Short Sales Restrictions and the Security Market Line

J A Schnabel, Unwerslty of Calgary

A reformulation of the CAPM IS dewed by taking account of short-sales restrlctlons on both r&y and safe assets The induced secunty market lme IS shown to be consistent with the emplncal security market lines of various researchers

The capital asset pncmg model, developed by Sharpe [ 181 and Lmtner [9], IS a posltlve theory of the valuation of nsky assets The model states that m equlhbnum the followmg equation obtains

-@T, > = RF + L%& > - RF I& >

where Rk, RF, and &, are one plus the rates-of-return, or simply returns, on the kfh asset, the nskless asset, and the market portfolio, respectively The market portfoho IS formed by an investment m every nsky asset m proportion to its market value Tildes denote random vanables Pk 1s the beta or systematic nsk of the k* asset E(e) IS the expectation operator Equation (1) defines the secunty market line which describes the linear relationship between the equlhbnum expected return on an asset and its beta

The capital asset pnclng model 1s based on numerous “unreahstlc” assumptions These include, among others, homogeneous expectations, the absence of taxes and transactions costs, infinite dlvlslblhty of all assets, atomlstlc markets, and, the focus of this note, the posslblhty of taking unlimited short positions in any asset Of course, the ultimate test of a positive theory 1s not m the realism of its assumptions but II-I the conformity of its emplncal lmphcatlons with observed phenomena The capital asset pncmg model has been empmcally tested by Douglas [6], Miller and Scholes [ 121, and Black, Jensen, and Scholes [3] on U S shares and, as a positive theory, it has been found wanting This paper prescmds from the econometnc problems raised by Roll [ 161 due to the mablhty to monitor the true market portfoho As investor expectations

Address correspondence to J A Schnabel, Faculty of Management, Unrversrty of Calgary, Calgary, Alberta T2N IN4, Canada

Journal of Bwness Research 12,87-96 ( 1984)

@ Elsewer Sctence Pubhshlng Co , Inc 1984

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88 J A Schnabel

are not directly observable, these authors examined the followmg relatlonshlp m lieu of Eq (1)

(2)

where Rk and & are the average expost returns on the k*h asset and the nskless asset, respectively, and Pk (1s the estimate of the k*) asset’s beta If the capital asset pncmg model 1s vahd, one would expect to obtain

. a0 = 0

as the regression estimates for Eq (2) Unfortunately, it turns out that q is significantly different from zero and &, , slgmficantly different from & - RF Thus, the secunty market line tends to have a different slope and intercept than what one would expect given the capital asset pncmg model Black, Jensen, and Scholes [3] noted the considerable penod-by- penod vanability m these slope and mtercept terms They concluded that any alternative hypothesis regarding capital market equlhbnum must be consistent with both the divergences of the slope and intercept terms from their capital asset pncmg model predictions and their vanability The empmcal studies of Modlghanl, Pogue, Scholes, and Solmk [ 131 for French, Italian, and Bntlsh shares, Ramsey, Lave, and Quay [15] for Japanese shares, and Monn [ 141 for Canadian shares amved at essen- tially the same conclusion Although their results may be viewed as consistent with these findings, Ball, Brown, and Officer [l] chose to ignore these aspects of the model’s predictions m then tests of the capital asset pncmg model for Australian shares

Various authors have relaxed some of the assumptions which underlie the capital asset pncmg model m attempts to explain these empmcal findings A useful compendium of most of these studies 1s provided by Tumbull [ 191 For example, Brennan [4] studied the ramlficatlons of the differential tax treatment of dlvldends and capital gams, Mayers [ 1 I] considered the unpact of nonmarketable assets, Chen, Kim, and Kon [S] introduced investor hquldrty needs and hquldatlon costs, and Levy [8] assessed the unpact of hmlted investor dlverslficatlon Some of these authors, specifically Black, Mayers, and Chen, Kim, and Kon, were able to explain the emplncal anomalies of the capital asset pncmg model, 1 e , the mtroductlon of the specific capital market lmperfectlons

Sales Restrzctzons 89

they exammed mduced secunty market lines with different intercepts and slopes than those predicted by the capital asset pncmg model Mayers was able to explam only the lower slope as his model predicts that the intercept of the secunty market lme m a market with nonmar- ketable assets 1s the return on the r&less asset ‘IINS note may be considered an addition to this list of studies The focus here 1s on the impact of short selling restnctlons on the secunty market lme

This note 1s closest m splnt to two other papers which successfully provided rationales for the above-mentioned anomalies These are the papers of Black [2] and Brennan [4] Black exammed the equlhbnum pncmg relationships which obtam m the context of two related capital market scenanos one scenano m which no nskless asset exrsts and where unhmtted short posltlons are allowed on all nsky assets and a second scenano m which, although a nskless asset exists, only long positions are allowed m the nskless asset whereas unlimited short posltlons may be held for any nsky asset On the other hand, Brennan [4], considered a situation where borrowing (1 e , taking a short position m the nskless asset) and lending (1 e taking a long position m the n&less asset) take place at divergent rates and where these rates differ from one investor to another However, Brennan assumed no restnctlons on short positions m any asset In contrast to the foregoing, this paper posits the existence of a nskless asset but disallows short positions m any asset, whether n&less or nsky Although the Black [2] and Brennan [4] papers invoke different vanatlons m assumptions, a common conclusion IS shared by them They all imply a secunty market lme which exhibits the followmg properties hneanty between the eqmhbnum expected return on a share and that share’s beta and slope and intercept terms which differ from the capital asset pncmg model’s predictions The ObJect of this note IS to show that a different but related set of assumptions leads to the same type of a secunty market lme

This comment 1s not the first to examme short-selling restnctlons m the context of the capital asset pncmg model Lmtner [IO], Ross [ 171, and Crube and Beedles [7] have assessed how portfoho choice and capital market equlhbnum are affected by the presence of such restnc- tlons However, they did not address the spectfic Issue which 1s the focus of this note, 1 e , the impact of short-selling restnctlons on the slope and intercept of the secunty market hne

The Individual’s Portfolio Selection Problem

The capital asset pncmg model assumes that there are no constraints on the use of the proceeds from a short sale In fact, various mstltutlonal

90 J A Schnabel

arrangements do not permit this In Australia short selling 1s illegal In the United States, Canada, Europe, and Japan mvestors must place the proceeds of a short sale m escrow and they must also deposit an additional amount equal to the margm requirement on the proceeds of the short sale Followmg the usual approach m the financial hterature, these restnctlons ~111 be modeled m the followmg sunple way The represent- ative investor examined m this paper will constram lus portfoho selec- tion activity so that only a long positlon or no investment will be undertaken m every asset

Thus, the lth mvestor’s optlmlzatlon problem 1s to

SubJect to the following constramt

e’x’ = 1. (3)

and, the no short sales condltlon,

x’ 2 0, (4)

where U,(e) ts the I* investor’s utility function with the Expected portfolio return and vanance of portfoho return as arguments, R,, 1s the return on hi portfoho, n’ 1s the column vector of his wealth allocations among the n assets avadable m the market, e 1s a vector of ones, and o*(s) is the vanance operator Normality of returns on all nsky assets 1s assumed Unless otherwise indicated by a pnme, which denotes trans- posltlon, all vectors are consldered column vectors Inequality (4) should be interpreted as the requirement that every element of the vector xl be nonnegative Definmg ~1 as the vector of expected returns on all assets and s1 as the vsance-covmance matnx of returns, observe that E@,,) = p’x’ and 02(Rpr) = xJ’i22xi

To solve the I* investor’s portfolio selectlon problem we form the following Lagrangean

L, =U,[E(k,,), a2(R,,)l + pz(e’x’ - 1) + K”x’, (5)

where p, 1s the Lagrange multlpher associated with constramt (3) and K’ 1s the vector of Lagrange multlphers associated with constramt (4) The Kuhn-Tucker theorem provides the followmg necessary condltlon for

Sales Restrzctlons 91

the solution of the I* mvestor’s portfoho selection problem

aL, au, -=

aE(E?,,) au,

aE(&, ,) ax’ +

au2 (Rp ,) ad au2 (ET, ,) ax1

+ pie + K’ = 0 (6)

Divide this equation through by i3U,/aa2@,,) The lmphclt function theorem states that

Applying this result to Eq (6) and defining the followmg

and

we obtain

-i&p + 2C2x’ + y,e + I? = 0, (7)

where $, = do2(I?pr)/dE(&,,), 1 e , the I* investor’s tradeoff between vanance of return and expected return at his optimal portfolio Assuming universal nsk aversion, 4, 1s stnctly greater than zero for all mvestors I

All the components of the vector 7~ must be less than or equal to zero This result may be demonstrated by mvokmg the Kuhn-Tucker theorem to show that all the components of K’ must be greater than or equal to zero The elements of K’ are the nonnegative shadow prices associated with the no short sales restnctlons denoted by inequality (4) The only time any element of K 1, say the k* component, equals zero 1s when a posltlve amount 1s invested by investor I m the correspondmg_asset, asset k Given the assumption of universal nsk aversion, iXJ,/a u2(Rpr) must be less than or equal to zero Thus, all the components of the vector ~1 must be less than or equal to zero

92 J A Schnabel

Capital Market Equilibrium

Consider the k* row of Eq (7)

-G(& ) + 2 cov(& ) ii,,) = -(yt + nkl), (8)

where nrkl IS the k* component of xl and cov( ;) 1s the covanance operator Multlplymg this equation through by WJW, the propotion of the I* investor’s wealth W, to the total wealth m the economy W = Z,W, and then summing across all investors we obtain

This implies that

where 4 1s defined as

1 e , as a weighted average of the nsk-return tradeoffs of all investors m the economy where the weights are the corresponding investors’ wealth proportions of total wealth m the economy Applying this equation to the nskless asset yields

This implies that Eq (10) may be rewntten as

E(i,) = RF + ; cov(& , & )

(11)

Sales Restnctzons 93

To stmphfy the notation, define for the n&less asset F

and, III general, for any asset k,

(13)

(14)

so that Eq (12) may be rewntten as

As 7rFTE1 and nkl are nonposmve, observe from Eqs (13) and (14) that or,- and flk must hkewise be nonpositrve Multlplymg Eq (15) through by Mk/&f, the market value of the k* asset i%!fk as a fraction of the market value of all assets ikt = C,M, and then summmg across all assets yields

E&)=RF +;u2(&)+(li, - nF)> (16)

where nM is defined as

F 2%

1 e , a weighted average of the n values, as defined by Eqs (13) and (14), of all assets where the weights are the market value proportlons of the correspondmg assets Equation (16) Implies that

1 E(&)-RF +(n,V -r,,f) -_= e o2 (%I )

Substltutmg 011s mto Eq (15) and recalhng that

Ok = cov(& ) ii, )/a2 (&.I )>

we obtain

E(&)=R, + [E(&r)-RF +(?TF-EM)]& +(nk

(17)

RF)>

(18)

94 J A Schnabel

where ?rF, TM, and rk for every aSSet k are all kSS than or equal t0

zero This equation displays the secunty market lme for the capital market scenano postulated m this note, one characterized by the absence of all nnperfectlons except for short sale restnctlons Observe that if there were no constraints on short selling or d all investors were motivated to maintain long positions m every asset, then, by the Kuhn- Tucker theorem, all the components of the vector of Lagrange multl- pliers associated with the no short sales constramt K’ would equal zero for all mvestors 1 Thus, nk’ would equal zero for all assets k and for all 1nVeStOrS I This implies that TI,, = nF = ?rM = 0 and Eq ( 18) would collapse to the capital asset pncing model’s secunty market line However, because of the presence of short sales restnctlons and be- cause, m the absence of these restnctlons, some investors would take short positions m some assets, the 7~ vanables are not necessanly equal to zero and these vanables induce changes m the slope and intercept of the secunty market lme

Consider first the slope of the secunty market line embodied m Eq (18) Because the quantity ( nF - nM) is, in general, not equal to zero, a rotation of the secunty market line with the return on the nskless asset R, as pivot IS induced Thus, the scope of the secunty market line would tend to differ from that predicted by the capital asset pncmg model, 1 e , E@,+, ) - RI. , by the amount ( 7rF - nM ) Note that this divergence term

(TF - nM) IS the difference between the aggregation of all shadow prices applicable to the no short sales constraint on the nskless asset, as given by Eq ( 13)) and the same term for a weighted average of all nsky assets, as given by Eq (14) and the definition of Al,+, In a multlpenod setting, penod-by-per& changes m portfoho decisions would impact on these shadow prices The result would be penod-by-penod changes m the amount by which the empmcally denved secunty market line’s slope would diverge from the capital asset pncmg model’s prediction This conclusion IS consistent with the Black, Jensen, and Scholes [3] finding, noted above, of considerable vanability m the deviations of the slope terms of their emplncally denved secunty market lines from the capital asset pncmg model’s predictions regarding these terms

What would happen to the vertical intercept? Because of the presence of the (nk - 7~~) terms, and because, in general, nk for asset k is not equal to n, for asset], the equlhbnum expected return-beta combmatlons would not form a single lme but would, instead, form a band of values clustenng about the lme

E(R,)=& + [E(&)--RF +br, -nM)lPk (19)

Sales Restrzctlons 95

However, the emplncally denved regression relationship would be linear as thus functional form would be unposed upon the band of values The extent by which the vertical intercept of this empmcal regression line exceeds or falls short of the vertical intercept of Eq (19), R, , would be an artifact of the extent by which the equlhbnum return-beta com- binations deviate from EQ (19) Here agam, note that this deviation 1s a function of the shadow prices associated with the no short sales re- stnctlons on both nsky and n&less assets Penod-by-penod changes m portfolio decisions would induce changes m these shadow prices which m turn would induce penod-by-penod vanatlons m the amounts by which the intercept terms of empmcally denved secunty market lines would diverge from the capital asset pncmg model’s predlctlons Thus, the Black, Jensen, and Scholes [3] finding of considerable vanatlon m deviations of the vertical intercept terms 1s corroborated

Conclusion

By reformulatmg the mdlvldual mvestor’s portfoho selection problem to take account of short selling restnctlons, the resulting secunty market lme would tend to have a different slope and intercept from those predicted by the capital asset pncmg model, 1 e , (E(R,) - RF) and R, , respectively These deviations of the slope and intercept terms should vary from penod to penod Thus, this capital market imperfection has the potential to explam the observed dlscrepancles between the capital asset pncmg model’s predicted secunty market lme and the empmcally denved secunty market lines of vanous researchers

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