4.1. Use of future contracts as hedging

For purpose of explanation, let´s introduce the following case: A NG distributor of an urban area acquires the fuel that it will subsequently sell to domestic or industrial users from PEMEX or an importer. The price of the NG is acquired at a fixed price at the entrance of the urban area (city gate), once the gas has been transported from the point of importation or from a processing terminal. For a month, the purchase price of the gas will be fixed in pesos and the distributor, in turn, must sell it at a fixed price to its users. The next month, the distributor will buy the NG at another price, which will depend on fuel prices in South Texas and the peso-dollar exchange rate, among others. To manage the risk represented by the variation of the NG in dollars and the exchange rate, the distributor may take positions of futures contracts for the gas and for the exchange rate. As was stated, the price volatility of NG in dollars may be higher than the volatility of the exchange rate so that the two hedging strategies, one for the price of NG in dollars and the other for the exchange rate in pesos, could be independent and intermittent.

The NYMEX market offers NG futures contracts with monthly maturities that span a decade ahead. For example, the December 2016 contract was last listed on November 28, had physical delivery on December 31, 2016 and each contract covers 10,000 MMBtu (ten billion Btu). The pulse (tick) of quotation prices is 0.001 US dollars. On the other hand, the contract of future peso-dollars in the Chicago Mercantile Exchange (MCE) covers Mx Ps 500,000, with a minimum fluctuation in the price of USD 0.00001 per peso, equivalent to 5 dollars per contract. The contracts have monthly maturities and cover a period of 18 months.

A distributor that estimates that the prices of NG in dollars will be on the rise and that the peso will depreciate in the coming weeks or months can buy NG futures in the NYMEX and buy dollar futures in the CME. To allow these operations, the distributor will need to open contracts and provide guarantees, and before the expiration of the contracts, he or she must revert them, unless the distributor wishes to reach the "physical delivery" of the goods. In case of the reversal, the distributor will take his or her profit or loss, and with it he or she will go to the spot exchange market to convert the dollars to pesos. With the possible benefit, the distributor can acquire NG from the new PVPM. If the hedging strategy was successful, the distributor will have the ability to acquire the same or a higher volume of NG as a result of good risk management. The operation would be contrary if the price expectation were down: Futures would be sold in the NYMEX and, if necessary, peso futures would be bought in the CME. In any case, the resulting dollar would be expected to be positive. Figure 1 shows the use of futures contracts as a hedging tool.

The classic theory of hedging with futures, see for example ^{Hull (2009)} and ^{Ghoddusi & Emamzadehfard (2017)}, consists of reducing or nullifying the price volatility of a spot position with the inclusion of a certain number of futures contracts in the portfolio. If we have a P portfolio with nS long asset positions and nF short futures positions, the hedge ratio is defined as the number of futures positions that are occupied to cover a unit of the spot position, that is, h  =nF/nS .

The value of the portfolio, considering nS units of assets to be covered and nF units of futures, would be given by equation (1),

Therefore, changes in the covered portfolio are given by equation (2),

The minimum variance hedge ratio is estimated by selecting the number of futures contracts that minimizes the conditional variance of changes in the value of the portfolio. The optimal hedging ratio is given by equation (3),

where I is the set of information in time t and h* and is the optimal hedging ratio. The hedge ratio h* can be easily estimated using ordinary least squares (OLS), as in equation (4), where ΔPt is the PVPM monthly growth rate and ΔFl,t is the monthly growth rate of an NG Futures Contract with l lags,

and thus, the optimal hedge ratio would be applicable to the hedging instrument Fj , as shown in equation (5),

The historical data can not only serve to determine an optimal hedging up to the last date of the data, it can also contribute to estimating the hedge that must be taken to face a risk that is expected in the immediate future through the prediction of conditional variances. Additionally, the initial strategy may change as new data is known that makes it necessary to rebalance the portfolio. In summary, in cases where there is a certain seasonality, historical data can be used to estimate future parameters, and it is convenient to update the information with newly available data that, in turn, will result in new estimates. Let us introduce the VCC multivariate GARCH model proposed to replicate the volatility of the underlying and suitable hedging instruments.

4.2. Multivariate GARCH VCC model

GARCH models are those in which the conditional variance of the errors can be explained through the variance of the previous errors and, usually, they are used together with the ARCH (Autoregressive Conditional Heteroscedasticity) models in which the conditional variance of the errors is explained through the behavior of the errors of the past periods. See ^{Engle (1982)} and ^{Bollerslev (1986)}.

Different authors have used and evaluated the use of GARCH models as predictive tools to estimate price volatility, particularly in energy. See ^{Agnolucci (2009)}, ^{Wei, Wang & Huang (2010)}, ^{Wang & Wu (2012)}, ^{Lv & Shan (2013)}, ^{Gannon & Liu (2013)}, ^{Scholtens & Van Goor (2014)}, and ^{Blazsek & Villatoro (2015)}.

The multivariate GARCH models (MGARCH), following the notation of ^{Orskaug (2009)}, are defined as:

μt in equation (6) can be modeled as a constant vector or as a time series; at is not correlated in time, which does not mean that it does not have a serial dependency, but that the dependency can be non-linear. On the other hand, Ht in equation (7) is a matrix of conditional variances, which needs to be inverted every period t. Besides, for Cholesky factorization to be possible, Ht must be positive and defined.

In the VCC multivariate GARCH model, conditional variances are modeled as univariate GARCH models and conditional covariances are modeled as non-linear functions of conditional variances. The parameters of the quasi-correlations involved in the non-linear functions of the conditional variances follow a GARCH model specified by Engel (2002). In the MGARCH VCC there is a revolving estimator of the covariance matrix of standardized residues, following the development of ^{Tse & Tsui (2002)}.

The optimal hedge ratios hj* of equation (5) can be calculated with the conditional variances and covariances obtained through the MGARCH VCC model. These optimal hedge ratios can correspond to the whole period of data or they can be estimated for subperiods, even on a daily basis, as conditional variances and covariances can be obtained dynamically, that is, the newest estimates considers the last historical data available, as new information arrives, a new set of conditional variances and covariances can be calculated, and thus, new optimal hedge ratios. This can be performed with in-sample or out-of- sample data.

In the following section, we discuss the data, the results from a single hedging strategy, the optimal hedging strategy from a MGARCH VCC model, and the suitability of the dynamic hedging proposal with the use of a backtesting tool.

5.1. Data

The data sample is from the beginning of 2012, until June 30, 2017 when CRE ended the publication of the PVPM. The NG price series in the United States are from the US Energy Information Administration (EIA) website; spot and futures market prices correspond to those of NYMEX; the PVPMs in Reynosa and Pemex City are those published by the CRE, and the exchange rates of the peso-dollar are those published by Banco de México (BANXICO). In its first part, as was stated, this study follows the methodology of ^{Barrera-Rivera & Valencia-Herrera (2019)} for an extended study period.

Figure 2 shows the graph of the daily and monthly PVPM in Reynosa during the study period. Notice that the monthly values do not correspond to the average of the daily values. The reason is that monthly PVPMs were determined one day before the beginning of the month and sustained throughout the period. However, daily prices were also calculated one day in advance and modified daily, which allowed them to reflect information more up to date on international prices. It should be noted that the PVPMs correspond only to business days, they exclude weekends and holidays.

Table 1 shows the statistics of the continuous growth rates of the Reynosa’s daily PVPM in dollars (USD PVPM), of the NG Spot Price at NYMEX (Spot NYMEX) and the Mexico United States exchange rate (Mx Ps - USD XR). The value of the skewness in the PVPM (-0.63395) indicates that the distribution is moderately biased and the value of kurtosis (57.51959) shows that the distribution is sharply leptokurtic; therefore, it is not normal. However, the elements of greatest interest for this work are those of volatilities: the standard deviation for the logarithmic variation of the daily PVPM was 0.04421 and the standard deviation of the logarithmic variations of the NG Spot Price in NYMEX was 0.03920, which means that, during the study period, the PVPM in Reynosa was more volatile than the NG spot price in NYMEX.

Table 1 also shows the statistics of the logarithmic exchange rate variations (Mex Ps - USD). During the study period, the standard deviation of the continuous daily growth rate was 0.0058. It should be noted that, although the prices and quotations of this study refer to the same period of analysis, the observations of the exchange rate include dates of weekends and others in addition to those of NG prices. During the period considered, the volatility of NG dollar spot prices (NYMEX), measured through the standard deviation of the logarithmic variation, was 6.7 times the volatility of the peso-dollar exchange rate.

Figure 3 shows graphically the peso-dollar exchange rate in the study period. As can be seen in the figure, the exchange rate experienced significant volatility from 2015 until the last date of the analyzed period.

Table 2 depicts numerically the differences in the standard deviations of the prices during two subperiods: from January 2012 to December 2014 and from January 2015 to the end of the sample period. Standard deviations of the second subperiod are larger than those of the first subperiod in the domestic markets. The opposite happens with the NG spot prices of NYMEX where the volatility is greater in the first subperiod. This shows that the local market had its own sources of variations. Notice that in both subperiods, the volatility of the NG spot price in the NYMEX was several times greater than the volatility of the peso-dollar exchange rate.

Since there is an open market to import NG to Mexico, the PVPM in Reynosa was the reference for the other local market prices, even for the PVPM in Pemex city, the main production center. Therefore, we will focus on the PVPM of Reynosa and, first, on its monthly version. During the study period, the monthly PVPM in Reynosa in dollars is highly correlated with the daily one-month future prices ‘Future # 1’ of the NYMEX NG. Figure 4 shows graphically the proximity of the monthly PVPM and the daily prices of the future contract.

5.2. The simple hedging strategy

In the case of an urban NG distributor and that of an industrial user of the product. We consider the CME lists NG futures contracts that take Henry Hub index prices as a reference. These futures contracts have a very close relationship with their underlying. Also, gas prices in Henry Hub have a very close relationship with those of Texas Eastern STX, Tennessee Zone 0 and Houston Ship Channel, as can be seen in ^{Barrera-Rivera & Valencia-Herrera (2019)}. Therefore, the hedge ratios consider PVPMs in Reynosa as spot prices and the CME NG Henry Hub futures.

In order to estimate the hedge ratios, we use equation (4), where ΔPt the monthly growth of the PVPM in Reynosa at month t and ΔF1,t is the one-month growth of the one-month Henry Hub gas future price at month t. Note that, due to the solution of the OLS method, the coefficient a1 in equation (4) is the same as the optimal hedge ratio h* in equation (3).

Since the estimation of PVPMs considers international NG previous prices, futures from previous periods can be useful for making PVPM hedges. Figure 5 shows the growth in the prices of the three-month futures and the growth of the PVPM prices in Reynosa in dollars. Notice that PVPM of Reynosa with one and two months of advance and delay have a statistically significant relationship with the futures at three months.

From Figure 5, it could also be stated that there may be more than one hedging instrument, for example, the Future # 3 with zero and one month of offset, so that equation (3) could be extended to more than one hedging instrument.

Table 3 shows the hedge ratios with NG futures of the CME for the period and previous periods. From the table, only the futures of one and two delayed periods to the PVPM offer hedging possibilities, since only in these cases hj* are statistically significant. The optimal hedging of a natural gas seller in the Mexican market could be structured with the instrument lagged one month by taking a short futures position for 54.7082% of the value of the position to be filled a month before the natural gas is sold to PVPM. The R2 is an indicator of the potential risk reduction using hedging, here 14.0259%. Since the optimal hedging ratio for futures with two months of delay is statistically significant, it could be hedged, for example, the purchase of PVPM buying futures for 36.4429% of the value of the position to be filled two months before it was made the purchase.

The hedging can be structured by acquiring multiple futures during several previous periods. Because the autocorrelation in the growth of futures with months of lag is very small and not statistically significant, it is possible to consider futures with arrears of one and two months as independent instruments. Therefore, the coefficients that are obtained when making a linear regression of the growth in PVPM with respect to the growth of futures with one and two months of lag can be considered as optimal hedge ratios with each instrument, in a multiple hedging. From Table 4, a position of gas subject to PVPM could be filled with futures of different maturities with one and two months of lag, acquiring futures at two months, with a value of 56.0542% of the position, one month before and 31.11712% of the value to cover two months before, for a risk reduction of 23.3381% ( R2 ).

To analyze the hedge with exchange rate futures for the purpose of analysis, synthetic futures prices were estimated using the interest rate parity Ft=St(1+rd)t/(1+rf)t , where Ft the price of the future quoted at period t , St is the exchange rate spot in direct quotation and rd y rf are the effective domestic and foreign rates at the future term in period t , respectively. In this case, the rates of the 91-day Cetes and the 90-day Treasury Bills were considered, adjusted for a period of one month. In a similar exercise, the PVPM in Reynosa can be covered with two- and three-month NG futures with one and two lags and one-month MXN-USD exchange rate futures, see Table 5.

5.3. Hedging under the MGARCH VCC model

In ^{Barrera-Rivera & Valencia-Herrera (2019)}, optimal hedging ratios hj* are estimated for a PVPM spot position with the exchange rate and one month and two months lagged NG futures. Once the hedge is determined, it is can be necessary to rebalance the hedge based on estimates of conditional variance forecasts and correlations, both variations in the PVPM in Reynosa and of the NG futures used, since these elements concentrate the risk.

Table 5 shows the results of the MGARCH VCC for the daily series of the variations of the PVPM in dollars (’reynosavpm’) and the two-month futures of the NG Henry Hub, with 20 and 40 days of lag (’lag20’ and ’lag40’, respectively). The 20 and 40 days of lag are equivalent, in the daily series of prices, to 1 and 2 months of lag in the monthly series used in section 3.4.1. For the model, 1,324 daily observations were used, distributed in a t-student manner and a Newton-Raphson optimization method. From the results of Table 6 it follows that the ARCH and GARCH coefficients are statistically significant at more than 99%; in the estimation of correlations, an acceptable statistical security was not achieved.

In the case of daily variations in Reynosa PVPM in dollars, the two-month futures with 20 days lag, and the two-month futures with 40 days lag, the conditional variance is estimated as in equations (8) to (10), respectively,

With these values of conditional variances, the new optimal hedge ratios hj* can be estimated and done on a recurring basis, as new information is obtained, in the manner of ^{Gannon and Liu (2013)}. The model in Table 6 and equations (8), (9) and (10) can be used both to forecast conditional variances of future periods, and to estimate conditional variances for the historical data period itself and, with conditional variances, calculate the optimal hedge ratios hj* by applying equation (5).

Figure 6 graphically shows the conditional covariances estimated in the study period obtained using the MGARCH VCC model. The estimation of conditional variances is dynamic, that is, even if the determined coefficients are applied to current data, the value of these coefficients is updated as new information is received and these new values are applied to the following current data.

Figure 7 shows the estimates of the conditional covariances between the variations of the PVPM in Reynosa and those of the two-month Henry Hub futures, with lags of 20 and 40 days, for the last 100 days of the series, which include 10 forecasted days. The predicted conditional covariances are those that appear after the vertical line. The covariances of the last 100 days are shown in Figure 8, however, the data of the 1,384 days of the study period were used to obtain them. The purpose of Figure 8 is to depict in greater detail the last part of the estimated conditional covariances.

As already stated from the data of the conditional variance matrix, the optimal hedge ratios hj obtained using equation (5). Figure 8 shows the graph of the hedge ratios between the spot position and the futures with lags of 20 and 40 days in the study period.

Table 7 lists the optimal hedge ratios hj* predicted for the 10 days following the last date with historical data and, as support, the conditional covariances between the spot position and the futures are detailed.

5.4. Backtesting in the VCC model

Forecasts of conditional variances and optimal hedge ratios hj* say little about the goodness of the estimate. Figure 8 above shows the conditional covariances predicted for a period of 10 days, however, how much do the out-of-sample covariances forecasts approximate the estimated in-sample covariances? In order to resolve this uncertainty, we performed a backtesting; first, we used the first 90% of historical data (in-sample) to forecast the last 10% of the information (out-of-sample). Figure 9 graphically shows the results in the forecast period; in that figure, the first current and forecast optimal hedge ratios appear within the ellipse.

Notice that the out-of-sample predicted hedge ratios hj* do not closely follow short-term changes in the in-sample ratios; however, the order of the predicted ratios is the same as that of the current ones, that is, the in-sample and out-of-sample h20* predicted hedge ratios are lower than the in-sample and out-of-sample h40* predicted ratios. Table 8 shows the backtesting statistics, both for in-sample data and in the out-of-sample forecast period.

The absolute differences between the optimal h20* and h40* in-sample and the forecasted out-of-sample hedge ratios are 0.05364 and 0.05894, respectively. Table 9 shows the “memory” that the estimated hedge ratios retain, since they do not fully reflect the decline in actual hedge ratios in the forecast period, within the backtesting.

In order to reduce this “memory” period in the estimated hedge ratios, we reduced the period of actual data to a minimum in which the MGARCH VCC estimates were convergent with the Newton-Raphson method and we sought to make forecasts for a shorter period (10 days). Table 8 shows the results for this shorter backtesting period.

The absolute differences between the optimal h20* and h40* in-sample hedge ratios and the out-of-sample forecasts in the 252 days period are 0.036685 and 0.001635, respectively, which implies reductions in the differences of the estimates of 31.61% and 97.23% for hedge ratios of 20 and 40 days. That is, the proximity of the actual data and the shortage of the predicted period result in better predictions if the historical data is enough for a convergent solution.