Apples and Oranges: Building a Better Dow

Transcription

1 Apples and Oranges: Buldng a Better Dow By Debraj Ray and Rajv Seth August Introducton The Dow Jones Industral Average s among the most closely watched stock market ndexes n the world. It s also an enormously valuable resource for research, wth an unnterrupted data seres stretchng back to Yet t has some glarng omssons. Apple, currently the largest publcly traded company n the world wth a market captalzaton of over $577 bllon, sn t a Dow component. Nor s Google, wth a hgher valuaton than all but a handful of frms n the ndex. Meanwhle, companes wth less than a tenth of Apple s valuaton, ncludng Alcoa and Hewlett-Packard, contnue to be ncluded. The reason for such seemngly arbtrary omssons has to do wth the manner n whch the Dow s computed. It s a prce-weghted ndex, and the average prce of ts thrty components s currently around $58. Both Apple and Google have share prces n excess of $600, and ther ncluson would cause day-to-day changes n the ndex to be drven largely by the behavor of these two securtes. As we show below, ther combned weght n the Dow would be about 43% f they were both to be ncluded n place of the two current components (Alcoa and Travelers) wth the lowest market captalzatons. Furthermore, the ndex would become consderably more volatle even f the ncluded stocks were ndvdually no more volatle than those they replace. As John Prestbo, Charman of the Index Oversght Commttee recently observed, such heavy dependence of the ndex on one or two stocks would hamper ts ablty to accurately reflect the broader market. 1 Indeed, prce-weghtng s decdedly an odd methodology. Smply addng Chevron s share prce to that of AT&T (as the Dow does) ensures that a one-dollar change n the prce of Chevron wll have the same effect on the ndex as a one-dollar change n the prce of AT&T. But a one-dollar change works out to well under a 1% change n the share prce of Chevron (0.89% as of August 3, 2012), whle that same dollar change means a 2.7% change n the share prce of AT&T. That means Chevron s three tmes as mportant n affectng the Dow as AT&T s, and for no legtmate reason. The market captalzaton of each company s just a shade over $218b; they are equally valuable. 1 John Prestbo, Why AAPL and GOOG Aren t n the Dow. Indexology, February

2 2 Ths ssue does not arse wth value-weghted ndexes such as the S&P 500. But an abrupt change of weghtng scheme would result n a methodologcal dscontnuty that would nterfere wth the hstorcal analyss of stock prce data. Prestbo contnues: The Dow s unque n that t has been contnuously calculated usng the same methodology for more than a century. The mportance of havng ths deep market hstory becomes abundantly clear when you start to analyze long-term market trends... Over The Dows 116-year hstory, there have been 23 busness cycles. Wonder how the market wll perform as we come out of a recesson? Or how t mght respond to changes n economc polcy? The Dow s the ndex that offers the most complete dataset. Changng the methodology would essentally oblterate ths hstory. Attenton n the fnancal press has therefore been focused on the need for a stock splt, whch would reduce the prce of Apple shares to a level that could be accommodated by the questonable methodology of the Dow. But an abrupt swtch to a value-weghted ndex or the flawed artfce of a stock splt are not the only avalable alternatves. We propose a modfcaton to the Dow formula that largely preserves the hstorcal ntegrty of the tme seres. At the same tme, t allows for the ncluson of securtes regardless of ther market prce and a smooth and gradual transton, as ncumbent stocks are replaced, to a fully value-weghted ndex n the long run. The proposed ndex s composed of two subndces, one prce-weghted to respect the nternal structure of the Dow, and the other value-weghted to apply to new entrants. There are two parameters, both of whch are adjusted whenever a substtuton s made. One of these mantans contnuty n the value of the ndex, whle the other ensures that the two subndces are weghted n proporton to ther respectve market captalzatons. Stock splts requre a change n parameters (as n the case of the current Dow dvsor) but only f the splt occurs for a frm n the prce-weghted subndex. Once all ncumbent frms have been replaced, the result wll be a fully value-weghted ndex. In practce ths could take several decades, as some ncumbent frms are lkely to be reman Dow components far nto the future. But frms n the prce-weghted component of the ndex that happen to have weghts roughly commensurate wth ther market captalzaton can be transferred wth no loss of contnuty to the value-weghted component. Ths procedure, whch we call brdgng, can accelerate the transton to a value-weghted ndex wth mnmal short term dsrupton. We llustrate ths set of prncples wth several examples.

3 3 2. Indexes Suppose that there are n companes. Company has share prce p and outstandng shares s, so ts market captalzaton s gven by v p s. An ndex based on these n companes s some functon f defned on the vector (p 1, s 1 ;... ; p n, s n ). f s a prce-weghted ndex f f(p 1, s 1 ;... ; p n, s n ) = A for some A > 0. It s a value-weghted ndex f for some B > 0. f(p 1, s 1 ;... ; p n, s n ) = B =1 p p s = B It wll help the exposton to thnk of f as beng defned on a larger doman, say, for all subsets of m stocks, where m n. =1 =1 v 3. Substtuton by Value-Weghtng at the Margn Let f be any ndex. The objectve s to nclude a new stock, call t n + 1 and drop an old stock, say stock 1. At the tme of the change, we are n a partcular envronment; call t ( p 1, s 1 ;... ; p n, s n ; p n+1, s n+1 ). Proceed as follows. Frst, recallng that f s defned on subsets of stocks, defne α > 0 such that (1) αf( p 2, s 2 ;... ; p n, s n ) = v. Now create the new ndex g(p 2, s 2 ;... ; p n, s n ; p n+1, s n+1 ) by defnng (2) g(p 2, s 2 ;... ; p n, s n ; p n+1, s n+1 ) δ [αf(p 2, s 2 ;... ; p n, s n ) + v n+1, where α s defned by (1), and δ s chosen so that (3) δ [αf( p 2, s 2 ;... ; p n, s n ) + v n+1 = f( p 1, s 1 ;... ; p n, s n ). Ths ensures contnuty of the ndex at the pont of substtuton. The procedure defned above takes care of a sngle addton to an arbtrary current ndex, whle value-weghtng at the margn. How about subsequent addtons? Suppose that we wsh to add a second securty n + 2 at some later tme, when the envronment s ( p 2, s 2 ;... ; p n+2, s n+2 ). Agan recallng that f s defned on a subset of n 2 stocks, defne α > 0 such that (4) α f( p 3, s 3 ;... ; p n, s n ) = ṽ. =3

4 4 Now defne a new ndex g (p 3, s 3 ;... ; p n+2, s n+2 ) as (5) g (p 3, s 3 ;... ; p n+2, s n+2 ) δ [ α f(p 3, s 3 ;... ; p n, s n ) + v n+1 + v n+2, where δ s chosen so that (6) δ [ α f( p 3, s 3 ;... ; p n, s n ) + ṽ n+1 + ṽ n+2 = g( p2, s 2 ;... ; p n+1, s n+1 ). As before, we have contnuty of the ndex at the tme of substtuton. Clearly the same prncple can be appled to add more stocks to replace the orgnal ncumbents as tme unfolds. Once all ncumbents have been replaced, we have a value-weghted ndex. Remarks. 1. Note that the stocks n the ndex at any pont n tme can be parttoned nto two groups: those that enter a subndex defned by f and those that enter a value-weghted subndex. All new addtons that replace any asset n the former group can be made n accordance wth the procedure dentfed above. But at some pont n tme t wll be necessary to replace an asset n the latter group. Ths s easly accomplshed. Suppose, for nstance, that stock n + 2 s to be replaced n the ndex (5) by a new stock, say n + 3. Then we replace v n+2 by v n+3 n the formula and adjust the parameters δ and α n order to satsfy two condtons: contnuty of the ndex value at the tme of substtuton, and proportonalty of each subndex wth ther respectve aggregate market captalzatons. 2. If a stock n the value-weghted subndex splts, t has no mpact on the ndex and no adjustment need be made. If a stock n the subndex defned by f splts then an adjustment may be necessary, but ts precse form wll depend on the nature of f. In the next secton we consder splts when f defnes a prce weghted ndex. 3. Our procedure yelds exactly the same answer even f we arbtrarly change f by a multplcatve constant on any subset of companes. To see ths, look at (1). Imagne that on the subset 2,..., n 1, we replace f by ˆf, where ˆf(p 2, s 2 ;... ; p n, s n ) = kf(p 2, s 2 ;... ; p n, s n ) for any k > 0. Then the correspondng value of α, call t ˆα, s obvously related to α by ˆα = α/k and so the subsequent equatons (2) and (3) are entrely unaltered. Exactly the same comments apply to the constructon descrbed n (4) (6). 4. Applcaton: Addng Apple to The Dow The Dow s gven by f(p 1, s 1 ;... ; p n, s n ) = 1 D p, =1

5 where D s the Dow dvsor pror to substtuton. The obvous extenson of f to lowerdmensonal subsets {m,..., n}, after droppng the frst m 1 stocks, s f(p m, s m ;... ; p n, s n ) = A m p, where A m s any postve constant (by Remark 3 n Secton 3, the precse choce of A m s unmportant). Usng ths n (1), we see that for any partcular vector ( p 2, s 2 ;... ; p n, s n ), α = 1 n p s A n 2 p, where we are agreeng to drop stock 1, say Alcoa. So the new ndex after addng Apple (our stock n + 1) s gven by [ n (7) δ p s n p p + p n+1 s n+1, where δ s chosen to satsfy (3), so that there s no jump n the current value of the Dow. More generally, once a total of m 1 ncumbents have been replaced, the ndex takes the followng two-parameter form: [ n+m δ α p + v. =m =n+1 The two parameters δ and α are adjusted each tme a substtuton s made: δ to mantan contnuty n the value of the ndex, and α to ensure that each of the two subcomponents receve weghts n proporton to ther market captalzatons. Stock splts requre changes n both parameters but only f they occur for frms n the prce-weghted subcomponent. Ths constructon allows us to add apples and oranges. At the tmes of substtuton, t s tantamount to addng valuatons just lke a value-weghted ndex would, but n subsequent movements of prces t respects the nternal structure of The Dow (projected onto the frst n companes). Of course the new ndex s nether prce-weghted nor value-weghted. But our teratve procedure allows us to keep addng new stocks. It s obvous that once the orgnal n stocks have dropped out, the procedure generates a fully value-weghted ndex. =m 5 5. An Implementaton Table 1 shows the 30 current Dow components n decreasng order of market captalzaton as of August 3, The ndex tself stood at at ths tme, and the Dow dvsor was approxmately D = The sxth column n the table contans the weghts of the respectve stocks n the computaton of the Dow, whch are smply proportonal to prces. Hence IBM, for example, has a weght n the Dow that s more than

6 6 Symbol Name Prce Shares Captalzaton P-Weght V-Weght AA Alcoa ,066,890,000 8,929,869, % 0.22% TRV Travelers ,358,000 24,551,158, % 0.62% HPQ Hewlett-Packard ,971,830,000 36,005,615, % 0.91% DD E.I. du Pont ,373,000 46,025,552, % 1.16% BA Boeng ,798,000 54,738,412, % 1.38% CAT Caterpllar ,500,000 55,475,550, % 1.39% MMM 3M ,320,000 63,394,044, % 1.59% AXP Amercan Express ,133,650,000 65,309,576, % 1.64% UTX Unted ,787,000 70,034,359, % 1.76% KFT Kraft Foods ,772,988,000 71,823,743, % 1.81% HD Home Depot ,530,660,000 80,038,211, % 2.01% BAC Bank of Amerca ,777,000,000 80,073,110, % 2.01% CSCO Csco ,356,880,000 87,584,988, % 2.20% DIS Walt Dsney ,787,360,000 88,956,907, % 2.24% MCD McDonald s ,016,240,000 91,044,941, % 2.29% VZ Verzon ,848,880, ,661,204, % 3.18% INTC Intel ,003,000, ,228,690, % 3.30% MRK Merck ,041,560, ,284,874, % 3.38% JPM JP Morgan Chase ,806,670, ,382,720, % 3.45% PG Procter & Gamble ,740,110, ,477,205, % 4.51% PFE Pfzer ,488,140, ,812,039, % 4.57% KO Coca-Cola ,250,960, ,945,096, % 4.57% JNJ Johnson & Johnson ,757,040, ,566,604, % 4.79% CVX Chevron ,962,140, ,032,996, % 5.48% T AT&T ,805,000, ,151,900, % 5.48% GE General Electrc ,558,800, ,312,448, % 5.56% IBM IBM ,142,780, ,864,685, % 5.70% MSFT Mcrosoft ,383,400, ,406,150, % 6.27% WMT Wal-Mart ,383,540, ,242,907, % 6.34% XOM Exxon Mobl ,615,940, ,125,547, % 10.16% Table 1. Dow Components as of August 3, 2012 nne tmes that of GE and almost seven tmes that of Mcrosoft, even though all three frms are of comparable market captalzaton. Meanwhle AA and BAC have neglgble weghts and hence lttle mpact on day-to-day movements of the ndex. Now consder the possblty of replacng Alcoa wth Apple, whch had a prce of and market captalzaton of 577 bllon at ths tme. If ths substtuton were made wthout alterng the method by whch the Dow s computed, then the new ndex would

7 be gven by 1 D 31 where the modfed Dow dvsor would be approxmately Apple would carry a weght of more than 26% n the new ndex, and both the volatlty and the day-to-day changes n value of the ndex would be heavly dependent on the behavor of ths one securty. And f Google, wth a share prce of , were also ncluded (n place of Travelers for nstance) we would have an even more absurd scenaro wth the two new entrants accountng for more than 43% of the Dow. Ths s precsely why the ncluson of these frms has been ressted to date. Yet Apple clearly belongs n any representatve stock market ndex. So does Google, whch would place comfortably n the top ten components by captalzaton were t to be ncluded. Ths could be done wthout serously compromsng the hstorcal ntegrty of the tme seres by adoptng the approach descrbed here. If Apple were to replace Alcoa under our proposed procedure, the new ndex would be [ 30 δ β p + v 31 where 30 β = v 30 p and δ = to ensure contnuty of the ndex at the tme of substtuton. The ntal weght of Apple n the new ndex wll smply equal ts market captalzaton relatve to total market captalzaton of the components, whch amounts to less than 13%. Ths wll change over tme wth changes n prces, and wll also change as addtonal substtutons are made. But note that the ntal weght of Apple n our proposed modfcaton to the Dow would be just slghtly greater than the weght of IBM n the current ndex, even though IBM has half the market captalzaton of Apple. If we can tolerate IBM on these terms, why not Apple? p, 7 6. Brdgng Reflect agan on just how the Dow s computed. In the Introducton, we used the example of Chevron and AT&T. We showed that Chevron s three tmes as mportant n affectng the Dow as AT&T s. The penultmate column of Table 1 shows the mplct prce weghts gven to the Dow 30 under the current method of computng the ndex. Chevron s prce weght s 6.42%, as opposed to AT&T s 2.17%. Indeed, the Dow s rfe wth such examples. Compare the prce weghts n the penultmate column of Table 1 wth the market value weghts n the very last column. Note how

8 8 Chevron and AT&T have dentcal (or near-dentcal) market value weghts, as do Intel and Merck, or Coca Cola and Pfzer, or Mcrosoft and Walmart. Yet the members n these pars of companes are accorded very dfferent roles n the Dow. As we ve dscussed, there s lttle we can do by way of radcal change wth such ncumbent companes for fear of destroyng the hstorcal contnuty of the ndex. But for companes that have the concdental property that ther prce weght happens to equal ther market value weght (or s wthn some tolerance threshold of the latter), we mght consder mgratng them over, or brdgng them, to a value-weghted system. Ths would speed up the transton to eventual value-weghtng. More formally, fx a partcular envronment ( p 1, s 1 ;... ; p n, s n ). Value weght equals prce weght for a company k f p k (8) p = v k v. In ths case, we can remove stock k from the prce-weghted component and brng t back n agan usng the value-weghted procedure wthout affectng the total value of the ndex or the mportance of that stock relatve to any of the other companes n the Dow. To see ths, suppose wthout loss of generalty that k = 1. Then followng the procedure leadng up to equaton (7), we have the new ndex (9) δ 1 [ n p s n p p + p 1 s 1 where δ 1 s chosen to satsfy (3) as before. Ths s exactly the same procedure except that we removed and added the same company (company 1 n ths case). Of course the value of the Dow s unaltered, just as before, by the choce of δ 1. But more mportantly, we ve left entrely untouched the relatve mportance of company 1 to every other company n the entre ndex, locally evaluated at the exstng envronment. Under prce weghtng, the relatve mpact of a one-percent change n stock 1 relatve to any other stock j was equal to p 1 /p j. After the change gven (9) that relatve mpact s n p 1 s 1 /p p s j n p. Generally speakng, ths rato wll dverge from p /p j as the envronment contnues to evolve, but the pont s that locally around the orgnal envronment ( p 1, s 1 ;... ; p n, s n ) when the change took place, the rato s no dfferent: n p 1 s 1 / p p s j n p = p 1 / p j. To see ths, use equaton (8). 2 2 Equaton (8) readly mples that p1/ n p = v1/ n v = p1 s1/ n p s, and now the desred equalty follows rght away.

9 The brdgng process speeds up convergence to a value-weghted ndex wthout compromsng the hstorcal contnuty of the Dow. To be sure, t can only be appled to ncumbent companes for whch the fortutous equalty descrbed n (8) happens to hold. Is that true today for any of the stocks n the Dow? The answer appears to be n the affrmatve for Coca Cola (prce weght 4.67% versus value weght 4.57%), and perhaps Dsney (2.88% versus 2.24%). We recommend brdgng them. 9