# Going for Derivatives or Forwards? Minimizing Cashflow Fluctuations of Electricity Transactions on Power Markets

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## Abstract

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## 1. Introduction

## 2. Minimum Variance Hedging Problems of Cashflow Fluctuations

#### 2.1. Minimum Variance Hedging Problem for Power Retailers

#### 2.2. Minimum Variance Hedging Problem for Solar PV Generators

#### 2.3. Minimum Variance Hedging Problem for Thermal Power Generators

#### 2.4. Electricity Transaction Market Including Derivatives

- Power retailers’ hedge. ${V}_{t}\equiv {V}_{t}^{demand}$: Total demand, ${W}_{t}\equiv {T}_{t}$: Temperature;
- Solar PV generators’ hedge. ${V}_{t}\equiv {V}_{t}^{solar}$: Total PV generation, ${W}_{t}\equiv {R}_{t}$: Solar radiation;
- Thermal generators’ hedge. ${V}_{t}\equiv {V}_{t}^{thermal}$: Total thermal power generation, ${W}_{t}\equiv {[{T}_{t},{R}_{t}]}^{T}$: Temperature and solar radiation.

## 3. Estimation and Test Procedures

#### 3.1. Variables Used for Hedging Problems

#### 3.2. Minimum Variance Hedging Using Derivatives

#### 3.3. Minimum Variance Hedging Using Forwards

#### 3.4. Empirical Test Procedure

- Step 1.
- Given observation data of ${V}_{t,m},{S}_{t,m}$ and ${W}_{t,m}$, split the data period into $t\in \left\{1,\dots ,{t}_{0}-1\right\}$ and $t\in \left\{{t}_{0},\dots ,{t}_{1}\right\};$
- Step 2.
- For each hourly period $m,$ apply GAM (6) (or GAM (10)) to find optimal smooth functions, ${f}_{m}$ and ${g}_{m}$ ($\mathrm{or}{\delta}_{m}$ and ${\gamma}_{m}$), and calendar trend function, $Calenda{r}_{m}$;
- Step 3.
- For the optimal smooth functions and $Calenda{r}_{m}$ obtained in Step 2, compute the out-of-sample hedge errors by$$\begin{array}{cc}& \overline{){\u03f5}_{t,m}^{out}}\u2254{V}_{t,m}{S}_{t,m}-\left({f}_{m}\left({S}_{t,m}\right)+{g}_{m}\left({W}_{t,m}\right)+Calenda{r}_{m}\left(t\right)\right),t\in \left\{{t}_{0},\dots ,{t}_{1}\right\}\\ \mathrm{or}& \overline{){\u03f5}_{t,m}^{out}}\u2254{V}_{t,m}{S}_{t,m}-\left({\delta}_{m}\left(t\right){S}_{t,m}+{\gamma}_{m}\left(t\right){W}_{t,m}+Calenda{r}_{m}\left(t\right)\right),t\in \left\{{t}_{0},\dots ,{t}_{1}\right\}\end{array}$$
- Step 4.
- For the out-of-sample data of $t\in \left\{{t}_{0},\dots ,{t}_{1}\right\}$, evaluate the out-of-sample hedge performance using the following variance reduction rate (VRR):$$\frac{\mathrm{Var}\left[{\u03f5}_{t,m}^{out}\right]}{\mathrm{Var}\left[{V}_{t,m}{S}_{t,m}\right]}$$$$\frac{\overline{\left|{\u03f5}_{t,m}^{out}\right|}}{\overline{\left|{V}_{t,m}{S}_{t,m}\right|}}$$

## 4. Empirical Hedge Simulations

#### 4.1. Data

- Electricity price ${S}_{t,m}$ [Yen/kWh]: JEPX spot price in Tokyo area (JEPX Tokyo area price) delivering 1 kWh of electricity from hours $m$ to $m+1$ (downloaded from http://www.jepx.org/market/index.html, accessed on 27 October 2021); since the length of delivery is 30 min for JEPX spot prices, we compute the average of two consecutive prices per hour; for example, we compute the average of 1:00–1:30 p.m. and 1:30–2:00 p.m. delivery prices for the 1:00–2:00 p.m. price;
- Volume ${V}_{t,m}$[kWh]: Hourly realized demand and supply data in the Tokyo area, including the total demand (${V}_{t,m}^{demand}$), the total solar power generation (${V}_{t,m}^{solar})$, and the total thermal power generation $({V}_{t,m}^{thermal})$ between hours $m$ and $m+1$ on day $t$ (downloaded from https://www.tepco.co.jp/forecast/html/area_data-j.html, accessed on 27 October 2021);
- Temperature ${T}_{t,m}$ [°C]: We use hourly realized temperature data on day $t$ in the Tokyo area (downloaded from https://www.data.jma.go.jp/gmd/risk/obsdl/, accessed on 27 October 2021). A temperature index is constructed using the electricity consumption-based weighted average of nine observation points (we used the yearly local electricity consumption data in Tokyo area as of the end of March 2016, obtained from https://www.tepco.co.jp/corporateinfo/illustrated/business/business-scale-area-j.html, accessed on 27 October 2021);
- Solar radiation ${R}_{t,m}$ [MJ/m
^{2}]: We use hourly realized solar radiation data on day $t$ in the Tokyo area (downloaded from https://www.data.jma.go.jp/gmd/risk/obsdl/, accessed on 27 October 2021). A solar radiation index is constructed using an installed capacity of local PV weighted average of seven observation points (we used the installation capacity data in Tokyo area as of the end of March 2019 (corresponding to the end of in-sample period), obtained from https://www.fit-portal.go.jp/PublicInfoSummary, accessed on 27 October 2021).

#### 4.2. Estimation Result for Power Retailers’ Hedges

#### 4.3. Estimation Results for Solar PV Generators’ Hedges

#### 4.4. Estimation Results for Thermal Generators’ Hedges

## 5. Reduction of Risks for Insurance Companies

#### 5.1. Basic Idea

#### 5.2. Evaluation of Insurance Company’s VRRs and NMAEs Using Empirical Data

## 6. Discussion

- For the hedging problems using derivatives for the power retailers and the thermal generators, the payoff functions of the electricity derivatives increase monotonically with the underlying electricity price, but a nonlinear dependence is observed when the electricity price is low during the day. This seems to reflect the relationship between the PV generation and electricity prices. In general, the electricity price increases with demand, but in the daytime solar radiation tends to increase, resulting in pushing the electricity price in the lower direction;
- The coefficients of the electricity forwards for power retailers’ and thermal generators’ hedges have two peaks in a year, which correspond to the demand increases in summer and winter. On the other hand, the coefficients of temperature forwards incorporate the correlation between the temperature and the demand. That is, the demand increases with a higher temperature and decreases with a lower temperature in summer, whereas in winter it increases with a lower temperature;
- Both derivatives and forwards are generally effective for reducing the cashflow fluctuations, but in the cases of power retailers’ and thermal generators’ hedging problems the out-of-sample VRRs and NMAEs were better for hedging problems using forwards. This may be explained by the fact that both cashflows for the power retailers and the thermal generators are largely dependent on the electricity demand, which may be better explained using the cyclic trend for the forwards than the spline functions for derivatives;
- On the other hand, in the case of the solar PV generators’ hedging problem, the hedge errors were smaller for derivatives in terms of the VRRs and NMAEs. The reason for this difference is that it seems that the radiation derivatives are more effective for reducing the cashflow fluctuations for the solar PV generations. The same phenomenon was observed in the hedging problem for the thermal generators, where the radiation derivatives are more effective for reducing the risk of cashflow fluctuations based on out-of-sample VRRs and NMAEs.

- 5.
- The fluctuations in the aggregate cashflow of the electricity derivative’s payoffs from the hedging problems for power retailers, solar PV generators, and thermal generators were reduced significantly compared to the sum of independent cashflow fluctuations. This indicates that the insurance company can take and cancel out the risk in electricity purchase by combining appropriate positions;
- 6.
- For temperature and radiation derivatives, the risk reduction effect for insurance companies is not as significant as in the case of electricity derivatives; however, their risks were reasonably reduced. Moreover, weather derivatives are useful products for insurance companies compared to other financial instruments because weather indexes are not affected by human activities, at least in a short period. Therefore, fair prices may be set using their mean values, and the risk of cashflow fluctuation may be averaged out if the transaction period is sufficiently long.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Daily average price in Tokyo and its 60 days moving average in the period of 1 April 2016 to 31 December 2019.

**Figure 3.**Daily fluctuations of thermal power and solar power generations, total demand, and their 60 days moving averages in the period of 1 April 2016 to 31 December 2019.

**Figure 4.**Daily average of temperature index in Tokyo and its 60 days moving average in the period of 1 April 2016 to 31 December 2019.

**Figure 5.**Daily average of solar radiation index in Tokyo and its 60 days moving average in the period of 1 April 2016 to 31 December 2019.

**Figure 6.**Results of minimum variance hedging with derivatives for electricity retailers’ cash flows, ${V}_{t,m}^{demand}{S}_{t,m}$, based on empirical data: (

**a**) optimal payoff functions of electricity derivatives; (

**b**) optimal payoff functions of temperature derivatives; (

**c**) out-of-sample VRR for each hour; (

**d**) out-of-sample NMAE for each hour.

**Figure 7.**Results of minimum variance hedging with forwards for electricity retailers’ cash flows, ${V}_{t,m}^{demand}{S}_{t,m}$, based on empirical data: (

**a**) coefficients of electricity forwards; (

**b**) coefficients of temperature forwards; (

**c**) out-of-sample VRR for each hour; (

**d**) out-of-sample NMAE for each hour.

**Figure 8.**Results of minimum variance hedging with derivatives for solar PV generators’ cash flows, ${V}_{t,m}^{solar}{S}_{t,m}$, based on empirical data: (

**a**) optimal payoff functions of electricity derivatives; (

**b**) optimal payoff functions of solar radiation derivatives; (

**c**) out-of-sample VRR for each hour; (

**d**) out-of-sample NMAE for each hour.

**Figure 9.**Results of minimum variance hedging with forwards for solar PV generators’ cash flows, ${V}_{t,m}^{solar}{S}_{t,m}$, based on empirical data: (

**a**) coefficients of electricity forwards; (

**b**) coefficients of solar radiation forwards; (

**c**) out-of-sample VRR for each hour; (

**d**) out-of-sample NMAE for each hour.

**Figure 10.**Results of minimum variance hedging with derivatives for thermal generators’ cash flows, ${V}_{t,m}^{thermal}{S}_{t,m}$: (

**a**) optimal payoff functions of electricity derivatives; (

**b**) optimal payoff functions of temperature derivatives; (

**c**) out-of-sample VRR for each hour; (

**d**) out-of-sample NMAE for each hour; (

**e**) optimal payoff functions of radiation derivatives; (

**f**) out-of-sample VRR & NMAE with or without radiation derivatives for each hour.

**Figure 11.**Results of minimum variance hedging with forwards for thermal generators’ cash flows, ${V}_{t,m}^{thermal}{S}_{t,m}$: (

**a**) optimal coefficients functions of electricity forwards; (

**b**) optimal coefficients of temperature forwards; (

**c**) out-of-sample VRR for each hour; (

**d**) out-of-sample NMAE for each hour; (

**e**) optimal coefficients of radiation forwards; (

**f**) out-of-sample VRR & NMAE with or without radiation forwards for each hour.

**Figure 12.**Cash flows (CFs) from derivatives payoffs: (

**a**) payoff of retailers, the sum of payoffs for thermal generators and solar PV generators, and their aggregate payoff from electricity derivatives for 10–11 a.m.; (

**b**) payoff of retailers, the sum of payoffs for thermal generators and solar PV generators, and their aggregate payoff from electricity derivatives for 2–3 p.m.; (

**c**) payoffs of retailers and thermal generators and their aggregate payoff from temperature derivatives for 10–11 a.m.; (

**d**) payoffs of retailers and thermal generators and their aggregate payoff from temperature derivatives for 2–3 p.m.; (

**e**) payoffs of thermal generators and solar PV generators and their aggregate payoff from radiation derivatives for 10–11 a.m.; (

**f**) payoffs of thermal generators and solar PV generators and their aggregate payoff from radiation derivatives for 2–3 p.m.

**Figure 13.**Variance reduction rates (VRRs) and normalized mean absolute errors (NMAEs) for insurance companies’ cash flows (CFs): (

**a**) VRRs for CFs of electricity derivatives’ payoffs; (

**b**) NMAEs for CFs of electricity derivatives’ payoffs; (

**c**) VRRs for CFs of temperature derivatives’ payoffs; (

**d**) NMAEs for CFs of temperature derivatives’ payoffs; (

**e**) VRRs for CFs of radiation derivatives’ payoffs; (

**f**) NMAEs for CFs of radiation derivatives’ payoffs.

Retailers | Solar PV | Thermal | |
---|---|---|---|

Average VRR (Derivatives) | 0.0378 | 0.1796 | 0.0603 |

Average NMAE (Derivatives) | 0.0493 | 0.1366 | 0.0716 |

Average VRR (Forwards) | 0.0218 | 0.1958 | 0.0401 |

Average NMAE (Forwards) | 0.0452 | 0.1506 | 0.0676 |

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## Share and Cite

**MDPI and ACS Style**

Yamada, Y.; Matsumoto, T.
Going for Derivatives or Forwards? Minimizing Cashflow Fluctuations of Electricity Transactions on Power Markets. *Energies* **2021**, *14*, 7311.
https://doi.org/10.3390/en14217311

**AMA Style**

Yamada Y, Matsumoto T.
Going for Derivatives or Forwards? Minimizing Cashflow Fluctuations of Electricity Transactions on Power Markets. *Energies*. 2021; 14(21):7311.
https://doi.org/10.3390/en14217311

**Chicago/Turabian Style**

Yamada, Yuji, and Takuji Matsumoto.
2021. "Going for Derivatives or Forwards? Minimizing Cashflow Fluctuations of Electricity Transactions on Power Markets" *Energies* 14, no. 21: 7311.
https://doi.org/10.3390/en14217311