Green exchange-traded fund performance evaluation using the EU-EV risk model

Green Exchange-Traded Fund Performance Evaluation Using the EU-EV Risk Model Irene Brito1  [0000−0002−7075−3265], Jos´e Manuel Azevedo2[0000−0001−6951−4278], and Ana Isabel Azevedo2[0000−0003−0882−3426] 1Center of Mathematics, Department of Mathematics, University of Minho, 4800-045 Guimar˜aes, Portugal, [email protected] 2CEOS.PP, ISCAP, Polytechnic of Porto, 4465-004 S. Mamede de Infesta, Portugal, [email protected], [email protected] Abstract. This work evaluates the performance of green exchange-traded funds (ETFs) using the expected utility, entropy and variance (EU-EV) risk model. Data from 14 green ETFs analysed in earlier literature in the in-sample period from January 2008 to December 2010 are used. The green ETFs are ranked according to their risk, considering the returns’ expected utility, entropy and variance, and the best-ranked ETFs are selected to construct equally weighted portfolios. Then, the performance of the green ETFs portfolios is evaluated and compared with those of the S&P500 Index. Cumulative returns in in-sample and out-of-sample periods and performance metrics, such as Maximum drawdown, Sharpe ratio, Sortino ratio, Beta and Alpha, are analysed. The results show that, in general, the equally weighted portfolios formed with half the number of best-ranked ETFs outperform the benchmark index in the in-sample period and for specific time ranges in the out-of-sample periods. Keywords: ETF ·Portfolio performance evaluation ·EU–EV risk model 1 Introduction Exchange-traded funds (ETFs) are popular financial instruments, made up of different securities (e.g. stocks, bonds), that can be traded on an exchange and that offer a great diversification, tax efficiency and low expenses (see e.g. [1], [2], [3]). Among these, green ETFs are funds that invest in companies that support environmentally responsible technologies or are involved with research, development, production and provision of alternative energy and possess positive environmental, social and corporate governance (ESG) characteristics. Due to the increasing interest in sustainable economic development and due to the increased financialization of green energy coupled with the interest of investors in this asset class, see e.g. the recent study in [4] and references therein, it is relevant to analyse in more detail the risk and performance of green ETFs, since one can find only few contributions addressing this issue. Sabbaghi [5] investigated the time-series behaviour of green exchange traded fund returns and their associated conditional volatility dynamics using the GARCH methodology. In [6], 2 Irene Brito et al. Sabbaghi constructed an equally-weighted portfolio of green ETFs and analysed its return performance (using e.g. mean, standard deviation, Jensen’s alpha) and compared it to that of the S&P500 index over different sub-periods in time from 2005 to 2010, before and after the financial market collapse of 2008. He found that the portfolio outperformed the S&P500 index prior to the financial collapse, however the portfolio was highly volatile and underperformed the S&P500 index in the period after the collapse. Tsolas and Charles [7] investigated the performance of green ETFs using data envelopment analysis. Rizvi et al. [4] studied the relationship between green and grey energy ETFs and concluded that green energy is more prominent in determining the returns in the US equity market. The objective of the present work is to analyse the risk and performance of green ETFs using the recently proposed EU–EV risk model and evaluate their performance using different metrics. The EU–EV risk model, developed in [8],[9], can be used for classifying stock risks and for the preselection of the best ranked stocks in order to construct optimum portfolios with a reduced number of stocks (see [10],[11]). The aim is now to apply the EU–EV risk model to green ETFs in order to assess their risk (using expected utility, entropy and variance) and in order to investigate the capability of the risk model to select the efficient funds for investment purposes. Data of a sample of 14 green ETFs from 2008 to 2010 are used, that were analysed in earlier literature ([6],[7]). The ETFs are ranked according to their risk and different equally-weighted portfolios are formed. The portfolios’ performances are analysed and compared with the S&P500 index considering different time periods. This paper is structured as follows. Section 2 contains the definitions of the EU–EV risk model and of the performance measures and explains how the ETF portfolios are constructed and their performances are evaluated. In section 3, the proposed method is applied to the sample of 14 green ETFs. The risks of the ETFs are analysed, different portfolios are formed with the best ranked ETFs and their performances are compared with those of the benchmark in the in-sample period and in different out-of-sample time intervals. The paper ends with the Conclusions in section 4. 2 ETF Portfolio Construction and Performance Evaluation In order to construct the ETF portfolios, the EU–EV risk model will be used to select the best ranked ETFs from an initial given set of ETFs. The selected ETFs will then be used to build equally weighted portfolios, whose performance is then analysed using different performance metrics. 2.1 EU-EV Risk Model Consider a set of ETFs S={S1, . . . , SI}and the action space A={a1, . . . , aI}, where ai= (xi1, pi1;xi2, pi2;. . . ;xiN , piN )∈A Green ETF performance evaluation 3 is the action of selecting the ETF Si,i= 1, . . . , I, yielding the frequency distribution of returns, where xin are the outcomes, occurring with probabilities pin,n= 1, . . . , N, represented by the random variable Xi. The EU–EV risk of the action aidepends on the corresponding returns’ distribution with random variable Xias follows. Definition 1 (EU–EV risk). The EU–EV risk for the action aiwith associated random variable Xiis defined by Rλ(Xi) = λ 2 H(Xi) + Var[Xi] max ai∈A{Var[Xi]} −(1 −λ)E[u(Xi)] max ai∈A{|E[u(Xi)]|}, where 0≤λ≤1,u(x) = ln(1 + x), x ≥0 −ln(1 −x), x < 0is the utility function and H(Xi) = −PN n=1 pin ln pin is the entropy. The constant λis a trade-off parameter that combines the expected utility and the uncertainty reflected by entropy [12] and variance [13] and it can be used to express the different risk attitudes of decision-makers (see [9] for more details). If λ < 0.5, then more weight is given to the expected utility term, which corresponds to a risk-averse attitude [14]. If λ > 0.5, more weight is given to the uncertainty component and this reflects a risk-seeking behaviour. The trade-off parameter is used in the model for: building portfolios that strike a balance between risk and return (portfolio risk models), understanding and navigating uncertainty in financial markets (entropy and uncertainty), evaluating the fluctuation and potential dangers associated with assets and derivatives (variance). Gaining a solid grasp of these concepts enables investors and financial professionals to make well-informed decisions, effectively manage risk, and maximise the potential of their portfolios. The ETFs are classified with the EU-EV risk model by ranking them according to the following rule. Given two ETFs Si1and Si2,i1, i2∈ {1, . . . , I}, if Rλ(Xi1)< Rλ(Xi2), then Si1is preferred over Si2(which can be written as Si1≻Si2), since Si1has lower EU–EV risk than Si2. For λ∈[0,1], we will select from the set Sthe I/2 best ranked ETFs and form subsets (with the initial number of funds reduced to the half) for the different values of λ. With these subsets we will construct equally weighted portfolios, whose performance will then be compared with those of the equally weighted portfolio consisting of all Ifunds and with a benchmark portfolio. According to results presented in earlier literature, for example in [15], [16], using equalweighted strategies lead to portfolios outperforming value-weighted strategies, therefore we opted to use equal weights in the portfolio construction. 4 Irene Brito et al. 2.2 Performance Measures The following performance measures (see e.g. [17],[18]) will be used to analyse the performance of the portfolios in different time periods. Definition 2 (Maximum Drawdown). The Maximum Drawdown (MDD) is the largest percentage drop in total returns from the start to the end of a period, computed over all intervals of time that can be formed within a specified interval of time, and it is defined as follows. Let xt,t= 1, . . . , T, represent the daily cumulative returns of the portfolio. The Maximum Drawdown is given by MDD =PV −LV PV , where LV =x∗ t, the lowest point value (trough value), and PV =x∗ t,max, the peak value, are the values that maximize the drawdown DDt=xt,max −xt xt,max , where xt,max = max{xs:s= 1, . . . , t}for t= 1, . . . , T . A lower MDD value indicates a lesser degree of risk. In the following definitions, rPrepresents the expected return of the portfolio, σPis the standard deviation of the portfolio returns and rfis the risk-free rate. We will consider rf= 0, meaning that the rate of return of a zero risk benchmark investment is taken to be equal to zero. Definition 3 (Sharpe ratio, Sortino ratio, Beta). The Sharpe ratio measures the excess return (the return of the portfolio less the risk-free rate of interest) per unit of total risk of the portfolio (the standard deviation of the portfolio’s returns) and is defined by Sharpe =rP−rf σP . The Sharpe ratio with rf= 0 quantifies the relation between the expected returns and the standard deviation of the returns of the portfolio. Portfolios with higher Sharpe ratios perform better according to this measure. The Sortino ratio is a modification of the Sharpe ratio, where only the downside deviation is taken into account, and is expressed by Sortino =rP−rf σ− P , where σ− Prepresents the standard deviation of the negative portfolio returns. A higher Sortino ratio indicates a better performance. The risk metric Beta determines the risk or volatility of a portfolio by comparing it to the market and is defined by Beta =Cov(rP, rS) σ2 S , Green ETF performance evaluation 5 where Cov(rP, rS)is the covariance between the expected return of the portfolio and the expected market return rSof the benchmark S, and σ2 Scorresponds to the variance of the market returns. Portfolios having Beta>1can be interpreted to be more volatile or riskier than the benchmark. If Beta<1, the portfolio is less volatile than the benchmark. If Beta= 1, it has the same volatility as the benchmark. Definition 4 (Alpha). Jensen’s Alpha is a performance metric that measures the portfolio return relative to the market return and is defined by Alpha =rP−[rf+Beta(rS−rf)], where Beta is given in Definition 3. A value of Alpha>0indicates that the portfolio has performed better than the market index. If Alpha<0, the portfolio has underperformed the market index. If Alpha= 0, the portfolio’s performance is in line with that of the market. 3 Application to Green ETFs The aim is to apply the EU-EV risk model to the selection of funds from a sample of 14 green ETFs in order to investigate if the EU-EV model adequately selects the relevant ETFs for an efficient portfolio construction with a reduced number of ETFs. Data of a sample of 14 green ETFs (with ticker symbols PBW, PHO, PUW, PKN, PIO, PZD, EVX, NLR, FIW, QCLN, CGW, DSI, KLD, PBD), described in [6] and [7], in the in-sample period from January 2008 to December 2010 are used. 3.1 Portfolio Construction From each green ETF Si,i= 1,...,14, the daily closing prices {Pi0, . . . , PiT }, T+ 1 = 756, from January 2008 to December 2010 are collected. The daily returns are calculated by rit = ln Pit Pi(t−1) ;i= 1,...,14; t= 1,...,755. The frequency distribution of returns is determined is follows. The interval [rmin, rmax] = [−1.0114,1.0205] where rmin = min 1≤i≤14{ri1, . . . , ri755}and rmax = max 1≤i≤14{ri1, . . . , ri755}, is divided into N= 19 subintervals Jn,n= 1, . . . , N, of length ∆=rmax−rmin N= 0.10694. Then, the relative frequency of the return of Siin the subinterval Jnis calculated by pin =|{rit ∈Jn:t= 1, . . . , T}| T 6 Irene Brito et al. and the expected return of Sifrom the subinterval Jnis estimated by xin =1 |{rit ∈Jn:t= 1, . . . , T}| X rit∈Jn t=1,...,T rit, where |·|represents the cardinality of a set. The EU–EV risks for each fund Si,i= 1,...,14, will then be determined using the frequency distributions. Table 1 contains the EU–EV risks for the 14 green ETFs for certain values of λ(λ= 0,0.5,0.75,0.85,1). For each value of λ, the 7 lower risk values are highlighted in bold. Table 1. EU–EV risks for λ= 0,0.5,0.75,0.85,1. Fund Ticker R(ai) λ= 0 λ= 0.5λ= 0.75 λ= 0.85 λ= 1 S1PBW 1.0000 0.5934 0.3901 0.3088 0.1868 S2PHO 0.0894 0.0989 0.1037 0.1056 0.1085 S3PUW -0.0072 0.0636 0.0991 0.1132 0.1345 S4PKN 0.1541 0.4380 0.5800 0.6367 0.7219 S5PIO 0.1917 0.1404 0.1147 0.1044 0.0890 S6PZD 0.2849 0.2022 0.1609 0.1443 0.1195 S7EVX -0.0386 0.0147 0.0414 0.0521 0.0681 S8NLR 0.3383 0.2189 0.1593 0.1354 0.0996 S9FIW -0.0410 0.0231 0.0551 0.0680 0.0872 S10 QCLN 0.6343 0.4011 0.2846 0.2380 0.1680 S11 CGW 0.2021 0.1407 0.1100 0.0977 0.0793 S12 DSI 0.0787 0.0757 0.0742 0.0736 0.0727 S13 KLD 0.0806 0.0657 0.0582 0.0552 0.0570 S14 PBD 0.8447 0.5029 0.3320 0.2636 0.1611 Considering the range of λ∈[0,1], one can form the following 5 different subsets G7,λ=0 ={PHO,PUW,PKN,EVX,FIW,DSI,KLD}, λ ∈[0,0.0561) G7,λ=0.5={PHO,PUW,PIO,EVX,FIW,DSI,KLD}, λ ∈[0.0561,0.5175) G7,λ=0.75 ={PHO,PUW,EVX,FIW,CGW,DSI,KLD}, λ ∈[0.5175,0.8139) G7,λ=0.85 ={PHO,PIO,EVX,FIW,CGW,DSI,KLD}, λ ∈[0.8139,0.9655) G7,λ=1 ={PIO,EVX,NLR,FIW,CGW,DSI,KLD}, λ ∈[0.9655,1], where the best 7 funds with lower risk were selected, with λbelonging to the intervals [0,0.0561), [0.0561,0.5175), [0.5175,0.8139), [0.8139,0.9655), [0.9655,1]. Each particular value of λin Table 1 belongs to one of these intervals, so that the notations G7,λ=0,G7,λ=0.5,G7,λ=0.75,G7,λ=0.85 and G7,λ=1 are used to represent the different sets. Green ETF performance evaluation 7 The portfolios are constructed as equal weighted combinations of the seven best ranked funds from the five sets. The portfolios will be denoted by G7,λ=0, G7,λ=0.5,G7,λ=0.75,G7,λ=0.85 and G7,λ=1. In the following analysis, we will use the S&P500 index as benchmark portfolio. We will also consider the equal-weighted portfolio G14 formed with all 14 green ETFs G14 ={PBW, PHO, PUW, PKN, PIO, PZD, EVX, NLR, FIW, QCLN, CGW, DSI, KLD, PBD} and the equal-weighted portfolio G4formed with the green ETFs of the set G4={EVX,NLR,FIW,CGW}, These ETFs were identified in an earlier study (see [7]) as top efficient ETFs using data envelopment analysis. 3.2 Performance Evaluation The performance of the five portfolios constructed with the best ranked ETFs will be analysed and compared with the performance of the G14 and G4portfolios and with the S&P500 benchmark portfolio, first in the in-sample period from January 2008 to December 2010 and then in different out-of-sample periods, from January 2011 to December 2020, with one-year, two-year, five-year and ten-year time horizons. For that puprpose, the cumulative returns are calculated in the mentioned time periods, along with the metrics presented in Section 2.2, using the S&P500 index as benchmark S. Considering the in-sample period from January 2008 to December 2010, Figure 1 contains the cumulative returns in that time interval and Table 2, the results of the different metrics (where in the following tables the values corresponding to the best performances are highlighted in bold). One can observe that G7,λ=0 is the best performing portfolio according to all performance indicators, yielding the highest cumulative returns and this with the lowest volatility according to Beta. The G14 portfolio is the second best performing portfolio taking into account the cumulative returns, the Alpha, Sharpe and Sortino metrics, however, achieved with the highest volatility in terms of Beta and it has the highest maximum drawdown. The evolution of the cumulative returns of the remaining portfolios is very similar to the evolution of the benchmark’s cumulative returns. Since the ETFs of G7,λ=0 were selected with the EU-EV model privileging higher expected utility and almost ignoring variance and entropy in the given period, one would expect a better performance this portfolio in terms of higher returns with respect to the other portfolios in the same period and this is consistent with the obtained results. In the one-year and two-year out-of-sample periods from 2011 to 2012, the results seem to indicate that G7,λ=0 followed by S&P500 are the best performing portfolios. This is evident from the performance measures in Table 3 and Table 4, 8 Irene Brito et al. Fig. 1. Cumulative returns from January 2008 to December 2010. Table 2. Performance measures from January 2008 to December 2010. MDD Sharpe Sortino Beta Alpha S&P500 0.5325 -0.0019 -0.0024 1.0000 0.0000 G14 0.5704 0.5363 0.7863 1.0164 0.2402 G40.5647 0.0510 0.0632 0.9447 0.0168 G7,λ=0 0.4890 0.9670 1.6931 0.8925 0.6630 G7,λ=0.50.5522 0.1161 0.1480 0.9771 0.0381 G7,λ=0.75 0.5514 0.1111 0.1396 0.9798 0.0365 G7,λ=0.85 0.5480 0.0807 0.1035 0.9409 0.0255 G7,λ=1 0.5504 0.0425 0.0538 0.9779 0.0133 where only the maximum drawdown is now the lowest one for S&P500. The portfolio G7,λ=0 achieves higher cumulative returns for a larger time interval in 2011 (see Figure 2). However, there are also time periods where S&P500 exhibits higher returns (in the first quarter and in the last quarter of that year). In 2012 (see Figure 3), the S&P500 benchmark portfolio outperforms the other portfolios in terms of cumulative returns for a wider time range in 2012 (only in the last quarter of 2012 the portfolio G7,λ=0 surpasses again the benchmark). Both in the one-year and two-year periods, the portfolios G14,G4and G7,λ=1 underperform the remaining portfolios. All portfolios are more volatile than S&P500. In the five-year out-of-sample period from 2011 to 2015, now, in contrast to the other out-of-sample periods, it is S&P500 that exhibits the best Sharpe and Sortino ratios and the lowest maximum drawdown (see Table 5), only the Alpha value is higher for G7,λ=0, which also yields the higher cumulative returns (see Figure 4). Green ETF performance evaluation 9 Fig. 2. Cumulative returns for 2011. Table 3. Performance measures for 2011. MDD Sharpe Sortino Beta Alpha S&P500 0.1939 0.0663 0.0850 1.0000 0.0000 G14 0.3032 -0.5402 -0.7855 1.1621 -0.1742 G40.2492 -0.6656 -0.0159 1.0523 -0.1741 G7,λ=0 0.2594 0.0820 0.1309 1.0721 0.0165 G7,λ=0.50.2606 -0.3128 -0.4320 1.0934 -0.0947 G7,λ=0.75 0.2476 -0.2355 -0.3283 1.0820 -0.0752 G7,λ=0.85 0.2326 -0.2671 -0.3708 1.0514 -0.0803 G7,λ=1 0.2349 -0.5135 -0.6958 1.0249 -0.1337 Table 4. Performance measures from January 2011 to December 2012. MDD Sharpe Sortino Beta Alpha S&P500 0.1939 0.3610 0.4598 1 0 G14 0.3090 0.0110 0.0159 1.1402 -0.0714 G40.2492 -0.1570 -0.2143 1.0661 -0.1003 G7,λ=0 0.2594 0.5742 0.9511 1.0131 0.1265 G7,λ=0.50.2606 0.1333 0.1797 1.0925 -0.0443 G7,λ=0.75 0.2476 0.1929 0.2620 1.0814 -0.0318 G7,λ=0.85 0.2326 0.1910 0.2595 1.0489 -0.0325 G7,λ=1 0.2349 -0.0527 -0.0707 1.0328 -0.0772 Considering the evolution of the cumulative returns in the ten-year out-ofsample period from 2011 to 2020 (see Figure 5), one can observe that from 2016 onwards the cumulative returns of G7,λ=0 increase notably. The portfolio G14