Nueva Época, año 28, número 53, julio-diciembre 2020

http://dx.doi.org/10.24275/etypuam/ne/532020/Santillan

** EGADE Business School, Tecnológico de Monterrey, Campus Monterrey, México. E-mail: roberto.santillan@tec.mx. ORCID: 0000-0001-5162-1403.

*** Tecnológico de Monterrey, Campus Monterrey, México. E-mail: jacob.escobar@tec.mx. ORCID: 0000-0001-9031-6815.

**** División de Investigación, Facultad de Contaduría y Administración, UNAM, México. E-mail: francisco_lopez_herrera@yahoo.com.mx. ORCID: 0000-0003-2626-9246.

This paper compares the performance of different hedging strategies using futures contracts on Mexico’s Stock Exchange Index (IPC), traded in the Mexican Derivatives Market (MexDer). The *ex-post* evaluation of each strategy is made with daily closing prices from December 30^{th}, 1999, through December 30^{th}, 2016. The strategies considered are a) a No-hedge; b) a Naive Hedge; c) Constant Hedge; and d) a Dynamic Hedge, using a Constant Conditional Correlation Asymmetric Bivariate GARCH model. Four structural breaks are identified during the sample period, suggesting a five subperiods analysis. The strategies are compared using different risk measures: a) Value at Risk; b) Expected Shortfall; and c) LAQ. In all cases, hedging strategies reduce the volatility of the portfolio relative to the no-hedge strategy, but the dynamic hedge ratio produces the best results.

Keywords: Futures contracts, hedging strategies, emerging derivatives markets.

JEL Classification: G11, G13.

Este articulo compara el desempeno de diferentes estrategias de cobertura con contratos de futuros sobre el indice de la Bolsa Mexicana de Valores (IPC) negociados en el Mercado Mexicano de Derivados (MexDer). La evaluacion *ex-post* de cada estrategia se realiza con precios de cierre diarios del 30 de diciembre de 1999 al 30 de diciembre de 2016. Las estrategias consideradas son: a) sin cobertura; b) razon de cobertura “naive”; c) razon de cobertura constante; y d) razon de cobertura dinamica, mediante un modelo GARCH bivariado asimetrico con correlacion constante. Se identifican cuatro rupturas estructurales durante el periodo de la muestra, lo que sugiere el analisis en cinco subperiodos. Las estrategias son comparadas usando diferentes medidas de riesgo: a) Valor en riesgo; b) Deficit esperado, y c) LAQ. En todos los casos, las estrategias de cobertura reducen la volatilidad de la cartera, pero la razon de cobertura dinamica produce los mejores resultados.

Palabras clave: Contratos futuros, estrategias de cobertura, mercados emergentes de derivados.

Clasificación JEL: G11, G13.

^{*} Fecha de recepción: 10/11/2018.

Fecha de aprobación: 16/01/2020.

The Efficient Market Hypothesis (EMH) (*i.e.*, markets are highly efficient in reacting to the arrival of new information, and that risk and return are highly positively correlated through long periods. While EMH opponents point out frequent evidence of under- and over-reaction episodes in securities markets, no conclusive rejection of the EMH exists. In any case, the EMH is the best-known description of financial securities’ price movements. If an analysis departs from the assumption that markets are efficient, investors can estimate the expected returns of individual securities using one or several well-known equilibrium models (such as CAPM, APT, among others), according to which returns are primarily a function of risk (total risk and systematic risk). With the expected return as a discount rate, the price of a security reflects the present value of its expected future cash flows. However, the estimation of the present value of those cash flows must incorporate a variety of risk factors, such as volatility, liquidity and bankruptcy. Portfolio diversification reduces the weighted average of the individual portfolio securities’ variance of returns, as long as they are less than perfectly positively correlated. However, diversification only dilutes the unsystematic risk component, *i.e.*, systematic risk cannot be eliminated through diversification. The most relevant implication is that there is a limit beyond which diversification can no longer reduce portfolio return risk.

The abandonment in 1973 of the Bretton Woods Agreement that unchained all countries’ currency exchange rates from fixed parities with respect to the U.S. dollar, allowing them to be henceforth determined by market forces, resulted in increasing market volatility not only in the exchange rates themselves, but also the prices in different commodity markets (oil, copper, etc.), thus creating the urgent need for hedging mechanisms for many types of economic agents. Modern financial markets responded swiftly, creating and making available to market participants a variety of financial contracts that help investors reduce their exposure to market risks. Nowadays, futures, forwards, swaps and options contracts on a wide diversity of underlying assets are available to deal with market risk in a disciplined and orderly fashion. The proliferation of derivative contracts on stock indices responded to the high costs associated to modifying the composition of widely diversified portfolios as market expectations about the future change. Some of these contracts convey the right and the obligation to buy or sell a given position in financial securities, commodities, currency or several other categories of underlying assets, and their effective cost is low (such as futures contracts, where the “round-trip” fee is around $15 US). Others convey the right, but not the obligation, to buy or sell, and the holder can decide when it is convenient to exercise the contract, albeit at a higher cost.

Following *et al*. (2018)^{th}, 1999, through December 30^{th}, 2016. The hedging strategies included in this experiment are: a) a No-Hedge strategy; b) a Naive Hedge Ratio Strategy; c) a Constant-Hedge Strategy (obtained using an OLS regression with a HAC Newey Covariance Matrix); and d) a Dynamic Hedge Strategy, based on a Constant Conditional Correlation Multivariate GARCH model with asymmetric variations. Finally, we compare and evaluate the above strategies using different risk measures to verify how closely the *ex-post* performance of each hedging strategy corresponds to the *ex-ante* investor risk tolerance choice. Our results suggest that the Asymmetric Constant Conditional Correlation M-GARCH (Asymmetric CCC) model (which recognizes the possibility of different responses in volatility depending on whether innovations are positive or negative) eliminates risk more efficiently than the No-Hedge or the Constant-Hedge strategies. The Asymmetric CCC proves to be the best choice from the point of view of *ex-post* performance in terms of correspondence with *ex-ante* risk tolerance choice. However, in terms of portfolio returns only, our results indicate that No-Hedge is the best performing strategy (consistent with the well-known trade-off theory between risk and return). The following section presents a sample of the most influential works in the literature on risk hedging. The third section introduces the data and the methodology used in the empirical section. The fourth section present the results of the estimations and their interpretation and, finally, the fifth section concludes.

The modeling of the second moment of financial asset return distributions has been a major field of study over the past few decades. Since *t* distributions, can lead to a rich model that allows for time-dependent conditional variances and leptokurtosis in the unconditional distribution of price changes.” Arguably, GARCH models had already proved to be useful in explaining the distribution of common stock prices, so these authors show that they are equally effective in describing the distributions of commodity cash and futures prices, and that they lead to a natural description of time-varying optimal hedge ratios in commodity futures markets. This latter strategy is implemented using bivariate GARCH models to compute the conditional variances under three alternative portfolio strategies: a) no hedging; b) hedging with a constant Optimal Hedge Ratio (OHR) estimated using a simple Ordinary Least Squares (OLS) regression; and c) hedging with a time-varying OHR. Some simple performance tests indicate that the usual assumption of a constant hedge ratio is quite costly in terms of a higher return variance for some commodities (coffee, corn and cotton), but not for others (beef, gold, and soybeans).

The inclusion of Multivariate Generalized Autoregressive Conditional Heteroskedastic (M-GARCH) models in the most frequently used econometric software packages represented a major step forward and gave an important impulse to time series volatility modeling. The most important and distinctive feature of M-GARCH models is their flexibility in incorporating time-varying conditional covariances and variances. Both can be of substantial practical use for modeling and forecasting the volatility of many diverse assets such as stocks, bonds, commodities, exchange rates, etc. But, among the many interesting applications of M-GARCH models in the field of finance and investments, these models represent a major improvement in the calculation of time-varying hedge ratios using futures contracts, including the possibility of discriminating the volatility response to an innovation depending on whether it is positive or negative in sign (*et al*. 2002

The most conventional method for estimating optimal hedge ratios is to use the slope coefficient from a simple OLS regression of spot prices on futures prices, where the slope coefficient reflects the ratio of the unconditional covariance to the unconditional variance of the futures prices. However, instead of applying a regression model, the OHR can be obtained from the second moments of the joint distributions of spot and futures prices.

Recognition that covariance matrix forecasts of financial asset returns are an important component of current practice in financial risk management led

*et al*. (2002)*et al*. (2002)

More recently, *e.g.*, *et al*. 2002

In the same line as

The hedging effectiveness of hedge ratio building models, like most fields in the discipline, progressively incorporates more realistic characteristics. For example, *et al*. (2018)*et al*. (2011)

For a recent complex study that combines different estimation techniques with time-series analysis and data from emerging markets, the paper by *et al*. (2019)

A representative example of the increasing robustness of testing approaches followed by studies interested in learning more about the optimal hedge ratio determination and some eye opening results, is the paper by *et al*. (2015)

In the context of Latin America, the liberalization of important economic sectors has created new financial need to support their daily operation, as in the case of Colombia where the development of the wholesale energy market originated a growing negotiating of electricity futures contracts in the local capital market. The work of *et al*. (2014)

There is an abundant literature that studies stock market indices’ hedging strategies using many different methodologies and contemplates a wide variety of countries, making it a challenge to cover them reasonably well in a short literature review, such as the one we seek to introduce in this section. Therefore, we only briefly discuss some illustrative cases in the following lines. *et al*. (2009)*et al*. (2018)

Emerging markets’ stock investments are attractive to the average investor not only because of their frequent above average returns, but also because of the diversification benefits they bring to investors’ portfolios. However, investors frequently consider disinvesting their Emerging Markets securities and converting their holdings into hard currency when they anticipate (or experience) the presence of turbulent international market conditions (*e.g.*, the 2007-2009 Global Financial Crisis, or the more recent European Sovereign Debt Crisis) or a period of domestic economic instability due to shocks in the main commodities’ exports markets, trouble in local banking systems, currency exchange rate problems, etc. An alternative open to investors is to use hedging strategies that protect their portfolio’s value from short-term adverse environmental conditions. However, not many Emerging Markets have domestic derivatives markets where investors can find specialized contracts, so international investors find the few emerging markets where that possibility exists relatively more attractive. The case of the relatively young MexDer (created in 1998) is an example of a derivatives markets in an emerging country that has a promising future, where a variety of futures and options contracts on the local stock market index, known as the IPC, and several other securities and commodities are traded. Our interest in this work is centered on the use of the MexDer Futures contract on the IPC as a hedging instrument to implement hedging strategies for diversified portfolios that contain Mexican stocks. More specifically, we aim to identify which, among several possible hedging strategies, is the best in terms of risk reduction, and from the point of view of the portfolio’s returns. The data used consists of daily observations of the IPC, and its futures contract, from December 30^{th}, 1999, through December 30^{th}, 2016. In total, the sample consists of 4,282 daily observations. All the series are retrieved from a Bloomberg terminal. Bloomberg provides a time series of rolling IPC futures quotations for the next four expiration dates at any given time, and labels them IS1, IS2, IS3 and IS4, with IS1 being the nearest expiration date and IS4 the most distant in time. The futures contract price time series was built with a rolling-hedge strategy that uses only IS1 contracts, by far the largest trading-volume contracts all the time. The rollover strategy assumes that an IS1 contract is rolled over at its expiration date, to the next expiration date contract, three months ahead. For our purposes, the futures contract series IS1 is hereafter called IPCF (IPC Future). *et al*. (2002)

This experiment consists of comparing four possible strategies in response to IPC volatility: a) a simple “no-hedge” strategy; and three different, statistically based, hedging strategies. The first statistically based strategy, also considered a “benchmark” strategy, is a so called b) “naive” strategy, and consists of using one unit of the IPCF to hedge each unit of the IPC; that is, a 1:1 fixed hedge ratio throughout the period of analysis. The second strategy, commonly used by practitioners, estimates c) a fixed hedge ratio, obtained by dividing the historical covariance of the IPC and the IPCF by its historical variance; that calculation is equivalent to obtaining the slope of an Ordinary Least Squares (OLS) regression of the IPC on the IPCF. The third and most sophisticated strategy considers a d) dynamically changing hedge ratio, estimated using conditional variances and covariances of the variables, as well as their asymmetric response to positive and negative innovations. The hedge ratio for this strategy is re-estimated each day as in equation [1] below, where βt* represents the optimal hedge ratio for period t from the conditional variance of the IPCF (Var_{LRIPCF,t}), and the conditional covariance between IPC and the IPCF (Cov_{LRIPC,LRIPCF,t}). In the OLS case, the time subscripts are dropped and a constant optimal hedge ratio (β*) is calculated from the unconditional variance and covariance, which does not account for possible asymmetry or cointegration.

The dynamic hedge ratio is calculated with a Bivariate GARCH^{1} model that uses a time-varying covariance matrix for the two time-series. GARCH models originated in response to the need to study and forecast the volatility of financial assets. The relevant literature includes several transcendental seminal papers, such as: *et al*. (1993)

Multivariate GARCH (M-GARCH) models are conceptually equivalent to the univariate GARCH models in the sense that both model the volatility of different series, but the former estimates the conditional variance of several series and add conditional covariance equations at the same time. As expected, their mathematical complexity is significantly greater compared to the univariate version. M-GARCH models have developed increasingly sophisticated variations that resemble the behavior of time series volatility in a more reliable way. The most popular include the VECH (

Following

*R _{it}:* the rate of return of asset I from time t-1 to time t.

*μ _{it}*: the expected return of asset i given all information at time
t-1.

*ε _{it}*: the unexpected return of asset i
(

*h _{iit}*: the conditional variance of

*h _{ijt}*: the conditional covariance between

*H _{t}*: the conditional covariance matrix.

The VECH model is represented as follows:

Where

The VECH model has two important estimation problems. The first one is that the number of parameters to be estimated grows very rapidly (a 20-asset model will have 630 parameters); and the second is that the model might not produce a positive definite covariance matrix (unless nonlinear inequality restrictions are imposed) (

The BEKK model proposes a solution to the covariance matrix positive definiteness problem, where the ijth covariance may be expressed as:

Where *ε*_{p} and
*ε*_{q} represent unexpected shocks
to series *p* and *q*.

The Constant Conditional Correlation model restricts the conditional covariance between two asset returns to be proportional to the product of the conditional standard deviation. This time, the conditional correlation coefficient of the two asset returns is time invariant. The model can be represented as follows:

The CCC model is positive definite if the correlation matrix [ϱ_{ij}]
is positive definite. In this case, the number of parameters is only (1/2)
N^{2}. For a portfolio of 20 assets, the number of parameters to be
estimated is 270.

The three different GARCH specifications described in the above estimations were attempted, but it was not possible to obtain convergence for the BEKK model, so only a DVECH (

As is the case with univariate GARCH models, the M-GARCH versions have also seen a proliferation of alternative versions that have become very popular in empirical work, in particular, the use of models where the conditional variances and/or covariances react differently to the positive or negative nature of innovations of the same magnitude (

The database used in this analysis includes daily closing price observations for the Mexican Stock Exchange Market Index (IPC) and its Future contract daily closing prices (IPCF), from December 30^{th}, 1999, through December 30^{th}, 2016, for a total number of 4,282 daily closing price observations. All the series are downloaded from Bloomberg Financial Services. The IPC Futures contracts have quarterly maturities in March, June, September and December.

Tables 1a and 1b show the descriptive statistics for the IPC and the IPCF, as well as for their logarithmic returns. Both series show a negative skewness in levels and a positive skewness in returns. Also, in both series there is positive kurtosis, a typical characteristic of financial returns. Such evidence also confirms that the distribution of the variables does not conform to a normal distribution according to the Jarque-Bera test. As expected, there is a very high correlation between the IPC and the IPCF, both in levels and in returns.

IPC | IPCF | |
---|---|---|

Mean | 25,668.70 | 25,768.97 |

Median | 28,459.33 | 28,614.50 |

Maximum | 48,694.90 | 48,812.00 |

Minimum | 5,081.92 | 5,090.00 |

Std. Dev. | 14,493.73 | 14,484.21 |

Skewness | -0.0910 | -0.0957 |

Kurtosis | 1.4839 | 1.4826 |

Jarque-Bera | 416.02 | 417.31 |

Probability | 0.0000 | 0.0000 |

Observations | 4282 | 4282 |

Correlation |
IPC |
IPCF |

IPC | 1.00000 | 0.99997 |

IPCF | 0.99997 | 1.00000 |

LRIPC | LRIPCF | |
---|---|---|

Mean | 0.00043 | 0.00042 |

Median | 0.00077 | 0.00069 |

Maximum | 0.1044 | 0.1095 |

Minimum | -0.0827 | -0.0803 |

Std. Dev. | 0.0132 | 0.0136 |

Skewness | 0.0225 | 0.0432 |

Kurtosis | 8.1215 | 8.3125 |

Jarque-Bera | 4,679.06 | 5,035.59 |

Probability | 0.0000 | 0.0000 |

Observations | 4281 | 4281 |

Correlation |
LRIPC |
LRIPCF |

LRIPC | 1.00000 | 0.96973 |

LRIPCF | 0.96973 | 1.00000 |

**Source:** Authors’ own, with data retrieved from Bloomberg.

Figure 1 is a graphical representation of the original series and their logarithmic returns, and confirms their highly-similar behavior in time, reaching levels of correlation close to 1 in the case of logarithms, and marginally superior to 0.9697 in the case of the log-returns.

**Source:** Authors’ own, with data retrieved from
Bloomberg.

Before proceeding to test if the series are stationary, a brief digression is required. During the sample period, different events had a strong impact on the stability of international financial markets. The most important period of very high volatility was the Global Financial Crisis (2008-2009). For that reason, the presence of breakpoints in the series that might impair the ability of conventional tests (ADF, PP, KPSS) to correctly diagnose the presence of unit roots was to be expected. So the series were first studied to detect the presence of break-point dates using a hybrid Global-plus-Sequential test (as described in

LIPC | LIPCF | Breakpoint dates | |
---|---|---|---|

Sequential F-statistic determined breaks: | 4 | 4 | 3/13/2003, 6/14/2006, |

Significant F-statistic largest breaks: | 4 | 4 | 3/10/2009, 4/12/2013 |

**Source:** Analysis formulated by the authors using data from Bloomberg.

Augmented Dickey Fuller (ADF) tests for the whole sample period and for the sub-periods detected by the break-point dates are reported in Table 3. The ADF tests are run for the variables in log-levels and in log-returns.

Period | LIPC | LIPCF | LRIPC | LRIPCF |
---|---|---|---|---|

Full sample (12/30/1999 12/30/2016) | 0.7340 | 0.7518 | 0.0001 | 0.0001 |

Subperiod 1 (12/30/1999 3/12/2003) | 0.0379 | 0.0320 | 0.0000 | 0.0000 |

Subperiod 2 (3/13/2003 6/13/2006) | 0.3186 | 0.2904 | 0.0000 | 0.0000 |

Subperiod 3 (6/14/2006 3/09/2009) | 0.5689 | 0.5224 | 0.0000 | 0.0000 |

Subperiod 4 (3/10/2009 4/11/2013) | 0.0002 | 0.0002 | 0.0000 | 0.0000 |

Subperiod 5 (4/12/2013 12/30/2016) | 0.0579 | 0.0430 | 0.0000 | 0.0000 |

Probabilities based on MacKinnon (1996) one-sided p-values |

**Source:** Analysis formulated by the authors using data from Bloomberg.

The ADF null hypothesis (Ho: there is no unit-root in the series) is not rejected for the whole sample and for subperiods 2 and 3 when the levels series are tested. However, in subperiods 1, 4 and 5 the null is rejected for both series at conventional significance levels. There is a marginal contradiction between the p values of the LIPC LIPCF series in Subperiod 5 as the LIPC p value is marginally above the 5 percent rejection criteria, but the LIPCF is clearly below that parameter. From a visual inspection of the LIPC series, Subperiod 5 looks like a typical lateral accumulation period with few deviations from a relatively stable trend, which may explain the marginal deviation of the ADF test parameter from the tolerance criterion of 5 percent. Accordingly, Johansen cointegration tests are reported only for the full sample and for subperiods 2 and 3 (see Table 4). The Johansen test confirms the presence of at least one cointegrating vector for the whole sample and for Subperiod 3. The results for Subperiod 2 suggest that the two variables are very highly correlated by as many as two cointegrating vectors, which is probably due to an oddity, since two series cannot be related by more than one cointegrating vector. This result is not reliable and, with evidence that both series during Subperiod 2 are non-stationary, they are treated accordingly in the ensuing analysis.

Period | CEs | Trace |
Max-eigen |
---|---|---|---|

Full-sample (12/30/1999-12/30/2016) | None | 0.0001 | 0.0001 |

At most 1 | 0.2839 | 0.2839 | |

Subperiod 2 (3/13/2003 6/13/2006) | None | 0.0001 | 0.0002 |

At most 1 | 0.0445 | 0.0445 | |

Subperiod 3 (6/14/2006 3/09/2009) | None | 0.0000 | 0.0000 |

At most 1 | 0.1864 | 0.1864 |

**Source:** Analysis formulated by the authors using data from Bloomberg.

A frequently used assumption when modeling the mean component of Multivariate GARCH models is that the relationship between the variables in the system may be represented by a Vector Autoregression (VAR) Model. Based on the results of the Unit Root and Cointegration tests presented above, a VAR model is appropriate to model the mean equations of the series in levels, in the case of Subperiods 1, 4 and 5.

A VAR model is not recommended when the variables are not stationary, but are simultaneously cointegrated, since there is a long-term relationship between them that needs to be recognized and incorporated in the form of a cointegration vector which corrects for the long-term relationship between the variables. In this case, the model is known as a Vector Error Correction Model (VECM).

The optimal lag length for the VAR (VECM) models is selected using Schwarz Information Criterion (SIC), as reported in Table 5.

Period | Levels | First Differences |
---|---|---|

Full-sample (12/30/1999-12/30/2016) | 2 | 2 |

Subperiod 1 (12/30/1999 - 3/12/2003) | 2 | 1 |

Subperiod 2 (3/13/2003 - 6/13/2006) | 2 | 1 |

Subperiod 3 (6/14/2006 - 3/09/2009) | 2 | 2 |

Subperiod 4 (3/10/2009 - 4/11/2013) | 2 | 2 |

Subperiod 5 (4/12/2013 - 12/30/2016) | 2 | 2 |

Level: LIPC, LIPCF | ||

First Difference: LRIPC, LRIPCF | ||

SIC: Schwartz Information Criterion |

**Source:** Analysis formulated by the authors using data from Bloomberg.

The Asymmetric DVECH model is very similar to the individual equation form already discussed
in

Equation terms: M_{i}: Long-term variance; G_{i}: GARCH; U_{i}:
Residuals.

Coefficients: A_{i}: ARCH_{i}; B_{i}: GARCHi; D_{i}:
TARCH_{i}.

Binary variables:
*Z _{LRIPC}*=1

The matrix form representation of the Asymmetric CCC is presented in

While the econometric analysis was performed for both the symmetric and asymmetric version of the DVECH and the CCC GARCH models, the following results (Tables 6-9) refer only to the Asymmetric DVECH and Asymmetric CCC models (the estimation output of the two symmetric models is available upon request from the authors), however, a graphical representation of the conditional variances of all four models is included as the Appendix. A visual representation of the effect of including the asymmetric response adjustment or not can be observed for both the DVECH and the CCC model in Figures A.1 through A.6. According to Figure A.1, conditional volatilities estimated for both the yields of the IPC and the yields of their futures contract are very similar for the full period, for the different models. They all capture the decline in volatility associated with the end of the dot.com bubble; the sharp rise in volatility associated with both the preamble and full manifestation of the Subprime Mortgages Crisis, as well as with the Sovereign Debt Crisis in the Eurozone. However, it is interesting to note that the maximum volatility is detected at different dates for each model, a situation that is observed in the same way for all the subperiods’ estimates. Also, the CCC and ACCC models show lower levels of conditional volatility and a smoother profile than their counterparts, the DVECH models. DVECH models estimated for the performance of the IPC and its futures contract during the first subperiod show very high conditional volatility levels compared to the other models, notably for the yields of the underlying. During subperiod 2, the opposite occurs, the estimated volatility of these models is the lowest, but, like the rest of the models, they capture the rising volatility at the end of that subperiod. The estimates for subperiods 3, 4 and 5 show similar behavior: all DVECH models produce higher conditional volatility estimates.

Equation | Variable | FS | SP1 | SP2 | SP3 | SP4 | SP5 | |
---|---|---|---|---|---|---|---|---|

Mean
Equations |
Coint. Eq. (CE) |
LIPC-1 | 1.000 | 1.000 | ||||

LIPCF-1 - | 1.006*** | -1.006*** | ||||||

C | 0.068 | 0.072 | ||||||

LRIPC |
CE | -0.033* | 0.324*** | |||||

LRIPC-1 | 0.174*** | 1.132*** | 0.053 | -0.187 | 0.891*** | 0.415*** | ||

LRIPC-2 | 0.260*** | -0.196*** | 0.115 | 0.050*** | 0.619*** | |||

LRIPCF-1 | -0.098** | -0.021* | 0.031 | 0.210 | 0.126*** | 0.659*** | ||

LRIPCF-2 | -0.256*** | 0.065*** | -0.151 | -0.076*** | -0.709*** | |||

C | 0.000*** | 0.175*** | 0.001*** | 0.000 | 0.086*** | 0.164*** | ||

LRIPCF |
CE | 0.119*** | 0.508*** | |||||

LRIPC-1 | 0.419*** | 0.410*** | 0.148 | -0.050 | 0.329*** | -0.169*** | ||

LRIPC-2 | 0.366*** | -0.378*** | 0.206 | -0.187*** | 0.350*** | |||

LRIPCF-1 | -0.331*** | 0.736*** | -0.031 | 0.073 | 0.687*** | 1.245*** | ||

LRIPCF-2 | -0.352*** | 0.213*** | -0.219 | 0.163*** | -0.443*** | |||

C | 0.000*** | 0.164*** | 0.001*** | 0.000 | 0.089*** | 0.184*** | ||

Variance
Equations |
LRIPC |
M | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** |

A1 | 0.037*** | 0.037*** | -0.011*** | 0.013 | 0.043*** | 0.044*** | ||

D1 | 0.075*** | 0.055*** | 0.050*** | 0.125*** | 0.030*** | 0.055*** | ||

B1 | 0.913*** | 0.866*** | 0.944*** | 0.885*** | 0.909*** | 0.898*** | ||

LRIPCF |
M | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** | |

A2 | 0.013*** | 0.007 | -0.011*** | 0.001 | 0.032*** | 0.036*** | ||

D2 | 0.109*** | 0.181*** | 0.051*** | 0.150*** | 0.049*** | 0.053*** | ||

B2 | 0.919 | 0.854 | 0.943 | 0.889 | 0.910 | 0.909 | ||

Covariance |
M | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** | |

A12 | 0.025*** | 0.022*** | -0.010*** | 0.008 | 0.038*** | 0.039*** | ||

D12 | 0.092*** | 0.105*** | 0.049*** | 0.136*** | 0.038*** | 0.054*** | ||

B12 | 0.916*** | 0.860*** | 0.941*** | 0.886*** | 0.908*** | 0.903*** | ||

AdjR2_ols |
0.940359 | 0.902991 | 0.938196 | 0.957329 | 0.962232 | 0.967173 | ||

AdjR2_As.dvech_1 |
0.003266 | 0.967051 | 0.010453 | 0.006116 | 0.995999 | 0.972142 | ||

AdjR2_As.dvech_2 |
0.004720 | 0.967591 | 0.021515 | 0.007639 | 0.995600 | 0.969190 |

**Note:** *** = 1% significance; ** = 5% significance; * = 10% significance.

**Source:** Analysis formulated by the authors using data from Bloomberg.

Adj Qstat Prob. | Full sample | Subperiod 1 | Subperiod 2 | Subperiod 3 | Subperiod 4 | Subperiod 5 |
---|---|---|---|---|---|---|

Lag 1 | 0.0001*** | 0.4695 | 0.7590 | 0.4249 | 0.2748 | 0.7632 |

Lag 2 | 0.0000*** | 0.7194 | 0.9453 | 0.3815 | 0.2881 | 0.0165** |

Lag 3 | 0.0000*** | 0.8512 | 0.9490 | 0.4274 | 0.1974 | 0.0249** |

Lag 4 | 0.0000*** | 0.2883 | 0.9629 | 0.6644 | 0.0859 | 0.0286** |

Lag 5 | 0.0000*** | 0.2852 | 0.6610 | 0.5584 | 0.1163 | 0.0444** |

Lag 6 | 0.0000*** | 0.4174 | 0.7403 | 0.5646 | 0.2157 | 0.0368** |

Lag 7 | 0.0000*** | 0.2328 | 0.7990 | 0.6698 | 0.3031 | 0.0306** |

Lag 8 | 0.0000*** | 0.2533 | 0.8553 | 0.7435 | 0.4356 | 0.0373** |

Lag 9 | 0.0000*** | 0.2715 | 0.9136 | 0.8691 | 0.3218 | 0.0321** |

Lag 10 | 0.0000*** | 0.2984 | 0.8688 | 0.8791 | 0.3643 | 0.0528* |

Lag 11 | 0.0000*** | 0.1929 | 0.9002 | 0.9219 | 0.3498 | 0.0609* |

Lag 12 | 0.0000*** | 0.2137 | 0.9555 | 0.8865 | 0.3579 | 0.0749* |

Conditional Correlation Orthogonalization (Doornik-Hansen)

**Note:** *** = 1% significance; ** = 5% significance; * = 10% significance

**Source:** Analysis formulated by the authors using data from Bloomberg.

Equation | Variable | FS | SP1 | SP2 | SP3 | SP4 | SP5 |
---|---|---|---|---|---|---|---|

Coint. Eq. |
LIPC_{-1} |
1.000 | 1.000 | ||||

(CE) |
LIPCF_{-1} |
-1.006*** | -1.006*** | ||||

C | 0.068 | 0.072 | |||||

LRIPC |
CE | 0.257*** | 0.336*** | ||||

LRIPC_{-1} |
-0.059 | 0.206** | 0.168 | -0.171 | -0.201 | -0.623*** | |

LRIPC_{-2} |
0.074 | -0.079 | -0.079 | 0.024 | -0.060 | -0.151 | |

LRIPCF_{-1} |
0.157*** | 0.177 | 0.240* | 0.667*** | |||

LRIPCF_{-2} |
-0.061 | -0.001* | 0.001*** | -0.070 | 0.084 | 0.142 | |

C | 0.000 | 0.000 | 0.000 | 0.000 | |||

LRIPCF |
CE | 0.340*** | 0.391*** | 0.302** | 0.496*** | ||

LRIPC_{-1} |
0.144*** | -0.241*** | -0.181 | -0.037 | 0.207 | -0.244 | |

LRIPC_{-2} |
0.123** | 0.103 | 0.122 | 0.024 | |||

LRIPCF_{-1} |
-0.035 | -0.001** | 0.001*** | 0.038 | -0.163 | 0.287 | |

LRIPCF_{-2} |
-0.107** | -0.131 | -0.102 | -0.038 | |||

C | 0.000 | 0.000*** | 0.000*** | 0.000 | 0.000 | 0.000 | |

Var_{LRIPC} |
M | 0.000*** | 0.000*** | 0.000*** | 0.000*** | ||

RESID_{-1} |
0.002 | -0.011 | -0.044*** | -0.020** | 0.008 | -0.008 | |

TARCH_{-1} |
0.082*** | 0.223*** | 0.099*** | 0.120*** | 0.039*** | 0.095*** | |

GARCH_{-1} |
0.952*** | 0.873*** | 0.896*** | 0.937*** | 0.952*** | 0.938*** | |

Var_{LRIPCF} |
M | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** | 0.000*** |

RESID_{-1} |
0.002 | -0.012 | -0.044*** | -0.019** | 0.001 | -0.001 | |

TARCH_{-1} |
0.086*** | 0.254*** | 0.089*** | 0.128*** | 0.049*** | 0.085*** | |

GARCH_{-1} |
0.950*** | 0.862*** | 0.931*** | 0.932*** | 0.951*** | 0.935*** | |

Correl |
R_{1,2} |
0.974*** | 0.955*** | 0.966*** | 0.979*** | 0.981*** | 0.986*** |

**Note:** *** = 1% significance; ** = 5% significance; * = 10% significance.

**Source:** Analysis formulated by the authors using data from Bloomberg.

Adj. Qstat Prob. | Full-sample | Subperiod 1 | Subperiod 2 | Subperiod 3 | Subperiod 4 | Subperiod 5 |
---|---|---|---|---|---|---|

Lag 1 | 0.0525* | 0.7386 | 0.9819 | 0.2492 | 0.9926 | 0.9612 |

Lag 2 | 0.0021** | 0.6908 | 0.9871 | 0.5606 | 0.9938 | 0.9592 |

Lag 3 | 0.0044** | 0.8553 | 0.9756 | 0.7232 | 0.6014 | 0.3644 |

Lag 4 | 0.0001*** | 0.1807 | 0.9850 | 0.9097 | 0.4549 | 0.3029 |

Lag 5 | 0.0000*** | 0.1057 | 0.7287 | 0.8706 | 0.6121 | 0.3128 |

Lag 6 | 0.0000*** | 0.2439 | 0.8571 | 0.8414 | 0.6735 | 0.0960* |

Lag 7 | 0.0000*** | 0.2520 | 0.9045 | 0.9160 | 0.7501 | 0.0912* |

Lag 8 | 0.0000*** | 0.2471 | 0.9482 | 0.9469 | 0.8301 | 0.1616 |

Lag 9 | 0.0000*** | 0.2526 | 0.9747 | 0.9817 | 0.7709 | 0.1070 |

Lag 10 | 0.0000*** | 0.3317 | 0.9357 | 0.9854 | 0.8347 | 0.1749 |

Lag 11 | 0.0000*** | 0.1436 | 0.9540 | 0.9891 | 0.7975 | 0.2471 |

Lag 12 | 0.0000*** | 0.1647 | 0.9810 | 0.9704 | 0.8662 | 0.3364 |

Conditional Correlation Orthogonalization (Doornik-Hansen).

**Note:** *** = 1% significance; ** = 5% significance; * = 10% significance.

**Source:** Analysis formulated by the authors using data from Bloomberg.

The decision to omit the tables that report the output of the symmetric version of both models responds to space limitations and to the fact that the results obtained with models that take into account a differentiated response of volatility to positive and negative innovations are considered to have a better adjustment (*e.g.*, *et al*., 2002

Table 6 shows the estimated Asymmetric-DVECH model’s coefficients for the full-period and the five subperiods determined by the analysis of structural breaks of the series. The coefficients of the cointegrating equation in the mean equation are significant for the full sample and for Subperiod 3, confirming the previous findings of cointegration between the variables, and fully consistent with previous studies that document the relationship between an underlying asset and its corresponding future contract. Also, the coefficients of the terms inside each cointegrating equation match the expectations mentioned earlier for equation [3], since β_{1} is close to -1 and β_{2} is close to 0 in both subperiods. Lastly, the GARCH, the Asymmetric term, and the Correlation parameter are significant in all the variance equations.

To test for the presence of autocorrelation in the residuals of the models presented in table 6 above, the results of the System Residual Portmanteau Test for Autocorrelation are presented in Table 7, below. For the full period, all the autoregressive terms are statistically significantly different from zero, *i.e.*, there is strong evidence of autocorrelation problems. However, when the same test is carried out for the subperiods, the first four are free from autocorrelation, while the fifth is affected by significant autocorrelation in eleven of the twelve lags.

e Asymmetric CCC models’ coefficients for the five sub-periods are reported in Table 8. The coefficients of the cointegrating equation as a term in the mean equations, α_{1} and α_{2}, are significant for the full sample and for subperiods 3, which confirms our previous findings of cointegration between the two variables when tested in those time periods. This is consistent with previous studies that document the relationship between the underlying asset and its corresponding future contract. Also, the coefficients of the terms inside each cointegrating equation match the expectations mentioned earlier for equation [3], since β_{1} is close to - 1 and β_{2} is close to 0 in both subperiods. Lastly, the GARCH, the Asymmetric term, and the Correlation parameter are significant in all the variance equations.

Table 9 shows the Portmanteau test for autocorrelation using Conditional Correlation orthogonalization. Similar to the case of the Asymmetric DVECH model, the Asymmetric CCC model autocorrelation tests show the coefficients are significant for the first twelve lagged residuals of the full sample. However, in all the subperiods, autocorrelations are not significantly different from zero at a 5 percent significance level. That fact makes the Asymmetric CCC model vastly superior to the Asymmetric DVECH model and justifies the decision to further develop the dynamic optimal hedge ratios using the Asymmetric CCC.

The conditional variance and covariance series from each sub-period’s estimated Asymmetric CCC
model are next used to calculate the dynamic hedge ratio as in

Period | β_{ols} |
---|---|

Full-sample (12/30/1999-12/30/2016) | 0.9448 |

Subperiod 1 (12/30/1999-3/12/2003) | 0.9419 |

Subperiod 2 (3/13/2003-6/13/2006) | 0.9752 |

Subperiod 3 (6/14/2006-3/09/2009) | 0.9391 |

Subperiod 4 (3/10/2009-4/11/2013) | 0.9366 |

Subperiod 5 (4/12/2013-12/30/2016) | 0.9508 |

**Source:** Estimation formulated by the authors using data from Bloomberg.

Next, we obtain the returns of theoretical portfolios that follow each of the three hedging strategies: the naive,^{2} constant and dynamic hedge ratios, in all three cases using

Table 11 reports the performance of the four strategies, including the no-hedge scenario, used as a benchmark to measure the benefits from each of the hedging strategies for the full sample and for each subperiod. Additionally, Table 11 reports the results of an out-of-sample performance evaluation of the four strategies for the next 20 trading days after December 30^{th}, 2016, the end of the sample period. The performance of each strategy during the forecast period is simulated as follows: a) using the previous sub-period (SP5) hedge ratio in the case of the OLS strategy; b) using the conditional hedge ratio obtained from the last observation of the SP5 for the first day of the forecast period, and the following daily changing hedge ratios based on the previous day conditional variance and covariance values obtained from the Asymmetric CCC model; or c) using the 1:1 ratio in the case of the naive hedge strategy.

Strategy | Period | σrp | Min. Return | Max. Return |
---|---|---|---|---|

No Hedge (LRIPC) | Full sample | 1.3226% | -8.3% | 10.4% |

SP1 | 1.6850% | -8.3% | 7.0% | |

SP2 | 1.0421% | -4.4% | 3.2% | |

SP3 | 1.8279% | -7.3% | 10.4% | |

SP4 | 1.1099% | -6.1% | 6.2% | |

SP5 | 0.8923% | -4.7% | 3.5% | |

Forecast horizon | 0.9389% | -1.4% | 2.2% | |

Naïve (1:1 hedge) | Full sample | 0.3315% | -4.8% | 2.5% |

SP1 | 0.5337% | -4.8% | 2.5% | |

SP2 | 0.2602% | -2.2% | 1.1% | |

SP3 | 0.3948% | -2.5% | 2.2% | |

SP4 | 0.2278% | -2.1% | 0.9% | |

SP5 | 0.1678% | -1.1% | 0.8% | |

Forecast horizon | 0.1313% | -0.2% | 0.3% | |

OLS (constant for each subperiod) | Full sample | 0.3226% | -4.4% | 2.2% |

SP1 | 0.5245% | -4.4% | 2.2% | |

SP2 | 0.2589% | -2.2% | 1.2% | |

SP3 | 0.3774% | -2.4% | 2.0% | |

SP4 | 0.2156% | -1.6% | 0.8% | |

SP5 | 0.1616% | -1.0% | 0.8% | |

Forecast horizon | 0.1254% | -0.2% | 0.3% | |

CCC (dynamic and asymmetric) | Full sample | 0.3213% | -4.1% | 2.8% |

SP1 | 0.5205% | -4.1% | 2.8% | |

SP2 | 0.2572% | -2.2% | 1.1% | |

SP3 | 0.3786% | -2.3% | 2.0% | |

SP4 | 0.2143% | -1.0% | 0.8% | |

SP5 | 0.1631% | -1.0% | 0.8% | |

Forecast horizon | 0.1300% | -0.2% | 0.3% |

**Source:** Estimation formulated by the authors using data from Bloomberg.

The volatility reduction achieved following each strategy, relative to the un-hedged case is reported in Table 12. The three hedge strategies prove very effective as they all significantly reduce volatility compared to the un-hedged scenario, but the Asymmetric CCC strategy is found to be more effective than the other two for the whole period. The naive strategy is always the least successful of the three strategies, and the Asymmetric CCC strategy shows better results than the OLS approach for three of the five sub-periods. In any case, the Asymmetric CCC strategy is also very close to the best performing strategy in the other two sub-periods, as well as during the forecast horizon. Any of the three optimal hedge ratio strategies seems to work much better for the forecast horizon than for any of the past five sub-periods; however, the length of the observation period is much shorter than in the case of any of the sub-periods and, of course, the whole period, so these results should be taken with some caution.

Naïve | OLS | Asymmetric CCC | |
---|---|---|---|

F. sample (12/30/1999 12/30/2016) | 74.93% | 75.61% | 75.70% |

Subperiod 1 (12/30/1999 3/12/2003) | 59.65% | 60.35% | 60.65% |

Subperiod 2 (3/13/2003 6/13/2006) | 80.33% | 80.42% | 80.56% |

Subperiod 3 (6/14/2006 3/09/2009) | 70.15% | 71.46% | 71.38% |

Subperiod 4 (3/10/2009 4/11/2013) | 82.77% | 83.70% | 83.79% |

Subperiod 5 (4/12/2013 12/30/2016) | 87.31% | 87.78% | 87.67% |

Forecast horizon (20 days) | 90.07% | 90.52% | 90.17% |

**Source:** Estimation formulated by the authors using data from Bloomberg.

The predominance of the Asymmetric CCC dynamic hedging strategy is, finally, corroborated using the following portfolio risk performance measures: a) Value at Risk (VaR), defined as the loss level that should not be exceeded with a certain confidence level (α) during a certain period of time (t); b) Expected Shortfall (ES), meaning the average loss over a certain time-period (t), assuming that it is greater than the (1- α) percentile of the loss distribution. VaR and ES complement each other, and, in both cases, a small magnitude is preferred. Two more risk measures considered are the c) mean and d) maximum Absolute Deviations (AD mean and AD max, respectively) between the observations and the quantiles, as in *et al*. (2004)

As reported in Table 13, according to all five risk measures, the Asymmetric CCC-based hedge ratio strategy formulation proves superior to the OLS and the Naive models estimated for the full sample, and for all sub-periods, except the last one. In subperiod 5, OLS results are marginally better according to the ES and AD mean risk measures. The Naive model is consistently the worst model in most cases, with only two instances in which it is superior to the OLS model, but not to the Asymmetric CCC model in any subperiod. Overall, we consider the Asymmetric CCC to provide the best hedge. Notwithstanding, the OLS model is a close second and has the advantage that, as a passive approach, its transaction costs should give it a practical advantage over the former.

Period | Model | VaR | ES | AD mean | AD max | LAQ |
---|---|---|---|---|---|---|

Full-sample | As. CCC | 0.0074 | 0.0085 | 0.0070 | 0.0338 | 0.0002 |

OLS | 0.0075 | 0.0086 | 0.0075 | 0.0367 | 0.0002 | |

Naïve | 0.0077 | 0.0085 | 0.0083 | 0.0400 | 0.0002 | |

SP1 | As. CCC | 0.0120 | 0.0137 | 0.0140 | 0.0292 | 0.0004 |

OLS | 0.0122 | 0.0139 | 0.0155 | 0.0321 | 0.0004 | |

Naïve | 0.0124 | 0.0142 | 0.0166 | 0.0354 | 0.0004 | |

SP2 | As. CCC | 0.0059 | 0.0068 | 0.0063 | 0.0157 | 0.0002 |

OLS | 0.0060 | 0.0069 | 0.0063 | 0.0157 | 0.0002 | |

Naïve | 0.0060 | 0.0069 | 0.0066 | 0.0163 | 0.0002 | |

SP3 | As. CCC | 0.0088 | 0.0101 | 0.0046 | 0.0144 | 0.0002 |

OLS | 0.0088 | 0.0101 | 0.0050 | 0.0149 | 0.0002 | |

Naïve | 0.0092 | 0.0105 | 0.0048 | 0.0155 | 0.0002 | |

SP4 | As. CCC | 0.0049 | 0.0056 | 0.0025 | 0.0055 | 0.0001 |

OLS | 0.0050 | 0.0057 | 0.0027 | 0.0109 | 0.0001 | |

Naïve | 0.0053 | 0.0061 | 0.0028 | 0.0156 | 0.0001 | |

SP5 | As. CCC | 0.0038 | 0.0044 | 0.0015 | 0.0063 | 0.0001 |

OLS | 0.0038 | 0.0043 | 0.0014 | 0.0067 | 0.0001 | |

Naïve | 0.0039 | 0.0045 | 0.0015 | 0.0072 | 0.0001 |

**Source:** Estimation formulated by the authors using data from Bloomberg.

As a general conclusion, it may be said that the hedge ratios built with the Asymmetric CCC model effectively reduce volatility and the complementary risk measures beyond the improvements obtained with a constant hedge ratio, either naive or OLS-based estimations, as could be expected in the implementation of hedging strategies from more sophisticated techniques. Our findings reinforce the confidence that practitioners may have in the use of futures contracts for hedging diversified portfolios of Mexican stocks, and provide authorities in charge of the MexDer with an empirical example of the usefulness of such contracts when hedging strategies are supported with adequate econometric estimations of the optimal hedge ratio at all times. As the country’s financial markets become more diversified and sophisticated, there will be a growing need for many other types of futures contracts, as well as for a careful analysis of different dynamic hedging strategies. That reality creates attractive areas of research that have great practical usefulness and provide theoretical challenges for the profession.

This paper compares the hedging performance of four different strategies in the combination of the IPC and its corresponding futures contract, the IPCF: a) a No-hedge strategy; b) a Naive hedge ratio (1 to 1); c) a Constant Hedge ratio, obtained with an OLS model; and d) a dynamic hedge ratio, obtained using a CCC Asymmetric Bivariate GARCH model. The estimation of the dynamic hedge ratio needed to implement strategy d) required the selection of a conditional variance/covariance model, so the initial analysis included a BEKK model, as well as asymmetric CCC and DVECH models. The choice of the asymmetric bivariate CCC-GARCH model to estimate dynamic hedge ratios for a portfolio that contains the IPC and its futures contract was based on the overwhelming evidence found in the literature on the better performance of asymmetric conditional variance models over their non-asymmetric counterparts (*et al*., 2002

The three hedging strategies prove to be very effective and to reduce volatility significantly when compared with the un-hedged alternative, but the Dynamic Asymmetric CCC strategy is confirmed to be more effective than the Naive or constant hedge ratio strategies for the whole period. The Naive strategy consistently proves to be the least successful of the three strategies, and the Asymmetric CCC strategy shows better results than the fixed hedge ratio (OLS) approach for three of the five sub-periods. Nevertheless, the Asymmetric CCC strategy is also very close to the best performing strategy in the other two sub-periods, as well as during the forecast horizon. The three statistically based strategies work much better for the forecast horizon than for the full sample period or any of the five sub-periods; however, the length of the forecast period is much shorter than in the case of any of the sub-periods and, of course, of the whole period, so these results should be taken with some caution. By contrasting different strategies that a typical investor can follow regarding the utilization of the IPCF to hedge a diversified portfolio of Mexican stocks, the conclusion of this work’s results is that any hedging strategy, including the naive strategy that assumes investors hedge on a one-to-one basis (one futures contract per unit of exposure), results in much less volatility in the returns of a Mexican stock portfolio than the no-hedge strategy.

From the comparison of two possible MGARCH modeling approaches, the DVECH and the CCC models, we confirm that both present serious autocorrelation problems for the full-sample period. However, once the sample is segmented according to previously identified structural breakpoint dates, while the DVECH model still presents autocorrelation in the estimation residuals, the CCC model (conventional and asymmetric) overcomes the autocorrelation problem. Finally, when comparing the two CCC versions of the model, the Asymmetric CCC is a better choice because its performance is superior in terms of volatility reduction, in-line with previous literature. To corroborate the high quality of the Dynamic Asymmetric CCC-built hedge strategy, we compare the different portfolios’ performance using several risk measures (VaR, ES and LAQ), and confirm that it is superior in most subperiods. However, the OLS approach is a close second and, as a passive approach, it may have a definitive advantage when considering transaction costs inherent to a dynamic hedging strategy.

The necessary conditions for an emerging capital market to thrive include the existence of increasingly sophisticated and liquid markets for financial securities and commodities. The institutional conditions for those markets to thrive are difficult to build but cannot be considered a matter of choice as they are an important building block that helps firms implement risk management strategies and control their exposure against unfavorable market conditions. The need for more complete derivatives markets that provide economic agents with a sufficient variety of contracts to implement investment risk management strategies and design portfolios that better respond to their risk-return preferences and objectives should clearly be a priority of emerging markets. At present, few emerging markets have modern and well-functioning derivatives markets. The MexDer represents a step in the right direction towards the modernization and sophistication of the Mexican financial market. Important environmental changes, including financial deregulation and the reform of the Mexican pension system, created individual retirement accounts managed by institutional investors (Afores) instead of the traditional “pay-as-you-go” system. This represents an exceptional opportunity for derivative contracts and the consolidation of the MexDer. There is a growing interest from other interested parties (agricultural product producers, mining companies, etc.) to have access to mechanisms that will help them to compensate uncertainty in the price of their product. Stimulating the development of a derivatives market is consistent with the objective of economic modernization and better conditions for producers across the economy. In addition, as part of that process, a better knowledge of the characteristics and functioning of the MexDer futures contracts represents a valuable insight that deserves further attention and study.

- Bai, Jushan and Perron, Pierre (1998), “Estimating and Testing Linear Models with Multiple Structural Changes”,
*Econometrica*, 66, pp. 47-78. - Baillie, Richard T. and Myers, Robert J. (1991), “Bivariate GARCH estimation of the optimal commodity futures hedge”,
*Journal of Applied Econometrics*, 6, pp. 109-124. - Billio, Monica; Casarin, Roberto & Osuntuyi, Anthony (2018), “Markov switching GARCH models for Bayesian hedging on energy futures markets”,
*Energy Economics*, 70, pp. 545-562. https://doi.org/10.1016/j.eneco.2017.06.001 - Bollerslev, Tim (1986), “Generalized autoregressive conditional heteroskedasticity”,
*Journal of Econometrics*, 31, pp. 307-327. - __________ (1990), “Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model”,
*Review of Economics and Statistics*, pp. 498-505. - Bollerslev, Tim; Engle, Robert F. and Wooldridge, Jeffrey M. (1988), “A Capital Asset Pricing model with Time-varying Covariances”,
*Journal of Political Economy*, pp. 116-131. - Bollerslev, Tim & Wooldridge, Jeffrey M. (1992), “Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances”,
*Econometric Reviews*, 11 (2), pp. 147-172 - Brooks, Chris; Henry, Olan T. and Persand, Gita (2002), “The effects of asymmetries on optimal hedge ratios”,
*The Journal of Business*, 75 (2), pp. 333-352. http://doi.org/10.1086/338484 - Brooks, Chris and Persand, Gita (2003), “The effect of asymmetries on stock index return value-at-risk estimates”,
*Journal of Risk Finance*, 4 (2), pp. 29-42. https://doi.org/10.1108/eb022959 - Brooks, Chris (2014),
*Introductory Econometrics for Finance*, 3rd ed., Cambridge University Press. - Chang, Chia-Lin; McAleer, Michael and Tansuchat, Roengchai (2011), “Crude oil hedging strategies using dynamic multivariate GARCH”,
*Energy Economics*, 33 (5), pp. 912-923. https://doi.org/10.1016/j.eneco.2011.01.009 - De Jesus Gutierrez, Raul (2016), “Estrategias dinamicas de cobertura cruzada eficiente para el mercado del petroleo mexicano: Evidencia de dos modelos GARCH multivariados con termino de correccion de error”,
*Economía Teoría y Práctica*, 44 (1), pp. 115-146. - Diaz-Contreras, Jhon A.; Macias-Villalba, Gloria I. and Luna-Gonzalez, Edgar (2014), “Estrategia de cobertura con productos derivados para el mercado energetico colombiano”,
*Estudios Gerenciales*, 30 (130), pp. 55-64. https://doi.org/10.1016/j.estger.2014.02.008 - Engle, Robert F. (1982), “Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation”,
*Econometrics*, 50, pp. 987-1008. - __________ (2002), “Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models”,
*Journal of Business & Economic Statistics*, 20, pp. 339-350. - Engle, Robert F. and Ng, Victor K. (1993), “Measuring and Testing the Impact of News in Volatility”,
*Journal of Finance*, vol. 48, issue 5, pp. 1749-78 - Engle, Robert F. and Kroner, Kenneth F. (1995), “Multivariate Simultaneous Generalized Arch”,
*Econometric Theory*, 11 (1), pp. 122-150. https://www.jstor.org/stable/3532933 - Engle, Robert F. and Manganelli, Simone (2004), “CAViaR: Conditional autoregressive value at risk by regression quantiles”,
*Journal of Business and Economic Statistics*, 22 (4), pp. 367-381. https://doi.org/10.1198/073500104000000370 - Fama, Eugene F. (1970), “Efficient Capital Markets: A Review of Theory and Empirical Work”,
*Journal of Finance*, 25 (2), pp. 383-417. - __________ (1991), “Efficient Capital Markets: II”,
*Journal of Finance*, 45 (5), pp. 1575-1617. - Glosten, Lawrence R.; Jagannathan, Ravi and Runkle, David E. (1993), “On the Relation Between the Expected Value and the Volatility of the Nominal Excess Returns in Stocks”,
*Federal Reserve Bank of Minneapolis*,*Research Department Staff Report*#*157*. - Gonzalez-Rivera, Gloria; Lee, Tae-Hwy and Mishra, Santosh. (2004), “Forecasting volatility: A reality check based on option pricing, utility function, value-atrisk, and predictive likelihood”,
*International Journal of Forecasting*, 20, pp. 629-645. - Hsieh, David A. (1993), “Implications of Nonlinear Dynamics for Financial Risk Management”,
*Journal of Financial and Quantitative Analysis*, 28 (1), pp. 41-64. - Koulis, Alexandros; Kaimakamis, George and Beneki, Christina (2018), “Hedging effectiveness for international index futures markets”,
*Economics and Business*, 32, pp. 149-159, doi: 10.2478/eb-2018-0012 - Kroner, Kenneth F. and Ng, Victor K. (1998), “Modeling Asymmetric Comovements of Asset Returns”,
*Review of Financial Studies*, 11, pp. 817-844. http://dx.doi.org/10.1093/rfs/11.4.817 - Lee, Cheng-Few; Wang, Kehluh and Chen, Yan L. (2009), “Hedging and Optimal Hedge Ratios for International Index Futures Markets”,
*Review of Pacific Basin Financial Markets and Policies*, 12 (04), pp. 593-610. https://doi.org/10.1142/S0219091509001769 - Lien, Donald and Luo, Xiangdong (1994), “Multiperiod hedging in the presence of conditional heteroskedasticity”,
*Journal of Futures Markets*, 14 (8), pp. 927-955. https://doi.org/10.1002/fut.3990140806 - Lien, Donald and Tse, Yiu K. (1999), “Fractional cointegration and futures hedging”,
*Journal of Futures Markets*, 19 (4), pp. 457-474. https://doi.org/10.1002/(SICI)1096-9934(199906)19:4<457::AID-FUT4>3.0.CO;2-U - Lien, Donald (2005), “A Note on Asymmetric Stochastic Volatility and Futures Hedging”,
*Journal of Futures Markets*, 25 (6), pp. 607-612. http://doi.org/10.1002/fut.20158 - Lien, Donald and Yang, Li (2008), “Asymmetric effect of basis on dynamic futures hedging: Empirical evidence from commodity markets”,
*Journal of Banking and Finance*, Elsevier, vol. 32 (2), pp. 187-198. - Lopez, Jose A. and Walter, Christian A. (2001), “Evaluating covariance matrix forecasts in a value-at-risk framework”,
*Journal of Risk*, 3 (3), pp. 69-97. http://doi.org/10.2139/ssrn.305279 - Mandelbrot, Benoit B. (1963), “The Variation of Certain Speculative Prices”,
*Journal of Business*, 36 (4), pp. 394-419. - McAleer, Michael and Da Veiga, Bernardo (2008), “Forecasting Value-at-Risk with a Parsimonious Portfolio Spillover GARCH (PS-GARCH) Model”,
*Journal of Forecasting*, 27, pp. 1-19. - Nelson, Daniel B. (1991), “Conditional Heteroskedasticity in Asset Returns: A New Approach”,
*Econometrica*, 59 (2), pp. 347-370. https://doi.org/10.2307/2938260 - Park, Sung Y. and Jei, Sang Y. (2010), “Estimation and Hedging Effectiveness of Time varying Hedge Ratio: Flexible Bivariate Garch Approaches”,
*The Journal of Futures Markets*, 30 (1), pp. 71-99. http://doi.org/10.1002/fut.20401 - R. Rabemananjara and J. M. Zakoian (1993), “Threshold arch models and asymmetries in volatility”,
*Journal of Applied Econometrics*, 8 (1), pp. 31-49. https://doi.org/10.1002/jae.3950080104 - Sheu, Her-Jiun and Lee, Hsiang-Tai (2014), “Optimal Futures Hedging Under Multichain Markov Regime Switching”,
*Journal of Futures Markets*, 34 (2), pp. 173-202. http://doi.org/10.1002/fut.21583 - Singh, Jitendra; Ahmad, Wasim and Mishra, Anil (2019), “Coherence, connectedness and dynamic hedging effectiveness between emerging markets equities and commodity index funds”,
*Resources Policy*, 61, pp. 441-460. https://doi.org/10.1016/j.resourpol.2018.03.006 - Terasvirta, Timo (2009), “An Introduction to Univariate GARCH Models”, in Andersen, T. G.; Davis, R.A; Kreiss, J. P. and Mikosch, Thomas (eds.),
*Handbook of Financial Time Series*, Springer, pp 17-42. - Tse, Yiu K. and Tsui, Albert K. C. (2002), “A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations”,
*Journal of Business and Economics Statistics*, 20 (3), pp. 351-362. - Wang, Yudong; Wu, Chongfeng and Yang, Li (2015), “Hedging with futures: Does anything beat the naive hedging strategy?”,
*Management Science*, 61 (12), pp. 2870-2889.

^{1} Generalized Autoregressive Conditional Heteroskedasticity (

^{2} Recall that βt = 1 for the naïve hedge.