ALL ETFs ARE NOT ALIKE

EDHEC DAYS - MARCH 2013 ALL ETFs ARE NOT ALIKE MEASURING THE EFFICIENCY OF ETF

ETF FAST MARKET EXPANSION GOES WITH MULTIPLICATION PRODUCTS TRACKING A SAME INDEXATION TEND TO MULTIPLY 2 European ETF assets 10 year CAGR: 40%, 2012: + 23.5% (source: ETFGI) 1332 ETFs listed in Europe in Dec. 2012 vs. 422 in 2007 Multiplication of ETFs tracking the same index 21 ETFs globally tracking the Euro STOXX 50, including 15 listed in Europe 700 NYSE Euronext European Markets - Number of listed ETF 600 500 400 300 200 100 0 592 578 449 491 348 184 26 38 44 50 84 131 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Number of ETFs listed at end of year LYXOR ETF MARCH 2013

ALL ETF ARE NOT ALIKE 3 PERFORMANCE RELATIVE TO THE BENCHMARK Euro STOXX 50 0.40% 2011 2012 0.70% 0.30% 0.20% 0.10% 0.60% 0.50% 0.40% 0.30% Up to 40 bps 0.00% -0.10% -0.20% Up to 45 bps 0.20% 0.10% 0.00% -0.10% Annual total return difference can represent p to 0.50% on European and US Equities PAST PERFORMANCE IS NO GUARANTEE OF FUTURE RESULTS. IT SHOULD NOT BE ASSUMED THAT THE PERFORMANCE OF THE INDEX IN THE FUTURE WILL BE COMPARABLE TO THE PERFORMANCE INFORMATION PRESENTED HERE. LYXOR ETF MARCH 2013

ALL ETF ARE NOT ALIKE 4 PERFORMANCE RELATIVE TO THE BENCHMARK MSCI USA 2011 2012 0.20% 0.20% 0.10% 0.15% 0.00% 0.10% -0.10% -0.20% Up to 45 bps 0.05% 0.00% Up to 25 bps -0.30% -0.05% -0.40% -0.10% Annual total return difference can represent p to 0.50% on European and US Equities PAST PERFORMANCE IS NO GUARANTEE OF FUTURE RESULTS. IT SHOULD NOT BE ASSUMED THAT THE PERFORMANCE OF THE INDEX IN THE FUTURE WILL BE COMPARABLE TO THE PERFORMANCE INFORMATION PRESENTED HERE. LYXOR ETF MARCH 2013

RELATIVE PERFORMANCE TO THE BENCHMARK ALL ETF ARE NOT ALIKE TRACKING ERROR VOLATILITY 5 Do all ETFs offer the same quality of tracking? Volatility of tracking error can cost up to 2% on MSCI Emerging Markets 1.00% 0.50% 0.00% Performance volatility impact close to 0bp -0.50% -1.00% -1.50% Performance volatility impact up to 2% -2.00% déc. 11 févr. 12 avr. 12 juin 12 août 12 oct. 12 déc. 12 LYXOR vs MSCI EM NTR COMPETITOR vs MSCI EM NTR PAST PERFORMANCE IS NO GUARANTEE OF FUTURE RESULTS. IT SHOULD NOT BE ASSUMED THAT THE PERFORMANCE OF THE INDEX IN THE FUTURE WILL BE COMPARABLE TO THE PERFORMANCE INFORMATION PRESENTED LYXOR HERE. ETF MARCH 2013

WHY DEVELOPING AN ETF EFFICIENCY MEASURE CONSISTENT WITH INITIATIVES FROM REGULATORS 6 ESMA guidelines, effective beginning of 2013 UCITS ETF providers are required to disclose predictions of Tracking Difference and Tracking Error The difference between the actual and the predicted tracking performance will need to be explained Pushes to more accuracy and predictability of Tracking Difference and Tracking Error Will highlight differences between ETF providers: a game changer? LYXOR ETF MARCH 2013

ETF ARE HETEROGENEOUS 7 ETFs tracking a same index are expected to be the same Indices are a theoretic construction, not a portfolio Many determinants = many outcomes Securities lending revenues Swap Cash drag TER ETF trading spreads Taxation Fund domicile ETF taxation Duration of investment Is there a simple way to capture all these differences? LYXOR ETF MARCH 2013

WHY DEVELOPING AN ETF EFFICIENCY MEASURE 8 Traditional tools are unadapted although frequently used Legacy from active management: measuring Alpha, measuring active risk budget Are they relevant for ETFs? Existing individual indicators mean nothing when taken separately Tracking Difference Tracking Error Volatility Trading Spreads Strong need for a comprehensive and accurate tool to assess and compare ETFs Investor goal: achieve investment results that correspond to the return of the index over its holding period How to select the right product that maximizes chances to achieve this goal? Until recently, there has been a lack of academic attention to the issue LYXOR ETF MARCH 2013

Measuring Efficiency of Exchange Traded Funds 1 An Issue of Performance, Tracking Error and Liquidity Thierry Roncalli Evry University & Lyxor Asset Management, France Joint work with Marlène Hassine March 18, 2013 1 The opinions expressed in this presentation are those of the author and are not meant to represent the opinions or official positions of Lyxor Asset Management. Thierry Roncalli Measuring Performance of Exchange Traded Funds 1 / 30

How to obtain the academic paper? Measuring Performance of Exchange Traded Funds Marlène Hassine ETF Strategy Lyxor Asset Management, Paris marlene.hassine@lyxor.com February 2013 Abstract Thierry Roncalli Research & Development Lyxor Asset Management, Paris thierry.roncalli@lyxor.com Fund selection is an important issue for investors. This topic has spawned abundant academic literature. Nonetheless, most of the time, these works concern only active management, whereas many investors, such as institutional investors, prefer to invest in index funds. The tools developed in the case of active management are also not suitable for evaluating the performance of these index funds. This explains why information ratios are usually used to compare the performance of passive funds. However, we show that this measure is not pertinent, especially when the tracking error volatility of the index fund is small. The objective of an exchange traded fund (ETF) is precisely to offer an investment vehicle that presents a very low tracking error compared to its benchmark. In this paper, we propose a performance measure based on the value-at-risk framework, which is perfectly adapted to passive management and ETFs. Depending on three parameters (performance difference, tracking error volatility and liquidity spread), this efficiency measure is easy to compute and may help investors in their fund selection process. We provide some examples, and show how liquidity is more of an issue for institutional investors than retail investors. Keywords: Passive management, index fund, ETF, information ratio, tracking error, liquidity, spread, value-at-risk. JEL classification: G11. 1 Introduction The market portfolio concept has a long history and dates back to the seminal work of Markowitz (1952). In that paper, Markowitz defines precisely what portfolio selection means: the investor does (or should) consider expected return a desirable thing and variance of return an undesirable thing. Indeed, Markowitz shows that an efficient portfolio is a portfolio that maximizes the expected return for a given level of risk (corresponding to the variance of return). Markowitz concludes that there is not only one optimal portfolio, but a set of optimal portfolios called the efficient frontier (represented by the solid blue curve in Figure 1). By studying liquidity preference, Tobin (1958) shows that the efficient frontier becomes a The academic paper can be downloaded from: Social Science Research Network (SSRN) http://ssrn.com/abstract=2212596 Munich Personal RePEc Archive http://mpra.ub.uni-muenchen.de/44298 Research Archive in Economics http: //ideas.repec.org/p/pra/mprapa/44298.html We are profoundly grateful to Bou Ly Wu for his support with data management and the computation of liquidity spreads on limit order books. We would also like to thank Arnaud Llinas, Raphaël Dieterlen, Valérie Lalonde, François Millet and Matthieu Mouly for their helpful comments. 1 Electronic copy available at: http://ssrn.com/abstract=2212596 Thierry Roncalli Measuring Performance of Exchange Traded Funds 2 / 30

Main results 1 Current rating systems are not adapted to index funds. 2 The information ratio can not be used to measure the performance of ETFs. 3 The efficiency measure of an exchange traded fund is a function of three main parameters: performance relative to the benchmark, tracking error volatility and liquidity spread. 4 The efficiency measure is the right statistic to measure the performance of ETFs. 5 For institutional investors and active managers, the efficiency measure is principally driven by the liquidity. Thierry Roncalli Measuring Performance of Exchange Traded Funds 3 / 30

Outline 1 Measuring the efficiency of exchange traded funds Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs 2 An application to European ETFs Different benchmarks 3 Choosing another risk measure The liquidity issue 4 5 Thierry Roncalli Measuring Performance of Exchange Traded Funds 4 / 30

Why index funds? Main result of Sharpe (1964): Tangency Portfolio = Market (Capitalization) Portfolio. Jensen (1968): No alpha in mutual funds. Wells Fargo Bank (1971): First (private) index fund. Wells Fargo/American National Bank in Chicago (1973): First S&P 500 index fund. Carhart (1997): No persistence in mutual fund performance. Thierry Roncalli Measuring Performance of Exchange Traded Funds 5 / 30

Performance or efficiency? Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs Fund picking process Current rating systems = measure the alpha and its persistence with respect to the right risk factors 1 How to define the universe of funds? 2 How to measure the alpha? Fund picking is different with passive management. 1 The categorization of funds is not an issue. 2 α is not the relevant measure to assess the performance of index funds. What is a good ETF? A fund that presents no risk wrt. to the index. Thierry Roncalli Measuring Performance of Exchange Traded Funds 6 / 30

Comparing benchmarked portfolios Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs x 2 is preferable to x 1 because it has a better excess return for the same tracking-error volatility. x 4 is a portfolio with 75% of x 2 and 25% of the benchmark. x 2 is preferable to x 3 because x 4 is better than x 3. Fundamental rule of benchmarked portfolios x is preferable to y IR(x b) IR(y b) Thierry Roncalli Measuring Performance of Exchange Traded Funds 7 / 30

Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs The irrelevance of the information ratio for ETFs Excess Tracking Error Information ETF Performance Volatility Ratio x 1 0.01 0.01 1.00 x 2 0.05 0.07 0.71 x 3 0.06 0.08 0.75 x 4 0.40 0.50 0.80 Selection criteria based on the information ratio x 1 is the worst ETF! x 4 is the best ETF! x 3 is better than x 2 but the difference is not statistically significant! Thierry Roncalli Measuring Performance of Exchange Traded Funds 8 / 30

Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs The irrelevance of the information ratio for ETFs Using the previous rule, x 1 is better than x 2. The problem is that we cannot replicate the benchmark exactly. The portfolio x 3 can not be reached. In the real life, we need to use an ETF x 0 to proxy the benchmark. In this case, x 2 is better than x 1 if we target a tracking error volatility equal to 15 bps. For benchmarked funds with low tracking-error volatility: IR(x b) > IR(y b) x y Investors, who consider the information ratio, will never chose the ETF x 0! Thierry Roncalli Measuring Performance of Exchange Traded Funds 9 / 30

The framework Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs The two-period trading model The investor buy the ETF at time t = 0 and sells it at time t = 1. The relative loss of the investor with respect to the benchmark is the bid-ask spread minus the realized tracking error. At time t = 0, the loss function is random and is not known. The ETF efficiency measure is a risk measure applied to the loss function of the investor. Thierry Roncalli Measuring Performance of Exchange Traded Funds 10 / 30

Illustration What is your best ETF? Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs Thierry Roncalli Measuring Performance of Exchange Traded Funds 11 / 30

Definition of the efficiency measure Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs We propose to use value-at-risk, which is today commonly accepted as a standard risk measure. In this case, the efficiency measure is defined as the quantile of the loss function with a confidence level α. Definition The efficiency measure of the ETF x with respect to the benchmark b corresponds to: eff α (x b) = er(x b) s(x b) c α tev(x b) where er(x b) is the expected value of the tracking error (or the excess return), s (x b) is the bid-ask spread, c α is a scalar that depends on the confidence level (c 95% = 1.65) and tev(x b) is the volatility of the tracking error ( ). (*) IOSCO terminology: er(x b) = Tracking Difference (TD) & tev(x b) = Tracking Error (TE). Thierry Roncalli Measuring Performance of Exchange Traded Funds 12 / 30

Computing the efficiency measure Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs We assume that er(x b) = 50 bps, tev(x b) = 40 bps and s (x b) = 20 bps. The confidence level α is set to 95%. The efficiency measure of the ETF eff α (x b) is 35.79 bps. Thierry Roncalli Measuring Performance of Exchange Traded Funds 13 / 30

Performance or efficiency? Information ratio as a selection criteria Efficiency indicator for ETFs Impact of parameters on the efficiency measure Thierry Roncalli Measuring Performance of Exchange Traded Funds 14 / 30

An application to European ETFs Different benchmarks Impact of parameters on the efficiency measure α = 95%. Study period: Nov., 30 th 2011 No., 30 th 2012. We compute the spread of the first limit order for each listing place and each trading day t. The daily spread is then the weighted average by considering the daily volume of the different listing places. The spread s (x b) is therefore the average of daily spreads. We rebuild the net asset value of the ETF by incorporating dividends in order to compute the excess return er(x b) and the tracking error volatility tev(x b). We consider e as the default currency. PERF: annualized performance (in %); ER: annualized relative performance of the ETF vs. the benchmark (in bps); SPREAD = spread (in bps); VOL: annualized volatility (in %); TEV: annualized tracking error volatility (in bps); EFF: efficiency measure (in bps). Thierry Roncalli Measuring Performance of Exchange Traded Funds 15 / 30

Results Measuring the efficiency of exchange traded funds An application to European ETFs Different benchmarks ETF PERF ER SPREAD VOL TEV EFF Amundi 15.29 62.56 9.84 22.33 11.97 32.98 db X-trackers 15.33 65.97 12.27 22.86 7.31 41.64 ishares (DE) 15.05 37.88 7.96 21.62 56.54 63.38 ishares 15.25 58.46 10.39 21.92 19.62 15.70 Lyxor 15.30 63.51 8.48 22.01 14.89 30.47 Source 14.90 23.51 15.38 22.23 7.25 3.83 Eurostoxx 50 14.67 22.09 ETF PERF ER SPREAD VOL TEV EFF Amundi 19.49 9.19 16.97 12.59 3.14 12.97 Credit Suisse 19.57 16.99 18.23 12.50 4.63 8.88 db X-trackers 19.56 16.04 18.26 12.79 4.65 9.90 HSBC 19.68 28.20 20.58 12.68 3.45 1.92 ishares 19.34 6.10 7.45 12.56 4.90 21.63 Lyxor 19.60 19.87 13.56 12.59 0.98 4.69 Source 19.34 5.30 17.81 12.87 1.78 26.04 UBS 19.40 0.02 41.13 12.85 0.59 42.09 S&P 500 19.40 12.76 Thierry Roncalli Measuring Performance of Exchange Traded Funds 16 / 30

Results Measuring the efficiency of exchange traded funds An application to European ETFs Different benchmarks ETF PERF ER SPREAD VOL TEV EFF Amundi 17.40 20.75 23.91 10.32 3.67 50.72 Commerzbank 17.42 18.25 18.72 10.83 3.56 42.85 db X-trackers 17.36 24.46 13.20 10.30 25.18 79.20 ishares 17.18 42.08 13.75 10.14 50.80 139.65 Lyxor 17.37 23.09 11.93 10.09 1.55 37.58 Source 17.08 52.64 24.83 10.31 1.69 80.25 UBS 17.36 23.98 31.25 10.39 14.51 79.16 MSCI World 17.60 10.18 ETF PERF ER SPREAD VOL TEV EFF Credit Suisse 13.20 205.79 30.05 13.18 150.68 484.46 db X-trackers 14.13 112.34 15.85 13.07 12.83 149.35 ishares 14.18 107.56 17.90 13.07 160.21 389.80 Lyxor 14.45 80.01 20.72 13.07 14.87 125.28 Source 14.16 109.20 50.30 13.22 3.96 166.02 MSCI EM 15.25 13.12 Thierry Roncalli Measuring Performance of Exchange Traded Funds 17 / 30

The case of different benchmarks An application to European ETFs Different benchmarks Problem ETF providers do not choose the same benchmark to give access to an asset class (e.g. Japanese equities with Topix and MSCI Japan). Answer Use the Amenc and Martellini (2002) PCA method to build a reference index. ETF PERF ER SPREAD VOL TEV EFF db X-trackers 0.72 35.87 15.82 16.58 256.73 475.29 ishares 0.76 40.64 19.98 15.97 66.84 170.90 Lyxor 1.29 92.96 14.98 16.10 62.81 211.58 Source 0.67 31.40 35.11 16.58 65.30 174.25 Japanese equities 0.36 15.84 Thierry Roncalli Measuring Performance of Exchange Traded Funds 18 / 30

Choosing another risk measure Choosing another risk measure The liquidity issue TEV: tracking error volatility. SV: semi-variance of tracking errors. VaR: historical value-at-risk. CF: Cornish-Fisher value-at-risk by considering the skewness and kurtosis of tracking errors. ES: expected shortfall. Thierry Roncalli Measuring Performance of Exchange Traded Funds 19 / 30

Choosing another risk measure Results Choosing another risk measure The liquidity issue Eurostoxx 50 ETF TEV SV VaR ES CF Amundi 32.98 44.72 47.75 47.25 62.54 db X-trackers 41.64 44.12 45.90 41.02 47.15 ishares (DE) 63.38 64.32 59.40 102.51 62.72 ishares 15.70 20.38 34.77 12.21 27.93 Lyxor 30.47 41.63 47.97 38.26 61.48 Source 3.83 3.74 5.00 4.63 19.17 S&P 500 ETF TEV SV VaR ES CF Amundi 12.97 10.64 10.24 10.55 8.04 Credit Suisse 8.88 8.26 6.60 10.37 7.35 db X-trackers 9.90 10.13 8.88 11.77 10.02 HSBC 1.92 2.92 4.24 2.23 3.73 ishares 21.63 21.15 19.10 23.59 20.23 Lyxor 4.69 4.73 4.88 4.25 4.71 Source 26.04 25.67 25.50 26.00 25.37 UBS 42.09 41.98 41.93 42.09 41.95 Thierry Roncalli Measuring Performance of Exchange Traded Funds 20 / 30

The liquidity issue Choosing another risk measure The liquidity issue Issue with the previous spread definition Institutional investors buy or sell a notional N, that can not generally be executed via the best first limit orders. Definition of the liquidity spread We then consider another spread measure s N (x b) corresponding to intraday spreads weighted by the duration between two ticks for a given notional. We have: s N (x b) = close j=open s j (t j+1 t j ) close j=open (t j+1 t j ) where s j is the spread of the j th tick in order to trade a notional N and t j+1 t j the elapsed time between two consecutive ticks. Thierry Roncalli Measuring Performance of Exchange Traded Funds 21 / 30

The liquidity issue Evolution of the spread of the Amundi Eurostoxx 50 ETF Choosing another risk measure The liquidity issue The liquidity spread increases with the notional: N 1 N 2 s N1 (x b) s N2 (x b) Thierry Roncalli Measuring Performance of Exchange Traded Funds 22 / 30

The liquidity issue Boxplot(*) of Eurostoxx 50 ETF spreads Choosing another risk measure The liquidity issue (*) The boxplot indicates the minimum value, the quartile range, the median and the last decile. Thierry Roncalli Measuring Performance of Exchange Traded Funds 23 / 30

Choosing another risk measure The liquidity issue The liquidity issue Impact of the liquidity on the efficiency measure (Eurostoxx 50) The efficiency measure becomes: eff α (x b) = er(x b) F 1 s N (α) c α tev(x b) where F sn is the distribution of the liquidity spread s N. Eurostoxx 50 (α = 95%) ETF 100 KEUR 1 MEUR 2 MEUR Amundi 30.19 25.45 15.00 db X-trackers 43.28 2.39 66.68 ishares (DE) 65.05 77.04 98.73 ishares 15.72 10.33 5.31 Lyxor 27.89 24.98 19.97 Source 8.80 64.29 193.50 The efficiency measure is not the same for retail investors and institutional investors! Thierry Roncalli Measuring Performance of Exchange Traded Funds 24 / 30

The liquidity issue What about active managers? Choosing another risk measure The liquidity issue Generalization to the multi-period model If we consider a multi-period model with m trades, the performance measure becomes: eff α (x b) = er(x b) m s N (x b) c α tev(x b) This formula highlights the importance of liquidity for active managers. Remark A highly active manager will only be interested in the spread measure because: lim m eff α (x b) = m s N (x b) Thierry Roncalli Measuring Performance of Exchange Traded Funds 25 / 30

1 Current rating systems are not adapted to index funds. 2 The information ratio can not be used to measure the performance of ETFs. 3 The efficiency measure of an exchange traded fund is a function of three main parameters: performance relative to the benchmark, tracking error volatility and liquidity spread. 4 The efficiency measure is the right statistic to measure the performance of ETFs. 5 For institutional investors and active managers, the efficiency measure is principally driven by the liquidity. Thierry Roncalli Measuring Performance of Exchange Traded Funds 26 / 30

Analytical expression of the spread s N (x b) We define the daily spread s N (x b) as a weighted average of intraday spreads: j=open s j close j=open s N (x b) = close ( tj+1 t j ) ( tj+1 t j ) where s j is the spread of the j th tick and t j+1 t j the elapsed time between two consecutive ticks: s j = c j ( P + j P j P 0 j ) We have also: P j = K k=1 Q j,k P j,k K k=1 Q j,k where P + j,k (resp. P j,k ) is the ask (or bid) price at t j for the k th limit order. The average mid price P 0 j corresponds to: P 0 j = P + j + P j 2 Thierry Roncalli Measuring Performance of Exchange Traded Funds 27 / 30

Analytical expression of the spread s N (x b) The quantity Q + j,k and Q j,k are defined as follows: Q j,k = max (0,min ( l =1 Q j,k,q j k 1 Q j,l )) Here, Q + j,k and Q j,k are the ask and bid volumes of the kth limit order. The reference quantity is the ratio between the trading notional N and the mid price: Q j Q j = N P 0 j Sometimes it may appear that the trading volume on the order book is lower than the notional N. That is why the factor c j may be greater than one: c j = max 1, Q j ( min K k=1 Q+ j,k, K k=1 Q j,k For instance, if we wish to execute an order of 2 MEUR and there is only a trading volume of 1 MEUR, we multiply the spread by two. ) For each trading day, we compute the daily spread for the different listing places using the previous formulas and we take the best spread. Thierry Roncalli Measuring Performance of Exchange Traded Funds 28 / 30

Example The limit order book k Buy orders Sell orders Q j,k P j,k Q + j,k P + j,k 1 900 85.90 600 86.05 2 200 85.85 300 86.06 3 57 85.82 400 86.20 4 18 85.75 213 86.21 5 117 85.74 73 86.22 6 1000 85.73 200 86.23 7 3000 85.72 1500 86.25 It corresponds to a notional N = Q P 0 j 85981 e. of Computing the spread for Q = 1000 k Buy orders Q j,k P j,k Sell orders Q + j,k P + j,k 1 900 85.90 600 86.05 2 100 85.85 300 86.06 3 0 85.82 100 86.20 K Q k=1 j,k 1000 1000 P j 85.89 86.07 We deduce that: and: P 0 j = s j = 85.89 + 86.07 2 86.07 85.89 85.98 = 85.98 = 20.12 bps Thierry Roncalli Measuring Performance of Exchange Traded Funds 29 / 30

Example Given a notional N, we find the optimal value of Q by solving the nonlinear inequality: Q = inf { Q N : Q P 0 j N } k N = 100 KEUR N = 500 KEUR Buy orders Sell orders Buy orders Sell orders Q j,k P j,k Q + j,k 1 900 85.90 600 86.05 900 85.90 600 86.05 2 100 85.85 300 86.06 200 85.85 300 86.06 3 57 85.82 263 86.20 57 85.82 400 86.20 4 6 85.75 0 86.21 18 85.75 213 86.21 5 0 85.74 0 86.22 117 85.74 73 86.21 6 0 85.73 0 86.23 1000 85.73 200 86.23 7 0 85.72 0 86.25 3000 85.72 1500 86.25 K Q k=1 j,k 1163 1163 5292 3286 P j 85.89 86.09 85.76 86.19 s j 23.24 bps 87.81 bps P + j,k Q j,k P j,k Q + j,k P + j,k Thierry Roncalli Measuring Performance of Exchange Traded Funds 30 / 30