Forecasted Hedge Ratios

97 obtained a total variance reduction of 19.19% and 15.30% over the full in-sample period by using the DCC-GARCH model when hedging with monthly and quarterly futures, respectively. The variance obtained by the DCC model is significantly lower compared to both the unhedged portfolio and the OLS model, but not compared to the CCC-GARCH model. The same is shown for VaR reduction in the same period. One interesting finding for both contracts is that the OLS model significantly outperforms the GARCH models in sub-period 3. Hence, the GARCH models struggle to capture the volatility characteristics in the return series during this period.

It is found that the differences in the hedging results are smaller when using the mean-variance utility function to assess the models’ performances. A notable finding from the results is that there is a substantial variation in the hedging effectiveness of the models across the sub-periods. One potential reason for this could be that the high and time-varying volatility in electricity markets makes it challenging to obtain good hedging results over time (Hanly et al., 2017). It was also found indications that the dynamic models obtain relatively better hedging results compared to the static models in periods with higher volatility in the spot market. Comparing the results for the monthly and quarterly futures, the ranking of the models is practically the same. However, the risk reductions with quarterly futures are significantly lower than the risk reductions obtained with monthly futures. This could be explained by the fact that the spot returns are more correlated to the monthly futures returns than the quarterly futures returns. An implication of this is that actors hedging a quarterly delivery of electricity obtain lower risk reductions in comparison to actors hedging a monthly delivery. The next sub-section will present the out-of-sample analysis to increase the robustness of the hedging results.

98 covariances and variances using the rolling window method explained in sub-section 6.3. Figure 17 shows the obtained hedge ratios from the forecasts for the dynamic models when using monthly futures.

Figure 17 – Forecasted dynamic hedge ratios - monthly futures – out-of-sample

Figure 17 shows that the hedge ratios from the CCC-GARCH and the DCC-GARCH model are very similar. This is found even though the correlation in the CCC-GARCH model out-of-sample is based on historical data up to the last week in the in-sample period, while the DCC-GARCH model forecasts the conditional correlation for each week by adding additional observations from the out-of-sample period.

The first noticeable difference between the models is that the CCC-GARCH hedge ratio is higher than the DCC-GARCH hedge ratio in the spike at the beginning of 2016. After the spike, the differences between the two models are negligible until mid-2017 where the CCC-GARCH model starts to produce somewhat higher hedge ratios. The mathematical reason for this is that forecasted conditional correlations for the DCC-GARCH model in this period become lower than the constant conditional correlation that is assumed for the CCC-GARCH model (see Appendix 3.3.1).

0 1 2 3 4 5 6 7 8

2015 2016 2017 2018

CCC-GARCH DCC-GARCH

99 Figure 18 – Forecasted dynamic hedge ratios - quarterly futures – out-of-sample

Figure 18 shows the hedge ratios for the quarterly futures. Just like for the monthly futures, the forecasted dynamic hedge ratios for the two GARCH models are closely linked, with small differences between them. This implies that the forecasted conditional correlations for the DCC-GARCH model do not deviate much from the constant correlation for the CCC-GARCH model (see Appendix 3.3.2). To further examine the characteristics of the models, and to be able to make comparisons, descriptive statistics for the time series of the dynamic hedge ratios are presented in Table 21.

0 1 2 3 4 5 6 7

2015 2016 2017 2018

CCC-GARCH DCC-GARCH

100 Table 21 - Descriptive statistics for out-of-sample forecasted dynamic hedge ratios

Monthly futures Quarterly futures

CCC-GARCH DCC-GARCH CCC-GARCH DCC-GARCH

Min 0.488 0.493 0.448 0.431

Max 7.428 7.651 6.424 6.708

Mean 1.110 1.054 1.095 1.056

No. of times higher than 1 49 / 163 47 / 163 49 / 163 49 / 163

No. of times higher than 𝛽𝛽_{𝑂𝑂𝑂𝑂𝑂𝑂}^∗ 53 / 163 52 / 163 55 / 163 51 / 163

ADF -5.591^*** -5.424^*** -5.839^*** -5.617^***

𝒬𝒬² 124.82^*** 132.97^*** 119.00^*** 134.62^***

Min and max are the lowest and highest hedge ratios observed in the time series, respectively. Mean is the average of the dynamic hedge ratios. No. of times above 1 represents all observations that lie above the naïve hedge ratio.

𝛽𝛽𝑂𝑂𝑂𝑂𝑂𝑂∗ represents the OLS estimated hedge ratio of 0.960 for the monthly contracts and 0.943 for the quarterly

contracts. ADF denotes the test statistic for the ADF test²⁶ for stationarity, with lags selected according to AIC. The critical value for the ADF test at the 1% significance level is -3.43. 𝒬𝒬² is the Ljung-Box test statistic²⁷ measuring autocorrelation with 15 lags. The critical value for the Ljung-Box test at the 1% significance level with 15 degrees of freedom is 30.58. *** indicates rejection of the null hypothesis at the 1% significance level.

In contradiction to the in-sample estimated hedge ratios, it is now shown that the CCC-GARCH model produces higher hedge ratios than the DCC-GARCH for both contract types, shown by the mean in Table 21. By comparing the hedge ratio for the monthly contracts to those for the quarterly contracts, one takeaway is that hedge ratio series for the quarterly contracts contain lower minimum and higher maximum hedge ratios than those found for the monthly contracts. This is the opposite results of those obtained from the in-sample analysis. All the hedge ratio series in the out-of-sample analysis are found to be stationary and to exhibit positive autocorrelation. These two properties imply that a high hedge ratio decided by the hedger in one week will lead to a high hedge ratio the next week in the absence of shocks, and that the hedge ratio will converge to the long-term mean.