The results from the static hedging strategy are shown in Table 7.1. As can be seen quite clearly in the static case, is that the effectiveness increases as one chooses to hedge with longer-lived futures contracts. The VaR increases with the TTM for all strategies except for Na¨ıve. The MV and mean-variance strategies outperform the na¨ıve strategy.
Static Hedge - ARA gasoil (Full Sample)
Maturity MV Quadratic U. Log U. Na¨ıve Spot
6 h0 0.339 0.339 0.339 1
SD HPR 0.042 0.042 0.042 0.135 0.138
Effectiveness 0.909 0.909 0.909 0.039 VaR -0.777 -0.777 -0.777 -2.372
9 h0 0.349 0.349 0.349 1
SD HPR 0.040 0.040 0.040 0.133 0.138
Effectiveness 0.915 0.915 0.915 0.072 VaR -0.789 -0.789 -0.789 -2.328
12 h0 0.355 0.355 0.355 1
SD HPR 0.039 0.039 0.039 0.133 0.138
Effectiveness 0.918 0.918 0.918 0.071 VaR -0.802 -0.802 -0.802 -2.324
Table 7.1: Static Hedge Results with ARA gasoil Futures using the full sample. The Effectiveness is calculated as in equation (6.1.12), and the SD of the HPR is simply the SD ofHP R= (Vt−V0)/V0. The VaR is at a 95%-level
We tested the static hedge using 6-months, 9-months and 12-months ARA gasoil
Static Rolling Hedge - ARA gasoil (Full Sample)
Maturity MV Quadratic U. Log U. Na¨ıve Spot
1 h0 0.319 0.319 0.319 1
SD HPR 0.380 0.380 0.380 0.173 0.138
Effectiveness 0.379 0.379 0.379 -0.577 VaR -1.845 -1.845 -1.845 -3.876
2 h0 0.208 0.208 0.208 1
SD HPR 0.412 0.412 0.412 0.171 0.138
Effectiveness 0.258 0.258 0.258 -0.540 VaR -1.398 -1.398 -1.398 -3.837
3 h0 0.205 0.205 0.205 1
SD HPR 0.411 0.411 0.411 0.168 0.138
Effectiveness 0.253 0.253 0.253 -0.491 VaR -1.369 -1.369 -1.369 -3.790
Table 7.2: Static Rolling Hedge Results with ARA gasoil Futures using the full sample.
The Effectiveness is calculated as in equation (6.1.12), and the SD of the HPR is simply the SD ofHP R= (Vt−V0)/V0. The VaR is at a 95%-level
futures. Compared to the static results in Chapter 6, we can clearly see that the hedge ratios are lower. This is also related to the lowered risk in the contract price compared to the spot price of German gasoil. Furthermore, the effectiveness increases and the SD of the HPRs decreases as the time to maturity increases for all strategies. Noteworthy is also the fact that all strategies except for Na¨ıve have equal results. This is again due to the martingale property of the futures prices.
The VaR increases with the TTM for the MV, Quadratic and Log Utility strategies, whereas it decreases for the na¨ıve strategy. The na¨ıve strategy performs the poorest with low effectiveness and high VaR. The effectiveness is highest and the SD lowest when going for the 12-months futures contract. Nevertheless, the VaR is highest for this maturity.
Also in the static rolling case, the na¨ıve strategy performs poorest, both with respect to effectiveness, VaR and SD of the HPRs. The effectiveness is even negative, which means that a na¨ıve hedging strategy not should be pursued. It therefore seems, as a lower hedge ratio is optimal in this case. The other strategies perform better, and are the most effective when using front-month futures. The SD is also lowest
for the shortest maturity. Not surprising is that the VaR is also highest when using front-month futures. Compared to the static case, the static rolling hedging strategy performs worse given overall lower effectiveness, higher SD of the HPRs and higher VaR.
In the semi-dynamic case, the MV and Quadratic Utility strategies clearly out-perform the Log Utility strategy. The VaR is lower, the effectiveness is substantially higher and the SD of the HPRs is much lower than in the Log Utility case. When comparing the MV strategy with the Quadratic Utility strategy, the Quadratic one slightly outperforms the former. It has a lower SD of the HPRs, a higher effectiveness and a lower VaR. The best performing strategy is thus based on a Quadratic Utility using 3-months ARA gasoil futures. However, when comparing the semi-dynamic hedging strategy with the static one - the semi-dynamic strategy comes short and is less effective with a higher risk involved. This might also be in relation the bid-ask spread, which will become more visible when futures are sold and bought. This might also very well be why the static hedging strategy outperforms both the static rolling and the semi-dynamic strategies.
The static strategy also outperformed the other strategies in the previous chap-ter. It therefore looks as though static hedging is more effective than the rolling or semi-dynamic. The 12-months futures also seem to be a bit more effective than the 6-months futures. This matches our results from Chapter 6, seen that we have not performed the PAR hedge in this case study. An interesting next step would naturally be to calculate and estimate a PAR hedge in this case as well. However, given the scope of this thesis, we decided not to.
Semi-Dynamic Hedge - ARA gasoil (Full Sample)
Maturity MV Quadratic U. Log U.
1 h0 0.191 0.191 0.239
h1 0.265 0.265 0.317
h2 0.326 0.327 0.379
h3 0.372 0.372 0.422
h4 0.405 0.406 0.452
h5 0.429 0.429 0.470
SD HPR 0.085 0.084 0.136
Effectiveness 0.620 0.628 0.027
VaR -1.517 -1.510 -1.542
2 h0 0.194 0.194 0.238
h2 0.335 0.336 0.386
h4 0.415 0.415 0.460
SD HPR 0.083 0.082 0.158
Effectiveness 0.638 0.645 -0.322
VaR -1.414 -1.408 -1.582
3 h0 0.197 0.198 0.239
h3 0.390 0.390 0.437
SD HPR 0.081 0.081 0.184
Effectiveness 0.650 0.656 -0.798
VaR -1.296 -1.290 -1.650
Table 7.3: Semi-Dynamic Hedge Results with ARA gasoil Futures using the full sample.
The Effectiveness is calculated as in equation (6.1.11), and the SD of the HPR is simply the SD ofHP R= (Vt−V0)/V0. The VaR is at a 95%-level
Conclusion
In this thesis, we have introduced different ways of hedging exposures in energy markets for which there do not exist any liquidly traded futures markets. Specifically we have showed ways of hedging a commitment to buy in the German gasoil markets with futures on Brent crude oil and ARA gasoil. The methodology used is based on a 3-factor model introduced by Bertus, Godbey, and Hilliard (2009). This model acted as the foundation for testing different hedging strategies while also taking into account different risk preferences.
We have presented how to build a stochastic model for spot prices and spreads, while taking into account characteristics of prices such as a stochastic convenience yield and mean reversion. The model also allowed for correlations between price, spread and convenience yield. The analysis started with testing whether spreads were stationary, and then an estimation of parameters using the Kalman filter fol-lowed. With this we were able to simulate spot prices for Brent crude oil and ARA gasoil, the associated convenience yield and the spread between German gasoil and futures prices on Brent crude oil and ARA gasoil. The next step was then to calcu-late futures prices and the illiquid German gasoil spot prices.
With the 3-factor model acting as the foundation of the study, we could move on with defining and testing different OHRs with different underlying strategies and risk preferences, and thus calculating cash flows. Finally, we used two different measurements to evaluate the effectiveness of each hedging strategies.
Despite some inconsistencies in our results we found that a static hedge outper-formed both a rolling and a semi-dynamic hedge. For the static hedge we observed a possible trade-off between correlation and volatility where the effectiveness increased
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with time to maturity. However looking at the VaR, it also increased with time to maturity. Thus the choice between hedging with futures of longer maturities or shorter ones eventually boils down to a choice between looking at potential returns or potential losses. Furthermore for all strategies, hedging with ARA gasoil futures seems a better choice, where we see in the static PAR case an effectiveness as high as 95.9% for a front month futures, than hedging with Brent crude oil futures. How-ever hedging with ARA gasoil may incur some higher liquidity risks. For the DONG Energy case we also saw that the static case seems to outperform all other strategies, showing some robustness in our results. However, in the DONG case we added a bid-ask spread to the prices to account for possible transaction costs, which would very likely reduce the effectiveness of both a rolling and a semi-dynamic hedge. The reason being that the more roll-overs and updates of the hedge ratio one do, the more transaction costs on incurs.
As a final note, we wish to emphasize the importance of considering different ways for a large energy company to hedge exposures in the spot market. For one energy prices are much more volatile than other asset types, thus effective risk management is highly important so as to ensure to some extent a more certain outlook for the company. Secondly, as hedging with futures within the energy industry is no trivial matter, research and empirical testing on strategies such as proxy hedging may become an important source for energy companies to turn to when choosing an optimal way of mitigating risks. For this reason we hope that, although further extensions should be considered, our analysis may contribute to existing research and spur interest within the field of proxy hedging.
Figures, Tables and Mathematical Proofs
A.1 Figures
All the log spreads between German gasoil and both Brent and ARA gasoil futures shown below.
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(a) GGO and BR1M (b) GGO and BR2M (c) GGO and BR3M
(d) GGO and BR4M (e) GGO and BR5M (f ) GGO and BR6M
(g) GGO and BR7M (h) GGO and BR8M (i) GGO and BR9M
(j) GGO and BR10M (k) GGO and BR11M (l) GGO and BR12M
Figure A.1: Log Spreads between German gasoil and Brent Futures. These twelve graphs show the log spreads between the illiquid German gasoil and the different Brent crude oil futures
(a) GGO and GO1M (b) GGO and GO2M (c) GGO and GO3M
(d) GGO and GO4M (e) GGO and GO5M (f ) GGO and GO6M
(g) GGO and GO7M (h) GGO and GO8M (i) GGO and GO9M
(j) GGO and GO10M (k) GGO and GO11M (l) GGO and GO12M
Figure A.2: Log Spreads between German gasoil and ARA gasoil Futures. These twelve graphs show the log spreads between the illiquid German gasoil and the different ARA gasoil futures