflaion rae. IMPLICATIONS FOR THE GOLD MARKET AND ITS VALUE In he long run, he gold mining indusry s real profi margin is consan and equals he real per capia produciviy. The price of gold, on average, mus be he average producion cos plus a consan mark-up. Furhermore, in order for he real value of gold o be mainained on a per invesor basis, he sock of gold has o grow a a rae ha can be no greaer han populaion growh in he long-erm. If he supply of gold grew a a lesser rae han populaion growh for reasons oher han depleion of he exhausible ore, gold price would grow faser han inflaion and he quaniy demanded for gold would drop. Evenually he supply of mined gold will dwindle, which will drive prices up unless world populaion experiences zero growh in he 8

19 foreseeable fuure. In ha circumsance, far off in he fuure, a subsiue medium of soring value may be discovered and used. Anoher predicion of our heory of gold pricing is ha he decrease in proporion of gold oal value as compared o world wealh is explained by RYT in he fac ha relaive o financial asses, he long-erm nominal value of gold mus increase a he inflaion rae, whereas he value of oher asses rise wih inflaion plus real produciviy. Thus, he proporion of invesable wealh declines a an annual rae equal o real per share earnings growh or GDP/capia growh. CONCLUSIONS We have exended he Required Yield Theory (RYT) developed by Faugere-Van Erlach [2003] o value gold and o deermine is reurn. RYT saes ha since global asses are priced o yield a global consan real reurn, and since gold is a global sore of value, is price will vary direcly wih he global required yield and he global inflaion rae. In he course of developing his asse valuaion model we inroduced a new exchange rule pariy based on required yields comparisons across counries. Specific predicions include: ) he real price of gold varies proporionaely o he change in long-erm economic produciviy as measured by GDP/capia growh. 2) Real gold prices vary proporionaely o changes in he foreign exchange rae (direc quoaion) when he domesic required yield is consan. 3) When he foreign exchange rae is consan and here are no major geopoliical or naural crises, real domesic gold price increases wih domesic inflaion. 4) When our new exchange rae pariy rule holds, hen effecively he real domesic price of gold is mosly deermined by he domesic required yield. This enails ha foreign exchange effecs will impac he domesic real gold price o he exen ha equalizaion of required yields is no aking place worldwide and/or ha PPP is violaed as well. 5) In he long-erm, he gold per-capia supply 9

20 remains consan. 6) The average long-erm absolue price of gold is marked-up cos where he profi margin is given by he global average long-erm per-capia rae of GDP growh. While we suspec ha cenral bank aciviies, hedging aciviies, supply/demand flucuaions, global real GDP growh changes or changes in global income and capial gains ax raes, affec gold prices as well, he valuaion approach developed here performs very well absen hese facors, wih over 92% accuracy in predicing US Gold reurns over a 23-year period. We leave an invesigaion of he role of hese oher facors for fuure research. 20

21 REFERENCES Barsky, Rober B. and Lawrence H. Summers, Gibson s Paradox and he Gold Sandard, Journal of Poliical Economy, vol. 96 (3) (988), pp Coyne, Herber J., Gold Use as a Hedge Preserves Values, Honors Rule of Prudence, The Money Manager, 976, New York. Faugere, Chrisophe and Julian Van Erlach, A General Theory of Sock Marke Valuaion and Reurn, Working Paper, June 2003, Universiy a Albany. Fisher, Irving, The Theory of Ineres, 930, New York McMillan. Ghosh, Dipak, Eric J. Levin, Peer Mc Millan and Rober E. Wrigh, Gold as an Inflaion Hedge?" Working Paper, 2002, Universiy of Sirling. Ibboson, Roger, Laurence Siegel and Kahryn Love, World Wealh: U.S. and Foreign Marke Values and Reurns," Journal of Porfolio Managemen, Fall 985, pp Lawrence, Colin, Why is Gold Differen from oher Asses? An Empirical Invesigaion, Working Paper, 2003, World Gold Council. Priche, Lan, 997, Divergence, Big Time, Journal of Economic Perspecives, vol. (997), pp Sherman, Eugene J., A Gold Pricing Model Journal of Porfolio Managemen, vol. 9 (3) (983), pp Sjaasad, Larry A. and Fabio Scacciavillani, The Price of Gold and he Exchange Rae, Journal of Inernaional Money and Finance, vol. 5 (6) (996), pp

22 Figure : Gold and Yields Indexes. U.S. Comparisons Quarerly Observaions yr T-Bond Yield SP 500 Fwd EY Real Gold Price N-78 D-79 J-8 F-82 M-83 A-84 J-85 J-86 A-87 S-88 O-89 N-90 D-9 J-93 M-94 A-95 M-96 J-97 J-98 A-99 O-00 N-0 D-02 J-04 Time 22

23 Figure 2: Annual Global Sock Marke Earnings Yield vs. Global Required Yield % 6.00% 5.50% 5.00% Global E/P Global RY 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% Year 23

24 Figure 3: Acual vs. RYT-Based Global Real Price of Gold Quarerly Observaions RYT based real gold price Global real gold price Nov-70 Feb-73 Apr-75 Jun-77 Sep-79 Nov-8 Jan-84 Mar-86 Jun-88 Aug-90 Oc-92 Jan-95 Mar-97 May-99 Jul-0 Oc-03 Time 24

25 Figure 4: Nominal USD$ Gold Price vs. RYT Pricing Formula Quarerly Observaions GOLD RYT based May-79 Jan-8 Sep-82 Apr-84 Dec-85 Aug-87 Apr-89 Nov-90 Jul-92 Mar-94 Oc-95 Jun-97 Feb-99 Oc-00 May-02 25

26 Figure 4-: Nominal USD$ Gold Price vs. RYT Pricing Formula Quarerly Observaions. U.S. inflaion expecaions May-79 Jan-8 Sep-82 Apr-84 Dec-85 Aug-87 Apr-89 Nov-90 Jul-92 Mar-94 Oc-95 Jun-97 Feb-99 Oc-00 May-02 GOLD RYT based 26

27 Figure 5: Gold Reurn: Acual vs. RYT based Quarerly Observaions 45% 40% 35% 30% 25% RYT formula Relaive Real Price 20% 5% 0% 5% 0% -5% -0% -5% -20% -25% May-79 Jan-8 Sep-82 Apr-84 Dec-85 Aug-87 Apr-89 Nov-90 Jul-92 Mar-94 Oc-95 Jun-97 Feb-99 Oc-00 May-02 27

28 Figure 6: Nominal USD$ Global Gold Price vs. Cos-Based RYT Formula Quarerly Observaions May-79 Jan-8 Sep-82 Apr-84 Dec-85 Aug-87 Apr-89 Nov-90 Jul-92 Mar-94 Oc-95 Jun-97 Feb-99 Oc-00 May-02 GOLD Cos Based Esimae 28

29 APPENDIX A Le P ~ and P ~ X respecively denoe he domesic and res of he world real gold prices a ime ; i.e. deflaed by each respecive region s GDP deflaor. Le π + and π X+ respecively denoe he domesic and res of he world expeced inflaion raes. The global real price of gold is defined as follows: ~ P W ~ ~ = θ P + ( θ ) P (A) X The global real price of gold is a weighed average of home and res of he world real prices weighed by each bloc s nominal GDP relaive weigh in he global economy. Given he Law of One Price given in Condiion 3) we mus have: leads o: ~ P ~ P X = ~ (A2) s Obviously when Purchasing Power Pariy (PPP) holds s~ =, and combining (A) and (A2) ~ P X ~ ~ = P = P (A3) w On he oher hand, if we choose no o assume PPP, and afer combining equaions (2), (A) and (A2) ogeher, he real domesic price of gold becomes a funcion of he global required yield as such: C RY ~ w+ P = ( + π w+ ) θ + ( θ ) ~ s (A4) 29

30 APPENDIX B An ineresing inerpreaion for he meaning of he consan C in equaion (2) arises if we consider he following argumen. Imagine a perpeuiy ha pays he holder one ounce of gold every year forever. The expeced value of an ounce of gold is given by E( P w+ ). Le he general price deflaor be denoed as Def w. Le ilde variables denoe real variables. For example, P ~ w represens he price P w divided by he global GDP deflaor a ime. I is sraighforward o see ha he inverse of he required yield is a required forward P/E raio, and hus ha forward (gold) annuiy paymens E( P w+ ) imes a forward P/E equals he ~ value of he above-described perpeuiy. Using equaion (2) o express he real value V of his w perpeuiy we ge: ~ E( P ) P ~ ( + π ) E( P V ~ ~ w+ w+ w+ w = = E( Pw+ ) = C Defw e w+ RYw+ Pw ) (B) Equaion (B) saes ha he real value of his perpeuiy increases wih he appreciaion of he real price of gold. Whenever inflaion and he real long-erm produciviy are sable, ha value urns ou o be our consan C. In oher words, he real price of gold is deermined in relaion o he required P/E in order o conserve he value of his perpeuiy. Equaion (B) also allows us o deermine he consan long-run real price of gold by muliplying he consan C equal $8929 by he real required yield of.5% plus LT global inflaion a 4.2% divided by ( +4.2%) o ge us a 5.47% yield. Based on our sample we ge an esimae of he long-erm real global gold price of $488 per ounce. 30

31 APPENDIX C Consider global mining profis. By definiion, acual ex-pos earnings per ounce of gold are given by: [ P ATC ] ) e τ (C) w+ = w w ( w+ Where ATCw sands for he global average oal cos per ounce, and τ w is he corporae ax rae in he global economy. Relaionship (C) is jus he sandard definiion of corporae ne income applied o he mining indusry. Dividing boh sides by he price and rearranging erms, we obain: Π w τ + e = ) ( τ w+ ( w+ w+ ) P w = ATC P w w (C2) The variable Π w+ represens he ex-pos profi margin. Rearranging erms, we ge he global price of gold as: P w ATCw = Π w+ ( τ w+ ) (C3) We posulae ha RYT holds for he componens of he profi margin, so ha he profi margin mus equal he required yield: Π e w+ w+ = = RYw+ Pw (C4) Equaion (C4) incorporaes RYT in he sense ha he earnings yield (earnings over lagged price) is assumed equal o he required yield (ex-pos). Thus, finally combining (C3) and (C4) gives us: P w ATCw = RYw ( τ + w+ ) (C5) 3

32 APPENDIX D DATA DESCRIPTION Unless oherwise specified, our daa covers a 33-year period on a quarerly basis from 979 QIII unil QII Nominal and real (CPI-adjused) spo direc exchange raes are obained from he US deparmen of Agriculure and compiled from he Inernaional Financial Saisics of he Inernaional Moneary Fund and Financial Saisics of he Federal Reserve Board. These are used o deermine he global real and nominal dollars gold prices from The represenaive group of non-u.s. counries is chosen o be: Euro zone, UK, Ausralia and Japan. U.S. Inflaion is USGDP deflaor (quarerly/annualized) and USGDP deflaor one-year ahead forecass from he BEA available only since 979. Prior o 979, acual GDP inflaion raes are used insead of GDP deflaor forecass. Global expeced inflaion esimaes are consruced by backing ou counry-specific inflaion raes from he raio of nominal vs. real exchange rae period o period change. Then, we assume ha each counry s expeced inflaion rae occurs in he same proporion as he raio of acual inflaion raes o U.S. inflaion (CPI-based). The GDP weighed average expeced inflaion rae for he baske of foreign currencies is derived by muliplying each period-wise proporionaliy coefficien by he corresponding period U.S. expeced inflaion. Finally, he global inflaion rae is obained as a GDP weighed average of U.S. vs. non-u.s. expeced inflaion raes. Gold prices are London pm Fix available monhly since 97, quoed in each naional currency: Euro, $USD, Yen, Ausralian $, and Pound Serling. For each naional currency, he GDP weighs are 980 real GDP weighs applied from QIII 979 unil QIV 989. The 997 real GDP weighs are applied from QI 990 unil QIII The GDP weighs daa is from he CIA s Handbook of Inernaional Economic Saisics. Real GDP 32

33 weighs are used insead of nominal GDP weighs in he implemenaion of our pricing formulas, since hey do no widely differ from nominal GDP weighs for hese wo years. The average oal mining $USD cos is from Brook Hun, which compiles annual dollarsconvered cos informaion for a se of mining companies consiuing beween 75% o 9% of he oal wesern world producion. The weighs used are relaive producion weighs, which for he sake of our analysis, we assume proxy GDP weighs. Quarerly average oal coss are inerpolaed beween years. U.S. hisorical op marginal raes from he IRS websie, serve as proxies for he global corporae ax raes. ENDNOTES The auhors hank Brook Hun and Mark Fellows for heir assisance wih providing aggregae coss daa for he gold mining indusry. These indexes are consruced so ha heir value is normalized o a he beginning of he firs quarer of In his paper, we will focus solely on socks as he alernae asse class. Faugere and Van Erlach [2003] show ha T-Bonds yields are essenially commensurae wih he sock marke earnings yield, and hus subjec o a common yield requiremen. 3 To show his, ake he forward P/E raio and divide i by he raio of curren o expeced GDP deflaor. 4 The condiion is slighly differen in Faugere-Van Erlach [2003], since i sipulaes ha he forward earnings yield equals he greaer beween he required yield as shown and he afer-ax Treasury bond rae. Here we absrac from he reamen of T-bonds as anoher asse class. 5 GDP weighs are used in our model, since a global inflaion rae mus be given by an average of counries specific inflaion raes where he dominan economies in erms of GDP carry more weigh. 6 The P/E raios are obained from he World Federaion of Exchanges. Sock markes represened are from hireen EU exchanges, London, Ausralian, Tokyo and he SP500 index. Daa was only available since We assume ha gold s real price is relaed o he afer-individual-income-and-capial-gains-axes forward marke earnings yield. In ha conex, formula (2) is consisen wih he resul of Faugere-Van Erlach [2003]. 8 I is compued as he mean of he raio of he absolue difference of acual minus model-based price over he acual price. Since inflaion expecaions are no hisorically available for mos indusrialized counries, we derive esimaes using he mehod described in Appendix D. If we assume ha global inflaion expecaions are idenical o US expecaions, he mean percenage error drops o 3.5%. We should noe ha hese resuls are dependen of he choice of he base period for he normalizaion. We argue hough ha a small racking error will resul when he normalizaion is done for a base period characerized by normal marke condiions, i.e. in he absence of bubbles or marke crashes. Noe ha all daa descripion is conained in Appendix D. 9 The -saisics on he slope has a value of 2.5, signifying ha he slope is no very fla. On he oher hand, when using U.S. inflaion expecaions we ge an R-square of 0.006% and a -saisics for he slope equal o

34 0 This definiion is consisen wih he required yield being an expeced nominal ineres rae. Noe hough ha he only possible way expeced fuure spo raes may increase is if inflaion expecaions rise. In ha case, a counry may pass along he effecs of is own inflaion by inducing a rise in he foreign price of gold, as long as hese expecaions are no refleced o a full measure in he curren spo rae. 2 This is because he expecaion is deermined wo periods ahead uncondiional on nex period s informaion se abou he new sae of he economy. 3 See Appendix D for a descripion of he currency baske. 4 Again, if we are assuming ha global inflaion expecaions are well approximaed by U.S. expecaions his leads o a smaller mean percenage absolue error of 6%. 5 Furhermore, he long-erm average global profi margin mus be he same as he long-erm produciviy growh per capia oherwise reurns in excess of GDP could be earned on a compound basis, which is impossible. 34