# Can One Reinforce Investments in Renewable Energy Stock Indices with the ESG Index?

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Empirical Methodology

#### 2.1. Modeling Marginal Distributions

#### 2.2. Copula Functions

#### 2.3. Generalized Autoregressive Score (GAS) Model in Time Varying Copula

## 3. Empirical Results

#### 3.1. Summary Statistics

- (1)
- The Wilder Hill New Energy Global Innovation Index (NEX (for more details on the weights of the NEX index, please refer to https://nexindex.com/whindexes.php.)) is weighted based on globally listed new energy innovation companies and is calculated by Solactive. The index focuses on renewable—solar (27.5% weight), renewable—wind (22.0% weight), energy conversion (5.5% weight), energy efficiency (23.1% weight), energy storage (6.6% weight), renewables—Biofuels % Biomass (8.8% weight), and renewables—others (6.6% weight).
- (2)
- The Wilder Hill Clean Energy Index (ECO (for more details on the weights of the ECO index, please refer to https://wildershares.com/about.php)) is mainly on US-listed clean energy companies and is calculated by the New York Stock Exchange (NYSE). The index focuses on renewable energy supplies (21% weight), energy conversion (21% weight), power delivery and conservation (20% weight), greener utilities (13% weight), energy storage (20% weight), and cleaner fuels (5% weight).
- (3)
- The S&P Global Clean Energy Index (SPGTCED (for more details on the weights of the SPGTCED index, please refer to https://us.spindices.com/indices/equity/sp-global-clean-energy-index)) is weighted based on 30 companies from around the world that are related to clean energy business. The index focuses largely, different from the other three indices, on information technology (24.6% weight). Other weights are allocated on utilities (52.4% weight), industrials (20.8% weight), and energy (2.1% weight).
- (4)
- The European Renewable Energy Total Return Index (ERIX (for more details on the weights of the ERIX index please refer to https://sgi.sgmarkets.com/en/index-details/TICKER:ERIX/)) tracks the stocks of largest European renewable energy companies that are highly involved in wind, water, solar, biofuels, geothermal, and/or marine investments. The index selects the largest companies, in which each component has a minimum weight of 5%. According to the most current weights, the companies are Verbund ag in Austria (21.75% weight), Vestas wind systems a/s in Denmark (20.52% weight), Siemens gamesa renewable ene in Spain (17.67% weight), Edp renovaveis sa in Spain (10.29% weight), and Meyer burger technology in Switzerland (5.95% weight).

#### 3.2. Results for the Marginal Distributions

#### 3.3. Results in the Copula Models

#### 3.4. Goodness-of-Fit Test

## 4. Portfolio Performance

- We generate dynamic rotated Gumbel copula parameters ${\left\{{\widehat{\gamma}}_{t}\right\}}_{t=1,2,\dots ,T}$ based on the estimated time varying pattern.
- For time t from 1 to T (total sample size), we generate uniform distributions for two targets ${U}_{u1,t}$ and ${U}_{u2,t}$ using ${\widehat{\gamma}}_{t}$ for S (= 5000) times, and hence the simulated standard residuals and returns based on the estimated marginal distribution parameters. Each element in marginal distributions at time t is stored in a vector of size $\mathrm{S}\times 1$.
- For time t, the portfolio returns are calculated based on each asset return and weight. For asset returns of size $\mathrm{S}\times 1$, a lower qth quantile (we selected 1%, indicating a confidence level of 99% for VaR) is regarded as the VaR (Equation (11)). CVaR can also be obtained (Equation (12)).$$Va{R}_{t}^{q}\equiv {F}_{t}^{-1}\left(q\right),q\in \left(0,1\right)$$$$CVa{R}_{t}^{q}\equiv E\left[{Y}_{t}|{F}_{t-1},{Y}_{t}\le Va{R}_{t}^{q}\right],q\in \left(0,1\right).$$

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Tests | ESG and NEX | ESG and ECO | ESG and SPGTCED | ESG and ERIX |
---|---|---|---|---|

Cointegration (index) | 0.9659 | 0.8583 | 0.8763 | 0.9660 |

ESG → others (returns) | 0.0001 | 0.0102 | 0.0702 | 0.0001 |

ESG ← others (returns) | 0.0001 | 0.0001 | 0.0111 | 0.0001 |

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**Figure 3.**Time varying lower tail dependence between ESG and NEX based on the rotated Gumbel copula (GAS) model.

**Figure 4.**Time varying lower tail dependence between ESG and ECO based on the rotated Gumbel copula (GAS) model.

**Figure 5.**Time varying lower tail dependence between ESG and SPGTCED based on the rotated Gumbel copula (GAS) model.

**Figure 6.**Time varying lower tail dependence between ESG and ERIX based on the rotated Gumbel copula (GAS) model.

**Figure 7.**Time varying conditional value-at-risk (CVaR) between the ESG and NEX portfolios ((

**a**) is the CVaR under the ‘naïve static’ strategy; (

**b**) is the CVaR under the ‘diversified risk parity’ and ‘optimal weights’ strategies).

**Figure 8.**Time varying CVaR between the ESG and ECO portfolios ((

**a**) is the CVaR under the ‘naïve static’ strategy; (

**b**) is the CVaR under the ‘diversified risk parity’ and ‘optimal weights’ strategies).

**Figure 9.**Time varying CVaR between the ESG and SPGTCED portfolios ((

**a**) is the CVaR under the ‘naïve static’ strategy; (

**b**) is the CVaR under the ‘diversified risk parity’ and ‘optimal weights’ strategies).

**Figure 10.**Time varying CVaR between the ESG and ERIX portfolios ((

**a**) is the CVaR under the ‘naïve static’ strategy; (

**b**) is the CVaR under the ‘diversified risk parity’ and ‘optimal weights’ strategies).

Copula Family | Name |
---|---|

$C\left({u}_{1},{u}_{2}\right)={{\displaystyle \int}}_{-\infty}^{{\varphi}^{-1}\left({u}_{1}\right)}{{\displaystyle \int}}_{-\infty}^{{\varphi}^{-1}\left({u}_{2}\right)}\frac{1}{2\pi \sqrt{1-{\theta}^{2}}}\mathrm{exp}(-\frac{{s}^{2}-2\theta st+{t}^{2}}{2\left(1-{\theta}^{2}\right)})dsdt$ | Normal |

$C\left({u}_{1},{u}_{2}\right)={\left({u}_{1}^{-\gamma}+{u}_{2}^{-\gamma}-1\right)}^{-1/\gamma},\gamma \in \left(0,+\infty \right),\tau ={2}^{-1/\gamma}$ | Clayton |

$C\left({u}_{1},{u}_{2}\right)=exp\{-{[{\left(-ln{u}_{1}\right)}^{\gamma}+{\left(-ln{u}_{2}\right)}^{\gamma}]}^{1/\gamma}\},\gamma \in \left(1,+\infty \right),\tau ={2}^{-1/\gamma}$ | Gumbel |

$C\left({u}_{1},{u}_{2}\right)={{\displaystyle \int}}_{-\infty}^{{t}_{v}^{-1}\left({u}_{1}\right)}{{\displaystyle \int}}_{-\infty}^{{t}_{v}^{-1}\left({u}_{2}\right)}\frac{1}{2\pi \sqrt{1-{\theta}^{2}}}{(1+\frac{{s}^{2}-2\theta st+{t}^{2}}{v\left(1-{\theta}^{2}\right)})}^{-\frac{v+2}{v}}dsdt$ | Student t |

Mean | S.E. | Min | Max | Skewness | Kurtosis | Jarque–Bera | |
---|---|---|---|---|---|---|---|

ESG | 0.0040 | 1.0701 | −7.2143 | 8.5659 | −0.4448 | 11.6885 | 8837 |

NEX | −0.0378 | 1.4811 | −10.4854 | 12.0705 | −0.4905 | 11.6112 | 8702 |

ECO | −0.0654 | 2.1128 | −14.4673 | 14.5195 | −0.3655 | 8.2339 | 3236 |

SPGTCED | −0.0706 | 1.9516 | −14.9729 | 18.0927 | −0.5195 | 16.6587 | 21733 |

ERIX | −0.0474 | 2.1309 | −16.9652 | 15.8214 | −0.3927 | 11.2607 | 7977 |

ESG | NEX | ECO | SPGTCED | ERIX | |
---|---|---|---|---|---|

Mean Model | |||||

${\varphi}_{0}$ | 0.0547 *** | 0.0358 * | 0.0138 | 0.0111 | 0.0486 |

S.E. | (0.0129) | (0.0212) | (0.0312) | (0.0249) | (0.0295) |

${\varphi}_{1}$ | 0.1073 *** | 0.1934 *** | 0.0501 *** | 0.1410 *** | 0.0381 ** |

S.E. | (0.0192) | (0.0193) | (0.0195) | (0.0195) | (0.0192) |

${\varphi}_{2}$ | −0.0241 | −0.0026 | 0.0260 | 0.0068 | 0.0201 |

S.E. | (0.0195) | (0.0194) | (0.0195) | (0.0193) | (0.0192) |

Variance Model | |||||

$\mathsf{\omega}$ | 0.0070 *** | 0.0075 ** | 0.0383 *** | 0.0140 * | 0.0316 * |

S.E. | (0.0025) | (0.0034) | (0.0136) | (0.0055) | (0.0131) |

$\mathsf{\alpha}$ | 0.1067 *** | 0.0738 *** | 0.0741 *** | 0.0727 *** | 0.0721 *** |

S.E. | (0.0151) | (0.0116) | (0.0118) | (0.0119) | (0.0125) |

$\mathsf{\beta}$ | 0.8923 *** | 0.9239 *** | 0.9160 *** | 0.9231 *** | 0.9219 *** |

S.E. | (0.0139) | (0.0115) | (0.0134) | (0.0120) | (0.0135) |

$\mathsf{\nu}$ | 6.2547 *** | 8.6623 *** | 11.6211 *** | 8.2568 *** | 6.7693 *** |

S.E. | (0.7348) | (1.3224) | (2.3456) | (1.2253) | (0.8643) |

Skewed t Density | |||||

$\mathsf{\lambda}$ | 6.5835 | 9.5779 | 12.6029 | 8.8251 | 7.0098 |

$\mathsf{{\rm N}}$ | −0.1340 | −0.1138 | −0.1813 | −0.1044 | −0.0999 |

Ljung–Box Test | |||||

$\mathrm{Q}\left(25\right)$ | 21.20 | 14.16 | 11.30 | 21.82 | 23.79 |

p value | (0.68) | (0.96) | (0.99) | (0.65) | (0.53) |

${Q}^{2}\left(25\right)$ | 25.15 | 27.32 | 32.40 | 18.03 | 20.23 |

p value | (0.45) | (0.34) | (0.15) | (0.84) | (0.73) |

GoF Tests on Skewed t Distribution Model (p-Value) | |||||

KS | 0.30 | 0.68 | 0.84 | 0.17 | 0.31 |

CvM | 0.34 | 0.32 | 0.72 | 0.15 | 0.30 |

ESG and NEX | ESG and ECO | ESG and SPGTCED | ESG and ERIX | |||||
---|---|---|---|---|---|---|---|---|

Parametric | Semi | Parametric | Semi | Parametric | Semi | Parametric | Semi | |

Normal Copula | ||||||||

$\widehat{\rho}$ | 0.7931 | 0.7933 | 0.6961 | 0.6951 | 0.6887 | 0.6887 | 0.6159 | 0.6159 |

S.E. | 0.0073 | 0.0075 | 0.0095 | 0.0111 | 0.0094 | 0.0100 | 0.0115 | 0.0127 |

Log Likelihood | 1375.56 | 1377.16 | 919.51 | 915.75 | 892.20 | 892.21 | 661.72 | 661.68 |

Clayton Copula | ||||||||

$\widehat{\gamma}$ | 2.0344 | 2.0684 | 1.4834 | 1.4972 | 1.4452 | 1.4601 | 1.0640 | 1.0804 |

S.E. | 0.0586 | 0.0862 | 0.0633 | 0.0618 | 0.0611 | 0.0591 | 0.0510 | 0.0570 |

${\widehat{\tau}}^{L}$ | 0.7113 | 0.7153 | 0.6267 | 0.6294 | 0.6190 | 0.6221 | 0.5213 | 0.5265 |

Log Likelihood | 1204.85 | 1227.97 | 861.37 | 865.30 | 816.19 | 827.34 | 564.71 | 571.87 |

Rotated Gumbel Copula | ||||||||

$\widehat{\gamma}$ | 2.3606 | 2.3754 | 1.9396 | 1.9434 | 1.9357 | 1.9401 | 1.6934 | 1.6984 |

S.E. | 0.0435 | 0.0472 | 0.0377 | 0.0313 | 0.0441 | 0.0386 | 0.0307 | 0.0314 |

${\widehat{\tau}}^{L}$ | 0.6587 | 0.6612 | 0.5704 | 0.5714 | 0.5694 | 0.5706 | 0.4942 | 0.4960 |

Log Likelihood | 1400.45 | 1415.23 | 963.50 | 963.81 | 941.20 | 946.61 | 662.66 | 666.58 |

Student’s t Copula | ||||||||

$\widehat{\rho}$ | 0.7986 | 0.7993 | 0.6997 | 0.6996 | 0.7000 | 0.7005 | 0.6228 | 0.6239 |

S.E. | 0.0085 | 0.0091 | 0.0116 | 0.0122 | 0.0120 | 0.0111 | 0.0149 | 0.0121 |

${\widehat{\nu}}^{-1}$ | 0.1637 | 0.1684 | 0.1340 | 0.1373 | 0.1789 | 0.1810 | 0.1261 | 0.1270 |

S.E. | 0.0234 | 0.0252 | 0.0192 | 0.0243 | 0.0204 | 0.0240 | 0.0217 | 0.0273 |

${g}_{T}\left(\widehat{\rho},\widehat{\nu}\right)$ | 0.4014 | 0.4085 | 0.2544 | 0.2597 | 0.3187 | 0.3218 | 0.1838 | 0.1859 |

Log Likelihood | 1429.05 | 1432.08 | 948.61 | 945.42 | 954.46 | 953.46 | 691.27 | 691.08 |

ESG and NEX | ESG and ECO | ESG and SPGTCED | ESG and ERIX | |||||
---|---|---|---|---|---|---|---|---|

Parametric | Semi | Parametric | Semi | Parametric | Semi | Parametric | Semi | |

Rotated Gumbel Copula (GAS) | ||||||||

$\widehat{\omega}$ | 0.0139 | 0.0146 | −0.0092 | −0.0078 | −0.0035 | −0.0037 | −0.0043 | −0.0042 |

S.E. | 0.0123 | 0.0151 | 0.0152 | 0.0088 | 0.0092 | 0.0115 | 0.0376 | 0.0593 |

$\widehat{\alpha}$ | 0.1364 | 0.1371 | 0.1790 | 0.1720 | 0.0966 | 0.1015 | 0.0545 | 0.0543 |

S.E. | 0.0329 | 0.0569 | 0.0287 | 0.0528 | 0.0335 | 0.0631 | 0.0329 | 0.0492 |

$\widehat{\beta}$ | 0.9522 | 0.9522 | 0.8588 | 0.8750 | 0.9685 | 0.9663 | 0.9899 | 0.9899 |

S.E. | 0.1367 | 0.0384 | 0.1170 | 0.0253 | 0.1346 | 0.0258 | 0.1413 | 0.1108 |

Log Likelihood | 1469.21 | 1485.20 | 992.11 | 992.89 | 983.49 | 989.66 | 697.50 | 702.23 |

Student’s t Copula (GAS) | ||||||||

$\widehat{\omega}$ | 0.0480 | 0.0507 | 0.1236 | 0.1212 | 0.0862 | 0.0829 | 0.0109 | 0.0109 |

S.E. | 0.0219 | 0.0225 | 0.0421 | 0.0373 | 0.0280 | 0.0278 | 0.0132 | 0.0125 |

$\widehat{\alpha}$ | 0.0921 | 0.0984 | 0.1328 | 0.1331 | 0.1354 | 0.1370 | 0.0442 | 0.0442 |

S.E. | 0.0146 | 0.0146 | 0.0245 | 0.0196 | 0.0184 | 0.0215 | 0.0115 | 0.0092 |

$\widehat{\beta}$ | 0.9782 | 0.9770 | 0.9288 | 0.9301 | 0.9501 | 0.9518 | 0.9926 | 0.9926 |

S.E. | 0.0101 | 0.0103 | 0.0243 | 0.0216 | 0.0157 | 0.0160 | 0.0087 | 0.0081 |

${\widehat{\nu}}^{-1}$ | 0.1154 | 0.1281 | 0.1185 | 0.1206 | 0.1467 | 0.1438 | 0.1108 | 0.1108 |

S.E. | 0.0206 | 0.0249 | 0.0201 | 0.0238 | 0.0223 | 0.0246 | 0.0216 | 0.0213 |

Log Likelihood | 1499.70 | 1501.32 | 982.72 | 979.94 | 1002.50 | 1002.36 | 730.02 | 730.04 |

ESG and NEX | ESG and ECO | ESG and SPGTCED | ESG and ERIX | |||||
---|---|---|---|---|---|---|---|---|

$\mathbf{K}{\mathbf{S}}_{\mathbf{C}}\left(\mathbf{K}{\mathbf{S}}_{\mathbf{R}}\right)$ | $\mathbf{C}\mathbf{v}{\mathbf{M}}_{\mathbf{C}}\left(\mathbf{C}\mathbf{v}{\mathbf{M}}_{\mathbf{R}}\right)$ | $\mathbf{K}{\mathbf{S}}_{\mathbf{C}}\left(\mathbf{K}{\mathbf{S}}_{\mathbf{R}}\right)$ | $\mathbf{C}\mathbf{v}{\mathbf{M}}_{\mathbf{C}}\left(\mathbf{C}\mathbf{v}{\mathbf{M}}_{\mathbf{R}}\right)$ | $\mathbf{K}{\mathbf{S}}_{\mathbf{C}}\left(\mathbf{K}{\mathbf{S}}_{\mathbf{R}}\right)$ | $\mathbf{C}\mathbf{v}{\mathbf{M}}_{\mathbf{C}}\left(\mathbf{C}\mathbf{v}{\mathbf{M}}_{\mathbf{R}}\right)$ | $\mathbf{K}{\mathbf{S}}_{\mathbf{C}}\left(\mathbf{K}{\mathbf{S}}_{\mathbf{R}}\right)$ | ||

Normal Copula | ||||||||

Parametric | 0.28 | 0.25 | 0.23 | 0.11 | 0.17 | 0.17 | 0.26 | 0.26 |

Semi | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.04 | 0.01 |

Clayton Copula | ||||||||

Parametric | 0.01 | 0.02 | 0.01 | 0.04 | 0.01 | 0.02 | 0.02 | 0.02 |

Semi | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 |

Rotated Gumbel Copula | ||||||||

Parametric | 0.28 | 0.21 | 0.58 | 0.30 | 0.27 | 0.13 | 0.21 | 0.07 |

Semi | 0.02 | 0.01 | 0.16 | 0.03 | 0.01 | 0.01 | 0.01 | 0.01 |

Student’s t Copula | ||||||||

Parametric | 0.26 | 0.34 | 0.35 | 0.09 | 0.10 | 0.15 | 0.36 | 0.29 |

Semi | 0.03 | 0.01 | 0.01 | 0.01 | 0.03 | 0.01 | 0.13 | 0.01 |

Rotated Gumbel Copula (GAS) | ||||||||

Parametric | 0.01 | 0.01 | 0.99 | 0.99 | 0.99 | 0.99 | 0.01 | 0.01 |

Semi | 0.42 | 0.86 | 0.99 | 0.99 | 0.99 | 0.99 | 0.54 | 0.41 |

Student’s t Copula (GAS) | ||||||||

Parametric | 0.41 | 0.20 | 0.47 | 0.22 | 0.11 | 0.08 | 0.16 | 0.21 |

Semi | 0.06 | 0.01 | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 |

Strategy | Measure | ESG and NEX | ESG and ECO | ESG and SPGTCED | ESG and ERIX |
---|---|---|---|---|---|

Naïve Static | ${\mathrm{RR}}_{\mathrm{com}}$ | 0.0118 | 0.0106 | 0.0130 | 0.0078 |

${\mathrm{SD}}_{\mathrm{com}}$ | −0.1672 | −0.2871 | −0.2640 | −0.2976 | |

${\mathrm{VaR}}_{\mathrm{com}}$ | −0.1404 | −0.2680 | −0.2406 | −0.2851 | |

${\mathrm{CVaR}}_{\mathrm{com}}$ | −0.1364 | −0.2562 | −0.2070 | −0.2770 | |

Diversified risk parity | ${\mathrm{RR}}_{\mathrm{com}}$ | 0.0167 | 0.0237 | 0.0254 | 0.0159 |

${\mathrm{SD}}_{\mathrm{com}}$ | −0.2066 | −0.4189 | −0.3777 | −0.4316 | |

${\mathrm{VaR}}_{\mathrm{com}}$ | −0.1852 | −0.4244 | −0.3379 | −0.4460 | |

${\mathrm{CVaR}}_{\mathrm{com}}$ | −0.1801 | −0.4039 | −0.2986 | −0.4305 | |

Optimal weights | ${\mathrm{RR}}_{\mathrm{com}}$ | 0.0291 | 0.0345 | 0.0403 | 0.0267 |

${\mathrm{SD}}_{\mathrm{com}}$ | −0.2722 | −0.4890 | −0.4502 | −0.4957 | |

${\mathrm{VaR}}_{\mathrm{com}}$ | −0.2501 | −0.4964 | −0.4052 | −0.5084 | |

${\mathrm{CVaR}}_{\mathrm{com}}$ | −0.2389 | −0.4712 | −0.3615 | −0.4903 |

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## Share and Cite

**MDPI and ACS Style**

Liu, G.; Hamori, S.
Can One Reinforce Investments in Renewable Energy Stock Indices with the ESG Index? *Energies* **2020**, *13*, 1179.
https://doi.org/10.3390/en13051179

**AMA Style**

Liu G, Hamori S.
Can One Reinforce Investments in Renewable Energy Stock Indices with the ESG Index? *Energies*. 2020; 13(5):1179.
https://doi.org/10.3390/en13051179

**Chicago/Turabian Style**

Liu, Guizhou, and Shigeyuki Hamori.
2020. "Can One Reinforce Investments in Renewable Energy Stock Indices with the ESG Index?" *Energies* 13, no. 5: 1179.
https://doi.org/10.3390/en13051179