Fundamentally Flawed Indexing

Transcription

1 Fnancal Analysts Journal Volume 63 Number 6 07, CFA Insttute PERSPECTIVES Fundamentally Flawed Indexng André F. Perold The captalzaton-weghted equty market portfolo holds a specal place n modernday nvestng and for good reason. The cap-weghted portfolo offers broad dversfcaton and low transacton costs. Captalzaton weghtng s also the only strategy that all nvestors can follow. Because the collectve holdngs of nvestors (by defnton) aggregate to the market portfolo, for every nvestor who s underweght a stock, another s overweght that stock, and between them, t s at best a zero-sum game. After fees and transacton costs, the average nvestor who devates from captalzaton weghts must underperform the market portfolo. Now, however, a new theory of fnance s beng advanced as provdng defntve proof that holdng stocks n proporton to ther market captalzatons s an nferor nvestment strategy. The clam s that captalzaton weghtng necessarly nvests more n overvalued stocks and less n undervalued stocks. Dubbed the nosy market hypothess, the theory s beng used to advocate nvestments n non-cap-weghted (sometmes called fundamental ) funds and ndces 1. Unfortunately, there s a fundamental flaw n the logc. As I wll explan n detal, the theory s seekng to poston an actve management strategy n a passve management framework. And t asserts rather than derves the nferorty of captalzaton weghtng. The asserton, moreover, s false. Bottom lne? Captalzaton weghtng does not mpose an nherent performance drag. The nosy market hypothess goes lke ths: Start wth the premse that the market errs n ts prcng of ndvdual stocks but that the prcng errors occur n a random fashon, so some stocks André F. Perold s George Gund Professor of Fnance and Bankng at the Harvard Busness School, Boston, Massachusetts. Edtor s Note: Professor Perold s a founder of HghVsta Strateges and char of ts Investment Commttee. Hgh- Vsta Strateges specalzes n endowment management for nsttutons and ndvduals and offers a fund that s broadly dversfed across marketable asset classes, hedge funds, and prvate nvestments. are overvalued whle others are undervalued. Overvalued stocks have nflated market captalzatons and wll have lower future returns; undervalued stocks have depressed market captalzatons and wll have hgher future returns. Accordngly, a cap-weghted strategy wll systematcally skew the portfolo toward nvestment n overvalued stocks, and such a strategy wll not, therefore, perform as well as an approach that avods the use of market captalzaton n determnng the ndvdual stock weghts. Robert Arnott, one of the prme archtects of fundamental ndexng, has wrtten: 2 No longer must nvestors suffer a performance drag by settlng for an ndex that nherently overweghts every overvalued company and underweghts every undervalued one. Wth due respect to the poneers n fnance theory and the cap-weghted ndexers, there s a better way. (Arnott 06, p. 41) Fundamental ndexng proponent Jeremy Segel has stated that we are on the edge of a revoluton and has called the nosy market hypothess a new paradgm for understandng how markets work. He stated that t can be rgorously proved that f stock prces are subject to nose, then captalzatonweghted ndexes wll offer nvestors rsk-andreturn characterstcs that are nferor to those of fundamentally weghted ndexes. (Segel 06, p. A14) 3 If vald, t would represent a profound fndng that captalzaton weghtng, by tself, creates a mathematcal headwnd aganst performance profound n that nvestors can beneft smply by avodng captalzaton weghtng and profound n that skll n dstngushng overvalued from undervalued stocks s not requred to obtan superor nvestment performance. But the proposton s not vald. Wth due respect to proponents of fundamental ndexng, I counter that t s not yet tme to rewrte the fnance text books. In what follows, I lay out what the nosy market hypothess s clamng and then explan why the concluson t reaches about the nferorty of captalzaton weghtng s ncorrect. November/December

2 Fnancal Analysts Journal The Nosy Market Hypothess and Its Fallacy The nosy market hypothess starts wth the assumpton that any gven stock s as lkely to be overvalued as undervalued. Ths statement may or may not be a good representaton of real-world captal markets, but that s not the ssue. The problem arses n gong from ths assumpton about market prces to the concluson that captalzaton weghtng systematcally skews nvestment toward overvalued stocks. The ssue s best llumnated by means of an example. A formal model s developed n Appendx A. Consder formng a portfolo of the shares of just two companes, Company A and Company B. To keep thngs smple, suppose that the two companes are actually smply rskless closed-end money market funds. Both have net asset values of $10 per share, and both yeld the market rate of nterest, whch s 10 percent per year. In a year, these funds wll each be lqudated and nvestors wll receve $11 per share rsk free. The market knows that Company A and Company B are closed-end money market funds, but t does not know ther net asset values per share. In accordance wth the nosy market hypothess, assume the market msprces these shares n ether drecton and wth equal probablty. Suppose further that the msprcng s percent of far value. A and B thus wll trade at ether $12 or $8 per share (as llustrated n Fgure 1). At $12 per share, nvestors wll lose 8.5 percent over the year because they wll get back only $11. At $8 per share, nvestors wll make 37.5 percent on ther shares over the year. Suppose that A and B have the same number of shares outstandng. Ther market captalzatons thus wll be n the proporton 60/40 ($12/$8) f A s overvalued and B s undervalued and wll be n the proporton 40/60 f A s undervalued and B s overvalued. If both are overvalued or both are undervalued, the cap-weghted ndex wll gve equal weght to A and B. The cap-weghted ndex thus dffers from the equally weghted portfolo when one stock s undervalued and the other s overvalued. Contnung wth ths lne of reasonng, the purported returns of the cap-weghted and equally weghted portfolos can now be calculated, as shown n Table 1. The analyss ndcates that the cap-weghted ndex underperforms the equally weghted ndex on average by 2.3 percent Fgure 1. Market Value Gven Far Value Market Value ($) Market Value = Far Value + % Market Value = Far Value 8 6 Market Value = Far Value % Far Value ($) Note: Stock randomly over/underprced by percent , CFA Insttute

3 Fundamentally Flawed Indexng Table 1. Cap-Weghted vs. Equally Weghted s: Perspectve of the Nosy Market Hypothess Equally Outcome Cap Weghts Probablty Cap-Weghted Weghted Both overvalued 50/50 25% 8.3% 8.3% Both undervalued 50/ One overvalued, one undervalued 60/40 or 40/ Expected return because captalzaton weghtng puts more weght on overvalued stocks than on undervalued stocks. Ths s the argument set forth by Arnott, Segel, and others. I wll now explan where the problem les. Take the case n whch Stock A s tradng at $12 per share, and consder what nvestors actually know about ths stock. They know ts current share prce s $12, and they know that A s ether overvalued or undervalued by percent wth equal probablty. They also know that A s a closed-end money market fund yeldng 10 percent per year that wll be lqudated n a year. Crucally, nvestors do not know that the far market value s $10/share. If they dd know far value was $10/share, the decson not to own any of A when t s tradng at $12/share would be easy. Gven that the stock s tradng at $12/share, there are only two possbltes: The far market value of A must be ether $10/share as a result of A beng overvalued by percent or $15 per share as a result of A beng undervalued by percent (as llustrated n Fgure 2). If the far value s $10, shareholders wll receve $11 n one year, thus sufferng a loss of 8.3 percent; f the far value s $15, shareholders wll receve $16.50 n one year, for a return of 37.5 percent. At $12 per share, and because undervaluaton and overvaluaton occur wth equal probablty, the expected return on the stock s 14.6 percent. Fgure 2. Far Value Gven Market Value Market Value ($) Possble Far Value Gven Market Value Far Value ($) Note: Stock randomly over/underprced by percent. November/December

4 Fnancal Analysts Journal A smlar analyss apples at $8/share. Gven ths prce level, there s a 50 percent chance that the stock s undervalued by percent and thus truly worth $10/share today and $11/share n a year; and there s a 50 percent chance that t s overvalued by percent and thus truly worth $6.67/share today and $7.33/share n a year. Once agan, the return on the stock s ether 8.3 percent or 37.5 percent, for an expected return of 14.6 percent. Importantly, n these calculatons, I am performng a Bayesan analyss: The nosy market hypothess tells us that nvestors have an unnformed pror dstrbuton on the far value of the stock, meanng that before observng the prce at whch the shares are tradng, nvestors have no opnon on what the stock s really worth. It s as lkely to have a hgh value as a low value as anythng n between. 4 The Bayesan analyss uses ths pror dstrbuton to conclude that after nvestors observe the $12 prce, the possbltes for far value are narrowed to just $10/share and $15/share wth equal probablty. Suppose now that Stock A s tradng at $12/ share and that Stock B s tradng at $8/share. Because A and B have the same number of shares outstandng, ther market captalzatons wll be n the rato 60/40. Investng 60 percent n A and 40 percent n B has four possble outcomes, as shown n Table 2, and the average return of the capweghted ndex s the same as that of the equally weghted ndex. The analyss shows that captalzaton weghtng mposes no drag on expected return because captalzaton weghtng does not cause one to nvest more n overvalued stocks and less n undervalued stocks. It nvests the same proportons, here 60/40, wthout regard to undervaluaton or overvaluaton of the shares. Provded that A and B are randomly overvalued or undervalued by percent, cap-weghted and equally weghted portfolos wll have the same expected return regardless of the prces at whch A and B are tradng. The Crux of the Issue The crux of the ssue s that the nosy market hypothess effectvely anchors on far value holdng far value fxed and usng the probablty dstrbuton of the prcng error to deduce the probablty dstrbuton of market prces. To do so s to presuppose systematc reversals n stock prces, an asserton that does not follow from stocks beng randomly msprced. The bg clam of the theory s that one can outperform cap-weghed ndces wthout knowng far value. If one does not know far value, then even though prces may move toward far value, the drecton of that movement s random. To anchor on far value s thus to contradct the gong-n assumpton of the nosy market hypothess that we do not know far value. If all that one knows about a stock s ts current prce, the correct analyss s to hold that prce fxed and use the probablty dstrbuton of the prcng error to deduce the probablty dstrbuton of the unknown far value. As llustrated n the example of Companes A and B and establshed more formally n Appendx A, such an analyss shows that a company s market captalzaton by tself does not predct the return on ts shares. Because market captalzaton does not reveal whether a stock s overvalued or undervalued, the random msprcng of stocks does not systematcally shft the portfolo weghts toward overvalued stocks. Correlaton of Prcng Error wth Far Value vs. wth Market Value Another way to state the precedng concluson s n terms of the correlaton of the prcng error wth far value and wth market value. Fundamental ndexng proponents argue that f a stock s prcng error s uncorrelated wth ts far value, the prcng error must be correlated wth ts market value, whch n turn, gves rse to captalzaton weghtng nducng a performance bas. Ths s not the case. The Bayesan analyss (wth unnformed pror belefs) shows that f the prcng error s uncorrelated wth Table 2. Cap-Weghted vs. Equally Weghted s: Bayesan Perspectve Cap- Weghted Equally Weghted Outcome Cap Weghts Probablty on Stock A on Stock B Both overvalued 60/40 25% 8.3% 8.3% 8.3% 8.3% Both undervalued 60/ A overvalued, B undervalued 60/ A undervalued, B overvalued 60/ Expected return , CFA Insttute

5 Fundamentally Flawed Indexng the stock s far value, the prcng error s also uncorrelated wth the stock s market value, condtonal on knowng market value. Ths lack of correlaton between the prcng error and condtonal market value s precsely why cap-weghted portfolo returns are not a pror based downward. The Nosy Market Hypothess and the Tme Seres of s A more elaborate verson of the nosy market hypothess relates to the tme seres of returns. The clam s that a share of stock that s randomly msprced from perod to perod wll have excess volatlty, whch mples mean reverson or negatve seral correlaton n ts stock returns. When asset returns are mean revertng over tme, a rebalancng strategy wll tend to outperform a buy-and-hold strategy (see Perold and Sharpe 1988). The reason s that a rebalancng strategy at the margn wll buy assets that have underperformed and sell assets that have outperformed, trades that are good, on average, because of subsequent mean reverson n returns. Captalzaton weghtng s a buy-and-hold strategy and, n the presence of mean reverson, s thus lkely to underperform a rebalancng strategy before takng nto account transacton costs. I take ssue wth ths argument on several grounds. Frst, stock returns beng mean revertng does not create a case aganst captalzaton weghtng per se but aganst all buy-and-hold strateges regardless of the ntal proportons n whch ndvdual assets are held. Second, t s hard to see how msprcng errors can be random from one perod to the next. Whatever s causng a stock to be overvalued today s lkely to cause t to be overvalued tomorrow. Random but persstent msvaluaton wll cause mean reverson n returns over perods lkely to be longer than a reasonable rebalancng nterval. Thrd, rather than causng mean reverson n returns, msprcng could go the other way and cause stocks to underreact to changes n fundamentals. In a world of underreactng stock prces (.e., postve seral correlaton/momentum of returns), buy-and-hold strateges wll outperform rebalancng strateges. Rebalancng strateges wll tend to be nferor n ths case because they sell a stock after t has moved up, only to fnd that t contnues to move up, on average, or they buy a stock after t has moved down, only to fnd that t contnues to move down, on average. And fourth, the emprcal evdence on seral correlaton of ndvdual stock returns s, at best, nconclusve. Stocks tend to exhbt momentum effects (f at all) over monthly and annual perods, and they tend to exhbt mean-reverson effects (f at all) over longer ntervals. Most mportantly, f an nvestor knows somethng about the seral correlaton of return of a partcular stock, why wouldn t the nvestor explot ths knowledge drectly? The approprate strategy to explot seral correlaton may be a far cry from a smple rebalancng rule, partcularly once transacton costs and taxes are taken nto account. 5 Fundamental Indexng As a Value Tlt The supposton that captalzaton weghtng nduces a mathematcal headwnd aganst performance s an mportant underpnnng for the argument proponents are makng for fundamental ndexng. Addtonally, fundamental ndexers are proposng that f one s not gong to nvest accordng to captalzaton weghts, a good strategy s to tlt the portfolo toward value stocks stocks wth such characterstcs as low P/Es and hgh dvdend yelds. 6 Clfford Asness (06) and Jack Bogle and Burton Malkel (06) have explaned eloquently how fundamental ndexng s smply a partcular packagng of quanttatve value nvestng. Hstorcally, value stocks have generated hgher-than-ndex returns, and the effect has been well documented and wdely debated (see Fama and French 1992). At ssue s whether value stocks have had hgh returns because they are rsker or because they are msprced. If the effect s about rsk, then fundamental ndexers (and quanttatve value nvestors generally) cannot expect to obtan hgh returns after adjustng for rsk. If the effect s about msprcng, fundamental ndexers wll need to rely on a contnuaton of that pattern of msprcng n order to obtan hgh future returns the pattern beng that the market does not fully account for companes book values, sales, earnngs, and other readly obtanable fundamental nformaton when determnng stock prces. If value stocks are systematcally msprced, fundamental ndexng may perform well along wth other value-orented strateges because t s explotng ths partcular neffcency, not because captalzaton weghtng, n and of tself, creates a performance bas. Concluson Holdng a stock n proporton to ts captalzaton weght does not change the lkelhood that the stock s overvalued or undervalued. The noton that captalzaton weghtng mposes an ntrnsc drag on November/December

6 Fnancal Analysts Journal performance s, accordngly, false. Fundamental ndexng s a strategy of actve securty selecton through nvestng n value stocks. It s a strategy not everyone can follow. Investors who have no skll n evaluatng value tlts and other actve strateges should hold the cap-weghted market portfolo. I thank Clff Asness, Jesse Barnes, Ken French, Jakub Jurek, Wa Lee, Bob Merton, Bll Sharpe, Erk Stafford, and Lus Vcera for helpful dscussons. Ths artcle qualfes for 0.5 PD credt. Appendx A. Formal Model The formal model behnd the example gven n the man text s as follows. 7 The equty market has N traded companes. The prce of the th stock s P, and ts far value s. The prcng error s P * P e = 1. (A1) Each company has one share outstandng. The share s nfntely dvsble, so nvestors can buy any fracton of the company they wsh. Because each company has one share outstandng, ts market captalzaton s smply ts stock prce, P. At current market prces, the captalzaton weght of stock s W P =. 1, Pj j= N (A2) Each stock f farly valued has the same known requred return r. Moreover, stock wll trade for certan at prce (1 + r) one perod from now. Investors observe the current prce P but do not know. Before observng P, nvestors have a pror dstrbuton of gven by the probablty densty functon, f ( ). Condtonal on, the probablty dstrbuton of the prcng error s g(e ) and s unrelated to. For the purpose at hand, t s not essental that the prcng error have expectaton zero or be symmetrcally dstrbuted. The probablty denstes f (.) and g(.) are common to all shares, but the prcng errors and far values are ndependently dstrbuted across shares. The pror dstrbuton f (.) s unnformed n that far value could be anythng (over the range > 0). Because stock prces tend to grow or shrnk geometrcally, t s sensble to assume that s equally lkely to le n ntervals that are geometrcally evenly spaced n other words, that s as lkely to le between $50 and $100 as t s between $100 and $0, between $0 and $400, and so on. Therefore, log( ) s unformly dstrbuted between ±. Let h(p ) denote the probablty dstrbuton of P condtonal on knowng. Thus, ( ( ) = ) gp 1 hp. (A3) Let k( P ) denote the probablty dstrbuton of condtonal on knowng P (the posteror dstrbuton of ). By Bayes Theorem, k( P ) s proportonal to h(p )f ( ). Wth log( ) beng unformly dstrbuted, k ( P ) evaluates to g(p / 1)P /( ) 2. Condtonal on knowng P but not, the return on stock s (1 + r)/p 1. Integratng over wth respect to the densty k( P ), and makng a change of varable from to e, shows that the expected condtonal return on stock, denoted m, can be expressed as m= ( 1 + r ) 1 E (A4) + e 1 1, where the expectaton n ths expresson s taken wth respect to the error densty, g(e ). A crucal pont s that a stock s condtonal expected return m s ndependent of ts stock prce P and, hence, of ts market captalzaton. Each stock has ths same expected return m, and thus any portfolo whether captalzaton weghted or otherwse wll also have ths expected return (14.6 percent n the example). Therefore, even though ndvdual stocks may have random prcng errors, market captalzaton does not predct returns and captalzaton weghtng, n and of tself, does not create a performance drag , CFA Insttute

7 Fundamentally Flawed Indexng Notes 1. The term nosy market hypothess was coned by Jeremy Segel n Fundamental Indexng s a trademark of Research Afflates, LLC. 3. The rgorous proofs Segel s referrng to are contaned n Treynor (05) and Hsu (06). 4. As dscussed formally n Appendx A, ths analyss actually assumes a unform pror on the log of far value. 5. For an analyss of the optmal rebalancng strategy n the presence of long-term mean reverson n ndvdual stock prces, see Jurek and Vcera (06). 6. See, for example, Arnott, Hsu, and Moore (05); Segel (06). 7. The basc setup here mrrors that n Hsu (06). References Arnott, Robert. 06. An Overwrought Orthodoxy. Insttutonal Investor (18 December): Arnott, Robert D., Jason C. Hsu, and Phlp Moore. 05. Fundamental Indexaton. Fnancal Analysts Journal, vol. 61, no. 2 (March/Aprl): Asness, Clfford. 06. The Value of Fundamental Indexng. Insttutonal Investor (19 October): Bogle, John C., and Burton G. Malkel. 06. Turn on a Paradgm? Wall Street Journal (27 June):A14. Fama, Eugene F., and Kenneth R. French The Cross- Secton of Expected Stock s. Journal of Fnance, vol. 47, no. 2 (June): Hsu, Jason. 06. Cap-Weghted Portfolos Are Sub-Optmal Portfolos. Journal of Investment Management, vol. 4, no. 3 (Thrd Quarter): Jurek, Jakub W., and Lus M. Vcera. 06. Optmal Value and Growth Tlts n Long-Horzon Portfolos. Workng paper, Harvard Unversty (September). Perold, André F., and Wllam F. Sharpe Dynamc Strateges for Asset Allocaton. Fnancal Analysts Journal, vol. 44, no. 1 (January/February): Segel, Jeremy J. 06. The Nosy Market Hypothess. Wall Street Journal (14 June):A14. Treynor, Jack. 05. Why Market-Valuaton-Indfferent Indexng Works. Fnancal Analysts Journal, vol. 61, no. 5 (September/October): November/December