6.2 Hedging Strategies
6.2.3 Semi-Dynamic Hedge
The out-of-sample results from hedging with ARA gasoil in a static rolling man-ner can similarly be found in Appendix A.2, Table A.5, and shows that the effec-tiveness increases with the time to maturity for every strategy. PAR is the most effective hedging strategy, but also has the highest VaR. The opposite holds for the na¨ıve strategy, which has the lowest effectiveness but also the lowest VaR.
are updated twice, after 2 and 4 months pass by. The cash flow in these cases is defined as
Vt=−m GGOt+h0 ∆F0 +h2 ∆F2 +h4 ∆F4, (6.2.1) whereh0 is determined in time t = 0, h2 in timet = 2 andh4 int = 4.
Looking at the cash flow it is fairly easy to understand the rationale behind the name semi-dynamic, as the hedge ratio is time varying yet it is updated in a discrete time manner. If the company were to choose a 3-months futures, the assumption is that it closes the position just prior to the settlement date, and thus only updates the hedge ratio once. In this case the cash flow looks like
Vt=−m GGOt+h0 ∆F0 +h3 ∆F3 (6.2.2) For a front-month futures contract, the updating is more frequent, where the cash flow is shown by
Vt =−m GGOt+h0 ∆F0 +h1 ∆F1 +h2 ∆F2 +h3 ∆F3 +h4 ∆F4 +h5 ∆F5,
(6.2.3) where the hedge ratio is updated in 5 points in time.
A semi-dynamic hedge is in many ways very similar to our static rolling hedge.
However, when the firm chooses to update the hedge ratio rather than keeping it fixed during the whole period, it avoids the risks of the market being in contango and not accumulating basis risk. Yet there is some risk associated with such a strategy.
If a sudden jump in prices should happen, but this high does not persist over time, the company stands the risk of overestimating the hedge ratio and the hedge may not be as efficient as initially hoped.
Tables 6.7, 6.8 and 6.9 present the full sample results from the semi-dynamic hedge using Brent crude oil futures. Each table gives the hedge results for two maturities. We have not reported the standard deviation of the holding period return for the Spot strategy in this section, but can inform the reader that the standard deviations of the spot strategy (with no hedge) is 0.308, 0.299, 0.293, 0.285, 0.288 and 0.295 for the front-month, 2-months, 3-months, 6-months, 9-months and 12-months maturities respectively.
As can be seen from the three tables, the effectiveness dramatically decreases with the time to maturity for both the MV and Log and Quadratic Utility strategies. In the Log Utility case, it even becomes negative. The VaR increases with respect to
TTM in each case. The MV and Quadratic Utility hedging effectivenesses are rather similar throughout, and are the best performing strategies. The Log Utility strategy performs poorly, and it goes to show that it is important to define the correct utility in order to obtain valuable results.
Overall, it seems as the front-month and 2-months futures perform better with respect to hedging performance than the other maturities, with the highest effective-ness being 69.8%. The MV and Quadratic Utility strategies also seem to outperform the Log Utility strategy. In the Log Utility case, the risk aversion parameter is much lower than it is for the Quadratic Utility case. This might of course have an impact on the hedge ratios and therefore the effectiveness, even though the return on the futures seem to be around zero (or that they are martingales). Bearing in mind the results from the na¨ıve strategy from the static and rolling case (can be found by looking at both Tables 6.3 and 6.5), the na¨ıve strategy seems to do better than the semi-dynamic updating strategy, that with respect to the SD of the HPRs, the effectiveness and lastly the VaR. That it is the futures contracts with the shortest maturities that seem to be performing the best, is no surprise. These contracts are more liquid, which can also be seen from Figure 3.1. There might therefore be some liquidity costs in hedging with the longer-lived futures contracts.
Tables 6.10, 6.11 and 6.12 present the full sample results from the semi-dynamic hedge for ARA gasoil. The standard deviations of the spot strategy (with no hedge) is 0.482, 0.478, 0.476, 0.471, 0.469 and 0.468 for the front-month, 2-months, 3-months, 6-3-months, 9-months and 12-months maturities respectively.
The effectiveness increases from front-month to 3-months maturities for the MV and Quadratic Utility strategies, then it decreases. In the Log Utility case, the effectiveness goes down until 6-months futures, but then it increases some. The VaR by following the MV strategy goes down until 3-months maturity, then it increases. The same happens to the Quadratic Utility strategy, whereas for Log Utility it decreases until 2-months maturity, and then it increases.
It also seems like here, for ARA gasoil futures, that the shorter maturities per-form best with the effectiveness being higher than 80% in most cases. Compared to Brent crude oil, the Log Utility strategy does perform very well with ARA gasoil futures. The VaR is, however, at its lowest in the MV strategy for 3-months futures.
The effectiveness and SD are also showing that this might be a competing strategy one could follow compared to the front-month Log Utility strategy. Nevertheless,
when comparing these results with the na¨ıve strategy results (which can be found in Tables 6.4 and 6.6), the na¨ıve strategy is still outperforming the semi-dynamic one. Yet again, we see that an ARA gasoil hedge, for all strategies, outperform a Brent crude oil hedge.
We have also tested the hedging strategies out-of-sample. Here we separate between two different cases. We first updated the hedge ratio based on historical data, and then on simulated data. We could do this, since we only used the first seven years of our data set to find the optimal hedge ratios. The latter will be commented on in the end of this section. All the out-of-sample results can be found in Appendix A.2.
The results from the historical testing when using Brent can be found in Tables A.6, A.7 and A.8. By pursuing the MV strategy, the effectiveness will decrease with the TTM. The same holds for the Quadratic and Log Utility strategies. The effectiveness is generally lower in the Log Utility case. The VaR increases with TTM in every strategy, which makes perfect sense given the decrease in effectiveness and increase in standard deviation of HPR.
The out-of-sample results for ARA gasoil using historical data when testing the hedging strategy, are shown in Tables A.9, A.10 and A.11. The effectiveness up to 3-months maturity increases for the MV and Quadratic Utility strategies, then it decreases some for 6-months futures, increases some for 9-months futures until it finally goes down for 12-months futures. All in all, the effectiveness decreases with TTM. In the Log Utility case, the effectiveness decreases with TTM. The VaR increases with the TTM for all strategies.
The out-of-sample results for Brent crude oil using simulated data to test the hedging strategy are shown in Tables A.12, A.13 and A.14. The Log Utility strategy is definitely the worst performing. The effectiveness does also here decrease with the TTM, whereas the VaR increases. The hedging performance is thus rather equal in the two cases where we have tested the hedging performance on both historical and simulated data.
Lastly, we document the out-of-sample hedging results using ARA gasoil futures and the simulated data to test the hedging strategy. The results are shown in Tables A.15, A.16 and A.17 in Appendix A.2. The effectiveness increases for MV and Quadratic Utility for maturities ranging from 1 to 3 months, then it decreases for maturities up to 12-months. The effectiveness decreases with the TTM of the
ARA gasoil futures in the Log Utility case. The VaR is higher if using 12-months futures than using shorter maturities, even though it varies and goes both up and down across different maturities.
Static Hedge - Brent crude oil (Full Sample)
MV Mean-Variance
Maturity REG PAR Quadratic Log Na¨ıve Spot
6 h0 1.278 1.245 1.278 1.278 1
SD HPR 0.171 0.169 0.171 0.171 0.165 0.285 Effectiveness 0.640 0.648 0.640 0.640 0.667
VaR -2.970 -2.893 -2.970 -2.970 -2.326
7 h0 1.290 1.246 1.290 1.290 1
SD HPR 0.164 0.162 0.164 0.164 0.161 0.285 Effectiveness 0.671 0.679 0.671 0.671 0.681
VaR -2.867 -2.770 -2.867 -2.867 -2.232
8 h0 1.300 1.248 1.300 1.300 1
SD HPR 0.157 0.156 0.157 0.157 0.158 0.286 Effectiveness 0.698 0.705 0.698 0.698 0.700
VaR -2.801 -2.689 -2.801 -2.801 -2.169
9 h0 1.310 1.249 1.310 1.310 1
SD HPR 0.152 0.151 0.152 0.152 0.156 0.288 Effectiveness 0.722 0.727 0.722 0.722 0.708
VaR -2.746 -2.639 -2.746 -2.746 -2.129
10 h0 1.318 1.250 1.318 1.318 1
SD HPR 0.148 0.146 0.148 0.148 0.153 0.289 Effectiveness 0.741 0.745 0.741 0.741 0.720
VaR -2.737 -2.616 -2.737 -2.737 -2.108
11 h0 1.325 1.252 1.326 1.325 1
SD HPR 0.144 0.143 0.144 0.144 0.152 0.292 Effectiveness 0.757 0.760 0.757 0.757 0.731
VaR -2.747 -2.611 -2.747 -2.747 -2.102
12 h0 1.333 1.253 1.333 1.333 1
SD HPR 0.142 0.141 0.142 0.142 0.150 0.295 Effectiveness 0.770 0.773 0.770 0.770 0.741
VaR -2.772 -2.623 -2.772 -2.772 -2.108
Table 6.3: Static Hedge Results with Brent crude oil 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
Static Hedge - ARA gasoil (Full Sample)
MV Mean-Variance
Maturity REG PAR Quadratic Log Na¨ıve Spot
6 h0 1.093 1.497 1.093 1.093 1
SD HPR 0.151 0.096 0.151 0.151 0.174 0.471 Effectiveness 0.898 0.959 0.898 0.898 0.863
VaR -4.896 -5.797 -4.896 -4.896 -4.665
7 h0 1.104 1.499 1.104 1.104 1
SD HPR 0.149 0.095 0.149 0.149 0.175 0.470 Effectiveness 0.899 0.959 0.899 0.899 0.861
VaR -4.916 -5.787 -4.916 -4.916 -4.659
8 h0 1.115 1.502 1.115 1.115 1
SD HPR 0.148 0.096 0.148 0.148 0.176 0.469 Effectiveness 0.901 0.958 0.901 0.901 0.859
VaR -4.939 -5.782 -4.939 -4.939 -4.656
9 h0 1.125 1.503 1.125 1.125 1
SD HPR 0.146 0.097 0.146 0.146 0.177 0.469 Effectiveness 0.903 0.957 0.903 0.903 0.857
VaR -4.964 -5.781 -4.964 -4.964 -4.655
10 h0 1.133 1.503 1.133 1.133 1
SD HPR 0.145 0.098 0.145 0.145 0.178 0.469 Effectiveness 0.904 0.957 0.904 0.904 0.856
VaR -4.987 -5.781 -4.987 -4.987 -4.657
11 h0 1.140 1.502 1.140 1.140 1
SD HPR 0.145 0.099 0.145 0.145 0.179 0.468 Effectiveness 0.905 0.956 0.905 0.905 0.855
VaR -5.006 -5.781 -5.006 -5.006 -4.659
12 h0 1.145 1.502 1.145 1.145 1
SD HPR 0.144 0.099 0.144 0.144 0.179 0.468 Effectiveness 0.906 0.955 0.906 0.906 0.854
VaR -5.023 -5.782 -5.023 -5.023 -4.663
Table 6.4: Static 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
Static Rolling Hedge - Brent crude oil (Full Sample)
MV Mean-Variance
Maturity REG PAR Quadratic Log Na¨ıve Spot
1 h0 1.180 1.248 1.180 1.180 1
SD HPR 0.161 0.166 0.161 0.161 0.158 0.308 Effectiveness 0.726 0.711 0.726 0.726 0.736
VaR -3.142 -3.327 -3.142 -3.142 -2.662
2 h0 1.208 1.246 1.208 1.208 1
SD HPR 0.162 0.165 0.162 0.162 0.156 0.299 Effectiveness 0.706 0.696 0.706 0.706 0.727
VaR -3.159 -3.258 -3.159 -3.159 -2.611
3 h0 1.231 1.244 1.231 1.231 1
SD HPR 0.164 0.165 0.164 0.164 0.157 0.293 Effectiveness 0.686 0.682 0.686 0.686 0.712
VaR -3.118 -3.151 -3.118 -3.118 -2.529
Table 6.5: Static Rolling Hedge Results with Brent crude oil 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
Static Rolling Hedge - ARA gasoil (Full Sample)
MV Mean-Variance
Maturity REG PAR Quadratic Log Na¨ıve Spot
1 h0 1.031 1.448 1.031 1.031 1
SD HPR 0.166 0.101 0.166 0.166 0.174 0.482 Effectiveness 0.882 0.957 0.882 0.882 0.870
VaR -4.361 -5.793 -4.361 -4.361 -4.257
2 h0 1.045 1.464 1.045 1.045 1
SD HPR 0.161 0.099 0.161 0.161 0.173 0.478 Effectiveness 0.886 0.957 0.886 0.886 0.869
VaR -4.382 -5.768 -4.382 -4.382 -4.232
3 h0 1.058 1.477 1.058 1.058 1
SD HPR 0.158 0.098 0.158 0.158 0.173 0.476 Effectiveness 0.889 0.958 0.889 0.889 0.867
VaR -4.396 -5.810 -4.396 -4.396 -4.203
Table 6.6: Static Rolling 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
Semi-Dynamic Hedge - Brent crude oil (Full Sample)
TTM 1-month Futures 2-months Futures
MV Quadratic U. Log U. MV Quadratic U. Log U.
h0 1.180 1.181 1.273 1.208 1.209 1.292
h1 0.964 0.965 1.084
h2 1.027 0.028 1.534 1.051 1.052 1.156
h3 1.073 1.074 1.196
h4 1.109 1.110 1.224 1.128 1.128 1.227
h5 1.135 1.136 0.239
SD HPR 0.170 0.169 0.232 0.168 0.168 0.217
Effectiveness 0.698 0.698 0.435 0.684 0.684 0.473
VaR -2.908 -2.907 -3.287 -2.981 -2.980 -3.148
Table 6.7: Semi-Dynamic Hedge Results with Brent crude oil 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
Semi-Dynamic Hedge - Brent crude oil (Full Sample)
TTM 3-months Futures 6-months Futures
MV Quadratic U. Log U. MV Quadratic U. Log U.
h0 1.231 1.232 1.318 1.278 1.278 1.370
h1
h2 1.031 1.032 1.121
h3 1.109 1.110 1.203
h4 1.083 1.084 1.165
h5
SD HPR 0.169 0.169 0.225 0.226 0.226 0.271
Effectiveness 0.667 0.667 0.408 0.374 0.374 0.095
VaR -2.983 -2.982 -3.147 -3.546 -3.546 -3.779
Table 6.8: Semi-Dynamic Hedge Results with Brent crude oil 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
Semi-Dynamic Hedge - Brent crude oil (Full Sample)
TTM 9-months Futures 12-months Futures
MV Quadratic U. Log U. MV Quadratic U. Log U.
h0 1.310 1.310 1.408 1.333 1.334 1.438
h1
h2 1.046 1.047 1.139 1.051 1.052 1.146
h3
h4 1.081 1.081 1.162 1.066 1.067 1.147
h5
SD HPR 0.251 0.251 0.295 0.280 0.280 0.324
Effectiveness 0.238 0.238 -0.050 0.098 0.098 -0.206
VaR -3.877 -3.876 -4.007 -4.301 -4.284 -4.371
Table 6.9: Semi-Dynamic Hedge Results with Brent crude oil 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
Semi-Dynamic Hedge - ARA gasoil (Full Sample)
TTM 1-month Futures 2-months Futures
MV Quadratic U. Log U. MV Quadratic U. Log U.
h0 1.031 1.031 1.079 1.045 1.045 1.089
h1 1.102 1.103 1.155
h2 1.247 1.247 1.300 1.271 1.271 1.320
h3 1.348 1.348 1.398
h4 1.422 1.422 1.468 1.444 1.444 1.488
h5 1.470 1.470 1.511
SD HPR 0.217 0.217 0.181 0.205 0.205 0.195
Effectiveness 0.796 0.797 0.858 0.815 0.816 0.833
VaR -5.268 -5.270 -5.714 -5.137 -5.140 -5.703
Table 6.10: Semi-Dynamic 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
Semi-Dynamic Hedge - ARA gasoil (Full Sample)
TTM 3-months Futures 6-months Futures
MV Quadratic U. Log U. MV Quadratic U. Log U.
h0 1.058 1.058 1.100 1.093 1.094 1.134
h1
h2 1.199 1.200 1.247
h3 1.394 1.395 1.442
h4 1.450 1.450 1.450
h5
SD HPR 0.194 0.193 0.218 0.369 0.368 0.422
Effectiveness 0.833 0.835 0.789 0.384 0.385 0.192
VaR -4.982 -4.988 -5.715 -7.602 -7.628 -7.809
Table 6.11: Semi-Dynamic 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
Semi-Dynamic Hedge - ARA gasoil (Full Sample)
TTM 9-months Futures 12-months Futures
MV Quadratic U. Log U. MV Quadratic U. Log U.
h0 1.125 1.125 1.153 1.145 1.145 1.161
h1
h2 1.253 1.253 1.278 1.285 1.285 1.299
h3
h4 1.506 1.506 1.524 1.542 1.543 1.553
h5
SD HPR 0.379 0.379 0.415 0.386 0.386 0.388
Effectiveness 0.344 0.345 0.212 0.315 0.316 0.309
VaR -8.008 -8.005 -8.193 -8.244 -8.243 -8.342
Table 6.12: Semi-Dynamic 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