Harry M. Markowitz. Investors Do Not Get Paid for Bearing Risk 1

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1 Investors Do Not Get Pad for Bearng Rsk Harry M. Markowtz The relatonshp between the excess return of each securty and ts beta, where beta s defned as ts regresson aganst the return on the market portfolo, s lnear n the Sharpe Lntner ( S L ) aptal Asset Prcng Model ( APM ). Ths lnear relatonshp s often nterpreted to mean that APM nvestors are pad for bearng systematc rsk. In ths artcle, I wll show 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, yet have dfferent excess returns. In fact, f the parameters of the APM are generated n a natural way, then securtes wth the same rsk structure almost surely wll have dfferent expected returns. THE MOSSIN VERSION OF THE SHARPE LINTNER MODEL Although the premses and conclusons of the S L APM were frst presented n Sharpe (1964), I wll use Mossn s (1966) systematc formulaton of Sharpe s verson of the S L model n ths analyss. Mossn states the premse of the Sharpe model explctly and draws vald conclusons from t, where Sharpe s verson s vague about the statement of hs premses and deducton of hs conclusons, and one of hs conclusons s ncorrect. 1 Inputs to Mossn s verson of the S L APM model ( Mossn S L model ) nclude the followng: the utlty functons of many nvestors; the number of shares each nvestor frst owns of each stock; and the expected returns per share (not per dollar) and covarances per share (not per dollar) upon whch all nvestors agree. There s also an nterest rate nput whch s only used n the excess return calculaton. Outputs of the Mossn S L model nclude market clearng prces, expected returns and covarances per dollar, the composton of the market portfolo, and the regressons aganst the market portfolo. Along the way, Mossn uses Tobn and Sharpe to prove varous thngs about the S L model. For example, the Mossn S L model nvolves equatons that nclude cash, expected returns, and a Lagrangan multpler. Wth a lttle algebra, the equatons can be expressed n excess returns rather than expected returns. Mossn also ctes that Sharpe s verson of the Tobn Separaton Theorem holds n equlbrum (Sharpe s verson of the Tobn Separaton Theorem nvolves borrowng as well as lendng, whle Tobn s verson nvolves lendng only). In Sharpe s verson of the Tobn Separaton Theorem, each nvestor s effcent set conssts of one partcular portfolo of rsky securtes as well as the ablty to borrow or lend. Ths much of the theorem holds for any mean varance nvestor who can borrow wthout lmt (even f no one else can do so), or has the same belefs, or even seeks mean varance effcency. When we assume that all nvestors have the same belefs and all seek mean varance effcency, then t follows that all nvestors mx the same portfolo of rsky assets and ths must be the market portfolo. 1 In connecton wth Fgure 6 of hs paper, Sharpe asserts that as prces adjust some combnatons of rsky assets can become perfectly correlated wth each other. Ths s assocated wth a lnear porton of the resultng expected return, standard devaton effcent set consderng rsky securtes only. Mossn assumes that the covarance matrx per share s nonsngular. It s not clear whether Sharpe assumes that ths covarance matrx of returns s nonsngular, but he certanly permts ths, so no matter what postve prces are present n the market, the covarance matrx of returns per dollar nvested as well as the covarance matrx per share s nonsngular. Ths mples that no two dstnct lnear combnatons of rsky assets can be perfectly correlated and there can be no lnear segment n the set of mean-standard devaton effcent combnatons. Investors Do Not Get Pad for Bearng Rsk 1

2 The Mossn S L model s lke a gant TV set: You set the dals (e.g., share avalabltes) and the screen shows outputs (e.g., prces and expected returns per dollar nvested). Lke Mossn, I won t worry about the exstence and unqueness of the solutons to the equatons nvolved. I assume that when I flp on the swtch the TV works. If I set the dals the same way as yesterday, I get the same output as yesterday. One noteworthy feature of the Mossn S L TV s that you can tell how the nputs are set by lookng at the outputs. Outputs nclude prces, and wth prces you can fgure the orgnal shares avalable as well as means and covarances per share. However, there are some concevable outputs one wll never see namely outputs that could only happen f some assumed gven shares were negatve (or zero for all nvestors because such securtes would be dropped from the analyss.) But, these would be the securtes wth zero or negatve percent demanded n the market portfolo. We wll only see output consstent wth any postve vectors X > n Equaton (1), where ν and σ j are the excess return and covarance per dollar nvested. σ11 σ12+ σ1n ν1 σ21 σ22 σ2n ν2.... X1+ X Xn = h (1) σ n1 σ n2 σ nn ν n So, how do we fnd all possble outputs that can occur on the Mossn S L TV? The queston practcally answers tself f we wrte the basc optmzaton equatons as n Equaton (1). We know that the covarance matrx per dollar nvested s nonsngular because the covarance matrx per share s nonsngular the only thng that Mossn tells us about the latter matrx. Gven any nonsngular covarance matrx, consder the set of excess return vectors, ν, that result f X n Equaton (1) s non-negatve, X (later we consder only X > ). These ν form the cone, K, generated by whch s drawn as follows. As n Equaton (1), look at each () column σ of n turn as f t were a pont n ν -space. Draw a ray from the orgn through σ for = 1,... n. Fll n the mddle of the dagram by takng convex combnatons of anythng you already have. Ths gves you K, the cone generated by. But we want X > only. We get ths by throwng away the exteror of K, keepng only the nteror K. Even though we toss the exteror of K, there wll be plenty of good stuff left as long as as assumed. I call K the compatble cone (.e., the cone full of ν -vectors whch are compatble wth the gven n the sense that only these wll ever appear together on the Mossn S L TV). In other words, gven any nonsngular, pck anyν -vector from K. Ths combnaton of and ν may appear on the Mossn S L TV. These are the only, ν combnatons whch can appear. Because n ths model s all thngs rsk and ν all thngs return, n an abstract way we now know the model s possble combnatons of rsk and return. But, we do not have a concrete pcture of what these K look lke. For starters let s assume that returns are uncorrelated (ths s not plausble, but easy to analyze). s then dagonal: () Investors Do Not Get Pad for Bearng Rsk 2

3 V1 V 2 = V3 V 4 The frst column, consdered as a ν -vector, les on the X -axs, the second on the 1 X -axs, etc. Because the locaton of a ray from the orgn through a pont does not depend on 2 whch pont on the ray we choose, we have That s, K = K I K s the same as the cone generated by the dentty matrx I, the entre postve orthantant, specfcally the entre frst quadrant, when n= 2. K s the nteror of ths regon. Thus, f returns are uncorrelated then any postve VV 1, 2,..., Vn and ν1,..., ν are permtted. In n partcular, two securtes can have dentcal V but dfferent ν, thus they can have dentcal rsk structures gven our assumpton of zero correlaton and yet have dfferng expected returns, therefore dfferng excess returns. From Equaton (1) wth dagonal we see that when returns are uncorrelated,.e., stocks wth hgh means and low varances are a large part of the market. X = kν / V (2) Equaton (2) may also be wrtten as XV = kν (3) Because σ j = for all j, the standard formula for the covarance between a securty and a portfolo yelds XV as the covarance between the return on securty and the return on the nvestor s portfolo. Thus, n ths case, Equaton (3) says that the covarance of each securty wth ts portfolo s proportonal to the securtes excess return. If you dvde both sdes of Equaton (3) by the varance of the nvestor s portfolo you obtan β = kν (4) where β s the regresson aganst the portfolo. Here k = k/ Var( R p ) and should be gven no other nterpretaton. Equaton (4) s a relatonshp between a securty and one ndvdual s portfolo. But f all nvestors are essentally the same, then Equaton (4) carres over to the market as well. It s useful to see where Equaton (1) and therefore Equaton (4) come from. For ths purpose let us further smplfy the Mossn S L by assumng the followng form of utlty functon E cv (5) where c may vary from one nvestor to the next. To maxmze Equaton (5), take partal dervatves and the set result to zero to get V E = (1/ c) = 1,..., n (6) X1 X In other words, each nvestor s advsed to push each securty nto the nvestor s portfolo to the pont where each securty has the same rato of margnal effect on portfolo varance to margnal effect on portfolo mean. Ths aggregates up to the portfolo level. Dvde by market varance and the well-known APM relatonshp s created. Investors Do Not Get Pad for Bearng Rsk 3

4 If returns are correlated, compatble cones are derved by plottng the columns of as f they were ν vectors, drawng straght lnes from the orgn through them, and then takng convex combnatons of these lnes usng and the nteror of the resultng cone. Ths s easest to do for n = 2. Note that the cone tends to close up as correlaton ncreases. If the two securtes have a correlaton of 1, then the two lnes are dentcal. Ths s the law of one prce. Identcally dstrbuted securtes must have proportonate expected returns. As long as the matrx s O O nonsngular, therefore K s not empty, two dfferent ponts can be found n K not on the same ray through the orgn. In ths case two securtes can have dentcal rsk structures but dfferng excess returns. If the ntal endowments of nvestors are drawn randomly from some contnuous dstrbuton, then the probablty dstrbuton of (, µ ) cases wll be contnuous. Thus, there s a zero probablty that two securtes wth the same covarances wll have the same µ. AFTERTHOUGHTS 1. Rather than use ( σ, σ,..., σ ) ʹ 1j 2 j nj as a pont n ν -space to help draw a ray from the orgn, we could use any other pont on the ray, lke ( σ / σ, σ / σ,..., σ / σ ) 1j jj 2j jj nj jj Recall that σ / σ s the regresson coeffcent the beta f you wll of the return of securty j jj aganst that of j. 2. Do not expect compatble cones to go away n multperod dscrete or contnuous-tme analyss. In partcular, there s not much dfference between a one-perod optmzaton and a many-perod optmzaton. Bellman (1957) tells us that to optmze a many-perod problem you just optmze a sequence of one-perod problems. It s a matter of gettng the objectves rght for the problem-wthn-a-problem. But often problems-wthn-a-problem are not all that dfferent from the genune one-perod problem. Therefore, f the genune one-perod model has compatble cones there should also be some lurkng around the model s many-perod verson. 3. The contnuous-tme model s a specal case of the dscrete tme model. If you do not beleve that t s because you are stuck wth an old-fashoned numberng system. The Greeks thought you could not squeeze any new numbers between the fractons (a.k.a. the ratonals because the Greeks thought that the new numbers were rratonal). In fact, n some sense there are more rratonals than there are ratonals. Now we have nfntesmals between each real (ratonal or rratonal) number. Gottfred Lebnz ( ) thought he had a handle on the nfntesmals, but t had to wat for Abraham Robnson to fgure t out rgorously n the md-2 th century. In a dynamc analyss, rather than have tme travel along the real contnuum, we can have t walk step-by-step along the hyperfnte tme lne. The latter s as smple as a garden path, but wth nfntesmal dstances between successve steppng stones. Wth the hyperfnte tmelne, Brownan moton s what comes out at a macro level f at each nfntesmal tme ncrement you flp a con and record an nfntesmal gan or loss. Thus, t s a small step to magne a sngle-perod Mossn S L APM where the perod s an nfntesmal tck long. After all, where does Mossn say how long the perod has to be n a sngle-perod APM? A day? A year? A mcrosecond? An nfntesmal ncrement? Because the analyss s ndependent of the tme ncrement, t apples to nfntesmal steps and, Investors Do Not Get Pad for Bearng Rsk 4

5 therefore, to contnuous models. Our concluson s the same as n the dscrete tme model: that possble combnatons of excess returns whch can go wth a gven covarance matrx wll probably form some knd of cone and that two securtes wth the same rsk structure wll probably have dfferent expected returns. Therefore, one clearly cannot say that the APM nvestor s pad for bearng rsk n ether n the sngle-perod, multple-perod, or contnuous case. WHERE DOES THIS LEAVE APM? The ponts made n ths paper do not change the major conclusons of APM. Gven the assumptons of APM, the market s an effcent portfolo and there s a lnear relatonshp between the expected return of each securty and ts regresson aganst the market. But, we must not nterpret ths as the bearng of rsk. Insofar as the world works lke APM at an aggregate level, ths lnear relatonshp s useful. Someone who wshed to ssue a new securty could estmate ts beta, whch would ndcate the expected return that the market would assgn to ths new securty. REFERENES Albevero,Sergo, Jens Erk Fenstad, Raphael Høegh-Krohn, and Tom Lndstrøm Nonstandard Methods n Stochastc Analyss and Mathematcal Physcs. Academc Press, Inc.: Orlando, FL. Bellman, Rchard E Dynamc Programmng. Prnceton Unversty Press: Prnceton, NJ. Mossn, J Equlbrum n a aptal Asset Market. Econometrca, vol. 34, no. 4 (October): Robnson, Abraham Nonstandard Analyss. North Holland Publ.: Amsterdam. Sharpe, Wllam F aptal Asset Prces: A Theory of Market Equlbrum under ondtons of Rsk. The Journal of Fnance, vol. 19, no. 3 (September): Investors Do Not Get Pad for Bearng Rsk 5