The tracking accuracy of S&P 500 index funds and its determinants

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The tracking accuracy of S&P 500 index funds and its determinants

June 2018

A bachelor’s thesis by Art de Vries

Supervised by R.C. Sperna Weiland MSc

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Statement of Originality

This document is written by Art de Vries who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

This research investigates the relation between the tracking accuracy of index funds that look to replicate the performance of the S&P 500 and their net asset values, expense ratios, management tenure, and fund structure as defined by being either a mutual fund or an exchange-traded fund (ETF). R2

-values that result from regressing index returns on fund returns are used as a measure of tracking accuracy. Funds generally tracked accurately, with an average R2_{-value of approximately 0.99 after}

adjusting for outliers. These R2_{-values were then regressed on net asset values, expense ratios,}

management tenure data, and a dummy variable indicating an ETF structure. The results indicate that management tenure and net asset value are positively related to tracking performance, and, in contrast to theory and previous findings, that expense ratio is also positively related to tracking performance. No significant difference in tracking performance was found between conventional mutual index funds and ETFs.

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1. Introduction

Investors who are interested in allocating a portion of their wealth to an index fund are regularly met with a strange phenomenon: on any given day, the return of the fund and that of the underlying index are different. Although these differences, or ‘tracking errors’, are generally small, a conundrum is posed to the investor: if a fund over- or underperforms its index on a daily basis, and if a fund tends to reverse to the average performance of the index over time, then timing matters, because buying when a fund outperforms its index may mean it will underperform in the days thereafter. If an investor buys regularly or with significant sums, these daily performance differences may start to add up. In order to reduce exposure to this risk, it might be best to invest in a fund for which the tracking error is small on a daily basis. Thus the investor wonders: by what characteristics can one recognize such a fund?

Whereas long-term average tracking errors are caused mainly by the expense ratio and dividend taxes (Blitz et al, 2012), daily tracking errors have been related not just to these, but to a much wider variety of sources. There is, for example, a negative relation between tracking accuracy (used interchangeably throughout this thesis as an inverse measure of tracking error) and the volatility of the underlying index (Chiang, 1998), broker fees, and securities regulations (Frino & Gallagher, 2001). Positive relations were found between tracking accuracy and management tenure (Fortin et al, 1999) and the amount of assets under management (Grinblatt and Titman, 1989). Also of influence are cash flows to and from the fund (Frino & Gallagher, 2001) and the replication strategy used by management (Fassas, 2014).

Most of this research concerned itself with mutual index funds (henceforth called conventional index funds). However, with the increasing popularity of the exchange-traded fund (ETF), a new vehicle for index investing has become available to the public. Low fees, continuous trading, and other benefits compared to conventional index funds have resulted in a swift expansion of the ETF market: over the past eight years, as the amount of ETFs doubled and their net asset value nearly quadrupled, they have overtaken the growth of the mutual fund industry (Investment Company Institute, 2018). This increasing popularity of ETFs raises another interesting question: can the ETF structure be added to the list of characteristics that indicate better daily tracking accuracy?

This thesis will seek to do two things: first, to contribute to existing literature by confirming, for a modern sample, the suggested relations between daily tracking error and the expense ratio, the net asset value, and management tenure, and to compare them with the findings of Cresson et al (2002), who performed this exact analysis for a set of conventional S&P 500 index funds for the year 1994. Second, to determine if an ETF structure also has an influence on daily tracking error, something which few researchers have yet investigated.

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In order to do so, this research will rely heavily on the methods described by Cresson et al (2002). The researchers used several regressions, composed of two major steps: the first was to determine the accuracy with which each fund in the sample was able to track the S&P 500. This was measured by using the R2_-values_{that resulted from regressing the returns of the index on the returns of}

each fund. The second step was to relate these R2_{-values to the characteristics of the funds by using}

both a multivariate and a set of univariate regressions. For this thesis, in addition to replicating these regressions for a modern data set, a dummy variable indicating whether the fund is an ETF or a conventional index fund will be added to the second regression.

The thesis is structured as follows: first a review of the existing literature will summarize some facts about the S&P 500 index and index tracking in general. Then, theoretical and empirically proven causes of index fund tracking errors will be examined, followed by an analysis of the differences between ETFs and conventional index funds, both of which may help form the basis for predictions about variations in tracking accuracy between different funds. After this, the sample data will be selected and modified to serve the purposes of this research, and the methodology as described in the previous paragraph will be applied. Results will then be presented and evaluated, followed by a discussion on the shortcomings of this paper, including recommendations for future research.

2. Literature Review

2.1 Tracking the market index: Standard & Poor’s Composite 500

Berk and DeMarzo (2014) describe a market index as a way to report the value of a specific stock portfolio. The Standard & Poor’s Composite 500 (S&P500) is perhaps the most famous of such indices: it consists of 500 of the largest U.S. stocks as measured by market capitalization, and is the standard proxy for the “market portfolio” as used in the Capital Asset Pricing Model (Berk and DeMarzo, 2014). Its 500 constituents, in number a mere 14% of the approximately 3500 actively traded listed firms in the U.S., represent some 70% of this group in terms of market capitalization (Standard and Poor’s, 2018), and represent them near perfectly in terms of weekly returns: between 1990 and 2009 this correlation was nearly 99% (Berk and DeMarzo, 2014).

Previously, a person seeking to invest in this representative portion of the American economy could only do so by constructing and maintaining a portfolio that held the same 500 stocks in the same percentages as does the S&P500. Such a venture would impose managerial demands far beyond the capabilities of the general public. The invention of the conventional index fund offered a solution. A conventional index fund’s board hires a management team to construct and maintain the index portfolio (Bodie et al, 2014). An individual investor then simply purchases a share of this portfolio from the

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issuing firm, and is thereby able to hold an asset that represents the entire index without any significant managerial responsibilities (Bodie et al, 2014).

2.2 The tracking error of index funds

There are many important differences between the indices and the funds that attempt to track them. While the index is simply a theoretical (or ‘paper’) portfolio, actually holding such a portfolio is subject to real-world frictions, which will cause a discrepancy between the returns of the index and those of the fund (Perold, 1988). This difference in performance is called the tracking error (Bodie et al, 2014), and is unavoidable for any index fund in a world without frictionless markets (Frino and Gallagher, 2001). Index fund managers must seek to minimize this error in order to achieve the performance of the benchmark that they are attempting to replicate.

There are numerous mathematical interpretations for tracking error. The earliest study on index fund tracking errors was done by Gruber (1996) and suggested using the Jensen alpha. The researcher found that S&P 500 index funds underperformed their benchmark by 0.202% per year from 1990 to 1994. Frino and Gallagher (2001) also investigated the tracking errors of S&P 500 index funds, but instead used three different methods simultaneously. The first method that they used was the monthly average of the absolute tracking error, as defined by the sum of all absolute monthly performance differences, divided by the amount of months under analysis. The second method was to use the standard deviation of these absolute performance differences over the period. The third method regressed the monthly returns of benchmark index on those of the portfolio, and used the resulting R2

-value as a measure of tracking accuracy, rather than error. The researchers found that between 1994 and 1999, fund underperformance compared to the S&P 500 was approximately 0.29% per year. Cresson et al (2002) chose to use this third method exclusively in their research on daily tracking accuracy, and found that this resulted in R2_{-values that were significantly smaller than those found in}

previous research based on monthly returns. This indicated that funds were much less able to track the index accurately on a daily basis than on a monthly basis.

2.2.1 Expense ratio

The potential sources for tracking errors are numerous. The most obvious one is perhaps management fees. Whereas the index does not need to be managed, merely calculated, an index fund is subject to management fees which reduce performance compared to that of the index. In addition to this, funds suffer from administrative costs, operating costs, and marketing costs. All these are summarized in the expense ratio of a fund, which expresses the percentage of the investor’s assets that is deducted annually to cover these costs. Frino and Gallagher (2001) found that, in the long-term, the underperformance of an index fund compared to its benchmark is approximately equal to the average expense ratio of the

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fund, confirming the suggestion by Chiang (1998) that the expense ratio should be the main source of long-term tracking errors.

2.2.2 The underlying index

Chiang (1998) also identified several other causes for tracking errors. One of them is related to the timing of index composition changes. The index assumes that such changes in the portfolio (for example, the deletion of the shares of one constituent and the addition of those of another) can be made immediately, in any quantity, and without cost (Perold 1988). In reality, however, stocks suffer from imperfect liquidity, meaning that such an adjustment is not immediate in a real portfolio, and the price at which issues are sold or bought according the index may be different from that which is realised by the fund (Chiang, 1998).

Furthermore, Frino and Gallagher (2001) suggest that not just composition changes of the index, but also share changes and corporate restructuring of its constituents may have a relation to tracking error, arguing in a manner similar to Chiang (1998) that these changes make it difficult for an index fund to replicate the target benchmark return. In the case of corporate restructuring, takeovers or mergers with companies that are not constituents of the index may cause a timing delay between the date when the fund receives the cash and when the target firm is removed from the index (Frino and Gallagher, 2001).

2.2.3 Dividends and dividend taxes

Another possible cause for tracking errors are dividends (Chiang, 1998): any index fund tracking a benchmark that is composed of a considerable amount of stocks may expect to receive dividends all year round as constituents will pay out on different dates; it would be impractical and costly to transfer this cash immediately to shareholders as it is received, and continual dividend reinvestment, as assumed by the S&P 500 (Standard & Poor’s, 2018), will again involve transaction costs. The timing of dividend distribution, in addition to dividend tax regulations that differ across geographical areas, can have a large influence on the tracking error. Blitz et al (2012) found, in fact, that dividend taxes may have a negative effect on long-term tracking accuracy that is nearly as large as that of the expense ratio.

2.2.4 Net asset value

There might also be a relation between tracking error and fund size (as measured by assets under management). Cresson et al (2002) found a significant negative relationship between the two, and Grinblatt and Titman (1989) provide a possible explanation by suggesting that funds with a larger amount of assets under management should, due to economies of scale, suffer less from the transaction costs that are unavoidably incurred when attempting index replication.

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2.2.5 Management

The aptitude and strategy of management may also have a large influence on the tracking error of the fund (Blume and Edelen, 2003, 2004). In order to track an index accurately, frequent trades must be made, for example when the composition or the weights of the index change. Higher turnover, however, also results in higher transaction costs, which in turn reduces performance and increases the after-cost tracking error (Frino and Gallagher, 2001). This double-edged effect poses a conundrum to management, and suggest that there is a significant role for managers to play in finding the optimal balance between the amount of transactions to make and the desired tracking accuracy (Frino and Gallagher, 2001).

In addition to this, Gastineau (2004) found that some conventional index funds outperform their benchmarks by deviating from the strategy of the index, which, although positive for the investor, may result in high (positive) tracking errors. Dickson and Shoven (1994) were able to show that tax minimization strategies are available to the management of conventional index funds, also allowing for higher post-tax returns, similarly affecting tracking error.

The effects caused by any management’s desire to minimize costs and optimize performance are complex and manifold, and its relation to the tracking error is not immediately clear from a theoretical perspective. However, Cresson et al (2002) found that the relation between management tenure and tracking error was significantly negative, suggesting that more experienced management teams are able to achieve lower tracking errors. These findings are supported by Fortin et al (1999), who also argue that experienced managers are in any case likely to be more efficient in replicating the benchmark.

2.3 Conventional index funds and ETFs

Since tracking errors are caused by market frictions that are not present in the case of index calculation, it may be assumed that whatever fund suffers the least from these frictions will track most accurately. In order to examine whether ETFs have an advantage in this area, a set of differences between the fund types is be presented in this section.

Index funds originally existed exclusively in the form of the passive mutual fund, a type of open-end investment company, where investors could only purchase shares once a day, directly from the management, at the net asset value (NAV) of the portfolio (Bodie et al, 2014). More recently, however, the exchange-traded fund (ETF) has become a popular vehicle with which to track indices (Investment Company Institute, 2018). In fact, as observed by Agapova (2011), many have argued that ETFs are more efficient investment vehicles and will therefore eventually replace the conventional index funds. Although the researcher dispelled this idea (findings indicated that the choice between conventional index funds and ETFs is more a matter of individual circumstance in terms of tax status

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and cash flow preference than it is of tracking efficiency), and while ETFs offer the same claim on underlying assets as conventional index funds do (Agapova, 2011) and suffer similarly from tracking error compared to their benchmark (Elton et al, 2002; Gastineau, 2002, 2004; Blume and Edelen, 2003, 2004), there are still several important differences that demand elaboration.

2.3.1 Liquidity, deviations from net asset value, and expenses

One of the prime differences is that ETFs trade continuously throughout the day, resulting in a higher liquidity compared to conventional index funds (Bodie et al, 2014). The disadvantage is that the market determines the price, allowing for deviations compared to the NAV of the ETF. The NAV of a conventional index fund, in contrast, is simply determined by the balance sheet at the end of the day, at the same time when shares are sold and purchased (Bodie et al, 2014). These disparities between the price and NAV of ETFs could increase tracking errors, and while these deviations are generally small, they may become exacerbated during periods of crisis, when high price volatility means it can be difficult to determine the net asset value of the underlying portfolio, especially when the ETFs track illiquid assets (Bodie et al, 2014).

An additional side-effect of being traded continuously on the market is that transactions with ETFs involve brokerage fees, unlike those with conventional index funds, which are purchased directly from the issuing firm (Bodie et al, 2014). As there is no broker involved when trading with the conventional index fund, however, this does mean that these funds must organize their own advertising, which might result in sales fees that could increase total fund costs (Bodie et al, 2014) and thus exacerbate tracking error.

2.3.2 Tax effects

The difference in fund structure also has tax effects, and Bodie et al (2014) note that ETFs may offer a potential tax advantage compared to conventional index funds. When a large amount of investors wish to redeem their positions in a conventional index fund during a small period of time, the fund must liquidate some of its holdings in order to gather the cash necessary to pay its investors. This may result in a capital gains tax being applied over the liquidated assets, which is passed through to the non-redeeming shareholders (Bodie et al, 2014), reducing performance and possibly increasing tracking error. This is not the case for ETFs, since redeeming shareholders will simply sell their shares to other investors, without the necessity of management having to liquidating part of the underlying portfolio (Bodie et al, 2014).

Poterba and Shoven (2002) performed an analysis to investigate this long-standing claim that ETFs are more tax efficient, and found that between 1994 and 2000, the pre- and post-tax returns of the largest S&P 500 ETF were not significantly different from those of the largest conventional S&P 500 index fund. One reason for this finding may be the fact the management team of conventional index

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funds has more flexibility in its cash-distribution policy, allowing for practices that may reduce the overall tax expense to the investor (Poterba and Shoven, 2002), whereas ETFs are not permitted this discretion by the SEC. Since both funds have tax benefits and drawbacks, it is difficult to predict whether the tracking accuracy of one fund type will suffer more from tax effects than the other.

2.4 Summary and predictions

Using this theoretical background, the following predictions may be made: first, Grinblatt and Titman (1989) suggested that larger net asset values should result in relatively lower transaction costs. Thus able to reduce this particular type of market friction, larger funds should be able to track the index more closely. Cresson et al (2002) confirmed this by showing that there was indeed a positive relation between tracking accuracy and net asset value. Furthermore, Fortin et al (1999) argued that management tenure should be positively related to tracking accuracy, as more experienced managers are able to replicate the index more efficiently, possibly through tax minimization strategies (Dickson and Shoven, 1994), or through optimizing the amount of trades and thus minimizing transaction costs (Frino and Gallagher, 2001). This was also confirmed by Cresson et al (2002). In addition, expense ratios, as indicated by Chiang (1998) and confirmed by Frino and Gallagher (2001) as well as Blitz et al (2012), should be negatively related to tracking accuracy. Finally, an analysis of differences between conventional index funds and ETFs determined several advantages and disadvantages for both fund types. No clear indication was found that one type of fund suffers less from market frictions in the aggregate than the other, so no predictions can be made as to the relation between fund type and tracking accuracy.

3. Methodology

In this section, the sample of index funds and their return data are selected and modified to suit the needs of this research. Following this, three regressions are ran: the first to determine the relation between the returns of the funds and the returns of the index, resulting in one measure of tracking accuracy (R2_{) for each fund for the year 2017. The remaining two regressions will determine the relation}

of this tracking accuracy to a set of independent variables; the first to confirm the findings of Cresson et al (2002), the second to see if ETFs also play a role in predicting daily tracking accuracy.

3.1 Sample and data selection

Mutual funds that track the S&P 500 are collected from the Kiplinger mutual fund database by using the search term “S&P 500”. Since some funds offer multiple share classes, this yields 88 share classes for approximately 31 different conventional index funds.

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To ensure that the sample consists only of funds that seek to replicate the performance of the index, certain selection criteria were put in place. First, funds that contain the ‘S&P 500’ name but are not designed to track the index are recognized by being designated by Kiplinger to anything other than the ‘large blend’ composition universe: in addition to one other leveraged fund that was identified to be of this category because of its name and its abnormal returns compared to the index (‘Gotham Enhanced S&P500 Index Fund’), all were removed, and reduced the sample size by 12 share classes. Removing the share classes of two funds that use a different weighting than the S&P 500, that of the ‘Invesco Equally-Weighted S&P 500’ fund and of the ‘Index Funds S&P 500 Equal Weight’ fund, reduces the sample by a further 6 shares. Share classes that cannot be identified due to lacking a ticker symbol are also excluded: there were 3 such instances in the original sample, all belonging to the ‘JNL’ or ‘JNL/Mellon Capital’ family. All 5 share classes of the ‘Ivy ProShares S&P 500 Dividend Aristocrats Index Fund’ were excluded because of an enhanced dividend distribution policy that would make it impractical to calculate and compare daily returns. The remaining data set contains 62 share classes for 22 mutual funds. In addition to this, three S&P 500 ETFs could be identified: the SPDR S&P 500 ETF, the iShares Core S&P 500 ETF, and the Vanguard S&P 500 ETF. Counting the ETFs as share classes, this brings the total sample size to 65 share classes for 25 funds.

Daily closing price data for each share class of each remaining conventional index fund, along with closing index values for the S&P 500, are downloaded for the year 2017 from The Wall Street Journal’s Market Data Center, resulting in 15.358 individual daily price data points (excluding those of the index). CRSP is used for the closing prices of the three ETFs for the same time period, for an additional 759 data points, bringing the total to 16.117.

3.1.1 Outliers

First, these closing values are converted into daily returns. In order to check for outliers, the daily returns are compared to those of the S&P 500, and descriptive analysis is performed. This analysis indicated that approximately 90% of the daily returns of the conventional index funds show a difference with the S&P 500 that is less than 0.1 percentage point; 95.55% of the data shows a return difference that is less than 1 percentage point (see Table 1). Most return differences that are larger than this are likely due to (1) stock splits, (2) extraordinary fees, such as load charges that occur at a specific time of the year, or (3) extraordinary distributions, such as annual or one-time capital gains distributions that are not made by the index but, as O’Neal (1999) argues, are used by some funds to satisfy particular tax- or investment objectives.

For the ETFs, no data point showed a return difference with the S&P 500 larger than 0.55 percentage points, and 98.68% of the returns showed a difference that was lower than 0.50 percentage points. This disparity in the presence of large return differences among ETFs and the share classes of conventional index funds is likely due to the fact that (1) ETFs do not charge the aforementioned

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extraordinary fees (Agapova, 2011), nor (2) distribute extraordinary gains (Agapova, 2011), and (3) no ETF stock splits took place for the three funds in the sample during the period under investigation. The distribution of return differences for the three ETFs is included separately in Table 1.

Table 1: histogram bin data describing the distribution of the absolute percentage-point

return differences between the S&P 500 (Rm) and the index funds (Rfund), on any given day i,

for all data points in the sample, before correcting for outliers.

bin, |Rm,i – Rfund,i | Frequencyconv. Cumulativeconv., % FrequencyETF CumulativeETF, %

0.0001 6338 35.59% 184 24.24% 0.0002 4089 58.56% 171 46.77% 0.0003 2535 72.79% 135 64.56% 0.0004 1462 81.00% 100 77.73% 0.0005 814 85.57% 49 84.19% 0.001 1025 91.33% 99 97.23% 0.0015 420 93.69% 6 98.02% 0.002 318 95.47% 3 98.42% 0.004 531 98.46% 2 98.68% 0.006 146 99.28% 10 100.00% 0.008 28 99.43% 0 100.00% 0.01 20 99.55% 0 100.00% More 81 100.00% 0 100.00%

For the share classes of the conventional index funds, there are a number of very large return differences that severely distort tracking accuracy, as measured by the R2_{-values that result from}

regressing index returns on fund returns. These R2_{-values are pertinent to the rest of this paper, as those}

are in turn to be regressed on the independent variables. Because of this, it is important that these R2

-values are somewhat in line with previous research. Cresson et al (2002), for instance, reported a mean R2_{-value of 0.935 in the period 1989-1994. Without correcting for outliers, the mean R}2_{-value of the}

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(2002). Furthermore, without outlier correction, the individual share classes show great disparity in R2

-values, with a high standard deviation among the sample, and some share classes showing virtually no correlation with the index (see Table 2).

Therefore, aided by the descriptive data of Table 1, the outlier threshold was chosen to be defined as any absolute performance difference with the S&P 500 that is larger than 1 percentage point on any given date. Using this as the threshold for identifying a particular return date as outlier and excluding those from the analysis ensures that (1) the amount of outliers is small enough so as not to significantly reduce the sample size of daily returns, (2) the tracking accuracy of the funds will be more in line with previous empirical findings, and (3) the removed outliers are unlikely to include ‘normal’ return differences.

In this manner 81 data points were excluded in total, or approximately 1.3 per conventional index fund share class. The least outliers removed per share class was 0; the most was 3. The effects of these adjustments on the tracking accuracy of the share classes are also included in Table 2.

Table 2: tracking accuracy of the 62 conventional index fund share classes before and after the removal of 81 outliers as defined by an absolute return difference with the index of 1 percentage point or greater. R2_{is the result of regressing the returns of the index on those of the fund, and is}

used here and throughout the paper as a measure for tracking accuracy.

mean R2 _{min R}2 _{max R}2 _{Std. Dev. R}2 _N

Before correction 0.735 0.042 0.993 0.237 15.358

After correction 0.990 0.966 0.999 0.009 15.277

3.1.2 Independent variables

Kiplinger’s database also provided the independent variables that will be regressed on the funds: net asset values, expense ratios, and management tenure data. For the ETFs the net asset value was obtained from Yahoo! Finance and the expense ratio from their respective prospectuses. No data on management tenure could be identified for the ETFs.

Unfortunately, no historical data for the start of 2018 was reliably available for a large part of the sample, only present figures. The datum for which the data for the independent variables is accurate is the time it was downloaded, June 2018, and is thus not precisely aligned with the return data, which ran from the start of 2017 to the start of 2018. The assumption is thereby made that expense ratios and

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net asset values did not change to a distortional degree during the time between the start of 2018 and the date of downloading the data.

A complete summary of all 62 mutual fund share classes, the three ETFs, and their independent variables can be found in Table 1 of the Appendix.

3.2 Calculating the tracking accuracy

In order to calculate the tracking accuracy for each individual index fund share class and ETF, we follow the methodology of Cresson et al (2002). First, market returns are regressed on index fund returns:

Rit = αi + βi Rmt + εit

where

Rit is the return on index fund i on day t;

Rmt is the return of the S&P 500 Index on day t;

αi and βi are ordinary least squares parameters for index fund i; and

εit is the error term.

After calculating αi and βi, we obtain an R2 value for each share class (the same measure as shown

earlier in Table 2). This is an approximation of the predictive ability of the S&P 500 in relation to the daily returns of the fund, and will be interpreted as the tracking accuracy, as suggested by Cresson et al (2002). The R2_{-values of each share class are also included in Table 1 of the Appendix.}

3.3 Adjustments to share classes

Before R2_{-values can be regressed on the independent variables, a remark must be made about share}

classes. Different share classes of the same index fund represent the same claim on the underlying assets, and vary only in relation to distribution arrangements. These variations in distribution arrangements are caused by management, who adjust the size and timing of charges and distribution fees in order to appeal to different types of investors (O’Neal, 1999). This has two consequences for this research. First, there will be a disparity among share classes of the same fund in both tracking accuracy and expense ratios. Second, because the share classes of a fund also have certain characteristics in common, such as net asset value and management tenure, sample bias becomes a major concern.

To prevent any particular value of management tenure or net asset value from becoming over-represented in the sample because a fund has a disproportionate amount of share classes, each fund in the sample should be represented only once. This means that all share classes of any particular fund must be merged into one single share class that represents the same asset pool and the same management. Because the R2_{-values and expense ratios differ among the share classes, this poses the}

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difficult problem of determining which values for these variables will provide an accurate representation of the fund in its entirety.

Problematically, there is no information available on the distribution of assets among the share classes, so that no weight can be given to any specific share class. In addition, no information was available from other researchers regarding a methodology for handling this issue. Because of this, a simple mean is used for both the R2_{-values and the expense ratios, and are thus assumed to be}

representative of the entire fund. In addition to affecting the independent variables significantly, this further reduces the sample size to the amount of index funds (25). The smaller, adjusted sample, including the new R2_{-values and expense ratios, can be found in Table 3.}

3.4 Regressing the tracking accuracy on independent variables

Before regressing the R2_{-values on possible causal factors, one additional adjustment is necessary.}

According to Amihud and Lev (1981), R2_{is not normally distributed, and in order to run the regression,}

the following transformation must be made (Freund, 1971):

NR = ln(1+R) / (1-R)

These NR (or ‘normal R2_{’) values then allow for the following regression, identical to that performed}

by Cresson et al (2002):

NRi = αi + β1i EXPi + β2i NAVi + β3i TENi + εi

where

NRi is the normalized R2 for index fund i;

EXPi is the expense ratio for index fund i;

NAVi is the net asset value in million USD at year-end for index fund i;

TENi is the tenure of management in days for index fund i;

αi and β(1,2,3)i are ordinary least squares parameters for index fund i; and

εit is the error term.

Because no management tenure data could be identified for ETFs, and to provide a more valid comparison with Cresson et al (2002), who did not include any ETFs in their sample, the regression above is ran only on the 22 conventional index funds. To include the ETFs, another regression must be performed:

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NRi = αi + β1i EXPi + β2i NAVi + β3i ETFi + εi

where all variables are the same, except that ‘ETFi’ is a dummy variable indicating whether the fund is

or is not an ETF. The sample size is increased to 25 to include the three ETFs.

Due to the relatively small sample size, univariate regressions are also presented in the results.

Table 3: Sample of funds, including ETFs, along with independent variables, after merging share classes.

A μ-subscript indicates mean values of the merged share classes were used.

Fund name R2

μ EXPμ (%) NAV ($M) TEN (days) ETF

BlackRock S&P 500 Stock 0.980 0.36 14,817 3814 0

California Investment Trust S&P 500 Index 0.990 0.86 181 5306 0

Deutsche S&P 500 Index 0.992 0.47 991 2018 0

Dreyfus Basic S&P 500 Stock 0.993 0.35 2,570 6005 0

DWS S&P 500 Index Fund 0.985 0.82 991 4059 0

Great-West S&P 500® Index Fund 0.982 0.20 3,211 801 0

Invesco S&P 500 Index 0.995 0.62 1,220 2932 0

Legg Mason Batterymarch S&P 500 Index 0.998 0.49 272 1502 0

MainStay S&P 500 Index Fund 0.999 0.55 1,177 7892 0

Maxim S&P 500 Index Portfolio 0.984 0.68 3,211 801 0

MM S&P 500® Index Fund 0.997 0.72 3,400 4089 0

Nationwide S&P 500 Index 0.992 0.65 3,085 2018 0

Principal LargeCap S&P 500 Index 0.991 0.65 5,448 2659 0

Rydex S&P 500 0.999 1.83 203 4424 0

Schwab S&P 500 Index 0.999 0.05 32,549 1956 0

SEI Inst. Inv. Trust S&P 500 Index Fund 0.966 0.06 4,336 1653 0 Shelton Capital S&P 500 Index Direct 0.983 0.36 181 5306 0

SIMT S&P 500 Index 0.973 0.54 874 2384 0

SSgA S&P 500 Index 0.973 0.16 1,551 4940 0

State Farm S&P 500 Index 0.996 0.50 1,536 2232 0

TIAA-CREF S&P 500 Index Fund 0.998 0.19 4,797 4575 0

USAA S&P 500 Index Fund 0.982 0.20 7,055 4210 0

SPDR S&P 500 ETF 0.973 0.09 267,474 n/a 1

Vanguard S&P 500 ETF 0.982 0.04 92,460 n/a 1

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4. Results and conclusions

The results are split up in two sections. First are the findings for the regressions that analyse the conventional index funds exclusively, and compare them with net asset values, expense ratios, and management tenure data, and which are presented in Table 4. Second are the findings for the regressions with the sample that also includes the ETFs, and which uses as the third independent variable not management tenure, but a dummy variable indicating the presence of an ETF structure. The results of this regression are summarized in Table 5.

4.1 The conventional index funds

The results of the first regression are partially in line with predictions. Net asset value is positively correlated to the NR-values (the ‘normal R2_{’) at the 5% level, which was predicted by Grinblatt and}

Titman (1989), who claimed that economies of scale would allow funds with larger net asset values to realize smaller operating fees in comparison to the total value of the fund. Management tenure also shows a positive correlation to NR-values, albeit at the 10% level. This, too, was expected, as Fortin et al (1999) suggested that more experienced managers were likely to be more efficient at replicating the index underlying their fund.

Table 4: the sample of conventional index fund NR-values is regressed in both its multivariate and

univariate forms on the three independent variables: expense ratio (EXP), net asset value (NAV), and management tenure (TEN); identically to the regression ran by Cresson et al, 2002. Standard errors are shown in parentheses.

Model 1 Model 2 Model 3 Model 4

Constant -115.953 (112.999) 93.567 (76.801) 145.131 (53.084) 37.890 (95.088) EXP 235.189* (116.136) 161.357 (121.537) NAV 0.0144** (0.0062) 0.00736 (0.00652) TEN 0.0438* (0.0225) 0.0403 (0.0246) Observations 22 22 22 22 F-value 3.43** 1.76 1.27 2.70 * p < 0.10; ** p < 0.05

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The data also suggests that the expense ratio is positively related to NR-values at the 10% level. This is against the expectations presented in the literature. Higher expense ratios have a direct negative impact on fund returns, and should thus increase the tracking error (Chiang, 1998; Frino and Gallagher, 2001), meaning lower NR-values.

Compared to the findings of Cresson et al (2002), the results appear to be similar: they, too, found significant positive relations for management tenure and net asset values. One difference is that they found a non-significant (yet, strangely, also positive) relation for the expense ratio. Cresson et al (2002) link this remarkable behaviour of the expense ratio to sample size issues.

Because Cresson et al (2002) likely used different (but undisclosed) measures for net asset value and management tenure, meaningful comparison between the coefficients of the significant results is difficult. It is interesting to note, however, that Cresson et al (2002) found significant relations for management tenure and net asset value only in their univariate analyses, whereas in the regressions of this paper the significant relations were present only in the multivariate regressions.

4.2 The conventional index funds and the ETFs

Table 5: the sample of index fund and ETF NR-values is regressed in both its multivariate and

univariate form on the three independent variables: expense ratio (EXP), net asset value (NAV), and the ETF dummy variable (ETF). Standard errors are shown in parentheses.

Model 1 Model 2 Model 3 Model 4

Constant 86.621 (75.576) 73.708 (62.158) 174.388** (44.478) 176.442** (43.534) EXP 164.389 (118.577) 185.798** (104.789) NAV 0.000561 (0.00157) -0.001 (0.001) ETF -164.082 (292.033) -145.790 (125.643) Observations 25 25 25 25 F 1.11 3.14* 0.86 1.34 * p < 0.10; ** p < 0.05

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The second regression (presented in Table 5 on the previous page) was ran on the expanded sample of 22 conventional index funds and three ETFs, and replaced the independent variable ‘management tenure’ by a dummy variable indicating the presence of an ETF structure. In this case, only the expense ratio showed statistical significance, and only in the univariate regression. There was no significant difference in tracking accuracy to be found between conventional index funds and ETFs. It must be noted, however, that (1) none of the three ETFs showed any outliers during the period, and generally displayed smaller performance differences with the index (see Table 1 in the Methodology section), possibly skewing the results in favour of conventional index funds whose R2_{-value was improved by}

the removal of outliers, and (2) there were only three S&P 500 ETFs available for inclusion in the sample. These two facts may have reduced the reliability of the results.

Similarly to the first regression, the expense ratio is significantly related to the NR-values, with a positive coefficient. This is contrary to what the literature predicts. Sample size may again be the cause of this distortion.

4.3 Conclusions

This thesis sought to do two things. Firstly, to determine whether the predictions of the literature and the findings of Cresson et al (2002) could be confirmed by replicating their research with a modern set of conventional index funds. The results indicated that, indeed, for the year 2017, net asset values and management tenure are positively related to the daily tracking accuracy of conventional index funds, just as they were in 1994. Neither the results of Cresson et al (2002) nor those of this thesis could confirm that expense ratio influences daily tracking accuracy in the manner described in the literature. Both studies, however, suffer heavily from sample issues (see the Discussion section), and the results, in turn, are suspect.

The second objective was to investigate whether ETFs tracked more accurately than

conventional index funds. No relation was found, and based upon the results of this research, it cannot be concluded that investors who are seeking an investment that tracks the S&P 500 as accurately as possible on a day-to-day basis should regard the ETF as the more efficient vehicle for achieving this objective. Similar to the first regression, issues related to sample size may have had a severe influence on the results, the problem possibly exacerbated by the fact that only three ETFs were available for use in the regression, making reliable comparison with the sample of conventional index funds difficult.

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5.

Discussion

This section summarizes the results, compares them to the predictions made in the literature review, discusses some of the many problems encountered during the production of this paper, and concludes with recommendations for further research.

5.1 Results and predictions

The correlations of the tracking accuracy of the index funds with net asset value and management tenure behaved as expected. As suggested by Grinblatt and Titman (1989), the correlation between tracking accuracy and net asset value should be positive for reasons related to economies of scale, and management tenure, according to Fortin et al (1999), should result in more efficient management, and thus also higher tracking accuracy. The expense ratio showed a significant but unexpected relation with the tracking error in both regressions. Theory and empirical research suggests that the expense ratio should be negatively correlated with tracking accuracy as measured by R2_{. Both regressions,}

however, showed positive coefficients for the expense ratio.

5.2 Difficulties

During the conducting of this research, many issues manifested. Several of them are presented here, along with suggestions for improvement.

5.2.1 Sample size

The strongly counter-intuitive result of the expense ratio coefficient may be an example of the many effects that sample size issues had on this research. The amount of funds that track the S&P 500 is very limited, and 22 or 25 funds is likely not enough to reliably perform either the univariate or the multivariate regressions as described in this thesis. In addition to the small overall sample size, the amount of ETFs compared to the amount of conventional index funds was very small, meaning that it would be unlikely for the regression that included the ETF dummy variable to show any significant differences between the conventional index funds and the ETFs, even if there were any.

One possible solution is to expand the sample to include funds that track a different index than the S&P 500. The literature suggests that the relations as investigated in this thesis are applicable to any index fund, and thus the methodology of this paper would need only small adjustments to deal with a more diverse sample. Another way to increase the sample size is to include more than one year of R2_{-values per fund, along with the independent variables for those years. This would also provide}

valuable insight into the development of tracking accuracy and index fund characteristics over the years, and the relationship between the two. Cresson et al (2002) attempted to do this by calculating R2_{-values for each year that fund return data was available, but unfortunately only performed the}

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5.2.2 Outliers

Another possible influence was the outlier threshold. Before adjusting for outliers, significant disturbances caused by stock splits, fees, and extraordinary capital gains distributions distorted R2

-values to such an extent that some funds showed very little correlation with the index. The 1 percentage point threshold was decided by using descriptive statistics of the return differences and preliminary regressions of correlations with the index, which suggested that, at a 1 percentage point threshold, (1) the amount of outliers was small enough so as not to significantly reduce the sample size of daily returns, (2) the R2_{values would be more in line with previous empirical findings, and (3)}

the removed outliers would not include ‘normal’ return differences. It might well be so that this manner of dealing with outliers influenced the R2_{-values of the funds strongly enough to bias the}

sample toward the funds that had more outliers than others. A more accurate, but significantly more laborious approach would be to investigate the returns of each individual fund and determine at what dates stock splits or extraordinary distributions were made, and to exclude only those data points. 5.2.3 Tracking error

In addition, the chosen measure for tracking accuracy may have been insufficient in describing the dependent variable. Pope and Yadav (1994) suggested that there are numerous ways by which to calculate tracking accuracy, among which standard deviations and average performance differences. Each method will yield slightly different results. In order to solve this issue, an approach similar to that of Chu (2011) may be used, who utilized all three methods of calculation for his research on the tracking performance of ETFs.

5.2.4 Independent variables

Finally, there were two issues with the independent variables. First, the data of the independent variables was available only for the date when the sample of index funds was downloaded from Kiplinger. The return sample, however, runs from the start of 2017 to the start of 2018. The values of the independent variables at either the start or the end of the period under investigation would have been suitable for the regressions, but neither was readily available. The independent variables used might have been significantly different at the start of 2017 or 2018. Surely this data is stored somewhere in historical form, possibly in the CSPR mutual fund database (to which this researcher had no access at the time of writing). More reliable results would likely follow from the usage of said data.

Second, as share classes had to be merged in order to prevent any one fund to be

overrepresented in the sample, mean values for R2_{and the expense ratio were chosen as representative}

characteristics of the single resulting share class. This method does not take into account the fact that certain share classes may carry a significantly smaller portion of the asset pool than other share

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classes, and thus biases the values for R2_{and the expense ratio to the characteristics of these (possibly}

irrelevant) share classes.

5.3 Recommendations for further research

It is the conviction of this researcher that many of these troubles can be overcome without increasing the complexity of the methodology by any noteworthy degree.

To do so, more reliable research ought to come in the form of a synthesis between the data collection scope of Agapova (2011), who posed little restriction on which index the funds tracked, nor on the year over which the returns were measured; the determination of tracking accuracy as

described in the methodology of Chu (2011), who used multiple different tracking measures and performed the analysis on each; and the secondary regressions as ran by Cresson et al (2002), which formed the basis of this thesis. Such a methodology would ensure a larger sample size, a more balanced and wide-ranging description and calculation of tracking errors, and still maintain the same essential investigative methods.

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Appendix Table 1: the sample of 62 conventional index fund share classes and three ETFs, including R2-values, expense ratios as a percentage of total asset value, net asset values, and management tenure in days (when available). Share classes of the same fund family can generally be identified by comparing name, ticker, net asset value in million USD, and/or management tenure in days. Extracted from the Kiplinger mutual fund database.

Fund and share class Ticker R2 _{Expense ratio (%)} _{Net asset value ($M)} _{Management tenure (days)}

BlackRock S&P 500 Stock Inv A BSPAX 0.98005 0.36 14817 3814

BlackRock S&P 500 Stock Fund Institutional Shares BSPIX 0.97471 0.11 14817 3814 BlackRock S&P 500 Stock Fund Investor C1 Shares BSPZX 0.99345 1.08 14817 3814

BlackRock S&P 500 Stock Fund Service Shares BSPSX 0.97738 0.23 14817 3814

BlackRock S&P 500 Stock WFSPX 0.97304 0.04 14817 3814

California Investment Trust S&P 500 Index K+ SPXKX 0.99016 0.86 181 5306

Deutsche S&P 500 Index A+ SXPAX 0.99343 0.59 991 4059

Deutsche S&P 500 Index Fund Class R6 SXPRX 0.99065 0.35 991 4059

Dreyfus Basic S&P 500 Stock DSPIX 0.98608 0.2 2570 6005

Dreyfus S&P 500 Index PEOPX 0.99907 0.5 2561 6676

DWS S&P 500 Index Fund/C+ SXPCX 0.97788 1.3 991 4059

DWS S&P 500 Index S SCPIX 0.99125 0.34 991 4059

Great-West S&P 500® Index Fund Institutional Class MXKWX 0.98243 0.2 3211 801

Invesco S&P 500 Index A SPIAX 0.99815 0.58 1220 2932

Invesco S&P 500 Index C SPICX 0.98579 1.31 1220 2932

Invesco S&P 500 Index Class R6 SPISX 0.99804 0.26 1220 2932

Invesco S&P 500 Index Y SPIDX 0.9983 0.33 1220 2932

Legg Mason Batterymarch S&P 500 Index D SBSDX 0.99773 0.39 272 1502

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Legg Mason QS Batterymarch S&P 500 Index A SBSPX 0.99793 0.59 272 1502

MainStay S&P 500 Index Fund Class A MSPIX 0.99914 0.6 1177 7892

MainStay S&P 500 Index I MSMAX 0.99911 0.35 1177 7892

MainStay S&P 500 Index Inv MYSPX 0.99904 0.7 1177 7892

Maxim S&P 500 Index Portfolio MXVIX 0.9869 0.54 3211 801

Maxim S&P 500 Index Portfolio Class L+ MXVJX 0.98065 0.81 3211 801

MM S&P 500® Index Fund Class A MMFFX 0.99695 0.72 3400 4089

Nationwide S&P 500 Index A GRMAX 0.99172 0.59 3085 2018

Nationwide S&P 500 Index Fund Class C+ GRMCX 0.99491 1.24 3085 2018

Nationwide S&P 500 Index Fund Class R2+ GRMRX 0.99275 0.88 3085 2018

Nationwide S&P 500 Index Institutional GRMIX 0.9888 0.17 3085 2018

Nationwide S&P 500 Index Institutional Service GRISX 0.98915 0.42 3085 2018

Nationwide S&P 500 Index Service GRMSX 0.99178 0.57 3085 2018

Principal LargeCap S&P 500 Index A+ PLSAX 0.99628 0.46 5448 2659

Principal LargeCap S&P 500 Index C+ PLICX 0.98177 1.3 5448 2659

Principal LargeCap S&P 500 Index Fund Class R5 PLFPX 0.99654 0.41 5448 2659

Principal LargeCap S&P 500 Index Institutional PLFIX 0.99595 0.16 5448 2659

Principal LargeCap S&P 500 Index J PSPJX 0.99641 0.37 5448 2659

Principal LargeCap S&P 500 Index R2 PLFNX 0.97644 0.9 5448 2659

Principal LargeCap S&P 500 Index R3 PLFMX 0.99599 0.72 5448 2659

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Principal LargeCap S&P 500Index R1+ PLPIX 0.98044 1.03 5448 2659

Principal LargeCap S&P 500Index R4 PLFSX 0.99585 0.53 5448 2659

Rydex S&P 500 H RYSPX 0.99895 1.58 203 4424

Rydex Series Trust S&P 500 A RYSOX 0.99889 1.57 203 4424

Rydex Series Trust S&P 500 C RYSYX 0.99869 2.33 203 4424

Schwab S&P 500 Index SWPPX 0.99875 0.05 32549 1956

SEI Inst. Invest. Trust S&P 500 Index Fund Class A SPINX 0.96604 0.06 4336 1653

Shelton Capital S&P 500 Index Direct SPFIX 0.98337 0.36 181 5306

SIMT S&P 500 Index A SSPIX 0.97135 0.43 874 2384

SIMT S&P 500 Index Institutional SPIIX 0.97475 0.65 874 2718

SSgA S&P 500 Index N SVSPX 0.97256 0.16 1551 4940

State Farm S&P 500 Index A+ SNPAX 0.9964 0.45 1536 2232

State Farm S&P 500 Index B+ SNPBX 0.99607 0.64 1536 2232

State Farm S&P 500 Index Fund/Legacy B SLIBX 0.99631 0.58 1536 2232

State Farm S&P 500 Index Institutional SFXIX 0.99665 0.39 1536 2232

State Farm S&P 500 Index Legacy A SLIAX 0.99623 0.35 1536 2232

State Farm S&P 500 Index R1+ RSPOX 0.99618 0.59 1536 2232

State Farm S&P 500 Index R2+ RSPTX 0.99673 0.53 1536 2232

State Farm S&P 500 Index R3+ RSPHX 0.99696 0.46 1536 2232

TIAA-CREF S&P 500 Index Fund Advisor Class TISAX 0.99815 0.19 4797 4575

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TIAA-CREF S&P 500 Index Institutional TISPX 0.99814 0.06 4797 4575

TIAA-CREF S&P 500 Index Retirement TRSPX 0.99812 0.31 4797 4575

USAA S&P 500 Index Fund Member USSPX 0.98315 0.25 7055 4210

USAA S&P 500 Index Reward Shrs USPRX 0.98104 0.15 7055 4210

SPDR S&P 500 ETF SPY 0.973 0.09 267474 n/a

Vanguard S&P 500 ETF VOO 0.982 0.04 92460 n/a

The tracking accuracy of S&P 500 index funds and its determinants