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Asset pricing and bid-ask pricing

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Asset pricing and bid-ask pricing Journal of Financial Economics 17 (1986) 223-219. North-Holland ASSET PRICING AIVD THE BID-ASK SPREAD* Received August 1985. tinal rsrsion received .~pril 1986 This paper studies the effect of the bid-ask spread on asset pricing. We analyze a model in w...

Asset pricing and bid-ask pricing
Journal of Financial Economics 17 (1986) 223-219. North-Holland ASSET PRICING AIVD THE BID-ASK SPREAD* Received August 1985. tinal rsrsion received .~pril 1986 This paper studies the effect of the bid-ask spread on asset pricing. We analyze a model in which investors with different expected holding periods trade assets with different relative spreads. The resulting testable hypothesis is that market-obsewed expected return is an increasing and concave function of the spread. We test this hypothesis. and the empirical results are consistent with the predictions of the model. 1. Introduction Liquidity. marketability or trading costs are among the primary attributes of many investment plans and financial instruments. In the securities industry, portfolio managers and investment consultants tailor portfolios to fit their clients’ investment horizons and liquidity objectives. But despite its evident importance in practice, the role of liquidity in capital markets is hardly reflected in academic research. This paper attempts to narrow this gap by examining the effects of illiquidity on asset pricing. llliquidity can be measured by the cost of immediate execution. An investor willing to transact faces a tradeoff: He may either wait to transact at a favorable price or insist on immediate execution at the current bid or ask price. The quoted ask (offer) price includes a premium for immediate buying, and the bid price similarly reflects a concession required for immediate sale. Thus. a natural measure of illiquidity is the spread between the bid and ask *We wish to thank Hans Stall and Robert W’haley for furnishing the spread data, and Manny Pai for excellent programming assistance. WC acknowledge helpful comments by the Editor. Clifford W. Smith. by an anonymous referee, by Harry DeAngelo. Linda DeAngelo, Michael C. Jensen. Krishna Ramaswamy and Jerry Zimmerman. and especiallv by John Long and G. William Schwert. Partial financial support by the lManageriaJ Economics Research Center-of the University of Roth .ster. the Salomon Brothers Center for the Study of Financial Markets. and the Israel Institute for Business Research is acknowledged. 0304~405X/86/53.50 S 1986. Elsevier Science Publishers B.V. (North-Holland) prices, which is the sum of the buying premium and the selling concession.’ Indeed. the relative spread on stocks has been found to be negatively corre- lated with liquidity characteristics such as the trading volume, the number of shareholders. the number of market makers trading the stock and the stock price continuity.’ This paper suggests that expected asset returns are increasing in the (rela- tive) bid-ask spread. We first model the effects of the spread on asset returns. Our model predicts that higher-spread assets yield higher expected returns, and that there is a clientele effect whereby investors with longer holding periods select assets with higher spreads. The resulting testable hypothesis is that asset returns are an increasing and concave function of the spread. The model also predicts that expected returns net of trading costs increase with the holding period, and consequently higher-spread assets yield higher net returns to their holders. Hence, an investor expecting a long holding period can gain by holding high-spread assets. We test the predicted spread-return relation using data for the period 1961-1980, and find that our hypotheses are consistent with the evidence: Average portfolio risk-adjusted returns increase with their bid-ask spread, and the slope of the return-spread relationship decreases with the spread. Finally, we verify that the spread effect persists when firm size is added as an explanatory variable in the regression equations. We emphasize that the spread effect is by no means an anomaly or an indication of market in- efficiency; rather, it represents a rational response by an efficient market to the existence of the spread. This study highlights the importance of securities market microstructure in determining asset returns, and provides a link between this area and mainstream research on capital markets. Our results suggest that liquidity- increasing financial policies can reduce the firm’s opportunity cost of capital, and provide measures for the value of improvements in the trading and exchange process.3 In the area of portfolio selection, our findings may guide investors in balancing expected trading costs against expected returns. In sum, we demonstrate the importance of market-microstructure factors as determi- nants of stock returns, In the following section we present a model of the return-spread relation and form the hypotheses for our empirical tests. In section 3 we test the ‘Demsetz (1968) first related the spread to the cost of transacting. See also Amihud and Mendelson (1980.1982). Phillios and Smith (1982). Ho and Stoll(1981.1983). Coueland and Galai (1983). and‘west and Tinic (i971). For an ‘analysis of transaction costs in the context of a fixed investment horizon. see Chen. Kim and Kon (1975). Levy (1978). Milne and Smith (1980). and Treynor (1980). ‘See. e.g.. Garbade (1982) and St011 (1985) ‘See, e.g., Mendelson (1982.1985,1986.1987), Amihud and Mendelson (1985.1986) for the interaction between market characteristics, trading organization and liquidity. predicted relationship, and in section 4 we relate our findings to the firm size anomaly. Our concluding remarks are offered in section 5. 2. A model of the return-spread relation In this section we model the role of the bid-ask spread in determining asset returns. We consider M investor types numbered by i = 1.2.. . . , M, and N + 1 capital assets inds.;ed by i = 0, 1,2,. . . . N. Each asset i generates a perpetual cash flow of $d, per unit time (d, > 0) and has a relative spread of S,, reflecting its trading costs. Asset 0 is a zero-spread asset (S, = 0) having unlimited supply. Assets are perfectl? divisible. and one unit of each positive- spread asset i (i = 1.2.. . ., N) is available. Trading is performed via competitive market makers who quote assets’ bid and ask prices and stand ready to trade at these prices. The market makers bridge the time gaps between the arrivals of buyers and sellers to the market, absorb transitory excess demand or supply in their inventory positions. and are compensated by the spread, which is competitively set. Thus. they quote for each asset i an ask price 7 and a bid price V,(l - S,). giving rise to two price vectors: an ask price vector (V,, V,, . . . . Vv) and a bid price vector (V,. V,(l - St) ,..., V,(l - S,v)).” A type-i investor enters the market with wealth U: used to purchase capital assets (at the quoted ask prices). He holds these assets for a random, exponentially distributed time q with mean E[T] = l/p,. liquidates his port- folio by selling it to the market makers at the bid prices, and leaves the market. We number investor types by increasing expected holding periods, PI -tQL~tl .** I/.$, and assets by increasing relative spreads, 0 = S, I S,I ... < S,v c 1. Finally, we assume that the arrivals of type-i investors to the market follow a Poisson process with rate X,, with the interarrival times and holding periods being stochastically independent. In statistical equilibrium, the number of type-i investors with portfolio holdings in the market ha, a Poisson distribution with mean m, = X,/p, [cf. Ross (1970, ch. 2)]. The market makers’ inventories fluctuate over time to accommodate transitory excess demand or supply disturbances, but their expected inventory positions are zero, i.e., market makers are ‘seeking out the market price that equilibrates buyin 0 and selling pressures’ [Bagehot (?57i, p. 14); see also Garman (1976)]. This implies that the expected sum of investors’ holdings in each positive-spread asset is equal to its available supply of one unit. Consider now the portfolio decision of a type-i investor facing a given set of bid and ask prices, whose objective is to maximize the expected discounted net 4Competition among market makers drives the spread to the level 3, of trading costs. In a different scenario. C; may be viewed as the sum of the market price and the buying transaction cost. and V, (1 - S, ) as the price net of the cost of a sell transaction. cash flows received over his planning horizon. The discount rate p is the spread-free, risk-adjusted rate of return on the zero-spread asset. Let Y,, be the quantity of asset J’ acquired by the type-i investor. We call the vector ( .Y,,. i = 0.1.2.. . . . LV} ‘portfolio i’. The expected present value of holding portfolio i is the sum of the expected discounted value of the continuous cash stream received over its holding period and the expected discounted liquida- tion revenue. This sum is given by =(p,+P)-‘ix~,[~,+P,Vi(‘-S,)l. J=o Thus, for given vectors of bid and ask prices, a type-i investor solves the problem max i x,,[d/+P,V,(l -s,)]. J-0 (1) subject to ,v c,y,,V,< W, and X,/LO forall j=O,1.2 ,..., N. J=o (2) where condition (2) expresses the wealth constraint and the exclusion of investors’ short positions.’ Under our specification, the usual market clearing conditions read : m,x,,=l, j= 1,2...., iv (3) r-1 (recall that m, is the expected number of type-i investors in the market). When an A4 x (N + 1) matrix X* and an (N + l)-dimensional vector V* solve the M optimization problems (l)-(2) such that (3) is satisfied. we call X * an equilibrium allocation matrix and V * - an equilibrium ask price vector [the corresponding bid price vector is ( VO*, F’t*(l - S,), . . . , Vf(l - S,- )]. The ‘In our context. the use of short sales cannot eliminate the spread effect. since short >aIes b! themselves entail additional transaction costs. Note that a constraint on short po,itions ib necessary in models of tax clienteles [cf. Miller (1977). Litzenberger and Ramaswamy (1980)]. Clearly. market makers are allowed to have transitory long or short positions. but are constrained to have zero expected inventory positions [cf. Garman (1976)]. above model may be viewed as a special case of the linear exchange model [cf. Gale (1960)]. which is known to have an equilibrium allocation and a unique equilibrium price vector. Our model enables us to derive and interpret the resulting equilibrium in a straightforward and intuitive way as foilows. We define the expected spread-adjusted return of asset j to investor-type i as the difference between the gross market return on asset j and its expected liquidation cost per unit time: r ,,=d,/v,-Ir& where d/V, is the gross return on security j, and p,S, is the spread-adjwt- ment, or expected liquidation cost (per unit time), equal to the product of the liquidation probability per unit time by the percentage spread. Note that the spread-adjusted return depends on both the asset j and the investor-type i (through the expected holding period). For a given price vector V, investor i selects for his portfolio the assets j which provide him the highest spread-adjusted return, given by r* = I max r, ;, (5) j-0.1.2 . . . . . N 1 with r,* I r2” I r3* I . . . I r;, since, by (4), r,, is a non-decreasing function of i for all j. These inequalities state that the spread-adjusted return on a portfolio increases with the expected holding period. That is, investors with longer expected holding periods will earn higher returns net of transaction costs.6 The gross return required by investor i on asset j is given by r,* + p(S,. which reflects both the required spread-adjusted return r,* and the expected liquidation cost p,.S,. The equilibrium gross (market-observed) return on asset j is determined by its highest-valued use, which is in the portfolio i with the minimal required return, implying that dj/V,* = mm r-1.2 .._.., v$* +PJJ. Eq. (6) can also be written in the form v,* = max r=l.Z . . . . . . M { d,/(r,* + p,s,)}9 (6) ‘This is consistent with the suggestions that while the illiquidity of investments such as real estate [Fogler (1984)] coins [Kane (1984)] and stamps [Taylor (1983)] excludes them from short-term investment portfolios, they are expected to provide superior performance when held over a long investment horizon (the same may‘apply to stock-exchange seats) [Schwert ( 1977)]. See also Day. Stall and Whaley (1985) on the clientele of small firms. and Elton and Gruber (1978) on tax clienteles. implying that the equilibrium value of asset j. V, *. is equal to the present value of its perpetual cash flow, discounted at the gross return (rI* + p,S,). Alterna- tively, VJ* can be written as the difference between (i) the present value of the perpetual cash stream d, and (ii) the present value of the expected trading costs for all the present and future holders of asset j. where both are discounted at the spread-adjusted return of the holding investor. To see this, assume that the available quantity of asset j is held by type-i investors: then (7) can be written as v/* = d/r,* - p,V,*S,/r,*, where the first term is, obviously. (i). As .for the second, the expected quantity of asset j sold per unit time by type-i investors is p,. and each sale incurs a transaction cost of <*S,; thus, /.~,y*S,/r,* is the expected present value (discounted at r,*) of the transaction-cost cash flow. The implications of the above equilibrium on the relation between returns, spreads and holding periods are summarized by the following propositions. Proposition I (clientele effect). Assets with higher spreads are allocated in equilibrium to portfolios with (the same or) longer expected holding periods. Proof. Consider two assets, j and k, such that in equilibrium asset j is in portfolio i and asset k is in portfolio i + 1 (recall that I*, 2 p,_ r). Apply- ing (5), we obtain rI, 2 rrk and r,_ 1. k 2 r,_ I.,; thus, substituting from (4), d,/V,*-I.r,S,2dk/V~-IL,Sk and dk/V~-~,,,SL)d,/~*-~,_lS,, im- plying that (IL, - p,*r)(Sk - S,) 2 0. It follows that if CL, > p,,r, we must have S, 2 S,. The case of non-consecutive portfolios immediately follows. Q.E.D. Proposition 2 (spread-return relationship). In equilibrium, the obserced market (gross) return is an increasing and concave piecewise-linear function of the (relative) spread. Proof. Let f,(S)= r,* + p,S. By (6) the market return on an asset with relative spread S is given by f(S) = min,,,.z..,.. ,&(S). Now, the proposition follows from the fact that monotonicity and concavity are presemed by the minimum operator, and that the minimum of a finite collection of linear functions is piecewise-linear. Q.E.D. Proposition 2 is the main testable implication of our model. Intuitively, the positive association between return and spread reflects the compensation required by investors for their trading costs, and its concavity results from the clientele effect (Proposition 1). To see this, recall that transaction costs are amortized over the investor’s holding period. The longer this period, the Market Investor type, i return in excess 1 2 3 4 of p. the Value of asset Relative return 00 j relative to bid-ask Length of holding period, p,-’ Asset, spread, l/12 l/2 1 5 the zero- that of the zero- spread spread asset. i s, Excess spread-adjusted return. r,, - p asset v//v, (1) (2) (3) (4) (5) (6) (7) (8) 0 0 101 0 0 0 0 1 1 0.005 1 0 L_l 0 0.05 0.055 0.059 0.06 0.943 2 0.01 3 0.015 - 0.05 l-l OS0 0.11 0.118 0.1’ 0.893 0.10 0.115 0.127 0.13 0.885 4 0.02 - 0.10 0.12 5 0.025 -0.155 W5 6 0.03 -0.21 0:09 0.12 7 0.035 - 0.265 0.085 8 0.04 - 0.324 0.076 D 0.136 0.11 0.877 0.140 0.145 0.873 0.12 0.144 0.15 0.870 0.12 0.148 0.155 0.866 ,116 0.148 0.156 0.865 9 0.045 - 0.383 0.067 0.112 0 0.148 0.157 0.864 aIovestors have the same wealth, and the expected number of investors of each type is 1. Y. A mhud and H. .Ylendelson. Asser prmng and the hrd-ash spread 22 9 Table 1 An example of the equilibrium relation between asset bid-ask spreads. returns and values (see section 2). There are 10 assets (j), each generating 51 per period, with relative bid-ask spreads S, ( = dollar spread divided by asset value) ranging from 0 to 0.045 (column 2). and 4 investor types (i) with expected holding periods, p;‘, of l/12. l/2, 1 and 5 periods.’ The return on the zero-spread asset is p; all re:urns are measured in excess of p. A type-i investor chooses the assets j which m aximixe his spread-adjusted return, r,,, given by the difference between the gross market return on asset j and its expected liquidation cost per unit time. The equilibrium solution gives the excess spread-adjusted returns, r,, - p, in columns 3-6, where the boxes highlight the assets with the highest excess spread-adjusted return for each investor-type. The equilibrium portfolio for each investor-type is composed of the boxed assets. Column 7 shows the assets’ equilibrium excess gross returns observed in the market. which include the expected liquidation cost to their holders. Co1um11 8 shows the resulting asset values, obtained by discounting the perpetuity by the respective equilibrium market return, as a fraction of the value of the zero-spread asset. smaller the compensation required for a given increase in the spread. Since in equilibrium higher-spread securities are acquired by investors with longer horizons, the added return required for a given increase in spread gets smaller. In terms of our model, !onger-holding-period portfolios contain higher-spread assets and have a lower slope ,LL, for the return-spread relation. A simple numerical example can illus,;ate the spread-return relation. Assume N = 9 positive-spread assets and M = 4 investor types whose expected holding periods are 1,‘~~ = l/12, l/p2 = l/2, l/p”, = 1, and l/p4 = 5. For simplicity we set X, = p,, implying that the expected number of investors of each type i is m, = 1. Assets yield d, = Sl per period, and all investors have equal wealth. The relative spread of asset j is S, = O.OOSj. j = 0, 1,2,. . . ,9; thus, asset percentage spreads range from zero to 4.5%. Using this data, we solve (l)-(3) and obtain the results in table 1 and figs. 1 and 2. Note that the additional excess return per unit of spread goes down 230 Y. A mlhud and H. .Men~elson .4sser pnwq und rhe hrd-usk spreud investor 0 0.01 0.02 0.03 0.04 0.05 RELATIVE BID-ASK SPREAD Fig. 1. An illustration of the relation between observed market return in excess of the return on the zero-spread asset (the excess gross return) and the relative bid-ask spread (see the numerical example of section 2 and table 1, column 7). There are 10 assets. each generating I$! per period. with relative bid-ask spreads (= dollar spread divided by asset va!ue) ranging from 0 to 0.045. and 4 investor types with expected holding periods ranging from l/l2 to 5 periods. Investors hav; equal wealth. and the expected number of investors of each type is 1. The relation between asset returns and bid-ask spreads is piecewise-linear. increasing and concave, with each linear section corresponding to the portfolio of a diRerent investor type. from p1 = 12 in portfolio 1 to p2 == 2 for portfolio 2, then to pj = 1 in portfolio 3, and finally to p4 = 0.2 in portiolio 4. The behavior of the excess markt t return as a function of the spread is shown in fig. 1, which demonstrates both the positive compensation for higher spread and the clientele effect which moderates the excess returns, especially for the high-spread assets. This figure summarizes the main testable implications of our model: The observed market return should be an increasing and concave function of the relative spread. The piecewise-linear functional form sug
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