# Estimating Portfolio Value at Risk in the Electricity Markets Using an Entropy Optimized BEMD Approach

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## Abstract

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## 1. Introduction

## 2. Entropy-Optimized BEMD-Based PVaR Methodology

#### 2.1. Bivariate Empirical Mode Decomposition

- Choose the number of directions k to calculate the envelope curve.
- Project the complex-valued signal z(t) on the direction ${\varphi}_{i},i\in 1\dots j:{p}_{{\varphi}_{j}}(t)=Re({e}^{-i{\varphi}_{j}}z(t))$.
- Calculate the maxima of ${p}_{{\varphi}_{j}}(t)$. Extract the locations ${t}_{i}^{j}$.
- Interpolate the set $({t}_{i}^{j},z({t}_{i}^{j}))$ using the chosen curve fitting algorithm, such as the cubic spline algorithm, to generate the envelop curve on the direction ${\varphi}_{j}:{e}_{{\varphi}_{j}}(t)$.
- Calculate the mean curves m(t) from all envelope curves in different directions: $m(t)=\frac{1}{k}{\displaystyle {\sum}_{j}{e}_{{\varphi}_{j}}(t)}$.
- Subtract the mean from the original signal z(t). The sifting elementary operators are defined as S
^{B}[z](t) = z(t) − m(t). - Test S
^{B}[z](t) for the conditions of bivariate IMF. If it qualifies, repeat the above steps on z(t) − S^{B}[z](t). Otherwise, repeat the above steps on S^{B}[z](t).

#### 2.2. Entropy Optimized BEMD Algorithm for PVaR Estimation

- We assume that the electricity market is influenced by market agents with different defining characteristics, such as normal and transient data characteristics. These market agents contribute equally to the market price movement and risk (fluctuation) level during the market price formation process. They are mutually independent, as their correlations and covariance are much smaller in scale and ignorable.
- We classify these characteristics into main groups, including investment strategies, time horizons and investment scales. The market is assumed to be dominated with some main investment strategies, stable at a particular scale over the period of analysis, with finite fluctuation bands at certain boundary values at each scale.

_{t}is returns from k assets and can be assumed to follow some time series models, such as the vector autoregressive moving average (VARMA) model or the vector autoregressive (VAR) model [31]. H

_{t}is the covariance matrix and is defined as H

_{t}≡ D

_{t}R

_{t}D

_{t}. R

_{t}is the correlation matrix at time t. D

_{t}is the standard deviations matrix with elements $\sqrt{{h}_{it}}$ at the i-th diagonal. The element h

_{it}, i = 1, …, k can be assumed to follow the univariate GARCH model as (1).

_{i}and S

_{i}are the lags of the GARCH model for each individual asset i. Parameters α and β need to satisfy the stationary condition ${\sum}_{p=1}^{{P}_{i}}{\alpha}_{i,p}}+{\displaystyle {\sum}_{s=1}^{{S}_{i}}{\beta}_{i,s}}<1$.

_{t}of ε

_{t}at time t is defined as in (2).

_{t}. Q

_{t}is N × N the symmetric positive definite unconditional correlation matrix of ε

_{t}as in (3).

_{t}∼ N(0, R

_{t}) are the standardized residuals. M and N are lags for the correlation specification. Both parameters α and β need to be positive and need to satisfy the stationary condition ${\sum}_{m=1}^{M}{\alpha}_{m}}+{\displaystyle {\sum}_{n=1}^{N}\beta}<1$.

_{t}is the aggregated conditional variance-covariance forecast matrix at time t. ∑

_{NB,t}and ∑

_{TB,t}are conditional variance-covariance forecast matrices for both returns at both normal and extreme phenomenon, respectively, at time t. ${\sum}_{t}=Cov({X}_{t},{Y}_{t})$ is the covariance matrix at time t, Cov(X

_{NB,t}, Y

_{TB,t}) = 0 and Cov(X

_{TB,t,}Y

_{NB,t}) = 0.

_{t}refers to the variance-covariance matrix. ω

_{t}refers to the weight matrix for the portfolio. Z

_{α}refers to the relevant quantile from the standard normal distribution. P is the invested portfolio value. h is the holding period.

^{n}, generated with the unknown data generating process (DGP) with unknown parameters, and the observation Y ∈ R

^{n}, the Shannon entropy of predictor is defined as (6).

## 3. Empirical Studies

_{i}, i ∈ (NSW, QLD) and r

_{i}, i ∈ (NSW, QLD) refer to the price and return in the i market, respectively. The electricity markets are subject to frequent and abrupt external shocks. Descriptive statistics in Table 1 show that the electricity prices in both markets significantly deviate from the normal distribution, as confirmed by the four statistical moments, as well as the rejection of both the Jarque–Bera (JB) test of normality and the Brock-Dechert-Scheinkman (BDS) test of independence [36,37]. Electricity prices in both markets exhibit significant fluctuations. The fluctuations in the NSW market are significantly higher than those in QLD, subject to different regional pricing mechanisms. As the autocorrelation and partial autocorrelation functions indicate the trend factors, the daily prices were log differenced at the first order to remove them as in ${y}_{t}=ln\frac{{p}_{t}}{{p}_{t-1}}$. We further calculated the descriptive statistics on the return data. The distribution of the return data appears to approximate the normal distribution, as indicated by the four moments. The kurtosis appears to deviate from the normal level, which indicates that the market exhibits a significant abnormal event. Besides, since the null hypotheses of both the JB and BDS tests are rejected, this further indicates that the market return contains unknown nonlinear dynamics, not easily captured by traditional linear models.

^{2}(1) distribution. MSE, defined as ${T}^{-1}{\displaystyle {\sum}_{i=1}^{T}{\epsilon}_{i}^{2}}$ where ${\epsilon}_{i}={\overline{y}}_{i}-{y}_{i}$, is used to evaluate the predictive accuracy. We fix the lags to one for the VARMA(1,1)-DCC-GARCH(1,1) model used, since it suffices for most of the situations in the empirical studies and represents the most parsimonious form.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Statistics | p_{NSW} | p_{QLD} | r_{NSW} | r_{QLD} |
---|---|---|---|---|

Mean | 34.4483 | 31.5751 | 0.0001 | 0.0002 |

Maximum | 291.9500 | 292.7700 | 2.2698 | 4.2669 |

Minimum | 13.8700 | 0.3300 | −2.3521 | −4.1527 |

Standard Deviation | 26.9090 | 24.1481 | 0.3203 | 0.3771 |

Skewness | 5.1524 | 4.7733 | −0.3634 | 0.2067 |

Kurtosis | 36.1707 | 33.9937 | 19.7392 | 30.0137 |

p_{JB} | 0.0010 | 0.0010 | 0.0010 | 0.0010 |

p_{BDS} | 0 | 0 | 0 | 0 |

scale | $\overline{{N}_{n}}$ | $\overline{{P}_{n}}$ | $\overline{MS{E}_{n}}$ | $\overline{Entrop{y}_{n}}$ | $\overline{{N}_{t}}$ | $\overline{{P}_{t}}$ | $\overline{MS{E}_{t}}$ | $\overline{Entrop{y}_{t}}$ |
---|---|---|---|---|---|---|---|---|

1 | 89 | 0 | 0.3415 | −203.9906 | 64.6667 | 0 | 0.4409 | −371.4122 |

2 | 37.3333 | 0.0505 | 0.2438 | 79.5904 | 25.6667 | 0.2556 | 0.3190 | 31.1773 |

3 | 37.6667 | 0.0178 | 0.2237 | 84.5989 | 23.6667 | 0.4295 | 0.2888 | 51.8779 |

4 | 26.3333 | 0.1188 | 0.2038 | 155.9300 | 20 | 0.2466 | 0.2644 | 156.2305 |

5 | 25 | 0.1928 | 0.2074 | 144.3604 | 18 | 0.2469 | 0.2693 | 137.4740 |

6 | 25.6667 | 0.1897 | 0.2148 | 154.1064 | 18 | 0.3788 | 0.2787 | 147.5416 |

7 | 20.3333 | 0.3003 | 0.2166 | 148.7923 | 15 | 0.2897 | 0.2828 | 135.7875 |

8 | 23.3333 | 0.1709 | 0.2250 | 149.4715 | 16.6667 | 0.2880 | 0.2943 | 133.2261 |

9 | 22 | 0.3025 | 0.2258 | 151.9358 | 16.6667 | 0.3279 | 0.2950 | 136.4991 |

10 | 22 | 0.3222 | 0.2248 | 151.9826 | 16.6667 | 0.3432 | 0.2939 | 136.6463 |

11 | 22 | 0.3328 | 0.2340 | 159.8013 | 16.3333 | 0.3683 | 0.3039 | 144.8537 |

Models | CL | N_{n} | P_{KT,n} | MSE_{n} | N_{t} | P_{KT,t} | MSE_{t} |
---|---|---|---|---|---|---|---|

MEWMA | 99% | 40 | 0.0867 | 0.2522 | 40 | 0.0867 | 0.2522 |

97.5% | 55 | 0 | 0.1915 | 55 | 0 | 0.1915 | |

95% | 67 | 0.4945 | 0.1477 | 67 | 0.4945 | 0.1477 | |

Average | 54 | 0.1648 | 0.1985 | 54 | 0.1648 | 0.1985 | |

DCC-GARCH | 99% | 11 | 0.6961 | 0.2822 | 7 | 0.0963 | 0.4187 |

97.5% | 20 | 0.0347 | 0.2106 | 11 | 0 | 0.2779 | |

95% | 31 | 0 | 0.1590 | 25 | 0 | 0.1931 | |

Average | 20.6667 | 0.2426 | 0.2173 | 25 | 0 | 0.1931 | |

BEMDPVaR_{exceedances} | 99% | 29 | 0 | 0.1719 | 12 | 0.9222 | 0.2466 |

97.5% | 39 | 0.1532 | 0.1326 | 30 | 0.8763 | 0.1695 | |

95% | 64 | 0.3003 | 0.1043 | 45 | 0.0222 | 0.1230 | |

Average | 44 | 0.3061 | 0.1363 | 29 | 0.6069 | 0.1797 | |

BEMDPVaR_{p} | 99% | 29 | 0 | 0.1741 | 27 | 0.0003 | 0.2029 |

97.5% | 43 | 0.0364 | 0.1348 | 58 | 0 | 0.1430 | |

95% | 66 | 0.5785 | 0.1064 | 90 | 0.0005 | 0.1071 | |

Average | 46 | 0.2050 | 0.1385 | 58.3333 | 0.0003 | 0.1510 | |

BEMDPVaR_{MSE} | 99% | 31 | 0 | 0.1577 | 16 | 0.3166 | 0.2247 |

97.5% | 51 | 0.0008 | 0.1226 | 31 | 0.9782 | 0.1556 | |

95% | 83 | 0.0081 | 0.0974 | 57 | 0.5343 | 0.1140 | |

Average | 55 | 0.0029 | 0.1259 | 34.6667 | 0.6097b | 0.1647 | |

BEMDPVaR_{Entropy} | 99% | 29 | 0 | 0.1741 | 16 | 0.3166 | 0.2247 |

97.5% | 43 | 0.0364 | 0.0364 | 31 | 0.9782 | 0.1556 | |

95% | 66 | 0.5785 | 0.1064 | 57 | 0.5343 | 0.1140 | |

Average | 46 | 0.2050 | 0.1385 | 34.6667 | 0.6097 | 0.1647 |

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## Share and Cite

**MDPI and ACS Style**

Zou, Y.; Yu, L.; He, K.
Estimating Portfolio Value at Risk in the Electricity Markets Using an Entropy Optimized BEMD Approach. *Entropy* **2015**, *17*, 4519-4532.
https://doi.org/10.3390/e17074519

**AMA Style**

Zou Y, Yu L, He K.
Estimating Portfolio Value at Risk in the Electricity Markets Using an Entropy Optimized BEMD Approach. *Entropy*. 2015; 17(7):4519-4532.
https://doi.org/10.3390/e17074519

**Chicago/Turabian Style**

Zou, Yingchao, Lean Yu, and Kaijian He.
2015. "Estimating Portfolio Value at Risk in the Electricity Markets Using an Entropy Optimized BEMD Approach" *Entropy* 17, no. 7: 4519-4532.
https://doi.org/10.3390/e17074519