Futures hedging in electricity retailing

Abstract

This paper is concerned with the risk management practices of an electricity retailer motivated by the Dutch electricity market. We examine the effectiveness of the existing base- and peak-load futures contracts as a risk management tool for the electricity retailers. We analytically characterize the retailer’s optimal hedging policy as a function of the serial correlation of the prices and the demand profiles of its customers. We find that the retailer typically over-hedges in the futures market, and the over-hedging amount increases when both base- and peak-load contracts are used. Our findings indicate that although the existing contracts in the futures market are quite efficient to replicate the exposure from profiled customers, when industrial consumers and renewable generation are included to the retailer’s portfolio, the effectiveness of such contracts decreases substantially. In our motivating example, hedging the risk of the profiled customers with base-load contracts, the firm may reduce the variance of its cash flows by 85.9%. In addition to the base-load contracts, including peak-load contracts into the hedging portfolio of the retailer increases the efficiency of hedging to 89.3%. However, when we consider the aggregate portfolio of the retailer including profiled customers, industrial consumers and renewable contracts, the efficiency of hedging through the existing futures contracts goes down as low as 32.8% during certain periods.

Appendices

Appendix 1 Proofs of Theorems in Sect. 3

Proof of Theorem 1 Taking the derivative of (2) with respect to $Q$ and setting it equal to 0, we get

$$ \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{j = 1}^{I} \left( {2Q^{*} - \left( {D_{i} + D_{j} } \right)} \right)Cov\left( {\tilde{S}_{i} ,\tilde{S}_{j} } \right) = 0. $$

Solving it for $Q^{*}$, we obtain (4). Finally, taking the second derivative with respect to $Q$, we have

$$ \frac{{\partial^{2} Var\left( {\tilde{X}} \right)}}{{\partial Q^{2} }} = 2\mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{j = 1}^{I} Cov\left( {\tilde{S}_{i} ,\tilde{S}_{j} } \right) = 2Var\left( {\mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} } \right) \ge 0 $$

for all values of $Q$ and hence, $Q^{*}$ is a global minimizer.

Proof of Theorem 2 Again, taking the partial derivatives of (6) with respect to $Q_{B}$ and $Q_{P}$, we get

$$ \begin{gathered} \frac{{\partial Var\left( {\tilde{X}} \right)}}{{\partial Q_{B} }} = \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{j = 1}^{I} Cov\left( {\tilde{S}_{i} ,\tilde{S}_{j} } \right)\left( {2Q_{B} + Q_{P} \left( {\alpha_{i}^{P} + \alpha_{j}^{P} } \right) - \left( {D_{i} + D_{j} } \right)} \right) = 0, \hfill \\ \frac{{\partial Var\left( {\tilde{X}} \right)}}{{\partial Q_{P} }} = \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{j = 1}^{I} Cov\left( {\tilde{S}_{i} ,\tilde{S}_{j} } \right)\left( {2\alpha_{i}^{P} \alpha_{j}^{P} Q_{P} + Q_{B} \left( {\alpha_{i}^{P} + \alpha_{j}^{P} } \right) - \left( {\alpha_{j}^{P} D_{i} + \alpha_{i}^{P} D_{j} } \right)} \right) = 0. \hfill \\ \end{gathered} $$

(8)

Solving these equations, we obtain (6). We now need to prove that the solution $(Q_{B}^{*} , Q_{P}^{*} )$ minimizes the variance by showing the convexity of variance with respect to these variables. We can calculate the Hessian of the variance as

$$ \nabla^{2} Var\left( {\tilde{X}} \right) = 2\left[ {\begin{array}{*{20}c} {Var\left( {\mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} } \right)} & {Cov\left( {\mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} , \mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} \alpha_{i}^{P} } \right)} \\ {Cov\left( {\mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} , \mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} \alpha_{i}^{P} } \right)} & {Var\left( {\mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} \alpha_{i}^{P} } \right)} \\ \end{array} } \right]. $$

The positive semi-definiteness of the Hessian matrix can then be proven by using the covariance inequality.

Proof of Theorem 3 The proof is similar to the proof of Theorem 1 and we take the first and second derivative of the variance of the cash flow.

$Var\left( {\tilde{X}} \right) = Var\left( {\mathop \sum \limits_{i = 1}^{I} \left( {P\tilde{D}_{i} + Q\tilde{S}_{i} - \tilde{D}_{i} \tilde{S}_{i} } \right)} \right)$.

Setting the first derivative equal to 0, we obtain the optimal $Q^{*}$ and similar to the proof of Theorem 1, we get the second derivative of the variance to be $2Var\left( {\mathop \sum \limits_{i = 1}^{I} \tilde{S}_{i} } \right)$ which proves the convexity.

Proof of Theorem 4 The proof of this theorem is similar to the proof of Theorem 2, where we take the first and second derivatives of the variance of the cash flow

$$ Var\left( {\tilde{X}} \right) = Var\left( {\mathop \sum \limits_{i = 1}^{I} \left( {Q_{B} \tilde{S}_{i} + \alpha_{i}^{P} Q_{P} \tilde{S}_{i} + P\tilde{D}_{i} - \tilde{D}_{i} \tilde{S}_{i} } \right)} \right). $$

The second derivative of the above variance is the same as in Theorem 2, which proves convexity with respect to $Q_{B}$ and $Q_{P} .$

Appendix 2 Basic distributional statistics for hourly price data

See Table

Table 9 The Basic Distributional Characteristics of Hourly Prices

9

Appendix 3 Simulation Results under Various Residual Distributional Assumptions

See Fig. 

Fig. 8

The daily price profiles in the period between 2005 and 2011 and the simulation results under various distributional assumptions on residuals of the price model

8.