Higher moments in the fundamental specification of electricity forward prices

Abstract

An extended specification for estimating the risk premia necessary for the forward pricing of wholesale electricity is developed in order to respond to the increasing need for more precise risk management of hedging positions in practice. Using Taylor expansions, we provide new specifications for the electricity forward premium including its dependency on all four moments of the expected wholesale price density as well as the higher moments of the demand density including skewness and kurtosis. Overall we argue that previous models have been underspecified and that the extended formulation proposed in this analysis is robust and worthwhile.

Keywords:

• Electricity prices
• Electricity demand
• Monte Carlo simulations
• Forward risk premium
• Student-t distribution
• Skew-t distribution
• Kurtosis

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We can arrive at (3(3) $\begin{array}{rl}h\left(x\right)& \approx {\text{β}}_1(x-{\mu }_W)+{\text{β}}_2(x-{\mu }_W{)}^2\\ & +{\text{β}}_3(x-{\mu }_W{)}^3+{\text{β}}_4(x-{\mu }_W{)}^4\end{array}$(3) ), just by taking the cubic Taylor approximation of a simpler function $g\left(x\right)=(x^{b+1}-c{P}_R{x}^b)$ and multiplying it by $(x-{\mu }_W)$.

2 For instance, consider $h\left(x\right)={\mathrm{e}}^x$ and the corresponding quartic Taylor polynomial (around x = 0): $p_4\left(x\right)=1+x+x^2/2+x^3/6+x^4/24$. If $X\sim N(0,1)$ we can compute $\mathbb{E}\left[{\mathrm{e}}^X\right]={\mathrm{e}}^{1/2}\approx 1.649$ and $\mathbb{E}\left[p_4\right(X\left)\right]=13/8=1.625$ and we see that the quartic Taylor approximation holds reasonably well also for the expectations. However, if X has a Laplace distribution with parameter $\lambda \in (0,1)$, that is $X=\lambda (Y_1-Y_2)$ where $Y_1$ and $Y_2$ are independent and exponentially distributed with mean 1, then we can compute $\mathbb{E}\left[{\mathrm{e}}^X\right]=(1-{\lambda }^2{)}^{-1}$. Next, we have $\mathbb{E}\left[X\right]=\mathbb{E}\left[X^3\right]=0$, $\mathbb{E}\left[X^2\right]=2{\lambda }^2$ and $\mathbb{E}\left[X^4\right]=24{\lambda }^4$, so that $\mathbb{E}\left[p_4\right(X\left)\right]=1+{\lambda }^2+{\lambda }^4$. We see that $\mathbb{E}\left[p_4\right(X\left)\right]$ can be far away from $\mathbb{E}\left[{\mathrm{e}}^X\right]$ if λ is close to 1. For instance, if $\lambda =0.9$, then the former expectation is 5.26, while the latter one is just 2.47.

3 They show from a detailed analysis of German prices that these prices, under a wide range of market conditions, can be characterized by two-, three- and four-moment distributions.

4 In other words we consider the distributions left-truncated at 0, which is in line with van Koten (2020). However, differently from him, we do not discard replicates outside the interval $\mu \pm 5\sigma$.

5 Convexity can be observed also in the case c = 2, 3 and $\lambda =1.2$ by widening suitably the range for ${\sigma }_D$. In figure 5 we kept the range $(1,40)$ for ${\sigma }_D$ in order to ease the comparison with results in van Koten (2020).

6 Data is from the website www.destatis.com accessed in February 2022.

7 Γ denotes the Gamma function.

Additional information

Funding

This work was supported by the Università Degli Studi di Modena e Reggio Emila [grant number FAR2021].