Download Why Fundamental Indexation Might

Home , Noisy market hypothesis, Value investing

Some proponents of fundamental indexation claim that the strategy is based on a new theory in which market prices of stocks deviate from fair values. A key assumption in this approach is that fundamental weights are unbiased estimators of fair value weights that are statistically independent of market values. This article demonstrates that, except in trivial cases, this assumption is internally inconsistent because the sources of the “errors” are also determinants of market values. The article shows under what conditions fundamental weights are better—or worse—estimators of fair value weights than are market value weights, thereby demonstrating that the new theory is merely a conjecture. A formula is developed for the value bias inherent in fundamental weighting, and two approaches to combining fundamental and market values are discussed.

ome proponents of fundamental indexation indexing, in contrast, assume that the market is claim that their strategy is based on a revolu- correct in its setting of different—sometimes tionary new paradigm in which market widely different—P/Es. S prices of stocks deviate from their fair val- Fundamental indexers are clearly wrong in ues.1 The new paradigm is called the “noisy market their assertion on first principles, because risk and 2 hypothesis.” And it is said to be in opposition to growth, the determinants of a fair P/E, do vary the efficient market hypothesis, in the sense that its among companies. Yet, market-cap proponents proponents believe that a fundamentally weighted also may be empirically wrong if the market errs index is a better representation of fair market val- in setting different P/Es for different companies— ues than is a capitalization-weighted index. as it does. Skeptics are quick to argue that fundamental Perold (2007) examined the logic of the noisy- indexation is nothing more than value investing in market hypothesis and found it flawed. He pointed a different guise (e.g., Asness 2006, 2007a, 2007b). out that a key assumption of the model is that inves- Proponents claim that their approach is different tors do not know the fair values of stocks; they know from value investing because it takes advantage of only the market prices and the probability distribu- the noise in stock prices rather than taking advan- tions from which the pricing errors are drawn. How- tage of value premiums. As Asness pointed out in ever, when the proponents of fundamental the works cited, however, previous researchers, indexation formulate the theoretical expected such as Lakonishok, Shleifer, and Vishny (1994), returns of market-cap-weighted portfolios in their had already shown that noise in stock prices can publications, they include the fair values in the serve as the rationale for value investing.3 information set. In other words, they implicitly, and Stated simply, the problem to be addressed is this: If the measure of fundamental value is, say, perhaps unknowingly, assume that the investor earnings, proponents of fundamental indexing knows the fair value of the stock. Not surprisingly, assume that a good approximation is that all com- they find a “drag” in the expected return of the panies should sell at the same price-to-earnings market-cap-weighted portfolio because stocks ratio (P/E). Proponents of market-cap-weighted known to be overvalued are overweighted and stocks known to be undervalued are under- weighted. When Perold formulated expected Paul D. Kaplan, CFA, is vice president of quantitative returns without knowledge of the fair prices, how- research at Morningstar, Inc., Chicago. ever, the drag disappeared. Note: Morningstar develops and licenses equity indices; In this article, I explore the logic of fundamen- it may use commercially some of the methods discussed tal indexation from another angle—the concept of in this article. fair value multiples. The fair value multiple is the

(unobservable) number M* that equates the (unob- Suppose we use some other measure of firm servable) fair value of a stock, V*, with some size instead of market capitalization to create observed measure of the stock’s fundamental portfolio weights. For simplicity, let us sup- value, F, as follows: V* = FM* or M* = V*/F. pose we do an admirable job at creating the weights so that Proponents of fundamental indexation assert ! * that fundamental weights can be unbiased estima- wi,t= w i,t ( 1+ vi,t) tors of the unobservable fair value weights with Where vi,t is a mean zero white noise uncorre- * “errors” that are statistically independent of mar- lated with other random variables. [wi,t is the ket values. I refer to this assertion as the “indepen- weight of stock i at time t based on fair values ! dence assumption” and demonstrate that it is and wi,t is the fundamental weight.] This is to internally inconsistent. The so-called errors are say that the selected portfolio weights may actually restatements of the fair value multiples of deviate significantly and across the board the stocks in question. For example, if the funda- from the “true-value-weight,” but these mis- mental weights are based on earnings, the errors in takes in assigning weights are not related to other variables, such as market prices or firm the fundamental weights are restatements of the capitalization. [Emphasis added.] fair P/Es of the stocks. Now, let us see if fundamental weights can A stock’s fair value multiple, by definition, satisfy these criteria. We use the following variables: reflects investors’ assessments of the stock’s risk Fi,t = fundamental measure of size used to and future growth prospects. Ideally, such factors ! should be fully taken into account in portfolio con- construct wi,t * struction. Because they are unobservable, however, V i,t = true fair value of stock i at time t they must be either taken into account through N = number of stocks in the portfolios being proxies or ignored. Market-cap weighting takes considered risk and expected growth into account by using the By definition, market values of stocks as proxies for their unob- ! F w it, servable fair values. If market prices contain noise, it, = N (1) F market-cap weights contain errors. Fundamental ∑ j=1 jt, weighting schemes introduce weighting errors of a and different type by ignoring risk and expected V * growth. The superior weighting scheme is the one * it, wit, = . (2) with the less egregious type of error. N * V jt, I demonstrate that, except in trivial cases, fun- ∑ j=1 damental weights cannot be unbiased estimators of Hence, fair value weights with errors that are statistically N V* independent of market values because the sources Fit, ∑ j=1 jt, 1+=νit, . (3) of those errors, risk and expected growth, are also * N Vit, Fjt, determinants of market values. I do so formally by ∑ j=1 * showing that the errors are restatements of fair value Note that the term Fi,t /Vi,t in Equation 3 is the multiples and that fair value multiples are correlated reciprocal of the fair value multiple of stock i at time with market values unless all stocks have the same t. So, the fair value multiple appears on the right- fair value multiple. Then, I show the conditions hand side of Equation 3 in spite of Hsu’s (2006) under which fundamental weights are better than assertion that the left-hand side is “white noise market value weights as estimators of fair value uncorrelated with other random variables . . . such weights and the conditions under which they are as market prices or company capitalization.” Hsu’s worse. I develop a formula for the value bias inher- assertion cannot hold unless fair value multiples ent in fundamental weighting. And finally, I con- are constant across stocks or market valuations are sider how fundamental weights and market value completely unrelated to fair values—two unimag- weights could be combined to form a better estima- inable scenarios. Fair value multiples should have tor of fair value weights than either one separately. some correlation with market values. After all, a stock’s fair market multiple is related to its risk and prospective growth, which should be to some The Independence Assumption extent reflected in its market price. One of the leading proponents of fundamental By way of analogy, imagine that we have a indexation, Hsu (2006, p. 49), stated the indepen- collection of numerous gems of various types, qual- dence assumption as follows: ities, and weights. We find out the market value of

January/February 2008 www.cfapubs.org 33 Financial Analysts Journal each gem, write it on a bag, place the gem in the Like Hsu, we assume that a stock’s market bag, and seal the bag. Once we have sealed the bags, value differs from its fair value by an independent we cannot tell which gem is in which bag. We do multiplicative error term, which Hsu denoted 1 + "i,t have a scale, however, so we weigh each bag and and we denote U. Hence, write the weight on the bag as well as the market VVU= * . (6) value. As with stocks, for these gems, we know the Thus, market value and have a fundamental measure of size (weight) but we do not know the fair value or V= FM* U. (7) fair value multiple (fair price per ounce) for each My critique of Hsu is that he assumed that M* variety of gem, each of which is likely to be differ- is independent of FM*U; that is, he assumed that ent. Unless all the gems are of the same type and cov(M*,FM*U) is zero. Because F, M*, and U are all quality, however, or market values are completely independent of one another, however, and because unrelated to fair values, there is a relationship E(U) is equal to 1, then between the market values written on the bags and the fair prices per ounce of the gems inside the bags. cov(M*** , FM U) = E() F var( M ) . (8) So, we do not want to rely on weight alone to assess Because fundamental variables take only pos- the value of each bag’s content. itive values, this covariance can be zero only if M* Proponents of fundamental indexing are ask- is a constant. Otherwise, it is positive and the inde- ing us to rely solely on weight, and they assure us pendence assumption is violated. that any errors in valuation that this practice causes To sum up, this mathematical exercise demon- are unrelated to what types of gems are in the bags. strates that Hsu’s arguments start with math and Why might such an idea seem plausible? logic that is internally inconsistent. Therefore, the Many investors believe that the ways in which the conclusions that he draws do not necessarily fol- market values companies are not perfectly related low. In particular, his conclusion that a fundamen- to the companies’ fair values. And these investors tally weighted index has a higher expected return want to exploit the difference. They also want to than a market-weighted index lacks the theoretical achieve approximate macroconsistency with the foundation that he claims. market as a whole, however, and keep tracking This is not to say that a fundamentally error versus a cap-weighted benchmark to an weighted index will not outperform a market- acceptable level. Hence, they need to weigh their weighted index. Fundamental indexing should portfolios by some measure of size while trying to outperform in periods when well-crafted value avoid the errors in market valuation. But doing so strategies outperform market-cap benchmarks— with fundamental weights ignores whatever addi- and such periods seem to be the rule rather than tional information about fair values is contained in the exception. The theoretical argument that its market prices. The independence assumption proponents put forth, however, is specious. made by the proponents of fundamental indexation asserts that market prices are useless to such Why Fundamental Indexation investors and can be safely ignored. Might Work To demonstrate the fallacy of the independence assumption formally, let us decompose a Having dismissed the notion that fundamental stock’s fair value, V*, into two independent compo- indexation must work, I will now show why it might nents: a fundamental measure of size, F, and the work (and why it might not). Like Treynor (2005), my analysis here focuses on the relationships corresponding fair value multiple, M*, as noted among fair prices, pricing errors, and market prices. earlier. Dropping the subscript i from the earlier Unlike Treynor, I assume that investors do not know notation, we have the fair value multiple of a stock the pricing errors of individual stocks.4 Rather, they related to Hsu’s “mistake” term as follows: know only the probability distributions of the N V * errors. Also, rather than trying to develop an 1 ∑ j 1 jt, M * = . explicit formula for how much value a non-market- = N (4) 1+ ν F weighting scheme might create for an investor ∑ j=1 jt, above that of a market-cap-weighted portfolio of the The fair value is simply the product of the same stocks, I develop a boundary condition that fundamental value and the fair value multiple: needs to be satisfied in order for a non-market- V**= FM . (5) weighting scheme to add positive value.

If a fundamentally weighted portfolio is to out- Equations 9 and 10, and the mutual independence perform a market-cap-weighted portfolio of the of f, m*, and u, it follows that the correlation of v same stocks, the fundamental variables used to con- and v* is struct the weights should contain more information 22 about the fair values of the stocks than the market σσ()fm+ ()* corr vv,.* ()= 2 2 2 (12) values of the stocks contain. My approach to assess- σ()f+ σ() m* + σ () u ing alternative weighting schemes is to compare (1) Note that this correlation is 1 if market values are the correlation between the fundamental values error free. and the fair values of the stocks in a portfolio with So, we will judge fundamental weighting to be (2) the correlation between the market values and superior to market-cap weighting if fair values. I then derive a necessary condition for the former correlation to exceed the latter correla- corr()f,,. v**> corr () v v (13) 5 tion. If the correlation between the fundamental From Inequality 13, we can calculate the lowest values and the fair values exceeds the correlation value of the variability of pricing errors, #(u), that is between the market values and fair values, then consistent with fundamental weighting being supe- fundamental indexation is the a priori superior rior to market-cap weighting. To do this, we use the 6 approach. If the reverse is true, then market-cap formulas for corr(f,v*) and corr(v,v*) given in Equa- weighting is superior. tions 11 and 12 so that Inequality 13 becomes I use the natural logarithms of the variables in 22 the analysis to simplify the formulas for the corre- σ ( f ) σσ( fm) + ( *) > . 22 2 2 2 (14) lations. Because the fair value of a stock is the σσ( fm) + ( *) σ( f) ++ σ( m*) σ ( u) product of a fundamental measure of size, F, and #(u) the fair value multiple with respect to that funda- Bringing to the left-hand side simplifies the inequality to mental measure, M*, using lower-case letters to denote the natural logarithm of each of these vari- 2 σ ()m* ables, we have σσ()um> ()* 1+ . (15) σ2 ()f v**=+ f m . (9) Similarly, because market value V is the prod- Inequality 15 is thus a boundary condition for fundamental weighting to be superior to market- uct of fair value V* and the multiplicative error cap weighting: The valuation errors that the term, U, we have market makes must be more variable than fair v=+ v* u value multiples. (10) =f + m* + u. Conversely, if the reverse of Inequality 15 is true, so that The three variables f, m*, and u are mutually independent random variables. Their variances are 2 2 2 2 σ ()m* # ( f ) # (m*) # (u) um* 1 , denoted , , and . σσ()< () + 2 (16) I assess the alternative weighting schemes by σ ()f the correlations of the logarithms of their weights then market-cap weighting is superior to funda- with the logarithms of fair value, v*. For the mental weighting. That is, if the market’s value fundamental-weighting scheme, this measure is errors across stocks are less variable than fair value the correlation of f and v*. From the definition of multiples, weighting by market value is better than correlation, the identity stated in Equation 9, and weighting by fundamental value. the independence of f and m*, it follows that the Because there is no theoretical basis for decid- correlation of f and v* is ing whether Inequality 15 or Inequality 16 holds, there is no theoretical foundation for fundamental σ()f corr ()fv,.* = indexation. Furthermore, because none of the vari- 22 (11) σσ()fm+ ()* ables in these inequalities are observable, one cannot devise empirical tests, apart from historical perfor- Note that this correlation is 1 if there is no variation mance, to see whether fundamental weighting or in fair value multiples. market-cap weighting is a better strategy. In other For market-cap weighting, the assessment is words, fundamental indexation is based on a con- based on the correlation between v and v*. From jecture about unobservable variables. There is no the definition of correlation, the identities stated in theory to support it.

January/February 2008 www.cfapubs.org 35 Financial Analysts Journal

The Value Bias This fact, then, is the source of the value bias in fundamental indexation. This analysis, however, As I mentioned in the introduction, some critics of fundamental indexation point out that fundamental has examined it only at the security level, but the weighting is simply an alternative way to introduce analysis can be extended to the portfolio level. a value bias into a portfolio. Because value-biased The yield on the fundamentally weighted portfolios historically have outperformed unbiased index is portfolios, it is no surprise that a fundamentally N weighted index outperforms a market-cap index of yF= ∑ x j y j. (22) the same stocks in a long-term backtest.7 In this j=1 section, I develop a formula for assessing the extent From Equations 21 and 22, we have of the value bias in a portfolio weighted by a single N 2 fundamental size measure. (The analysis can be wyjj ∑ j=1 (23) extended to weighting schemes that use multiple yF = . yV fundamental measures.) The market-value-weighted variance of yields Again, N is the number of stocks in the portfo- across the stocks is lio, Vi is the market value of stock i, and Fi is the fundamental size measure of stock i. The yield N N 2 2 2 2 (dividend yield, earnings yield, etc.) on stock i is σ y=∑ w j() y j− y V=∑ w j y j − yV . (24) j=1 j=1 Fi yi = . (17) From Equations 22 and 23, we have Vi 2 Let wi be the weight of stock i in the market- σV yyFV−= . (25) value-weighted portfolio and xi be the weight of yV stock i in the fundamentally weighted portfolio. Hence, the yield on a fundamentally weighted Hence, index exceeds the yield on a market-value- V weighted index of the same stocks by the ratio of w = i i N (18) the variance of yield across the stocks in the index Vj ∑ j=1 to the yield on the market-weighted index. and Note that yF $ yV. This is not a boundary condition; it is true by construction (because #2 must F V x = i . be greater than or equal to zero). Thus, if a value i N (19) F j ∑ j=1 bias is defined as a portfolio yield higher than that of the market-cap-weighted index, the conclusion The yield on the market-value-weighted that fundamental indexation contains a value bias index is does not depend on circumstances (that is, on the N observed values of variables). It is always true. yV= ∑ w j y j j=1 N (20) Toward a Better Weighting F ∑ j 1 j = . Method = N V ∑ j=1 j Because fundamental data and market capitaliza- tions both contain information about fair market From Equations 18, 19, and 20, we have value, the best thing to do would be to combine them yi to form estimates of fair market value. In this sec- xi = wi. (21) yV tion, I briefly discuss two approaches to combining In other words, as Asness (2007b) pointed out, the methods. One is based on a theoretical optimi- the fundamental weight is simply the market-cap zation. The other is a pragmatic approach that tries weight multiplied by the ratio of the yield on the to combine the best features of both approaches security to the yield on the market-cap-weighted while avoiding the disadvantages of each. portfolio. Hence, the fundamentally weighted index underweights stocks that have lower yields Optimization Approach. The first approach than the market-cap-weighted portfolio and over- builds off the previous analysis of the components weights stocks that have higher yields than the of fair value, fundamental value, and market value. market-cap-weighted portfolio. The underweight- Recall that the criterion for ranking weighting- ing and overweighting are in proportion to each scheme variables is the correlation of the logarithm stock’s yield. of the variables with the logarithm of fair value, v*.

Because the logarithms of fundamental value f and folio diversified. In a similar fashion, by rebalanc- market value v are both correlated with v*, there ing a portfolio of stocks to a set of market-invariant could be linear combinations of the two that have weights, fundamental indexation promotes diver- a higher correlation than either one separately. In sification (to be precise, it reduces the volatility of Appendix A, I show that the linear combination of the level of diversification). f and v that has the highest correlation with v* is To retain low turnover but control the level of diversification, Arya and Kaplan used fundamen- v" =+θθ f()1,− v (26) tal weights to set boundaries on portfolio weights where rather than to be the portfolio weights themselves. σ2 u These boundaries (or collars) are fixed multiples of () . θ = 22 (27) the fundamental weights, such as a lower bound of σσ()mu* + () half the fundamental weight and an upper bound The formula in Equation 27 echoes my previous of twice the fundamental weight. Stocks with mar- observation that if there is no variation in fair value ket value weights that fall within the boundaries multiples across stocks [so that #(m*) = 0], funda- are held at the market value weights, whereas mental weighting is optimal (% = 1). It also implies stocks with market value weights that fall outside that if market values are error free [#(u) = 0], market- the boundaries are held at the boundary weights. cap weights are optimal (% = 0). If neither condition The idea is to hold most of the portfolio at market holds, the optimal weights fall somewhere between value weights most of the time while avoiding con- these two extremes, with the blended weights tak- centration in a few large stocks during market run- ing the form ups, such as occurred during the late 1990s. The effect would be similar to the way in which strate- FVθθ1− w" ii . gic asset allocation models kept overall portfolios i = N (28) FVθθ1− ∑ j=1 jj balanced during that period. Of course, the true value of % is unobservable, so selecting it in practice would largely be a matter Conclusion of judgment. Proponents of fundamental indexation claim that the noisy-market hypothesis implies that funda- “Collared” Approach. Arya and Kaplan mental weighting must be superior to market-cap (2006, 2007) presented another approach to com- weighting. Perold (2007) and this article demon- bining market value and fundamental variables, an strate, in different ways, that these proponents’ rea- approach that is not based on optimization. They soning is flawed. Perold showed that they err in developed a weighting methodology intended to their formulation of the expected return of a market- combine the best practical features of the market cap-weighted portfolio. I have shown that one of value and fundamental approaches while avoiding their key assumptions, the independence assump- the disadvantages of each. tion, must be false except in the most trivial cases. The advantage of market-cap weighting that The proponents of fundamental indexation, rather Arya and Kaplan sought to preserve is low turn- than having a clear theory on which to base their over. A great advantage of market value weighting claims, have only a conjecture that market valuation over any non-cap-weighted approach is that market errors are more variable than fair value multiples. value weighting does not need to be continuously They may have been correct over long historical rebalanced, so it keeps transaction costs to a mini- periods, as the successful backtests of their strate- mum. In contrast, any non-cap-weighted scheme gies seem to demonstrate, but they should be much requires regular rebalancing and thus generates more modest in their claims. In particular, they transaction costs that must be overcome by superior could argue that it is better to introduce a value bias performance. into a portfolio by using their weighting scheme Weighting solely on market value, however, than by excluding low-yield stocks. But they have can produce portfolios with volatile levels of diver- not, to date, pursued this more humble line of rea- sification. Cap-weighted portfolios are never soning. They may have a successful investment rebalanced, so a stock that keeps rising has an strategy, but they have not produced a revolution in increasingly large portfolio weight. Most investors investment theory. would never allow this decreasing diversification Finally, I showed that if market capitalization at the asset-class level; rebalancing to a set of and fundamental variables both contain informa- market-invariant asset-class weights keeps a port- tion about fair market values, they can be combined

January/February 2008 www.cfapubs.org 37 Financial Analysts Journal to form portfolio weights that reflect the information v" =+θθ f()1.− v (A4) contained in each. I discussed two ways to do so, but Therefore, other approaches are possible and I encourage this line of research. v" =+ f()1− θ () m* + u . (A5) Recall that f, m*, and u are mutually indepen- The main idea of this article grew out of discussions with dent. Hence, James Knowles and Clifford Asness. I am grateful to 22 them and to Eric Jacobson and Laurence Siegel for their cov v", v**= σ() f+()()1− θ σ m (A6) many helpful comments. ( ) and

This article qualifies for 1 CE credit. 22 2 22 σv" = σ f+1− θ σ m* + σ u . (A7) ( ) () ()() () Replacing the terms that are on the left-hand Appendix A. Derivation of sides of Equations A6 and A7 with the expressions Optimal Combination of on the right-hand sides of these equations in the definition of Q(.) given in Equation A1 yields Fundamental and Market 22 Q()()()()θ=ln  σ f +1− θ σ m*  Values   1 2 2 22 Recall the following definitions: −ln σf+1− θ σ m* + σ u  (A8) ()()()()  v = natural logarithm of market value 2 {} 2 v* − ln σ m*  . = natural logarithm of fair value  () f = natural logarithm of fundamental value Differentiating the right-hand side of Equation A8 m* = natural logarithm of fair value multiple produces u = natural logarithm of the multiplicative 2 pricing error in market value −σ m* Q () ′()θ = 22 vˆ = linear combination of v and f (chosen to σ()fm+()()1− θ σ * have maximum correlation with v*) 22 (A9) #(x) = standard deviation of x (x being one of ()()1− θ σmu* + σ () +   . the preceding variables) 2 2 22 σ ()f + ()()1− θ σmu* + σ () % = weight on f in the construction of vˆ   The aim is to select % so as to maximize the A necessary condition for a given choice for % correlation between vˆ and v*. To simplify the alge- to yield the maximum or minimum value of Q(%) bra, we define the objective function of this problem, is that Q&(%) = 0. From Equation A9, we see that Q(.), as the natural logarithm of this correlation. (We Q&(%) = 0 only once for finite choices of %, when the can do this because the correlation between vˆ and v* following condition holds: at and around the optimal choice for % is positive.) 2 σ m* We have 1 () . −θ = 22 (A10) σσ()mu* + ()  "  cov()vv , * Hence, Q ()θ = ln   σσvv" *  2  ( ) () σ ()u (A1) θ = (A11) 1 2 22 = ln covvv" ,*  − ln σ v"  σσ()mu* + ()  () 2  () is the unique finite choice for % that could maxi- − lnσ()v ∗  . mize Q(%). Recall that, by definition, the following three From Equation A9, we see that Q&(0) $ 0 and identities hold Q&(1) ' 0. From Equation A11, it is evident that our choice for % is between 0 and 1. Hence, Q&(.) is v**=+ f m (A2) decreasing around the choice for % so that we have v= f + m* + u (A3) a maximum value rather a minimum.

Notes

1. I use the term “fundamental indexation” from the title of 4. Treynor assumed that “MVI [market-valuation indiffer- Arnott, Hsu, and Moore (2005) to refer to any indexing ent] investors spend the same number of dollars on the strategy in which fundamental values of the securities, such underpriced as they spend on the overpriced stocks” as earnings, book value, dividends, and/or other variables (p. 67). Of course, they can do this only if they know the not related to market capitalization, are used for weighting relative true values of all stocks, a condition that propo- the portfolio holdings. There are a number of such strategies nents of fundamental indexation (including Treynor in the in the market. Fundamental Index is a registered trademark same paper) deny is necessary for their systems to work. of Research Affiliates, LLC. See Perold (2007). 2. As Perold (2007) noted, Siegel (2006) coined the term “noisy 5. Strictly speaking, I compare the correlations of the natural market hypothesis.” Arnott, Hsu, and Moore (2005), Hsu logarithms of these variables. See the discussion that follows. (2006), and Treynor (2005) presented the ideas that this term 6. The a priori superior approach is the one that has the higher denotes. The idea of using fundamental measures of com- expected return. The realized returns of each strategy will, pany size as alternatives to market capitalization goes back of course, differ from the expected returns. at least to Berk (1995) and Grobowski and King (1996), albeit 7. Wiandt (2007) examined the recent performance of in a different context. Those researchers were questioning exchange-traded funds that track fundamentally weighted whether there is a “size effect” in stock returns apart from indices and found that that they have been lagging their a market-cap effect. market-cap-weighted counterparts. He attributed this per- 3. Arnott, Hsu, Liu, and Markowitz (2007) demonstrated that formance lag to the value bias in the fundamentally noise can explain the size and value effects. weighted indices during a time when value is out of favor.

References

Arnott, Robert, Jason Hsu, and Philip Moore. 2005. “Fundamen- Treynor, Jack. 2005. “Why Market-Valuation-Indifferent Index- tal Indexation.” Financial Analysts Journal, vol. 61, no. 2 (March/ ing Works.” Financial Analysts Journal, vol. 61, no. 5 (September/ April):83–99. October):65–69. Arnott, Robert, Jason Hsu, Jun Liu, and Harry Markowitz. 2007. Wiandt, Jim. 2007. “Do They Really Beat the Market?” Exchange- “Does Noise Create the Size and Value Effects?” Working paper, Traded Funds Report, no. 83 (October):1, 7, 9. University of California, San Diego (rady.ucsd.edu/faculty/ directory/liu/docs/size-value.pdf). Arya, Sanjay, and Paul Kaplan. 2006. “Collared Weighting: A New Hybrid Approach to Indexing.” Working paper, Morningstar (17 July): http://corporate.morningstar.com/US/ documents/Indexes/CollaredWeighting.pdf. ———. 2007. “Collared Weighting: A Hybrid Approach.” Morningstar Indexes Yearbook, vol. 3:22–25. Asness, Clifford. 2006. “The Value of Fundamental Indexing.” Institutional Investor, vol. 40, no. 10 (October):94–99. ———. 2007a. “Letter to the Editor.” Institutional Investor, vol. 41, no. 1 (January):14. ———. 2007b. “Non-Cap Weighted Indexes.” Presentation to the Institute for Quantitative Research in Finance (Q-Group), Sea Island, GA (27 March). [ADVERTISEMENT] Berk, Jonathan. 1995. “A Critique of Size-Related Anomalies.” Review of Financial Studies, vol. 8, no. 2 (Summer):275–286. Grobowski, Roger, and David King. 1996. “New Evidence on Size Effects and Rates of Return.” Business Valuation Review, vol. 15, no. 3 (September):103–115. Hsu, Jason. 2006. “Cap Weighted Portfolios Are Sub-Optimal Portfolios.” Journal of Investment Management, vol. 4, no. 3 (Third Quarter):44–53. Lakonishok, Josef, Andrei Shleifer, and Robert W. Vishny. 1994. “Contrarian Investment, Extrapolation, and Risk.” Journal of Finance, vol. 49, no. 5 (December):1541–1578. Perold, André F. 2007. “Fundamentally Flawed Indexing.” Financial Analysts Journal, vol. 63, no. 6 (November/December): 31–37. Siegel, Jeremy J. 2006. “The ‘Noisy Market’ Hypothesis.” Wall Street Journal (14 June):A14.

January/February 2008 www.cfapubs.org 39