# Modeling Electricity Price and Quantity Uncertainty: An Application for Hedging with Forward Contracts

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Methodology

#### 2.1. Spot Price and Energy Generation

#### 2.2. Forward Price

#### 2.3. Pay-Off Function

#### 2.4. Risk Indicators

#### 2.5. Methodology

- (i)
- Parameter estimation of spot price and energy generation

- (ii)
- Bivariate distribution estimation for price and energy generation

- (iii)
- Monte Carlo simulation of bivariate SNP distribution

## 3. Data Description

^{6}kWh).

## 4. Results and Discussion

#### 4.1. Parameter Estimation of the Spot Price and Energy Generation Series

_{4}is positive and significant, reflecting the excess kurtosis of this series. However, skewness parameter (d

_{3}) seems not to be relevant and is excluded from the model. On the other hand, the energy generation series exhibit negative and significant skewness, and thus, a third-order SNP is enough to account for non-normality. ENDG series seems to be an exception since, in this case, the coefficient d

_{3}is not statistically significant, and thus, a normal distribution fits data accurately, as indicated by the Jarque–Bera statistic. The marginal distributions are depicted in Figure A1 of the Appendix B. These plots illustrate how the SNP distributions adequately capture the non-normality of the data. It is also noteworthy that the estimates for marginals are used as initial seeds for the maximum likelihood estimation of the bivariate SNP for every pairwise series.

#### 4.2. Sensitivity of the Risk Indicators

#### 4.3. Effect of SNP Parameters

_{3}and d

_{4}values for the hypothetical generator’s energy generation. The first graph of Figure 9 plots the behavior of the CVaR when the marginal distribution of the energy generation only contains the third-order HP, and parameter d

_{3}takes values of 0.5, 0.14, 0.07, 0, −0.07, and −0.14. The second graph shows a similar sensitivity analysis, but for the fourth-order HP. It is worth noting that d

_{3}= 0, and d

_{4}= 0 indicates a normal distribution.

_{3}and d

_{4}coefficients affect the CVaR. If we take the case of a normal distribution as a reference, negative coefficients tend to decrease the CVaR levels, while positive ones tend to increase them and move the maximum point to the right. Normal conditions do not allow us to properly represent the sensitivity of risk indicators to different hedging levels.

#### 4.4. Optimal Forward Contracting Level

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Energy generators | |

EPMG | Generator Agent, Empresas Públicas de Medellín |

ISGG | Generator Agent, Isagen |

CHVG | Generator Agent, Chivor |

ENDG | Generator Agent, Endesa |

Risk Measures: | |

Rho | Correlation |

Sd | Standard deviation |

VaR | Value-at-risk |

CVaR | Conditional value-at-risk |

Eta | Hedge ratio-η. |

Market: | |

$\mathit{T}$ | Maturity time |

${\mathit{P}}_{\mathit{T}}$ | Spot price at T |

${\mathit{p}}_{\mathit{t}}$ | Log spot price at t |

${\mathit{F}}_{\mathit{T}}^{{\mathit{t}}_{\mathit{o}}}$ | $\mathrm{Forward}\mathrm{contract}\mathrm{price}\mathrm{negociated}\mathrm{at}{\mathit{t}}_{\mathit{o}}$ and maturity time T |

FRP | Forward risk premium |

${\mathit{Q}}_{\mathit{T}}^{\mathit{i}}$ | Energy production of agent i at moment T |

${\mathit{q}}_{\mathit{t}}^{\mathit{i}}$ | $\mathrm{Natural}\mathrm{logarithm}\mathrm{for}{\mathit{Q}}_{\mathit{T}}^{\mathit{i}}$ |

${\mathit{I}}^{\mathit{i}}$ | Income due energy sales of the agent i |

GWh | Energy production unit (Gigawatt hours) equivalent to 10^{6} kWh |

MCOP | One million Colombian Pesos (one million COP) |

SNP modeling: | |

SNP | Semi-nonparametric |

${\mathit{Z}}_{\mathit{t}}$ | Vector that contains J variables distributed with zero mean and multivariate SNP distribution |

${\mathit{G}}_{\mathit{Z}}\left({\mathit{Z}}_{\mathit{t}}\right)$ | $\mathrm{Multivariate}$ |

${\mathit{f}}_{\mathit{p}}\left({\mathit{\u03f5}}_{\mathit{t}}^{\mathit{p}}\right)$ | Marginal distribution function |

${\mathit{H}}_{\mathit{m}}\left({\mathit{v}}_{\mathit{j}\mathit{t}}\right)$ | $\mathrm{m}-\mathrm{order}\mathrm{Hermite}\mathrm{polynomial}\left(\mathrm{HP}\right)\mathrm{for}\mathrm{the}\mathrm{variable}{\mathit{v}}_{\mathit{j}}$$$ |

${\mathit{d}}_{\mathit{j}\mathit{m}}$ | j-order weighted parameters for HP |

$\mathit{E}\left[\cdot \right]$ | Expected value operator |

$\mathit{g}\left(\cdot \right)$ | Standard normal pdf |

${\mathit{F}}_{\mathit{Z}}$ | Joint probability density function |

$\mathit{f}\left(\cdot \right)$ | |

${\mathit{x}}_{\mathit{t}}$ | $\mathrm{AR}\left(1\right)\mathrm{process}\mathrm{with}\mathrm{parameter}{\mathit{\varphi}}_{\mathit{p}}$ |

${\mathit{e}}_{\mathit{t}}$ | White noise with zero mean |

## Appendix A. Proof for the Marginal SNP Pdfs

## Appendix B. Illustrations for Univariate and Bivariate SNP Data Fitting

**Figure A1.**Marginal density functions of residuals ${\u03f5}_{\mathrm{t}}$. Marginal distributions of the spot price and energy generation for spot price and energy generation series. Density functions (shaded area), normal distribution (solid line), and SNP distribution (dashed line).

**Figure A2.**Bivariate PDF and CFF for spot price and energy generation residuals ${\u03f5}_{\mathrm{t}}.$ Joint probability density (PDF) (

**left**column) and cumulative distribution (CDF) (

**right**column) functions for the stochastic component.

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**Figure 1.**Monte Carlo simulation of bivariate distribution. This figure illustrates the method used to simulate random numbers based on [33].

**Figure 2.**Spot price time series of energy in Colombia. The Q-Q plots of the spot price and of its natural logarithm are compared to the normal distribution at a 95% confidence interval.

**Figure 3.**Energy generation time series of different electricity generators in Colombia. The Q-Qplots of the energy generation are compared to the normal distribution at a 95% confidence interval.

**Figure 4.**Scatterplots of the residuals of spot price versus those of each energy generation. Linear correlations among the variables is depicted with the fitted lines.

**Figure 5.**Sensitivity of the mean and the standard for Eta and FRP. Simulation of the behavior of the mean and standard deviation at different contracting levels (Eta) and forward risk premium (FRP).

**Figure 6.**Sensitivity of the VaR and the CVaR to the contracting level (Eta). Simulation of the behavior of the value-at-risk (VaR) and conditional VaR (CVaR) at different contracting levels (Eta).

**Figure 7.**Sensitivity of the VaR and the CVaR to the Eta and FRP. Sensitivity of the value-at-risk (VaR) and conditional VaR (CVaR) to different contracting levels (Eta) and forward risk premium (FRP) values.

**Figure 8.**Sensitivity of the standard deviation, the VaR, and the CVaR to the Eta and Rho. Sensitivity of the standard deviation, the value-at-risk (VaR), and the conditional VaR (CVaR) of the net income from the sale of energy to different contracting levels (Eta) and correlation values (Rho).

**Figure 9.**Sensitivity of the CVaR to different SNP parameters. Sensitivity of the conditional value-at-risk (CVaR) to skewness (d

_{3}) and kurtosis (d

_{4}) parameters. The y axis in both graphs is measured in million Colombian pesos (MCOP).

Type | Generator | Unit | Mean | SD | Skewness | Kurtosis | Percentile | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

5th | 25th | 50th | 75th | 95th | |||||||

Series without transformations | |||||||||||

Spot | Spot | COP/kWh | 124.1 | 125.8 | 4.38 | 27.3 | 40.9 | 64.6 | 85.5 | 145.8 | 249.2 |

Energy Generation | EPMG | GWh | 1038 | 209 | −0.25 | 2.26 | 686.2 | 866 | 1068 | 1198 | 1359 |

ISGG | GWh | 798 | 258 | 0.34 | 2.92 | 391.7 | 620 | 794 | 949 | 1267 | |

CHVG | GWh | 337 | 130 | 0.80 | 3.22 | 160.3 | 246 | 319 | 409 | 616 | |

ENDG (1) | GWh | 1110 | 169 | 0.38 | 3.57 | 882.4 | 1002 | 1093 | 1190 | 1468 | |

Natural logarithm of the series (2) | |||||||||||

Spot | Spot | log | 4.58 | 0.62 | 1.01 | 4.51 | 3.71 | 4.17 | 4.45 | 4.98 | 5.52 |

Energy Generation | EPMG | log | 6.92 | 0.22 | −0.68 | 2.91 | 6.53 | 6.76 | 6.97 | 7.09 | 7.21 |

ISGG | log | 6.63 | 0.35 | −0.55 | 3.06 | 5.97 | 6.43 | 6.68 | 6.86 | 7.14 | |

CHVG | log | 5.75 | 0.38 | −0.09 | 2.60 | 5.08 | 5.51 | 5.76 | 6.01 | 6.42 | |

ENDG | log | 7.00 | 0.15 | −0.22 | 3.86 | 6.78 | 6.91 | 7.00 | 7.08 | 7.29 | |

Relative to the mean. Series with transformation: x/E(x) (3) | |||||||||||

Spot | Spot | pu (4) | 1.00 | 1.01 | 4.38 | 27.25 | 0.33 | 0.52 | 0.69 | 1.17 | 2.01 |

Energy Generation | EPMG | pu | 1.00 | 0.20 | −0.25 | 2.26 | 0.66 | 0.83 | 1.03 | 1.15 | 1.31 |

ISGG | pu | 1.00 | 0.32 | 0.34 | 2.92 | 0.49 | 0.78 | 0.99 | 1.19 | 1.59 | |

CHVG | pu | 1.00 | 0.38 | 0.80 | 3.22 | 0.48 | 0.73 | 0.94 | 1.21 | 1.83 | |

ENDG | pu | 1.00 | 0.15 | 0.38 | 3.57 | 0.79 | 0.90 | 0.98 | 1.07 | 1.32 |

Lag | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Series | |||||||||||||||

Spot | 1.0 | 0.83 | 0.68 | 0.6 | 0.56 | 0.53 | 0.37 | 0.29 | 0.26 | 0.25 | 0.25 | 0.23 | 0.23 | 0.21 | 0.18 |

EPMG | 1.0 | 0.86 | 0.78 | 0.73 | 0.71 | 0.70 | 0.68 | 0.65 | 0.61 | 0.59 | 0.60 | 0.62 | 0.62 | 0.59 | 0.58 |

ISGG | 1.0 | 0.88 | 0.77 | 0.71 | 0.69 | 0.69 | 0.66 | 0.62 | 0.57 | 0.54 | 0.55 | 0.57 | 0.58 | 0.52 | 0.47 |

CHVG | 1.0 | 0.65 | 0.29 | 0.00 | −0.16 | −0.24 | −0.26 | −0.26 | −0.20 | −0.03 | 0.22 | 0.39 | 0.44 | 0.31 | 0.12 |

ENDG | 1.0 | 0.61 | 0.37 | 0.23 | 0.12 | 0.04 | −0.08 | 0.01 | 0.15 | 0.21 | 0.28 | 0.35 | 0.40 | 0.33 | 0.19 |

First differences (Delta of x) | |||||||||||||||

Spot | 1.0 | −0.03 | −0.24 | −0.12 | 0.00 | 0.35 | −0.20 | −0.17 | −0.06 | −0.04 | 0.05 | −0.03 | 0.05 | 0.04 | −0.09 |

EPMG | 1.0 | −0.25 | −0.11 | −0.14 | 0.00 | 0.02 | 0.05 | 0.02 | −0.05 | −0.11 | −0.01 | 0.03 | 0.13 | −0.07 | 0.08 |

ISGG | 1.0 | −0.14 | −0.17 | −0.16 | −0.04 | 0.12 | 0.03 | −0.01 | −0.05 | −0.17 | −0.05 | 0.05 | 0.27 | −0.05 | −0.11 |

CHVG | 1.0 | 0.04 | −0.09 | −0.20 | −0.11 | −0.07 | −0.04 | −0.09 | −0.18 | −0.12 | 0.11 | 0.18 | 0.24 | 0.10 | 0.02 |

ENDG | 1.0 | −0.19 | −0.13 | −0.02 | −0.06 | 0.06 | −0.27 | −0.07 | 0.11 | −0.01 | −0.01 | 0.04 | 0.15 | 0.08 | −0.07 |

Parameter | Spot Price | Energy Generation | ||||
---|---|---|---|---|---|---|

EPMG | ISGG | CHVG | ENDG (1) | |||

Beta 0 | Coeff | 3.84 | 6.61 | 6.14 | 5.68 | 6.75 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | |

Beta 1 (2) | Coeff | 0.0065 | 0.0027 | 0.0043 | 0.0006 | 0.0016 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | |

Phi 1 (3) | Coeff | 0.829 | 0.652 | 0.673 | 0.608 | 0.544 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | |

Residuals (4) | ||||||

Statistics | Mean | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

SD | 0.25 | 0.09 | 0.15 | 0.30 | 0.12 | |

Skewness | 0.51 | −0.60 | −0.65 | −0.43 | 0.04 | |

Kurtosis | 5.64 | 4.27 | 3.76 | 3.64 | 3.35 | |

JB test (5) | Statistic | 72.1 | 28.2 | 21.2 | 10.6 | 0.5 |

p-value | <0.0001 | <0.0001 | <0.0001 | 0.00495 | 0.77195 | |

Ho | rejected | rejected | rejected | rejected | Accepted |

Epsilon | Descriptive Statistics | Multivariate SNP Estimation (2) | |||||||
---|---|---|---|---|---|---|---|---|---|

Standardized Series | |||||||||

Bivariate | Mean | SD | Skewness | Kurtosis | d3 | d4 | Correlation | ||

Spot Price | 0.00 | 0.25 | 0.51 | 5.64 | Coeff. | 0.166 | |||

p-value | 0.001 | Coeff. | 0.278 | ||||||

EPMG | 0.00 | 0.09 | −0.60 | 4.27 | Coeff. | −0.224 | p-value | 0.106 | |

p-value | 0.002 | ||||||||

Spot Price | 0.00 | 0.25 | 0.51 | 5.64 | Coeff. | 0.178 | |||

p-value | <0.0001 | Coeff | −0.676 | ||||||

ISGG | 0.00 | 0.15 | −0.65 | 3.76 | Coeff. | −0.410 | p-value | <0.0001 | |

p-value | <0.0001 | ||||||||

Spot Price | 0.00 | 0.25 | 0.51 | 5.64 | Coeff. | 0.160 | |||

p-value | 0.001 | Coeff | −0.427 | ||||||

CHVG | 0.00 | 0.30 | −0.43 | 3.64 | Coeff. | −0.251 | p-value | 0.002 | |

p-value | 0.003 | ||||||||

Spot Price | 0.00 | 0.25 | 0.51 | 5.64 | Coeff. | 0.146 | |||

p-value | 0.023 | Coeff | −0.420 | ||||||

ENDG (1) | 0.00 | 0.12 | 0.04 | 3.35 | Coeff. | −0.041 | p-value | 0.029 | |

p-value | 0.700 |

FRP | EPMG | ISGG | ||||||

Mean (1) | SD | VaR | CVaR | Mean | SD | VaR | CVaR | |

−20 | 2 | 1.13 | 1.20 | 0.96 | 2 | 0.61 | 0.62 | 0.49 |

−15 | 2 | 1.13 | 1.16 | 0.95 | 2 | 0.61 | 0.62 | 0.48 |

−10 | 2 | 1.13 | 1.16 | 0.95 | 2 | 0.61 | 0.61 | 0.48 |

−5 | 2 | 1.13 | 1.16 | 0.95 | 2 | 0.61 | 0.56 | 0.46 |

−2 | 2 | 1.13 | 1.16 | 0.95 | 2 | 0.61 | 0.56 | 0.45 |

0 | 0 | 1.13 | 1.06 | 0.93 | 0 | 0.61 | 0.56 | 0.43 |

2 | 0 | 1.13 | 1.04 | 0.90 | 0 | 0.61 | 0.44 | 0.43 |

5 | 0 | 1.13 | 1.04 | 0.85 | 0 | 0.61 | 0.44 | 0.41 |

10 | 0 | 1.13 | 1.00 | 0.83 | 0 | 0.61 | 0.26 | 0.40 |

15 | 0 | 1.13 | 0.95 | 0.76 | 0 | 0.61 | 0.26 | 0.40 |

20 | 0 | 1.13 | 0.95 | 0.76 | 0 | 0.61 | 0.24 | 0.40 |

FRP | CHVG | ENDG | ||||||

Mean | SD | VaR | CVaR | Mean | SD | VaR | CVaR | |

−20 | 2 | 0.88 | 0.81 | 0.67 | 2 | 0.87 | 0.84 | 0.74 |

−15 | 2 | 0.88 | 0.81 | 0.66 | 2 | 0.87 | 0.84 | 0.73 |

−10 | 2 | 0.88 | 0.81 | 0.66 | 2 | 0.87 | 0.84 | 0.72 |

−5 | 2 | 0.88 | 0.79 | 0.65 | 2 | 0.87 | 0.77 | 0.72 |

−2 | 2 | 0.88 | 0.69 | 0.64 | 2 | 0.87 | 0.76 | 0.71 |

0 | 0 | 0.88 | 0.69 | 0.64 | 0 | 0.87 | 0.76 | 0.71 |

2 | 0 | 0.88 | 0.69 | 0.64 | 0 | 0.87 | 0.76 | 0.71 |

5 | 0 | 0.88 | 0.69 | 0.63 | 0 | 0.87 | 0.76 | 0.7 |

10 | 0 | 0.88 | 0.69 | 0.63 | 0 | 0.87 | 0.76 | 0.69 |

15 | 0 | 0.88 | 0.69 | 0.62 | 0 | 0.87 | 0.76 | 0.67 |

20 | 0 | 0.88 | 0.68 | 0.61 | 0 | 0.87 | 0.68 | 0.66 |

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**MDPI and ACS Style**

Trespalacios, A.; Cortés, L.M.; Perote, J.
Modeling Electricity Price and Quantity Uncertainty: An Application for Hedging with Forward Contracts. *Energies* **2021**, *14*, 3345.
https://doi.org/10.3390/en14113345

**AMA Style**

Trespalacios A, Cortés LM, Perote J.
Modeling Electricity Price and Quantity Uncertainty: An Application for Hedging with Forward Contracts. *Energies*. 2021; 14(11):3345.
https://doi.org/10.3390/en14113345

**Chicago/Turabian Style**

Trespalacios, Alfredo, Lina M. Cortés, and Javier Perote.
2021. "Modeling Electricity Price and Quantity Uncertainty: An Application for Hedging with Forward Contracts" *Energies* 14, no. 11: 3345.
https://doi.org/10.3390/en14113345