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1 Economcs Alumn Workng Paper Seres An Economc Analyss of the Compettve Rsk-Return Paradgm Peter C. Dawson Workng Paper January Farfeld Way, Unt 1063 Storrs, CT Phone: (860) Fax: (860) Ths workng paper s ndexed on RePEc,

2 An Economc Analyss of the Compettve Rsk-Return Paradgm Ttle: An Economc Analyss of the Compettve Rsk-Return Paradgm * Author: Peter C. Dawson Author Contact Informaton: petercdawson@yahoo.com Date: January 27, 2013 Abstract: Key Words: JEL Codes: The tradtonal CAPM s wdely cted as an authortatve foundaton for usng a postve rsk-return relatonshp n practcal applcatons. Gven ts wde use, the tradtonal CAPM s a relevant theoretcal context to analyze a compettve asset market s equlbrum rsk-return relatonshp. An easly understandable Economcs market model s used, wth substantve detal and suffcent analytcal background to get readers up-tospeed, so that ts rsk-return analyss wll be accessble to a broad, mult-dscplned readershp. Its negatve rsk-return concluson s based on supplementng the CAPM wth two assumptons: (1.) The nvestor populaton explctly ncludes rsk-preferrng nvestors (2.) whose tradng actvty can domnate the asset market. An ncomplete nformaton assumpton, whch would contradct the compettve asset market assumpton, s not needed to obtan the negatve rsk-return result. A unversal concluson that nvestors are (or should expect to be) rewarded wth hgher returns for nvestng n hgher rsk assets s not supported. Ths s consstent wth Markowtz (2008), who concludes one should not nterpret the CAPM s postve, lnear relaton between expected beta rsk and expected return to ndcate CAPM nvestors are pad for bearng systematc rsk, and wth Sharpe (1991), whose mathematcal CAPM equaton shows the rsk-return relatonshp, n general, depends on socetal rsk tolerance. Fnance, fnancal economcs, captal asset prcng model, capm, asset prcng, nvestment theory, portfolo theory, theory of portfolo choce, portfolo selecton, wllam sharpe, harry markowtz, compettve asset market. D4 Market Structure and Prcng, G10 General Fnancal Markets, G11 Portfolo Choce; Investment Decsons, G12 Asset Prcng; Tradng Volume; Bond Interest Rates, G31 Captal Budgetng; Fxed Investment and Inventory Studes; Capacty, G32 Fnancng Polcy; Fnancal Rsk and Rsk Management; Captal and Ownershp Structure; Value of Frms, M20 Busness Economcs. * Ths paper s developed from Chapters 14, 15 and 16 n Dawson (2010). Peter C. Dawson earned a Ph.D. n Economcs at the Unversty of Connectcut n 1999 (M.A. Economcs 1990, B.A. Economcs 1988). As an Economst wth the U.S. Internal Revenue Servce n Farmers Branch, Texas, USA ( ), he worked tax audt cases nvolvng transfer prcng valuatons, closely-held company and famly lmted partnershp valuatons, and ntangble asset valuatons. He s the author of 2010 book The Economcs of Busness Valuaton Dscounts and The Compettve Rsk-Return Paradgm (ISBN ) and "The Economcs of Valung Covenants Not to Compete Under the Far Market Value Standard" n the Journal of Legal Economcs (2007, 14(2): pp.25-60). He s the co-author, wth Stephen M. Mller, Ph.D., of "Optmal Negotated Transfer Prcng and Its Implcatons for Internatonal Transfer Prcng for Intangbles" n the Internatonal Journal of Intellectual Property Management (2011, 4(4): pp ). Page 1 of 40

3 An Economc Analyss of the Compettve Rsk-Return Paradgm I. Introducton There s broad agreement wth Fnance s postve rsk-return relatonshp: few people quarrel wth the dea that nvestors requre some extra return for takng on rsk. That s why common stocks have gven on average a hgher return than U.S. Treasury blls. Who would want to nvest n rsky common stocks f they offered only the same expected return as blls? We would not, and we suspect you would not ether (Brealey, Myers and Allen 2008, p.217; talcs orgnal). The present author suggests that ths quote reflects an ncomplete underlyng compettve asset market analyss. The man contrbuton of Fnance s Captal Asset Prcng Model (CAPM), or the Tradtonal CAPM, whch s attrbuted to Sharpe (1964) (among others), s a postve relatonshp between an asset s rsk and expected rate of return apples to an asset s undversfable rsk, or systematc rsk, not to an asset s total rsk (whch ncludes both systematc and unsystematc rsk): We have argued that n equlbrum there wll be a smple lnear relatonshp between the expected return and standard devaton of return for effcent combnatons of rsky assets. Thus far nothng has been sad about such a relatonshp for ndvdual assets. Typcally the E R, σ r values assocated wth sngle assets wll le above the captal market lne, reflectng the neffcency of undversfed holdngs. Moreover, such ponts may be scattered throughout the feasble regon, wth no consstent relatonshp between ther expected return and total rsk (σ r ). However, there wll be a consstent relatonshp between ther expected returns and what mght be best called systematc rsk (Sharpe 1964, p.436; talcs orgnal). Sharpe s (1964) analyss, whch les at the heart of the present-day nterpretaton of the Tradtonal CAPM, concludes there s a consstent relatonshp between ther expected returns and what mght be best called systematc rsk, where the lnear captal market lne (or securty market lne) llustrates that consstent relatonshp. The captal market lne s postve slope reflects the demand-sde assumpton that ratonal, well-dversfed, rsk-averse nvestors prefer Page 2 of 40

4 An Economc Analyss of the Compettve Rsk-Return Paradgm (or requre, or demand) to be compensated wth a hgher rate of return on an nvestment n assets wth hgher systematc rsk. The postve rsk-return relatonshp s not a general concluson, however, because t s based on the assumpton that rsk-averse nvestors requre or demand a hgher rate of return for hgher-systematc-rsk n the absence of (explct) supply consderatons and the consderaton of rsk-preferrng nvestors. The predcton that rsk averse nvestors requre full compensaton for an asset s addtonal ncrement of systematc rsk reflects the (mplct) assumpton of a vertcal asset market supply curve. The Tradtonal CAPM s accurately characterzed as follows: (1) In a compettve asset market, (2) systematc rsk drves asset portfolo choce rather than total rsk and (3) there s a predctable, lnear relatonshp between an ndvdual asset s systematc rsk and expected rate of return, where (4) that postve relatonshp s based on the assumpton that the relevant populaton of nvestors s rsk averse and, therefore, demands to be compensated wth a hgher rate of return for an asset s addtonal ncrement of systematc rsk (5) n an amount that fully compensates for that addtonal rsk because a vertcal market supply curve s (mplctly) assumed. The fact that rsk-averse nvestors prefer hgher rates of return for hgher-systematc-rsk assets, and therefore demand a hgher rate of return, s not at ssue here. The ssue s whether the typcal nvestor (or nvestors n general) expects (expect) the compettve asset market wll reward hm (them) wth a hgher rate of return as compensaton for bearng hgher systematc rsk. The market, not ndvdual nvestors, determnes the market rsk-return relatonshp. Although an nvestor prefers a hgher rate of return for hgher rsk, ths does not necessarly mean the compettve asset market wll provde a hgher return. The ratonal, well-nformed nvestor understands that he has no control over the rsk premum provded by a compettve asset market because he s just one of many nvestors, such that hs buyng and sellng actvty Page 3 of 40

5 An Economc Analyss of the Compettve Rsk-Return Paradgm has neglgble mpact, f any, on the market prce (.e., he s a prce taker ). Wthholdng hs asset purchase or sale can have only a neglgble mpact on the market prce, whch leaves each nvestor n a compettve asset market no avenue by whch to cause, or requre, the market to provde hm a hgher rate of return (let alone a hgher rate of return that would fully compensate hm for an asset s addtonal systematc rsk). A pror, there s no reason to expect a compettve asset market to exhbt a postve equlbrum relatonshp between systematc rsk and return. The present author s not the frst to queston the valdty of the concluson that hgher systematc rsk assets wll, on average, provde a hgher market rate of return, ceters parbus. Emprcal studes of the market relatonshp between rsk and return have shown mxed results regardng the valdty of a predctable postve rsk-return relatonshp. 1 The poor emprcal record has tended to undermne the valdty of usng the Tradtonal CAPM 2 n many practcal applcatons. Accordng to Fama and French (2004), the emprcal record of the model s poor poor enough to nvaldate the way t s used n applcatons (p.25). In the end, we argue that whether the model s problems reflect weaknesses n the theory or n ts emprcal mplementaton, the falure of the CAPM n emprcal tests mples that most applcatons of the model are nvald (p. 26). For example, fnance textbooks often recommend usng the Sharpe-Lntner CAPM rsk-return relaton to estmate the cost of equty captal. But emprcal work, old and new, tells us that the relaton between beta and average return s flatter than predcted by the Sharpe-Lntner verson of the CAPM (p.43; talcs added). Mayers (1976) shows that, under the assumpton of constant absolute rsk averson, the market prce of rsk and the prce of assets are neutral wth respect to changes n asset marketablty. 3 Stapleton and Subrahmanyam (1979) extend Mayers (1976) to show that, For a range of utlty functons, the degree of marketablty has no effect on the market prce of rsk or on the level of [asset] prces (p.9). In a general equlbrum overlappng generatons analyss that assumes Page 4 of 40

6 An Economc Analyss of the Compettve Rsk-Return Paradgm nvestors n general have a large degree of rsk-averson, Vayanos (1998) concludes that an asset s market prce may ncrease when ts transacton costs 4 ncrease: Our results are surprsng and contrary to conventonal wsdom about the effect of transacton costs on asset prces. the prce of a stock may ncrease n ts transacton costs (p.26). McCown (1999) asks Should rsk premums always be postve? : Are ex ante expected returns on rsky assets always hgher than the certan return on a rsk-free asset? The results of Boudoukh, Rchardson, and Smth (1993), Ostdek (1998), and ths research show that ths s not necessarly the case (p.111). Shefrn (2001) hypotheszes that, even though nvestors may state that[,] n prncple, rsk and expected return are postvely related, n practce they form judgments n whch the two are negatvely related (p.176). He further hypotheszes that the nverse relatonshp found n hs study s based on nvestors errors (Shefrn 2001, p.179), or the fact that nvestors rely on crude heurstcs that predspose ther belefs to bas (Shefrn 2005, p.3). 5 That s, the nverse rsk-return relatonshp n Shefrn s (2001) study s hypotheszed to be based on a form of ncomplete or mperfect nformaton. 6 Accordng to Shefrn (2005), Although a central tenet of modern fnance s that the relatonshp between rsk and return s postve, many nvestors appear to form judgments to the contrary. Evdence suggests that nvestors relance on the representatveness heurstc s the key reason why they expect hgh returns from safe stocks. Investors who judge that good stocks are stocks n good companes wll assocate good stocks wth both safety and hgh future returns (p.277). The present author s analyss wll not be founded on ncomplete or mperfect nformaton, however. In a compettve asset market, by defnton market partcpants (.e., nvestors) are assumed to be (at least) well-nformed. An nverse equlbrum market relatonshp between a gven asset s systematc rsk and market rate of return wll be shown to be possble when the aggregate nvestment/tradng actvty of rsk-takng nvestors domnates that of rsk- Page 5 of 40

7 An Economc Analyss of the Compettve Rsk-Return Paradgm averse nvestors. That s not to say that the aggregate trade of rsk-lovng nvestors, as an emprcal matter, always domnates the aggregate trade of rsk-averse nvestors n a compettve asset market. On the contrary, the present author hypotheszes that, dependng on any number of factors, on any gven day aggregate asset trade may, on net, reflect the trades of rsk-takng nvestors, whch would make the market relatonshp between rsk and return negatve when rsklovng nvestors tradng actvty domnates the market. The mplcaton s there s no blanket, a pror reason for the average or typcal well-nformed, ratonal nvestor to requre, or expect, a compettve asset market to provde hm a hgher rate of return for nvestments n hgher systematc rsk assets. Consstent wth the present author s vew, Markowtz (2008) concludes that t s naccurate to nterpret the Tradtonal CAPM s postve, lnear relaton between expected beta rsk and expected return as ndcatng or meanng that CAPM nvestors are pad for bearng systematc rsk (p.91). Markowtz (2008) explans that ths s not a correct nterpretaton, because two securtes may have dentcal rsk structures n terms of ther covarances wth other securtes n the market, and yet have dfferent excess returns. In fact, f the parameters of the CAPM are generated n a natural way, then securtes wth the same rsk structure almost surely wll have dfferent expected returns (p.91). one clearly cannot say that the CAPM nvestor s pad for bearng rsk (p.94). The ponts I make do not change the major conclusons of the captal asset prcng model. Gven the assumptons of the CAPM, the market s an effcent portfolo, and there s a lnear relaton between the expected return of each securty and ts regresson aganst the market. But we must not nterpret ths [relaton] as [reflectng compensaton for] the bearng of rsk (p.94). Although Markowtz s (2008) reasonng may dffer from that of the present author, the present author s analyss s not nconsstent wth Markowtz (2008) and wll arrve at the same general concluson. The present author wll show that the compettve asset market need not necessarly Page 6 of 40

8 An Economc Analyss of the Compettve Rsk-Return Paradgm fully compensate the margnal nvestor, or compensate hm at all, for the addtonal ncrement n systematc rsk assocated wth nvestng n a rsky asset compared to a rsk-free asset. In the present author s analyss, the combnaton of the structure of the compettve asset market (wth a postve-sloped asset supply curve) and the ncluson of the nvestment behavor of rsk-takng nvestors leads to a general concluson that there need not be a stable or predctable postve compettve asset market relatonshp between a gven rsky asset s expected systematc rsk and expected rate of return. II. The Tradtonal CAPM: An Equlbrum Model? Sharpe (1964) descrbes the typcal vew (at least at the tme) of captal asset prcng, or of the determnaton of captal asset prces under condtons of uncertanty or rsk (p.425). Sharpe (1964) brngs the reader s attenton to the fact that there had not been an attempt to construct a market equlbrum theory of asset prces under condtons of rsk (p.427; talcs orgnal). Rather, the typcal explanaton of captal asset prcng under uncertanty or rsk assumed that the captal market prced an asset s rsk (.e., added a market rsk-premum to the pure rate of nterest or tme value of money) drectly n relaton to an asset s total rsk (Sharpe 1964, p.425). Moreover, lackng such a theory, t s dffcult to gve any meanng to the relatonshp between the prce of a sngle asset and ts rsk. Through dversfcaton, some of the rsk nherent n an asset can be avoded so that ts total rsk s obvously not the relevant nfluence on ts prce; unfortunately lttle has been sad concernng the partcular rsk component whch s relevant (Sharpe 1964, p.426). Sharpe s (1964) prmary contrbuton les n hs dstncton between total rsk and systematc rsk. Today, t s accepted that the non-systematc rsk component of an asset s total rsk s dversfed away when the ratonal nvestor s well-dversfed, and therefore non-systematc rsk s not part of a compettve asset market s equlbrum rsk-return relatonshp. Page 7 of 40

9 An Economc Analyss of the Compettve Rsk-Return Paradgm The Tradtonal CAPM s characterzed as an equlbrum market model, ncludng the equlbrum models of Sharpe (1964) and Lntner (1965[a]), n whch rsk and expected return are related to each other through the forces of supply and demand expected return beng the necessary reward to cause nvestors to bear undversfable rsk (Lo 2000, p.x). The equlbrum process through whch the captal market assgns a rsk premum to ndvdual assets (or to asset portfolos) was not explaned n a way that was rooted frmly n an equlbrum market theory, however. Sharpe (1964) dd not provde a suffcently-detaled analyss to see the underlyng adjustment process toward equlbrum asset prces, smply statng asset prces wll adjust untl a new equlbrum s reached: In equlbrum, captal asset prces have adjusted so that the nvestor, f he follows ratonal procedures (prmarly dversfcaton), s able to attan any desred pont along a captal market lne. He may obtan a hgher expected rate of return on hs holdngs only by ncurrng addtonal rsk (p.425; talcs orgnal). Varan (1990) provdes a more detaled descrpton of the underlyng adjustment process: Accordng to our model, all assets that are held n equlbrum have to le along. the market lne. What f some asset s expected return and beta ddn t le on the market lne? What would happen? Suppose that you found an asset whose expected return, adjusted for rsk, was hgher than the rsk-free rate [.e., that was hgher than the market lne]: r β( rm rf ) > rf. Then ths asset s a very good deal. It s gvng a hgher rsk-adjusted return than the rsk-free rate. snce t s offerng a better tradeoff between rsk and return than exstng assets, there s certanly a market for t. as people attempt to buy ths asset they wll bd up today s prce: p 0 wll rse. Ths means that the expected return r = ( p1 p0 ) p0 wll fall. How far wll t fall? Just enough to lower the expected rate of return back down to the market lne (pp ; bold orgnal). 7 Although Sharpe (1964) attempts to extend the theory of asset prcng to construct a market equlbrum theory of asset prces under condtons of rsk [p.427], Hs analyss stops short of provdng such an extenson (Berwag and Grove 1965, p.89; brackets orgnal). Sharpe (1964) does not explctly consder an excess demand equaton as a market equlbrum Page 8 of 40

10 An Economc Analyss of the Compettve Rsk-Return Paradgm condton, or closure condton (Berwag and Grove 1965, p.91). Presumably, snce Sharpe s (1964) analyss s consdered an equlbrum analyss, t s mplct that the ndvdual excess demands for assets sum to zero. Stone (1970) agrees: If markets are not cleared, then prces must adjust so that securtes not n the tangency portfolo wll be held (p.9). By explctly requrng that supply equal demand for every securty, we have extended Sharpe s model of equlbrum for completeness. Sharpe s only explct requrement was that captal asset prces must, of course, contnue to change untl a set of prces s attaned for whch every asset enters at least one combnaton lyng on the captal market lne [26, p. 435]. However, market clearng s necessary for market equlbrum; t s mplct n Sharpe s concept of equlbrum. Thus, t s requred that prces adjust untl markets are cleared (p.9n; brackets orgnal, talcs added). Nevertheless, based on the Tradtonal CAPM s smplfyng assumptons, t s possble to derve a few key characterstcs of equlbrum asset and portfolo returns wthout detalng the underlyng structure, that s, the demand for and supply of assets, or dscussng ther prces (Danthne and Donaldson 2005, p.123). III. The Tradtonal CAPM: A Demand-Sde Model? The Tradtonal CAPM focuses on the demand-sde of a compettve asset market. Danthne and Donaldson (2005) support ths vew: as a theory of fnancal equlbrum t makes the assumpton that the supply of exstng assets s equal to the demand for exstng assets and, as such, that the currently observed asset prces are equlbrum ones. There s no attempt, however, to compute asset supply and demand functons explctly. Only the equlbrum prce vector s characterzed. Let us elaborate on ths pont. Under the CAPM, portfolo theory nforms us about the demand sde. If ndvdual nvests a fracton w of hs ntal wealth j Y n asset j, the value of hs asset j holdng s w Y j 0. Absent any nformaton that he wshes to alter these holdngs, we may nterpret the quantty w Y j 0 as hs demand for asset j at the prevalng prce vector. If there are I ndvduals n the economy, the total value of all 0 Page 9 of 40

11 An Economc Analyss of the Compettve Rsk-Return Paradgm holdngs of asset j s I w jy 0 ; by the same remark we may nterpret ths quantty as aggregate demand. At equlbrum one I must have w jy0 = p jq j where p j s the prevalng equlbrum prce per share of asset j, Q j s the total number of shares outstandng and, correspondngly, p Q j s the market captalzaton of asset j. The CAPM derves the mplcatons for prces by assumng that the actual economy-wde asset holdngs are nvestors aggregate optmal asset holdngs (pp ). Although the typcal (or margnal) nvestor may demand, or requre, a hgher rate of return to fully compensate hm for an asset s hgher systematc rsk, that fully rsk-adjusted rate of return s not necessarly the compettve equlbrum asset market outcome because t does not consder the other sde of the market.e., the supply sde. A focus on the requred rate of return, whch s a demand-sde concept, underles, the present author beleves, the accepted noton that the expected rate of return for an asset that s more rsky should be rsk-adjusted by an amount that fully compensates an nvestor for the asset s greater rsk. Lkewse, the Tradtonal CAPM s focus on the rate of return that rsk-averse nvestors read rsk-averse buyers or demanders requre or demand reflects a focus on the demand-sde of a compettve asset market. Jensen (1972) descrbes the demand-sde focus of the Tradtonal CAPM as follows: The man result of the orgnal papers [of Treynor (1961), Sharpe (1964), Lntner (1965a, 1965b), Mossn (1966), Fama (1968, 1971)] n ths area s the demonstraton that one can derve an ndvdual s demand functon for assets, aggregate these demands to obtan the equlbrum prces (or expected returns) of all assets, and then elmnate all the ndvdual utlty nformaton to obtan market equlbrum prces (or expected returns) solely as a functon of potentally measurable market parameters (p.363). Regardng the Sharpe-Lntner (S-L) captal asset prcng model (CAPM), Markowtz (2008) prefers Mossn s [1966] verson of the S-L CAPM model (Mossn S-L model) because he vews t as a systematc formulaton of Sharpe s verson of the S-L model (p.91): Mossn Page 10 of 40

12 An Economc Analyss of the Compettve Rsk-Return Paradgm [1966] states the premse of the Sharpe model explctly, and draws vald conclusons from t, whle Sharpe s verson s vague on the statement of hs premses and deducton of hs conclusons (p.91; brackets orgnal). Mossn (1966) agrees wth Berwag and Grove (1965) that Sharpe s (1964) model s ncomplete n the sense that t does not explctly nclude the market equlbrum condton(s): The paper by Sharpe [1964] gves a verbal-dagrammatcal dscusson of the determnaton of asset prces n quas-dynamc terms.. But hs lack of precson n the specfcaton of equlbrum condtons leaves parts of hs arguments somewhat ndefnte. The present [.e., Mossn (1966)] paper may be seen as an attempt to clarfy and make precse some of these ponts (p.769). Mossn (1966) shows that a market lne can be derved from the condtons for general equlbrum (f t exsts) (p.778). Mossn (1966) assumes a fxed supply of the subject asset, asset j: To determne general equlbrum, we must also specfy equalty between demand and supply for each asset. These market clearng condtons can be wrtten (7 ) ( x x j ) = 0 ( j = 1, n) m = 1 j..., [where] also the nth equaton of (7 ) must hold. We may therefore m nstead wrte: (7) x = x j ( j = 1,..., n 1) = 1. j, where x j denotes the gven total supply of asset j: x j = = x 1 j (p.773) [where x j are the quanttes of asset j that he brngs to the market; these are gven data (p.772)]. An ncrease n asset j s systematc rsk would lead rsk-averse nvestors to demand less of asset j, ceters parbus, thereby shftng the market demand curve for asset j downward (or to the left). Wth a fxed or vertcal market supply curve for asset j, ths would lead to a reducton n asset j s market prce and correspondngly an ncrease n asset j s expected rate of return by an amount that would fully compensate nvestors for the added ncrement n systematc rsk. Mossn m Page 11 of 40

13 An Economc Analyss of the Compettve Rsk-Return Paradgm (1966) refers to the ncrement n expected rate of return as the rsk compensaton per unt of rsk on a unt of asset j, or, to put t dfferently, the gan n expected yeld per unt s ncrease n the rsk of one unt of asset j (p.783; talcs orgnal). Full compensaton for bearng an addtonal ncrement of systematc rsk s founded on the Tradtonal CAPM s (mplct) assumpton that there exsts a vertcal market supply curve for an asset (or, at least, the market s operatng along the vertcal porton of the market supply curve). Therefore, ncreases n an asset s systematc rsk whch lead rsk-averse nvestors to requre an addtonal ncrement of rate of return that would fully compensate for the addtonal rsk, as measured by the magntude of the decrease n the market demand curve translates nto an equlbrum market prce (and expected rate of return) that fully compensates for the addtonal ncrement n systematc rsk. IV. Illustratng the Tradtonal CAPM s Demand-Sde Under the Tradtonal CAPM, the equlbrum market expected rate of return on an nvestment n Asset, E [ r ], s: E [ r ] r + β ( E[ r ] r ) =. (1) f m f In terms of market prce, P, the expected rate of return on an nvestment n Asset s: E r [ ] [ ] P + E[ TO ] E P =, (2) P where E [ P ] s expected future market prce at the end of the nvestment perod and E[ ] expected cash throw off (e.g., dvdends). Therefore, under the Tradtonal CAPM, E [ r ] where, by rearrangng, [ ] P + E[ TO ] E P ( E[ r ] r ) = = rf + β m f, (3) P E[ P ] + E[ TO ] [ r + β ( E[ r ] r )] P = > 0. (4) 1+ f m f TO s Page 12 of 40

14 An Economc Analyss of the Compettve Rsk-Return Paradgm However, the prce functon n Equaton 4 s not the equlbrum market prce; t s the Tradtonal CAPM s demand-sde relatonshp between the relevant exogenous shft factors (ncludng systematc rsk, β ) and the poston of the demand curve when nvestors are rskaverse. For example, an ncrease n Asset 's systematc rsk, β, leads to a downward shft n Asset 's demand curve by a magntude of ( E[ P ] + E[ TO ]) ( rf E[ rm ]) 2 ( 1+ [ r + β ( E[ r ] r )]) f m f β, 8 as shown n Fgure 1. Fgure 1: The CAPM Equaton Represents the Demand-Sde [Fgure 1 here] Quantty suppled and quantty demanded are a functon of the market prce, or, stated S D mathematcally, Q = f ( P ) and g( P ) Q =. However, t s usual practce for economsts to 1 S llustrate a market graphcally usng nverse supply and demand functons.e., P = f ( Q ) 1 D and P g ( Q ) = because t s n lne wth the mathematcal conventon of placng the dependent varable on the vertcal axs (Chang 1984, p.38; talcs orgnal). Customary nverse supply and demand curves are depcted n Fgure 1, wth prce (P ) on the vertcal axs and quantty (Q ) on the horzontal axs. Sharpe (1964), and the Tradtonal CAPM n general, focus, at least mplctly, on the nverse demand functon as reflected n Equaton 4, E[ P ] + E[ TO ] [ r + β ( E[ r ] r )] P = > 0, whch s an nverse demand functon that s a functon of Asset 's 1+ f m f attrbutes (or demand curve shft factors) rather than Asset s quantty demanded. The quantty demanded s mplct, however. That s, more generally, Equaton 4 s D ( Q ( β,e[ P ],E[ TO ],E[ r ] r )) 1 P g m =. (5) f Page 13 of 40

15 An Economc Analyss of the Compettve Rsk-Return Paradgm Berwag and Grove (1965) pont out that Sharpe s (1964) CAPM would need to specfy the orgnal (non-nverse) supply and demand functons n order to properly specfy the necessary market equlbrum condton, whch s that, n the aggregate, the ndvdual excess demands for Asset sum to zero: Sharpe s [1964] approach begns wth a dsequlbrum. [, where] He then postulates a dynamc model whch he asserts wll lead to a market equlbrum. He fals, however, to state the necessary condtons for such an equlbrum to exst. If he had focused hs analyss on the quanttes of assets themselves rather than upon the attrbutes that these assets possess, the necessty of the closure property, [as n Berwag and Grove s] equaton 10, would have been more obvous (p.92). 9 In a sngle market model, for example the market for Asset, the necessary condton for an equlbrum s the attanment of a market prce that equates the aggregate quantty suppled wth the aggregate quantty demanded, such that the aggregate excess demand for Asset equals zero, D. Substtute Q g( P ; β,e[ P ],E[ TO ],E[ r ] r ) D S.e., Q Q 0 = = whch s the orgnal, non- S nverse demand functon for Equaton 5 and f ( P ) m f Q = nto ths excess demand equaton, whch produces one excess demand equaton wth one unknown endogenous varable, P : Q S ( P ;,E[ P ],E[ TO ],E[ r ] r ) Q ( P ) 0 β. (6) D m f = Equaton 6 s Asset s market equlbrum condton specfed n general functonal form. A basc premse of the Tradtonal CAPM s rsk-averse nvestors requre a rsk premum, β ( E[ r ] r ), n the expected rate of return, E [ ] m f r, to be persuaded to purchase Asset (n the frst place) and t s E [ r ] and β that nvestors use, as asset attrbutes, to determne whether to purchase Asset, and how much of Asset to purchase. Therefore, t appears that a proper characterzaton of the Tradtonal CAPM equaton s that t reflects and represents the demandsde of an asset s market, rather than an equlbrum n an asset s market supply and demand Page 14 of 40

16 An Economc Analyss of the Compettve Rsk-Return Paradgm curves. For relevancy and smplcty, only the exogenous attrbute varables,.e., [ ], E[ TO ], and E[ rm ] rf β, E P such as the expected future prce, E [ ], are ncluded n Asset s demand equaton (exogenous varables, P, may also be determnants n the supply functon, but for smplcty they are not ncluded). In terms of Economcs smple market model, then, the Tradtonal CAPM s assumpton of rsk-averse nvestors whch, n regard to the shft parameter D Q β, s modelled by the assumpton < 0 leads to the specfc concluson that there s a β postve relatonshp between Asset 's systematc rsk and ts expected rate of return, ceters parbus. An exogenous ncrease n Asset s systematc rsk, whch causes Asset s market demand curve to shft leftward, leads to a decrease n Asset s market prce, and, by Equaton 2, a correspondng ncrease n ts expected rate of return. Ths s the Tradtonal CAPM s postve rsk-return relatonshp. The supply-sde of Asset 's market s ether not consdered, or t s mplct that (the relevant porton of) the market supply curve s vertcal, allowng the margnal nvestor to be fully compensated for Asset s hgher systematc rsk than the rsk-free rate of return. That s, the leftward shft n Asset s market demand curve occurs along the vertcal porton of Asset s market supply curve, thereby leadng to a fall n Asset s market prce by an amount that rases Asset s rate of return by an amount that fully compensates the margnal nvestor for Asset s hgher systematc rsk. The present author s vew that the CAPM equaton reflects the demand-sde of an asset s market s consstent wth Mshkn s (1995) Theory of Asset Demand, n whch factors that nfluence nvestors decsons to buy and hold dfferent assets (p.104), ncludng an asset s expected rate of return and rsk, cause shfts n an asset s demand curve (Chapter 5). Wealth (W) Page 15 of 40

17 An Economc Analyss of the Compettve Rsk-Return Paradgm and lqudty (L) also affect asset demand, where Mshkn s (1995) Theory of Asset Demand can be summarzed n the followng equaton: ( ) ( + ) ( ) ( ) ( + ) + D Q = g P ;W,E[ r ], β, L. (7) Assumng rsk-averse nvestors (p.98), an ncrease n Asset 's systematc rsk (whch s measured by β ) reduces the quantty of Asset demanded at every prce, whch s llustrated by a leftward shft n Asset 's demand curve (refer to Fgure 2). We have seen that an asset contrbutes rsk to a well-dversfed portfolo n the amount of ts systematc rsk as measured by beta. When an asset has a hgh beta, meanng that t has a large amount of systematc rsk and s therefore less desrable, we would expect that nvestors would be wllng to hold ths asset only f t yelded a hgher expected return. Ths s exactly what the CAPM tells us (Mshkn 1995, p.103; talcs added). Fgure 2: Mshkn s (1995) Theory of Asset Demand and the CAPM; An Exogenous Increase n β [Fgure 2 here] If, however, we assume that nvestors are rsk-lovng, then the sgn of the relatonshp between β and D D Q Q would be postve nstead of negatve (.e., > 0 ). In that case, an ncrease n β Asset 's systematc rsk would ncrease (rather than decrease) ts quantty demanded at every prce, whch s llustrated by a rghtward shft n Asset 's demand curve (refer to Fgure 2). Wth an assumpton of rsk-lovng nvestors, an ncrease n β rases Asset 's market prce, and therefore the Tradtonal CAPM equaton would have, unlke Equaton 1, a negatve sgn before the beta: E [ r ] r β ( E[ r ] r ) =. (1') f m f Page 16 of 40

18 An Economc Analyss of the Compettve Rsk-Return Paradgm An ncrease n Asset 's market prce corresponds to a decrease n Asset 's expected rate of return, ceters parbus; there s a negatve rsk-return relatonshp when nvestors are rsk-lovng. V. Explctly Includng the Supply Sde Hrshlefer (1966) characterzes the Tradtonal CAPM as a theory of asset prcng that assumes a fxed supply of fnancal assets, whch s an analyss of the very short run : If mean-return and varablty-of-return are to be regarded as commodtes, the analyss must go beyond the ndvdual level of decson to show how the relatve prces for mean-return and varablty-of-return are determned n the market. There seem to be rather consderable dffcultes facng theorsts who attempt to fulfl ths program (p.252). The furthest development to date seems to be that of W. F. Sharpe, Captal Asset Prces: A Theory of Market Equlbrum Under Condtons of Rsk, Journal of Fnance, XIX (Sept. 1964). Ths may be regarded as a theory of prces for mean and varablty n the very short run, wth fxed supples of productve and fnancal assets (p.252n). Black (1976) agrees the Tradtonal CAPM s a demand-sde model: the statc Sharpe- Lntner-Mossn model of captal asset prces s unnecessarly restrctve n ts neglect of the supply sde (p.776). The Tradtonal CAPM assumes a fxed outstandng supply of a gven asset n the short-run: The Sharpe-Lntner-Mossn theory of captal asset prcng assumes fxed supples of securtes (Black 1976, p.769). Wth a fxed outstandng supply of an asset that s bndng n the short-run, the Tradtonal CAPM assumes the market for a gven asset, Asset (whch s a rsky share n the outstandng equty n a company, for example), s operatng along the vertcal porton of ts supply curve, as llustrated n Fgure 3. Therefore, an ncrease n the systematc rsk of Asset, whch n the context of rsk-averse nvestors reduces the market demand for Asset, causes the market prce of Asset to declne by an amount that rases the expected rate of return for Asset by an ncrement that would fully compensate the margnal nvestor for Asset s hgher Page 17 of 40

19 An Economc Analyss of the Compettve Rsk-Return Paradgm rsk. In that analytcal context, the margnal nvestor would expect to earn hs requred rate of return. The Tradtonal CAPM can focus on the demand-sde of a compettve asset market when the market s operatng along the vertcal porton of ts supply curve. Fgure 3: The Supply Assumpton Underlyng the Tradtonal CAPM [Fgure 3 here] Black (1976) adopts a more general assumpton regardng the supply-sde. The market supply curve for a gven asset, Asset, has a normal postve slope throughout. Unlke the Tradtonal CAPM s (mplct) supply curve, whch has a vertcal slope begnnng at the shortrun fxed, outstandng supply of Asset, SR Q S, Black (1976) assumes that, although SR Q S s bndng n the short-run, the market supply curve for Asset does have a postve slope at quanttes greater than (and equal to) SR Q S because he assumes asset supples are flexble n the short-run (as well as n the long-run). 10 The dashed porton of the long-run supply curve, as seen n Fgure 3, becomes an extenson of the short-run supply curve, as ndvdual frms can respond, n the short-run, to changes n a gven asset s prce by ssung or retrng shares n the company s equty (p.770). Implct n Black (1976) s the assumpton that the outstandng supply of Asset s bndng n the short-run. Otherwse, the market for Asset would be operatng along the lower, postve porton of the supply curve n the short-run, whch would make the shape of the upper porton of the market supply curve rrelevant to an analyss n the short-run. In contrast to the analyses of Black (1976) and the Tradtonal CAPM, the present author assumes the outstandng market supply of Asset s fxed n the short-run, but that fxed outstandng supply s not bndng. The market for Asset operates along the lower porton of the short-run supply curve, such that the fxed outstandng supply of Asset, SR Q S, s not bndng on demand (regardless of one s assumpton about the upper porton of the short-run supply curve) Page 18 of 40

20 An Economc Analyss of the Compettve Rsk-Return Paradgm (refer to Fgure 4). When an asset market operates along the lower, postve-sloped porton of ts short-run supply curve, the quantty suppled n response to prce changes comes from exstng owners of outstandng shares of Asset, such that the company (or entty) that ssued Asset need not be assumed to be able to ssue new (or retre outstandng) shares of ts equty n the short-run. When operatng along the postve-sloped porton of the short-run supply curve, a leftward shft n the market demand curve (reflectng rsk averse nvestors aggregate response to Asset s hgher systematc rsk) decreases Asset s market prce by an amount less than that whch would fully compensate the margnal nvestor for Asset s hgher rsk. Unlke the vertcal market supply curve mplct n the Tradtonal CAPM, the decrease n market demand occurs along the postve-sloped porton of the market supply curve. Ths, the present author beleves, s an approprate analytcal context for evaluatng the market s response to an asset s hgher systematc rsk because, on any gven day, not all exstng owners of outstandng shares n a gven asset (and perhaps not even a substantal porton of them) would be n the market offerng ther shares for sale. Therefore, t s reasonable to assume the fxed quantty of shares outstandng, SR Q S, s not bndng on market demand n the short-run. Fgure 4: The Present Author s Supply-Sde Assumpton [Fgure 4 here] VI. Includng a Broad Populaton of Heterogeneous Investors Hrshlefer (1964) hypotheszed that observed dscrepances between actual asset prces and asset rsk were not consstent wth rsk averson, referrng to the complex queston of the consstency of observed market behavor wth the rsk-averson assumpton (p.79). Hrshlefer (1966) recognzed that there was a need to nclude rsk-lovng nvestors, along wth rsk-averse nvestors, n the theory of asset prcng, and that t had not yet been determned how the Page 19 of 40

21 An Economc Analyss of the Compettve Rsk-Return Paradgm nvestment behavor of both types of nvestors combnes to create a market rsk premum or dscount: It remans unclear how the rsk-lovng or rsk-avodng propenstes of ndvduals are composed nto an overall market premum or dscount for rsk (p.276). In the Tradtonal CAPM, nvestors are assumed to be rsk averse (at least n general), whch s llustrated by the postve-sloped Securty Market Lne. Sharpe s (1991) more general presentaton of the Securty Market Lne expresses the fact that the Securty Market Lne s slope can depend on socetal rsk tolerance, whch s the wealth-weghted rsk tolerance of the nvestors n the market : A graphcal portrayal s termed the securty market lne. The CAPM mples that all securtes and portfolos wll plot along such a lne. Many would argue that ths relatonshp s the most mportant sngle concluson derved from the CAPM. It shows that expected returns wll be lnearly related to market rsk, but not, as often beleved, to total rsk [pp ; talcs orgnal]. Sharpe (1991) derves a more general equaton for the CAPM as follows: E 2 = R f + Vm βm. 11 (8) τ m Snce R f > 0, V m > 0, and β 0, when rsk-averse (rsk-preferrng) nvestors nvestment m actvty domnates the asset market.e., when τ > m 0 ( τ < 0 m ) the slope of the Securty Market Lne s postve (negatve). 12 The present author s analyss n the next secton develops Sharpe s (1991) result further, wthn the context of an explct compettve asset market (.e., supply and demand) analytcal framework. VII. A General Rsk-Return Paradgm The present author s extenson of the Tradtonal CAPM s presented n the followng dscusson. A compettve equlbrum market model s adopted, wth normally-sloped market supply and Page 20 of 40

22 An Economc Analyss of the Compettve Rsk-Return Paradgm demand curves. 13 In a compettve asset market, there are a great number of assets, wth varyng degrees of smlartes and/or dssmlartes, but, for smplcty and ease of understandng, assume the compettve asset market conssts of one subject asset, Asset 1, whle all other assets are bundled nto a second asset, Asset 2. Ths smplfyng assumpton does not contradct the assumed compettve asset market, because Asset 2 s a composte of all other assets (or all alternatve or substtute assets for Asset 1) n the compettve market (or n the compettve asset market segment that ncludes Asset 1). The two assets ntally are dentcal, ncludng dentcal rsk characterstcs. At the ntal equlbrum between Market 1 and Market 2, Asset 1 and Asset 2 s market prces have cleared, such that they are equal. Even wth heterogeneous nvestors (who can have heterogeneous asset and rsk preferences), market prces for these ntally dentcal assets are equal. At equal prces, nvestors are happy wth ther holdngs of Asset 1 and Asset 2, regardless of whether or not each nvestor has the same mx of Asset 1 and 2 n hs well-dversfed asset portfolo. Wth dentcal asset characterstcs, the ntal poston n ths compettve asset market s represented by an equlbrum where 1 P2 [ r ] [ ] P E[ P ] [ ] = E r2 P1 P2 P =, [ P ] E[ ] E =, and 1 P 2 E P P E = =. 14 Although t s assumed that nvestors need not have dentcal asset preferences, snce both Asset 1 and 2 are dentcal ther prces must be equal n equlbrum. In a compettve asset market, a dsturbance to Market 1 has a neglgble affect on Market 2. There are a large number of market partcpants (.e., sellers and buyers), assets, and quanttes traded n the compettve asset market, and therefore a change n Asset 1 s quantty traded has a neglgble effect on the quantty traded of the composte Asset 2. In economc terms, Page 21 of 40

23 An Economc Analyss of the Compettve Rsk-Return Paradgm market partcpants are prce takers who have neglgble or no market power to nfluence the market prce of ether asset. The dsturbance to the market s an unexpected ncrease n Asset 1 s systematc rsk. The relevant queston s How does an asset s equlbrum rate of return change n response to a change n ts systematc rsk, ceters parbus? An adjustment process toward a new asset market equlbrum, n response to an ncrease n Asset 1 s systematc rsk, s explctly llustrated and explaned. Asset 1 s ncrease n systematc rsk s reflected by an ncrease n the varaton of ts rates of return (.e., ts normal probablty dstrbuton becomes wder). Snce Asset 1 s probablty dstrbuton s normal, the dstrbuton becomes wder by equal amounts on both sdes (.e., n both the up-sde and the down-sde). 15 In response to the ncrease n Asset 1 s systematc rsk, whch makes Asset 1 less desrable (or marketable) to rsk-averse nvestors, rsk-averse nvestors generally buy (.e., demand) less Asset 1 (at every market prce), whch s llustrated by a decrease n the market demand for Asset 1. In Fgure 5, market demand shfts leftward (or downward) from D 0 to D 1, creatng an excess supply at Asset 1 s ntal market prce, P 0, whch n turn puts downward pressure on Asset 1 s market prce. The resultng decrease n Asset 1 s market prce s llustrated graphcally by a movement down along the exstng supply curve, S 0, to P CAPM n Fgure 5. By Equaton 2, Asset 1 s decrease n market prce leads to a correspondng ncrease n ts expected rate of return, ceters parbus. If ths were the end of the adjustment process, the relatonshp between systematc rsk and expected rate of return would be postve, whch s the Tradtonal CAPM concluson as expressed n terms of normally-sloped market supply and demand curves. Rsk-averse nvestors are generally happy wth ther relatvely fewer holdngs of Asset 1, and greater holdngs of Asset 2, because Asset 2 s now a relatvely safer asset, all else equal. Page 22 of 40

24 An Economc Analyss of the Compettve Rsk-Return Paradgm Fgure 5: Market for Asset 1; The Tradtonal CAPM s Postve Rsk-Return Relatonshp [Fgure 5 here] When one consders rsk-lovng nvestors n addton to rsk-averse nvestors, the adjustment process descrbed thus far under the Tradtonal CAPM s ncomplete. A less restrctve (and more general) populaton of nvestors s assumed, whch ncludes both rskaverse and rsk-lovng nvestors (or, more realstcally, nvestors who may have a broad spectrum of rsk preferences). Sharpe (2007) agrees that a more complete understandng of the rsk-return relatonshp can be obtaned by ncludng a broad spectrum of nvestors who possess dverse, and varyng degrees of, preferences for rsk: Tradtonal models are often forced to assume that all nvestors are of one type or, at the least that the market acts as f there were only one nvestor, whose preferences are representatve of those of the entre populaton of nvestors. [, but] At the very least, t must be admtted that nvestors have dverse types of preferences and that real captal markets reflect such dversty (p.61). Black (1976), for example, consders the omtted short-run speculatve factors nfluencng the demand to hold fnancal assets (p.767). Presumably, these short-run speculatve factors nclude the nvestment behavor of rsk-lovng nvestors. Because rsk-lovng nvestors exst alongsde rsk-averse nvestors, a heterogeneous populaton of nvestors s assumed here. The nvestment behavor of rsk-lovng nvestors s assumed to domnate the compettve asset market, at least n the short-run. The domnant aggregate tradng actvty of rsk-lovng nvestors pushes Asset 1 s expected rate of return back down, whch contnues to adjust downward untl t has, after all the adjustment process has played tself out, decreased n response to the ncrease n Asset 1 s systematc rsk. Ths s consstent wth the fndngs of Berwag and Grove (1965): Page 23 of 40

25 An Economc Analyss of the Compettve Rsk-Return Paradgm the nfluence of any one nvestor s opnon n determnng the market prce vares drectly wth hs audacty [.e., the greater s hs rsk tolerance] and wth hs confdence n hs forecast [of the future market prce]. Thus, an nvestor who s both very audacous and very confdent wll be very nfluental n the determnaton of the equlbrum market prce (p.91). In the most general stuaton n whch there may be dverse expectatons and a mxture of rsklovers and rsk-averters, the equlbrum condton (10) [.e., Berwag and Grove s Equaton 10] suggests that the functon of the market s to blend these dfferences n a manner that wll determne a prce whch wll permt all traders to reach an equlbrum (p.92). 16 In a compettve asset market that ncludes a broad spectrum of nvestors wth varyng degrees of rsk tolerance, there can be a negatve or nverse relatonshp between an asset s systematc rsk and ts equlbrum market rate of return. Rsk-lovng nvestors consder Asset 1 s hgher systematc rsk to be a desrable asset characterstc 17, because there s now an opportunty to earn a hgher rate of return on an nvestment n Asset 1. Asset 1 s varance of rates of return s greater, so there s an opportunty to earn a rate of return n the upsde tal of ts wder probablty dstrbuton (as well as an opportunty to earn a rate of return n the downsde tal of ts wder probablty dstrbuton). The opportunty can only become realty for those few (rsk-lovng) nvestors who are quck enough to purchase some of Asset 1 before ts market prce has had a chance to ncrease to a level that would extngush that opportunty. In a compettve asset market, the wndow to purchase an opportunty to earn a greater-than-compettve rate of return s very short, snce ratonal and well-nformed nvestors act very quckly, whch s facltated by a flud market that ncludes quck communcaton and tradng technology and the fact that asset markets vrtually never close n hghly-ntegrated worldwde asset markets. In a compettve asset market, nformaton about changes n asset characterstcs (as well as about the qualty and relablty of that new nformaton), such as the ncrease n Asset 1 s systematc rsk, travels quckly, and, gven hgh-tech communcatons and tradng mechansms, the aggregate purchases Page 24 of 40

26 An Economc Analyss of the Compettve Rsk-Return Paradgm and sales of large quanttes of assets can occur almost nstantaneously. As such, an ncremental ncrease n Asset 1 s attractveness or desrablty due to ts ncrease n systematc rsk drves many rsk-lovng nvestors to swtch, as reflected n ther relatvely large aggregate tradng response, quckly away from other assets (.e., the composte Asset 2) toward Asset 1. Some rskpreferrng nvestors even purchase large quanttes, f possble, of Asset 1. Because the asset market s compettve, changes n the demand for Asset 1 has a neglgble (f any) countervalng nfluence on the composte Asset 2 s market demand. The aggregate nvestment response of rskpreferrng nvestors quckly ncreases the market demand for Asset 1, creatng an excess demand, an ncrease n Asset 1 s market prce, and a decrease n Asset 1 s expected rate of return (whle not nfluencng Asset 2 s demand, market prce, or expected rate of return). Once a new equlbrum s reached, there s no further opportunty to earn a hgher rate of return on Asset 1, because the prce-to-rsk rato for Asset 1, Z 1, returns to the compettve level. In ths alternatve rsk-return analyss, rsk-lovng nvestors are assumed to have a greater aggregate response (and perhaps a correspondng greater ntensty n ther aggregate reacton) to the ncrease n Asset 1 s systematc rsk compared to the aggregate response by rsk-averse nvestors. The aggregate effect of the rsk-lovng nvestors nvestment actvty on Asset 1 s market s assumed to outwegh the aggregate effect of rsk-averse nvestors nvestment actvty. On any gven day ncludng the tme perod assumed for the analyss the aggregate dollar value of trade volume by short-term speculatve nvestors 18 s larger than that of long-term, more rsk-averse nvestors (even though on any gven day, the aggregate dollar value of assets owned by long-term nvestors may be larger than that owned by short-term nvestors). 19 In the aggregate, rsk-averse nvestors tradng actvty tends to reduce the excess demand for Asset 1 (as the market demand curve tends to shft leftward), whch n turn puts downward Page 25 of 40