IMPLICATIONS OF A FIRM S MARKET WEIGHT IN A CAPM FRAMEWORK

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1 IMPLICATIONS OF A FIRM S MARKET WEIGHT IN A CAPM FRAMEWORK Marti Lally School of Ecoomics ad Fiace Victoria Uiversity of Welligto* ad Steve Swidler J. Staley Macki Professor of Fiace Departmet of Fiace Aubur Uiversity ** # * PO Box 600, Welligto, New Zealad Phoe: Fax: Marti.Lally@vuw.ac.z ** 303 Lowder Busiess Buildig Aubur Uiversity, AL Phoe: (334) Fax: (334) swidler@aubur.edu # Steve Swidler will atted the symposium ad preset the paper Ackowledgemets: The commets of Gle Boyle, Mike Sher ad Mark Tippett are gratefully ackowledged. The secod author would like to thak the Office of the Comptroller of the Currecy for providig release time for this research. However, the paper s cotet does ot reflect the official view of the Office of the Comptroller of the Currecy, ad ay remaiig errors are the resposibility of the authors.

2 IMPLICATIONS OF A FIRM S MARKET WEIGHT IN A CAPM FRAMEWORK Abstract This paper derives the relatioship betwee a stock's beta ad its weightig i the portfolio agaist which its beta is calculated. Cotrary to ituitio the effect of this market weight is i geeral very substatial. We the suggest a alterative to the covetioal measure of abormal retur, which requires a estimate of a firm s beta whe its market weight is zero. We argue that the alterative measure is superior, ad show that it ca differ substatially from the covetioal measure whe a firm has o-trivial market weight. The differece i abormal returs may be disaggregated ito a market retur effect ad a beta effect. JEL classificatio: G1 Keywords: Beta; CAPM; Market Weight

3 IMPLICATIONS OF A FIRM S MARKET WEIGHT IN A CAPM FRAMEWORK 1. Itroductio Sice the developmet of the risk measure covetioally termed beta, by Sharpe (1963, 1964) ad Liter (1965), a umber of papers have developed theoretical relatioships betwee it ad various uderlyig variables. These iclude Hamada (197) with respect to fiacial leverage, Rhee (1986) cocerig operatig leverage, ad Ehrhardt ad Shrieves (1995) for the impact of warrats ad covertible securities. Others, such as Roseberg ad Guy (1976) have empirically idetified variables correlated with beta. Give that the portfolio agaist which a asset's beta is calculated icludes that asset, the beta must also be sesitive to the weightig of that asset i the portfolio. This paper models the relatio betwee a asset's beta ad its market weight, ad the discusses its implicatios for measures of abormal returs. The sigificace of this issue will deped upo the extet to which sigle assets achieve otrivial market weights. Assumig that betas are defied agaist domestic "market" portfolios, a o-trivial market weight occurs for a umber of assets i differet fiacial markets 1. Furthermore, the smaller the market, the more importat is the pheomeo. A example of this is Hog Kog's Hag Seg Idex, i which Chia Telecom represets 4%, ad two others exceed 10%. Similar situatios arise i New Zealad's NZSE40 Idex, with Telecom NZ represetig 3%, ad the ext two lyig i the 5-10% rage, ad i The Netherlads where Royal Dutch Petroleum costitutes 0% of the CBS Idex ad the ext four lie i the 8-10% rage. Most remarkable of all is Filad, i which Nokia represets 69% of the HEX Geeral Idex ad a secod firm represets a further 10%. Eve i the UK, which costitutes the world's secod largest equity market, the two largest stocks each represet aroud 10% of the FTSE 100 (all data courtesy of Ord Miett). 1 It is stadard practice to estimate betas agaist domestic market portfolios rather tha the world market portfolio. A commo motivatio for doig so is estimatio of a firm s cost of equity usig some versio of the Capital Asset Pricig Model that assumes that atioal capital markets are segregated. This assumptio of segregatio is broadly cosistet with the observatio that ivestor portfolios are strogly tilted towards domestic assets (see, for example, Cooper ad Kaplais (1994) ad Tesar ad Werer (1995)).

4 The paper begis by developig the relatioship betwee a stock's beta ad its weightig i the market proxy. Clearly, as a stock's weight i the market goes to 1, its beta agaist that market also approaches 1. Ituitio might suggest that the progressio is mootoic, ad possibly eve proportioal to the weight. However subsequet examiatio shows that this ituitio would be icorrect, ad istead fids a o-liear relatioship. Moreover, the aalysis suggests that eve if a firm's set of risk characteristics remais costat, chages i market weight ca cause dramatic shifts i its beta. The paper the goes o to explore this pheomeo's implicatios for the process of measurig abormal returs. The aalysis offers a alterative method to usig the stadard market model, ad shows how the differece i estimated abormal returs ca be disaggregated ito a "market retur effect" ad a "beta effect.". Modelig the Relatioship betwee a Asset's Beta ad Market Weight The followig aalysis derives how the beta of a stock varies as its market weight chages. Let R deote a asset's retur, m the market proxy, w the weight of i m, the market proxy exclusive of, ad the beta of agaist. The, the beta of agaist m (i.e., the stadard defiitio of a asset's ) is: Var ( R R ) Cov, Cov ( R ) m m [ R, wr + (1 w) R ] [ wr + (1 w) R ] Var w + (1 w) (1) w + (1 w) + w(1 w) Dividig through by yields

5 w + (1 w) () w + (1 w) + w(1 w) Assumig that, ad are idepedet of w, it follows that the relatioship betwee ad w is either proportioal or mootoic. To prove this, we show that the slope of chages sig betwee the two market weightig extremes. Defiig V as the deomiator o the right had side of (), the differetiatig with respect to w ad usig the quotiet rule yields d dw V w + (1 w) w V + w + 4w (3) If asset costitutes the etire market (w 1), the this reduces to d dw 1 (4) < 0 for > ad > 0 Also, if w 0 d dw + + (5) > 0 for 3

6 < < (6) If, as i (4), it is assumed that > 0, the (6) simplifies to < (7) The above derivatios make it clear that the exact curvature of the path for betwee w 0 ad w 1 depeds upo the boudary coditios ivolvig,, ad. A typical stock has variace four times that of the market (Fama, 1976, pp. 5-54). So, if the (7) becomes < Eve if the (7) reduces to < 1.8, a requiremet likely met by may stocks. The other requiremets oted i (4) are implied by > ad > 0, ad these should be satisfied by most stocks. Thus, for the typical stock, 4 d dw w 0 > 0 ad d dw w 1 < 0 Sice the slope chages from positive to egative as the market weight of asset icreases, it follows that reaches its maximum at 0 < w < 1. This implies o-mootoicity, as claimed. A illustratio of the o-mootoic relatioship betwee ad w appears i Figure 1. Cosider a stock where.5 ad. From equatio () 4 4w 4w +.5(1 w) + (1 w) + w(1 w)(.5) 4

7 As the firm's market weight icreases from zero to.15, doubles from.50 to 1. As the market weight icreases further, reaches its maximum value of 1.53 (at a market weight of.41) ad the falls to 1 as w approaches 1. The o-mootoic patter illustrated i Figure 1 has at least two importat implicatios. First, although goes to 1 as w goes to 1, it will diverge from 1 over some rage of values for w. Secod, the o-mootoic patter implies that a stock with a low value for (i.e., less tha 1) whe its market weight is low will have a high value for (i.e., greater tha 1) at certai higher market weights. All of this implies that modest chages i w ca produce quite dramatic shifts i. This is most proouced whe is less tha 1. The full thrust of the secod implicatio is that, depedig o a firm's market weight, it may be a high or low beta stock. Put aother way, whe market weight is o-trivial, a give set of firm risk characteristics (i.e., a give ad ) does ot imply a uique value for systematic risk. Thus, it is possible for a firm to go from a low beta stock to a high beta stock with little chage i it's volatility ( ), the volatility of the rest of the market ( ) or its correlatio with other stocks ( ). At the same time portfolio (i.e., the rest of the market portfolio) also has a beta agaist the market, which complemets that of, i.e. w ( 1 w) 1 (8) + m which implies 1 w m (9) 1 w Thus, m also varies with w, i.e., the average stock withi experieces a market weight effect govered by this complemet law. For the values from the example, Figure 1 plots the path of m. At w.5, asset 's beta has icreased from.5 to while the beta of the remaiig assets, i aggregate, has falle from 1 to

8 3. Implicatios for Abormal Returs The effect of a asset's market weight o its beta has importat empirical implicatios for measurig abormal returs. The covetioal measure of the abormal retur o asset i period t is the excess of the asset retur i that period over its market model couterpart, AR, t R t + [ ˆ α ˆ R ] (10) mt where αˆ ad ˆ are estimated from a time series regressio of R o market retur R m (see, for example, Brow ad Warer, 1985). The "ormal" retur [. ] is typically iterpreted as a estimate of R t i the absece of evets specific to firm, so that the abormal retur is a measure of firm specific evets. However, if asset is icluded i the market idex m, the R mt ad hece the ormal retur icludes R t. Cosequetly the ormal retur icludes firm specific evets, ad so the abormal retur fails to represet firm specific evets. If asset 's weight i m is o-trivial, this error may be substatial. A possible solutio to this problem is to replace R mt by R t, which does ot iclude R t ad hece does ot iclude the evet specific to firm. As a further cosequece, αˆ ad ˆ must be estimated by a time-series regressio of R t o R t. This leads to a measure of abormal retur ivolvig a alterative versio of the market model, AR, t R t + [ ˆ α ˆ R ] (11) t where the time-series regressio of R t o R t yields the estimated coefficiets αˆ ad ˆ. The stadard market model is iferior i two respects. First, it uses a market idex, ad hece a measure of ormal retur, that partly icludes the very shock oe is tryig to measure. Thus the measure of abormal retur must be biased. Secod, i so far as a firm experieces a otrivial chage i its market weight over the beta estimatio period, the estimate of beta i the covetioal market model will be biased, ad this flows through to the estimate of the abormal retur. By cotrast, the proposed abormal retur measure is free of both cocers. The firm s shock is ot icluded i the measure of ormal retur. Furthermore, if the firm experieces a 6

9 o-trivial shift i market weight over the beta estimatio period, this has o effect upo the estimate of its beta, because the latter is ivariat to the firm s market weight. The differece i these two abormal retur measures is the AR ˆ α ˆ ˆ α, t AR, t + Rt ˆ R mt ( ˆ α ˆ α ) ˆ ( R R ) + R ( ˆ ˆ ) (1) mt t t The first of the three terms o the right had side of (1) is geerally trivial because estimated alphas are typically close to zero for the short periods used i abormal retur aalysis. The secod term (the market retur effect ) arises from the differece i market retur measures ad the third term (the beta effect ) arises from the differece i betas. Sice the first of these two betas i the last term ( ) ca be viewed as the covetioal beta whe the asset s market weight is zero, ad the previous sectio has show that the covetioal beta ca vary sigificatly with market weight shifts, the the beta effect ca be substatial if the asset s market weight is otrivial. To illustrate this issue we examie the tradig history of Telecom Corporatio of New Zealad, the largest firm o the New Zealad Stock Exchage. From its iitial public offerig o July 18, 1991 to Jue 30, 1997, Telecom's market weight i the New Zealad idex (NZSE40) fluctuated betwee 16 ad 9 percet. To further appreciate Telecom's sigificace i the New Zealad stock market, its correlatio coefficiet with the NZSE40 over the six-year period is.74. If Telecom is excluded from the idex, which is the deoted the NZSE39, the correlatio coefficiet drops to.49. Across this period of time we select the two largest absolute daily returs for Telecom, ad these are show i Table 1. The first of these is February 16, 1993, whe Telecom aouced strog earigs, well above its profits from the previous year, ad its stock price icreased 11.4% (compared with the retur for the rest of the market of.9%). The stadard measure of abormal retur i equatio (10) yields a abormal retur equal to 8.5%, whereas estimatig AR yields If expectatios are applied to the market models i equatios (10) ad (11), the the alphas are the itercepts i expected retur models. Over short periods of time (such as days, as examied below), expected returs are very small; accordigly, so too are the alphas. The estimated alphas show i Table 1 are cosistet with this. 7

10 the more appropriate result of 10.9%. The differece of.4% i these abormal retur measures is due primarily to the market retur effect (%). Here the beta effect is modest (.3%), i spite of the substatial differece i the two beta estimates (.513 versus.891), simply because R t is close to zero. The secod of these days is Moday, November 8, 1993, ad this provides a eve more strikig example of the differece i the two abormal retur methodologies. Over the precedig weeked, a atioal electio took place i New Zealad ad a party which was widely perceived as ati-busiess, ad which had threateed to buy back recetly spu-off public assets, performed well eough to potetially hold the balace of power. Ivestors iterpreted the electio result as bad ews for the market, ad particularly bad ews for Telecom, the largest of the recetly sold public assets. Table 1 shows that Telecom's retur for the day was 9.3% compared to the rest of the market's retur of 5.0%. Usig the traditioal measure of abormal retur i equatio (10), Telecom's abormal retur was 1.5%. However, AR uses a market retur that sigificatly depeds upo Telecom's retur, ad therefore does ot yield a abormal retur measure that disetagles the electio's separate effect o Telecom. AR provides a better measure of the electio's specific effect o Telecom ad equals 5.7%, early four times the size of AR. The differece of 4.% i these abormal retur measures is ow largely due to the beta effect (.7%) with most of the remaiig differece attributable to the market retur effect (1.3%). This beta effect arises because the two beta estimates are agai substatially differet (.769 versus 1.31). All of these calculatios presume that the betas are accurately estimated. However, as we have oted earlier, the estimate of will be biased if the firm experieces a o-trivial shift i this over the estimatio period. Estimatio of is free of this problem. The two examples provide evidece that the two abormal retur models ca geerate substatially differet results, that the differece i betas may be substatial, ad that the latter ca cotribute sigificatly to the differece i abormal returs. I additio to all of this, the differece i betas coforms to the predictios of the theoretical aalysis, i.e., if beta is sigificatly less tha oe whe market weight is zero (the beta estimates of this kid are.513 for the first evet i Table 1 ad.769 for the secod) the the effect of sigificatly icreasig market weight is to sigificatly icrease the beta (to.891 for the first evet i Table 1 ad 1.31 for the secod). Fially, with respect to the secod evet i Table 1, the shift i beta from.769 8

11 to 1.31 illustrates the poit made earlier that a substatial chage i market weight ca alter a stock from a low beta oe to a high beta oe. 4. Coclusio This paper derives the relatioship betwee a stock's beta ad its weightig i the market idex. Cotrary to ituitio, the relatioship is i geeral o-mootoic, with the result that market weight variatios withi the observed rage ca have dramatic effects o a firm's beta. We the suggest a alterative to the covetioal measure of abormal retur, which requires a estimate of a firm s beta whe its market weight is zero. We argue that the alterative measure is superior, ad show that it ca differ substatially from the covetioal measure whe a firm has a o-trivial market weight. The differece i abormal returs may be due i large part to the differece i betas, which will arise if the firm's market weight is o-trivial. The fact that market weight chages ca substatially affect a stock s beta also has implicatios for the cost of capital whe the Capital Asset Pricig Model is ivoked. I particular, the aalysis suggests that the traditioal measure of systematic risk, ad thus the opportuity cost of capital, may chage eve whe a firm's lie of busiess remais the same. We leave the implicatios of market weight o a firm's cost of capital for future research. 9

12 Refereces Brow, S., Warer, J., Usig Daily Stock Returs: The Case of Evet Studies, Joural of Fiacial Ecoomics 14, Cooper, I., Kaplais, E., Home Bias i Equity Portfolios, Iflatio Hedgig ad Iteratioal Capital Market Equilibrium, The Review of Fiacial Studies 7, Ehrhardt, M., Shrieves, R., The Impact of Warrats ad Covertible Securities o the Systematic Risk of Commo Equity, The Fiacial Review 30, Fama, E., Foudatios of Fiace. Basic Books, New York. Hamada, R., 197. The Effect of the Firm's Capital Structure o the Systematic Risk of Commo Stocks, Joural of Fiace 7, Liter, J., The Valuatio of Risky Assets ad the Selectio of Risky Ivestmets i Stock Portfolios ad Capital Budgets, Review of Ecoomics ad Statistics 47, Rhee, G., Stochastic Demad ad a Decompositio of Systematic Risk, Research i Fiace 6, Roseberg, B., Guy, J., Predictios of Beta From Ivestmet Fudametals: Part II, Fiacial Aalysts Joural July-August, Sharpe, W., A Simplified Model For Portfolio Aalysis, Maagemet Sciece 9, , Capital Asset Prices: A Theory of Market Equilibrium Uder Coditios of Risk, Joural of Fiace 19, Tesar, L., Werer, I., Home Bias ad High Turover, Joural of Iteratioal Moey ad Fiace 14,

13 Figure 1 The Relatioship Betwee Beta ad Market Weight m w This figure shows the relatioship betwee asset 's market weight w ad its beta agaist the market portfolio,, ad also the relatioship betwee w ad the beta of portfolio (the market exclusive of ) agaist the market portfolio, deoted m. The figure assumes that asset 's beta agaist,, is 0.5 ad that 's variace is four times that of. 11

14 Table 1 Differeces i Abormal Retur Measures: AR versus AR Examples from Telecom NZ Stock Returs Tradig Telecom NZSE40 αˆ ˆ AR,t NZSE39 αˆ ˆ Date Retur Retur Retur AR,t Feb Nov This table computes the abormal returs for Telecom o two dates, ad applies two measures of the market idex - the NZSE40 ad the NZSE39. The resultig abormal returs are desigated AR,t ad AR,t respectively. Desigatig the tradig date as t 0, the α ad coefficiets are estimated from OLS regressios based o daily returs from t 160 to t 11. 1

An Empirical Study of the Behaviour of the Sample Kurtosis in Samples from Symmetric Stable Distributions

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