Market efficiency and index fund flow
An empirical study of the relationship between passive investment and
broad-market efficiency
Authors: Erik Larsson
Jacob Wergeland
Supervisor: Taylan Mavruk
University of Gothenburg
Department of Economics
Centre for Finance
Graduate School
Master Thesis in Finance
Gothenburg
June 2020

iAbstract
An observable rise in the popularity of index funds have caused the index funds to, in 2017,
capture 20% of total fund assets globally. A cornerstone of such passive investment is a
belief in an efficiently priced security market. This paper aims to relate index fund flows
with market efficiency during the period 2000-2019. Using S&P500 returns we estimate a
market efficiency measurement called the Hurst exponent, using two accredited methods:
the rescaled range analysis (RS) and the detrended fluctuation analysis (DFA). We find
similar estimations as previous studies, wherein the S&P500 index have exhibited a slight
mean-reverting return process, close to theoretical market efficiency. We further relate
this time-varying market efficiency measurement of S&P500 to its index fund flows. Using
a correlation filtering method to find index funds in the US targeting the S&P500 index,
and aggregating these mutual funds individual flow, we obtain aggregate index fund flow.
Conducting a Granger causality test on both fractional flow and dollar flow, we find a
causality that market efficiency Granger cause index fund flow. We further estimate that
a lesser degree of market efficiency have a negative impact on flow: the more long-term
memory the index experience, the smaller level of flow. These results hold stronger for
dollar flow rather than fractional flow.
Suggested keywords: market efficiency, Hurst exponent, mutual fund flow, passive
investments
ii
Acknowledgement
The classic pirate saying goes: It is not the treasure chest at the X-marked spot that is
most valuable; the real treasure is the friends acquired on the journey. When we now find
ourselves at the crossroads between academic school years and a professional career, we
say this hold with absolute truth. Whether this friendship is characterized by knowledge,
companionship, or everlasting-friends, does not matter; every student should cherish every
moment.
In such dire times as the present with COVID-19 ravaging the world, where a smile
might be rare, we would like to start this paper with a small and playful joke.
Question:
What is the difference between a government bond and a man?
Answer:
The government bond matures!
Before we let the reader dive into the text, we would like to direct our incomparable
thanks towards our supervisor Taylan Mavruk for his valuable and helpful guidance. Af-
ter all, the mason stands powerless without the forge. With that, we wish you a pleasant
reading experience and hope this paper provides both interesting and educational aspects.
Best regards,
Erik & Jacob
CONTENTS iii
Contents
Abstract i
Acknowledgement ii
1 Introduction 1
2 Literature review 3
2.1 Market efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Fund flows and passive investments . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Fees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Data 6
3.1 Data management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.1 Total net assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.2 Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Handling of fees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Methodology 12
4.1 General method setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Long-term dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 The Hurst exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.1 GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.2 Rescaled range analysis . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.3 Detrended fluctuation analysis . . . . . . . . . . . . . . . . . . . . . 18
4.4 Endogeneity concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.5 Estimation and control variables . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Results 25
6 Conclusion 33
7 References 34
A Appendix 37
A.1 Data adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.2 Hurst histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.3 Regressions - naked data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
LIST OF TABLES iv
List of Figures
1 Histogram of returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Flow time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Evolution of Fees over time . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Hurst exponent results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Hurst exponent and CI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6 Hurst histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
List of Tables
1 Descriptive statistics for returns . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Flow descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 AR(1)-GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 VAR specification - fractional flow . . . . . . . . . . . . . . . . . . . . . . . 20
5 VAR specification - dollar flow . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Granger Causality test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Annual statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8 Regressions - fractional flow . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9 Regressions - dollar flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10 Regressions (lagged Hurst) - dollar flow . . . . . . . . . . . . . . . . . . . . 30
11 Standardized Hurst coefficients . . . . . . . . . . . . . . . . . . . . . . . . 32
12 Data adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
13 Regressions, naked data- fractional flow . . . . . . . . . . . . . . . . . . . . 38
14 Regressions, naked data - dollar flow . . . . . . . . . . . . . . . . . . . . . 39
1 INTRODUCTION 1
1 Introduction
Over the last decade, in the wake of the great financial crisis, the financial markets have
experienced an unprecedented rise in index funds as a popular investment vehicle; a rise
supported by their cost-efficiency (Malkiel, 2003; Sirri & Tufano, 1998; Weissensteiner,
2019). Passive investments, typically referring to a broad market index investment sce-
nario, 1 did in june 2017 grow up to 20% of total fund assets globally, compared to 8%
in 2007 (Sushko & Turner, 2018). Although there may be considerable amounts of moti-
vations for utilizing passive investing, one cornerstone, which should not be diminished,
is that the market must be considered efficient for passive investing to perform (Wer-
mers, 2000).2 If the market would not be efficient, prowess investors would not choose
passive investing (with its corresponding investment into something incorrectly priced)
but instead change strategy and actively invest, improving the performance of their in-
vesting (Garleanu & Heje Pederson, 2019). As such, in the case of inefficient markets,
some investors would deviate from this passive investing and would therefore change their
allocation from the market distribution; assuming informed decisions and sufficient and
adequate market-moving power, the efficiency of the market would thus improve.
Fama (1970, p.383) famously described market efficiency as a market where ”security
prices always ’fully reflect’ all available information”. Correctly priced securities thus
bring a no-arbitrage trading or investing environment, wherein no parties suffer informa-
tional disadvantages. One important aspect of this no-arbitrage system of an efficient
market is that no investor can be expected to persistently and systematically realize ab-
normal returns; and, as such, in an efficient market, only random (or rather unpredictable)
fluctuations of the returns are possible (Kristoufek & Vosvrda, 2013). As such, it seems
that the key motivational tool for a passive investor is a belief in an efficient market;
where upon such a realization of an efficient market, inducing an inflow of passive money.
Fund inflow and outflows have been a thoroughly researched topic within the finance lit-
1Explicitly, passive investments are investments with a purpose of replicating or obtaining the return
of a market benchmark or index. Such an investment could be an ETF (exchange-traded fund) or
mutual fund targeting an index. Most mutual funds replicate target index return by holding the index
composite assets weighted by their index proportion (traditional passive investing), while other funds
use derivative instruments (e.g. a synthetic ETF) to artificially replicate the return (synthetic passive
investing). Important to note here is the implication for market efficiency and how the all-encompassing
passive investing relates to it. As the synthetic passive investment vehicles or instruments have no market
moving-impact on the underlying index, it should not be equalized with traditional passive investments,
which posses such powers, with regard to market efficiency. It is self explanatory that a derivative
transaction should have no price effect on the underlying asset, while a transaction of the underlying
asset should affect the price of the underlying asset. Finally is the distinction between passive investment
and passive management (an investment with no intent of actual monitoring), where the latter is often
categorized as dumb money. Although passive investment infers a non-monitoring scenario as well, it is
not identical to passive management and should not be confused as such.
2Such motivations could be time-constraints and similar hobby-related causes. For such investors,
passive investing might prove easiest and most well-performing choice, regardless of the efficiency of the
market.
1 INTRODUCTION 2
erature. Understanding catalysts for fund flows have been essential for both practitioners
and researchers. Previous studies has mainly focused on the impact of fees (Huang, Wei,
& Yan, 2008; Sirri & Tufano, 1998), fund returns (Edelen & Warner, 2001) and volatility
(Cao, Chang, & Wang, 2008).
So how do we start by examining this causality between market efficiency and index
fund flows? And maybe more importantly, what is the direction of the causality? Several
event studies have examined the phenomenon of abnormal returns realized by index com-
position changes (see, for example, Belasco, Finke, & Nanigian (2012) or Petajisto (2011)),
clearly outlining an impact on prices, compared to non-constituents, due to index fund
flows. Such impact suggests predictability of abnormal returns of single constituents, but
does it indicate direction of the index or broad market efficiency? Intuitively, the overall
index return cannot be predicted from abnormal returns realized by index constituents
addition or deletion. As such, we have to utilize another type of measurement to deter-
mine broad market efficiency and how it relates to index fund flows. There exists various
such ways to measure and test broad market efficiency: Approximate Entropy [ApEn]
(Pincus, 1991; Pincus, 2008) to measure the irregularity and unpredictability of fluctua-
tions in time-series data; Variance Ratio (Lo & MacKinlay, 1988); the Efficiency Index
(Kristoufek & Vosvrda, 2013); or the Hurst exponent (Bariviera, 2011; Eom et al., 2008)
for measuring long-term memory. Notably, as Belasco, Finke, & Nanigian (2012) suggest,
the abnormal single constituents returns are in fact the liquidity premium corresponding
to index inclusion. Maybe this hints that it is in fact liquidity that is related to market
efficiency (it may be reasonable to assume that liquidity should be higher for an efficient
market, i.e. unpredictable returns), but Bariviera (2011) find only a partial relationship
between liquidity and market efficiency.
The majority of these previous studies have focused on testing, measuring, and rank-
ing the efficiency of large indices of various securities; but, in light of the above discussion
about the rise of passive investing, we will try to expand the existing body of litera-
ture by investigating the relationship between market efficiency and index fund flows,
utilizing the Hurst exponent as a measurement for market efficiency. The expansion is
two-fold: extending the econophysics literature by relating market efficiency to important
market characteristics, building upon the liquidity-linkage by Bariviera (2011) and the
predictability-linkage by Eom et al. (2008); and, further developing the fund flow litera-
ture by widening the studies from Huang, Wei, & Yan (2007), and Cao, Chang, & Wang
(2008), and partly Sirri & Tufano (1998).
The vast majority of research within both market efficiency and flow have been con-
ducted for the US-market, due to being among the most active and liquid markets. Further
reasons to use the US-markets is the possibility to compare our findings to similar previous
studies.
We aim to examine whether the S&P500 have exhibited market efficiency over last
2 LITERATURE REVIEW 3
two decades. We expect to find similar degree of efficiency for S&P500 as other developed
markets.3 We further aim to explain if, and subsequently how, index fund flow relates to
market efficiency: the direction of the causality and the following impact. We hypothesize
the causal direction that the degree of market efficiency affects the level of index fund
flows, with the magnitude that a less efficient market would induce less flow.
We identify 633 index funds targeting S&P500 between 2000 and 2019. Using monthly
index fund return and total net asset, we compute monthly aggregate flow values, both
fractional flow and dollar flow. We estimate the Hurst exponent for S&P500, acting as
a market efficiency measurement proxy. Overall, the S&P500 experienced a slight mean-
reverting return process, close to market efficiency during our sample period. Identifying
a causal relationship wherein a lesser degree of market efficiency negatively affects aggre-
gate index fund flow. This relationship is characterized by a magnitude of one standard
deviation change in the market efficiency measurement indicating a change in dollar flow
by approximately 15% of the standard deviation in dollar flow.
2 Literature review
2.1 Market efficiency
In 1970 (Fama) famously postulated the efficient market hypothesis (EMH) which has
been the dominant view of market functionality. Ever since creation, the EMH has been
challenged. Grossman & Stiglitz (1980) examined the aspects of information accessibility
and how this affects the EMH. Costless information is a requirement for efficient markets,
and, for a functioning marketplace, this is per se impossible (Grossman & Stiglitz, 1980).
With respect to the rise of passive investing in the recent decade, Fama (1991) posited that
efficient markets gave rise to passive investing, since markets were theoretically already
fully efficient. More recently, Weissensteiner (2019) instead theorized a reverse relation-
ship wherein market-wide forecast errors were reduced by passive investing and efficiency
improved. Kristoufek & Vosvrda (2013) found that, by estimating their Efficiency Index,
the most efficient markets were Japan, Denmark, and Germany and the least efficient
markets were Peru, Sri Lanka, and Slovakia. Suggesting that maybe the more globally in-
tegrated markets exhibit a larger degree of efficiency; nevertheless, the more local and less
developed markets often exhibit more long term memory (i.e. a Hurst exponent greater
than 0.5) while the US, UK, and other similar global markets were characterized by a
reverse condition wherein they experienced anti-persistence (i.e. a Hurst slightly lower
3The expectation of a similar degree of efficiency for S&P500 as other developed market is based on
previous studies findings, wherein such a similarity is found. Furthermore, Eom et al. (2008) conclude
that the degree of efficiency is highly related to predictability (average hit-rate); thus, global markets
should intuitively not exhibit easily available predictability.
2 LITERATURE REVIEW 4
than 0.5)(Kristoufek & Vosvrda, 2013)4. Bariviera (2011) found a similar result, wherein
the Thai stock market nearly consequently experienced positive memory from 1975 until
2005. Cajueiro & Tabak (2004) elaborated further on emerging markets and computed
Hurst coefficients for a wide range of market and compared them to the US market. They
found over time mainly positive long term memory for all the examined markets, hence
the US was closest to a Hurst of 0.5, concluding it to be the most efficient, whereas the
emergent markets were less efficient.
Likewise, Eom et al. (2008) found a difference in market efficiency between emerging
and more established markets. They utilized the Hurst coefficient and find that many of
the less develop market places to be above market efficiency level of 0.5, hence a positive
correlation. This implies that emerging markets exhibit more long-term memory than
their developed counterparts. Their result is in general in line with previous studies,
implying less efficiency in emerging markets than more developed markets; suggesting
that a higher Hurst exponent corresponds to a lower degree of efficicency, rather than a
Hurst exponent disconnected from 0.5. However, the Hurst levels estimated by Eom et al.
(2008) for emerging markets were, in general, lower than other studies (see for example
Cajueiro & Tabak (2004)). An explanation for this discrepancy could be due to differing
underlying computational statistics between the papers,5 which will be covered in this
thesis method section.
2.2 Fund flows and passive investments
Numerous studies have examined factors affecting mutual funds flow. Generally, the mu-
tual fund flows are affected by previous returns. While all mutual funds flow are sensitive
to recent returns (Sirri & Tufano, 1998; Sapp & Tiwari, 2004), the sensitivity increases
with lower participation costs (Huang, Wei, & Yan, 2007). Similarly, Warther (1995)
found that market-wide aggregate inflows are strongly affected by simultaneous aggregate
price movements. Cao, Chang, & Wang (2008) find that high frequency market volatil-
ity is negatively affected by aggregate flow, entailing evidence that a positive (negative)
shock in flow decreases (increases) market volatility. Sirri & Tufano (1998) found that
the flow into high-performing funds were disproportionately larger than the outflow from
poor-performing funds. The large body of mutual fund flow studies mainly deal with
panel data and intra-competition between the funds, and are often comprised entirely of
4The long term memory can be resembled by a momentum-factor (increases are likely to be followed
increases), and anti-persistence by a more negative correlation than randomness prescribes (increases are
more likely to be followed by decreases). The Hurst exponent is a method stemming from engineering,
to find repeating patterns within a data. A Hurst of 0.5 indicates no memory whereas a higher or lower
signals memory, either positive or negative memory.
5Eom et al (2008) uses detrended fluctuations analysis (DFA) to compute their Hurst coefficient
whereas Cajeiro & Tabak (2004), Bariviera (2011) and Kristoufek & Vosvrda (2012) utilizes rescaled
range statistics (RS). DFA is excluding short term memory in data at a greater level compared to the
RS. Yielding a difference in results. This will be further explained in the method section
2 LITERATURE REVIEW 5
actively managed funds. Index funds flow studies widely concerns miss-pricing due to
the flow, rather than trying to explain the flow. An inclusion into the substantial main
indices may distort prices due to the large share of passive capital inflow caused by the
relatively unconscious passive investing. Abnormal returns, both negative with the dele-
tion from index and positive with the addition to index, affects securities bordering an
index; between 1990 and 2005, the average excess (abnormal) return was from an index
addition 8.8% and from index deletion -15.1% (Belasco, Finke, & Nanigian, 2012).
Nonetheless, Hortacsu & Syverson (2004) identify the importance of non-portfolio
attributes in attracting investors for index funds. They further note that the low partic-
ipation costs of index funds cause the fund with the lowest fee not to capture the entire
homogeneous index fund market. On the same note, passive investing have shown to be
the most cost efficient and optimal choice for investors irregardless of market condition
due to the inability of fund managers to over time outperform their benchmark index
(Malkiel, 2003). This introduces an important aspect of index fund flows. There is a
fundamental difference between studying fund flows on a micro-level (fund specific flow)
and on a macro-level (aggregate flow)(Cao, Chang, & Wang, 2008). Micro-level studies
on mutual funds may examine differences between fund categories, often categorized the
funds having different objectives and returns, where the outflow from one fund might
be offset by inflow into another fund. Macro-level studies disregard specific fund flow,
thus only examine market flows (Warther, 1995). Hortac¸su & Syverson (2004) found that
index fund intra-competition were mainly driven by non-portfolio attributes, suggesting
that micro-level studies on index funds relationship to some index measurement makes
less sense since, generally, the reason of one index funds outflow should not be a another
index funds’ better return (after all index funds targeting the same index should have
similar returns). The low sensitivity of fees for index funds combined with their cost
efficiency suggest that index funds do not, in a performance setting, compete intravenous,
like actively managed fund do. To conclude, when examining funds consisting of index
holdings a macro-level approach is more suitable (Warther, 1995).
2.2.1 Fees
The subject of fees and its impact on flow has been revisited in the literature, Sirri &
Tufano (1998) and Huang, Wei, & Yan (2007) are among the more noticed. The many
properties of the fee component is not in the main scope for this thesis, although it is
essential to keep in mind when discussing funds. The general effect found in previous
literature indicate that fees has a negative impact on the fund flows, higher fees results
in lower inflow Sirri &Tufano, (1998). This relationship comes with exceptions, the most
prominent is the participation cost, including both search and marketing costs (Sirri
&Tufano, 1998). Funds pay to gain investors and therefore raise the fees, however still
attracting flow.
3 DATA 6
The fee structure is usually complex and includes several different fees, where the
most common are the front and rear load fees, management fees and 12b fees.6 For index
funds the main aspect of differentiating is how to streamline operations in regards to fees,
since the assets are bound to be index replicating. Cash management of the inflows and
outflows are crucial when evaluating the overall performance (Elton, Grauber & Busse,
2004).
3 Data
From CRSP (WRDS), we obtain monthly data of total net assets (TNA) and fund return
for 43 939 funds during the period 2000-2019, as well as both daily and monthly data for
S&P500 for the period 1996-20197. The daily data is used for the calculation of Hurst
exponent and the monthly S&P500 data is utilized to generate returns for index funds,
when specific fund returns are missing. To identify index funds tracking the S&P500
amongst all the acquired data we use a method suggested by WRDS (the provider of
CRSP). By checking whether the funds have a return with a correlation of at least 99.5%
(ρ ≥ 0.995) to the S&P500, we isolate all index funds tracking the S&P500.8 By utilizing
this filtering method, we reduce the number of mutual funds down to 633; in other words,
we identify 633 index funds which at some period between 2000 and 2019 target S&P500.9
We present introductory summary statistics for the identified index funds and S&P500
in Table 1. In figure 1 we show histograms over the S&P500 and identified index funds
returns. We observe similar means and medians, while the index funds exhibit a wider
distribution with more a larger outliers, resulting in a higher kurtosis and fatter tails.
The lowest reported TNA from CRSP is 0.10, i.e. $100 000, and is often reported the
first period the index fund became active: why both the minimum and 1% value are the
same.
3.1 Data management
Regarding missing values in both the return and total net assets (TNA) data of these
identified index funds, numerous actions has been undertaken. As the remaining funds
now target S&P500, we simply replace missing return values with the relevant return for
S&P500. Although we noted previously, and as can be observed in Table 1, the index
612b fees are costs associated with distribution and marketing. In general this fee combined with the
management fee is acknowledge as the expense ratio.
7The return data rt and the total net assets data TNAt indicate the return and total net assets for
period t as of month-end.
8The index fund flag indicator provided by CRSP is deemed insufficient for isolating index funds
targeting specific indices, why we instead use the suggested correlation filter approach.
9For some of these identified index funds we had ”inadequate” number of monthly returns, down to as
few as only 2 monthly returns. Although these sparse monthly returns produced a sufficient correlation
against S&P500, for robustness, we did a name check and confirmed index funds targeting S&P500.
3 DATA 7
Summary statistics
Panel A: returns Panel B: total net assets
S&P 500 (D) S&P 500 (M) Index funds (M) Index funds (M)
N 240 5031 62111 70158
Max 11.55% 10.77% 34.06% 319624.1
99% 9.42% 3.43% 9.73% 70468.7
75% 2.97% 0.56% 3.22% 667.9
50% 0.055% 0.96% 1.11% 148.8
25% -1.78% 0.47% -1.70% 23.3
1% -10.99% -3.32% -10.91% 0.1
Min -16.94% -9.03% -35.33% 0.1
Mean 0.023% 0.42% 0.56% 2847.8
Std 1.129% 4.18% 4.46% 15304.9
Skew -3.83 -59.9 -60.70 10.4
Kurt 11.8169 4.1068 5.1379 136.3
Table 1: Descriptive statistics for S&P500 and identified index funds from 2000-01-01 until 2019-12-31.
Index funds were identified by filtering with a correlation of at least 99.5% towards S&P500. The TNA
are in millions of USD. (D) and (M) represents daily and monthly data, respectively. The difference in
the number of observations between the returns and TNA is due to that the missing values do not always
align for the two variables.
funds and S&P500 does not exhibit identical distributions, the high correlation threshold
should be sufficient to not cause structural deficiencies. Below we outline procedures
undertaken for the TNA and flow data.10
3.1.1 Total net assets
Several funds had reported a value of 0 for total net asset (TNA) for some months, which
is unreasonable, as that would pertain to a non-existing fund; hence these values were
treated as missing values. The missing TNA values were filled by three techniques: back-
ward filling, forward filling and linear interpolation11. We utilize the Linear Interpolation
method, and the other two methods for robustness check. Gaps of missing values ranging
from 1 month up to 12 months were filled. Gaps of more than 12 months were treated as
missing values and were not filled. To ensure this would not cause inconsistencies in the
data, the number of gaps that would be filled from 13 months up to 24 months were one
gap of 16 months and four gaps of 17 months. Limiting the fill gap method to 12 months
does not greatly inhibit the data and suggest a removal of the possibility of an eventual
10In appendix A.1 we outline these procedures as well, but in a table format instead.
11For clarification, the linear interpolation fill method is simply a equal step-by-step increase to reach
the end of the gap from the beginning of the gap. Explicitly, if Ystart is the value before the gap and Yend
is the value after the gap, and n is the number of missing values between these two data points, then
the ni:th missing value Yni is given by Yni = Ystart + ni ×
Yend−Ystart
n+1 . The backward and forward filling
techniques are simply the next value after the gap carry backward and the previous value before the gap
carry forward, respectively.
3 DATA 8
(a) S&P500 (b) Index funds
Figure 1: Histogram over the returns of S&P500 and identified index funds from 2000-01-01 until
2019-12-31. The return data depicted are in percentage and can be seen in Table 1.
structural bias in the data set. This procedure increases the number of total TNA data
points by 2782, from 67 341 to 70 158.
3.1.2 Flow
In an attempt to not reinvent the wheel, and thus by convention of previous research
(Sirri & Tufano, 1998; Huang, Wei, & Yan, 2007), we utilize a standard measurement
variable for mutual funds flow. Explicitly, this can be defined as
FLOWi,t =
TNAi,t − TNAi,t−1 × (1 + ri,t)
TNAi,t−1
(3.1)
where TNAi,t is the total net assets for the fund i for time period t, and ri,t is the return
for fund i for the time period t. This standard procedure of estimating fund flows removes
the intrinsic returns of the funds constituents, thus cleaning the total assets change from
general price movements. This procedure for the funds flow computation is normally
done for estimating cross-sectional relationships. We base our aggregate index fund flow
computation formula on equation 3.1, but, as expected on an aggregate level, we modify
the formula to estimate flow on a macro-scale. We adopt a similar method as Cao, Chang
& Wang (2008) for calculating aggregate flow. The computation becomes fairly simple,
where, for each time period, we estimate the dollar value flow for each fund and obtain a
aggregate dollar value flow for each period. We subsequently adjust the aggregate dollar
value flow by dividing by previous periods aggregate total net assets, creating a fractional
3 DATA 9
flow measurement. Mathematically, this procedure is equivalent to
FLOWAGGt =
Nt∑
i=1
FLOWi,t × TNAi,t−1 ×
Nt−1∑
j=1
1
TNAj,t−1
(3.2)
where Nt is the number of index funds for period t. Explicitly, the dollar value flow is
thus given by
FLOWAGGt =
Nt∑
i=1
FLOWi,t × TNAi,t−1 (3.3)
As these flow equations rely on a difference operator, the resulting number of FLOWAGGt
values is one less than the number of
∑Nt
i=1 TNAi,t values. As such, we compute two
different flow measurements: fractional flow using equation 3.2, and dollar flow using
equation 3.3.
The reason why we utilize equation 3.2 and 3.3 for calculating FLOWAGGt values is
due to that some adjustments need to be done on individual fund flow level. Due to this
nature of the flow data, we obtain a total of 70 255 final individual flow values, after all
adjustments described below. Before any such adjustments to the flow data, we have 69
516 data points (i.e. naked individual flow data).12 As the data exhibits irregularities
regarding when different funds start and stop reporting data (we explain these as the
inception and death of the fund); following Sirri & Tufano, 1998, we manually insert a
100% flow value for the month of the first reported TNA value (the Inception) and a -
100% flow value for the month succeeding the last reported TNA value (the Death). This
Inception/Death procedure adds 739 new flow values, raising total data points to 70 255.
Likewise, some funds have long time periods of non-reported data in between data points;
we treat funds with such data structure as dead if the gap is longer than 12 months. To
reduce the impact of large outliers in the flow data, especially occurring in the second
period of a funds reported data, wherein the first period TNA is typically a very low
value and then a huge influx of money in the second time period’s TNA value, causing an
enormous flow value. We have winsorized the flow values in the first and the ninety ninth
percentile. We winsorize the entire data set, and not per fund or per month, replacing
1405 number of Flow values.13 Finally, we utilize equation 3.2 and 3.3 to compute our
FLOWAGG values.
In Table 2 we present descriptive statistic for our calculated FLOWAGG values, pro-
viding us with several insights, and in figure 2 we show the evolution of FLOWAGG over
time. The fractional flow data experience a clear positive mean of 0.35% (4.28% annu-
alized) very similar to the median, possibly due to the much larger number of inflow
12In appendix A.3 we present regressions with the naked aggregate flow data. We advise the reader
to read section 4 and 5 before engaging the naked data regressions.
13It is obvious that the ordering of the data adjustments matter. Naturally, we winsorize last as to
not cause any ’damage’ to the dataset.
3 DATA 10
Flow descriptive stats
Panel A: Aggregate mutual fractional fund flow
N Max 75% 50% 25% Min Mean Std Skew Kurt
Flow 239 155 49 32 15 -70 35 33 0.78 4.76
Inflow 212 155 51 36 22 00 41 29 1.43 4.45
Outflow 27 -1 -6 -12 -19 -70 -15 14 -2.45 9.87
Panel B: Aggregate mutual dollar fund flow
N Max 75% 50% 25% Min Mean Std Skew Kurt
Flow 239 8.48 3.81 2.21 0.89 -1.73 2.47 2.06 0.49 2.82
Inflow 212 8.48 3.99 2.54 1.46 0.00 2.86 1.85 0.72 2.96
Outflow 27 -0.02 -0.30 -0.44 -0.57 -1.73 -0.56 0.40 -1.51 4.78
Panel C: partial autocorrelations of aggregate mutual fund flow
Lag 1 2 3 4 5
Fractional flow 0.1105 0.0817 0.1525 -0.0501 0.1205
Dollar flow 0.5527 0.2733 0.2244 0.0609 0.1138
Table 2: Descriptive statistics for monthly aggregated flow values from Feb 2000 until Dec 2019. The
values of fractional flow are in basis points and the values for dollar flow are in billions of dollar. Flow
represent all flow data, while Inflow and Outflow represent the flow data corresponding to positive and
negative flow, respectively.
observations. We observe that only (approximately) 11.3% of the flow data are outflows,
and the last outflow occurs on Oct 2011. The extreme values are clearly observed for
inflow, for both the fractional and dollar measurement. The fractional flow data seems to
exhibit leptokurtic properties, meaning it contains fat tails and are more prone to outliers
in comparison with a standard normal distribution, whereas the dollar fund flow show
signs of the opposite, platykurtic, implying fewer and lower extreme values. We also ob-
serve a positive skewness for flow (more so for fractional flow than dollar flow), somewhat
contradicting the daily S&P500 flow values computed by Cao, Chang & Wang (2008)
over the period Feb 1998 to Dec 2003, who obtained a mean of -0.20 basis points with
more outflows than inflows. They attribute their negative skewness to the large amounts
of outflows that occurred during the dot-com bubble. Nevertheless, the evolution of our
monthly aggregate fractional flows appear similar to Cao, Chang & Wang (2008)(the evo-
lution of our fractional flow is showed in figure 2), suggesting that for an increasing fund
sector the volatility in mutual fund flows occur more early than late for a given time span;
why we also compute and test dollar flow. We contribute this decreasing fractional flow
to the somewhat stable dollar flow. The large returns attributed to S&P500 during 2010-
2019 is causing a size effect for fractional flow, wherein the total net assets is increasing
more than dollar flow, resulting in a smaller relative flow.
3 DATA 11
Figure 2: Estimated monthly aggregate Flow values from Feb 2000 until Dec 2019. Fractional flow
(left hand axis) are presented in basis points, and dollar value Flow (right hand axis) are presented in
billion of dollars.
3.2 Handling of fees
Data of different fees were obtained, including expense ratio, management fees and 12b
fees. The return data obtained from CRSP handles the management and 12b fees, as
they are included in the net asset value (NAV) of which the return is calculated (Center
for research in security prices (CRSP), 2019). Equation 3.4 is used by CRSP to calculate
the return for a specified fund.
rt =
NAVt ∗ Cumfact
NAVt−1
− 1 (3.4)
where Rt is the return for time t, Cumfact is a factor consisting of various distributions,
such as dividends and splits that occur in the holdings, and NAVt is the net asset value as
of period-end t. This procedure is estimated to capture the effect of these events and yield
a fair fund return. In conclusion, we are left with the expense ratio paid by investors,
which will be utilized as the fee variable throughout this thesis.
Hortac¸su & Syverson (2004) find that index fund intra-competition were mainly related
to other factors than fund fees. This suggests that the aggregating the index fund fee
data does not cause harmful inconsistencies in the data, but rather provide us with a
useful tool estimate the aggregate index fund flows to the general index fund fee level. As
such, we collect yearly fee data for the identified index funds. Fee values are collected as
percentage points and in the report fees are on average for all fund, i.e non weighted to
capture the entire market of index fund and not suffer from large impact of leading index
funds. The fee data experienced missing values in connection to inception and liquidation.
These missing values were either filled by the previous years reported fee, or, if that value
does no exist, the next years reported fee. Since the fees reported are yearly fees, they
4 METHODOLOGY 12
Figure 3: The evolution of average index fund yearly fee for the period 2000-01-01 to 2019-12-
31. The Fee data are presented in basis points.
were divided by 12 to represent the cost for each month of holding the fund. Thus, we
obtain piece wise monthly fee data, readjusting on a yearly basis. Finally we calculate
the mean of each month to obtain an aggregated fee per month for all the index funds.
Figure 3 displays a clear pattern in fee reduction over the past 20 years. The average
cost of a S&P500 index fund has dropped roughly 30 basis points. Interestingly, a small
upwards rebound is observed during periods of crisis (in 2002 and 2008-2010).
4 Methodology
Following the recent conventional papers on the evolution of market efficiency (i.e. time
indexed/varying), we will utilize the Hurst exponent as a proxy for market efficiency
(Bariviera, 2011; Eom et al, 2008; Kristoufek & Vosvrda, 2012). Explicitly, this thesis
thus aims to examine the relationship between aggregate index funds flow (estimated on
aggregated fund level) and the markets Hurst exponent (estimated on market level, i.e.
the target index). This section is organized as follows: we will first setup up the general
methodology as a martingale sequence for testing EMH, then introduce the concept of
long-term dependency, and finally we will conduct a Granger causality test and specify
our estimation model.
4 METHODOLOGY 13
4.1 General method setup
As is custom (see for example Kristoufek & Vosvrda (2013) or Huang, Wei, & Yan,
(2007)), we can define an efficient market in accordance to security prices conforming to
a martingale sequence. Let C be the securities market expressed by the probability space
(Ω,F , P ), where Ω is the sample space, F the event space (a subset of Ω). Further, let P
be the real probability measure and the conditional probability as P[Xt|Ft]. As such,
A securities market C = (Ω,F , P ) is efficient if there exists P such that the time-series
of prices S = (St)t≥1 is a martingale process; i.e.,
P[|St|] <∞ P[St+1|Ft] = St t ≥ 0 (4.1)
From here we can redefine the martingale price process to incorporate a random error
term. Let (t)t≥1 be random IID. variables with mean zero, that is P[t] = 0. This
incorporates innovations with no autocorrelation. Thus, we can reformulate the security
price series into
P[St+1|Ft] = St + P[t+1|Ft] (4.2)
Such a security market, where the security prices are described by the above, would be
efficient as the market does not exhibit any pattern or memory to exploit (McCauley,
Bassler, & Gunaratne, 2008): there exists no systematic way to beat the market. By
utilizing this martingale feature of security prices, we obtain a robust model and avoid the
random walk assumption of homoskedasticity (Kristoufek & Vosvrda, 2013). Converting
price data to return data, and let the 1-period return be rt =
St−St−1
St−1
and µ be the drift
of the return process, we obtain
P[rt+1|Ft] = µ+ φrt−1 + P[t|Ft] (4.3)
Modelling security returns in this manner (according to equation 4.3), the prices would
adhere to a random walk, suggesting a weak form of EMH is present. Subsequently, such
a return series residuals can be estimated to follow a white noise process by modelling the
long-term dependency measurement the Hurst exponent, where a Hurst exponent equal
to 0.5 corresponds to a random walk (Hsieh, 1993). This produces a variable effective
as a test of the martingale sequence and long-term dependency. Important to note, is
that a martingale sequence can exhibit memory in the sense we know the previous values
and that the expected value is just this previous value (McCaule, Bassler, & Gunaratne,
2008). When we, in this paper, talk about a no-memory scenario, we rather mean that the
innovation defining a step in the sequence cannot be predicted (i.e. it should be random).
4 METHODOLOGY 14
4.2 Long-term dependency
The Hurst exponent is a popular estimate for assessing long-term memory of a time series.
The method was originally invented by a hydrologist named Hurst who struggled to pre-
vent the Nile River Dam of overflowing. The method was utilized within finance roughly a
century after its first appearance and is today widely used to detect long term memory in
data (Peters, 1991). The Hurst exponent generally has a support of H ∈ (1, 0) with three
general outcome estimations: H = 0.50 indicates a random and uncorrelated series with
no memory, H < 0.50 indicates that the series exhibit anti-persistence or anti-memory,
and H > 0.50 indicates long-term memory in the series Peters (1991).14 Unfortunately,
no known distribution exists for Hurst exponent, but these general outcomes have been
asymptotically proven. The computation of Hurst exponent can be done by many different
statistical techniques, the two most commonly used within finance and therefore utilized
in this paper are; the rescheduled range (RS) and detrended fluctuations analysis (DFA).
Both techniques are more thoroughly explained later in this chapter. One drawback of
the RS Hurst exponent is the estimation of existing long-term memory due to short-term
dependency in the data series (Grau-Carles (2000); Di Matteo, (2007)). The two main
methods to avoid this issue is (i) to filter the return data through a GARCH-process, and
(ii) to use the Detrended Fluctuation Analysis (DFA) method.
Bariviera (2011) and Cajueiro & Tabak (2004) [among others] bypass this issue by
filtering the return series through an AR(1)-GARCH(1,1) process. This procedure re-
moves the short-term dependencies (memory) found in the time-series (by construction of
the autoregressive element), which, if left undisturbed, may instill long-term dependency
(where none actually exist) in the Hurst exponent (Tabak & Cajueiro, 2004; Di Matteo,
2007). The lagged component of the main equation in the GARCH leaves the residuals as
the ”true and random” returns for which we aim to estimate long-term memory. Bariv-
iera (2011) and Grau-Carles (2000) estimate the Hurst exponent using the DFA method,
wherein the residuals of a locally detrended integrated time-series are used to compute
the global fluctuations and repeated over multiple window lengths.15 Di Matteo (2007)
further criticize the conventional Hurst RS method for its sensitivity to outliers and pres-
ence of heteroskedasticity in the returns time series. In this paper, to avoid this spurious
detection of long-term memory, we will estimate the Hurst exponent using both the RS
14The reason why the Hurst exponent, with its support of H ∈ (0, 1), can act as a proxy for market
efficiency is due to the measurement of the average fluctuation and how it relates to time periods.
Unfortunately When the average fluctuation for a long window sizes is roughly equal to the average
fluctuation for short window sizes, the return data is exhibiting frequent sign-changes (+ and−). Likewise,
when the average fluctuation is larger for longer window sizes, the return data, by necessity, exhibit sign-
trends wherein subsequent returns change sign less frequently. A white noise process have been empirically
asymptotically discovered to equal a Hurst exponent of 0.5 (Anis & Lloyd, 1976).
15We here use the word integrated in the sense of the original paper from Peng et al. (1995), wherein
a detailed description of the DFA method can be found, meaning a mean adjusted cumulative sum; i.e.
a cumulative deviation from the mean.
4 METHODOLOGY 15
approach (with filtered returns) and the DFA method. Furthermore, previous studies
consistently estimate a higher Hurst exponent using the RS method than by using the
DFA method (see for example Bariviera (2011)).
4.3 The Hurst exponent
The Hurst exponent is formally defined as a power law function (for reference, see for
example Peters (1991)) of the type
Φn = Cn
H as n −→∞ (4.4)
where H is the Hurst exponent, C is a constant, n is the number of observations in a
partial time series, and ΦRSn = E(RS)n for RS (rescaled range analysis), or Φ
DFA
n = Fn
for DFA (detrended fluctuation analysis). As such, distinguishing these methods is the
computation of the left hand side (both of which will in detail be described in the following
sections). To obtain the Hurst exponent H, a log-transformation is utilized16, converting
equation 4.4 into
log(Φn) = log(C) +H × log(n) (4.5)
where, once Φn is computed, a simple linear regression can be run to estimate H. Let’s
define this inverse function for running equation 4.5 and estimating H, by inputting Φn
on the left hand side, as Θ(Φn). Explicitly, Φn is thus a vector with equal size as n.
When estimating a time varying Hurst exponent, reasonable assumptions regarding
window size are of importance. Eom et al. (2008) used a window size of 60 months with 12
months rolling. In accordance with similar studies (see Cajuerio & Tabak, 2004; Bariviera,
2011), we are arguing for a Hurst window size corresponding to political cycles. As such,
we utilize a window size of N = 1008 observations (252 trading days multiplied by four
years)(let’s call this the global window length[GWL]).17 Calculating the Hurst exponent
Ht thus requires N = GWL − 1 of previous observations including the observation on
period t. As such, to compute the first Hurst exponent for Jan 2000, data is needed from
Jan 1996; why we download a longer data set for S&P500.
4.3.1 GARCH
The Generalized Autoregressive Conditional Heteroskedasticity model, or GARCH, is a
process to estimate the conditional variance of a time-series which exhibits heteroskedas-
ticity. The GARCH-model is constituted by two equations: the mean equation, which
16By standard of convention, the log-operator is defined as the natural logarithm.
17The Hurst window size profoundly affects the estimated Hurst exponent. Thus, choosing the correct
window length is a sensitive issue, why we follow previous studies. As with any econometric computation,
more observations generally produce a more robust estimation; nonetheless, when estimating current day
market efficiency, far historical returns should have a diminishing effect.
4 METHODOLOGY 16
makes assumptions and models the underlying time-series, and the variance equation,
which models the conditional variance of the underlying time-series. As previously dis-
cussed, we filter the return data through an AR(1)-GARCH(1,1) model for a better input
variable into the RS analysis.
Thus, our model is specified as an AR(1)-GARCH(1,1) process
rt = µ+ Φrt−1 + t (4.6)
t = σtzt, zt ∼ N(0, 1) (4.7)
σ2t = ω + α
2
t−1 + βσ
2
t−1 (4.8)
where equation 4.6 is the mean equation with the specific AR(1)-term, and equation 4.8
is the variance equation, wherein the rt is the return for time t and σ2t is the conditional
variance of the returns for time t, and µ, ω, α, and β are unknown model parameters that
need to be estimated. For a reasonable behaving GARCH-process, some of the parameters
require some restrictions: ω > 0, α > 0, and β > 0 to ensure positivity of the conditional
variance, and α + β < 1 to ensure stationarity. In Table 3, we present our estimation of
the AR(1)-GARCH(1,1) process. We observe a clear significant autoregressive process in
the S&P500 returns.
AR(1)-GARCH(1,1) estimation
Value SE T-stat P-value
AR(1)
Constant 0.0008 0.0001 7.4025 0.00
AR(1) -0.0494 0.0139 -3.5559 0.0004
GARCH(1,1)
Constant 0.00 0.00 6.3747 0.00
GARCH(1) 0.8745 0.0061 142.34 0.00
ARCH(1) 0.1050 0.0049 21.311 0.00
Table 3: Model estimation for AR(1)-GARCH(1,1), estimated using daily S&P500 returns from Jan
4th 1999 to Dec 31 2019; where the AR-value is the coefficient for the lagged return term in the mean
equation, the ARCH-value is the coefficient for the lagged error term in the variance equation, and the
GARCH-value is the coefficient for the lagged variance term in the variance equation.
4.3.2 Rescaled range analysis
The computation of the rescaled range analysis follows several steps, outlined below, and
is performed for multiple partial series of the original full length series. That is, we first
divide the full time series of length N into j = 1, 2, ..., k non-overlapping partial time
series with lengths ni. There exists several ways to conduct this division: for example,
ni = N,N/2, N/4..., or ni equal to the factors of N , where i = 1, 2, ..., v. For our
4 METHODOLOGY 17
calculations we utilize the latter approach, due to its computational simplicity (following
Weron (2002). We then execute the following steps for each ni
1. Calculate the mean: E[Xnij ] =
1
ni
∑ni
t=1X
ni
j,t
2. Calculate mean-adjusted series: Yj,t = X
ni
j,t − E[X
ni
j ]
3. Calculate cumulative deviations: Znij,t =
∑ni
t=1 Y
ni
j,t
4. Calculate the range: Rnij = max(Z
ni
j )−min(Z
ni
j )
5. Calculate the variances: (σnij )
2 = E[(Xnij,t)
2]− E[Xnij,t]
2
6. Compute the rescaled range and the average (expectation) for all partial time-series:
R
ni
j
σ
ni
j
and subsequently E(RS)n = E
[
Rnj
σnj
]
= 1k
∑k
j=1
Rnj
σnj
After obtaining E(RS)n, it is now possible to estimate the Hurst exponent in Θ(Φn).
However, such an estimation will have a significant deviation from its theoretical value
for small window sizes ni (Weron, 2002). To correct for this, we subtract the window
sizes theoretical white noise approximation (this modification to the RS analysis was
introduced by Anis & Lloyd (1976) with some slight modifications by Peters (1991)). As
such, we obtain the true deviations from the white noise slope; the Hurst exponent can
thus be calculated as 0.5 plus this deviation. The theoretical white noise approximation
is given by (keeping the original left hand side notation from Anis & Lloyd (1976))
E(R∗∗n ) =



(
n− 12
n
)
Γ{ 12 (n−1)}√
piΓ{ 12n}
∑n−1
i=1
√
n−i
i for n ≤ 340
(
n− 12
n
)
1√
1
2pin
∑n−1
i=1
√
n−i
i for n > 340
(4.9)
where Γ{λ} is the gamma function evaluated at λ, and n are the window sizes ni. Ex-
plicitly, the corrected version of the RS Hurst is thus calculated as18
HRS = 0.5 + Θ (E[RS]n)−Θ(E[R∗∗n ]) (4.10)
Following Cajueiro & Tabak (2004) and Bariviera (2011), before calculating the Hurst
exponent, we employ the AR(1)-GARCH(1,1) process to filter the returns for short-term
dependency (see Table 3); the estimated residuals from the mean equation 4.6 are then
divided by the conditional standard deviation from the variance equation 4.8. The result-
ing fraction is our filtered returns and is thus used to complete the Hurst calculation.
18The attentive and enlightened reader might here realize that E(R∗∗n ) is not time-varying, but only
a function of window sizes n. As such, in equation 4.10, we are subtracting a constant number 0.5 −
Θ(E[R∗∗n ]) from all estimated RS Hurst exponents. For a global window size of 1008 returns, and local
window sizes n = divisor(1008), this constant adds up to approximately -0.0688. This means that for
our local window sizes n, Θ (E[RS]n) consistently overestimate the Hurst exponent by this value.
4 METHODOLOGY 18
Formally we can define the filtered returns explicitly as:
Ωt =
t
√
σ2t
(4.11)
where Ωt are the filtered returns, t are the residuals from the AR(1)-GARCH(1,1) mean
equation, and σ2t are the conditional variances from the AR(1)-GARCH(1,1) variance
equation. That is, we conduct the RS approach on the filtered returns Ωt.
4.3.3 Detrended fluctuation analysis
Detrended fluctuation analysis (DFA) is a method to detect long term memory in data
similar to the previously introduced RS approach, but differing in some aspects, which
makes DFA a useful variable along the rescaled range analysis for robustness purposes.
DFA was introduced by Peng et al. (1995) and foremost used within the medical field
to find long term correlations in heart rate and DNA data. DFA was popularized within
financial data by Kantelhardt et al. (2001) and is frequently used to examine long range
dependencies. When detecting long range memory it is essential to filter out disturbances
possibly causing spurious memory. In the RS Hurst approach short term memory was
removed from the data by the filtering process. DFA instead identifies trends within each
local window size, which can cause false dependencies, both long and short term (Eom et
al, 2008) caused by externalities (Kantelhardt et al, 2001).
The computation of the detrended fluctuation analysis is in principal similar to the
rescaled range analysis, but differs in some methodological aspects. Exactly alike, we first
divide the full time series of length N into k non-overlapping partial time series (windows)
with equal lengths ni (utilizing the divisors approach here as well). For each ni, we do
the following. For each partial time series j = 1, 2, ..., k, we compute the mean x¯j = E[xj]
and subsequently the integrated series of Xj,t (a cumulative mean-centered sum)
Xj,t =
t∑
i=1
[xj,i − x¯j] (4.12)
Withing each integrated series Xj,t, a fitted straight line is located to find the trend
for each window. Let Yˆj,t be this straight line fit, obtained from Xj,t = α + bτ , where
τ = 1, 2, ..., ni. Then, the fluctuation Fn is computed as follows for each window length
ni
Fn =
√
√
√
√ 1
N
N∑
t=1
(Xj,t − Yˆj,t)2 (4.13)
Thus, we obtain an average fluctuation Fn for each integrated and detrended time se-
ries of length ni, and we can now estimate the Hurst exponent H as Θ(Fn) (equation
4 METHODOLOGY 19
4.5).19 In contrast with the rescaled range analysis, the Hurst exponent estimated from
the detrended fluctuation analysis have a support of (0 < d ≤ ∞). Similarly, a Hurst
coefficient of H < 0.5 indicates negative memory or anti-persistence and H > 0.5 positive
memory, and d ≈ 0.5 is a sign of no memory. Although normal behaving return series
still lie within H ∈ (0, 1), a Geometric Brownian Motion process (a cumulative sum of
the returns) would theoretically produce a Hurst exponent equal to 1.5.
4.4 Endogeneity concerns
The causal relationship between passive investing and market efficiency exhibits uncer-
tainty regarding direction of causality. This duality of whether passive investing exists
because of efficient markets or does passive investing affect market efficiency, raises con-
cerns about the dependency in the structural regression.
To examine whether a variable has a causal relationship with another variable we
choose to conduct a Granger causality test. The test procures a measurement of how well
the dependent variable exhibits the same pattern as the key variable. By lagging the key
variable, where p is the minimum and q the maximum lag of significance, we observe how
well they can describe or predict the dependent variable. The model is thus given by a
VAR(p,q)-model, specified explicitly as


Ft
Ht

 =


b1
b2

+


ψ11,p ψ12,p
ψ21,p ψ22,p




Ft−p
Ht−p

+ ...+


ψ11,q ψ12,q
ψ21,q ψ22,q




Ft−q
Ht−q

+


1,t
2,t

 (4.14)
where, Ft is the flow variable, Ht is the Hurst exponent, i,t is a standard error term
i,t ∼ N(0,Σ) where Σ is a 2×2 covariance matrix. The off-diagonal elements in equation
4.14 represent the cross-impact of flow and Hurst, while the diagonal elements are the
autoregressive terms. We produce VAR-models for dependent variables fractional flow
and dollar flow, with dependent variables RS Hurst as well as DFA Hurst. To determine
19 We believe it is here appropriate to clarify the DFA-method with an example. Lets say the integrated
series Xt have a length of 252 (let us also define this as the global window size). From this we choose a
our window lengths n to be from 12 up to 1008 in increments of 1. That is we first divide Xt into non-
overlapping windows of length 5, giving us b 2525 c = 50 windows (let us call these windows local windows).
For each local window we locate the straight line fit Yˆt. Next, for the calculation of F5 we detrend Xt by
its corresponding Yˆt and complete the computation. We then repeat this process for F6,F7,...,F252 with
local window sizes of 6, 7, ..., 252. For a financial time series it is reasonable to set the minimum local
window size to 5, corresponding to 1 trading week. This DFA technique is quite computational intensive
as it estimates a vast number of regression. Explicitly, lets denote the total length of our required series
to T , the global window length as GWL, and the maximum local window length as LWLN , then the
total number of estimates Ψ are equal to Ψ = (T −GWL+1)×GWL×
(∑LWLN
n=LWL1
n
LWLN
)
. To compute
daily DFA in the preiod 2000-2019 with GWL = 1008 and LWLn = 5, 6, ..., 252, we need T=6038 return
data, from 8th Jan 1996 to 31th Dec 2019, resulting in approximately 641 million estimations. A more
simple approach established by Weron (2002), which we utilize, is to choose LWL equal to the factors of
the GWL, as well as limiting minimum LWL to 8. This is far less computational intensive and avoids the
problem of unregular LWL causing last window to have fewer observations.
4 METHODOLOGY 20
VAR specification - fractional flow
DFA-VAR(3) RS-VAR(1)
Dependent → Flow Hurst Flow Hurst
Response ↓
Constant 0.0055** 0.0127** 0.0089** 0.0161*
(2.1027) (2.1772) (2.2137) (1.8786)
AR(1)
Flow 0.0982 0.0812 0.1056 0.0994
(1.5474) (0.5761) (1.6418) (0.721)
Hurst 0.0003 1.4905*** -0.0106 0.9699***
(0.0104) (23.35) (-1.4457) (61.646)
AR(2)
Flow 0.0488 0.0954 - -
(0.7768) (0.6841)
Hurst 0.0453 -0.7232*** - -
(0.9483) (-6.8149)
AR(3)
Flow 0.1437** 0.0326 - -
(2.2949) (-0.2343)
Hurst -0.0528* 0.2021*** - -
(-1.8322) (3.1607)
NumParam, k 14 6
LogLikelihood 1860.43 1868.38
AIC -3692.86 -3724.75
Table 4: VAR model specification for fractional flow and Hurst exponent calculated using both rescaled
range (RS) and detrended flucuation analysis (DFA). The best fit for DFA was a VAR(3) model and for
RS a VAR(1) model. Flow was computed as specified in equation 3.2. Daily DFA and RS Hurst exponents
was computed accordingly to section 4.3.2 and 4.3.3, respectively, and subsequently averaged on monthly
basis. The values in parenthesis represent the above coefficients t-statistic.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.
the length of which to lag the key variable in the Granger Causality test (that is, to find
optimal q), we conduct an Akaike Information Criterion-test (AIC). The AIC determines
”information loss” given a specific model, and the best fit is the model which minimizes
the AIC. The AIC is given by
AIC = −2(logL) + 2k (4.15)
where logL is the loglikelihood value of the estimated VAR(p,q)-model and k is the
number of estimated parameters for the model. We run the AIC estimation for p = 1 and
q = 1, 2, ..., 12. In Table 4 and 5 we present the VAR models, and their corresponding
LogLikelihood and AIC values, for fractional flow and dollar flow, respectively. The
best fit (lowest AIC value) obtained for fractional flow is for a DFA specification q = 3
and for a RS specification q = 1; and, the best fit (lowest AIC value) obtained for
dollar flow is for a DFA specification q = 3 and for a RS specification q = 2. The off-
diagonal elements in equation 4.14 is our main interest of study, where, for example, ψ12,1
4 METHODOLOGY 21
VAR specification - dollar flow
DFA-VAR(3) RS-VAR(2)
Dependent → Flow Hurst Flow Hurst
Response ↓
Constant 6.4417*** 0.0213*** 7.0607*** 0.0172**
(3.9386) (2.8912) (3.5449) (1.9896)
AR(1)
Flow 0.2875*** 0.00 0.3591*** 0.00
(4.5409) (0.3181) (5.734) (0.0446)
Hurst -11.296 1.4818*** -27.307* 0.9613***
(-0.7989) (23.27) (-1.8522) (14.809)
AR(2)
Flow 0.1414** -0.0003 0.2309*** -0.0002
(2.1943) (-0.8633) (3.7402) (-0.587)
Hurst 27.11 -0.7083*** 14.573 0.0034
(1.1539) (-6.693) (0.9791) (0.0515)
AR(3)
Flow 0.1761*** -0.0004 - -
(2.8273) (-1.3749)
Hurst -28.433** 0.1803*** - -
(-2.0007) (2.817)
NumParam, k 14 10
LogLikelihood 399.08 391.15
AIC -770.16 -762.30
Table 5: VAR model specification for dollar flow and Hurst exponent calculated using both rescaled
range (RS) and detrended flucuation analysis (DFA). The best fit for DFA was a VAR(3) model and
for RS a VAR(1) model. Flow was computed as specified in equation 3.3. Daily DFA and RS Hurst
exponents was computed accordingly to section 4.3.2 and 4.3.3, respectively, and subsequently averaged
on monthly basis. The values in parenthesis represent the above coefficients t-statistic. Dollar value flow
are in billions of dollar.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.
correspond to dependent variable flow and independent response AR(1) Hurst in their
respective tables.20
We note that HDFAt−3 have a significant impact on Flowt (although only partially for
fractional flow), possibly causing the best fit to be a VAR(3)-model for both fractional
and dollar flow. Nonetheless, HRSt−1 is not significant on fractional flow, it is partially
significant at 10% on dollar flow. Moreover, all of the AR-terms on Hurst are highly
significant. This is most likely due to that both DFA and RS are stationary processes.
Economically this is intuitive, as one state of market efficiency highly depend on previous
states of market efficiency.
Next, we run Granger Causality tests with the specified VAR models presented in
Table 4 and 5. A Granger causality test estimates if either of the series Granger cause the
20To ensure the causality we also conduct a BIC-test, similar to the AIC. The BIC penalizes more for
complex models than the AIC, and is given by BIC = −2(logL) + k ∗ ln(n), where n is the number of
observations and otherwise the same notation as in equation 4.15 applies. The BIC-test yields similar
results and confirm the direction of the causality.
4 METHODOLOGY 22
Granger Causality test
Null hypothesis Statistic P-value Causal direction
Panel A: fractional Flow
DFA does not Granger cause FLOW 1.6538 0.1778
DFA = FLOW
FLOW does not Granger cause DFA 0.2892 0.8332
RS does not Granger cause FLOW 2.0638 0.1522
RS = FLOW
FLOW does not Granger cause RS 0.5133 0.4744
Panel B: dollar value Flow
DFA does not Granger cause FLOW 4.6735 0.0034
DFA → FLOW
FLOW does not Granger cause DFA 1.3794 0.2498
RS does not Granger cause FLOW 6.1413 0.0025
RS → FLOW
FLOW does not Granger cause RS 0.20499 0.8148
Table 6: Granger causality test results for the VAR specified models. Panel A is conducted using
VAR specification in Table 4 and Panel B is conducted using VAR specification in Table 5. The null
hypothesis tests whether the VAR coefficients are jointly difference from zero, see equations 4.16 and
4.17 for explicit specifications. A significant p-value indicates rejection of the null hypothesis. The test
statistics are computed using a F-test.
.
other one; i.e. if predictive power exists. The null hypothesis are stated as the off-diagonal
elements (in equation 4.14) provide no joint significance.21 Explicitly, this is
HF0 : ψ12,p = ψ12,p+1 = ... = ψ12,q = 0 (4.16)
and
HH0 : ψ21,p = ψ21,p+1 = ... = ψ21,q = 0 (4.17)
where HF0 is interpreted as HURST does not Granger cause FLOW, and H
H
0 as FLOW
does not Granger cause HURST. In Table 6 we present test statistics for our Granger
Causilty test. We find no predictive power for either fractional flow on RS or DFA, and
neither for RS or DFA on fractional flow. Nevertheless, both RS and DFA seem to Granger
cause dollar value flow, indicating a a causality running from market efficiency towards
index fund flow. Furthermore, no observable simultaneity endogeneity is indicated, sug-
gesting a one-direction causality relationship. This forecasting/predictive power dictate
the direction of the relationship we want to study. We will as such continue this paper
by examining the effect market efficiency have on aggregate index fund flow. A possible
21Another approach to test for Granger causality is the null hypothesis that the summed off-diagonal
coefficients are equal to zero,
∑
ψi,j = 0 for j 6= i. Such testing procedure examine if the predictor variable
have any overall impact on the effect variable; in contrast with our current method which examines if
at some lag p to q, the other series can be predicted. Although this particular sum-test approach could
be of interest, we consider it out this papers scope, and to therein not conduct one. For the interested
reader, this sum-procedure is performed by Kadapakkam, Krause, & Tse (2015) on ETFs.
4 METHODOLOGY 23
explanation for why market efficiency Granger cause dollar flow, and not fractional flow,
could be due to the size effect of fractional flow discussed in section 3.1.2.
4.5 Estimation and control variables
We base our estimation model on the causality found in the Granger causality test. Thus,
to examine the relationship between passive investments and market efficiency, we regress
Hurstt on FLOWt, while controlling for variables that might affect Flow levels. Several
variables beside flow and Hurst are included in the regression to increase explanatory
power. The variables are generally consistent with the regression of Sirri & Tufano (1998),
and Huang, Wei, & Yan (2007) as well as Edelen & Warner (2001), but adapted for index
funds. Hence, variables related to specific fund returns (typically a ranking based on
performance of the mutual funds) are omitted since they are very similar. We instead
use market returns of the index, as it is reasonable to conclude that it is the index return
which base an investment choice rather than the specific index fund return. When an
investor analyzes whether to invest in an index fund, it is sensible to assume that the
investor inquire the general index return and then buys the index fund available from an
institution where they are existing customers. As such, we regress flow on the following
variables:
• Hurst - is included to examine the impact market efficiency have on index fund
flow. Both contemporanous and lagged Hurst exponents are estimated.
• Fees - lagged expense ratio is included in the regression on an aggregated level, to
capture effects of fee levels differing for index funds over the time scope of the study.
The fee value is equally weighted among all the index funds.
• Log(TNA) - lagged logged aggregate total net assets is included to capture changes
in the total market size of index funds.
• Mkret - is included to capture effects of previous performance of the index fund
market. We include contemporaneous as well as both 1-period and 2-period lagged
market returns.
• CumV ol - lagged 12-month cumulative standard deviation is included to cover the
effect of overall total riskiness of the market. This variable is created as a 12-month
rolling window computation of the standard deviation.
Our specified final estimation thus arrive at
FLOWt = α + β1Hurstt + β2Mkrett + β3log(TNAt−1) + β4Feet−1 + β5CumV olt−1 + t
(4.18)
We conduct two series of regressions: the first with fractional flow as dependent variable,
and the second with dollar value flow as dependent variable. Several regressions are
4 METHODOLOGY 24
estimated per flow variable with varying regressors. These regressions are presented in
section 5.
Annual statistics
Year Flow (frac) Flow (dollar) TNA Return Fee Hurst(RS ) Hurst(DFA)
2000 0.58% 1.8764 319.25 -0.7674% 6.29 0.4471 0.3965
2001 0.56% 1.7750 294.99 -0.9904% 6.41 0.4760 0.4348
2002 0.39% 1.7325 268.86 -2.0061% 6.34 0.4884 0.4452
2003 0.59% 0.8870 286.08 2.0593% 6.19 0.4889 0.4344
2004 0.48% 2.0230 359.74 0.7786% 5.71 0.5202 0.4691
2005 0.39% 1.4814 406.18 0.3123% 5.31 0.5014 0.4747
2006 0.08% 1.4125 458.34 1.1246% 5.17 0.4814 0.4393
2007 0.31% 0.6684 532.33 0.3641% 5.31 0.4519 0.4072
2008 0.51% 1.6435 480.24 -3.7423% 5.48 0.4362 0.4283
2009 0.23% 2.3668 415.74 2.0805% 5.48 0.4739 0.4797
2010 0.22% 0.8026 501.18 1.2312% 5.25 0.4998 0.4802
2011 0.22% 1.2997 596.47 0.1488% 5.13 0.5089 0.4817
2012 0.27% 1.0359 670.56 1.1846% 4.84 0.5075 0.4545
2013 0.41% 2.1999 870.87 2.3160% 4.56 0.4614 0.4107
2014 0.35% 3.4926 1094.3 0.9964% 4.47 0.4468 0.4062
2015 0.35% 3.9248 1264.3 0.0592% 4.10 0.4321 0.3703
2016 0.38% 4.2773 1423.1 0.8755% 3.95 0.4349 0.4086
2017 0.26% 5.3130 1833.1 1.5807% 3.55 0.4381 0.3935
2018 0.22% 4.4228 2138.6 -0.4070% 3.62 0.4752 0.4014
2019 0.18% 4.5541 2435.4 2.2948% 3.65 0.4916 0.4194
Table 7: Annual statistics of regression inputs. The values showcased are averaged per month on
a yearly basis. Reported TNA and Flow (dollar)-values are in billions of US-dollars. The returns are
averaged using S&P500 returns. The fees are in basis points and based on monthly holdings.
In Table 7, we present a summary of the main data used in the regressions averaged
yearly. We observe a clear decreasing trend in the flow data. Looking at the TNA
values, which instead portray an apparent increase, it seems reasonable to assume that
the declining Flow values stem from the growing TNA, considering flow is a relative
value. Cao, Chang & Wang (2008), find similar pattern in their flow data, for multiple
fund categories between 1998 and 2003. The time period in this thesis suffers two majors
set backs, return wise. Firstly, the dot-com crash in the early 2000s, and secondly, the
great financial crisis in 2008-2009. The averaged monthly return of -3.7423% in 2008
would yield an annualized loss of almost 37%. Fees, as already mentioned, have declined
in the last 20 years and are substantially lower in 2019. As argued by Huang, Wei, & Yan
(2007) and Sirri & Tufano (1998) there is a connection between fund flow and the charged
fee. The Hurst RS estimations vary around 0.45-0.5, while the Hurst DFA estimations
vary around 0.4-0.45.22 This discrepancy between RS and DFA values are in line with
findings of Bariviera (2011).
22In appendix A.2 we preset histograms over the daily estimated Hurst exponents.
5 RESULTS 25
5 Results
Before we begin discussing the relationship between broad market efficiency and index
fund flow, we will conduct a small examination of our estimated proxy for market efficiency
(the Hurst exponent). In figure 4 we show graphically the evolution of the Hurst expo-
nent during the period 2000-2019. As we can observe, HRS is nearly consistently higher
than HDFA, where HRS is hovering around 0.45-0.5, and HDFA between 0.35 and 0.5.
Explicitly, this means that S&P500 returns exhibits signs of unpredictable returns and
market efficiency according to HRS, while HDFA rather indicates a slight mean-reverting
mechanism of the returns. This procures a strange task to analyze such measurements
impact. Notably, the discrepancy between the RS and DFA values mirror those of previ-
ous research (see for example Bariviera (2011)). Furthermore, these result are consistent
with the theory behind both of these market efficiency measurements (see Grau-Carles,
2000).
Figure 4: Hurst exponents and the underlying fluctuations (Fn) used in the computations. The Hurst
exponents, RS and DFA, are computed accordingly to sections 4.3.2 and 4.3.3, respectively. Daily Hurst
exponents are estimated, which then are averaged monthly. The second and third plots show the average
fluctuations Fn and E(RS)n used in the computations. The 24 lines represent the various window sizes,
where lines higher up correspond to larger window sizes (see footnote 19).
Interestingly, in boom periods, the Hurst exponent seems to be trending downwards,
5 RESULTS 26
and vice versa, in periods of recession, the Hurst exponent is rising. Mathematically this
makes sense, since a steady growing equity market exhibits small average fluctuations Fn
(especially for larger window sizes), resulting in a lesser increase in average fluctuation
for increasing window sizes. Likewise, for periods of high volatility and turbulence, the
average fluctuation naturally increases causing an upward trend in the Hurst exponent.
Economically, this response in the market efficiency is less intuitive. The asymptotically
proven white noise series of a Hurst exponent equal to 0.5 seemingly appears arbitrary
in an economic sense. Intuitively, it is reasonable that periods of high volatility should
induce non-market predictability in returns, which we observe for the increasing of HDFA
closer to 0.5 during such periods. Conversely, such equity market volatility normally
arises from intervals of steady market conditions followed by subsequent intervals of sharp
downturns (e.g. the famous great financial crisis of 2008). Sufficiently long periods of
downturns are thus periods of same-sign returns, exhibiting economic predictability and
long-term dependence. Such reasoning further purports the notion that developing mar-
kets experience a lower degree of efficiency (Bariviera, 2011; Cajueiro & Tabak, 2004;
Eom et al., 2008) given a historically higher volatility (Harvey, 1995; Umutlu, Akdeniz,
& Altay-Salih, 2010).
In Figure 5 we show the estimated Hurst coefficients as well as the asymptotic em-
pirical 95% confidence interval.23 As such, Hurst exponents lying inside the confidence
interval corresponds to an empirical estimation of market efficiency. We thus observe
that the market efficiency measured by RS are for multiple periods non-distinguishable
from efficiency, while market efficiency measured by DFA are consistently below market
efficiency except for a brief period after the great financial crisis.
To examine the relationship between index fund flow and market efficiency we esti-
mate three different set of regressions. In Table 8 we present our estimations on fractional
flow and in Table 9 we present our estimations on dollar flow.24 We also run a series of
regressions with lagged Hurst exponents; these are presented in Table 10. The reasoning
behind the lagged Hurst regressions is to provide robustness. We observe the same neg-
ative significance for the lagged Hurst as for the contemporaneous Hurst. This suggests
that there exists no reverse causality relationship between flow and Hurst. For fractional
flow we observe that both market efficiency measurements coefficient are most significant
in regressions (B) and (F), whereas for dollar flow, all market efficiency measurement
(both contemporaneous and lagged) are significant.
Noteworthy, we find a negative relationship between flow and contemporaneous market
return. This is not in line with Edelen & Warner (2001), where daily fund flow are
23There exists no known distribution for the Hurst exponent, but Weron (2002) estimated an empirical
confidence interval for both RS and DFA around a Hurst exponent eqaul to 0.5.
24In appendix A.3 we also present regressions run on naked aggregate flow data, wherein no adjust-
ments on either retunr, TNA, or flow data has been performed. As we observe, we lose significance
probably due to the large outliers in the naked flow data.
5 RESULTS 27
Figure 5: The evolution of the Hurst exponent over time, calculated using both DFA and RS. The red
dotted lines represent a 95% empirical asymptotic confidence interval around Hurst = 0.5, estimated by
Weron (2002).
positively correlated with concurrent market returns. They further conclude that it is
not an attribute of simultaneous feedback trading, but rather a causality running from
flow to returns within a day. Notably is the weak positive significance for Feest−1; a
contra-intuitive relationship wherein flow increase with increasing fees. We suggest this is
due to the fee variable acting as a time-proxy as we observe a clear negative trend in both
fractional flow (see figure 2) and fees (see figure 3).25 The same significant impact cannot
be found in the dollar flow regressions. This is in contrast with studies done by Sirri &
Tufano (1998) and Huang, Wei, & Yan (2007) who find a general negative relationship
between fund flows and fees. A clear distinction from their work is that our thesis only
analyze index funds flow whereas Sirri & Tufano (1998) and Huang, Wei, & Yan (2007)
studied a broader spectrum of funds, which possibly are more sensitive to fees. Moreover,
the logged and lagged TNA values are consistently positively significant in the regressions
run on dollar flows, and partly negatively significant in the fractional flow regression. This
observed difference could stem from the previous mentioned time factor as fractional flow
is slightly decreasing over time whereas the dollar flow is steadily increasing. Overall we
find no evidence that past year market volatility impacts current flow values.
25Including a trend variable in these regressions cause Feest−1 to become a negative non-significance.
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Linear regression 1: fractional flow
Panel A: DFA Panel B: RS
N = 239 (A) (B) (C) (D) (E) (F) (G) (H)
Intercept 0.0045** 0.02754*** -0.0251 -0.0211 0.0064** 0.0251*** -0.0354 -0.0327
(2.17) (3.43) (-0.79) (-0.67) (2.17) (3.86) (-1.36) (-1.26)
Hurstt -0.0023 -0.0147** -0.0109 -0.0122 -0.0061 -0.0135** -0.0113 -0.0126*
(-0.47) (-2.12) (-1.31) (-1.47) (-0.97) (-2.08) (-1.64) (-1.85)
Mkrett - -0.0124** -0.0133** -0.0132** - -0.0126** -0.0131** -0.0130**
(-1.99) (-2.12) (-2.11) (-2.01) (-2.10) (-2.09)
Mkrett−1 - - - 0.0037 - - - 0.0037
(0.56) (0.56)
Mkrett−2 - - - 0.0024 - - - 0.0026
(0.40) (0.44)
Log(TNAt−1) - -0.0013*** 0.0016 0.0014 - -0.0011*** 0.0023 0.0022
(-3.15) (0.95) (0.83) (-3.32) (1.60) (1.50)
CumV olt−1 - - .0048 0.0055 - - -0.0009 -0.0009
(0.28) (0.32) (-0.06) (-0.06)
Feest−1 - - 22.0462* 20.9689 - - 26.6578** 26.1407**
(1.66) (1.58) (2.23) (2.19)
Adjusted R2 -0.38% 8.24% 9.45% 8.99% -0.13% 7.66% 9.54% 9.09%
F-statistic 0.22 6.24*** 4.72*** 3.91*** 0.95 7.58*** 5.38*** 4.69***
Table 8: Linear regressions estimated on the dependent variable fractional FLOW aggt . FLOW
agg
t is calculated as the sum of all index funds dollar value flow
for month t adjusted by the previous month sum of all index funds total net assets (see equation 3.2). Mkrett is the S&P500 return from period t. CumV olt are
the cumulative standard deviation from t− 11 up to t using monthly S&P500 returns. The Log(TNAt), is the logged value of aggregated total net assets for all
the index funds for month t. Feest are the average fee level (expense ratio) of the index funds for month t. The Hurst exponents in panel A are calculated using
the DFA-method and the Hurst exponents in Panel B the RS-method, both with a rolling window size of 4 years (1008 observations). The Hurst exponents are
computed on a daily basis and then averaged over each month to obtain monthly estimates. The regression are estimated on a monthly basis and the numbers
in parenthesis represent the t-statistic for the estimated coefficient above.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.
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Linear regression 2: dollar flow
Panel A: DFA Panel B: RS
N = 239 (A) (B) (C) (D) (E) (F) (G) (H)
Intercept 13.7845*** -16.0782*** -31.4213** -31.2826** 14.9674*** -15.7781*** -37.4288*** -37.1942***
(10.31) (-4.83) (-2.09) (-2.08) (7.84) (-4.64) (-3.02) (-2.99)
Hurstt -26.1927*** -9.1362*** -8.3138** -8.3271** -26.4069*** -11.1122 ** -10.3066*** -10.3651***
(-8.77) (-3.04) (-2.30) (-2.26) (-6.64) (-3.11) (-2.81) (-2.75)
Mkrett - -9.7575*** -10.0412*** -10.1331*** - -9.5229*** -9.7557*** -9.8287***
(-3.83) (-3.94) (-3.96) (-3.68) (-3.78) (-3.78)
Mkrett−1 - - - 0.8797 - - - 0.9735
(0.37) (0.41)
Mkrett−2 - - - -.8349 - - - -.5472
(-0.37) (-0.24)
Log(TNAt−1) - 1.6843*** 2.5533*** 2.5455*** - 1.7598*** 2.9901*** 2.9771***
(9.02) (3.05) (3.04) (9.98) (4.31) (4.28)
CumV olt−1 - - 3.4860 3.4207 - - -0.3548 -0.4249
(0.48) (0.48) (-0.05) (-0.06)
Feest−1 - - 6392.223 6341.537 - - 9550.218* 9487.402*
(1.08) (1.07) (1.88) (1.87)
Adjusted R2 19.73% 43.42% 43.51% 43.08% 13.54% 43.77% 44.18% 43.75%
F-statistic 76.91*** 52.91*** 31.88*** 23.26*** 44.04*** 53.82*** 32.99*** 23.66***
Table 9: Linear regressions estimated on the dependent variable dollar FLOW aggt . FLOW
agg
t is calculated as the sum of all index funds dollar value flow for
month t (see equation 3.3). Mkrett is the S&500 return for period t. CumV olt are the cumulative standard deviation from t− 11 up to t using monthly S&P500
returns. The Log(TNAt), is the logged value of aggregated total net assets, expressed in $ billions, for all the index funds for month t. Feest are the average fee
level (expense ratio) of the index funds for month t. The Hurst exponents in panel A are calculated using the DFA-method and the Hurst exponents in Panel B
the RS-method, both with a rolling window size of 4 years (1008 observations). The Hurst exponents are computed on a daily basis and then averaged over each
month to obtain monthly estimates. The regression are estimated on a monthly basis and the numbers in parenthesis represent the t-statistic for the estimated
coefficient above.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.
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Linear regression 3: dollar flow, lagged Hurst
Panel A: DFA Panel B: RS
N = 239 (A) (B) (C) (D) (E) (F) (G) (H)
Intercept 13.9142*** -15.5915*** -29.1311* -28.7852* 15.7408*** -14.5046*** -35.3008*** -34.8036***
(10.07) (-4.72) (-1.93) (-1.91) (8.70) (-4.41) (-2.92) (-2.88)
Hurstt−1 -26.4940*** -9.8331*** -9.0674** -9.1521** -28.0499*** -13.0412*** -12.1971*** -12.4051***
(-8.60) (-3.25) (-2.48) (-2.46) (-7.44) (-3.82) (-3.48) (-3.46)
Mkrett - -9.6418*** -9.9188*** -10.0051*** - -9.1592*** -9.3951*** -9.4662***
(-3.80) (-3.91) (-3.93) (-3.68) (-3.66) (-3.66)
Mkrett−1 - - - 1.0370 - - - 1.3526
(0.44) (0.57)
Mkrett−2 - - - -0.7389 - - - -0.523741
(-0.33) (-0.24)
Log(TNAt−1) - 1.6703*** 2.4362*** 2.4173*** - 1.7326*** 2.9127*** 2.8873***
(9.03) (2.91) (2.88) (9.99) (4.28) (4.24)
CumV olt−1 - - 3.7741 3.4207 - - -0.5538 -0.6273
(0.53) (0.53) (-0.08) (-0.10)
Feest−1 - - 5579.195 5469.133 - - 9167.343* 9050.544*
(0.94) (0.92) (1.83) (1.82)
Adjusted R2 20.55% 43.74% 43.74% 43.31% 15.73% 44.59% 44.92% 44.53%
F-statistic 73.89*** 62.67*** 38.00*** 26.98*** 55.40*** 64.84*** 39.83*** 28.30***
Table 10: Linear regressions estimated on the dependent variable FLOW aggt . FLOW
agg
t is calculated as the sum of all index funds dollar value flow for month
t (see equation 3.3). Mkrett is the S&500 return for period t. CumV olt are the cumulative standard deviation from t−11 up to t using monthly S&P500 returns.
The Log(TNAt), is the logged value of aggregated total net assets, expressed in $ billions, for all the index funds for month t. Feest are the average fee level
(expense ratio) of the index funds for month t. The Hurst exponents in panel A are calculated using the DFA-method and the Hurst exponents in Panel B the
RS-method, both with a rolling window size of 4 years (1008 observations). The Hurst exponents are computed on a daily basis and then averaged over each
month to obtain monthly estimates. The regression are estimated on a monthly basis and the numbers in parenthesis represent the t-statistic for the estimated
coefficient above.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.
5 RESULTS 31
Although the RS coefficients are on average more significant than the DFA coefficients,
we find very similar coefficients for both, suggesting that choice of method does not
significantly alter the market efficiency’s estimated effect on flow.26 As such, we find
that a lesser degree of market efficiency negatively impacts flow. But how do we further
examine this phenomena? This market efficiency measurement is both stationary and
continuous; flow and market efficiency are ultimately two completely different variables
in terms of structure. Market efficiency should be seen as a current state of the market,
a continuous condition of how inter-dependent the returns are. Flow, on the other hand,
is an event, a one period happening. This touches upon an important issue; the levels of
index fund flow should be affected by whether the market returns show signs of long-term
dependency or short-term anti memory. Both states should indicate an inefficient market,
but are otherwise completely different in nature.
In the regressions in Table 9 and 10, we observe that the higher the Hurst exponent
is, the lower Flow becomes. This is reasonable, as the Hurst exponent generally increases
when estimated over periods of high volatility with a justifiable simultaneous liquidation
of index fund assets. This reasoning seems strengthened when concurrent market returns
are included [(B) and (F)], which reduces the market efficiency coefficient. Nonetheless,
the economic interpretation is more difficult. Although, that irregardless of the current
state of the market efficiency, it exhibits a negative impact on flow.27 Important here is
the findings of Eom et al. (2008), that markets with higher Hurst exponent tends to be
more predictable and exhibit a lower degree of efficiency. We thus argue that our findings,
of a negative impact on Flow from a higher Hurst exponent, reinforce this phenomena.
The more inefficient the market is (corresponding to a higher Hurst exponent), the lower
the index fund Flow is. For clarification, our estimation of the degree of market efficiency
for S&P500 is in line with other studies estimation of the degree of market efficiency in
developed markets (Bariviera, 2011; Cajueiro & Tabak, 2004; Eom et al., 2008), wherein
these developed markets generally obtain a Hurst estimation of around 0.40 and 0.50
(that is, a slight mean reversing process of returns).28 Previous studies further find that
26Regarding the discrepancy between the significance of the coefficients between the DFA and RS
approach, wherein RS coefficients overall are more significant, the conclusion should not be that the RS
Hurst is a better measurement for market efficiency. Nonetheless, it seems that the RS market efficiency
measurement more accurately describes concurrent index fund flows.
27Unfortunately, transforming the Hurst exponent in various ways in accordance to the 95% confidence
interval, to try and isolate whether significant long-term memory or anti-persistence yield different effects
on flow, produce non-significant results. Such non-significance in the current position of the market
efficiency in respect to true efficiency (i.e. whether the Hurst exponent is below, inside, or above the
confidence interval), strengthens our argument of a lesser degree of efficiency above than below the
confidence interval.
28As of the writing of this paper, we have not found any other studies that show an equity market
consistently exhibiting a Hurst exponent less than 0.4. This suggests that, while the theoretical support
of the Hurst exponent is H ∈ (0, 1), the empirical Hurst exponent support in finance applications is
different. Maybe such a strong mean-reverting return process indicated by sub-0.30 Hurst exponent is not
a plausible security market phenomena. Rather, the slight mean-reversing return process characterized
by H ≈ 0.4 appear standard for the current developed and considered most efficient markets.
5 RESULTS 32
Standardized coefficients
Regression (A) (B) (C) (D) (E) (F) (G) (H)
Table 8 -0.0243 -0.1559** -0.1152 -0.1289 -0.0537 -0.1183** -0.0987 -0.1101*
Table 9 -0.45*** -0.16*** -0.14** -0.14** -0.37*** -0.16** -0.15*** -0.15***
Table 10 -0.45*** -0.17*** -0.16** -0.16** -0.40*** -0.18*** -0.17*** -0.18***
Standard deviations
RS DFA Flow (dollar) Flow (fractional)
Std 0.0287 0.0348 2.06 33
Table 11: Standardized coefficients of the Hurst exponent in the estimated regressions for each table.
The standardized coefficients are obtained by standardizing all variables in each regression to have a
mean 0 and standard deviation of 1. These coefficients thus measure a one standard deviations change
in the dependent variable with a one standard deviation change in the dependent variable multiplied by
the coefficient. For reference, we also present the standard deviation of each measurement.The standard
deviation of fractional flow is in basis points and the standard deviation for dollar flow is in billions of
dollar.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.
developing markets display a higher Hurst exponent (lesser degree of efficiency). This
touches upon state vs. event problematic, and further purports the notion that, irregard-
less of where a security market lies on the Hurst support scale, an increase in the Hurst
exponent indicates a movement towards inefficiency; and, as of the findings of this paper,
a reduction in index fund flow.
To examine the magnitude of the impact of market efficiency on Flow we utilize stan-
dardized coefficients (also called beta coefficients). In Table 11 we present the transformed
Hurst coefficients into standardized coefficients. We observe that the standardized coef-
ficients for Table 9 and 10 are similar with only small differences between the regressions
(except for regressions (A) and (E)). Examining the more significant coefficients against
dollar flow indicates that an increase in the Hurst exponent of one standard deviation
would result in an approximately decrease in dollar flow by 0.15 to 0.17% multiplied by
one standard deviation in dollar flow. With a HRS standard deviation of 0.0287 and dollar
flow standard deviation of $2.06 billion, an increase in HRS by one standard deviation
roughly translates into an index fund outflow of $309 million. To put into perspective,
the maximum one month difference in HRS was an increase by 0.0247 that occurred in
September 2009, corresponding to a simultaneous outflow of $997 million, the fifth largest
outflow in our sample period.
The significant lagged Hurst exponents provide evidence that lead index fund flow
can be predicted from current state of market efficiency. Ceteris paribus, a movement
towards market efficiency, corresponding to a reduction in the Hurst exponent, indicates
that aggregate index fund flow should be larger (in comparison with current month) in
the impending month.
6 CONCLUSION 33
6 Conclusion
We find evidence that the S&P500 market index have exhibited signs a slight mean-
reverting process (anti-persistence, or negative memory), close to market efficiency, be-
tween 2000 and 2019; characterized by HurstRS ≈ 0.5 and a HurstDFA ≈ 0.4. These
findings are similar to previous studies of time-varying Hurst exponents. We further find
a causality direction between market efficiency and index fund flows, where the degree of
market efficiency negatively affect the level of flow. We find no indication of causality in
the reverse direction. Our findings suggests that equity markets which are characterized
by a higher Hurst exponent, a lower degree of efficiency, experience lower levels of aggre-
gate index fund flow. For dollar flow, this relationship is characterized by a one standard
deviation change in the market efficiency measurement corresponding to a change in dol-
lar flow of (roughly) 15% of the standard deviation in dollar flow. The same relationship
exists for fractional flow, albeit less significant, wherein the effect is (roughly) 12% of the
standard deviation in fractional flow. Moreover, we find a similar significant relationship
for previous month’s Hurst exponent, indicating predictability for impending index fund
flow from current month market efficiency. These results generally hold stronger for dollar
flow than fractional flow.
Our findings suggests that dollar flow provides greater insight than fractional flow into
the dynamics of macro-scale mutual fund flows. This discrepancy is not found in previous
mutual fund flow studies, and highlights the size-effect present in a bull-markets, wherein
the return dwarfs the dollar flow growth. Nonetheless, our estimated causality expands the
possible significant control variables when evaluating mutual fund flows. The estimated
market efficiency’s effect on index fund flow bridge the theoretical gap between passive
investing and degree of efficiency, suggesting that a random walk market (weak form
EMH) induce larger flow. These implications are powerful for further studies regarding
both fund flows and market efficiency, but also for practitioners trying to model mutual
fund flows.
Generally, we encourage further studies examining the same relationship in other secu-
rity markets in order to determine if our findings is a general index fund flow mechanism,
or if it holds specifically for S&P500. As this paper studied broad-market efficiency, a
natural continuation is to study the index fund flow and the efficiency of specific index
constituents; i.e., the market impact of index fund flows on single company equity. Rea-
sonably, large institutional actors providing index funds need to adjust their holdings to
track the index (say once a month), and therein cause abnormal price movements. Such a
study would prove useful for traders, highlighting arbitrage opportunities. Furthermore,
as has been noted by Tiwari, Albulescu, & Yoon (2017), the stock market seems to exhibit
signs of a multi-fractal nature. We therein propose further studies of multi-fractal Hurst
exponents (MF-DFA) and its relation to index funds.
7 REFERENCES 34
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A APPENDIX 37
A Appendix
A.1 Data adjustments
Variable Step Procedure Effect
Return
0) Naked data 62 111
TNA
0) Naked data 67 341
1) Fill 67 341 → 70 158
Flow
0) Naked data 69 516
1) Inception & Death 69 516 → 70 225
2) Winsorize (1 and 99%) 70 255
Table 12: Data adjustments for downloaded mutual fund data. Data for a total of 43 938
funds were obtained from CRSP. These steps outline our data management procedure explained
in section 3.1.
A.2 Hurst histogram
Figure 6: Histogram over daily Hurst estimates computed using a window size of 1008 during
the period 2000-2019.
A.3 Regressions - naked data
A
A
P
P
E
N
D
IX
38
Linear regression, naked data: fractional flow
Panel A: DFA Panel B: RS
N = 239 (A) (B) (C) (D) (E) (F) (G) (H)
Intercept 0.0143* 0.0187 -0.2024** -0.2333** 0.0216* 0.0252 -0.1720* -0.1915*
(1.75) (0.83) (-1.98) (-2.31) (1.79) (0.92) (-1.73) (-1.97)
Hurstt -0.0230 -0.0260 -0.0017 0.0092 -0.0365 -0.0399 -0.0298 -0.0196
(-1.23) (-1.19) (-0.07) (0.39) (-1.46) (-1.36) (-0.96) (-0.69)
Mkrett - 0.0149 0.0125 0.0122 - 0.0164 0.0151 0.0147
(0.66) (0.56) (0.53) (0.73) (0.69) (0.66)
Mkrett−1 - - - -0.0378 - - - -0.0358
(-1.57) (-1.50)
Mkrett−2 - - - -0.0181 - - - -0.0156
(-0.78) (-0.69)
Log(TNAt−1) - -0.0002 0.0121** 0.0137** - -0.0002 0.0120** 0.0120**
(-0.20) (2.08) (2.40) (-0.13) (1.97) (2.18)
CumV olt−1 - - -0.0496 -0.0554 - - -0.0410 -0.0407
(-0.82) (-0.93) (-0.72) (-0.71)
Feest−1 - - 95.66** 103.72** - - 89.72** 93.10**
(2.21) (2.38) (2.12) (2.17)
Adjusted R2 0.09% -0.46% 0.85% 2.44% 0.46% -0.03% 1.35% 2.61%
F-statistic 1.50 0.68 1.42 1.57 2.13 1.24 1.68 1.49
Table 13: Linear regressions estimated on the dependent variable fractional FLOW aggt . FLOW
agg
t is calculated as the sum of all index funds dollar value flow
for month t adjusted by the previous month sum of all index funds total net assets (see equation 3.2). No adjustments are done on the return, TNA, or flow
data. Mkrett is the S&P500 return from period t. CumV olt are the cumulative standard deviation from t − 11 up to t using monthly S&P500 returns. The
Log(TNAt), is the logged value of aggregated total net assets for all the index funds for month t. Feest are the average fee level (expense ratio) of the index
funds for month t. The Hurst exponents in panel A are calculated using the DFA-method and the Hurst exponents in Panel B the RS-method, both with a rolling
window size of 4 years (1008 observations). The Hurst exponents are computed on a daily basis and then averaged over each month to obtain monthly estimates.
The regression are estimated on a monthly basis and the numbers in parenthesis represent the t-statistic for the estimated coefficient above.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.
A
A
P
P
E
N
D
IX
39
Linear regression, naked data: dollar flow
Panel A: DFA Panel B: RS
N = 239 (A) (B) (C) (D) (E) (F) (G) (H)
Intercept 28.2045*** -42.2071** -173.7096** -196.0168** 29.2642*** -41.7072 -164.0102* -178.5299*
(3.33) (-2.04) (-2.19) (-2.20) (3.67) (-1.39) (-1.95) (-1.95)
Hurstt -56.7264*** -17.4416 -3.4558 4.3331 -54.0161*** -20.9468 -14.7685 -7.2979
(-3.09) (-1.42) (-0.26) (0.29) (-3.31) (-0.95) (-0.64) (-0.28)
Mkrett - 2.3508 0.8832 1.1187 - 2.7874 1.9706 2.1526
(0.16) (0.06) (0.07) (0.19) (0.13) (0.14)
Mkrett−1 - - - -31.5603 - - - -30.7215
(-1.23) (-1.17)
Mkrett−2 - - - -8.8723 - - - -7.8688
(-0.80) (-0.71)
Log(TNAt−1) - 3.9991*** 11.3171** 12.5377** - 4.1395** 11.0443** 11.7807**
(2.69) (2.39) (2.37) (2.46) (2.30 (2.27)
CumV olt−1 - - -26.3973 -30.0667 - - -24.4713 -23.7961
(-0.81) (-0.91) (-0.81) (-0.77)
Feest−1 - - 56850.13** 62667.51** - - 55589.42** 58118.02**
(2.00) (2.02) (1.99) (1.97)
Adjusted R2 2.98% 7.30% 7.32% 8.13% 1.68% 7.30% 7.44% 8.15%
F-statistic 9.58*** 3.67** 2.27** 2.61** 10.98*** 6.52*** 4.03*** 3.86***
Table 14: Linear regressions estimated on the dependent variable dollar FLOW aggt . FLOW
agg
t is calculated as the sum of all index funds dollar value flow for
month t (see equation 3.3). No adjustments are done on the return, TNA, or flow data. Mkrett is the S&500 return for period t. CumV olt are the cumulative
standard deviation from t− 11 up to t using monthly S&P500 returns. The Log(TNAt), is the logged value of aggregated total net assets, expressed in $ billions,
for all the index funds for month t. Feest are the average fee level (expense ratio) of the index funds for month t. The Hurst exponents in panel A are calculated
using the DFA-method and the Hurst exponents in Panel B the RS-method, both with a rolling window size of 4 years (1008 observations). The Hurst exponents
are computed on a daily basis and then averaged over each month to obtain monthly estimates. The regression are estimated on a monthly basis and the numbers
in parenthesis represent the t-statistic for the estimated coefficient above.
***, **, and * indicates significance at 1%, 5%, and 10%, respectively.