# Quantifying Risk in Traditional Energy and Sustainable Investments

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model and Methodology

#### 3.1. Modelling Asset Returns

_{t+}

_{1}= −100log (P

_{t}

_{+1}/P

_{t}), where P

_{t}represents the corresponding index prices. As it is commonly employed in the literature (see, e.g., [49]), we suppose that conditional on the location-scale parameters ${\mu}_{t+1}$ and ${\sigma}_{t+1}$, negative log-returns follow ${L}_{t+1}={\mu}_{t+1}+{\epsilon}_{t+1}$ and the innovations are ${\epsilon}_{t+1}={\sigma}_{t+1}{Z}_{t+1}$. The random variables ${Z}_{t+1}$ are assumed to be independently distributed with a common cumulative distribution function (CDF) $G$ that, for certain cases, depends on unknown parameters. We discuss several possibilities for $G$ in the next section. The parameter ${\mu}_{t+1}$ is modelled by an ARMA (1,1) process and a GARCH (1,1) process is employed for ${\sigma}_{t+1}$, that is,

- For the $\alpha $-stable distribution, the ML approach is employed by using the direct integration method in Reference [51].
- For the generalized Pareto distribution, the peaks over threshold (POT) method is employed to estimate the parameters. According to [49], the VaR or $\alpha $-quantile is obtained from$${q}_{\alpha}\left(Z\right)=u+\frac{\beta}{\xi}\left[{\left(\frac{1-\alpha}{{T}_{u}/T}\right)}^{-\xi}-1\right],$$

#### 3.2. Backtesting ES

_{i}= α

_{i}for i = 1, …, N

_{i}≠ α

_{i}for at least one i ∈ {1, …, N},

## 4. Data

^{st}‘CL’ Futures (CL1), Generic 1

^{st}‘NG’ Futures (NG1) and Richards Bay Coal Futures (XO1) are obtained for oil, gas and coal prices, respectively). The abovementioned data are obtained from Bloomberg terminal (Bloomberg Professional Service is an information service that, through subscription, provides economic and financial data at the level of individual securities and the entire market. The workstations with the installed service are traditionally called Bloomberg terminals).

## 5. Statistical Results

## 6. Discussion, Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

VaR | Value-at-Risk |

ES | Expected Shortfall |

CVaR | Conditional Value-at-Risk |

TI | Traditional Index |

SI | Sustainability Index |

ARMA | Autoregressive Moving Average |

GARCH | Generalized Autoregressive Conditional Heteroskedasticity |

S&P | Standard and Poor’s |

FTSE | Financial Times Stock Exchange |

CDF | Cumulative Distribution Function |

Probability Distribution Function | |

QML | Quasi Maximum Likelihood |

EVT | Extreme Value Theory |

GPD | Generalized Pareto Distribution |

POT | Peaks-Over-Threshold |

MN | Multinomial distribution |

ESG | Environmental, Social and Governance |

CAGR | Compound Annual Growth Rate |

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**Table 1.**Descriptive statistics for daily stock returns of the SI (sustainable index) and TI (traditional oil and gas index).

Statistics | SI Returns | TI Returns | Oil Returns | Gas Returns | Coal Returns |
---|---|---|---|---|---|

Mean | 0.024 | 0.007 | −0.008 | −0.019 | 0.016 |

Median | 0.070 | 0.000 | 0.000 | 0.000 | 0.000 |

Standard deviation | 1.002 (15.85) | 1.638 (25.91) | 2.324 (36.75) | 3.034 (47.97) | 1.500 (23.72) |

Skewness | −0.489 | −0.261 | 0.134 | 0.617 | 0.697 |

Excess Kurtosis | 9.003 | 13.747 | 4.968 | 5.873 | 44.624 |

Minimum | −6.785 | −16.294 | −13.065 | −18.054 | −20.729 |

Maximum | 8.648 | 17.208 | 16.410 | 26.771 | 23.841 |

Panel A: ARMA-GARCH Estimation | ||
---|---|---|

ARMA-GARCH with Gaussian innovations | TI Index | SI Index |

${\theta}_{0}$ | 0.036 (0.067) | 0.058 (0.000) |

${\theta}_{1}$ | −0.082 (0.887) | 0.023 (0.857) |

${\theta}_{2}$ | 0.028 (0.961) | 0.101 (0.435) |

${\beta}_{0}$ | 0.023 (0.000) | 0.009 (0.000) |

${\beta}_{1}$ | 0.084 (0.000) | 0.105 (0.000) |

${\beta}_{2}$ | 0.907 (0.000) | 0.888 (0.000) |

ARMA-GARCH with Student’s t innovations | TI Index | SI Index |

${\theta}_{0}$ | 0.050 (0.007) | 0.071 (0.000) |

${\theta}_{1}$ | −0.040 (0.938) | −0.067 (0.656) |

${\theta}_{2}$ | −0.013 (0.980) | 0.188 (0.206) |

${\beta}_{0}$ | 0.019 (0.002) | 0.006 (0.004) |

${\beta}_{1}$ | 0.080 (0.000) | 0.101 (0.000) |

${\beta}_{2}$ | 0.913 (0.000) | 0.898 (0.000) |

$\nu $ | 6.923 (0.000) | 5.414 (0.000) |

Panel B: ARMA-GARCH Estimation Considering External Regressors in the variance equation | ||

ARMA-GARCH with Gaussian innovations | TI Index | SI Index |

${\theta}_{0}$ | 0.035 (0.667) | 0.059 (0.000) |

${\theta}_{1}$ | −0.081 (0.887) | 0.024 (0.855) |

${\theta}_{2}$ | 0.027 (0.961) | 0.102 (0.429) |

${\beta}_{0}$ | 0.023 (0.003) | 0.005 (0.144) |

${\beta}_{1}$ | 0.083 (0.000) | 0.109 (0.000) |

${\beta}_{2}$ | 0.907 (0.000) | 0.881 (0.000) |

${\gamma}_{1}$ | 0.000 (0.999) | 0.000 (0.425) |

${\gamma}_{2}$ | 0.000 (0.999) | 0.000 (0.277) |

${\gamma}_{3}$ | 0.000 (0.999) | 0.000 (0.999) |

ARMA-GARCH with Student’s t innovations | TI Index | SI Index |

${\theta}_{0}$ | 0.050 (0.007) | 0.071 (0.000) |

${\theta}_{1}$ | −0.040 (0.938) | −0.068 (0.652) |

${\theta}_{2}$ | −0.013 (0.980) | 0.189 (0.205) |

${\beta}_{0}$ | 0.019 (0.003) | 0.006 (0.093) |

${\beta}_{1}$ | 0.080 (0.000) | 0.103 (0.000) |

${\beta}_{2}$ | 0.913 (0.000) | 0.898 (0.000) |

${\gamma}_{1}$ | 0.000 (0.999) | 0.000 (0.999) |

${\gamma}_{2}$ | 0.000 (0.999) | 0.000 (0.999) |

${\gamma}_{3}$ | 0.000 (0.999) | 0.000 (0.999) |

$\nu $ | 6.923 (0.000) | 5.311 (0.000) |

**Table 3.**Comparison of 99%-VaR and 97.5%-ES (implicit) backtesting for the Sustainable Index (SI) and the Traditional Oil and Gas Industry (TI).

Model | 99% VaR | 97.5% ES | |||||||
---|---|---|---|---|---|---|---|---|---|

EE | Violations | ${\mathit{O}}_{0}$ | ${\mathit{O}}_{1}$ | ${\mathit{O}}_{2}$ | ${\mathit{O}}_{3}$ | ${\mathit{O}}_{4}$ | Pearson | Nass | |

Panel a) Sustainable Index (SI): | |||||||||

Normal | 27 | 33 (0.270) | 2644 | 13 | 17 | 10 | 25 | 7.60 | 7.39 |

Student’s-t | 27 | 11 (0.000) | 2658 | 9 | 21 | 10 | 11 | 9.71 | 9.45 |

Stable | 27 | 32 (0.357) | 2631 | 21 | 19 | 16 | 12 | 2.84 | 2.76 |

GPD-POT | 27 | 15 (0.011) | 2654 | 17 | 20 | 10 | 8 | 8.17 | 7.94 |

Panel b) Traditional Oil & Gas Industry (TI): | |||||||||

Normal | 27 | 32 (0.357) | 2648 | 9 | 17 | 14 | 21 | 5.22 | 5.07 |

Student’s-t | 27 | 14 (0.005) | 2657 | 14 | 8 | 16 | 14 | 5.87 | 5.71 |

Stable | 27 | 32 (0.357) | 2638 | 9 | 21 | 17 | 24 | 7.65 | 7.44 |

GPD-POT | 27 | 20 (0.151) | 2660 | 13 | 13 | 9 | 14 | 6.18 | 6.01 |

**Table 4.**Comparison of 99%-VaR and 97.5%-ES (implicit) backtesting for the Sustainable Index (SI) and the Traditional Oil and Gas Industry (TI). Considering external regressors in the variance equation of GARCH model.

Model | 99% VaR | 97.5% ES | |||||||
---|---|---|---|---|---|---|---|---|---|

EE | Violations | ${\mathit{O}}_{0}$ | ${\mathit{O}}_{1}$ | ${\mathit{O}}_{2}$ | ${\mathit{O}}_{3}$ | ${\mathit{O}}_{4}$ | Pearson | Nass | |

Panel a) Sustainable Index (SI): | |||||||||

Normal | 27 | 33 (0.270) | 2641 | 15 | 16 | 13 | 24 | 4.13 | 4.02 |

Student’s-t | 27 | 17 (0.036) | 2655 | 12 | 21 | 10 | 11 | 7.40 | 7.20 |

Stable | 27 | 32 (0.357) | 2633 | 19 | 17 | 17 | 23 | 2.45 | 2.39 |

GPD-POT | 27 | 20 (0.151) | 2649 | 15 | 21 | 11 | 13 | 4.21 | 4.10 |

Panel b) Traditional Oil & Gas Industry (TI): | |||||||||

Normal | 27 | 35 (0.144) | 2643 | 11 | 18 | 14 | 23 | 4.83 | 4.70 |

Student’s-t | 27 | 23 (0.417) | 2650 | 18 | 10 | 18 | 13 | 3.91 | 3.81 |

Stable | 27 | 34 (0.199) | 2630 | 15 | 21 | 18 | 25 | 5.16 | 5.02 |

GPD-POT | 27 | 25 (0.683) | 2655 | 14 | 10 | 11 | 19 | 5.75 | 5.59 |

**Table 5.**Exceptions obtained for each VaR level for the Sustainable Index (SI) and the Traditional Oil and Gas Index (TI).

Model | 97.5%VaR | 98.125%VaR | 98.75%VaR | 99.375%VaR |
---|---|---|---|---|

Panel a) Sustainable Index (SI): | ||||

[52;84] EE = 68 | [37;65] EE = 51 | [23;45] EE = 34 | [9;25] EE = 17 | |

Normal | 65 | 52 | 35 | 25 |

Student’s-t | 51 | 42 | 21 | 11 |

Stable | 78 | 57 | 38 | 22 |

GPD-POT | 55 | 38 | 18 | 8 |

Panel b) Traditional Oil & Gas Industry (TI): | ||||

[52;84] EE = 68 | [37;65] EE = 51 | [23;45] EE = 34 | [9;25] EE = 17 | |

Normal | 61 | 52 | 35 | 21 |

Student’s-t | 52 | 38 | 30 | 14 |

Stable | 71 | 62 | 41 | 24 |

GPD-POT | 49 | 36 | 23 | 14 |

**Table 6.**Exceptions obtained for each VaR level for the Sustainable Index (SI) and the Traditional Oil and Gas Index (TI). Considering external regressors in the variance equation of GARCH model.

Model | 97.5%VaR | 98.125%VaR | 98.75%VaR | 99.375%VaR |
---|---|---|---|---|

Panel a) Sustainable Index (SI): | ||||

[52;84] EE = 68 | [37;65] EE = 51 | [23;45] EE = 34 | [9;25] EE = 17 | |

Normal | 68 | 53 | 37 | 24 |

Student’s-t | 54 | 42 | 21 | 11 |

Stable | 76 | 57 | 40 | 23 |

GPD-POT | 60 | 45 | 24 | 13 |

Panel b) Traditional Oil & Gas Industry (TI): | ||||

[52;84] EE = 68 | [37;65] EE = 51 | [23;45] EE = 34 | [9;25] EE = 17 | |

Normal | 66 | 55 | 37 | 23 |

Student’s-t | 59 | 41 | 31 | 13 |

Stable | 79 | 64 | 43 | 25 |

GPD-POT | 54 | 40 | 30 | 19 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Díaz, A.; García-Donato, G.; Mora-Valencia, A. Quantifying Risk in Traditional Energy and Sustainable Investments. *Sustainability* **2019**, *11*, 720.
https://doi.org/10.3390/su11030720

**AMA Style**

Díaz A, García-Donato G, Mora-Valencia A. Quantifying Risk in Traditional Energy and Sustainable Investments. *Sustainability*. 2019; 11(3):720.
https://doi.org/10.3390/su11030720

**Chicago/Turabian Style**

Díaz, Antonio, Gonzalo García-Donato, and Andrés Mora-Valencia. 2019. "Quantifying Risk in Traditional Energy and Sustainable Investments" *Sustainability* 11, no. 3: 720.
https://doi.org/10.3390/su11030720