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Green exchange-traded fund performance appraisal using slacks-based DEA models

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Abstract

This paper appraises the performance of a sample of green exchange-traded funds (ETFs) using two types of data envelopment analysis (DEA) metrics. The first type is based on slacks-based DEA models, namely, the range-adjusted measure (RAM) and its variant the RAM-BCC model; the second type is based on a common set of weights of RAM. The appraisal is performed under the assumption that there are value stocks on the green equity market and the potential investors prefer ETFs that put emphasis on value stocks. In the first stage of the analysis, ETF efficiency ratings are derived, whereas in the second stage, ordinary least squares, censored Tobit, and bootstrapped-truncated regression are employed to model the fund ratings. The results are acceptable, indicating that four or five out of the sixteen sample funds depending on the model employed can be candidates for value investors. Moreover, although there is not much evidence for systematic effects of the beta coefficient on fund rating, the findings of the analyses entail implications for potential investors by using the models as an investment pick and for fund managers by considering the mitigation of risk and bringing higher selectivity to their portfolios.

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Acknowledgments

The authors acknowledge the constructive comments of two anonymous reviewers which helped them to significantly improve the quality of this research article.

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Appendix

Appendix

In DEA efficiency calculation of DMUs, different sets of weights are obtained. The following model (7) formulated to simultaneously maximize the ratio of outputs to inputs for all of the DMUs, is used to find CSW. For the DEA efficiency calculation using fractional programming, the interested reader is referred to Cooper et al. (2007a) and Tsai et al. (2006).

$$ \begin{aligned} &Max\,\left\{ {\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{r1} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{1} } }}, \ldots ,\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rk} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{im} } }}} \right\} \hfill \\& s.t. \hfill \\& \frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rj} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} } }} \le 1,\quad j = 1,2, \ldots ,n, \hfill \\& - \mu_{r} \le - 1/R_{r}^{ + } (m + k),\quad r = 1,2, \ldots ,k, \hfill \\& - v_{i} \le - 1/R_{i}^{ - } (m + k),\quad i = 1,2, \ldots ,m, \hfill \\& \omega \;free\;on\;sign. \hfill \\ \end{aligned} $$
(7)

For solving this model, we use a goal-programming formulation [model (8)] based on the L norm. This approach takes into account the efficiency ratio of all of the DMUs to calculate and find a CSW so that the efficiency ratio of all of the DMUs becomes better as the ratio gets larger (Tsai et al. 2006).

$$ \begin{aligned} &Max\left( {Min\left\{ {\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{r1} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{1} } }}, \ldots ,\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rk} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{im} } }}} \right\}} \right) \hfill \\& s.t. \hfill \\& \frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rj} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} } }} \le 1,\quad j = 1,2, \ldots ,n, \hfill \\& - \mu_{r} \le - 1/R_{r}^{ + } (m + k),\quad r = 1,2, \ldots ,k, \hfill \\& - v_{i} \le - 1/R_{i}^{ - } (m + k),\quad i = 1,2, \ldots ,m, \hfill \\& \omega \;free\;on\;sign. \hfill \\ \end{aligned} $$
(8)

By introducing a positive goal achievement variable, z, model (8) can be converted into the following model (Jahanshahloo et al. 2005; Tsai et al. 2006):

$$ \begin{aligned} &Max\;z \hfill \\& s.t. \hfill \\& \sum\limits_{r = 1}^{k} {\mu_{r} y_{rj} } - \sum\limits_{i = 1}^{m} {v_{i} x_{ij} + \omega \le 0} ,\quad j = 1,2, \ldots ,n, \hfill \\& \sum\limits_{r = 1}^{k} {\mu_{r} y_{rj} } - z\sum\limits_{i = 1}^{m} {v_{i} x_{ij} + \omega \ge 0} ,\quad j = 1,2, \ldots ,n, \hfill \\& - \mu_{r} \le - 1/R_{r}^{ + } (m + k),\quad r = 1,2, \ldots ,k, \hfill \\& - v_{i} \le - 1/R_{i}^{ - } (m + k),\quad i = 1,2, \ldots ,m, \hfill \\& \omega \;free\;on\;sign. \hfill \\ \end{aligned} $$
(9)

Model (9) is identical to model (5). A set of (μ r *, v i *), i.e., CSW, can be calculated according to Eq. (10) and the efficiency score \( p_{j} \) of a DMU can be calculated with the CSW.

$$ p_{j} = \frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r}^{*} y_{rj} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i}^{*} x_{ij} } }},\quad j = 1,2, \ldots ,n. $$
(10)

In case that according to Eq. (10) a complete ranking of the DMUs using CSW is not achieved, a procedure by omitting the corresponding constraints of efficient DMUs can be employed (Jahanshahloo et al. 2005).

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Tsolas, I.E., Charles, V. Green exchange-traded fund performance appraisal using slacks-based DEA models. Oper Res Int J 15, 51–77 (2015). https://doi.org/10.1007/s12351-015-0169-x

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