What drives the performance difference between cap- weighted indexation and fundamental indexation ?

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(1)What drives the performance difference between capweighted indexation and fundamental indexation ?. Tim M. Kremer (1576356). Supervisor: Auke Plantinga Faculty of economics and Business. June 23, 2009. Abstract In this research we investigate the performance difference between fundamental indexation and capweighted indexation on the Dutch stock market in the period 1984-2008. In this paper the returns of five individual fundamental indexes (dividend, book value, free cash flow, sales and earnings) and two composite indexes (the composite index of all five indexes and a composite index without the book value index) are related to the adjusted benchmark index. We analyze which variables are good proxies for noise and determine the performance difference between fundamental indexation and capweighted indexation. JEL classification: G11 and G14 Keywords: Fundamental indexation, Cap-weighted indexation, noise, performance difference.

(2) I. Introduction The asset pricing model of Sharpe (1964), Lintner (1965), and Black (1972) has long shaped the way academics and practitioners think about average returns and risk. The central hypothesis of the model is that the market portfolio of invested wealth is mean-variance efficient in the sense of Markowitz (1959). When the market portfolio is the mean-variance efficient portfolio, an indexation strategy based on capweighted indexation is representative for the market portfolio. The advantages of cap-weighted indexation are: it is a relatively objective portfolio selection criteria, it saves costs in fund operations, it needs limited portfolio direction, the trading costs are low to portfolio turnover and it has a high degree of natural tax efficiency (Hsu, 2006). Arnott, Hsu, Moore (2005), Hsu (2006) and Treynor (2005) conclude that cap-weighted indexation is not mean-variance efficient and cap-weighting may lead to suboptimal portfolio return characteristics because prices are too noisy relative to fundamentals. If stock markets are inefficient, cap-weighting assuredly gives additional weight to stocks that are currently overpriced relative to their (unknown) discounted future cash flows (true fair value) and reduces weight in stocks that are currently trading below their fair value. This mismatch leads to performance drag in cap-weighted and other priceweighted portfolios. Arnott, Hsu and Moore (2005) demonstrate that cap-weighted portfolios suffer from a return drag if prices are noisy relative to movements in company fundamentals and that capitalization weighting is sub-optimal. Treynor (2005) shows that random pricing errors lead to a negative alpha for any price-weighted or cap-weighted portfolio relative to a price-indifferent portfolio. (see Hsu (2004) and Treynor (2005), for different derivations of this result). Arnott and Hsu (2005) are the pioneers of a new strategy named fundamental indexation. They claim that cap-weighted indexation is representative for the market portfolio and therefore mean-variance inefficient. They propose to create a portfolio based on fundamental values of a firm as proxies for size instead of market capitalization. The fundamentals used are gross revenues, equity book value, gross sales, gross dividends, cash flows and total employment (Arnott, Hsu and Moore, 2005). The fundamental index of Arnott, Hsu and Moore (2005) shows that the fundamental-weighted, non capitalization-based indexes consistently provide higher returns and lower risks than the traditional capweighted equity market indexes while retaining many benefits of traditional indexation. They conclude that fundamental indexation delivers a superior mean variance performance. The resulting portfolios based on fundamental indexation outperformed the S&P 500 by an average of 1,97 pps. a year over a 43-year span tested. The performance is robust over time, across stages of the business cycle, across bear and bull stock markets, and across rising- and falling-interest-rate regimes. Furthermore, Hsu and Campollo (2006) built fundamental indexes for 23 countries over the 1984-2004 period and find that these indexes outperform their respective MSCI (cap weighted) benchmarks in every country and on average by 2,8% a year. Estrada (2006) links fundamental indexation and 2.

(3) international diversification. Considering 16 benchmarks of countries that make up over 93% of the world market capitalization and a 32-year (1974-2005) sample period, the results show that a dividendweighted fundamental index outperform a cap-weighted index by the substantial margin of 1,9% a year. Hemminki and Puttonen (2008) examine the benefits of fundamental indexation using European data. Their findings based on European data suggest that by re-weighting a capitalization-weighted index by fundamental values, it is possible to produce consistently higher returns and higher risk adjusted returns. This result is in line with Arnott, Hsu and Moore (2005) for the US markets. The excess returns of the fundamental index portfolios over the cap-weighted index could arise from (1) superior market portfolio construction, (2) price inefficiency, (3) additional risk to exposure to distress risk, (4) a mixture of these three (Arnott, Hsu and Moore, 2005). As the numbers above show, fundamental indexation seems to generate a positive alpha compared to cap-weighted indexation. Alphas are used repeatedly in the academic literature to reject (1) the S&P 500 as a good market proxy, (2) the link between noise in asset pricing and the factor returns observed for value and size, (3) the CAPM’s single factor framework, and (4) price efficiency (Arnott, Hsu and Moore, 2005). According to Arnott, Hsu and Moore (2005), Hsu and Campollo (2006), Estrada (2006), Hemminki and Puttonen (2008), a strategy based on fundamental indexation is less noisy as compared to a strategy on market cap weighting. Therefore, the alpha of fundamental indexation is positive. Roll (1981, 1983) and Blume and Stambaugh (1983) argue that the observed price is the bid or the ask, not the fair value, thus price is different from fair value by a random noise. This random noise is the amount by which the ask price exceeds the bid price. This is essentially the difference in price between the highest price that a buyer is willing to pay for an asset and the lowest price for which a seller is willing to sell it. Price can be different from the fair value if investors under- or over-react. Black (1986) proposes that financial markets are noisy (that prices are different from fair values) due to trading by investors without information. Hsu (2006) shows that a mispricing premium may exist because there are investors with liquidity needs. Arnott, Hsu, Liu and Markowitz (2007) suggest that noise can create the size and value effect. Arnott, Hsu and Moore (2005), Hsu and Campollo (2006), Estrada (2006), Hemminki and Puttonen (2008) conclude that fundamental indexation is less noisy than cap-weighted indexation. In this paper we want to analyze which variables drives noise and thus the performance difference between capweighted indexation and fundamental indexation. In the academic world this topic is not yet described. The problem is that it is unclear what noise exactly drives. In this paper cross sectional standard deviation, volatility and volume are analyzed as proxies of noise. The other six variables we use in this paper are variables that can explain the performance difference between fundamental indexation and cap-weighted indexation but are not proxies for noise. The aim of this research is to find out what drives 3.

(4) the difference between cap-weighted indexation and fundamental indexation. Arnott (2005) and Siegel (2006) have spent a lot of time heralding the supposed return benefits of their respective approaches, but very little time explaining the source of these benefits or discussing the inherent systematic frictions (Ambruster, 2006). In this paper our objective is to add evidence for the noise trading explanation and find out what the performance difference drives. This results in the following research question: What drives the performance difference between cap-weighted indexation and fundamental indexation? The methods used to answer the research question are a regression analysis and a bilinear t-test. The regression analysis is performed to analyze whether the nine variables drive the performance difference between fundamental indexation and cap-weighted indexation. The difference in performance will be calculated by comparing two portfolios. The first portfolio is the cap-weighted indexation portfolio, it replicates the stock index of the Netherlands and is called the adjusted benchmark in this paper. The second portfolio is the fundamental indexation portfolio and is created with the variables from Arnott, Hsu and Moore (2005). The objective of the bilinear test is to analyze if the circumstances of the variables (upward/downward or above average/below average) drive the performance difference between fundamental indexation and cap-weighted indexation. All the observations on monthly basis are split up in two groups based if the variable is above or below average. Then we will test with a twopaired t-test whether fundamental indexation outperforms cap-weighted indexation and thus drives the performance difference between fundamental indexation and cap-weighted indexation in that specific circumstance. The regression analysis and the bilinear t-test are performed on the stock exchange of the Netherlands named the AEX-index in the period from 1983 up till 2008. The data related to fundamental indexation are collected from Datastream. The quotes for the stocks are collected on daily basis from Datastream. The data of the nine variables is collected on monthly basis. The structure of the paper is as follow: section II presents the methodology, section III shows the data and the descriptive statistics. In section IV we show the results. The conclusion is presented in section V.. 4.

(5) 2. Methodology For the calculation of the composite index, first the five fundamental indexes (dividend, earnings, sales, book value, free cash flow) have to be calculated. The weight of stock i in month t depends on the fundamental value in this period. The fundamental value of stock i on month t is divided by the sum of the 25 stocks with the highest fundamental value. The composite index is the average of the five fundamental indexes (dividend, earnings, sales, book value, free cash flow). When a stock did not pay out dividends, the composite index is the average of the indexes : earnings, sales, book value and free cash flow.. Return of the indexes The formula used to calculate the weight of stock i in year t is the following: . .

(6) . Where, . weight of stock i in the adjusted benchmark on date t.

(7) . total market value of all stock in the adjusted benchmark on date t. . market value of stock i on date t. The return of the AEX index in a particular month is calculated with the formula: ∑ Where, . return of the adjusted benchmark on date t. 5.

(8) To construct the fundamental indexes for all of the five fundamental indexes the data of all stocks quoted at moment i are collected. The return of the stocks is calculated with the same formula as the return formula of the adjusted benchmark. The weight of stock i on date t depends on the fundamental value in this period. The fundamental value of stock i on date t is divided by the sum of the 25 stocks with the highest fundamental value. The weight of the relevant stocks can be calculated with the following formula: . .

(9) . Where, . weight of stock in the fundamental index on date t.

(10) . value of the 25 highest fundamental value stocks on date t. . value of fundamental value of stock i on date t. The return of a fundamental index in a particular month is calculated with the formula: . . Where, . return of the fundamental index on date t. The return of the composite index in a particular month is calculated with the formula: . ! . ". ". ". ". ". # + $ %! + &' #( + !(! + ) ) . . . . . We use nine explanatory variables and test whether these variables explain the performance difference. Below the motivation of the nine variables is described and the methodology we use to calculate the nine variables if necessary is given.. 6.

(11) Cross sectional standard deviation The first variable is cross sectional standard deviation, Hwang and Salmon (2004) conclude that cross sectional standard deviation is a statistical measures to test for herding or contagious behaviors in different stock markets and is an indirect proxy for noise. Yu and Sharaiha (2007) observe that crosssectional return dispersion can be an attractive metric for alpha granularity in markets. Quantitative analysis of alpha granularity yields insights into optimal portfolio construction. If cross-sectional return can be used to create a positive alpha, this indicates that there is noise in the market. Cross-sectional standard deviation measures the dispersion of stock returns at one point in time, the formula that is used to calculate the cross sectional standard deviation is the following:. *$!! ! ( + , - . -/ . Where, -/. average monthly return across all assets. ,. weight of asset i in the computation of the volatility. -. monthly return of asset i. Volume The second variable is volume, one important relationship between returns and volume is transaction costs. Kramer (1999) concludes that the volume of noise trading influences the marginal cost of transacting, it can also influence equilibrium prices. Thus a relationship between price and volume may exist is consistently with rational pricing, albeit rational pricing that takes noise trading into account as a risk factor. The rational agent operates in a market with noise traders, whose activities affect the marginal cost of transaction. This indicates that trading volume can be expected to be a proxy for noise. Trading volume of the adjusted benchmark in month i is calculated by multiplying the trading volume of stock i in month t by the weight of this stock in the adjusted benchmark.. Volatility of the AEX-index The third variable is volatility: if noise traders affect price, the noisy signal is sentiment, and the risk they cause is volatility. Brown (1997) show that, in fact, unusual levels of individual investors sentiment are associated with greater volatility of closed-end investment funds. This implies that volatility can be a proxy for noise and drives the performance difference between fundamental indexation and cap7.

(12) weighted indexation. Brown (1997) observes that irrational investors acting in concert on a noisy signal can influence asset prices and generate additional volatility. Deviations from the average level of sentiment are associated with increases in fund volatility only during trading hours. This result was expected because noise traders should affect prices only through trading hours.. Volatility is a statistical measure of the dispersion of returns of the stocks in the adjusted benchmark. Commonly, a higher volatility results in a riskier security. The formula used to calculate the volatility of the adjusted benchmark is in month i is: 012341435 +. ∑7. 8" . 6 9. Where,. return of stock i on day t. N. number of days in the month. 6. mean return of the stocks in the adjusted benchmark on day t. The other six variables we use in this paper are variables that can explain the performance difference between fundamental indexation and cap-weighted indexation but are not proxies for noise. The first three variables are gross domestic product, inflation and interest. In table one below we show all the relationship previous studies find between the variables and stock returns.. Gross domestic product The growth of the economy of a country is based on the volume change of the gross domestic product. This the sum of the national products and the valued added to those products. In this paper the we use the growth in gross domestic product as variable in the regression analysis and the bilinear t-test. Table I displays the gross domestic product variable applied in previous researches with their findings.. 8.

(13) Table I. Macro economic variable: Gross domestic product Gross domestic product. Positive. Insignificant. Chen, Roll and Ross (1986) Cutler, Poterba, Summers (1989) Schwert (1999) Lee (1992) Adelberger and Lockert (1999), Young (2006) Diacogiannis (1986) Poon and Taylor (1991) Chen (1995) Chan, Karceski and Lakonishok (1998). Inflation rate Inflation is a rise in the general level of prices of goods and services in an economy over a period of time. Price inflation is the inflation rate, which is mainly measured by the percentage change in a consumer price index (CPI) over time. In this study the CPI will be applied. Table II displays the variable inflation applied in previous researches with their findings. Table II. Macro economic variable: inflation Inflation. Negative. Insignificant. Fama and Eugene (1981) Chen, Roll and Ross (1986) Kaul (1987) Adelberger and Lockert (1999) Gunsel and Cuker (2007) Diacogiannis (1986) Poon and Taylor (1991) Lee (1992) Chen (1995). Interest rate The interest rate is measured by the long-term Dutch government bond rate (10 year). Table III displays the variable interest applied in previous researches with their findings. Table III. Macro economic variable: interest Interest. Negative Positive Insignificant. Chen, Roll and Ross (1986) Clare and Thomas (1994) Gunsel and Cuker (2007) Flannery and Protopapadakis (2002). The following two variables we use in this paper are the prices earnings ratio and the market to book value. Danielson and Dowdell (2001) conclude in their paper that the price earnings ratio and market to book value are good proxies for analyzing the real price of a company. The return-stages model can help managers gain a first-cut estimate of the type of future performance an acquired unit must produce (or the level of synergies that must be realized) to justify the purchase price. By doing so, the return-stages model can help guide a more detailed discounted cash flow analysis of the value of a company. 9.

(14) Price/earnings ratio The price earnings ratios is a valuation ratio of a company's current share price compared to its pershare earnings. In this paper the following formula is used to calculate the price earnings ratio of the AEX-index:. L. P Price per share. Ratio E Annual earnings per share. M. Where, N O. Ratio. price earnings ratio of the adjusted benchmark in month t. . weight of stock i in the adjusted benchmark in month t. Annual earnings per shareMP. annual earnings of stock i in month t. Price per shareMP. price per share of stock i in month t. Market/book ratio The market/book ratio is used to find the value of a company by comparing the book value of a firm to its market value. In this paper the following formula is used to calculate the market/book ratio of the adjusted benchmark: L. Market value of stock. M RatioP B Book value of stock. M. Where V W. RatioP. market to book value of the adjusted benchmark in month t. . weight of stock i in the adjusted benchmark on date t. Book value of stock MP. book value of stock i in month t. Market value of stock MP. market value of stock i in month t. Bull/bear market Finally the bull/bear market variable is used to analyze if the performance difference can be explained by the fact if the market is bearish or bullish. Bull markets are characterized by optimism, investor. 10.

(15) confidence and expectations that strong results will continue. In this paper the adjusted benchmark is labeled bull if the average monthly return is positive. A market condition in which the prices of securities are falling, and widespread pessimism causes the negative sentiment to be self-sustaining is called a bear market. In this paper the adjusted benchmark is labeled if the average monthly return is negative.. Regression analysis To analyze the performance difference between fundamental indexation and cap-weighted indexation, the monthly excess returns of the fundamental index related to the adjusted benchmark are regressed to the nine variables that could drive the performance difference. The regression equation of the variables driving the performance difference between fundamental indexation and adjusted benchmark used in this paper have the following form: . ! . . XY Z X" [" Z X [ Z X\ [\ Z X] [] Z X [ Z X^ [^ Z X_ [_ ZX`[` Z Xa [a Z b (1). In this study the following variables are employed: . . ! . return of the composite index in month t return of the cap-weighted index in month t. XY =. Constant term. c =. Trade volume in month t. c" =. c\ = c] =. c =. c^ =. c_ = c` =. ca =. b . Cross sectional standard deviation in month t. Volatility of the AEX index in month t Gross domestic product in month t Inflation rate in month t Interest rate in month t Bull/bear market in month t Price/earnings ratio in month t Market/book ratio in month t Residual error. The X" , X , X\ , X] , X , X^ , X_ , X` and Xa are the coefficients indicating the change in excess return by the variables x1, x2, x3, x4, x5, x6, x7, x8, x9.. In this paper also the individual fundamental indexes which together create the fundamental index are regressed to the nine variables. This result in six sub-regressions, which are almost the same as. 11.

(16) equation number one. The only adaptation is that . ! . is changed into the return of the. individual index.. Where, # . Is the return of the dividend index in month t. &'#( . Is the return of the book value index in month t. $ %! . Is the return of the earnings index in month t. !(! . Is the return of the sales index in month t. . Is the return of the fundamental index without the book value index in. ) . ! ef . Is the return of the Free cash flow index in month t. month t. Bilinear test The second method we use in this paper to test what the performance difference between capweighted indexation and fundamental indexation drives is a bilinear test. We discern two groups of variables, potentially moderating the excess return between the adjusted benchmark and the fundamental index. We will test whether the nine variables are bilinear. For the noise variables an average is created based on all the observations. All the observations are split up in two groups. The first group are the observations in which the variable is above average. The second group are the observations in which the variable is below average. The other variables are split up based on the trend of the observation in month i. The first group are the observation in which the variables are upward trended and group two are the observation in which the variables are downward. For each of the variables the monthly excess return related to the benchmark index is calculated. Furthermore, we test whether this excess return significantly differs from zero.. 12.

(17) 3. Data description All total returns of the stocks used in this paper are collected from Thomson’s Datastream. The data is collected from the period from January 1983 - November 2008. For a full coverage of all the stocks listed on the Dutch Stock exchange, two constituent lists are integrated, the all share index of the Netherlands and the Dead list index. This new combined list presents all stocks that have been traded in the period between January 1983 and November 2008. In this paper we use an adjusted version of the AEX-index as the benchmark, because of missing data. First, since market value and price data are not available for every stock, these stocks are excluded from the fundamental indexes and therefore also from the adjusted benchmark. Second, since data on fundamentals is not available for every stock in every year, these stocks are excluded from the fundamental index and should therefore also be excluded from the benchmark. The adjusted benchmark is created with information from www.behr.nl on this website the constituent list of the AEX-index is published and these constituents lists are adjusted. The adjusted AEX-index is. simply named the adjusted benchmark, in the remaining of this paper. In this paper fundamental indices are based on dividend, book value, earnings, sales and the total free cash flow. For all stocks that have been traded in the period between 1983-2008 the relevant measures are collected from Thomson’s Datastream. In this paper also a composite index is created. This index is an equally weighted index of the five fundamental metrics by size (when a company was not paying dividend, the averaged of the four size metrics are used instead of the full five). In the first 12 years the composite index is created by the book value, earnings, dividend and sales. In the other years the free cash flow is also included. All the fundamentals are equally weighted. The indexes are only balanced once a year on the last trading day, following Arnott, Hsu and Moore (2005). In their study they used a monthly, yearly and semiannual rebalancing index turnover, but they did not find an applicable return over monthly rebalancing. At the fundamental index free cash flow, the world scope free cash flow, the first 12 years are missing because there is not enough data in Datastream to make a representative fundamental index. 13.

(18) In table IV the descriptive statistics from the AEX-index and the fundamental indexes are presented. From table IV can be derived that the index of the book value is significantly different from the other fundamental indexes. This is the reason that a composite index is created in this study, which excludes the book value index. Table IV. Descriptive statistics Descriptive statistics of the adjusted benchmark and the fundamental indexes. Adj. Comp Dividend Earnings Bookvalue Sales FCF Mean 1,230% 0,855% 0,926% 1,061% -0,048% 1,380% 0,653% Median 1,921% 1,693% 1,684% 1,280% 0,433% 1,928% 1,915% Standard deviation 6,174% 5,347% 5,675% 6,541% 6,156% 5,824% 7,463% Skewness -0,865 -1,819 -1,408 0,576 -1,061 -1,130 -1,966 Kurtosis 2,227 9,030 6,355 12,196 7,515 3,544 9,345 Observations 287 287 287 287 287 287 155 Min. monthly -28,491% -36,910% -34,532% -32,526% -41,449% -29,987% -46,055% Max. monthly 15,477% 12,707% 15,193% 49,121% 21,268% 13,545% 15,755%. Comp1,106% 1,846% 5,730% -1,483 6,584 287 -35,770% -13,563%. Adj. = Adjusted benchmark Comp = Composite index FCF = Free cash flow index Comp- = Composite index without the book value index. Table IV shows that the fundamental indexes have a slightly stronger skewness and kurtosis. This suggests modestly more outliers in the historical returns of the fundamental indexes. Table IV also shows that the fundamental indexes are exposed to more extreme monthly outliers. The exposure to outliers is less extreme in the composite indexes. The most extreme outliers are found in the book value index. Remarkable is that the negative outliers as well as the positive outliers are extreme in the book value index and the monthly return of the index is extremely low. In table V the correlation coefficients among the fundamentals indexes and the AEX-index are presented. The fundamental index based on book value shows the lowest correlation with the other indexes. All the indexes have a correlation with each other above the 80% except the book value which correlation with the other indexes is far below the 80%. As mentioned previously in this paper this is another reason why we create a composite index without the book value index.. 14.

(19) Table V. Correlation between the indexes Adj. Comp Dividend Earnings Bookvalue Sales FCF Comp-. Adj. 100,00% 90,37% 77,21% 51,99% 93,94% 92,39% 90,39% 92,42%. Comp 100,00% 82,52% 51,74% 91,41% 96,41% 94,19% 96,97%. Dividend. 100,00% 54,97% 77,91% 80,49% 89,82% 90,73%. Earnings Bookvalue. 100,00% 52,01% 59,67% 71,82% 55,77%. Sales. FCF. Comp-. 100,00% 95,56% 100,00% 92,71% 95,17% 100,00% 95,12% 97,12% 97,77% 100,00%. Adj. = Adjusted benchmark Comp = Composite index FCF = Free cash flow index Comp- = Composite index without the book value index. Table VI presents the performance and volatility of the AEX-index and fundamental indexes in two different periods. Table VI. Returns indexes sub-divided in two periods. Mean Median St. dev Observations. Adj. 1,72% 1,80% 4,84% 132. Mean Median St. dev Observations. Adj. 0,815% 1,988% 7,090% 155. Mean Mediaan St dev. Adj. -0,903% 0,192% 2,250%. Panel A: period 1985-1996 Dividend Earnings Bookvalue 1,27% 1,36% 0,62% 1,25% 1,29% -0,24% 3,94% 3,97% 4,34% 132 132 132 Panel B: period 1996-2008 Comp Dividend Earnings Bookvalue 0,469% 0,631% 0,805% -0,619% 1,857% 2,307% 1,280% 0,791% 6,570% 6,798% 8,102% 7,308% 155 155 155 155 Difference between Panel B and Panel A Comp Dividend Earnings Bookvalue -0,840% -0,641% -0,555% -1,243% 0,421% 1,057% -0,013% 1,030% 3,242% 2,854% 4,129% 2,966% Comp 1,31% 1,44% 3,33% 132. Adj. = Adjusted benchmark Comp = Composite index FCF = Free cash flow index Comp- = Composite index without the book value index. 15. Sales 1,98% 1,64% 4,57% 132. FCF. Comp1,54% 1,71% 3,84% 132. Sales FCF Comp0,873% 0,653% 0,740% 2,111% 1,915% 2,187% 6,665% 7,463% 6,926% 155 155 155 Sales -1,104% 0,473% 2,090%. FCF. Comp-0,796% 0,480% 3,089%.

(20) It becomes clear that the first period has a higher average return, each of the indexes performed better in the first period (1984 till 1996) than during the second period (1996 till 2008). Remarkable is that the higher average return in the first period coincides with a lower standard deviation. After the description of the adjusted benchmark and the fundamental indexes in this part the descriptive statisics of the nine variables( X1, X2…. X9) is presented. In table VII the descriptive statistics of the nine variables are presented. Table VII. Descriptive statistics of the nine variables Mean Median Standard deviation Skewness Kurtosis Observations. CSSD. Volume. Volatility. 6,77% 5,86% 3,42% 1,91 5,14 287. 165.851,02 162.669,41 55.611,48 0,96 1,92 155. 0,46% 0,40% 0,27% 2,48 12,06 287. GDP. Inflation Interest Bull/Bear. 2,80 3,00 1,22 (0,52) (0,39) 276. 2,07 2,10 1,17 (0,01) 0,75 287. 4,92 4,48 2,11 0,85 (0,17) 273. 15,01% 26,05% 0,22 (0,88) 0,19 287. P/E. 17,79 14,30 10,04 1,35 0,94 287. M/B. 4,76 3,83 2,28 0,54 (1,10) 287. CSSD = Cross sectional standard deviation GDP = Gross domestic product. In table VIII the correlation coefficients between the nine variables are presented. For the power of the regression it is necessary that the independent variables have a relative low correlation with each other. (Brooks, 2002). From table VIII can be observed that only two variables have a positive correlation coefficient above 60%, namely volume and market to book and cross sectional standard deviation and volatiliy. Table VIII. Correlation between the independent variables CSSD CSSD 100,000% Volume 43,086% Volatility 82,718% GDP -17,290% Inflation 29,324% Interest -8,066% Bull/Bear -19,491% P/E 12,049% M/B 29,657%. Volume. Volatility. GDP. Inflation. 100,000% 46,701% -22,322% 32,653% 30,238% -45,015% -8,423% 60,272%. 100,000% -13,671% 13,523% -7,379% -22,849% -3,411% 21,755%. 100,000% -20,880% 11,720% 40,658% 30,276% -46,385%. Interest. Bull/Bear. P/E. M/B. 100,000% 22,684% 100,000% -33,106% 9,718% 100,000% 15,311% -42,706% 4,715% 100,000% 37,545% -32,617% -64,913% 81,213% 100,000%. CSSD = Cross sectional standard deviation GDP = Gross domestic product. In table IX the descriptive statistics are presented. The desecriptive statistics are sub-divided in two different periods. The first period is drom 1985-1996 and the second period is from 1996-2008. Table IX also presents the difference between the two periods. 16.

(21) Table IX. Descriptive statistic independent variables sub-divided in two periodes Panel A: period 1985-1996 Mean Median St. dev Observations. Mean Median St. dev Observations. Mean Median St. dev. CSSD. Volatility. 5,90% 5,27% 2,36% 132. 0,43% 0,38% 0,20% 132. CSSD. Volatility. 7,51% 6,49% 3,97% 155. GDP. Inflation. Interest. 2,80 1,87 6,73 2,80 2,20 5,80 0,93 1,38 1,80 132 132 120 Panel A:period 1996-2008 GDP. Inflation. Interest. 0,48% 2,80 2,25 3,50 0,41% 3,40 2,00 3,40 0,31% 1,44 0,91 0,89 155 144 155 153 Difference between Panel B and Panel A. Bull/Bear. P/E. M/B. 0,21 0,28 0,18 132. 12,08 10,44 3,02 132. 3,57 2,80 1,53 132. Bull/Bear. P/E. M/B. 0,10 0,13 0,23 155. 22,64 21,05 11,30 155. 5,77 5,54 2,33 155. CSSD. Volatility. GDP. Inflation. Interest. Bull/Bear. P/E. M/B. 1,62% 1,23% 1,61%. 0,05% 0,03% 0,11%. 0,00 0,60 0,51. 0,38 -0,20 -0,47. -3,23 -2,40 -0,91. -0,10 -0,15 0,06. 10,56 10,62 8,29. 2,20 2,74 0,80. CSSD = Cross sectional standard deviation GDP = Gross domestic product. Table IX shows that the cross sectional standard deviation and the volatility are higher in de second period than in the first period. This indicates that investing in the second period is more risky than in the first period. It also becomes clear that the average inflation in the second period is higher than in the first in contrast with the interest which decreases in the second period.. 17.

(22) 4. Results First we show the results of the analysis of the performance difference in returns between the fundamental indexes and the adjusted benchmark. We proceed in the second part presenting the results of the regression analyses. Finally in the third section, we present the results of our bilinear test. In table X we present the average excess return of the seven fundamental indexes related to the adjusted benchmark over the period from 1983-2008. The t-test is performed on the paired monthly returns of the fundamental indexes and the benchmark index. Table X. Cumulative test results The returns are total returns including reinvestment of dividend. The returns are presented on yearly basis. The t-tests are performed on the monthly returns of the fundamental indexes and the adjusted benchmark over the period 1984-2008. Adj. Comp Dividend Earnings Bookvalue Sales FCF CompAverage Return 14,818% 10,303% 11,190% 12,805% -0,653% 16,643% 8,001% 13,355% Standard deviation 21,828% 21,453% 21,513% 24,216% 30,207% 22,218% 25,969% 21,864% Excess return -4,515% -3,627% -2,013% -15,470% 1,825% -1,934% -1,462% Significance 0,014 0,042 0,387 0,000 0,183 0,371 0,292 Sharpe ratio 0,475 0,273 0,314 0,345 -0,169 0,549 0,137 0,408 Adj. = Adjusted benchmark Comp = Composite index FCF = Free cash flow index Comp- = Composite index without the book value index. Table X presents the yearly returns and standard deviation of the eight indexes. In the first two rows we show the yearly average return per index and the standard deviation of the index. In the third row we present the cumulative average excess returns of the period 1983-2008. We test whether the cumulative excess returns of the fundamental indexes outperforms the adjusted benchmark index. Every index except the sales index shows a negative cumulative excess return related to the benchmark index. The dividend index, composite index and the book value index are significantly different than the adjusted benchmark. The other average excess returns are not significantly rejected. In the fifth row the Sharpe ratio is presented on yearly basis. The Sharpe ratio is the return of the “index” minus the risk free rate, divided by the standard deviation of the index. The average risk free rate in the period 19832008 is 4,44%. It becomes clear that the sales index has the highest Sharpe ratio and that the book value index results in the lowest Sharpe ratio.. 18.

(23) Results regression analyses In this section we present the results of the regression analyses, which measures if the nine variables drive the performance difference between fundamental indexation and cap-weighted indexation. First, we present the assumptions that are required to proof that the estimation technique and the hypothesis tests regarding the coefficient estimates are conducted in a valid way. We test for heteroscedasticity, auto correlation and whether data are normally distributed. We find evidence for the presence of heteroscedasticity in three of the nine indexes. We solve this problem by estimating the regression analysis with a heteroskedasticity consistent covariance matrix estimator which provides correct estimates of the coefficient covariance’s in the presence of heteroskedasticity of unknown form. Because the Durbin Watson values are in the range between 1.65-2.35 (Brooks, 2002, page:674), see table XI, we can observe that our data does not suffer from auto correlation. In table XI also the JarqueBera statistic is presented and shows that only the book value- and earnings regression equations are normally distributed. Therefore it is necessary to be extra cautious with conclusions/observations related to the other equations which are not normally distributed. In table XI, we present the results of the regression analysis. Without repeating the whole methodology, we give the main regression used in this paper with a short description of the variables included in this main regression.. 19.

(24) Table XI. Results regression analysis The dependent variables in the regression analysis are the seven indexes. The regression analysis is adjusted with White Heteroskedasticity-Consistent Standard Errors & Covariance. The estimators provides correct estimates of the coefficient covariance in the presence of heteroskeasticity. The coefficients that are significant at a 5% lever are market bold.. F-statistic Prob(F-statistic) Constant CSSD Trade volume Volatility GDP Inflation rate Interest rate Bull/Bear P/E ratio M/B ratio Durbin-Watson Jarque Berra Skewness Kurtosis R Square. Comp. 1,787 0,076 2,295 0,023 -0,914 0,362 0,839 0,403 -0,622 0,535 -0,413 0,681 0,161 0,873 -0,147 0,883 -1,256 0,211 -1,073 0,285 -1,628 0,106 2,103 3,921 1,595 1,043 0,107. Dividend 2,544 0,010 1,660 0,099 -0,306 0,760 0,719 0,473 -1,232 0,220 -0,524 0,601 0,001 0,999 0,051 0,959 -1,218 0,225 0,004 0,997 -0,912 0,364 1,986 2,062 0,492 4,571 0,146. Earnings 0,628 0,772 1,235 0,219 -0,739 0,461 0,870 0,386 -0,757 0,451 -0,184 0,854 0,191 0,849 0,163 0,871 -0,387 0,699 -0,389 0,698 -0,981 0,328 1,653 2,452 6,615 6,554 0,040. CSSD = Cross sectional standard deviation GDP= gross domestic prodcut of the netherlands. 20. Book value 1,146 0,335 1,991 0,049 -0,840 0,402 0,722 0,471 0,308 0,759 0,032 0,974 -0,581 0,562 -0,507 0,613 -1,456 0,148 -1,548 0,124 -1,176 0,242 2,316 3,835 0,082 3,782 0,009. Sales 1,763 0,081 1,177 0,241 -0,595 0,553 0,248 0,805 -1,160 0,248 -0,267 0,790 0,902 0,369 0,117 0,907 -0,535 0,593 -0,240 0,811 -0,995 0,322 2,105 2,260 0,355 4,806 0,106. FCF 2,286 0,020 1,967 0,050 -1,219 0,225 0,231 0,817 -0,712 0,478 -0,955 0,342 1,073 0,285 0,209 0,835 -0,807 0,421 -0,367 0,714 -1,583 0,116 2,132 2,345 -0,097 3,594 0,133. Comp-BV 2,056 0,038 1,680 0,095 -0,823 0,412 0,703 0,483 -1,136 0,258 -0,525 0,600 0,620 0,536 0,158 0,875 -0,779 0,437 -0,337 0,737 -1,367 0,174 1,879 3,863 1,545 1,041 0,121.

(25) In the first two rows of table XI we show the significance of the whole regression model. From the seven regression models, the earning and book value models are not significant, meaning that all independent variables do not significantly contribute to the variation of the dependent variables. The sales index and composite index are significant with 90% certainty and the free cash flow index, the composite index – book value index and the dividend index are significant with 95% certainty. Looking at the R-squared values we can observe that these values are low. A low R- squared indicates that the regression models explain a low percentage of the excess returns of the regression equations. The reason is that the variable ‘risk premium’ which explains a large part of the returns is not included in this regression model. From the regression model showed in table XI can be observed that the constant terms in the composite, book value and free cash flow regression equation are significant. This significance indicates an excess return, which is constituted independently from the independent variables in these regression equations.. Results Bilinear test In this part of the paper we test which variables are bilinear distributed and drives the performance difference between fundamental indexation and cap-weighted indexation. The results are presented in table XII.. 21.

(26) Table XII. Results bilinear t-test The t-tsts are performed on the montly excess returns of the fundamental index related to the adjusted benchmark over the period from 1984-2008. Table nine shows the excess return and the significance of the t-value. The excess returns that are bold are significant at 5% significance level. Name Comp Dividend Earnings Bookvalue Sales FCF CSSD Cum. Excess -75,26% -50,88% -40,77% -232,85% 1,92% -39,46% + Significance 0,030 0,109 0,396 0,003 0,654 0,152 CSSD Cum. Excess -32,52% -36,53% -7,90% -133,88% 41,19% 14,32% Significance 0,185 0,170 0,593 0,030 0,049 0,352 Volume Cum. Excess -34,39% -21,63% 3,57% -99,02% -10,35% -44,50% + Significance 0,141 0,206 0,655 0,056 0,465 0,094 Volume Cum. Excess -19,30% -6,88% -5,04% -123,25% 19,31% 19,36% Significance 0,276 0,499 0,608 0,030 0,267 0,291 Volatility Cum. Excess -46,04% -37,43% -12,85% -165,75% 12,82% -20,18% + Significance 0,115 0,102 0,594 0,014 0,437 0,365 Volatiliy Cum. Excess -61,75% -49,97% -35,82% -200,98% 30,29% -4,96% Significance 0,046 0,148 0,285 0,007 0,216 0,589 GDP Cum. Excess -42,72% -59,11% -3,43% -169,07% 50,26% -131,51% + Significance 0,150 0,026 0,660 0,024 0,045 0,647 GDP Cum. Excess -65,07% -28,30% -45,24% -197,65% -7,15% -23,12% Significance 0,025 0,257 0,140 0,003 0,418 0,241 Inflation Cum. Excess -23,41% -8,68% 37,96% -153,56% 54,24% 8,09% + Significance 0,331 0,543 0,405 0,030 0,020 0,478 Inflation Cum. Excess -77,48% -91,79% -86,81% -155,23% 10,11% -36,54% Significance 0,011 0,005 0,016 0,013 0,522 0,163 Interest Cum. Excess -23,02% -23,03% 59,06% -178,05% 45,45% 8,95% + Significance 0,270 0,227 0,198 0,010 0,027 0,372 Interest Cum. Excess -64,93% -50,98% -91,47% -142,43% -1,37% -20,13% Significance 0,021 0,063 0,016 0,025 0,487 0,237 Bull/Bear Cum. Excess -95,22% -73,43% -21,90% -332,16% 27,76% -250,41% + Significance 0,010 0,038 0,545 0,001 0,281 0,444 Bull/Bear Cum. Excess -12,57% -13,98% -26,77% -34,56% 15,35% -1,49% Significance 0,456 0,445 0,241 0,421 0,333 0,488 P/E Cum. Excess -69,02% -64,02% -34,09% -206,29% -2,61% -4,00% + Significance 0,024 0,008 0,441 0,006 0,642 0,618 P/E Cum. Excess -35,54% -20,16% -8,66% -153,93% 43,00% -21,14% Significance 0,195 0,407 0,583 0,021 0,025 0,308 M/B Cum. Excess -24,04% -34,98% 12,78% -112,71% 36,81% -57,69% + Significance 0,369 0,157 0,598 0,126 0,111 0,269 M/B Cum. Excess -80,51% -49,20% -55,53% -247,51% 3,57% -200,54% Significance 0,037 0,044 0,453 0,001 0,159 0,593 Cum. Exces = cumulative excess return compared to the adjusted benchmark CSSD = Cross sectional standard deviation GDP= gross domestic prodcut of the netherlands. 22. Comp-bv -33,66% 0,230 -1,87% 0,651 -18,23% 0,350 6,69% 0,521 -13,92% 0,457 -21,61% 0,356 -7,87% 0,562 -27,65% 0,209 11,92% 0,471 -56,40% 0,034 19,80% 0,265 -42,61% 0,078 -26,55% 0,332 -8,98% 0,504 -32,42% 0,194 -0,97% 0,667 -2,10% 0,652 -31,29% 0,431.

(27) The only fundamental index that outperforms the benchmark index under certain circumstances is sales. This is the case when the average cross sectional standard deviation is below average, the average BBP above average, the inflation raises and the interest raises. In case the average price earnings ratio decreases the sales index outperforms the benchmark index (> 95% significance). The book value index underperforms the adjusted index in almost all circumstances. Remarkable is that when the inflation decreases most of the fundamental indexes underperforms the benchmark index. Oppositely, when the inflation rate increases almost all fundamental indexes perform better than the benchmark index. When the interest rate increases, the fundamental indexes perform better in comparison with the situation of decreasing interests.. 23.

(28) 5. Conclusion The efficiency of cap-weighted indexation is subject of an important debate in de academic world. The central prediction that the asset pricing model is mean-variance, and thus a strategy based on capweighted indexation is the mean-variance efficient, is rejected by Arnott, Hsu and Moore (2005), Hsu (2006) and Treynor (2005). They conclude that a portfolio based on cap-weighted indexation leads to suboptimal portfolio return characteristics because prices are too noisy. As an alternative for capweighted indexation Arnott, Hsu and Moore (2005) introduce indexes based on fundamental indexation. In this paper we analyze which variables are good proxies for noise and determine the performance difference between fundamental indexation and cap-weighted indexation. In this research we investigate the performance of fundamental indexes and the cap-weighted index of the Dutch stock market in the period 1984-2008. In this paper the returns of five individual fundamental indexes (dividend, book value, free cash flow, sales and earnings) and two composite indexes (the composite index of all five indexes and a composite index without the book value index) are related to the benchmark index. We use nine variables subdivided in two groups, the noise variables and five variables, to explain the excess returns of the seven indexes. The research question, we answer using the results of the regression analysis and bilinear test, is: ‘what drives the performance difference between cap-weighted indexation and fundamental indexation?’ The answer of this research question is that the noise proxies, indicators of noise, do not drive the performance difference between fundamental indexation and cap-weighted indexation. Only in one case the noise indicator drives the performance difference between the two indexes. The only variable significantly explaining the excess return of the dividend index is volatility. Bilinear tests show that the circumstances of the macroeconomic indicators drive the performance difference. When the inflation rate increases almost all fundamental indexes perform better than the adjusted benchmark index. When the interest rate increases, the fundamental indexes perform better in comparison with the situation of decreasing interests. Gross domestic product did not drive the performance difference between fundamental indexation and cap-weighted indexation. So the overall conclusion is that noise does not drive the performance difference, but the circumstances of the macroeconomic indicators drive performance differences between fundamental indexation and capweighted indexation.. 24.

(29) In contrast to Arnott, Hsu and Moore (2005), Hsu and Campollo (2006) and Estrada (2008), who show that fundamental indexation consistently provides a higher return and lower risk than traditional capweighted indexation, we find the cap-weighted indexation to outperform the dividend index, book value index and the composite fundamental index. The limitation of this study is that the Dutch stock market is a relative small stock market. The relative small number of stocks on this market results in a great fluctuation in the composition of the index. In the paper this problem is partly solved by increasing the number of observations by using a relatively long research period. Other limitations of the study are that not all regression models are significant, one regression equation has many auto correlated variables, and five of the fundamental indexes do not provide a normal distribution. Because of these limitations the results of this study should be carefully interpreted. This study can be extended by an analysis of value and size effects of the stock presented in the indexes, for example using the three factor model of Fama and French (1993). These variables could be valuable in explaining the difference in excess return. Another interesting extension is the analysis of the influence of mean inversion in explaining the performance of the different indexes.. 25.

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