Department of Banking and Finance
Performance Measurement of Exchange
Traded Funds
Master Thesis in Banking and Finance
Lodged with Prof. Dr. Markus Leippold
Under the supervision of Nikola Vasiljevic
Lodged at Chair of Financial Engineering
Author Thomas Schär
Alte Bernstrasse 75b
3075 Rüfenacht BE
E-Mail: [email protected]
Matriculation number: 08-125-684
Zurich, August 18th, 2014
i
Executive Summary
The selection of actively managed funds aims at identifying an investment that is likely to
outperform a selected benchmark. In contrast, excess performance is not a focus in choosing
Exchange Traded Funds (ETFs). This paper argues that the performance measures of active
funds are often not only irrelevant, but may be misleading when being applied to passively
managed ETFs. Investors of ETFs want to buy and sell a diversified bundle of assets with the
same return profile as the benchmark which is replicated. Since investing directly into the
benchmark is not possible, unlevered, passive ETFs aim to mirror, or track the performance of
the benchmark. Therefore the ETFs’ performance relative to the benchmark and the
competing ETFs is more significant than absolute returns. The key research question of this
study is therefore how to comprehensively measure and assess the tracking ability of ETFs. The
thesis aims to design an intuitive efficiency measure that helps selecting the most efficient ETF
amongst its peers replicating the same benchmark. The most distinctive feature of this paper is
that it does not only take several statistical considerations into account, but also adjusts for a
selection of typical trading strategies on the ETFs primary and secondary market.
As suggested by Hassine and Roncalli (2013), the theoretical framework of the ETF efficiency
measure is based on the Value-at-Risk (VaR) methodology and combines tracking difference,
tracking error and the bid-ask spread. The theoretical processing of the efficiency measure is
completed with empirical research conducted on unlevered, passively managed ETFs listed on
SIX. ETFs replicating the Swiss Market Index and the EURO STOXX 50 over the observation
period from May 2013 to May 2014 are considered. The VaR framework is extended
throughout this study by considering both the underlying assumptions about the distributions
of relative returns and the measurement methods of ETF risk-factors. Alternative calculation
techniques, robust and unilateral measures of tracking error are applied. Gaussian and non-
Gaussian distribution assumptions as well as intra-horizon risk are taken into account by
considering the historical VaR, Cornish-Fisher VaR, Expected Shortfall and Intra-Horizon VaR.
The empirical evidence in this study suggests that the ETF rankings according to the efficiency
measures are largely consistent across the data sample. More efficient funds tend to perform
better in most of the methods considered. However, the results of strongly depend on the
underlying statistical assumptions of the risk metrics as well as the trading characteristics of
the investor. The sample data is found to suffer from strong non-normality and data outliers.
The findings show that investors need to consider several ETF performance measurement
alternatives in their ETF selection process while adjusting the efficiency measures to their
underlying investment objectives and trading circumstances.
ii
Contents
List of Figures ............................................................................................................................... iv
List of Tables .................................................................................................................................. v
Chapter 1 Introduction ................................................................................................................ 1 1.1 Outline ........................................................................................................................................................ 2
Chapter 2 Exchange Traded Funds .............................................................................................. 3 2.1 Delimitation .............................................................................................................................................. 3 2.2 Historical Background ............................................................................................................................ 4 2.3 Replication Strategies ............................................................................................................................ 5
2.3.1 Physical Replication .............................................................................................................................. 6 2.3.2 Synthetic Replication ............................................................................................................................ 7 2.3.1 Net Asset Value ...................................................................................................................................... 8
2.4 ETF Costs .................................................................................................................................................... 8 2.4.1 Internal Costs ........................................................................................................................................... 9 2.4.2 External Costs .......................................................................................................................................... 9
2.5 Extra Revenue – Securities Lending ................................................................................................ 10 2.6 Indices ....................................................................................................................................................... 11 2.7 Market Environment ............................................................................................................................ 12
2.7.1 Providers ................................................................................................................................................. 12 2.7.2 Authorized Participants .................................................................................................................... 13 2.7.3 Creation- Redemption Process ...................................................................................................... 13 2.7.4 Investors and Trading Strategies ................................................................................................. 15
Chapter 3 ETF Risk Metrics ........................................................................................................ 16 3.1 Delimitation ............................................................................................................................................ 16 3.2 Tracking Efficiency: Tracking Difference and Tracking Error .................................................. 17
3.2.1 Tracking Difference ............................................................................................................................ 17 3.2.2 Tracking Error - Based on the Standard Deviation ............................................................... 18 3.2.3 Sources of Tracking Difference and Tracking Error .............................................................. 18
3.3 Liquidity Metrics .................................................................................................................................... 20 3.3.1 Delimitation: Relative versus Absolute Liquidity ................................................................... 20 3.3.2 AuM and Trading Volume ............................................................................................................... 21 3.3.3 Bid-Ask Spread ..................................................................................................................................... 21 3.3.4 Market Impact Costs and the Notional Traded ...................................................................... 23
3.4 Pricing Efficiency .................................................................................................................................... 24
Chapter 4 Literature Review ...................................................................................................... 25 4.1 ETF Performance ................................................................................................................................... 25 4.2 Tracking Efficiency, Liquidity and Pricing Efficiency ................................................................... 27
Chapter 5 Performance Measurement ...................................................................................... 30 5.1 ETF Selection Principles ....................................................................................................................... 30 5.2 Sharpe Ratio and Information Ratio ............................................................................................... 31
5.2.1 Pitfalls of the Information Ratio ................................................................................................... 33 5.3 The ETF Efficiency Measure by Hassine and Roncalli ................................................................ 35
iii
Chapter 6 Empirical Research .................................................................................................... 39 6.1 Data Sample ............................................................................................................................................ 39 6.2 Data Treatment ...................................................................................................................................... 40 6.3 Sample Statistics .................................................................................................................................... 41 6.4 Results for the ETFs on SMI ............................................................................................................... 42 6.5 Results for the ETFs on EURO STOXX 50 ........................................................................................ 45
6.5.1 Information Ratio versus Efficiency Measure ......................................................................... 48
Chapter 7 Adjustments to the Efficiency Measure .................................................................... 49 7.1 Pricing Efficiency .................................................................................................................................... 49 7.2 Alternative Tracking Error Measures .............................................................................................. 50
7.2.1 TE – Based on Correlation of Returns ......................................................................................... 50 7.2.2 Tracking Error based on the Residuals of a Linear Regression ........................................ 53 7.2.3 Tracking Error based on Robust Measures .............................................................................. 54 7.2.4 Tracking Error based on Semi-Variance .................................................................................... 57 7.2.1 Autocorrelation ................................................................................................................................... 61
7.3 Alternative Bid-Ask Spread Measurement .................................................................................... 63 7.4 Alternative Value-at-Risk Measures ................................................................................................ 66
7.4.1 Cornish-Fisher Value-at-Risk .......................................................................................................... 70 7.4.1 Historical Value-at-Risk .................................................................................................................... 72 7.4.2 Intra-horizon Value-at-Risk ............................................................................................................ 74 7.4.1 Expected Shortfall ............................................................................................................................... 78
7.5 Alternative Interpretation of the Efficiency Measure ............................................................... 80
Chapter 8 Conclusion and Outlook ............................................................................................ 81
iv
List of Figures
Figure 1: ETP Classification ............................................................................................................ 4
Figure 2: Global ETP Numbers and AuM ....................................................................................... 5
Figure 3: ETF Replication Strategies .............................................................................................. 6
Figure 4: Internal and External Costs ............................................................................................ 8
Figure 5: Dividend Distribution ................................................................................................... 12
Figure 6: Creation - Redemption Process .................................................................................... 14
Figure 7: Tracking and Pricing Efficiency ..................................................................................... 16
Figure 8: Sources of Tracking Error and Tracking Difference ...................................................... 19
Figure 9: Impact Factors on Bid-Ask Spread................................................................................ 22
Figure 10: Information Ratio based on Benchmark .................................................................... 32
Figure 11: Information Ratio based on Tracker .......................................................................... 34
Figure 12: Illustration of the Efficiency Measure ........................................................................ 36
Figure 13: Larger Tracking Difference ......................................................................................... 37
Figure 14: Larger Bid-Ask Spread ................................................................................................ 37
Figure 15: Larger Tracking Error .................................................................................................. 37
Figure 16: Percentage Spread ETF 2# SMI................................................................................... 43
Figure 17: Tracking Difference ETF 2# SMI .................................................................................. 43
Figure 18: Tracking Difference ETF 1# SMI .................................................................................. 44
Figure 19: Tracking Difference ETF 5# SMI .................................................................................. 45
Figure 20: Tracking Difference ETF 7# EURO STOXX 50 .............................................................. 46
Figure 21: Percentage Spread ETF 2# EURO STOXX 50 ............................................................... 47
Figure 22: Sample Regression with Data Outliers ....................................................................... 52
Figure 23: Sample Regression without Data Outliers ................................................................. 52
Figure 24: Tracking Difference ETF 5# EURO STOXX 50 .............................................................. 56
Figure 25: Tracking Difference ETF 3# EURO STOXX 50 .............................................................. 58
Figure 26: Autocorrelation Function ETF 3# SMI ........................................................................ 62
Figure 27: Autocorrelation Function ETF 1# EURO STOXX 50 ..................................................... 62
Figure 28: Relative Tracking Difference Distribution .................................................................. 66
Figure 29: Absolute Tracking Difference Distribution ................................................................. 69
Figure 30: Cumulative Tracking Error ETF 1# and 3# on SMI ...................................................... 74
Figure 31: Decision Tree of ETF Efficiency Measures .................................................................. 85
Figure 32: Time Series ETF Tracking Difference ........................................................................ 107
Figure 33: ETF Autocorrelation Function .................................................................................. 113
Figure 34: Percentage Bid-Ask Spreads ..................................................................................... 117
v
List of Tables
Table 1: Information Ratio .......................................................................................................... 31
Table 2: Results for the ETFs on SMI ........................................................................................... 42
Table 3: Results for the ETFs on EURO STOXX 50 ........................................................................ 46
Table 4: Information Ratio ETF SMI............................................................................................. 48
Table 5: Information Ratio ETF EURO STOXX 50 ......................................................................... 48
Table 6: Pricing Efficiency ETF SMI .............................................................................................. 49
Table 7: Pricing Efficiency ETF EURO STOXX 50 .......................................................................... 49
Table 8: Tracking Error ................................................................................................................ 51
Table 9: Alternative TE ETF SMI .................................................................................................. 53
Table 10: Data Outliers ETF SMI .................................................................................................. 54
Table 11: Data Outliers ETF EURO STOXX 50 .............................................................................. 54
Table 12: Robust TE ETF SMI ....................................................................................................... 55
Table 13: Robust TE ETF EURO STOXX 50 .................................................................................... 56
Table 14: IQR ETF SMI ................................................................................................................. 57
Table 15: IQR ETF EURO STOXX 50 .............................................................................................. 57
Table 16: Semi-Variance ETF SMI ................................................................................................ 59
Table17: Semi-Variance ETF EURO STOXX 50 ............................................................................. 60
Table 18: Adjusted Spread ETF SMI ............................................................................................. 64
Table 19: Adjusted Spread ETF EURO STOXX 50 ......................................................................... 65
Table 20: Normality Test ETF SMI ............................................................................................... 68
Table 21: Normality Test ETF EURO STOXX 50 ............................................................................ 69
Table 22: Cornish-Fisher VaR ETF SMI ......................................................................................... 70
Table 23: Cornish-Fisher VaR ETF EURO STOXX 50 ..................................................................... 71
Table 24: Historical VaR ETF SMI ................................................................................................. 72
Table 25: Historical VaR ETF EURO STOXX 50 ............................................................................. 72
Table 26: Intra-horizon VaR ETF SMI ........................................................................................... 76
Table 27: Intra-horizon VaR ETF EURO STOXX 50 ....................................................................... 77
Table 28: Expected Shortfall ETF SMI .......................................................................................... 79
Table 29: Expected Shortfall EURO STOXX 50 ............................................................................. 79
Table 30: EURO STOXX 50 ......................................................................................................... 100
Table 31: Swiss Market Index .................................................................................................... 101
Table 32: ETF Sample ................................................................................................................ 102
Table 33: Fund Information ....................................................................................................... 103
Table 34: Trading Information .................................................................................................. 105
Table 35: Efficiency Measures Overview .................................................................................. 121
Table 36: Efficiency Measures Overview without Spread ......................................................... 123
vi
Abbreviations
AP Authorized Participant
AuM Asset under Management
Bps Basis Points
CHF Swiss Franc
CESR Committee of European Securities Regulators
ES Expected Shortfall
ESMA European Securities and Markets Authority
ETF Exchange Traded Fund
ETI Exchange Traded Instrument
ETN Exchange Traded Note
ETP Exchange Traded Product
FINMA Swiss Financial Market Supervisory Authority
iid independent and identically distributed
iNAV indicative Net Asset Value
IQR Interquartile Range
IOSCO International Organization of Securities Commissions
IR Information Ratio
LOB Limit Order Book
LPM Lower Partial Moment
MAD Median Absolute Deviation
MiFID Markets in Financial Instruments Directive
NAV Net Asset Value
OTC Over-the-Counter
PnL Profit and Loss
SIX SIX Swiss Exchange
SMI Swiss Market Index
TD Tracking Difference
TE Tracking Error
TER Total Expense Ratio
UCITS Undertakings for Collective Investment in Transferable Securities Directives
USD United States Dollar
VaR Value-at-Risk
Chapter 1 Introduction
1
Chapter 1 Introduction
Over the past decade, Exchange Traded Products (ETP) have experienced a surge in popularity
whereas the possibilities of their application has increased. iShares, being the largest ETP
provider in the world with more than 440 funds and over USD 480 billion of Assets under
Management (AuM), estimates that the ETP industry grew from worldwide 106 products with
USD 79,4 billion AuM in 2002 to 5’025 ETPs with USD 2’300 billion AuM in January 2014
(iShares, 2014, p.5). This trend seems to continue as the industry grows across all markets and
segments.
Above all, the subcategory of Exchange Traded Funds (ETF) was responsible for a large amount
of the growth in ETPs (Blackrock, 2012, p.4). In Switzerland, the number of ETFs increased from
888 listed products in 2012 to 940 in 2013 with a rise in volume of roughly 20% (SIX, 2013a,
p.3). Moreover, during the financial crisis in 2008 ETFs boosted their volume unlike any other
investment class (SIX, 2009, p.5). Institutional as well as retail investors gained interest in these
products, which offer index tracking, the main purpose of ETFs, at relative modest costs.
In accordance with the enormous increase in size, came an increase in variety and complexity
of ETFs. Nevertheless, ETFs are praised for their high level of transparency and are thus
awarded a special role in comprehensive due diligence procedures and risk management to
investors. Guidelines from regulatory bodies such as the Undertakings for Collective
Investment in Transferable Securities Directives (UCITS) in Europe on the one hand and
enhanced due diligence procedures by the ETF providers on the other hand, helped to increase
the investment knowledge on ETFs. Nevertheless investors need to be aware of several factors
when looking for ETFs suiting their portfolio and investment needs. They not only need to take
into account investor-related factors such as the underlying investment objective, horizon and
universe, they also need to compare and contrast ETFs from a range of providers according to
their structure, tradability, risk and performance.
As selecting the most excellent fund is core for any investor, the efficiency and performance1
of an ETF is a primary concern. Where traditional mutual funds are judged by how much they
outperform their opponents and benchmark, the performance measurement of ETFs is not
straightforward, as their aim is not to beat the performance of their underlying benchmark,
1 ETF Efficiency and performance are used as synonyms in the context of this thesis. In order to achieve a consistent terminology
throughout the study, the official ETF terminologies by the European Securities and Markets Authority (ESMA) and the naming of
the ETFs according to SIX Swiss Exchange are applied. If not stated otherwise, the term ETF refers to long only, passively managed,
unlevered exchange traded funds. ETF performance and ETF efficiency measurement are used conterminously.
Chapter 1 Introduction
2
but to replicate it as close as possible. Since many tools developed for traditional funds fail
when being applied to ETFs, standalone measures such as the tracking error and the tracking
difference are used to compare the ETF’s tracking ability. However, little is known about their
consolidation and joint interpretation as one comprehensive ETF performance measure.
1.1 Outline
The overall structure of the study takes the form of eight chapters, including this introductory
chapter. In order to acquire a fundamental knowledge about the ETFs history, replication
strategies, costs, revenues, market environment and trading characteristics, Chapter 2
presents thorough discussion of the specific features of the ETFs. As for every section of the
thesis, the theoretical outlay in Chapter 2 focuses on factors relevant for efficiency
measurement.
Chapter 3 holds the processing of both qualitative and quantitative performance
measurement and provides the full mathematical framework of the ETFs performance metrics.
Chapter 4 will complement the theoretical insight on the performance of ETFs with an
extensive literature review.
Looking at existing performance figures, Chapter 5 presents both empirical as well as
mathematical assessment on existing funds evaluation methods. Ultimately, the framework
for the efficiency measurement of ETFs, initially designed by Hassine and Roncalli (2013), will
be set up.
Chapter 6 presents the data sample processed in this study and calculates the efficiency
measure for the ETFs selected. Moreover, comprehensive insight on the data sample and data
treatment is given.
The analysis in Chapter 7 enhances the basic model by applying Gaussian and non-Gaussian
efficiency measures, as well as liquidity risk and alternative tracking error measures.
The conclusion Chapter 8 summarizes and critically evaluates the empirical and mathematical
findings. As a final point, areas for further ETF research are identified.
Chapter 2 Exchange Traded Funds
3
Chapter 2 Exchange Traded Funds
According to the European Securities and Markets Authority (ESMA), ETFs are open-ended
collective investment schemes that trade throughout the day like a stock on the secondary
market, which takes place on the exchange. Generally, ETFs seek to mirror the performance of
a target benchmark and are structured and operate in a similar way. Like operating companies,
ETFs register subscriptions and redemptions of shares and list their shares for trading (ESMA,
2011, p.9). From a legal perspective, ETFs are considered to be special assets not included in
the bankruptcy assets, should the ETF provider become insolvent. Unlike index funds which
are priced only once at the end of each trading session, ETF prices adjust throughout the day
and can be bought without any direct subscription and redemption fee on the secondary
market (Picard & Braun, 2010, p.2).
2.1 Delimitation
Since the inception of the first ETP, a large number of products have been introduced in the
financial market. Whereas ETFs have to pursue strict guidelines the widening of the ETP
product universe put forth many, partially regulated products such as Exchange Traded Notes
(ETNs). For the sake of clarity, it is crucial to consistently distinguish these instruments
according to their types. The classification in this paper generally follows the suggestion by
iShares (2014, p.17).
ETPs are understood as an umbrella term of three subcategories. Besides the mentioned ETFs,
the second subcategory subsumes ETNs as well as Exchange Traded Commodities (ETCs),
whereas the third subcategory covers the remaining Exchange Traded Instruments (ETIs). ETNs
are structured products that are issued as non-interest paying debt instruments, whose prices
fluctuate with an underlying index or an underlying basket of assets. Because they are debt
obligations, ETNs are backed by the issuer and subject to the solvency of the issuer (ESMA,
2012b, p.10). The remaining ETP-spectrum that is neither defined as funds nor as notes, is
classified as ETIs. Here included are listed options, warrants and hybrid instruments (iShares,
2014, p.11). Figure 1 highlights the important distinction of these product classes.
Chapter 2 Exchange Traded Funds
4
Figure 1: ETP Classification
The distinction of ETPs is illustrated. The term Exchange Traded Product covers Exchange Traded Funds, Exchange Traded Notes/ Exchange Traded Commodities and Exchange Traded Instruments. (Source: Own illustration following iShares, 2014)
The subcategories, ETNs, ETCs and ETIs will not be subject to analysis in this paper.
2.2 Historical Background
Eugene Fama set the cornerstone for passive management in 1965 with his study on market
efficiency. He suggested that, since the prices of securities instantly and fully reflect all public
and inside information available, price movements cannot be predicted using past prices. In
consequence, investors on average can only perform the same as the market. They come off
best by simply buying and holding a diversified basket of stocks, whilst minimizing fees and
taxes (Fama, 1965). Following the idea by Fama, Wells Fargo, an American banking and
financial services company, commercially adapted a passive strategy and launched the first
institutional index fund in 1976. The first index fund available for retail investor was launched
by Vanguard only five years later (Hehn, 2006, p.120).
Nonetheless, it was not until 1993 that ETFs initially became a viable investment opportunity.
Indeed, the first registered index-tracker was launched that year by State Street Global
Advisors under the name of SPDR tracking the US-Stock index S&P 500. Today, this fund
remains one of the most heavily traded funds in the world, with more than USD 37 billion AuM
(Picard & Braun 2010, p.17).
Whereas the story of success of ETFs in Europe began in the 2000, the total amount of ETFs
available globally had grown to 169 a year later only, whereby 103 of ETFs were traded
exclusively on U.S. markets (Wiandt & McClatchy, 2001, p.73). Apart from Switzerland, it was
Germany, Great Britain and Sweden who were the vanguards by offering ETFs in Europe
(Picard & Braun, 2010). Deutsche Börse emerged as the most important trading platform for
ETFs in Europe, significantly contributing to its success story (Hehn, 2006, p.120). In 2004 the
first ETFs on Emerging Markets, Real Estate and Commodities were introduced, whereas two
Exchange Traded Products (ETPs)
Exchange Traded Funds (ETFs)
Exchange Traded Notes (ETNs) /
Exchange Traded Commodities
(ETCs)
Exchange Traded Instruments (ETIs)
Chapter 2 Exchange Traded Funds
5
years later, the first ETFs taking short positions have been launched (Picard & Braun, 2010,
p.17). Figure 2 presents the global development of number and AuM of both ETFs and ETPs
since 1993. Only the financial crisis in 2008 resulted in a yearly decrease AuM of both ETFs and
ETPs. The amount of products on the market however steadily increased since the first
inception of an ETF.
Figure 2: Global ETP Numbers and AuM
The development of ETP and ETF AuM and the number of products are illustrated over the period from 1993 to June 2014. The ETF AuM figures for June 2014* are estimations, as official figures are not available. (Source: Own illustration following Blackrock 2009, 2012 & 2014)
2.3 Replication Strategies
ETFs can be categorized by their way of replicating a benchmark. Two main types of index
replication, namely physical and synthetic replication, can be differentiated. Physically
replicating ETFs hold all or a selection of constituents of an index, whereas synthetic
replication refers to the usage of derivatives in order to achieve benchmark returns. Synthetic
ETFs deliver the performance of a benchmark through the use of swaps and other derivatives
(IOSCO, 2013, p.2).
The style of replication has a significant influence on the ETFs tracking ability and subsequently
is explained in more detail. The explanations follow the principles for the regulation of ETFs,
issued in 2013 by the board of the International Organization of Securities Commission
(IOSCO), which regulates more than 95% of the world’s securities markets. Figure 3 gives an
overview on the commonly used segmentation of ETFs according to their replication strategy.
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013June
2014*
ETF AuM 0.8 1.1 2.3 5.3 8.2 17.6 39.6 74.3 105 142 212 310 412 566 796 711 1036 1311 1351 1644 2010 2207
ETP AuM 0 0 0 0 0 0 2 79.4 109 146 218 319 428 598 851 772 1156 1483 1525 1944 2396 2632
# of ETFs 3 3 4 21 21 31 33 92 202 280 282 336 461 713 1170 1595 1944 2460 3011 3297 3490 3650
# of ETPs 0 0 0 0 0 0 2 106 219 297 300 357 524 883 1541 2220 2694 3543 4311 4759 4988 5217
0
1000
2000
3000
4000
5000
0
500
1000
1500
2000
2500
3000
Nu
mb
er
of
pro
du
cts
Au
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n m
illi
on
US
D
Chapter 2 Exchange Traded Funds
6
Figure 3: ETF Replication Strategies
The ETF Replication Strategies are depicted. Physical and synthetic replication is differentiated. Physical replication can be further split into the full replication and the sampling methods, which furthermore comprises the representative and optimized sampling. Synthetic replication includes the unfunded and the fully funded swap model. (Source: Own illustration following the classification by IOSCO, 2013; iShares, 2014; Picard & Braun, 2010)
2.3.1 Physical Replication
Physically replicating ETFs can be segmented according to the share of benchmark constituents
they hold. ETFs which contain all elements of a benchmark are referred to as fully replicating.
Whereas ETFs that hold a selection of the underlying constituents are denoted as optimized or
partially replicating (IOSCO, 2013, p.2). If the physically replicating ETF does not conduct
securities lending, a process described later on in more detail, the strategy generally does not
expose an investor to counterparty risk (Hehn, 2006, p.16). Derivatives are only used in order
to equitize cash dividends of the constituents. In this process, the ETF avoids building up
unprofitable cash-positions by reinvesting the cash trough futures and other derivatives
(Wiandt & McClatchy, 2001, p.42).
2.3.1.1 Full Replication
A fully replicating ETF generally invests in the component securities of the underlying
benchmark in the same approximate proportions as in the benchmark itself. In consequence,
this type of ETF commonly displays a high degree of transparency (IOSCO, 2013, p.2).
Besides offering relatively close tracking of the benchmark, certain methodical problems might
arise with full replication as Benchmark constituents often are published with rounded down
decimal places only. Furthermore, dividend distributions may result in a higher cash
component of the ETFs. The reason is that the ETF distributes dividends each quarter or
semester only, whereas the benchmark may assume to distribute dividends on a daily basis.
Replication
strategies
Physical replication
Full replication
Sampling methods
Representative sampling
Optimized sampling
Synthetic replication
Unfunded swap model
Fully funded swap model
Chapter 2 Exchange Traded Funds
7
Replicating broad indices such as the MSCI World furthermore is costly, as more securities
have to be bought in order to fully replicate the index. On the one hand this increases
corresponding transaction costs, on the other hand, it leads to frequent rebalancing within the
index. The cancellation and admission of new benchmark constituents increases the
rebalancing costs for the ETF. Finally, if the index holds illiquid constituents, prices might be
driven up by the additional demand triggered by the ETF (Picard & Braun, 2010, p.47-48).
2.3.1.2 Replication by Sampling
With sampling techniques, the ETF overcomes some of the problems previously mentioned.
The ETFs thus acquires only a subset of the underlying indexes constituents whilst adding
securities that exhibit similar return patterns as the constituents omitted (IOSCO, 2013, p.2).
This method has the advantage of lower management fees and administrative costs but the
ETF may suffer from more inaccurate return tracking (Picard & Braun, 2010, p.42).
2.3.2 Synthetic Replication
Synthetic or derivative replicating ETFs invest in a diversified basket of assets while entering
into a derivative contract, typically through a total return swap (IOSCO, 2013, p.2). The swap
counterparty guarantees to deliver the return of the index in exchange for a variable swap
spread (Picard & Braun, 2010, p.52).
Although this replication strategy avoids the high rebalancing costs and tracking inaccuracy
associated with physical replication, it exposes the investor to counterparty risk. Regulatory
requirements reduce the risk arising. A UCITS fund in Europe e.g. is allowed to maximally
allocate 10% of its total assets into derivatives such as swaps (ESMA, 2012b, p.7). In case of
default of the swap counterparty, the ETF should be fully covered by the collateral of the swap
contract (Picard & Braun, 2010, p.61).
According to IOSCO Guidelines, the unfunded and the funded structure of a synthetic
replicating ETF can be further differentiated. In an unfunded structure, the ETF manager
invests in a substitute or reference basket of securities. This baskets return is used as collateral
in the derivative contract in exchange for the return of the index (IOSCO, 2013, p.3). In the
funded model, synthetic ETFs engage in a swap in exchange for cash without the creation of a
substitute basket (IOSCO, 2013, p.3). The fund transfers the cash proceeds from investors to
the counterparty, which in return provides collateral in excess of the subscription value and
further guarantees the performance of the benchmark (iShares, 2014, p.14).
Chapter 2 Exchange Traded Funds
8
2.3.1 Net Asset Value
The value of an ETFs underlying constituents is described by its Net Asset Value (NAV). The
NAV is calculated by summing the value of all constituents of the fund including cash positions
and deducting all liabilities. The sum is divided by the amount of outstanding ETF shares on a
daily basis, in order to receive the NAV (iShares, 2014, p.50). A unique feature of ETFs is that a
so-called indicative Net Asset Value (iNAV) is issued. It is calculated throughout the day from
current market prices and is published every 15 seconds (iShares, 2014, p.50). This process
allows for real-time tradability of an ETF. Conversely, traditional mutual funds are prices once
a day (Picard & Braun, 2010, p.6).
2.4 ETF Costs
ETFs attract a broad range of investors due to their diversification benefits coming at low fees.
Depending on the provider, the benchmark and the replication method, ETF costs may
however diverge and therefore need to be watched closely (Picard & Braun, 2010, p.8). This
section therefore assesses the total costs of ETF ownership. In line with the classification of
iShares (2014, p.44), internal cost, being ongoing charges to the ETF, are differentiated from
external costs, which are costs incurred at the time of trading the ETF. Figure 4 presents the
classification of costs for both physically and derivative replicating ETFs.
Figure 4: Internal and External Costs
The internal cost of holding the ETF and the external cost from trading the ETF are listed for both physical and synthetic replicating ETFs. (Source: Own illustration following iShares, 2014, p.44)
Fund structure
Physical replicating
synthetic replicating
Internal costs
- Total expense ratio
- Rebalancing Costs
-Additional factors
- Total expense Ratio
- Swaps Spread
-Additional factors
External costs
-Bid- / Ask -Spreads
-Transaction Costs
- Taxation
Chapter 2 Exchange Traded Funds
9
2.4.1 Internal Costs
The set of internal costs include all expenses of the ETF on an ongoing basis. The most
prominent, being the Total Expense Ratio (TER) and the rebalancing costs, are discussed
subsequently.
TER is expressed in percentages and includes all ongoing fees charged to the ETF (Picard &
Braun, 2010, p.27). The TER must be officially published by the ETF provider and has to
comprise the following indicative, but not exhaustive list of charges:
It has to include all payments to the management company, directors, depositary, custodians,
investment adviser as well as any outsourced services of the ETF. The management fee hereby
covers all costs arising from ETF administration, maintenance and management (Picard &
Braun, 2010, p.27). Furthermore, registration and regulatory fees as well as audit fees must be
included in the TER. Entry and exit charges, as well as performance related fees are not
covered by the TER (Committee of European Securities Regulators, 2010). The TER is deducted
from a fund’s NAV on a daily basis and therefore influences an ETFs daily tracking performance
(iShares, 2014, p.45).
Rebalancing cost arise when a change in the benchmark requires a reweighting of the ETFs
constituents. Equivalently, costs arising from a change in the variable swap spread are
regarded as internal costs for synthetic ETFs (iShares, 2014, p.45). Both cost types negatively
influence an ETFs tracking performance.
2.4.2 External Costs
External costs are defined as costs that are charged to the investor only at the time of ETF
trading. They include trading costs such as the bid-ask spread and transaction costs as well as
any taxes levied on the ETF trade.
Bid and ask prices indicate the best price at which a security can be sold and bought at a given
point in time (Picard & Braun, 2010, p.88). The bid-ask spread of an ETF is defined as the
difference between the bid price and the ask price. The bid-ask spread is always positive as the
maximal price for which an investor is willing to sell his ETF shares is always larger than the
price for which he is willing to buy them (Picard & Braun, 2010, p.28). The spread does not
directly influence the tracking performance of the ETF. However, it is crucial factor in ETF
liquidity measurement and will be subject to profound evaluation in Chapter 3.
Chapter 2 Exchange Traded Funds
10
Transaction costs comprise all additional costs that arise when an ETF is sold or bought. Such
costs include brokerage and custody feed and provisions for banks and brokerage dealers
(Picard & Braun, 2010, p.102).
The taxation of an investment in an ETF may occur on the ETF, the underlying securities and
the investor level.
On a fund level the applicable jurisdiction of the domicile country as well as the legal structure
of the ETF are relevant (iShares, 2014, p.21). For all ETF structures in Switzerland, income and
capital taxes have to be paid only on a dividend and investor level, as ETFs do not represent a
legal entity (Picard & Braun, 2010, p.83). Withholding taxes on income or on capital gains
received by the fund are charged at the ETF constituents’ level. They arise in the country
where the underlying securities are situated and strongly depend on the treaty between the
country of the securities and the country in which the ETF is domiciled. Those two layers of
taxation become especially important as the benchmark and the ETF may have diverging
taxation principles. Benchmarks such as gross total return indices assume that dividends are
reinvested without any tax deductions, whereas the ETF will have to pay withholding taxes
when disbursing dividends. Net total return indices may assume that withholding taxes are
paid on dividends, whereas the ETF is able to reclaim its taxes partially. This practice, known as
dividend tax enhancement, may boosts ETF returns relative to the benchmark (Johnson et al.,
2013, p.6). An additional example of tax optimization is called dividend tax arbitrage. In this,
ETFs lend stocks that are subject to dividend withholding taxes to counterparties located in
more tax-efficient jurisdictions during dividend season (Bioy & Rose, 2013, p.6).
Taxes levied on an investor level are not subject to ETF performance measurement. In
consequence the taxation of investors is not discussed in this thesis.
2.5 Extra Revenue – Securities Lending
Apart from the revenues generated trough the rise in value of the underlying constituents, an
ETF can generate additional revenue from securities lending. Securities lending refers to
additional revenue trough the transfer of securities from the ETF to a third party, who will
provide collateral to the lender and pay a fee. The extra revenue can be used to partially offset
the internal costs the ETF. It even can lead to an over performance of the ETF with respect to
the benchmark, which does not exert securities lending (Picard & Braun, 2010, p.68). However,
since lending activities can be executed in direct agreements between the lender and the
borrower, true magnitude is difficult to assess (Bioy & Rose, 2013, p.3). In Europe, the UCITS
Chapter 2 Exchange Traded Funds
11
regulation requires all revenues, net of direct and indirect operational costs, to be returned to
the ETF (ESMA, 2012a, p.7).
Securities lending may be undertaken in physical as well as synthetic replicating ETFs. As the
lending in physically replication ETFs is more straightforward, securities lending possibilities for
synthetic ETFs are more complex. Physically lending the securities inherits the risk that the
borrower of the security becomes insolvent and is unable to return the loaned securities.
Therefore physical ETFs engaged in lending expose investors to counter party risk.
In synthetic ETFs, the lending process takes place within the reference basket and thus does
expose the investor to counterparty risk. (iShares, 2014, p.49). Securities lending is an
important factor in total return assessment of an ETF and has to be kept in mind when
evaluating the tracking performance of an ETF.
2.6 Indices
Even though ETFs are found to replicate a variety of benchmarks, this thesis is confined to ETFs
replicating equity indexes only. Picard and Braun (2010, p.33) define an index as a statistical
figure, representing a basket of financial assets, which can only be bought by investors through
the use of ETFs or index-certificates.
The underlying assumptions of the index thereby have an important impact on the relative
performance of the ETF. In this context, the indexes’ most relevant feature is their dividend
reinvestment assumption. Divergent assumption on the dividend treatment of ETF and index
result in tracking dissimilarity. In general, price and total return indices are differentiated. Price
indices only return the prices of the underlying assets whilst assuming that all dividends of the
constituents are distributed. Total return indices on the other hand assume that all dividends,
interest payments and other income are reinvested (Picard & Braun, 2010, p.36). They can be
further separated according to their assumption on taxation on the dividends reinvested. A net
total return index assumes that dividends are taxed at the biggest available rate, whereas
gross total return indexes assume that no taxes are levied on the reinvested dividends (iShares
2014, p.22). These assumptions are particularly important as they results in performance
differences of the ETF and the index whenever dividends are paid or reinvested (Hassine &
Roncalli, 2013).
The timing of dividend reinvestment can lead to further performance differences between ETF
and its benchmark. Whereas many indices assume dividends to be disbursed at ex-dividend
dates of the corresponding constituents, the dividend payouts by ETFs are often cumulatively
administered (Picard & Braun 2010, p.24).
Chapter 2 Exchange Traded Funds
12
From the time series of the SMI price index and the DB X-TRACKERS SMI UCITS ETF in Figure 5,
it can for instance be seen that the ETFs dividend payment on July 25th, 2013 results in an
instant drop of NAV compared to the NAV of the benchmark. By the end of March 2014
however, the NAV of the ETF and the benchmark return to similar values, as the SMI price
index distributes dividends throughout the investment horizon.
Figure 5: Dividend Distribution
The development of the NAV of the SMI price index and the DB X-TRACKERS SMI UCITS ETF are illustrated. The period covers May 2013 to May 2014. The red circle indicates the period of the dividend payment of the ETF. (Source: Own calculations/ illustrations)
2.7 Market Environment
Due to its size, distinctive structure and easy accessibility, the ETF market consists of various
types of participants. In the following, the most relevant, market participants and market
characteristics are presented.
2.7.1 Providers
The ETF provider is in charge of all defining aspects of an ETF such as the replication strategy,
pricing and dividends attribution. Therefore, he is the main responsible for the ETF
performance (Picard & Braun 2010, p.29).
Following a recent statistic by Deutsche Bank (2014, p.37), by the end of 2013, a total of 180
ETF providers existed across the globe. However, 85.8% of the ETF industries total assets were
concentrated amongst the top 10 providers only. Blackrock, the mother company of iShares, is
the largest ETP provider with 40.3% of total market share, followed by State Street GA (17%)
and Vanguard (15.1%). Holding a total of 72.4% of the global ETF market in 2013, those three
big players provide nine of the ten biggest ETFs by AuM.
Deutsche Bank AG (2.3%) is the biggest provider located in Europe, whereas UBS, the biggest
Swiss provider, holds USD 16 billion AuM (0.7%). The ETF market is mainly dominated by banks
7000
7200
7400
7600
7800
8000
8200
8400
8600
8800
9000
SMI
ETF SMI
Chapter 2 Exchange Traded Funds
13
and bank-owned providers such as Deutsche Bank, Lyxor of Société Général, UBS and Zürcher
Kantonalbank (Deutsche Bank, 2014, p.41). In Switzerland iShares covers 53.36% of the Swiss
ETF market by the end of 2013. Out of a total of 18 ETF providers, the five biggest providers
including UBS (19.3%), Zürcher Kantonalbank (6.4%), Lyxor (6.25%) and db x-trackers (6%)
covered 97.5% of the ETF market (SIX, 2013a, p.3).
2.7.2 Authorized Participants
Authorized participants (AP), also referred to as Market Makers, govern the ETFs creation-
redemption process in which ETF shares are issued or redeemed. Typically being large
investment banks or brokerage businesses, APs conclude participation agreements with the
ETF, which allow the AP to subscribe and redeem units of the ETF on an in-kind basis. APs may
act as distributors of the ETF shares on various stock exchanges as well (Hehn, 2006, p.96). The
process in its details will be explained in chapter 2.7.3.
On the Swiss exchange platform SIX, at least one AP is employed for each ETF. APs are obliged
provide continuous liquidity in the ETF (Picard & Braun, 2010, p.30). APs have to provide bid
and ask prices for a fixed minimum of trading volume and have to avoid prices, which exceed a
maximum bid-ask spread of 5% in the case of an inexistent Over-the-Counter (OTC)2 market or
2% where a functioning OTC market exists (SIX, 2014).
As the spread is going to be integral part of the performance analysis, it’s important to
understand the role of APs in the ETF market. By providing liquidity through their so-called
Limit Order Books (LOB) 3, APs have an important influence on the prices that need to be paid
when trading an ETF. According to Roncalli and Zheng (2014), during a trading day, all selling
and buying orders are matched against the best order of their counterpart. If the volume of
the market order is larger than the quantity available at the corresponding best limit order, the
best limit price will be adapted and executed on the second limit order. Therefore the spread
between the best bid and the best ask price increases with the ETF notional traded (Roncalli &
Zheng, 2014).
2.7.3 Creation- Redemption Process
The creation- redemption process of ETF shares is administered through several steps.
At the beginning of the trading day, the portfolio manager of the ETF designates the APs with
the basket of securities to be taken into the fund at the closing of the market in exchange for
new units of the ETF. The AP will buy the corresponding basket of securities on the capital
2 OTC trades are administered on the primary market directly between the AP and the ETF. The final price for the ETF share is negotiated directly between the two counterparties. 3 LOB is the set of all active trading order of buyers and sellers of ETFs listed on any electronic system (Gould et al., 2013).
Chapter 2 Exchange Traded Funds
14
markets. The AP exchanges this basket against the equivalent amount of ETF-shares, which he
now is able to sell on the secondary market or OTC (Picard & Braun, 2010, p.63-65). This
process eliminates transaction fees and avoids tax events borne by the ETF, as the ETF does
generally not need to buy and sell component securities directly (Hehn, 2006, p.96). The main
benefit of the creation-redemption process is that it causes the market price of an ETF to
remain closely linked to its NAV, despite being priced continuously (Charupat & Miu, 2011,
p.968). If the price of the ETF moves sufficiently far outside the bid and ask band of the
underlying basket of securities, an arbitrage opportunity arises. If for example the price of the
ETF is sufficiently below the price of the underlying basket, the AP takes advantage of this
opportunity by buying ETF shares and selling the basket of securities short. At the end of the
day the AP uses the ETF shares to receive the underlying basket of securities and employ those
to cover the short position previously entered. By doing so, the AP receives the difference
between the two prices. The AP exploits the arbitrage opportunity until the price difference
between ETF and the underlying basket erode. As a consequence, the ETF is traded within the
bid-ask-spread of the underlying basket in normal market conditions. Figure 6 gives an
overview on how the primary and secondary markets are interlinked and how the creation-
redemption process is administered through the AP.
Figure 6: Creation - Redemption Process
The process of creating and redeeming ETF shares is depicted. The AP administers the process and acts as an intermediary of the capital markets, the ETF management and the Investors. Institutional investors can trade directly with the AP, on the primary as well as on the secondary market. Retail investors generally are only able to trade on exchange. (Source: Own Illustration following Hehn, 2006, p.123)
Institutional
Investor
Retail
Investor
Au
tho
rize
d P
arti
cip
ant
Capital
markets:
Stocks, ETFs,
Futures etc.
ETF / Fund manager
Cash
ETFs
ETFs
Securitie
s Secu
riti
es
ETFs
Redemption
Subscription
Cash
Securiti
OTC trading
Secondary Market Primary Market
Bro
kers
/ B
anks
Exch
ange
Cash
ETFs
Chapter 2 Exchange Traded Funds
15
2.7.4 Investors and Trading Strategies
Retail as well as institutional investors gained interest in ETFs as they use ETFs as an effective
mean of asset allocation by combining investments throughout different markets and asset
classes. Pension funds, insurance companies, foundations and other corporates typically use
ETFs in order to track selected markets precisely and transparently. As indicated in Figure 6,
institutional investors furthermore have the possibility to acquire ETF directly on the primary
market through OTC transactions. By trading OTC at the NAV of the ETF, an investor can get
ETF units created or redeemed at a premium only, without paying the bid-ask spread. The
investor however bears the risk that the constituents of the ETF might lose in value by the
official close of trading (Picard et al., 2014, p.16).
ETFs can be used as building blocks in a variety of strategies such as in core-satellite portfolios.
Hereby, ETFs are suitable for the core portfolio, which tries to achieve the broadest possible
diversification across the various asset classes. In the satellite portfolio the ETFs can be used to
build up diversified satellite positions in emerging markets, specific sectors or alternative asset
classes (Picard et al., 2014, p.10). ETFs can furthermore be used as a hedging tool against
declining markets, as they either can be sold short or short-ETFs can be bought (Picard &
Braun, 2010, p.11). Similar to the derivatives used to equitize cash, ETFs can be used to
manage short or medium-term cash holdings.
The variety of trading strategies indicates that investors may have completely heterogonous
interest in ETFs. The span reaches from buy-and-hold investors, who profit from a ETFs low
fees, tax efficiency and broad diversification, to a frequent trader who benefits from the low
volatility of ETFs compared to individual equities (Wiandt & McClatchy, 2001, p.98).
Having discussed the ETFs history, replication strategies, costs, revenues and market
environment, the following chapter will focus on the most prominent ETF performance
metrics. Both tracking and liquidity measures are considered.
Chapter 3 ETF Risk Metrics
16
Chapter 3 ETF Risk Metrics
As indicated in the preceding chapters, ETFs combine many advantageous features such as
security, good transparence, permanent pricing and trading, diversification as well as dividend
participation for relatively low costs. However, ETFs do also bear risks. In fact, many of the
existing statistical performance measures of ETFs are in fact risk figures due to the fact that
ETFs do not aim to outperform, but replicate a benchmark as closely as possible (Hassine &
Roncalli, 2013). This section hence provides the mathematical framework of key ETF risk
figures and reviews their sources. The most common metrics are the tracking difference
(subchapter 3.2), tracking error (subchapter 3.2), liquidity measures (subchapter 3.3) and the
pricing efficiency (subchapter 0). Besides pricing efficiency, those risk factors jointly depict the
building blocks for the efficiency measure derived in this thesis.
3.1 Delimitation
Much of the literature considered does not clearly distinguish tracking efficiency from pricing
efficiency. The term tracking efficiency refers to how closely the NAV of an ETF corresponds to
the NAV of the benchmark, whereas pricing efficiency measures how closely the price of the
ETF follows the NAV of the ETF. Figure 7 presents the difference between pricing and tracking
efficiency.
Figure 7: Tracking and Pricing Efficiency
The difference of pricing efficiency, measuring the equality of the ETFs price and NAV, and the tracking efficiency, measuring the equality of the ETFs NAV and benchmarks NAV are illustrated. (Source: Own illustration following Charupat & Miu, 2011)
By analyzing daily or even intraday data, pricing efficiency is concerned about the prevalence
of market inefficiencies such as failures in the creation-redemption process and its underlying
arbitrage mechanisms. However, resulting premiums or discounts over the NAV are not
expected to persist over the long run and are likely to lie within transaction costs (Charupat &
Miu, 2011). While tracking difference may be accounted to the ETF management, pricing
deviations are more likely to be market inefficiencies caused by the AP and therefore need to
be distinguished strictly.
Pricing Efficiency Tracking Efficiency
Price ETF NAV ETF NAV
Benchmark
Chapter 3 ETF Risk Metrics
17
3.2 Tracking Efficiency: Tracking Difference and Tracking Error 4
The ESMA Guidelines on ETFs and other UCITS defines Tracking Difference (TD) as the total
return difference of the annual return of the ETF and the annual return of the tracked
benchmark. Tracking Error (TE) is defined as the average daily volatility of the difference
between the return of the ETF and the return of the tracked benchmark (ESMA, 2012, p.4).
Both backward-looking coefficients measure the quality of benchmark tracking, whereas the
closer their values are to zero, the better is the tracking. There exist several methodologies to
calculate TE, whereas the calculation of TD is generally consistent across studies. To begin
with, Subchapter 3.2.2 provides the most prominent calculation method of TE following Pope
and Yadav (1994). In a later chapter, additional methods to compute TE are presented.
Calculating both the TD and TE requires the calculation of ETF and benchmark returns. The
mathematical formulas applied in the herein thesis are illustrated below, where and
denote the NAV of the ETF and of the benchmark at the time respectively. The returns
of the ETF and of the benchmark compute as indicated below.
(1)
(2)
3.2.1 Tracking Difference
The difference in ETF and benchmark returns, denoted by in formula (3) is calculated
according to Pope and Yadav (1994). The vector of weights of the ETF and the benchmarks
constituents are denoted with and respectively, whereby it is assumed that the number
of ETF and benchmark constituents is the same for the funds considered in this thesis.
(3)
4 Tracking Error is labeled differently in academic literature. Pope and Yadav (1994) as well as Frino and Gallagher (2001) define TE as the standard deviation of the difference between ETF returns and benchmark returns. Roll (1992) as well as Hassine and Roncalli (2013) define TE as the difference in portfolio and benchmark returns. In the herein thesis, the term TD will be used to describe the difference in returns, whereas TE refers to the standard deviation of the return differences.
Chapter 3 ETF Risk Metrics
18
The annual TD in returns computes as the return difference of ETF and the benchmark over the
time period of one year. The expression in the below formula denotes the vector of expected
return differences
| (4)
3.2.2 Tracking Error - Based on the Standard Deviation
The most widely used methodology used in the ETF industry is based on the day-to-day
variability of the difference in returns between a fund and its benchmark (Frino & Gallagher,
2001; Pope & Yadav, 1994). The TE | is calculated as the standard deviation of the
difference in daily returns as indicated below. The term hereby denotes the covariance
matrix of asset returns.
| √ (5)
Pope and Yadav (1994) indicate an estimation bias arising from the usage of the above formula
with daily or weekly data, inflating the TE measured. The issue can be resolved by either using
monthly data (Pope & Yadav, 1994) or make use of the correction based on the Lo and
MacKindlay (1988) analysis. iShares (2014, p.27) suggest the usage of at least 30 observation
points in order to receive a significant TE coefficient. In consequence the calculation of TE
based on monthly data requires a track record of at least two and a half years. As this trading
horizon may not reflect the average holding period of an ETF, this paper calculates TE on a
daily basis, as the average over one trading year.
3.2.3 Sources of Tracking Difference and Tracking Error
In order to get a sense for what may cause TD and TE, the sources ETF and benchmark return
differences are discussed in the following. Figure 8 lists the causes of TD and TE for both
physically and synthetically replicating ETF. The illustration additionally indicates the direction
of impact of the various factors. Whereas it can be seen from formula (3) and (4) that TD can
take both positive and negative values, TE, by squaring the return differences, only takes
positive values. A positive TD is desirable as it indicates an outperformance of the ETF over the
benchmark, whereas a high TE indicates many deviations in the returns. Thus Figure 8
allocates the direction negative to all factors increasing TE, whereas the classification negative
(positive) indicates a resulting excess performance of the benchmark (ETF).
Chapter 3 ETF Risk Metrics
19
Figure 8: Sources of Tracking Error and Tracking Difference
The causes of TD and TE for both physical and synthetic ETFs are depicted, indicating the direction of their impact. Negative impact indicates that the TD and TE increase. No influence* for synthetic ETFs indicates that the impact may be compensated indirectly through the swap price quoted by the swap counterparty. (Source: Own illustration following Johnson et al., 2013)
The illustration indicates an unambiguous relationship between the method how an ETF
replicates his benchmark and the quantity of potential TD and TE sources. Continuous factors
such as the daily deduction of the TER influence TD, but do generally not inflate TE. Non-
recurring and impermanent sources as for instance transaction and rebalancing costs, taxes
and diverging dividend reinvestment assumptions may influence both measures.
Many of the factors listed and their direction of impact on tracking have been discussed in the
previous chapters and thus will not be discussed again. An additional source of tracking
deviation comes from different cash holdings due to time lags during index composition
changes, where the ETF is forced to hold returns from cancelled constituents in cash (Johnson
et al., 2013, p.5). Another reason is that the ETF daily deducts a fraction of the annual
management fee. This will result in the fund holding smaller amounts of cash until payout.
Finally cash drag can arise from the lag between dividend or coupon payments by the index
constituents and the distribution of dividends to the ETF investors. Cash drag influences TE
negatively in up- and positively in down markets (Picard & Braun, 2010, pp. 22 - 23).
Cause
TER
Transaction and Rebalancing Costs
Dividend reinv. assumption
Taxation
Securities Lending
Swap Spread
Sampling method
Cash Drag
Tracking Difference
physical ETFs
Negative
Negative
Negative / Positive
Negative / Positive
Positive
No Influence
Negative / Positive
Negative / Positive
Tracking Difference
Synthetic ETFs
Negative
No influence*
No influence*
No influence*
No influence*
Negative / Positive
No influence*
No influence*
Tracking Error
physical ETFs
No Influence
Negative
Negative
Negative
Negative
No Influence
Negative
Negative
Tracking Error
Synthetic ETFs
No Influence
No influence*
No influence*
No influence*
No influence
Negative
No influence*
No influence*
Chapter 3 ETF Risk Metrics
20
3.3 Liquidity Metrics
Not only do the ETF constituents and their sizes have an important impact on an ETFs liquidity,
but it is likewise important to look at the property of the platforms on which the ETFs are
traded and the functioning of the underlying arbitrage mechanism. In the following, common
liquidity metrics and their applicability in the context of efficiency measurement are discussed.
Firstly, it is important to distinguish the absolute and the relative liquidity of ETFs (Roncalli &
Zheng, 2014).
Depending on the investment horizon of an investor, liquidity becomes a key criterion. The
investment of the investor determines whether to invest in a liquid or illiquid asset. A short
investment horizon may require an asset to be regularly traded at no or little discount,
whereas a long-time investor may benefit from higher returns by investing in illiquid assets
(Amihud & Mendelson, 1991). In addition, liquidity is important for risk-management purposes
in order to react quickly to changing market situations (Picard & Braun, 2010, p.9).
The subscription and redemption of ETF shares may become at risk in times of financial
distress. Recent events such as the downgrading of Greece to a non-investment grade, as well
as the earthquake in Japan in 2011 showed that ETF benchmarked to the corresponding
indices became difficult to trade (Hassine & Roncalli, 2013). Finally the liquidity of an ETF may
indicate the liquidity cost at which the ETF is traded.
3.3.1 Delimitation: Relative versus Absolute Liquidity
Relative liquidity refers to a comparison of both the ETFs and the underlying indexes liquidity.
As the liquidity of the index may be crucial when deciding for a benchmark, relative liquidity is
not relevant for intra-provider comparison as the benchmarks’ liquidity is same for all ETFs and
providers. Absolute liquidity on the other hand refers to the liquidity of the ETF himself and
does not relate to the liquidity of the underlying benchmark (Roncalli & Zheng, 2014). As
absolute liquidity is relevant in the context of this study, a selection of absolute liquidity
measures are presented.
In practice, there exist a range of measures to approximate the absolute liquidity of a fund,
however not all are appropriate when it comes to measuring the liquidity of an ETF. Why some
traditional liquidity metrics such as the AuM and the trading volume may be misleading in the
context of ETFs is explained subsequently. In order to design a comprehensive measure in
combination with TE and TD, the liquidity measure should furthermore allow for precise
scaling of ETF liquidity cost. The bid-ask spread and market impact costs are hereby found to
be the most relevant.
Chapter 3 ETF Risk Metrics
21
3.3.2 AuM and Trading Volume
The simplest approach to approximate the liquidity of an ETF is to look at total AuM.
Comparing the proportion of trades in the ETF allows to a certain extend to approximate how
liquid and costly trading the ETF is. Practitioners however argue, that the fund size and trading
volume allow little inference on the true liquidity of an ETF, as it is the constituent’s tradability
and their availability on the exchange that determines the liquidity of an ETF (iShares, 2013,
p.34; Justice & Rawson, 2012, p.4). Especially for ETFs, it is advisable to apply those two
measures with caution. As previously mentioned, the so-called on-screen activity coming from
trades on the secondary market which is measured and published by the exchange platforms
may only be a fraction of what is actually traded in the ETF due to OTC trades on the primary
market. Calamia, Devilla and Riva (2013) and iShares (2013, p.45) indicate that significant
portions of the ETF turnover is traded on the primary market. As those trades are not
consistently reported, overall ETF liquidity measurement with trading volume and AuM may be
misleading and are therefore not used to draw inference about the ETFs liquidity in this thesis.
3.3.3 Bid-Ask Spread
It is important to understand, that depending on whether a trade is executed on the primary
or secondary market, different liquidity costs arise. OTC trading orders can be entered at a
fixing with no bid-ask spread, whereas the price is at the discretion of the AP and the investor.
Such orders are generally handled at the NAV, meaning on the closing price of the benchmark
index plus a fee negotiated with the AP. While these fees certainly depend on the taxes, the
hedging costs as well as any operational costs of the AP, it certainly may also depend on the
negotiation power the investors.
The bid-ask spread to be paid on the secondary market is calculated throughout the day and is
generally not negotiable. Amundi (2011) stated, that the bid-ask spread for buying the ETF
MSCI World for the year 2010 on NYSE Euronext was 0.20%, whereas set-up and redemption
fees at the NAV were only 0.045%. Figure 9 lists again some of the most common factors which
influence the size of the bid-ask spread.
Chapter 3 ETF Risk Metrics
22
Figure 9: Impact Factors on Bid-Ask Spread
The factors influencing the bid-ask spread are illustrated. They include the costs of buying/- selling the securities, taxes levied, foreign exchange and hedging costs, the costs of holding the ETF shares and the margin of the AP. Larger trading volume and AuM furthermore are expected to have an impact on the size of the bid-ask spread. (Source: Own illustration following Amundi, 2011; Amihud & Mendelson, 1991)
From a mathematical point of view, the bid-ask spread is calculated as the difference of the bid
price and the ask price
In order to incorporate the bid-ask spread into the design of
an efficiency measurement, the relative bid-ask spread has to be calculated. This spread
measures the percentage difference in prices. Following the suggestions by Hassine and
Roncalli (2013), the relative bid-ask-spread | is calculated as the difference between the
quoted bid price and the quoted ask price
divided by their mid-price :
|
(6)
The mid-price is thereby defined as the average of both prices.
(7)
Formula (6) indicates the costs of a full market cycle, meaning purchasing an ETF share for the
bid price and consequently selling it for the ask price on the market. Whenever an investor is
more concerned with just the cost of only one transaction side, meaning either only selling or
only buying the ETF, the measure can be reduced to the half-spread by dividing the spread
| by two. In the herein thesis a full market cycle is assumed.
Impact Factors
Buying/ - selling the securitites
Taxation
Foreign Exchange/ Hedging costs
Cost of carrying the ETF
AP Margin
Trading Volume / Notional
Assets under Management
Chapter 3 ETF Risk Metrics
23
3.3.4 Market Impact Costs and the Notional Traded
Market impact cost arises when large trading volumes cause price in- or decreases in the
underlying assets of the ETF due to changed demand and supply on the markets (Amihud &
Mendelson, 1991). Practitioners such as Morningstar measures market impact cost by the
average market price movement in percent caused by a USD 100’000 trade in the ETF. Market
impact costs are expected to be larger the more volatile market prices are, the smaller and less
traded the ETF is and the more liquid the underlying securities are (Justice & Rawson 2012,
p.4). This costs impact becomes a key criterion for larger investors, who trade large amounts.
Not only the prices of the constituents, but also the bid-ask spread of the ETF may widen with
larger trading amounts. The best bid-ask spread may not be available as a large order generally
cannot be executed via the first best limit order in limited order book of an AP. Market impact
costs and the bid-ask spreads therefore are both influenced by the notional that is traded on
the market. The theoretical relationship between the bid-ask spread and the notional traded is
positive, meaning an increase in the notional traded results in an increase in the spread. In a
solely theoretical setting, the spread becomes infinite once the order size becomes large
enough to not be executable (Hassine & Roncalli, 2013).
In this paper however, market impact cost as well as bid-ask spreads for second or lower limit
orders will not be computed due to several reasons. Firstly, measuring those costs requires
accurate high frequency data on the bid- ask prices of the ETFs constituents, the ETF itself and
on the limited order book of the APs. Secondly, measuring the price impact of an ETF trade is
not straightforward due to many additional influential factors such as the liquidity of the ETF
and benchmark, the market microstructures as well as the capacity of the AP (Justice &
Rawson, 2012). Complementary mathematical background and exemplary calculation methods
are given in Stoll (2000), Roncalli and Zheng (2014) as well as Hassine and Roncalli (2013) and
references therein.
Chapter 3 ETF Risk Metrics
24
3.4 Pricing Efficiency
Closing Chapter 3 on ETF Risk metrics, this subchapter will discuss how to measure the pricing
efficiency of an ETF. Pricing efficiency, being part of microstructure analysis of an ETF,
measures how well the price of the ETFs mirrors the value of the NAV. It focuses on the
efficient functioning of exchange platforms and the products traded thereon. In this concept,
the APs’ role of providing liquidity and efficient pricing by acting as a dealer, broker and
exchange official is important.
According to Charupat and Miu (2011) the pricing efficiency of an ETF can be computed as the
percentage deviation of the end-of-day prices of an ETF from its NAV
(8)
According to Flood (2010), an issue with the above risk factor arises as closing prices are of the
noisiest prices in the day due to arbitrage trades at closing. A remedy is to use the midpoint of
closing quotes instead of the day-end prices as defined in formula (8).
(9)
Frictionless pricing is ensured by the arbitrage mechanism in the creation-redemption process
described in chapter 2.7.3. As this study assumes normal market conditions and frictionless
pricing, the pricing efficiency is anticipated to be within transaction costs. How robust pricing
and arbitrage mechanisms of ETFs are in volatile markets is yet subject to research.
This chapter mathematically evaluated the methods used to assess the risks and performance
characteristics of an ETF relative to its benchmark. Turning now to the empirical evidence on
the metrics discussed, the next chapter will review the literature on ETF performance
measurement.
Chapter 4 Literature Review
25
Chapter 4 Literature Review
A selection of authors providing comprehensive background on ETF structure, risks, trading
characteristics and costs has been mentioned in the previous chapters (Picard & Braun, 2010;
Wiandt & McClatchy, 2001; Hehn, 2006). Likewise, there is a large body of studies describing
specific fields of ETFs. The herein literature review accordingly is confined on most relevant
topics for performance measurement of unleveraged, passively managed ETFs. In particular,
the focus is laid on ETF characteristics that facilitate intra-provider comparison. As a
complement to the theoretical examinations of ETFs in the previous and subsequent chapters,
this chapter takes a look at the existing empirical evidence. Section 4.1 gives an overview of
ETFs overall performance and Section 4.2 evaluates the specific risk metrics previously
discussed.
4.1 ETF Performance
Being a relatively new financial product, there has not only been an increasing amount of
literature on the performance of different ETFs types, but also lots of comparisons of ETFs to
similar instruments in recent years. Several studies investigate the performance of ETFs
respectively to traditional mutual funds. Due to their similarity, ETF have however most
prominently been contrasted to index funds. Rompotis (2005) presents evidence that the two
instruments produce very similar results with respect to average return and mean risk levels as
well as tracking ability. Considering German passively managed ETFs, the author deems ETFs to
be hybrids between index funds and equities. Blitz, Huij and Swinkels (2010) find that both
European index funds and ETFs underperform their benchmarks due to dividend taxes, but not
necessarily due to expense ratios. Harper, Madurab and Schnusenberg (2006) on the other
hand, report from their cross-country studies higher mean returns and sharp ratios than
closed-end funds due to the lower expense ratios of ETFs. The extensive study by Svetina
(2008), which evaluates domestic equity, international equity as well as fixed income ETFs,
reports that ETFs deliver better performance than retail index funds and similar performance
than institutional index funds. Roncalli and Zheng (2014) consider the liquidity of an ETFs to be
the main advantage compared to index mutual funds.
Several authors however find that ETFs display poorer returns than index funds as their
structure and management processes are reluctant not to recapture transaction costs during
benchmark changes (Gastineau, 2004; Elton, Gruber & Busse, 2002). Looking at the relative
performance of US ETF from 2011 to 2013, Qiao (2013) discovers excess performance, but
Chapter 4 Literature Review
26
larger TE of ETFs compared to mutual funds. In consequence ETFs did not outperform mutual
funds, despite having lower management fees.
Raising the question whether the two instruments are complements rather than substitutes,
Svetina (2008) suggests that only 83% of ETFs in the US track the same index as index funds.
Guedj and Huang (2010) similarly find by virtue of their equilibrium model, that for well-
diversified, broad indices, ETFs and index funds indeed can be regarded as substitutes,
whereas for narrower and less liquid indexes ETFs are more suitable. These results are
consistent with the findings by Agapova (2011), who compares flows into the two instruments
and does not find cannibalization effects of ETF and index funds. Contrarily, Roncalli and Zheng
(2014) suggest that ETFs are not used as pure substitutes of index funds as the trading activity
of ETFs is spread throughout the day and not centered at the market close.
Comparing European and Swiss ETFs with their benchmarks, Rompotis and Milonas (2006)
identify a positive correlation of the management fees and the risk of the ETF. The authors
report that ETFs perform poorer and encumber investors with greater risk than their
benchmark. Gallagher and Segara (2006) examine ETFs traded on the Australian stock
exchange and argue that the ETFs produce the same return as their underlying benchmark
before costs.
Studying the cost features of ETFs and index funds, Dellva (2001) reports a cost advantage of
ETFs compared to index funds due to lower management fees. Gastineau (2001) confirms
those findings and identifies the size and the lack of transfer agency functions as the reason for
the ETFs cost efficiency. Elton et al. (2004) investigate the informational efficiency of index
funds and conclude that higher expenses such as a higher management fees does not increase
performance adequacy. Justice and Rawson (2012) incorporate holding costs as well as market
impact costs of ETFs in their total cost analysis and provide a range of results. By grouping ETFs
according to their size, they show that total costs are lower the larger and more heavily traded
the ETFs are. Moreover, they find that older ETFs have lower estimated holding costs and
tracking volatility.
Securities lending is found to be consequently used to enhance or completely offset TER
(Picard & Braun, 2010, p.55). According to a survey conducted by Morningstar, roughly 45% of
physical replicating ETFs in Europe engage in securities lending. The survey reveals that
revenue sharing arrangements and disclosure to investors vary greatly across providers. The
portion of revenues returned to the fund hereby ranges from 45% to 75% of gross revenue
(Bioy & Rose, 2013).
Chapter 4 Literature Review
27
4.2 Tracking Efficiency, Liquidity and Pricing Efficiency
Frino, Gallagher, Neubert & Oetomo (2004) point out that causes of mismatches in tracking
may have exogenous as well as endogenous reasons. Exogenous TD arises from index rules and
preservation procedures such as revision of the index composition, share issuance and
repurchases and spin-offs and are beyond the fund manager’s control. Endogenous TD is
induced from the individual activities of the funds. Despite being evaluated on be basis of
index funds, those distinctions are valid for ETFs as well. Research by Johnson et al. (2013, p.9)
suggests that TD of ETFs is usually negative as an ETF should underperforms its benchmark by
its TER when assuming perfect tracking.
A number of authors compare the TE of various ETF types based on their replication style.
Meinhardt, Mueller and Schoene (2012) argue in their study about the German ETF market,
that contrary to the general belief, synthetic ETFs regardless of their appropriation and asset
class, do not have a smaller TE and do not offer superior tracking compared the full replicating
ETFs. Synthetic fixed income ETFs however, seem to have persistently lower TE. Johnson et al.
(2013, pp.5-7) suggest, that despite the reduced set of potential source for TD in synthetic
ETFs, the tracking quality may however be influenced indirectly by the swap pricing. The
authors observe lower TE with synthetic replicating ETFs tracking the S&P 500 index. Physical
replication however offers less underperformance, meaning a lower TD, than the synthetic
funds. Furthermore TD tends to vary significantly over the time period considered. Due to the
fact that the index return is guaranteed by the swap agreement, TE is genuinely lower than
with physically replicating ETFs. According to the authors, benefits from lower TE are offset by
the exposure to counterparty risk. Meinhardt et al. (2012) argue in their study about the
German ETF market, that contrary to the general belief, synthetic ETFs regardless of their
appropriation and asset class generally do not have a smaller TE and do not offer superior
tracking compared the full replicating ETFs. Synthetic fixed income ETFs however seem to have
persistently lower TE. This result stands in contrast to the findings by Johnson et al. (2013),
which find lower TE with synthetic replicating ETFs tracking the S&P 500 index.
Several studies examine the liquidity in ETFs and try to conceptualize the rather impalpable
term. According to the survey by Ernst and Young (2014), the depth of the secondary market,
approximated by the tightness of bid-ask spreads and the level of trading turnover are often
used as approximations of liquidity. To some extent, the size of the fund can be used to draw
inference about liquidity as well. The bid-ask spread and price impact are the two measures
mostly focused on in statistical literature on liquidity (Brennan & Subrahmanyam, 1996).
Chapter 4 Literature Review
28
A number of studies however suggest that as the AuM and liquidity do not show high
correlation, simply looking at the ETFs’ total assets may be misleading. According to Roncalli
and Zheng (2014) this is especially true for the European ETF market, as it is mainly
institutional investors who hold large volumes of ETF for their strategic asset allocation. Other
authors contrarily find that the fund size and the amount of trading volume actually is an
indication of tighter spreads (Calamia et al., 2013). Looking at the traded volume on exchanges
as a proxy for liquidity, may however be misleading as it does not reveal the true liquidity, of
the underlying constituents of the ETF. Moreover it is the APs obligation to provide liquidity
which is relevant (Picard & Braun, 2010, p.9). Furthermore Roncalli and Zheng (2014) show
that for the EURO STOXX 50 index, trading activity is concentrated in only a small number out
of all available ETFs.
Aggrawal and Clark (2009) develop a five-factor scoring model for ETF liquidity and
consequently rank over 500 ETFs according to their underlying liquidity in the secondary
market. They find that a lower bid-ask spread, a higher market capitalization, a lower expense
ratio and higher average trading volume are the strongest indicators for high liquidity in the
ETF. Hassine and Roncalli (2013) find that the changes in spread in relation to the notional
traded depend on the markets. In their two dimensional study, Sanchez and Wei (2010)
examine bid-ask spreads and holding periods of ETFs. They use the inverse of the turnover
ratio as a proxy for the trading intensity since for ETFs the amount of shares in the market is
not fixed. The main two result of their research are, that ETFs replicating broad benchmarks
have lower spreads and that the overall liquidity of an ETF is not necessarily better than the
liquidity of their top holdings. Calamia et al (2013) find in their extensive study, that the
liquidity of ETFs not only depends on the liquidity of the benchmark, but trading volumes,
market fragmentation and AuM of ETFs result in tighter bid-ask spreads.
Furthermore, Roncalli and Zheng (2014) observe that the intraday spreads of the ETF and the
index are not related, yet there exists a positive relationship between the liquidity of the
underlying constituents and the ETF. Additionally, the competition and number of APs in the
ETF market are found to be important as they narrow the quoted bid-ask spreads (iShares,
2014, p.37). Petersen and Fialkowski (1994) show that posted and effective spreads on the
New York Stock Exchange often diverge, implicating that the quoted spreads are not what the
investors can expect in reality.
With respect to the pricing of ETFs, Ackert and Tian (2008) demonstrate that the relationship
between liquidity and pricing efficiency generally is positive for domestic ETFs as more liquid
markets tend to be more efficient. For international ETFs however, this relationship is found to
Chapter 4 Literature Review
29
be non-linear. Moreover, they find that holdings of international ETFs often are not listed at
domestic markets, which hinders the efficient creation-redemption process and harms pricing
efficiency of ETFs. The authors find that mispricing of country ETFs is related to momentum,
illiquidity as well as the ETFs size. Petajisto (2013) discovers that the prices of ETFs may
fluctuate significantly, reaching a difference of 260 bps, despite the prevalence of arbitrage
mechanisms in the creation-redemption process.
Engle and Sarkar (2006) look as the daily as well as intraday transaction of US listed domestic
and international ETFs. They conclude that price deviations of domestic ETFs are generally
small and temporary, while international ETFs suffer from larger and more persistent discounts
or premiums.
Overall, the results from the comprehensive body of literature on ETF performance suggests
that the empirical evidence often is inconclusive. The purpose of this chapter was to review
the literature on ETF performance measurement, whereas the subsequent chapter will
thoroughly discuss how to measure ETF performance quantitatively.
Chapter 5 Performance Measurement
30
Chapter 5 Performance Measurement
Markowitz (1952) was the first to provide a methodology of portfolio evaluation by maximizing
the expected return of a portfolio for a given level of market risk. Subsequent theories such as
the Capital Market Line by Tobin (1985) and the Two Fund Separation Theorem by Sharpe
(1994) laid the cornerstone for passive management. The concept of Jensen’s Alpha
established by Jensen (1968) ultimately questioned the value added of active management. In
his research on mutual funds, the author investigated the performance of 115 actively
managed mutual funds. He concluded that they do not only underperform a simple buy-and
hold approach, but found little evidence that an actively managed fund was able to perform
better than a random choice strategy. Alongside the alpha measure, other figures with the
purpose of measuring the performance of mutual funds such as the Sharpe- and Information
Ratio have been established.
This chapter will therefore illustrate why many of the tools developed up till now are not
suited to evaluate passively managed funds and may moreover return incorrect results in the
context of ETFs. The first subsection will open up this chapter b evaluating the quality features
important for an ETF investor. The subsequent sections prove why traditional metrics are
unsuited to evaluate ETF performance. Finally a performance measure based on the
framework by Hassine and Roncalli (2013), which is specifically tailored to ETFs is elaborated.
5.1 ETF Selection Principles
Fund picking for passive funds largely deviates from fund picking for actively managed funds.
Investors in ETFs generally are neither concerned about over-performing their benchmark, nor
do they need to narrow down their investment universe, as they simply try to replicate the
benchmark selected (Hassine & Roncalli, 2013). In order to understand which performance
measures are useful to compare ETFs, it is important to understand that passive investors do
not seek absolute performance, but want to buy and sell a diversified bundle of instruments at
the exact same return as the benchmark. Furthermore close-up tracking is a necessity when
ETFs are used for hedging purposes or mandates of an asset manager (Justice & Rawson,
2012).
In a global survey in 2014 representing 87% of the ETF industries global assets, Ernst and
Young (2014) determined the most important selection criteria when choosing an ETF.
Amongst the most relevant factors were the promoter’s reputation, which roughly 30% of the
participants chose to be relevant, the management fee (17%), tracking error (20%), the
Chapter 5 Performance Measurement
31
liquidity/size (23%) and the level of spreads (13%) were chosen to be the most relevant (Ernst
and Young, 2014, p,9). Whereas an isolated assessment of those features may be practicable,
their joint evaluation is less straightforward. An ideal ETF therefore is a fund that tracks the
benchmark perfectly and at the same time exposes the investor with no risk of suffering from
bigger losses than the benchmark at any time. The following subchapter does explain why the
Sharpe- and Information Ratio are not suited to evaluate ETFs.
5.2 Sharpe Ratio and Information Ratio
Combating both the dimension of risk as well as performance, the Sharpe Ratio and
Information Ratio are often used to directly compare the performance comparable funds. They
both capture the excess return per unit of risk associated with this excess return. The Sharpe
Ratio measures the excess return of a portfolio over the risk-free rate (Sharpe, 1994) whereas
the Information Ratio measures the excess return of the investment with respect to a given
benchmark (Grinold & Khan, 2000). Both measures can be applied either to a single asset as
well as a portfolio of assets such as a fund or an ETF. The Sharpe Ratio has little indicative
value for ETF performance comparison as the ETF does not try to outperform the benchmark.
The Information Ratio may be a better measure as it captures how much the returns of the ETF
deviate from the benchmark returns and puts it into relation of the risk associated risk with
this deviation. Expressed in the mathematical terms previously established, the Information
Ratio calculates as the ratio of the TD and TE.
| |
| (10)
To illustrate the concept, the following fictional funds and their values for the Information
Ratio are considered.
Table 1: Information Ratio
Fictional examples of ETF TD, TE and Information Ratio are presented. The metrics take the value zero for the benchmark, as they measure the relative tracking with respect to the benchmark itself. (Source: Own calculations/ illustrations)
ETF Number Symbol | | |
Benchmark 0% 0% n/a
1 3.6% 3% 1.2
2 4.5% 3% 1.5
3 2.4 % 1.6% 1.5
4 2% 2.5% 0.8
Chapter 5 Performance Measurement
32
Taking the Information Ratio as a measure, fund two is preferred to fund number one as it
displays a higher excess return | for the same level of risk | . On the other hand,
fund number one offers better performance than fund number three, but exhibits almost
double the volatility in TD. In consequence, fund three is more favorable as indicated by the
higher Information Ratio. If fund number one and four are compared, fund number four offers
a lower excess return over the benchmark and a lower risk, but nevertheless is inferior to fund
number one. Finally fund three offers a lower excess return, but also lower TE than fund
number two but has the same Information Ratio of 1.5. According to the Information Ratio, an
investor will be indifferent between the two most favorable funds number two and three.
The graphical proof of the examples considered is presented in Figure 10 below.
Figure 10: Information Ratio based on Benchmark
The Information Ratio for the four fictional Funds calculated relative to the benchmark is illustrated. The vertical axis depicts the TD and the horizontal axis indicates the TE. (Source: Own illustration following Hassine & Roncalli, 2013)
It becomes evident that fund number two is preferable to fund one as it offers better TD for
the same TE. Furthermore it can be seen that fund number three, having the same Information
Ratio of 1.5, can be constructed as a linear combination of fund number two and the
benchmark. In consequence fund number two is superior to funds number one and four as
well. Mathematically, the linear combination of fund number two and the benchmark is
defined by formula (11).
Benchmark
X3
X1
X2
X4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
Tra
ckin
g D
Iffe
ren
ce
Tracking Error
IR 1.2
IR 0.8
IR 1.5
Chapter 5 Performance Measurement
33
(11)
The Information Ratio then can be calculated as a linear combination of fund number two and
the benchmark.
| | (12)
The ability to combine benchmark and the fund presumes that the benchmark can be
replicated perfectly. If this requirement is fulfilled, the following proposition about the
performance of two funds and and their Information Ratio can be made according to
Hassine and Roncalli (2013).
(13)
5.2.1 Pitfalls of the Information Ratio
As it is neither is possible to invest in the benchmark directly nor to replicate it perfectly, a
tracker has to be used in order to approximate the benchmark.
(14)
In consequence the proposition (13) does no longer hold and the calculation of the
Information Ratio of fund number three becomes less trivial.
( | ) | |
√ | | | (15)
The mathematical proof of the above equation is given in Appendix 1. Whenever the TD of the
tracker is negative, due to management fees or other costs for example, the Information
Ratio of fund three is smaller than the Information Ratio of fund two , as the TE is
positive by its mathematical definition (Hassine & Roncalli, 2013). In consequence, the
subsequent relationship holds for | and .
Chapter 5 Performance Measurement
34
( | ) | (16)
Graphically it can be seen from Figure 11 that the Information Ratio of fund three
being a linear combination of the fund and a tracker may be misleading as it
indicates to be inferior to fund . Due to the fact that could not be reached, fund
conversely is superior to fund , even though its Information Ratio is lower.
Figure 11: Information Ratio based on Tracker
The Information Ratio is calculated using a tracker, which serves as a proxy for the benchmark. The vertical axis depicts the TD and the horizontal axis indicates the TE. (Source: Own illustration following Hassine & Roncalli, 2013)
According to Hassine and Roncalli (2013) this issue becomes more severe for benchmarked
funds with low levels of TE. By looking at the increasing gap between the dotted and the
drawn-trough line in Figure 11, the authors’ proposition becomes evident. The linear
approximation gets less accurate for lower TD and TE levels.
An additional drawback of the Information Ratio in the context of ETFs is that it ignores the
magnitude of the TE. Whereas the investor may be interested in a ETF which replicates its
benchmark as close as possible. A fund that has lower TE and at the same time lower TD levels
does a better replication job than a fund with the same Information Ratio but higher TD and TE
values.
X1
X2
X3
X3 proxy
X0
-0.05
0
0.05
0.1
0.15
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6
Tra
ckin
g D
iffe
ren
ce
Tracking Error
Chapter 5 Performance Measurement
35
5.3 The ETF Efficiency Measure by Hassine and Roncalli
Hassine and Roncalli (2013) incorporate the most important ETF metrics, the TE, TD and the
bid-ask spread into a single performance measure, allowing for consistent intra-ETF
comparison. The mathematical outlay of their efficiency measure is presented in the following.
Assuming a model with two time periods, the authors define an investor’s Profit and Loss (PnL)
function of buying an ETF at time and selling it at as the difference of the TD and
the bid-ask spread of the ETF.
| | (17)
The loss function of the investor is defined as all negative events in | .
| | { | } (18)
The efficiency measure finally is a risk measure applied to the loss function of the ETF based on
the Value-at-Risk (VaR) for a given probability level .
| { { | } } (19)
VaR is defined as the threshold value, such that the loss over the given time horizon does not
exceed this value for the given probability . In other words, the investor has a probability of
of losing an amount greater than - | . By switching sings, the authors make use of
the opposite of the loss in order to achieve an ascending order of the efficiency measure.
Therefore, opposite to traditional VaR interpretation, higher values of the efficiency are more
favorable.
The relevant percentiles do depend on the probability level selected. depicts the
probability distribution function of | and is assumed to be normal in this base-scenario.
The efficiency measure is then described by the inverse of the probability distribution function
at a given level of probability.
| (20)
Chapter 5 Performance Measurement
36
Assuming that asset returns are normal distributed, the probability distribution function takes
the values of the inverse of the standard normal distribution denoted by the Greek letter phi
.
| | | | (21)
The closed-form expression of the efficiency measure is computed by taking the TD |
and subtracting both the costs of trading the ETF measured by the bid-ask spread | and
the inverse of the standard normal probability distribution function multiplied with the TE
| . The full mathematical derivation of formula (21) is presented in Appendix 2.
In order to illustrate the intuition of the efficiency measure, Figure 12 exemplifies the
efficiency measure at a confidence level of 95%. At this quantile, the expression
takes the rounded value of For the computation, the following fictional values are
assumed: | , | and | .
Figure 12: Illustration of the Efficiency Measure
The efficiency measure is illustrated. A positive TD shifts the mean of the normal curve to the right, the bid-ask spread shifts it to left. The inverse of the inverse of the standard normal probability distribution function at the 95% percentile, multiplied by the TE finally indicates the VaR on the horizontal axis. (Source: Own illustration following Hassine & Roncalli, 2013)
As it can be seen from Figure 12, the graph takes the form of a normal distribution. The
difference between the TD and the spread shifts the curve to the right. Consequently the PnL
distribution is centered around 20 bps. Looking at the 95% percentile, the efficiency measure
takes the value of | . The investor in this example therefore has a 95%
s(x │ b)
μ(x │ b)
1.65*σ(x │ b)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-1.0
0
-0.9
0
-0.8
0
-0.7
0
-0.6
0
-0.5
0
-0.4
0
-0.3
0
-0.2
0
-0.1
0
0.0
0
0.1
0
0.2
0
0.3
0
0.4
0
0.5
0
0.6
0
0.7
0
0.8
0
0.9
0
1.0
0
1.1
0
1.2
0
1.3
0
Value-at-Risk
Chapter 5 Performance Measurement
37
chance of not losing an amount greater than 30 bps or reciprocally a 5% chance of losing an
amount greater than 30 bps.
Figure 13: Larger Tracking DifferenceFigure 13 to Figure 15 illustrate how the efficiency measure
adapts to changes in the underlying risk factors.
Figure 13: Larger Tracking Difference Figure 14: Larger Bid-Ask Spread
The dashed line illustrates an increase in the TD. The horizontal axis depicts the VaR. The dashed lines indicate the shift in the probability distribution function as compared to the initial fund. (Source: Own illustration following Hassine & Roncalli, 2013)
The dashed line illustrates an increase in the spread. The horizontal axis depicts the VaR. The dashed lines indicate the shift in the probability distribution function as compared to the initial fund. (Source: Own illustration following Hassine & Roncalli, 2013)
In Figure 13 the TD has been increased to 50 bps, meaning that the fund outperforms his
benchmark by an additional 10 bps and therefore is favorable to the initial fund. Consistent
with this interpretation, the efficiency measure takes a value of -10 bps. Figure 13 illustrates a
widening in the spread of 20 bps as the efficiency measure deteriorates to -50 bps, the initial
fund is preferred.
Figure 15: Larger Tracking Error
The dashed line illustrates an increase in the TE. The horizontal axis depicts the VaR. The dashed lines indicate the shift in the probability distribution function as compared to the initial fund. (Source: Own illustration following Hassine & Roncalli, 2013)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-1.0
0
-0.8
5
-0.7
0
-0.5
5
-0.4
0
-0.2
5
-0.1
0
0.0
5
0.2
0
0.3
5
0.5
0
0.6
5
0.8
0
0.9
5
1.1
0
1.2
5
1.4
0u1(x|b) < u2(x|b)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-1.0
0
-0.8
5
-0.7
0
-0.5
5
-0.4
0
-0.2
5
-0.1
0
0.0
5
0.2
0
0.3
5
0.5
0
0.6
5
0.8
0
0.9
5
1.1
0
1.2
5
1.4
0
s1(x|b) < s2(x|b)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-1.0
0
-0.8
5
-0.7
0
-0.5
5
-0.4
0
-0.2
5
-0.1
0
0.0
5
0.2
0
0.3
5
0.5
0
0.6
5
0.8
0
0.9
5
1.1
0
1.2
5
1.4
0
σ1(x|b) < σ2(x|b)
Chapter 5 Performance Measurement
38
In Figure 15, the altered fund has a greater TE of 40 bps, which means that the distribution of
TD is more scattered and exhibits fatter tails. In consequence the fund is more likely to deliver
bigger. The lower value of the efficiency measure -46 bps confirms this presumption and
therefore the initial fund is preferred.
According to Hassine and Roncalli (2013), it can be concluded that if and are two ETFs
benchmarked against the same index, is preferred to if and only if the efficiency measure
of fund | is larger than the efficiency measure of fund | .
| | (22)
This chapter described which factors are most relevant for an ETF investor and argued why
traditional performance metrics may fail in the context of ETFs. It suggested that a
comprehensive ETF performance measure should include the bid-ask spread, TD and TE in
order to cover all relevant ETF performance dimensions. The next chapter will apply the
procedures and methods previously derived to the ETF data sample.
Chapter 6 Empirical Research
39
Chapter 6 Empirical Research
This chapter follows on from the previous chapter, which laid out the mathematical framework
of the efficiency measure | , turning now to the empirical evidence on Swiss and
European ETFs. The first subsection provides details on the ETF sample and data treatment.
The second subchapter prepositions the sample statistics and the last subchapter calculates
the efficiency measure based on the ETF data sample.
6.1 Data Sample
The sample period covers one trading year of an ETF. It starts on May 1st, 2013 and ends on
April 30th, 2014. For all calculations the perspective of a Swiss investor is adopted. In
consequence, solely ETF which were active on SIX Swiss Exchange throughout the entire study
period are eligible. ETFs liquidated or not incepted at the beginning of the study period are
excluded from the analysis. Even though investors are able to trade ETFs internationally,
additional expenses from on-site storage or costly transfer to the home stock exchange justify
the above confinement of only considering the funds quoted on the Swiss exchange.
The ETF sample is restricted to all funds replicating the Swiss Market Index (SMI) and the EURO
STOXX 50, whereas currency hedged share classes are not part of the data sample. For the
SMI, ETFs benchmarked against both the prices as well as the net total return index are
considered. For EURO STOXX 50, due to tax-efficiency reasons, only ETFs on the net total
return index are on the market. Regardless of the fact that certain providers issue multiple
ETFs with diverging share classes on the same benchmark, all ETFs are considered. The
different share classes are specially designed for either institutional or retail investors and in
consequence have varying management fees, TER, minimum investment and finally tracking
efficiency.
All calculations are held in the corresponding base currency of the benchmark, which is Swiss
Francs (CHF) for the SMI and Euros (EUR) for the EURO STOXX 50. As iShares (2014, p.29)
indicates, calculating TE in a different currency results in marginally different volatility and
additionally comprises currency risk. Such risk is not part of the analysis in this study.
In total, five ETFs benchmarked against the SMI and seven ETFs replicating the EURO STOXX 50
are evaluated. Further information regarding the constituents and weights of the considered
benchmarks are given in the Appendix 3. An overview of the sample ETFs, including additional
Chapter 6 Empirical Research
40
information on e.g. management fees, inception dates, fund sizes and dividend appointments
can be found in Appendix 4, 5 and 6.
6.2 Data Treatment
The relevant data is extracted from Bloomberg and merged with the information available on
the provider websites. If not stated otherwise, all data is measures at the closing of the
corresponding trading day. The bid (ask) prices available on Bloomberg are defined as the
highest (lowest) price a seller will accept for a security. They correspond to the last bid (ask)
price from the last day the market was open. As receiving the data from an index may be
costly for ETF providers, they often avoid costs by purchasing data either from the total return
or the price index. In the herein data sample, three distributing ETFs are therefore
benchmarked against the SMI total return index. In the case of EURO STOXX 50, due to fiscal
optimization, all providers of the sample chose to replicate the EURO STOXX 50 total return
Index. In consequence three distributing ETFs have deviant dividend assumptions from their
benchmark.
In order to receive significant and consistent statistics, the following definitions and numerical
adjustments are applied to the data sample:
For each ETF a number between 1# and 7# is assigned in order to obtain anonymous and
provider-independent results. The concrete ETF names can be found in Appendix 4, whereby
the order of the funds in the appendix does not correspond to the order of the numbers
assigned.
With the purpose of receiving consistent and comparable results, the confidence level for the
VaR is set at throughout the whole analysis.
Whenever an ETF treats dividend payments differently than the benchmark, TD and TE arise
and distort calculations. In order to compute commensurable metrics, the daily performance
of the ETF is corrected every time a dividend is paid. Hereto the dividends are added back to
the NAV of the ETF at the dividend date and are assumed to exhibit the same percentage
return as the NAV for the remaining observation period.
Chapter 6 Empirical Research
41
Due to divergent holidays of the ETF and the benchmark, partially no NAV is quoted. In
consequence, such days are excluded in the calculations of both ETF and benchmark returns.
Additionally, whenever there was no trading activity in an ETF and thus no bid and ask prices
placed, those days are not considered for the calculations of the spread.
6.3 Sample Statistics
The study period is denoted by [ ] and includes daily observations. The corresponding
value of the inverse of the 95%-percentile of the standard normal probability distribution
is 1.645. For the data sample, the efficiency measure takes the following closed-form
expression
| | | | (23)
where the sample mean | , TD | the bid-ask spread | and the TE |
are calculated according to Hassine and Roncalli (2013) over the whole observation period.
The annual TD | is calculated as the annual return difference between ETF and
benchmark returns, whereas corresponds to one year.
| (
)
(
)
(24)
The sample mean | is measured as the average of the daily return difference of the ETF
and the benchmark.
|
(25)
The calculation of the percentage spread is based on the day end bid and ask,
prices. It is assumed, that all the trading orders are executed at the best available prices,
indicated by the superscripted number one.
|
(26)
Chapter 6 Empirical Research
42
The sample annual average spread is the sum of all spreads divided by the amount of trading
days observed in the sample.
|
|
(27)
The sample TE calculates as the annualized daily standard deviation from the sample mean.
| √
|
√ (28)
In order to consistently calculate the VaR for all ETFs, the daily TE is annualized by being
multiplied with the factor √ As suggested by Hassine and Roncalli (2013), the number 260
depicts the standardized amount of trading days a year. Although only being an appropriate
procedure, when assuming Brownian motion in returns, it is a universally accepted rule to
annualize the daily TE by multiplying it with the square root of the amount of trading days
(Duffie & Pan, 1997).
6.4 Results for the ETFs on SMI
Table 2 presents the descriptive statistics of all sample variables as well as the result of the
efficiency measure for the ETFs on SMI.
Table 2: Results for the ETFs on SMI5
The summary annual statistics of the TD, the sample mean, the bid-ask spread, the TE and the efficiency measure are presented. The measures are calculated on a daily basis from May 2013 to May 2014, whereas the numbers depict annualized results. The statistics are rounded to two decimal places and measured in basis points (bps), where one percentage corresponds to 100 bps. (Source: Own calculations/ Illustrations)
ETF on SMI | | | | |
1# -30.39 0.67 15.34 78.15 -174.28
2# -112.12 -0.43 62.21 46.97 -251.59
3# -69.35 -0.26 10.92 39.04 -144.47
4# -43.30 -0.16 4.95 4.40 -55.49
5# -32.98 -0.12 6.94 7.95 -53.01
Average -57.63 -0.06 20.07 35.30 -135.77
5 The empirical data analysis and computation framework are held in Microsoft Excel and are enclosed to this thesis. If not stated otherwise, all subsequent results are measured in basis points. For all calculations the explanations laid out in subchapter 6.1 and 6.2 apply.
Chapter 6 Empirical Research
43
According to the efficiency measure for the ETFS on the SMI, ETF 5# is the most efficiently
tracking ETF amongst all funds. With fund number 5#, an investor has a 95% chance of not
losing an amount greater than 53 bps by the end of the study period. ETF 4# exhibits similar
results. Looking at the magnitude of the sample calculations, ETF 2# not only underperforms
his benchmark more than its peers, but comes at the highest bid-ask spread of 62.2 bps. The
evolution of the bid-ask spread of fund number 2# can be seen in Figure 16. In comparison to
the spread development of the other ETFs, fund 2# exhibits much higher and step-like spread
values. One of the potential reasons for this distortion could be the previously discussed
problem of taking day-end prices, instead of e.g. intra-day bid and ask prices.
Figure 16: Percentage Spread ETF 2#6 SMI
The percentage spread development for ETF 2# on SMI is illustrated. The observation period covers May 2013 to May 2014. (Source: Own calculations / illustrations)
The second reason for the large underperformance of fund number 2#, is the large TD of more
than 112 bps caused by three large negative spikes in on the February 27th 2014, March 6th,
2014 and April 14th 2014. These negative outliers are illustrated in the figure below.
Figure 17: Tracking Difference ETF 2# SMI
The TD development for ETF 2# on SMI is illustrated. The observation period covers May 2013 to May 2014. The red circle indicates the three negative outliers towards the end of the observation period. (Source: Own calculations / illustrations)
6 For graphical intra-ETF analysis it is important to bear in mind that scales of the vertical axis of the subsequent illustrations diverge across the exhibits. An overview of all graphs scaled consistently is given in Appendix 7 and 10.
0.20%0.30%0.40%0.50%0.60%0.70%0.80%0.90%1.00%
-0.25%-0.20%-0.15%-0.10%-0.05%0.00%0.05%0.10%0.15%
Chapter 6 Empirical Research
44
Interestingly, the other ETFs on SMI exhibit similar deviations in returns around the same
dates. The reason is, that as the SMI is a market capitalization-weighted index, the three big
weights Nestlé, Roche and Novartis each make out about 18-21% of the index. Whenever one
of these companies pays dividends, (Roche on March 4th and 5th, 2014; Novartis on February
25th, 2014; Nestlé on April 10th, 2014) TD as well as TE arise. Curiously, the synthetic replicating
ETF 2# seems not to be protected against distortions from dividend payments in the index. The
physically replicating ETFs number 4# and 5# adjust to the index dividend schedule by
distributing their dividends at the exact same dates. Presumably those dates are chosen in
order to avoid cash drag from the constituents’ dividends. Fund 1#, being a distributing and
physically replicating fund on the other hand exhibits large positive TD as he is benchmarked
against the SMI price index. In consequence fund number 1# exhibits the largest TE of all funds
considered, but also the most favorable TD characteristics as depicted in Figure 18.
Figure 18: Tracking Difference ETF 1# SMI
The TD development for ETF 1# on SMI is illustrated. The observation period covers May 2013 to May 2014. The red circle indicates the four positive outliers towards the end of the observation period. (Source: Own calculations / illustrations)
An example of the TD over the observation period without large outliers can be seen from
fund 5# in Figure 19. Due to the lack of large data outliers, the annual TE of fund 5# is the
second lowest. In consequence, the fund is rated the most efficient tracking fund out of the
ETF on SMI data sample.
-0.10%
0.00%
0.10%
0.20%
0.30%
0.40%
0.50%
Chapter 6 Empirical Research
45
Figure 19: Tracking Difference ETF 5# SMI
The TD development for ETF 5# on SMI is illustrated. The observation period covers May 2013 to May 2014. (Source: Own calculations / illustrations)
In conclusion, none of the ETFs on the SMI considered are able to outperform their benchmark
and attain a positive TD. ETF 1# was the only fund which attained a positive value for the
sample mean, indicating the large proportion of positive TE.
From the sample calculations, it becomes apparent that the TE has the biggest influence on
the efficiency measure and the ETF tracking performance. On average, the ETFs tracking the
SMI exhibited an annualized TE of roughly 35.3 bps. Nevertheless both TE and the bid-ask
spread play an important role by making out the difference amongst ETFs with similar TE
values.
6.5 Results for the ETFs on EURO STOXX 50
Largely consistent with the results reported by Hassine and Roncalli (2013) for the study period
of November 2011 to November 2012, the TD for all ETFs on EURO STOXX 50 suggest an
outperformance of the benchmark. The average TD takes a value of 43 bps. Fund number 7#
performs best by beating the benchmark by 73 bps over the course of one trading year. The
results for the ETFs on EURO STOXX 50 are reported in Table 3. Astonishingly, fund 7# does
expose an investor with a 95% chance of not gaining an amount less than 26.65 bps by the end
of the observation period. According to the efficiency measure, fund 1# is ranked the most
efficient ETF, while part of its excess-performance may be explained by securities lending. ETFs
4#, 5# and 7# profit from securities lending as well and consequently outperform the
benchmark. ETF 1# is ranked second by exhibiting the second lowest TE of all ETFs. Fund 4# is
the least favorable ETF by exposing an investor to a loss not greater than 102 bps. The reason
of the by far weakest performance is the low TD, large bid-ask spread and highest TE.
-0.03%-0.02%-0.02%-0.01%-0.01%0.00%0.01%0.01%0.02%
Chapter 6 Empirical Research
46
Table 3: Results for the ETFs on EURO STOXX 50
The summary annual statistics of the TD, the sample mean, the bid-ask spread, the TE and the efficiency measure are presented. The measures are calculated on a daily basis from May 2013 to May 2014, whereas the numbers depict annualized results. The statistics are rounded to two decimal places and measured in basis points (bps), where one percentage corresponds to 100 bps. (Source: Own calculations/ Illustrations)
ETF on EURO STOXX 50
| | | | |
1# 45.97 0.15 15.30 10.95 12.67
2# 41.76 0.14 59.75 9.95 -34.35
3# 12.86 0.23 25.86 12.10 -32.90
4# 17.87 0.24 52.73 41.20 -102.62
5# 67.28 0.22 13.82 32.45 0.09
6# 42.34 0.14 14.01 11.17 9.95
7# 73.01 0.24 15.68 18.71 26.55
Average 43.02 0.20 28.17 19.50 -17.23
Johnson et al. (2013) which investigated on the excess-performance of ETFs on EURO STOXX
50 find that much of the outperformance can be explained by the choice of the benchmark.
The reason is that the net return version of the index assumes certain withholding taxes on the
dividends paid, whereas most ETFs in fact achieve lower withholding tax rates in their country
of domicile. The authors furthermore report significant and higher than average performance
deviation around May of each year as a result of dividend optimization. Those findings are
consistent with most of the results obtained from the sample analysis. As it can be seen from
ETF 7# in Figure 20, the TD is especially volatile around the period of Mai 2013. With the
exception of ETF 4#, the other funds exhibited similar in TD.
Figure 20: Tracking Difference ETF 7# EURO STOXX 50
The TD development for ETF 1# on EURO STOXX 50 is illustrated. The observation period covers May 2013 to May 2014. The red circle indicates the higher volatility at the beginning of the observation period. (Source: Own calculations / illustrations)
Finally, the ETFs outperformance partially helps to explain the higher than average TE due to
the positive deviation from the benchmark. Overall, results show evidence that synthetic
replicating ETFs such as funds number 1#, 2# & 6# indeed produce lower TE than those using
-0.04%
-0.02%
0.00%
0.02%
0.04%
0.06%
0.08%
0.10%
0.12%
Chapter 6 Empirical Research
47
physical replication. In the sample average, physical replicating ETFs had a greater TE of about
43 bps.
With respect to the bid-ask spread, funds 2# and 4# show a particularly high spread. Fund 2#
hereby exhibits an inexplicable development in the spread, as the bid-ask spread level takes a
leap around September 05th and December 1st, 2013 as illustrated in Figure 21. Interestingly,
the development of the percentage spread does exhibit two breaks in the trend line at the
dates where no spread was quoted on Bloomberg.
Figure 21: Percentage Spread ETF 2# EURO STOXX 50
The percentage spread development for ETF 2# on EURO STOXX 50 is illustrated. The observation period covers May 2013 to May 2014. The red circles illustrate the inexplicable leap in the percentage spread levels. (Source: Own calculations / illustrations)
The illustration of the spread development of the other funds in the sample can be found in
appendix 10, whereas they do not exhibit such anomalies in the development of the spread.
The jumps and high spread levels raise the suspicion, that the data on the quoted spreads may
be incorrect. For the subsequent calculations it is important to bear in mind the potential bias
stemming from the exceptionally large spread of ETF 2#.
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
Chapter 6 Empirical Research
48
6.5.1 Information Ratio versus Efficiency Measure
This subchapter evaluates whether the ranking according to the previously discussed
Information Ratio corresponds to the ranking according to the efficiency measure. The bid-ask
spread is therefore added back to the efficiency measure, as it is not accounted for in the
Information Ratio. The sample results are reported in Table 4 and Table 5.
Table 4: Information Ratio ETF SMI Table 5: Information Ratio ETF EURO STOXX 50
The Information Ratio in absolute terms and the efficiency measure in bps are presented. (Source: Own calculations / illustrations)
The Information Ratio in absolute terms and the efficiency measure in bps are presented. (Source: Own calculations / illustrations)
ETF on SMI | |
# -0.39 -174.28
2# -2.39 -251.59
3# -1.78 -144.47
4# -9.83 -55.49
5# -4.15 -53.01
Average -3.71 -135.77
ETF on EURO STOXX 50
| |
1# 4.20 27.96
2# 4.20 25.40
3# 1.06 -7.04
4# 0.43 -49.89
5# 2.07 13.91
6# 3.79 23.96
7# 3.90 42.23
Average 2.81 -10.93
The ranking of the ETFs according to the Information Ratio is different than the ranking
according to the efficiency measure values. This highlights the drawback of the Information
Ratio in connection with passive fund evaluation, as it not account for the absolute levels of TE
and TD but only looks at their ratio. From the Information Ratio of ETF 2# and 3# on EURO
STOXX 50, it becomes evident that even though fund 3# performs worse than fund 2# by
exhibiting a higher TE, the ETF compensates by having a much larger TD. In conclusion, the
funds 2# is favored when looking at the Information Ratio, whereas fund 3# performs slightly
better when looking at the ETF efficiency.
In this chapter, the empirical evidence on the ETF efficiency measure has been presented for
the sample ETFs on the SMI and the EURO STOXX 50. As the sample calculations indicate
asymmetric in the TE distribution, the VaR method applied needs to be further evaluated. The
viability of the model and its underlying assumptions thus will be further evaluated in the
subsequent chapter.
Chapter 7 Adjustments to the Efficiency Measure
49
Chapter 7 Adjustments to the Efficiency Measure
Building on the basic framework presented in Chapter 6, this chapter enhances the
mathematical assumptions and calculation methods of the ETF efficiency measure by taking
into account a set of statistical considerations and adjustments. To begin with, the first
subchapter discusses pricing efficiency. Comparing alternative ways of calculating TE,
subchapter 7.2 evaluates on the one hand whether the results differ significantly across the
calculation methods and on the other hand how robust those results are to data outliers.
Additionally, the concept of semi-volatility is introduced. Finally, the second subchapter will
cut into potential errors arising from autocorrelation in TD. The third subchapter discusses
alternative approaches to measure the bid- ask spread. Adjusting the underlying assumptions
of the VaR framework, subchapter 7.4 applies alternative risk measures such as the historical
VaR, Cornish-Fisher VaR, intra-horizon VaR as well as the Expected Shortfall.
7.1 Pricing Efficiency
Before looking at potential adjustments of the factors and the framework of the efficiency, this
subsection will look at the previously discussed pricing efficiency. It evaluates, whether the
efficiency measure can be enhanced by including pricing efficiency of an ETF. As day end prices
tend to be the most volatile, the average daily pricing deviation is calculated as the
percentage difference of the midpoint spread and the NAV of the ETF. Table 6 and Table 7
present the sample calculations.
Table 6: Pricing Efficiency ETF SMI Table 7: Pricing Efficiency ETF EURO STOXX 50
The average daily pricing efficiency measured in bps is presented. (Source: Own calculations / illustrations)
The average daily pricing efficiency measured in bps is presented. (Source: Own calculations / illustrations)
ETF on SMI
1# 0.37
2# 0.60
3# -0.25
4# 0.10
5# 1.41
Average 0.45
ETF on EURO STOXX 50
1# -0.50
2# -2.89
3# 1.56
4# 0.98
5# 3.40
6# -1.54
7# 1.20
Average 0.32
Negative values for the pricing efficiency indicate that the NAV is larger than the midpoint
spread, whereas positive values suggest that the midpoint spread on average was higher.
Chapter 7 Adjustments to the Efficiency Measure
50
The results show that the daily prices do not deviate more than half a basis point from their
NAV. Only ETF 2# and 5# on EURO STOXX 50 deviate significantly, which indicates that the ETF
pricing and markets function efficiently. Furthermore, APs seem to generally step in, which
closes arbitrage opportunities from deviating prices.
Pricing efficiency will not be included in the efficient measure for three reasons. Firstly the
above values indicate that at normal market conditions, pricing deviations tend to be small
and within transaction costs. Secondly, due to the usage of day-end prices, it is uncertain that
the data on the bid and ask prices is representative. Finally, as the pricing efficiency highly
depends on the efficient functioning of the markets rather than on the ETFs tracking ability
itself, the measures will not be included.
7.2 Alternative Tracking Error Measures
Besides the computation method previously presented, there exist alternative ways to
calculate TE. The methods vary either on what kind of volatility is measured or how TE is
computed. In order to keep the various measures apart, the different TE measures will be
labeled with a subscript. TE from formula (28), based on the standard deviation of return
differences is referred to as . Measuring the same type of volatility, but with alternative
calculation methods, TE based on the correlations of returns will be denoted as and TE
based on the residuals of a linear regression is labeled as The robust measure considered is
based on the Median Absolute Deviation (MAD) and is labeled . Finally TE measured as
the standard deviation below a certain threshold will be referred to as semi-volatility .
7.2.1 TE – Based on Correlation of Returns
According to Hwang and Satchell (2001), the daily TE can be calculated based on the values of
covariance of the benchmark and the ETF returns as well as their variances
and .
| √
(29) The covariance can be further disseminated into the product of correlation and standard
deviations. The annualized sample TE is calculated based on the values of correlation of
ETF and benchmark returns as well as their variances and standard deviations .
| √
√ (30)
Chapter 7 Adjustments to the Efficiency Measure
51
Again the annualized TE is received by multiplying the daily TE with the factor √ . The
sample values for the alternative TE measure | can be seen in Table 8 and Table 9.
Table 8: Tracking Error
The summary statistics of the ETF and benchmark variances and the covariance of ETF and benchmark are reported in absolute values. The TE measures based on the standard deviation and the correlation of returns in bps are presented. (Source: Own calculations / illustrations) Panel A: ETF on SMI
ETF on SMI | |
1# 0.70515 0.70932 0.99834 78.31 78.15
2# 0.73433 0.73798 0.99942 47.07 46.97
3# 0.68211 0.70095 0.99967 39.12 39.03
4# 0.70088 0.70095 0.99999 4.41 4.40
5# 0.69968 0.70095 0.99998 7.97 7.95
Average 0.70 0.71 1.00 35.38 35.30
Panel B: ETF on EUTO STOXX 50
ETF on EURO STOXX 50
| |
1# 0.9390 0.9392 0.99998 10.97 10.95
2# 0.9485 0.9514 0.99998 9.97 9.95
3# 0.9649 0.9701 0.99997 12.12 12.10
4# 0.9176 0.9392 0.99971 41.28 41.20
5# 0.9358 0.9356 0.99978 32.51 32.45
6# 0.9383 0.9356 0.99997 11.20 11.17
7# 0.9340 0.9389 0.99993 18.75 18.71
Average 0.9407 0.9452 0.99990 19.85 19.50
From the tables above it can be seen that the correlation coefficient of ETF and
benchmark returns is approximately one for all ETFs, indicating an approximately perfect
correlation of their returns. When jointly plotting the ETF and benchmark returns, their
positive linear relationship becomes evident. Taking ETF 1# on SMI as an example, Figure 19
illustrates the match of ETF and benchmark returns. Except for the four previously discussed
outliers from dividend payment in the SMI, all returns lie perfectly aligned on the regression
line.
Chapter 7 Adjustments to the Efficiency Measure
52
Figure 22: Sample Regression with Data Outliers
The regression of ETF and benchmark returns for ETF 1# on SMI is presented. The vertical axis depicts the ETF returns and the horizontal axis depicts the index returns. The equation of and the coefficient of determination in the top left corner indicate that the ETF and benchmark returns fit the linear regression very well. The red circle indicates the four outliers in the data sample. (Source: Own calculation / illustrations)
Figure 23 exemplifies an approximately perfect correlation of 0.99997 between the ETF and
benchmark returns of fund 3# on EURO STOXX 50.
Figure 23: Sample Regression without Data Outliers
The regression of ETF and benchmark returns for ETF 1# on SMI is presented. The vertical axis depicts the ETF returns and the horizontal axis depicts the index returns. The equation of and the coefficient of determination in the top left corner indicate that the ETF and benchmark returns fit the linear regression very well. (Source: Own calculation / illustrations)
Looking at the difference between measure | and | , there is no significant
change in TE values associated with the alternative calculation method. Overall, the resulting
values for | differ from the original calculations for less than one bps only. As the
values of the efficiency indicator do not change significantly, the ranking of the ETFs is not
altered as well.
y = 0.9954x + 7E-05 R² = 0.9967
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
-0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
ET
F R
etu
rns
Index Returns
y = 0.9996x + 2E-05 R² = 0.9999
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
-0.04 -0.02 0 0.02 0.04
ET
F R
etr
un
s
Index Returns
Chapter 7 Adjustments to the Efficiency Measure
53
7.2.2 Tracking Error based on the Residuals of a Linear Regression
Using the concept of correlation alike, Treynor and Black (1973) measure the daily TE by
analyzing the regression of the return of the ETF on the return of the benchmark. The standard
deviations of the residuals of the linear regression are then used as an estimate of TE.
(31)
| (32)
| √ (33)
The daily TE of a portfolio can be computed from both formula (32) and (33) yielding the same
results. The annualized sample TE is calculated by multiplying the daily TE with the square root
of the standardized amount of trading days 260.
In Table 9 the sample statistics for the alternative TE based are listed. As expected, |
and | do return exact same TE values. Furthermore, strong evidence is found that
none of the three alternative calculation methods does significantly deviate. The ranking of the
ETFs according to the efficiency measure remains the same, independent of the TE measure
considered.
Table 9: Alternative TE ETF SMI
The summary statistics of the ETF TE measures based on the standard deviation, the correlation of returns and the residuals of a linear regression are presented. The final row depicts the alternative way of calculating the TE based on a linear regression. All results are reported in basis points. (Source: Own calculation / illustration)
Panel A: ETF on SMI
ETF on SMI | | | |
1# 78.15 78.31 78.06 78.06
2# 46.97 47.07 46.88 46.88
3# 39.04 39.12 34.35 34.35
4# 4.40 4.41 4.41 4.41
5# 7.95 7.97 7.87 7.87
Average 35.30 35.38 34.32 34.32
Chapter 7 Adjustments to the Efficiency Measure
54
Panel B: ETF on EURO STOXX 50
ETF on EURO STOXX 50
| | | |
1# 10.95 10.97 10.97 10.97
2# 9.95 9.97 9.68 9.68
3# 12.10 12.12 12.10 12.10
4# 41.20 41.28 36.98 36.98
5# 32.45 32.51 32.37 32.37
6# 11.17 11.20 11.17 11.17
7# 18.71 18.75 18.28 18.28
Average 19.50 19.54 19.79 18.79
7.2.3 Tracking Error based on Robust Measures
TD and TE are strongly affected by data outliers from e.g. diverging dividend payments
schedules, holidays, missing or misaligned data and rounding errors. Such data outliers may
inflate the TE of an ETF, despite its fairly good tracking of the benchmark. In the herein study,
data outliers are understood according to the three-sigma rule following Duda, Hart and Stork
(1997). The authors state, that in a normal distribution, 99.73% of all values lie within three
standard deviations of the sample mean. Table 10 and Table 11 indicate how many outliers
were found according to the three-sigma rule over the whole study period.
Table 10: Data Outliers ETF SMI Table 11: Data Outliers ETF EURO STOXX 50
The amount of data outliers in the data sample according to the three-sigma rules. (Source: Own calculations / illustrations)
The amount of data outliers in the data sample according to the three-sigma rules. (Source: Own calculations / illustrations)
ETF on SMI Number of
outliers
1# 4
2# 1
3# 1
4# 2
5# 1
Average 1.80
ETF on EURO STOXX 50
Number of outliers
1# 7
2# 8
3# 6
4# 2
5# 1
6# 3
7# 2
Average 4.14
Simply removing outliers may erroneously distort the TE test statistics. Additionally data
outliers may mask other deviants, which would fall under the three-sigma rule after the
removal of the initial outliers. Furthermore, accurate data outliers from e.g. different dividend
appointment should be included in the efficiency measure whenever they reflect the actual
ETF tracking.
Chapter 7 Adjustments to the Efficiency Measure
55
Instead of removing outliers from the data sample, a robust measure of TE is applied. The
measure allows to assess the TD volatility of an ETF in its essence. It furthermore may be a
better metric to draw long term inference about the TD volatility, not affected by singular
outlier points. For the herein data sample, it is expected that all ETFs will display improved
sample statistics, due to a reduction in TE.
Following the definition by Hwang and Satchell (2001), a way of measuring robust TE is based
on the Median Absolute Deviation (MAD). The resulting TE | is defined as the
median of the absolute deviations from the data’s median . The median is defined as the
middle value, separating the upper from the lower half of the empirical sample TD.
|
| |
(34)
Compared to the previous efficiency measure the is no longer calculated with respect
to the sample mean | but to the median in absolute terms. Data outliers drag the
mean towards them, away from the true center of the data. By arranging all observations
according to their magnitude and picking the middle value, the median is robust to data
outliers.
The results, illustrated in Table 12 and Table 13, indicate an overall decrease of TE based on
the calculations on MAD. On average, the switch to the robust measure improves the TE of the
ETFs on SMI for approximately 30 bps.
Table 12: Robust TE ETF SMI
The summary statistics of the TE measures based on the standard deviation and on MAD as well as the corresponding efficiency measures are presented for the ETFs on SMI. (Source: Own illustrations/ calculations)
ETF on SMI | | | |
1# 78.15 20.74 -174.28 -79.85
2# 46.97 13.47 -251.59 -196.48
3# 39.04 18.06 -144.47 -109.96
4# 4.40 2.07 -55.49 -51.66
5# 7.95 6.14 -53.01 -50.02
Average 35.30 12.09 -135.77 -98.08
Nevertheless, the reduction in TE did not affect all ETFs alike. ETF 5# is still the most favorable
fund with respect to the TE as well as the corresponding efficiency indicator | . ETF 4#
and ETF 1# become the second and third most efficient fund. As suggested by the amount of
Chapter 7 Adjustments to the Efficiency Measure
56
data outliers, the TE of ETF 1# is reduced by more than 70%. Fund number 3#, previously
having a lower VaR of roughly 30bps than fund number 1#, dismounts in the rating.
Table 13: Robust TE ETF EURO STOXX 50
The summary statistics of the TE measures based on the standard deviation and on MAD as well as the corresponding efficiency measures are presented for the ETFs on EURO STOXX 50. (Source: Own illustrations/ calculations)
ETF on EURO STOXX 50
| | | |
1# 10.95 4.63 12.67 23.06
2# 9.95 4.52 -34.55 -25.62
3# 12.10 6.30 -32.90 -23.36
4# 41.20 27.51 -102.62 -80.11
5# 32.45 6.66 0.09 42.50
6# 11.17 3.72 9.95 22.20
7# 18.71 8.72 26.55 42.99
Average 19.503 8.87 -17.26 0.24
Similarly, the rankings for the ETFs on EURO STOXX 50 change. On average, the robust
measure | reduces TE of all funds for about 11 bps. Fund 5# profits the most from
an improvement in the TE and is now ranked the second most efficient fund, only half a bps
behind fund 7#. Although ETF 5# only exhibits one data outlier, it can be seen from the
distribution of TE in Figure 24 that the magnitude of the outlier causes the mean as well as the
TE to inflate. This exemplifies that the robust TE measure based on MAD does not only account
for the amount, but the magnitude of outliers as well.
Figure 24: Tracking Difference ETF 5# EURO STOXX 50
The TD development for ETF 5# on EURO STOXX 50 is illustrated. The observation period covers May 2013 to May 2014. (Source: Own calculations / illustrations)
-0.05%
0.00%
0.05%
0.10%
0.15%
0.20%
0.25%
0.30%
0.35%
Chapter 7 Adjustments to the Efficiency Measure
57
In order to supplement the intuition of the results received from | and to get a
feeling about the symmetry of the TD distribution, a complementary robust measure called the
Interquartile Range (IQR) is introduced. The IQR measures the difference between the 75th
percentile and the 25th percentile of the empirical TD data sample. Therefore it captures 50%
of the distribution of the TD.
(35)
The larger the IQR, the more spread is the TD. IQR can be regarded as a nonparametric
equivalent to the standard deviation (Hyndman & Fan, 1996). However, the IQR only considers
half of the data sample and therefore will not be included in the design of the efficiency
measure. The resulting IQR values are reported in Table 14 and Table 15 and are compared
with the | values.
Table 14: IQR ETF SMI Table 15: IQR ETF EURO STOXX 50
The IQR in absolute terms and the efficiency measure measured based on MAD are presented. (Source: Own calculations / illustrations)
The IQR in absolute terms and the efficiency measure measured based on MAD are presented (Source: Own calculations / illustrations)
ETF on SMI |
1# 1.28 0.97
2# 0.84 0.78
3# 1.12 1.14
4# 0.13 0.17
5# 0.38 0.60 Average 0.75 0.73
ETF on EURO STOXX 50
|
1# 0.29 0.17
2# 0.28 0.17
3# 0.39 0.32
4# 1.71 2.55
5# 0.41 0.28
6# 0.23 0.06
7# 0.54 0.47
Average 0.55 0.57
Interestingly, the ranking of the ETFs based on MAD and the IQR are not entirely consistent for
the ETFs on SMI. Furthermore, a lower | does not necessarily indicate a lower IQR.
For the ETFs on EURO STOXX 50 however, the ranking according to the robust TE measure
correspond to the ranking according to the IQR.
For a symmetric distribution of TD, | should be equal to half of the IQR. Since this
property is not given for any of the ETFs, it is expected that they exhibit an asymmetric
distribution in TD. The asymmetry property will be accounted for the subsequent chapters.
7.2.4 Tracking Error based on Semi-Variance
Following the suggestions of Markowitz (1959), Fishburn (1977), Harlow (1991) and Estrada
(2007), Hassine and Roncalli (2013) suggest measuring risk by using semi-variance, instead
Chapter 7 Adjustments to the Efficiency Measure
58
variance. The underlying principle of semi-variance questions the usage of variance as a risk
measure, since it depends on both positive and negative values of TD. The investor however
might only be interested in the negative deviations from a certain threshold. In the traditional
setting of the efficiency measure, positive deviations are punished by a higher TE and may
results in a lower efficiency measure score. Accounting for only the negative deviations from
e.g. the mean or zero, semi volatility is defined as a special case of lower partial moments. The
mathematical derivation following Bawa (1975) as well as Casella and Berger (2001) is given in
appendix 8.
Contrary to the application of the mean suggested by Hassine and Roncalli (2013), the median
is assumed to be a more suitable threshold in this thesis. The reason is the previously
discussed robustness of the median to outliers. The corresponding formula for the semi-
variance with respect to the median is illustrated by formula (36). It calculates the expected
variance by looking at the negative deviations from the median only.
| [ ] (36)
The semi-variance in relation to the threshold of zero calculates identically.
| [ ] (37)
The semi-standard deviation with respect to the median becomes the square root of the semi-
variance. In order to illustrate the intuition of the efficiency measure graphically Figure 25
presents the TD captured by the semi-variance measure.
Figure 25: Tracking Difference ETF 3# EURO STOXX 50
The TD development for ETF 3# on EURO STOXX 50 is illustrated. The observation period covers May 2013 to May 2014. The dotted line illustrated all the TD values not accounted for by the semi-variance TE measure and the green line depicts the threshold zero. (Source: Own calculations / illustrations)
-0.02%
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
Chapter 7 Adjustments to the Efficiency Measure
59
With respect to the adjustment of the efficiency measure, Hassine and Roncalli (2013) indicate
that the ratio of the standard deviation and semi standard deviation is equal to √ , whenever
the distribution of TD is symmetric around the threshold.
√ (38)
The modified ETF efficiency measure based on the semi-standard deviation becomes:
| | | √ | (39)
The results for the ETFs on SMI for both semi-variance with respect to the median and zero are
reported in Table 16. In order to compare the results to the initial TE measures, | and
| are listed as well.
Table 16: Semi-Variance ETF SMI
The TE and the corresponding efficiency measure based on the standard deviation, the semi-variance with threshold median and the semi-variance with threshold zero are presented. (Source: Own calculations / illustrations)
threshold = median threshold = zero
ETF on SMI
| | | | ) |
|
1# 78.15 -174.28 6.80 -61.57 7.29 -62.69
2# 46.97 -251.59 43.88 -276.40 44.10 -276.91
3# 39.04 -144.47 21.21 -129.61 22.26 -132.04
4# 4.40 -55.49 3.60 -56.61 3.80 -57.09
5# 7.95 -53.01 4.73 -50.93 5.36 -52.40 Average 35.30 -135.77 16.48 -115.02 16.56 -116.23
Looking at the magnitude of the results, the semi TE measure generally reduces the TE values
for ETF 1#, 3# and 5#, whereas it increases the efficiency indicator for ETF 2#, 4# on SMI. Apart
from the move of ETF 5#, the relative rankings for the ETFs based on the semi standard
deviation do not change. Furthermore, the rankings do not depend on whether the median or
zero is selected as the threshold. Since ETF 2#, 3# 4# and 5# exhibit a negative median, they
reveal lower TE at the median as compared to the threshold zero. Thus, the efficiency measure
for ETF 2# and 4# deteriorate given that multiplying the semi-standard deviation with √
penalizes negative outliers over-proportionally.
Fund 1#, having four large positive TD values at the end of the study period, profits the most
from the change to the downside risk measure. The fund now is ranked the third most efficient
Chapter 7 Adjustments to the Efficiency Measure
60
ETF as the four outliers are not captured by the semi-volatility. ETF 2# on the other hand,
exhibits three large negative TD values, which are included in the semi-variance measure and
thus the ETF keeps his lowest ranking.
Generally, the results from the ETFs on SMI suggest applying the semi-volatility efficiency
measure carefully. Whereas the concept of downside risk may be suitable in e.g. alternative
pricing models based on downside beta (Estrada, 2007), it is less suitable in the context of ETF
tracking efficiency measurement. The short time horizon as well as the persistence of data
outliers makes the application of VaR based on semi-variance cumbersome and potentially
misleading.
Turning to the results for the ETFs on EURO STOXX 50, Table17 lists the sample statistics for
the efficiency measure based on the semi standard deviation with threshold zero and the
median, as well as the initial volatility measure | .
Table17: Semi-Variance ETF EURO STOXX 50
The TE and the corresponding efficiency measure based on the standard deviation, the semi-variance with threshold median and the semi-variance with threshold zero are presented. (Source: Own calculations / illustrations)
threshold = median threshold = zero
ETF on EURO
STOXX 50 | | |
| | |
1# 10.95 12.67 1.06 28.20 1.30 27.64
2# 9.95 -34.55 2.00 -22.85 2.09 -23.06
3# 12.10 -32.90 2.05 -17.77 2.01 -17.66
4# 41.20 -102.62 19.40 -79.98 18.86 -78.73
5# 32.45 0.09 1.86 49.13 2.07 48.65
6# 11.17 9.95 1.71 24.35 1.77 24.22
7# 18.71 26.55 3.95 48.14 4.12 47.76 Average 19.50 -17.26 4.58 28.20 4.60 4.12
With the new TE measure, the ranking for the ETFs change due to a large improvement of fund
5# whereas the relative position of the other funds remain the same. The results show that
ETF 5# now ranks as the most efficient ETF, followed by fund 7# and 1#. Fund 5# exhibits a
large positive TD on January 09th, 2014 and thus profits from an improved TE. Overall, all funds
reach lower TE values with the semi-variance measure. The choice of the threshold does not
influence the ranking of the ETFs. Since all ETFs reveal positive outliers only, the efficiency
measures | improves for all sample ETFs.
Chapter 7 Adjustments to the Efficiency Measure
61
In addition to the previously mentioned drawbacks of the TE based on semi-volatility, investors
concerned about the tracking ability of the ETF may dislike both the negative as well as the
positive deviations from the benchmark. For those investors, the efficiency measure based on
the semi-volatility is an inaccurate risk concept, especially if TD is not centered on the
threshold.
7.2.1 Autocorrelation
To conclude the section about alternative TE measures, it is tested whether the return
difference of benchmark and ETFs are subject to serial correlation, also called autocorrelation.
However, it is not the purpose of this study to statistically correct the efficiency measure for
autocorrelation. Instead, it should give an indication of how to interpret the empirical results
in the prevalence of autocorrelation.
Autocorrelation refers to the correlation of a value with its own past or future value. It is
computed based on the general correlation function. Autocorrelation is measured as the
correlation of TD between two dates, TD at time and TD at time , where is referred
to as the number of lagged days tested for. In the herein sample, a total of 25 lags are
considered in order to incorporate the time of a full trading month.
(40)
According to Chatfield (2004), if the time series is random and the sample size is large, the
autocorrelation coefficients are approximately normal distributed with mean zero and
variance , where depicts the amount of trading days. The null hypothesis of no
autocorrelation is rejected at the 95% level whenever exhibits values above (below)
the upper (lower) test statistics, given by
√ (41)
Although formula (41) gives a rough estimation of the true statistics only, it is sufficient to give
an indication about the prevalence of autocorrelation in the ETF data sample.
Out of the whole ETF sample, all ETFs except for fund 5# on EURO STOXX 50 and fund 3# on
SMI exhibit significant autocorrelation. Figure 26 illustrates the different autocorrelation
Chapter 7 Adjustments to the Efficiency Measure
62
values for ETF 3# on SMI as well as the upper and lower critical values. The values in the graph
suggest that the null hypothesis of no autocorrelation cannot be rejected.
Figure 26: Autocorrelation Function ETF 3# SMI
The autocorrelation function for the first 25 lags for the ETF 3# on SMI is illustrated. The dotted red lines correspond to the test statistics at which the null hypothesis of no autocorrelation is rejected. (Source: Own calculations / illustrations)
Figure 27 gives the autocorrelation value for ETF 5# on SMI for which the null hypothesis of no
autocorrelation is rejected at the first seven lags. This indicates that the TD values from up to
seven days ago, influence present values and thus bias the results. The complementary graphs
for all ETFs on SMI and EURO STOXX 50 are presented in appendix 9.
Figure 27: Autocorrelation Function ETF 1# EURO STOXX 50
The autocorrelation function for the first 25 lags for the ETF 3# on SMI is illustrated. The dotted red lines correspond to the test statistics at which the null hypothesis of no autocorrelation is rejected. (Source: Own calculations / illustrations)
Pope and Yadav (1994) show that autocorrelation in TE can bias the estimate of TE upwards.
The authors argue that delayed adjustments of prices due to changes of the investors’
expectations and obligations by APs to stabilized prices, will lead to positive serial correlation.
Although the sources for autocorrelation are difficult to identify in this data sample, it is
furthermore assumed that autocorrelation is caused by lagged portfolio adjustments lagged
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
Chapter 7 Adjustments to the Efficiency Measure
63
dividend payment schedules of the ETF. In order to receive unbiased results, the so called
heteroskedasticity and autocorrelation consistent estimations of covariance matrices
suggested by Andrews (1991) or the long run variance suggested by Phillips (1991) may be
applied.
7.3 Alternative Bid-Ask Spread Measurement
This subsection evaluates an alternative method of measuring the bid-ask spread | . The
discussion provides the mathematical framework and reasoning behind the adjusted spread
measure.
Roncalli and Zheng (2013) as well as Flood (2010) argue that using intraday prices instead of
day-end prices is more relevant in the context of ETFs, as they tend to be less volatile.
Furthermore the previously discussed bid-ask spread assumes all trades to be executed at the
first limit order, meaning the best spread available. As Hassine and Roncalli (2013) indicate,
this may be an appropriate assumption for retail investors, since they generally trade smaller
amounts. For large institutional investors however, trading large notional in ETFs may result in
an increase in the spread, especially when liquidity in the ETF is limited. Therefore, using day
end prices for the first limit order may not reflect the true trading conditions. Hassine and
Roncalli (2013) in consequence weight each spread by the number of trades executed at that
spread. They compute the intraday spreads weighted by the duration between two ticks for a
given notional, where a tick is defined as the minimum change in price of the ETF.
|
) (42)
In formula (44), depicts the spread of the tick in order to trade a notional N and
is the elapsed time between two consecutive ticks. The ETF efficiency measure
adjusts according to the distribution of the spread .
| | | (43)
Hassine and Roncalli (2013) report, that depending on the notional traded, ranking for their
ETFs evaluated changes. However, in order to empirically calculate the adjusted spread
measure, high frequency data on the spreads at the microsecond level in combination with
changes in the limit order book of APs and the amount of trades is needed. Although
Chapter 7 Adjustments to the Efficiency Measure
64
Bloomberg does provide historical intraday data going back six month, the required data is not
available for the whole observation period. Furthermore, the processing of the data requires
large computational capacity. For example, the intraday data for fund 4# on SMI counts
486’892 price data points in April 2014 only. Analyzing the full trading year in order to receive
comprehensive and significant results would require processing roughly 5.8 million bid and ask
prices. In consequence, the adjusted spread measure is not empirically calculated in this
thesis. Exemplary calculations and statistical analysis are explained in Roncalli and Zheng
(2014), Degryse, De Jong and Van Kervel (2011) as well as Hassine and Roncalli (2013).
Instead of calculating the adjusted spread measure, this thesis looks the influence spread on to
the ETF ranking. Table 18 compares the initial spread and efficiency measure to the efficiency
measure where the spread was calculated based on the median instead of the daily average
spread. Additionally the efficiency indicator is calculated without including the bid-ask spread.
As discussed previously, certain investors are able to trade ETF shares OTC and thus do not pay
the full bid-ask spread. For an investor trading on the primary markets, the spread becomes
irrelevant and should be neglected in the ETF ranking.
Table 18: Adjusted Spread ETF SMI
The summary statistics of the efficiency measure incorporating the initial spread measure, the spread measure based on the median and no spread are presented. (Source: Own calculations / illustrations)
median No Spread
ETF on SMI | | | | |
1# 15.34 -174.28 7.45 -166.39 -158.94
2# 62.21 -251.59 64.34 -253.73 -189.39
3# 10.92 -144.47 9.02 -142.58 -133.56
4# 4.95 -55.49 4.80 -55.34 -50.54
5# 6.94 -53.01 6.15 -52.21 -46.06
Average 20.07 -135.77 18.35 -134.05 -115.70
This study does not reveal an altered ranking for the ETFs on SMI, neither when the median
spread is included nor when the spread is omitted completely. Except for fund 1#, the spread
values based on the mean or on the robust median do not indicate the prevalence of data
outliers as they are alike. Even though ETF 2# does exhibit an average spread which is almost
four times as high as the second highest average spread, the ETF furthermore exhibits the
largest negative TD and large TE. In consequence it is still ranked the least efficient tracker,
even when omitting the spread.
Table 19 illustrates the results for the ETF on EURO STOXX 50.
Chapter 7 Adjustments to the Efficiency Measure
65
Table 19: Adjusted Spread ETF EURO STOXX 50
The summary statistics of the efficiency measure incorporating the initial spread measure, the spread measure based on the median and no spread are presented. (Source: Own calculations / illustrations)
threshold = median No Spread
ETF on EURO STOXX 50
| | | | |
1# 15.30 12.67 11.52 16.45 27.96
2# 59.95 -34.55 49.22 -23.81 25.40
3# 25.86 -32.90 16.26 -23.30 -7.04
4# 52.73 -102.62 56.17 -106.06 -49.89
5# 13.82 0.09 14.09 -0.18 13.91
6# 14.01 9.95 12.87 11.09 23.96
7# 15.68 26.55 14.21 28.02 42.23 Average 28.19 -17.26 24.91 -13.97 10.93
For those ETFs, excluding the bid-ask spread changes the rankings. Fund 2#, previously having
the largest bid-ask spread of all ETFs, now is ranked the third most efficient ETF. Even though
fund 4# exhibits a large bid-ask spread as well, the other risk metrics for ETF 4# suggest inferior
tracking. Therefore the ETF does not improve its ranking.
Taking the median spread, does not alter the ranking according to the efficiency measure. The
similarity of the mean and the median spread in the data sample indicates that the distribution
of daily spreads is approximately symmetrical.
To conclude this section about the importance of the spread, it is to say that this thesis
assumes one trade throughout the whole observation period. It therefore assumes that the
ETF is bought at the beginning and sold at the end of the observation period. However, the
bid-ask spread has to be paid each time a full trade cycle is accomplished. A cycle comprises
buying and selling the ETF and is depicted by the factor in the below formula.
| | | | (44)
Despite playing a minor role in the initial efficiency measure setting, the spread measure may
become the most important cost factor for a high frequency ETF trader. Hassine and Roncalli
(2013) therefore suggest that in the limit, the investor is only interested in the spread. TD and
TE become less relevant the more often an investor trades the ETF on the secondary market.
| | (45)
Chapter 7 Adjustments to the Efficiency Measure
66
7.4 Alternative Value-at-Risk Measures
The underlying assumptions of the VaR framework have a significant influence on which
possibility of loss is essentially accounted for. This subsection extends the mathematical
assumptions of the basic VaR framework with four alternative measurement methods of VaR.
Each method takes a different point of view of VaR and eliminates some of the drawbacks of
the other methods considered. The discussion starts with assessing the so-called delta-normal
VaR. The initial efficiency measure makes use of the delta-normal VaR, proposed by Jorion
(2007, pp.249-251). In this method, the calculations are based on the assumption that the TD
are normally distributed | | with mean | and
variance | .
The empirical distributions of TD as well as the figures for the IQ and the median suggest that
the normality assumption most likely does not hold. Out of the entire ETF sample, only fund 5#
on SMI exhibits an approximate normal distribution. Figure 28 illustrates the relative TD
distribution in comparison to an arbitrary normal distribution with the same mean and
variance.
Figure 28: Relative Tracking Difference Distribution
The relative frequency distribution of TD (blue bars) for the ETF 5# on SMI and the corresponding normal distribution (red line) are presented. The vertical axis depicts the relative frequency of TD and the horizontal axis depicts the absolute TD values. (Source: Own calculations / illustrations)
The sample distribution of TD from all other funds show clear signs non-normality. This is a
severe issue, as the delta-normal VaR based on a normal distribution is likely to misestimate
the true potential loss on the empirical distribution. Furthermore, the efficiency measure
0
0.01
0.02
0.03
0.04
0.05
0.06
Chapter 7 Adjustments to the Efficiency Measure
67
based on VaR underestimated the potential loss, as it measures the percentile, but not the
property of the tail below the confidence interval. In order to statistically test whether the
normality assumption holds, the third and fourth moments as described in appendix 8 are
examined. Being commonly referred to as the skewness of a distribution, the sample statistics
based on the third moment | is calculated according to Bai and Ng
(2005).
| [ ]
[ ]
|
| (46)
The skewness measures how symmetric the distribution is around the mean. It can take both
positive and negative values. A negative skew indicates that the tail on the left side of a
distribution is longer whereas a positive skew suggests that the distribution is tailed to the
right. In order for the normality assumption to hold, the values of the skewness should be
close to 0 (Bai and Ng, 2005).
The fourth moment | is used to calculate the kurtosis of a distribution.
| [ ]
[ ]
|
| (47)
Formula (47) shows how peaked the distribution TD is and furthermore indicates the heaviness
of the tails in the distribution. Even though the interpretation of the absolute kurtosis values is
rather cumbersome, higher values indicate a more pointed and fat tailed distribution. As the
normal distribution exhibits a kurtosis of three, the above formula often is reduced by three to
receive the excess kurtosis | . For the normality assumption to hold, | should
take values close to 0 (Bai & Ng, 2005).
In order to statistically test whether the normality assumption holds for the ETF sample, the
Jarque-Bera-Test is applied. The test statistics by Jarque and Bera (1987) is based on the
skewness | , excess kurtosis | as well as the sample size .
|
( | )
(48)
Chapter 7 Adjustments to the Efficiency Measure
68
The Jarque-Bera test statistics has a chi-square distribution with two degrees of
freedom. The corresponding null hypothesis suggests, that the sample distribution is normally
distributed. The alternative hypothesis states that the TD are not normally distributed. The
quantile of the chi-square distribution at the 95% probability level takes the value of 5.99. For
values of above 5.99, the null hypothesis is rejected at the 95% significance level. Large
values indicate that the null hypothesis of normal distribution in TD can be rejected at even
higher probability levels. Table 20 presents the relevant sample statistics for the ETFs on SMI.
Table 20: Normality Test ETF SMI
The summary statistics of the skewness, the kurtosis and the Jarque-Bera test statistics for the ETFs on SMI are presented. (Source: Own calculations / illustrations)
ETF on SMI | |
1# 7.36 57.69 1969993.28
2# -6.60 57.97 1958435.53
3# 4.03 55.11 1729827.64
4# -2.57 30.91 305387.57
5# -0.19 0.47 2.54
Average 0.41 40.43 1192729.31
It becomes evident that except for ETF 5# on SMI, for all funds the null hypothesis of normal
distribution is rejected at the 95% probability level. Furthermore it can be seen, that fund
number 1# and 3# exhibit positive skewness, suggesting that the distribution of TD is tailed to
the right. Consequently, the delta normal VaR may in fact overestimates the risk for those two
funds, whereas it underestimates the risk for the funds 2#, 4# and 5#. The large excess kurtosis
suggests a leptokurtic distribution, meaning a strong peak around the mean and fat tails.
Those fat tails suggest that extreme TD are likely to be milder in the normal compared to the
fat-tailed distributions.
The intuition of a negative skew (-2.57) and a large excess kurtosis (30.91) can be seen from
ETF 4# in Figure 29. It is assumed that the large negative outliers inflate the kurtosis whereas
more TD on the left of the sample mean cause negative skewness.
Chapter 7 Adjustments to the Efficiency Measure
69
Figure 29: Absolute Tracking Difference Distribution
In order to illustrate skewness and kurtosis, the absolute frequency distribution of TD for the ETF 4# on SMI is presented. The vertical axis depicts the absolute frequency of TD and the horizontal axis depicts the TD values. (Source: Own calculations / illustrations)
The ETFs on EURO STOXX 50 all display a positive skewness, meaning that the TD is skewed to
the left. This is in favor of the investor as it indicates the prevalence of more positive TD on the
right side of the sample mean. Those results are consistent with the findings by Hassine and
Roncalli (2013) who find high skewness in their data sample. The null hypothesis of normally
distributed TDs can be rejected for all funds according to the Jarque-Bera Test.
Table 21: Normality Test ETF EURO STOXX 50
The summary statistics of the skewness, the kurtosis and the Jarque-Bera Test for the ETFs on EURO STOXX 50 are presented. (Source: Own calculations / illustrations)
ETF on EURO STOXX 50
| |
1# 4.49 27.41 213717.53
2# 4.33 28.92 244629.92
3# 2.96 11.67 16531.69
4# 2.55 18.73 68121.69
5# 14.20 214.92 102596534.30
6# 6.93 69.00 3396863.42
7# 5.02 36.98 519455.95
Average 5.91 61.78 17756066.42
Together these results provide important insight on the ETFs TD distribution, suggesting that
the distribution of TD is non-normal. The next section discusses an alternative method of VaR
which does relax the statistical assumption of normality.
0
5
10
15
20
25
30
Chapter 7 Adjustments to the Efficiency Measure
70
7.4.1 Cornish-Fisher Value-at-Risk
Cornish and Fisher (1937) propose an extension of the delta-normal VaR which approximates
the percentiles of a normal distribution function of TD adjusted to its primary four moments,
TD | , TE | , skewness | and excess kurtosis | .
The Cornish-Fisher expansion allows the assessment of the VaR even if the normality
assumption is violated. The adjusted factor hereby replaces the previously applied inverse
of the standard normal distribution function and is calculated as below:
|
( ) |
( ) | (49)
As we are consistently interested in the 95% percentile, takes the value 1.645. From
formula (49) it can be seen that whenever the skewness as well as the excess kurtosis takes
the value zero, as it is the case in the normal distribution, collapses to .
Given and the standard deviation of TD | , the efficiency indicator then adjust as
depicted below.
| | | | (50)
In Table 22 the Cornish-Fisher expansion is used to compute the efficiency measure | .
Counterintuitively, the measure indicates an advantage to fund 2# and 4# which both
exhibited negative sample skewness and high kurtosis, indicating that the tail on the loss side
is longer. Fund 4# now even is regarded to be the best tracker, raising suspicion that the
expansion may not be suitable for distributions with great skewness and kurtosis.
Table 22: Cornish-Fisher VaR ETF SMI
The summary statistics for the sample skewness, kurtosis and the Cornish-Fisher factor as well as the corresponding efficiency measure based on the Cornish-Fisher expansion are presented. The last row lists the efficiency measure based on the delta-normal VaR. (Source: Own calculations / illustrations)
ETF on SMI | | | |
1# 7.36 57.69 1.56 -167.29 -174.28
2# -6.60 57.97 -2.22 -70.01 -251.59
3# 4.03 55.11 1.37 -133.87 -144.47
4# -2.58 30.91 0.16 -48.97 -55.49
5# -0.19 0.47 1.58 -52.50 -53.01
Average 0.40 40.43 0.49 -94.52 -135.77
Chapter 7 Adjustments to the Efficiency Measure
71
The most surprising aspect of the data is that even though large kurtosis values suggested the
existence of fat tails, the Cornish-Fisher expansion translates into an improvement in the
efficiency measure score | score even for the funds with negative skewness. For
example, the large negative skewness and the high kurtosis resulted in a negative value for
ETF 2# on SMI.
Table 23: Cornish-Fisher VaR ETF EURO STOXX 50
The summary statistics for the sample skewness, kurtosis and the Cornish-Fisher factor as well as the corresponding efficiency measure based on the Cornish-Fisher expansion are presented. The last row lists the efficiency measure based on the delta-normal VaR. (Source: Own calculations / illustrations)
ETF on EURO STOXX 50
| | | |
1# 4.49 27.41 1.99 8.90 12.67
2# 4.33 28.92 1.94 -37.48 -34.55
3# 2.96 11.67 2.09 -38.23 -32.90
4# 2.55 18.73 1.87 -111.84 -102.62
5# 14.20 214.92 -2.44 132.71 0.09
6# 6.93 69.00 1.32 13.58 9.95
7# 5.02 36.98 1.85 22.67 26.55
Average 5.91 61.78 1.13 1.04 -24.56
For the ETFs on EURO STOXX 50, the results seem somewhat more consistent with the theory
on the Cornish-Fisher VaR. Even though funds exhibit positive skewness, they suffer from the
large kurtosis values and in consequence inferior efficiency measure values. Nevertheless, the
results suggest that those implausibly large kurtosis values positively influence the efficiency
measure values. ETF 5# exhibits the most extreme and dubious result by exhibiting negative
value and improving his ranking to the first position.
The empirical results not only lay in strong contrast to the findings by Hassine and Roncalli
(2013), which found nearly consistent rankings for both, the delta normal as well as the
Cornish-Fisher efficiency measure, but they suggest that the Cornish-Fisher expansion fails
when the TD distribution is strongly non-normal. Jaschke (2002) confirms this suspicion and
points out that monotonicity and the convergence are not certain for the Cornish-Fisher
expansion. The author points out that the approximation is only suitable when the returns are
sufficiently close to being normal. Maillard (2012) further calculates the domain of validity of
the transformation which is | | (√ ) | | (√ ) . Except
for ETF 5# on SMI, none of the sample ETFs fall within domain of validity.
The fact that neither the skewness nor the kurtosis is a robust measure may be the reason why
such high values are found in the empirical analysis. Even though the Cornish-Fisher approach
Chapter 7 Adjustments to the Efficiency Measure
72
adjusts for the shape of the distribution of TE, it may actually fail to estimate the true VaR in
the existence of data outliers. A solution to this problem would be to use robust skewness and
kurtosis measures as discussed by Moors (1988).Another alternative is to consider the
historical VaR which does estimate the ETF risk from the empirical distribution of TD.
7.4.1 Historical Value-at-Risk
The historical simulation of the VaR is a remedy to the delta-normal and the Cornish-Fisher
method, as it makes no specific assumption about the distribution of TD, but draws samples
from historical data (Jorion, 2007, p.253). The historical VaR uses empirical percentiles from
the observation period by arranging the historical TD over the last trading days according to
their size. The value representing the -% percentile will be the corresponding historical
VaR. Following Hassine and Roncalli (2013) the efficiency measure based on the historical VaR
is defined as follows, whereas depicts the historical probability distribution function of
centered TD.
| | | (51)
The approach generally requires a larger data sample than the delta normal method in order
to return significant results. A 99% daily VaR estimated over a time period of 100 days only
produces one observation in the tail. For the ETF 3# on SMI for example, the observation
period counts 252 observations, which means that the 95% percentile is the 12.4th largest loss
observed in the data sample.
The efficiency measures for the ETFs on SMI derived from the delta-normal as well as the
historical VaR are reported in Table 24 and for the ETF on EURO STOXX 50 in Table 25.
Table 24: Historical VaR ETF SMI Table 25: Historical VaR ETF EURO STOXX 50
The efficiency measure based on the delta-normal VaR and the historical VaR are presented. (Source: Own calculation / illustration)
The efficiency measure based on the delta-normal VaR and the historical VaR are presented. (Source: Own calculation / illustration)
ETF on SMI |
1# -174.28 -64.59
2# -251.59 -193.48
3# -144.47 -121.68
4# -55.49 -54.54
5# -53.01 -55.05
Average -135.77 -98.02
ETF on EURO STOXX 50
|
1# 12.67 26.84
2# -34.55 -24.26
3# -32.90 -17.72
4# -102.62 -85.67
5# 0.09 47.83
6# 9.95 26.00
7# 26.55 48.53 Average -24.56 -5.18
Chapter 7 Adjustments to the Efficiency Measure
73
For the ETFs on SMI especially fund 1# improves his efficiency measure values. The ETF profits
from a low negative TD values at the 95% percentile and now is ranked the third most efficient
ETF. Except for fund 5#, all ETF exhibit better efficiency measure values. This suggests that the
VaR based on a normal distribution overestimates the risks in the ETFs.
For the ETFs on EURO STOXX 50, fund 7# remains the most efficient ETF. Compared to the
measurement based on the delta-normal method, all ETFs profit from improved measures.
Especially ETF 5# improved his ranking and is now the second most efficient ETF. The low
ranked funds 2#, 3# and 4# remain to be the worst performing funds.
From the above described method it can be seen that some issues of historical VaR remain.
Firstly, historical VaR does not indicate how large the losses are below the 95% percentile.
Secondly, historical VaR generally assumes that returns are independent and identically
distributed (iid), meaning that every TD has the same probability distribution and that they are
mutually independent. Historical VaR thus does not account for time-varying volatility. Diebold
et al. (1998) show that high frequency and daily returns are often not iid. The shortcoming is
an issue when the daily historical VaR is scaled to an annualized measure. According to
Hendricks (1996) historical simulations cannot be easily transformed between multiple
percentiles and study periods. The author states that it is not uncommon to use up to five
years of data in order to circumvent the matter.
Another key shortcoming of the efficiency measure based on historical VaR is that historical
scenarios are assumed to fully represent future observations. The problem is, that the
approach weights recent data the same as outdated observations. If current market trends
such as higher volatility in returns occur, the historical VaR is an inappropriate measure to
evaluate performance. This issue can partly be resolved by weighting recent returns more
strongly.
In conclusion, the absolute values of the efficiency measure based on the historical VaR need
to be interpreted with caution. A reasonable approach to tackle the issues discussed is to
simply consider the corresponding ranking of ETFs, but not the efficiency measures absolute
values. An alternative is to use a statistical improvement of the delta-normal method, which
does not rely on the empirical distribution of returns, but adjusts the normal distribution
function in the delta-normal Var for the sample distribution function.
Chapter 7 Adjustments to the Efficiency Measure
74
An overall weakness of all the VaR methods considered so far is, that they generally fail to
indicate how the TD behaves over the observation horizon. Therefore the next subchapter will
introduce the concept of intra-horizon risk.
7.4.2 Intra-horizon Value-at-Risk
The issues of intra-horizon risk was already under discussion in the context of minimum capital
requirement by the Basel 3 committee (1996, p.4). The Overview of the amendment to the
capital accord to incorporate market risks (1996) finally suggested the usage of a multiplication
factor of three in order to take into account the weaknesses of end-of-horizon VaR. The
reports suggest the importance of a continuous measure of risk for minimum capital
requirements. Amongst others, Kritzman and Rich (2002) and Bakshi and Panayotov (2010)
argue that the traditional calculation of VaR allows estimating the probability of loss at the
end, but not along the investment period. The authors suggest a measure which incorporates
information about the dynamic path of potential losses, taking into consideration the losses
incurred before the end of a specified horizon.
Intra-horizon risk is relevant for any investor. However, it is particularly important for stop-loss
investors who follow the strategy to sell their assets whenever their value falls below a
predefined hurdle rate. Furthermore intra-horizon risk may be essential for pension funds or a
portfolio manager, who needs to rebalance of his portfolio, once the value of an investment
falls below a certain threshold. In the context of ETFs, previous VaR methods do not allow
inference about the likelihood and the maximum negative tracking outcome throughout the
whole study period. In a situation where the ETF investor is forced to sell his fund before the
due date, he may suffer from the current low return. The intuition behind the measurement of
intra-horizon risk is illustrated in Figure 30 at the examples of ETF 1# and 3# on SMI.
Figure 30: Cumulative Tracking Error ETF 1# and 3# on SMI
The evolution of the cumulative TE for ETF 1# and ETF 3# on SMI are illustrated. The observation period covers May 2013 to November 2013. (Source: Own calculations / illustrations)
-0.20%
-0.15%
-0.10%
-0.05%
0.00%
0.05%
0.10%
0.15%
1#
3#
Chapter 7 Adjustments to the Efficiency Measure
75
The investment horizon for the herein illustration is arbitrarily chosen from May 2nd until
October 2nd 2014. It can be seen, that according to the cumulative TD, both funds perform
equally well at the end of the observation period, as both end up at the same cumulative TD
value, indicating that VaR is concerned about the likelihood of the distribution of returns at
the end of the investment period. In the above example it is expected that the volatility partly
punishes the higher deviations of fund 3# as compared to ETF 1#. If fund 1# would however
exhibits positive deviations, but end up at the same cumulative TD level at the end of the
investment horizon, the efficiency measure would fail to indicate which fund is more efficient.
If the investor is forced to sell either ETF 1# or 3# one month prior to the end of the
investment horizon, he would experience a much greater loss when being invested with ETF 3#
than with ETF 1#.
Kritzman and Rich (2002) discuss a measure of intra-horizon VaR that is calculated based on
the probability of maximum loss at any time during the investment period at some given
confidence level. In order to obtain an intuition of this method, Kritzman and Rich (2002) firstly
derive the end-of-horizon VaR ( ), based on the formula below.
√ (52)
depicts the probability of loss, calculated by the difference between the cumulative
percentage loss in continuous units and the annualized expected return , meaning the
annual TD, divided by the annualized standard deviation of continuous returns √ , being the
annualized TE. is measured in terms of years. Finally the normal distribution function is
applied to convert the standardized distance from the mean to a probability estimate. As we
are considering the probability of loss, is the 5% percentile in order to correspond to the
95% probability of not losing an amount greater than . The end-of-horizon VaR then is
derived by rearranging the formula (52) and solving for .
√ (53)
The efficiency measure with the above VaR including the bid-ask spread is calculated as:
| | (54)
Chapter 7 Adjustments to the Efficiency Measure
76
In order to capture the probability of intra-horizon loss , Kritzman and Rich (2002) apply a
statistic called first-passage time probability. The method captures the probability that the
return will hit a particular hurdle rate during the investment horizon. Alternatively it estimates
the maximum loss during the period at the given confidence level of 95%. The probability
that an investment will fall under a particular value while being constantly monitored is
(
√ )
(
√ )
(55)
The first part of the equation is equal to the end-of-horizon VaR from formula (53), whereas
the second part of the equation only takes positive values, making the probability of intra-
horizon loss always larger than end-of-horizon loss . In contrast to the tradition VaR
probability measure, furthermore increases for a longer investment horizon, thus arguing
against the benefit of time diversification of risk.
As solving for cannot be administered analytically, numerical optimization is applied in
order to derive intra-horizon VaR . The efficiency measure finally measures the worst trade
at a chosen probability to any time during the investment horizon.
| | (56)
The results of the efficiency measure based on the delta-normal VaR | , the alternative
method of calculating the end-of horizon VaR | as well as the intra-horizon VaR
| are presented in Table 26 and Table 27.
Table 26: Intra-horizon VaR ETF SMI
The sample statistics for the efficiency measure based on the delta-normal, the end of horizon and the intra-horizon VaR are presented. (Source: Own calculations / illustrations)
ETF on SMI | |
|
1# -174.28 -173.07 -193.14
2# -251.59 -250.43 -256.53
3# -144.47 -143.83 -149.86
4# -55.49 -55.46 -55.65
5# -53.01 -52.95 -53.68
Average -135.77 -135.15 -141.77
Chapter 7 Adjustments to the Efficiency Measure
77
As expected, the end-of-horizon VaR calculation method by Kritzman and Rich (2002),
returned efficiency measure values that deviate by not more than one basis point from the
traditional delta-normal method | . Considering the intra-horizon risk | does
however change absolute values significantly. As suggested by the definition of formula (55),
intra-horizon risk is greater for all ETFs than end-of-horizon risk. Particularly fund 1# exposes
an investor with greater risk throughout the study period, increasing the continuously
measured loss for about 20 bps. The reason can be found in its TE, being the largest of all ETFs
considered. Intuitively, it is the volatility of the TD that has the biggest influence weather a
fund exposes an investor to greater risk throughout the investment horizon.
Table 27: Intra-horizon VaR ETF EURO STOXX 50
The sample statistics for the efficiency measure based on the delta-normal, the end of horizon and the intra-horizon VaR are presented. (Source: Own calculations / illustrations)
ETF on EURO STOXX 50
| |
|
1# 12.67 12.60 -19.21
2# -34.55 -34.60 -63.51
3# -32.90 -32.91 -40.08
4# -102.62 -102.51 -118.89
5# 0.09 -0.13 -37.02
6# 9.95 9.89 -18.43
7# 26.55 26.37 -22.89
Average -17.26 -17.33 -45.72
Similarly the ETFs on EURO STOXX 50 do not exhibit larger deviations by more than half a basis
point for the delta normal | and the end-of-horizon VaR | . The intra-horizon
VaR again reduces the efficiency measures, suggesting that the risk during the trading horizon
is substantially larger than at the end of the observation period. Other than for the ETFs on
SMI, the ranking of the efficiency measure | largely changes for the ETFs on EURO
STOXX 50. Fund 6# exposes an ETF investor to the smallest loss at the 95% probability level
during the whole study period. ETF 7# and 8# are ranked second and third. Especially ETF 7#
deteriorated for more than 40 bps. This indicates that even though the fund gives the best
result at the end of the investment horizon, it exposes an investor to substantial intra-horizon
risk.
Unfortunately, the intra-horizon VaR does not indicate how large the loss will be in the worst
cases occurring outside of the probability scenario. The method furthermore assumes
approximate normality in the distribution of TD.
Chapter 7 Adjustments to the Efficiency Measure
78
Although being largely accepted as a market risk measure, the VaR calculation methods
considered so far suffer from an additional shortcoming with respect the magnitude of the loss
in the tail. The VaR methods give e.g. the maximal loss at the 95% probability level, but do not
indicate how big the loss is in the remaining 5% cases. The next subchapter therefore discusses
a method which addresses this issue.
7.4.1 Expected Shortfall
Although being largely accepted as a market risk measure, VaR suffers from additional
shortcomings with respect the magnitude losses in the tail and the fact that it is generally not
considered to be a coherent measure of risk. The herein subchapter discusses those drawbacks
and evaluates a method that provides a remedy.
According to Atzner, Delbaen, Eber and Heath (1999), VaR is not a coherent measure as it
considers the risk of a portfolio to be higher than the sum of risk of its individual asset. In other
words, the VaR concept implicitly revokes the benefits of diversification and is largely criticized
for not being sub-additive. Furthermore, VaR is concerned about the probability level at which
a certain loss will not be exceeded only, but does not indicate how big the loss in the
remaining events will be. Expected Shortfall (ES), alternatively referred to as the conditional
VaR, is a measure that explicitly includes tail risk (Acerbi & Tasche, 2002). ES measures how big
the expected loss will be on average, in case the worst events occur. In the herein
setting, ES looks at the average of the losses that fall outside the 95% percentile of the VaR.
[ | ] (57)
According to Hassine and Roncalli (2013), the efficiency measure adjusts as follows:
| | | [ | ] (58)
In Table 28 and Table 29 the results for | on the ETFs from both benchmarks are
reported. Furthermore the efficiency measure based on the historical VaR is listed. The
comparison to the historical VaR is especially informative, as it indicates how big the negative
TEs are, that fall out of the 95% empirical quantile, which is depicted by the historical VaR.
Chapter 7 Adjustments to the Efficiency Measure
79
Table 28: Expected Shortfall ETF SMI Table 29: Expected Shortfall EURO STOXX 50
The sample statistics of the efficiency measure based on the historical VaR and on the ES are presented. (Source: Own calculations / illustrations)
The sample statistics of the efficiency measure based on the historical VaR and on the ES are presented. (Source: Own calculations / illustrations)
ETF on SMI
1# -64.43 -72.52
2# -193.48 -301.76
3# -121.68 -164.34
4# -54.26 -62.11
5# -55.05 -58.94
Average -97.78 -131.93
ETF on EURO STOXX 50
1# 26.84 26.15
2# -24.26 -26.76
3# -17.72 -19.83
4# -85.67 -102.63
5# 47.83 46.09
6# 26.00 22.64
7# 48.53 41.85 Average -4.50 -9.06
Large deviations of the efficiency measure based on the historical VaR and the ES indicate that
the loss in the tails indeed reach significantly higher values. For the ETFs on SMI, it is fund 2#
and 3# that exhibit large losses in the tail. The ranking of the bottom ETFs however is not
changed. Only ETF 5# performs relatively better and ranks as the most efficient ETFs amongst
all ETFS on SMI. By definition the measure based on ES is expected to reveal lower efficiency
measure scores as the historical VaR is lower than ES. For the ETFs on SMI, the average
measures deteriorate for roughly 23bps.
The ranking for the ETFs on EURO STOXX 50 did not change by any means. The reason is that
none of those funds exhibits large negative TD and thus large losses in their empirical tails. On
average, the measure corrupts the tracking efficiency for about 4bps only as compared to
the historical VaR. Those results are largely consistent with the findings by Hassine and
Roncalli (2013), which observe the ETFs on EURO STOXX 50 over the period of November 2012
until November 2013.
Similarly to the drawbacks of the historical VaR, the ES values have to be interpreted with
caution as again the √ rule is applied in order to receive annual results. It is advisable to
use the results for ranking purposes only. In order to receive significantly absolute VaR figures,
entropic VaR suggested by Ahmadi-Javid (2011) can be applied. Entropic VaR is a coherent risk
measure, which is defined as an upper bound for the delta-normal VaR and the ES.
Furthermore the efficiency measure based on ES indicates the maximal loss at the end of the
period, but does fail to indicate how big the maximal loss will be at any time during the
observation period. The same drawback can be found in the other VaR methods considered so
far.
Chapter 7 Adjustments to the Efficiency Measure
80
It can be concluded that even thought the computation methods jointly account for the main
shortcomings of VaR, none of them is able to completely offset all drawbacks. It is thus
important to consider several efficiency measures. The closing section of Chapter 7 will discuss
the interpretation of the efficiency measure.
7.5 Alternative Interpretation of the Efficiency Measure
To conclude, the adjustments of the ETF efficiency measure, the interpretation suggested by
Hassine and Roncalli (2013) is discussed critically. The authors propose that if the efficiency
measure of ETF is larger than the efficiency indicator for ETF , fund should be preferred to
fund . A major limitation of this interpretation is that it does not indicate how efficient the
fund actually mimics the returns of the benchmark. Large but steady excess performance
increases the efficiency measure and makes a fund more favorable in the traditional
understanding. The question that needs to be asked is which value for the efficiency measure
actually indicates efficient tracking. A perfectly replicating ETF has no TD, no TE and preferably
comes at no bid-ask spread. In consequence, the most efficient tracker would have an
efficiency measure of zero.
Whereas negative values in the TE certainly are undesirable, positive values indicate an over
performance possibly coming from securities lending. Close up tracking becomes a necessity
when ETFs are used for example hedging purposes or in a mandate of an asset manager.
Depending on the objective of the investor, it makes sense to consider the negative absolute
values of the tracker and evaluate how much the value deviates from zero.
| | | (59)
The above application may have the side benefit that providers are less likely to choose an
inaccurate benchmark in order to boost the excess-performance of the ETF. When applying
the above proposition to the ETFs in the herein sample, 5# on SMI and fund 5# on EURO
STOXX 50 are generally regarded as the most efficiently tracking ETFs. Overall, they exhibit the
least deviation from the benchmark.
From the previous discussion, it can be seen that the design of an appropriate efficiency
measure is not straightforward. The mathematical assumptions of the models and the
backdrop of the efficiency measure have to be borne in mind. The subsequent conclusion
therefore again sums the main results derived in this thesis and gives recommendations on the
application of the efficiency measure.
Chapter 8 Conclusion and Outlook
81
Chapter 8 Conclusion and Outlook
This research paper provides profound investigation of ETF performance measurement
following the research by Hassine and Roncalli (2013). After gaining in-depth insight on the
structure and functioning of ETFs, the thesis shows that traditional performance not only are
inappropriate, but may be misleading when being applied to passively managed ETFs.
Subsequently, the efficiency measure based on VaR is established. Empirical research on
unlevered, passively managed, equity ETF on SMI and the ETFs on EURO STOXX 50 is
conducted. The subsequent analysis on TE applies alternative computation methods of TE and
evaluates robust and semi-variance measures. After discussing the liquidity metrics ETFs, the
efficiency measure is adjusted for normal and non-normal TD distribution as well as ES and
intra-horizon risk. Finally alternative interpretations of the efficiency measure are discussed.
To conclude the analysis from the previous chapters, the subsequent sections give case-by-
case recommendations for the design and application of the efficiency measure. Suggestions
are given by taking into account the different types of ETF investors and trading strategies as
well as the underlying mathematical assumptions of the efficiency measure. To begin with, the
ETF performance characteristics are discussed. Subsequently, some of the rule-in and rule-out
criteria’s of ETF investors are highlighted. The conclusion elaborates on the insights from the
theoretical and empirical analysis of this thesis and gives concrete recommendations on which
efficiency measure is most suitable in a particular scenario. As a final point, an outlook on
potential fields of ETF performance measurement is held.
Before looking at any ETF efficiency measure, the investors need to determine the risk metrics
most relevant in the context of their underlying ETF investment strategy. The span of ETF
investors may reach from a buy-and-hold investor, who profits from the ETFs low fees and tax
efficiency, to a frequent trader who benefits from the ETFs liquidity, lower volatility and broad
diversification. Moreover, eligible investors are able to trade ETF shares OTC, whereas others
only are able to trade on the exchange. The corresponding trading circumstances and the
underlying investment objective therefore jointly influence which efficiency measure is most
suitable for the investor.
Chapter 8 Conclusion and Outlook
82
How ETF performance is measured largely depends on whether the ETF is traded on the
secondary or the primary market. The major difference in ETF performance measurement
either market stems from the different liquidity costs of trading the ETF. Analyzing the ETF
liquidity costs by looking at the spread and pricing efficiency only, fails to take into account
that investors with access to the primary market in fact trade OTC directly with the AP. Large
institutional investor for example may subscribe or redeem their shares at the NAV plus
spread, while potentially negotiating favorable conditions. As the costs of OTC trades are not
publicly available and may depend on the investor’s negotiation power, it is advisable to
evaluate the ETFs based on the efficiency measures without the spread. Appendix 12 presents
all efficiency measure values without including the bid-ask spread. For the ETFs on SMI,
especially fund 4# and 5# perform consistently well. ETF 1# exhibits most favorable results for
the semi-volatility VaR and ES efficiency measure. For the ETFs on EURO STOXX 50, its fund 5#
and 7# that constantly probe to be the most efficient.
The efficiency measure including the bid-ask spread most likely is suitable for secondary
market trading only. In order to approximate to cost of trading the ETF on the exchange,
investors should look at the bid-ask spread measure which takes the average of intraday prices
and not day-end prices as reference. For investors, who trade ETF volumes, it additionally
advisable to adjust the spread for the notional traded and estimate the market impact costs
caused by their trades in the ETF and
Finally investors from both the primary and secondary market are interested in how closely
the ETF replicates its benchmark. Evaluating both the TE and the TD therefore is vital for
efficiency measurement. Nevertheless, the relative importance of each of these metrics
depends on the investment purpose of the investor. A high frequency trader, investing short
term in order to e.g. equitize cash positions or profit from short term market developments,
may be more concerned about the relative deviation of the ETF returns on a daily basis,
measures by TE. Additionally, the bid-ask spread becomes highly important for the frequent
trader, as for every ETF trade the spread has to be paid. In the limit, the investor may solely
care about the ETF trading costs.
A buy and hold investor may be mostly concerned about the long term performance difference
of the ETF relative to the benchmark measured by TD, but not so much about the short term
volatility measured by TE. A buy and hold investor cares tracking deviation in the long run,
caused by fees and additional charged, instead of the short term deviations caused by e.g.
dissimilar dividend payment schedules.
Chapter 8 Conclusion and Outlook
83
After eliminating unsuitable ETFs and determining which risk metrics are the most important,
the investor is faced with selecting the best performing product. As this thesis has shown ETFs
may in fact track the same benchmark, however, differ largely regarding their performance
relative to the benchmark as well as the underlying investment risks. In order to compare like
with like, it is crucial to have a close look at the ETFs and the benchmarks underlying dividend
assumptions. Quite frequently ETFs are benchmarked against different types of the same
benchmark, as ETF providers generally acquire data from one benchmark type only. An
accumulating ETF benchmarked against a net total return index may track less accurately than
a distributing ETF benchmarked against price index version of the same benchmark. Correcting
the daily performance of the ETF every time a dividend is paid is at risk of exhibiting return
abnormalities due to diverging taxation assumptions or timing. Investors need to be aware
that data outliers from any source strongly influence ETF efficiency measurement. One large
outlier may inflate the TE and indicate low quality tracking, even though the ETF mirrors his
benchmark fairly well. Especially non-robust measures as the mean, volatility, Skegness and
the kurtosis have to be interpreted carefully.
Finally the underlying assumptions of the various VaR methodologies have to be kept in mind.
The decision tree in Figure 31: Decision Tree of ETF Efficiency MeasuresFigure 31 assists
investors in choosing the most suitable VaR method in the ETF performance measurement
process.
Chapter 8 Conclusion and Outlook
84
After determining which risk metrics are the most important, the investor is faced with
selecting the best performing ETF. As this thesis has shown, ETFs may track the same
benchmark, however, differ largely regarding their performance and underlying investment
risks. In order to compare like with like, it is therefore crucial to have a close look at the ETFs
and the benchmarks underlying dividend assumptions. Quite frequently ETFs are benchmarked
against different types of the same benchmark. An accumulating ETF benchmarked against a
net total return index may track less accurately than a distributing ETF benchmarked against
the price index version of the same benchmark. Correcting the performance of the ETF every
time a dividend is paid is at risk of resulting in return abnormalities due to diverging taxation
assumptions. Investors need to be aware that data outliers strongly influence ETF efficiency
measurement. One large outlier may inflate the TE and indicate low quality tracking, even
though the ETF mirrors his benchmark fairly well in 99% of the cases. Especially non-robust
measures as the mean, volatility, Skegness and the kurtosis have to be interpreted carefully.
Finally the underlying assumptions of the various VaR methodologies have to be kept in mind.
The decision tree in Figure 31: Decision Tree of ETF Efficiency MeasuresFigure 31 assists
investors in choosing the most suitable VaR method for ETF performance comparison.
Chapter 8 Conclusion and Outlook
85
Figure 31: Decision Tree of ETF Efficiency Measures
The decision tree for the efficiency measures based on the different VaR methods is presented. Starting from the square on the top left, the drawn-out lines represent eligible options. The rhombuses represent the corresponding VaR measure on which the efficiency measure should be based on. (Source: Own calculations / illustrations)
If the relative return of the ETF is not allowed to fall below a certain threshold at any time
during the investment horizon, the efficiency measure based on the intra-horizon risk is
appropriate. If investors are only concerned about the probability of loss at the end of the
observation period and the TD are normally distributed, the efficiency measure based on the
delta-normal VaR is most suitable. When the distribution of TD is sufficiently close to a normal
distribution and the skewness as well as the kurtosis are within their domain of validity, the
ETF performance measure based on the Cornish-Fisher VaR complements the delta-normal
VaR. The historical VaR based on the empirical distribution of TD is suitable whenever large
skewness and kurtosis and finally non-normality in TD bias the statistical VaR approach. To
complement any VaR efficiency measure, the efficiency measure based on the expected
No
Are the ETFs tracking differences
normally distributed? Delta-normal
VaR
Are the ETFs tracking differences
approximately normally distributed? Cornish-Fisher
VaR
Are the ETFs tracking differences not
normally distributed? Historical VaR
Is the investor concerned about the
loss in the tails?
Is the investor concerned about the
expected loss at any time along the
investment horizon?
Intra-Horizon
VaR
Expected
Shortfall
Yes
No
Yes
Yes
Yes
Yes
No
No
Chapter 8 Conclusion and Outlook
86
shortfall indicates the losses in the tails and is relevant for investors which are concerned
about the losses occurring outside of the 95% scope.
Despite the fact that the branches Figure 31 indicate that each option is mutually exclusive, it
is highly advisable to consider several measures in order to select the most sufficient ETF. The
reason is that none of the measures considered is entirely free of drawbacks, and no single
measure can account for the full range of investor’s objectives.
The distinctive feature of this thesis is that it does not only evaluate single risk factors, but
presents an integral performance measure, incorporating the most important metrics in ETF
performance measurement. The analysis conducted is independent of any product provider
and ads to the considerations by Hassine and Roncalli (2013) by introducing robust measures
of risk as well as intra-horizon risk. This is the first study to look at ETF performance
measurement for both, ETFs traded on exchange and OTC. Therefore, the thesis not only takes
several statistical considerations into account, but furthermore adjusts to a selection of typical
ETF trading strategies on the ETFs primary and secondary market.
Whereas many studies focus on either the US or the European ETF market, this thesis is one of
the few to analyze parts of the Swiss ETF market. The randomized results suggest that the
rankings according to the various efficiency measures are largely consistent across the data
sample, where more efficient funds tend to perform better for most methodologies applied.
For the ETFs on EURO STOXX 50, the analysis generally confirms the results received by other
authors. However it does not find the same consistency across ETF ratings than suggested by
Hassine and Roncalli (2013). Especially the results for the Cornish-Fisher VaR deviate from the
results in their study, indicating larger skewness and excess kurtosis in the data sample. The
results in this thesis show that the efficiency measure is susceptible to the statistical
assumptions of the underlying TD distribution and suffers from other drawbacks such as non-
robustness. Overall, the measurement of ETF tracking efficiency is very precarious. It is
essential for investors not only to have transparent measures sourcing from high quality data,
but moreover to receive in-depth education about the risks and the shortcomings of ETF
efficiency measurement.
Chapter 8 Conclusion and Outlook
87
It is subject to further research whether the methods in the herein thesis give consistent
results when being extrapolated to ETF replicating e.g. fixed income, real estate, commodity or
hedge fund indexes. Additionally, research has to be conducted in order to test the viability of
the measures in different markets such as the US or Asia.
The ETF-market is one of the most growing markets within the asset management industry.
Not only new providers and ETF variations enter the market on a regular basis, but also the set
of potential index benchmarks increases. The rise in complexity jeopardizes the highly praised
simplicity and transparency of ETFs. The applicability of the efficiency measures framework to
different types of ETFs and indexes needs to be investigated going forward. With new products
such as leveraged and actively managed funds, the variety of ETFs increases. This further
evokes the need to have expedient efficiency indicators. As measures such as the TE and TD
may fail in the context of these products, additional risk metrics need to be developed.
88
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Appendix 1 – Proof of Equation (15)
Proof of equation (15) following Hassine and Roncalli (2013):
|
The weights of the linear combination of the tracker and are
Where the weights with respect to the benchmark adjust as follows
Consequently the TD is
| | |
And the TE calculates as follows
|
| |
| | |
| | |
98
The Information Ratio therefore is the following:
| |
|
| |
√ | | |
99
Appendix 2 – Derivation of the Efficiency Measure
Derivation of the efficiency measure based on the VaR framework following Hassine and
Roncalli (2013) as well as Jorion (2007).Starting from the general expression for VaR
| { { | } }
We have
{ | }
{ }
{ }
{ |
|
|
| }
Or
{ |
|
|
| }
( |
| )
|
|
Solving for
| |
And since
|
We get
| |
100
Appendix 3 – Index Information
Table 30: EURO STOXX 50
The constituents of the EURO STOXX 50 net total return index are depicted. The reference date is May 30
th 2014 and the name, the corresponding industry, the country of domicile and the percentage of the
constituents are presented. (Source: STOXX, 2014) No. Name Industry Country Percentage as of 30.05.2014
1
AIR LIQUIDE Chemicals FR 1.60% 2 AIRBUS GROUP NV Industrial Goods & Services FR 1.58% 3 ALLIANZ Insurance DE 2.84% 4
ANHEUSER-BUSCH INBEV Food & Beverages BE 2.88% 5 ASML HLDG Technology NL 1.35% 6
ASSICURAZIONI GENERALI Insurance IT 1.11% 7 AXA Insurance FR 2.02% 8
BASF Chemicals DE 3.75% 9 BAYER Chemicals DE 4.24%
10 BCO BILBAO VIZCAYA RGENTARIA Banks ES 2.64% 11 BCO SANTANDER Banks ES 3.86% 12 BMW Automobiles & Parts DE 1.49% 13
BNP PARIBAS Banks FR 3.15% 14 CARREFOUR Retail FR 0.90% 15
CRH Construction & Materials IE 0.76% 16 DAIMLER Automobiles & Parts DE 3.39% 17 DANONE Food & Beverages FR 1.51% 18 DEUTSCHE BANK Banks DE 1.72% 19 DEUTSCHE POST Industrial Goods & Services DE 1.27% 20
DEUTSCHE TELEKOM Telecommunications DE 1.83% 21 E.ON Utilities DE 1.45% 22
ENEL Utilities IT 1.34% 23 ENI Oil & Gas IT 2.52% 24
ESSILOR INTERNATIONAL Healthcare FR 0.83% 25 GDF SUEZ Utilities FR 1.56% 26 GRP SOCIETE GENERALE Banks FR 1.87% 27 IBERDROLA Utilities ES 1.30% 28 Industria de Diseno Textil SA Retail ES 1.20% 29
ING GRP Insurance NL 2.03% 30 INTESA SANPAOLO Banks IT 1.68% 31
L'OREAL Personal & Household Goods FR 1.44% 32 LVMH MOET HENNESSY Personal & Household Goods FR 1.80% 33 MUENCHENER RUECK Insurance DE 1.30% 34 ORANGE Telecommunications FR 1.03% 35 PHILIPS Industrial Goods & Services NL 1.19% 36
REPSOL Oil & Gas ES 0.76% 37 RWE Utilities DE 0.73% 38
SAINT GOBAIN Construction & Materials FR 1.01% 39 SANOFI Healthcare FR 4.70% 40
SAP Technology DE 2.81% 41 SCHNEIDER ELECTRIC Industrial Goods & Services FR 1.87% 42 SIEMENS Industrial Goods & Services DE 4.25% 43 TELEFONICA Telecommunications ES 2.32% 44 TOTAL Oil & Gas FR 5.91% 45
UNIBAIL-RODAMCO Real Estate FR 0.96% 46 UNICREDIT Banks IT 1.83% 47
UNILEVER NV Food & Beverages NL 2.32% 48 VINCI Construction & Materials FR 1.46% 49 VIVENDI Media FR 1.31% 50 VOLKSWAGEN PREF Automobiles & Parts DE 1.35%
101
Table 31: Swiss Market Index
The constituents of the SMI price index are depicted. The reference date is December 12th
2013 and the name, the corresponding industry and the percentage of the constituents are presented. (Source: SIX, 2013b)
No. Name Industry Percentage as of
30.12.2013
1 ABB Industrials 5.44%
2 Actelion Health Care 0.84%
3 Adecco Industrials 0.95%
4 Credit Suisse Financials 4.13%
5 Geberit Industrials 1.02%
6 Givaudan Basic Materials 1.06%
7 Holcim Industrials 1.51%
8 Julius Baer Group Financials 0.96%
9 Nestlé food products 21.08%
10 Novartis Health Care 19.29%
11 Richemont clothing and accessories 4.13%
12 Roche Health Care 17.53%
13 SGS Industrials 1.13%
14 Swatch Group clothing and accessories 1.82%
15 Swiss Re Financials 2.82%
16 Swisscom Telecommunications 1.06%
17 Syngenta Basic Materials 3.31%
18 Transocean Oil & Gas 1.47%
19 UBS Financials 6.09%
20 Zurich Insurance Group Financials 3.85%
102
Appendix 4 – ETF Sample
Table 32: ETF Sample
The sample fund names, the International Securities Identification Number (ISIN) as well as the corresponding product provider are presented. The funds are arranged in a random order and do not correspond to the sequence of numbers used in the main body of the thesis as well as the listing in Appendix 5. (Source: Provider Factsheets available on provider websites and Bloomberg)
Panel A: ETF on SMI
Names ISIN Product provider
ISHARES SMI DE DE0005933964 iShares
COMSTAGE SMI UCITS ETF LU0392496427 ComStage ETF
ISHARES SMI CH CH0008899764 iShares
DB X-TRACKERS SMI UCITS ETF LU0274221281 db x-trackers
UBS ETF CH-SMI CHF A CH0017142719 UBS
DB X-TRACKERS SMI UCITS ETF LU0943504760 db x-trackers
UBS ETF CH-SMI CHF I CH0200721360 UBS
SMI Price Index CH0009980894 SIX Swiss Exchange
SMI Total Return Index CH0000222130 SIX Swiss Exchange
Panel B: ETF on EURO STOXX 50
Names ISIN Product provider
COMSTAGE ETF DJ EUR STOXX 50 NR UCITS ETF LU0378434079 ComStage ETF SICAV
LYXOR UCITS ETF (FCP) EURO STOXX 50 FR0007054358 Lyxor International Asset Management
ISHARES EURO STOXX50 UCITS ETF (DE) DE0005933956 BlackRock Asset Management Deutschland AG
UBS ETF (LU) EURO STOXX 50 UCITS ETF LU0136234068 UBS ETF
ISHARES Core EURO STOXX 50 UCITS ETF IE00B53L3W79 iShares VII plc
HSBC EURO STOXX 50 UCITS ETF IE00B4K6B022 HSBC ETFs PLC
AMUNDI ETF EURO STOXX 50 UCITS ETF FR0010654913 AMUNDI
EURO STOXX 50 Net Total Return Index EU0009658145 STOXX
103
Appendix 5 – ETF Facts
Table 33: Fund Information
The ETF facts as of April 2014 are depicted. The illustration covers the ETF status, share class, the ETF domicile, inception date, replication strategy, whether the fund distributes or accumulates dividends and whether the ETF conducts securities lending. (Source: Provider Factsheets available on provider websites and Bloomberg)
Panel A: ETFs on SMI
ETF on SMI Status Share class Domicile Inception
Date Replication Acc./ Dist.
Securities lending
1# active 1D Luxembourg 14.05.2007 Full Distributing No
2# active I Luxembourg 04.11.2009 synthetic Accumulating Unknown
3# active A Germany 29.08.2001 Full Distributing No
4# active A Switzerland 15.03.2001 Full Distributing Yes
5# active A Switzerland 05.12.2003 Full Distributing Yes
104
Panel B: ETFs on EURO STOXX 50
ETF on EURO STOXX 50
Status Share class Fund
Domicile Inception
Date Replication Acc./ Dist.
Securities lending
1# active C France 13.04.2010 synthetic Accumulating No
2# active I Luxembourg 01.06.2010 synthetic Accumulating Unknown
3# active A Ireland 11.04.2011 full Distributing No
4# active A Germany 12.09.2003 full Distributing Yes
5# active B Ireland 26.01.2010 full Accumulating Yes
6# active D France 15.09.2004 synthetic Accumulating
and /or Distribution
No
7# active A Luxembourg 13.11.2001 full Distributing Yes
105
Appendix 6 – ETF Trading Information
Table 34: Trading Information
The trading facts as of April 2014 are presented. The illustration covers the ETF trading information symbol, Bloomberg ticker and benchmark. In addition, the responsible APs on the SIX Swiss Exchange and the current registrations within the European Union and Switzerland are listed. Furthermore the AuM, the trading currency, the management fees and the total expense ratio are depicted. (Source: Provider Factsheets available on provider websites and Bloomberg)
Panel A: ETFs on SMI
ETF on SMI Symbol Bloomber
g ticker Benchmark
(Symbol) APs on SIX
Current Listing
EU & CH
AuM in Mio
Trading Currency
Mgmt. fee
TER
1# XSMI XSMI SW SMI
Deutsche Bank AG London Branch; Flow Traders B.V.; Commerzbank AG; Optiver V.O.F; Susquehanna; KCG
Europe Limited
AT, CH, DE, ES,
FR, IE, IT, LU, UK
465 CHF 0.30% 0.30%
2# CBSSMI CBSSMI SW SMIC Commerzbank AG;
Susquehanna
AT, CH, DE, LU
59.64481
CHF 0.25% 0.25%
3# SMIEX SMIEX SW SMIC Flow Traders B.V.;
Commerzbank AG; Optiver V.O.F; Susquehanna
CH, DE, LI 255 CHF 0.50% 0.50%
4# CSSMI CSSMI SW SMIC
Credit Suisse AG; Susquehanna; Commerzbank
AG; Flow Traders B.V.; Optiver V.O.F; KCG Europe Limited;
Zürcher Kantonalbank
CH, DE, LI 3319.99
1 CHF 0.39% 0.39%
5# SMICHA SMICHA
SW SMIC
UBS; Commerzbank AG; Optiver V.O.F; Flow Traders
B.V.; Credit Suisse AG; Timber Hill (Europe) AG; Zürcher
Kantonalbank; Susquehanna
CH, LI 1050.05
7 CHF 0.20% 0.20%
106
Panel B: ETFs on EURO STOXX 50
ETF on EURO STOXX 50
Symbol Bloomberg
ticker Benchmark
(Symbol) APs on SIX
Current Listing EU &
CH
AuM in Mio
Trading Currency
Mgmt. fee
TER
1# C50 C50 SW SX5T Flow Traders B.V.; BNP Paribas; Susquehanna
CH, DE, ES, FR, IT, UK
980.8394 EUR 0.15% 0.15%
2# CBSX5E CBSX5EEU SW SX5T Commerzbank AG;
Susquehanna AT, CH, DE, LU 234.3195 EUR 0.10% 0.10%
3# H50E H50EEUR SW SX5T HSBC Bank Plc;
Susquehanna; Flow Traders B.V.
AT, CH, DE, SP, FR, UK, IT, NL,
SW 69.42362 EUR 0.15% 0.15%
4# DJSXE SX5EEX SW SX5T Commerzbank AG; Flow
Traders B.V.; Susquehanna
AT, CH, DE FR, LI, LU
4955.333 EUR 0.15% 0.16%
5# CSSX5E CSSX5E SW SX5T
Credit Suisse AG; Timber Hill (Europe)
AG; Flow Traders B.V.; Susquehanna
AT, CH, DE, ES, FR, IE, IT, UK
122.3061 EUR 0.20% 0.20%
6# MSE MSE SW SX5T
Société Générale; Commerzbank AG; Flow
Traders B.V.; Susquehanna
AT, BE, CH, DE, ES, FR, IT, UK
4780.819 EUR 0.20% 0.20%
7# E50EUA E50EUA SW SX5T UBS; Commerzbank AG;
Flow Traders B.V.; Susquehanna
At, CH, DE, FR, IT, LI, LU, SG,
UK 614.0051 EUR 0.15% 0.15%
107
Appendix 7 – Tracking Differences Figure 32: Time Series ETF Tracking Difference
The TD development for the sample ETFs is illustrated. The observation period covers May 2013 to May 2014 and is illustrated on the horizontal axis. In order to receive comparable illustrations, the vertical axis is scaled consistently across all ETFs and depicts the TD in percentage. (Source: Own calculations / illustrations)
Panel A.1: ETF 1# on SMI
Panel A.2: ETF 2# on SMI
Panel A.3: ETF 3# on SMI
-0.05%
-0.03%
-0.01%
0.01%
0.03%
0.05%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
18
.04
.14
-0.05%
-0.03%
-0.01%
0.01%
0.03%
0.05%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
18
.04
.14
-0.05%
-0.03%
-0.01%
0.01%
0.03%
0.05%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
18
.04
.14
108
Panel A.4: ETF 4# on SMI
Panel A.5: ETF 5# on SMI
Panel B.1: ETF 1# on EURO STOXX 50
-0.05%
-0.03%
-0.01%
0.01%
0.03%
0.05%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
18
.04
.14
-0.05%
-0.03%
-0.01%
0.01%
0.03%
0.05%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
18
.04
.14
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
0.06%
3.0
5.1
3
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
9.0
8.1
3
23
.08
.13
6.0
9.1
3
20
.09
.13
4.1
0.1
3
18
.10
.13
1.1
1.1
3
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
7.0
2.1
4
21
.02
.14
7.0
3.1
4
21
.03
.14
4.0
4.1
4
109
Panel B.2: ETF 2# on EURO STOXX 50
Panel B.3: ETF 3# on EURO STOXX 50
Panel B.4: ETF 4# on EURO STOXX 50
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
0.06%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
0.06%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
-0.03%
-0.02%
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
0.06%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
110
Panel B.5: ETF 5# on EURO STOXX 50
Panel B.6: ETF 6# on EURO STOXX 50
Panel B.7: ETF 7# on EURO STOXX 50
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
0.06%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
0.06%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
-0.01%
0.00%
0.01%
0.02%
0.03%
0.04%
0.05%
0.06%
03
.05
.13
17
.05
.13
31
.05
.13
14
.06
.13
28
.06
.13
12
.07
.13
26
.07
.13
09
.08
.13
23
.08
.13
06
.09
.13
20
.09
.13
04
.10
.13
18
.10
.13
01
.11
.13
15
.11
.13
29
.11
.13
13
.12
.13
27
.12
.13
10
.01
.14
24
.01
.14
07
.02
.14
21
.02
.14
07
.03
.14
21
.03
.14
04
.04
.14
111
Appendix 8 – Semi-Variance as a Special Case of LPM
According to Bawa (1975) semi-variance is a special case of lower partial moments (LPM).
Starting from as a random variable with distribution , the mathematical expectation of is
defined according to Casella & Berger (2001)
[ ]
∫ ∫
Casella and Berger (2001) provide the full mathematical framework the moments of a
distribution. The centered moment of order , which essentially measures the shape of a set
of points around the media is defined as:
[ ]
∫
The first four moments are then defined as follows:
Formula Mathematical Sign Numerator for
[ ]
[ ]
[ ]
[ ]
According to Bawa (1975), the lower Partial Moment (LPM) of the above distribution is defined
as follows, where denotes the threshold:
[ ]
∫
Semi-variance then is defined as the second lower partial moments with respect to
the threshold of the mean
112
[ ]
∫
Whenever the distribution of is symmetric around the theshold, the semi-variance is half of
the variance.
∫
∫
∫
∫
113
Appendix 9 – Autocorrelation
Figure 33: ETF Autocorrelation Function
The autocorrelation function for the first 25 lags is presented. The dotted red lines correspond to the coefficients in order to reject the null hypothesis of no autocorrelation. (Source: Own calculations / illustrations).
Panel A.1: ETF 1# on SMI
Panel A.2: ETF 2# on SMI
Panel A.3: ETF 3# on SMI
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
114
Panel A.4: ETF 4# on SMI
Panel A.5: ETF 5# on SMI
Panel B.1: ETF 1# on EURO STOXX 50
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
115
Panel B.2: ETF 2# on EURO STOXX 50
Panel B.3: ETF 3# on EURO STOXX 50
Panel B.4: ETF 4# on EURO STOXX 50
-0.15
-0.05
0.05
0.15
0.25
0.35
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
116
Panel B.5: ETF 5# on EURO STOXX 50
Panel B.6: ETF 6# on EURO STOXX 50
Panel B.7: ETF 7# on EURO STOXX 50
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Lag k
Autocorrelation
U-Critical Val
L Cirtical Val
117
Appendix 10 – Bid-Ask Spread
Figure 34: Percentage Bid-Ask Spreads
The percentage spread development for all ETFs are presented. The observation period covers May 2013 to May 2014. (Source: Own calculations / illustrations).
Panel A.1: ETF 1# on SMI
Panel A.2: ETF 2# on SMI
Panel A.3: ETF 3# on SMI
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
17
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
17
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
17
.04
.14
118
Panel A.4: ETF 4# on SMI
Panel A.5: ETF 5# on SMI
Panel B.1: ETF 1# on EURO STOXX 50
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
17
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
17
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
119
Panel B.2: ETF 2# on EURO STOXX 50
Panel B.3: ETF 3# on EURO STOXX 50
Panel B.4: ETF 4# on EURO STOXX 50
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
120
Panel B.5: ETF 5# on EURO STOXX 50
Panel B.6: ETF 6# on EURO STOXX 50
Panel B.7: ETF 7# on EURO STOXX 50
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
02
.05
.13
16
.05
.13
30
.05
.13
13
.06
.13
27
.06
.13
11
.07
.13
25
.07
.13
08
.08
.13
22
.08
.13
05
.09
.13
19
.09
.13
03
.10
.13
17
.10
.13
31
.10
.13
14
.11
.13
28
.11
.13
12
.12
.13
26
.12
.13
09
.01
.14
23
.01
.14
06
.02
.14
20
.02
.14
06
.03
.14
20
.03
.14
03
.04
.14
121
Appendix 11 – Efficiency Measure Values
Table 35: Efficiency Measures Overview
The summary statistics of all efficiency measure calculation methods is presented. The measures are calculated in bps on an annual basis from May 2013 to May 2014. For each efficiency measure, the number of the corresponding formula is indicated. The coloring in the cells illustrates the ranking of the fund according to the efficiency measure considered. Orange coloring indicates a lower ranking and yellow coloring suggest a better ranking amongst the ETFs considered. (Source: Own calculations / illustrations)
Panel A: ETFs on SMI
Formula 1# 2# 3# 4# 5#
| with | 23 & 28 -174.28 -251.59 -144.47 -55.49 -53.01
| with | 29/30 -174.54 -251.76 -144.60 -55.51 -53.03
| with | 32/33 -174.13 -251.44 -136.77 -55.51 -52.87
| 34 -79.85 -196.48 -109.96 -51.66 -50.02
| 36 -61.57 -276.40 -129.61 -56.61 -50.93
| 37 -62.69 -276.91 -132.04 -57.09 -52.40
Median Spread 43 -166.39 -253.73 -142.58 -55.34 -52.95
No Spread 43 -158.94 -189.39 -133.56 -50.54 -46.06
49 -64.43 -193.48 -143.83 -54.26 -55.05
| 51 -167.29 -70.01 -133.87 -48.97 -52.50
52 -72.52 -301.76 -164.34 -62.11 -58.94
| 53 -173.07 -250.43 -219.21 -55.46 -52.95
| 58 -193.14 -256.53 -149.86 -55.65 -53.68
Average -132.53 -232.30 -144.98 -54.94 -52.64
122
Panel B: ETFs on EURO STOXX 50
Formula 1# 2# 3# 4# 5# 6# 7#
| with | 23 & 28 12.67 -34.55 -32.90 -102.62 0.09 9.95 26.55
| with | 29/30 12.63 -34.58 -32.94 -102.76 -0.02 9.91 26.49
| with | 32/33 12.63 -34.11 -32.91 -95.68 0.22 9.95 27.26
| 34 23.06 -25.62 -23.36 -80.11 42.50 22.20 42.98
| 36 28.20 -22.85 -17.77 -79.98 49.13 24.35 48.14
| 37 27.64 -23.05 -17.67 -78.73 48.65 24.22 47.76
Median Spread 43 16.45 -23.81 -23.30 -106.06 -0.18 11.09 28.02
No Spread 43 27.96 25.40 -7.04 -49.89 13.91 23.96 42.23
49 26.84 -24.26 -17.72 -85.67 47.83 26.00 48.53
| 51 8.90 -37.48 -38.23 -111.84 132.71 13.58 22.67
52 26.15 -26.76 -19.83 -102.63 46.09 22.64 41.85
| 53 -19.21 -63.51 -40.08 -118.89 -37.02 -18.43 -22.89
| 58 -3.91 -3.55 -14.21 -66.15 -23.20 -4.42 -7.21
Average 15.38 -25.29 -24.46 -90.85 24.67 13.46 28.64
123
Appendix 12 – Efficiency Measure without Bid-Ask Spread
Table 36: Efficiency Measures Overview without Spread
The summary statistics of all efficiency measure versions without incorporating the bid-ask spread is presented. The measures are calculated on an annual basis from May 2013 to May 2014 and are measured in Basis points. For each efficiency measure, the number of the corresponding formula is indicated. The coloring in the cells illustrates the ranking of the fund according to the efficiency measure considered. Orange coloring indicates a lower ranking and yellow coloring suggest a better ranking amongst the ETFs considered. (Source: Own calculations / illustrations)
Panel A: ETFs on SMI
Formula 1# 2# 3# 4# 5#
| with | 23 & 28 -158.94 -189.39 -133.56 -50.54 -46.06
| with | 29/30 -159.20 -189.55 -133.69 -50.56 -46.09
| with | 32/33 -158.79 -189.24 -125.85 -50.56 -45.92
| 34 -64.51 -134.27 -99.05 -46.71 -43.07
| 36 -46.22 -214.20 -118.69 -51.66 -43.99
| 37 -47.35 -214.71 -121.12 -52.14 -45.46
49 -49.09 -131.28 -143.83 -49.31 -55.05
| 51 -151.95 -7.80 -122.95 -44.02 -45.55
52 -57.18 -239.55 -153.42 -57.16 -51.99
| 53 -157.73 -188.23 -208.29 -50.51 -46.01
| 58 -177.80 -194.33 -138.95 -50.71 -46.73
Average -111.71 -172.05 -136.31 -50.35 -46.90
124
Panel B: ETFs on EURO STOXX 50
Formula 1# 2# 3# 4# 5# 6# 7#
| with | 23 & 28 27.96 25.40 -7.04 -49.89 13.91 23.96 42.23
| with | 29/30 27.93 25.37 -7.08 -50.02 13.80 23.93 42.16
| with | 32/33 27.93 25.84 -7.04 -42.95 14.04 23.96 42.93
| 34 38.36 34.33 2.50 -27.37 56.33 36.21 58.66
| 36 43.50 37.10 8.09 -27.25 62.95 38.36 63.82
| 37 42.94 36.90 8.20 -26.00 62.47 38.23 63.44
49 42.14 35.69 8.15 -32.93 61.65 40.01 64.20
| 51 24.20 22.47 -12.37 -59.11 146.53 27.59 38.35
52 41.44 33.19 6.03 -49.90 59.91 36.65 57.53
| 53 -3.91 -3.55 -14.21 -66.15 -23.20 -4.42 -7.21
| 58 11.38 56.40 11.65 -13.42 -9.38 9.59 8.47
Average 15.38 -25.29 -24.46 -90.85 24.67 13.46 28.64
125