Open Access

The volatility of returns from commodity futures: evidence from India

Financial Innovation20173:15

https://doi.org/10.1186/s40854-017-0066-9

Received: 23 September 2016

Accepted: 2 September 2017

Published: 5 September 2017

Abstract

Background

This paper examines the pattern of the volatility of the daily return of select commodity futures in India and explores the extent to which the select commodity futures satisfy the Samuelson hypothesis.

Methods

One commodity future from each group of futures is chosen for the analysis. The select commodities are potato, gold, crude oil, and mentha oil. The data are collected from MCX India over the period 2004–2012. This study uses several econometric techniques for the analysis. The GARCH model is introduced for examining the volatility of commodity futures. One of the key contributions of the paper is the use of the β term of the GARCH model to address the Samuelson hypothesis.

Result

The Samuelson hypothesis, when tested by daily returns and using standard deviation as a crude measure of volatility, is supported for gold futures only, as per the value of β (the GARCH effect). The values of the rolling standard deviation, used as a measure of the trend in the volatility of daily returns, exhibits a decreasing volatility trend for potato futures and an increasing volatility trend for gold futures in all contract cycles. The result of the GARCH (1,1) model suggests the presence of persistent volatility and the prevalence of long memory for the select commodity futures, except potato futures.

Conclusions

The study sheds light on significant characteristics of the daily return volatility of the commodity futures under analysis. The results suggest the existence of a developed market for the gold and crude oil futures (with volatility clustering) and show that the maturity effect is only valid for the gold futures.

Keywords

Commodity futuresDaily returnVolatilitySamuelson hypothesisGARCH

Background

Volatility plays a vital role in derivative pricing, hedging, risk management, and optimal portfolio selection. The concept of volatility relates to the uncertainty or risk about an asset’s value. A higher volatility means that an asset can assume a large range of values, while a lower volatility implies that an asset’s value does not fluctuate dramatically, even though it changes over time. Accurate modeling and forecasting of volatility in asset returns are major issues in financial economics. Derivative markets, particularly commodity futures markets, have become more sophisticated now a day. The futures price depends on the availability of information. A small change in price may have large effects on the trading results across futures markets. Researchers around the world showed increasing interest in the volatility of commodity futures. In the present analysis, an attempt is made to examine the trend and pattern of the volatility of daily returns of few select commodity futures in the Indian context.

As a first step, we examine the characteristics of the commodity futures. In particular, we analyze whether the price variability of a future increases or decreases when the contract approaches maturity. The Samuelson hypothesis for the selected commodity futures is tested. Samuelson (1965) argued that the volatility of the change in futures price increases as the contract approaches maturity. This phenomenon is also called the “Maturity Effect.” The purpose of testing the Samuelson hypothesis is to assess the degree of maturity of Indian commodity futures. From the view point of the Samuelson hypothesis, the prediction of price volatility is very useful for all participants in the futures market, such as hedgers, speculators, and traders. We also address the trend in daily return’s volatility across the contract cycles to decipher the volatility characteristics of the select commodity futures. To this end, we introduce the concept of rolling standard deviation.

We, then, proceed to examine the volatility aspects of the commodity futures. The steps involved in this exercise are the graphical plotting of the daily returns series, followed by its descriptive statistics. The daily returns are tested for stationarity. Then, we explored the GARCH (1, 1) model for the return volatility of the select futures.1

The present paper derives its motivation from the following considerations. First, commodity futures as a financial asset is gaining prominence in the Indian capital market. The uninterrupted transactions in futures contracts from 2004, with a volume of trade surging from Rs 1.29 lakh crore in 2003–2004 to a peak of Rs 181 lakh crore in 2011–2012,2 confirms the phenomenal importance of commodity futures. Second, empirically testing the Samuelson hypothesis as an indicator of developed and mature futures market seems necessary for the Indian commodity futures market. One of the key contributions of this paper is to use the GARCH (1,1) process for testing the Samuelson hypothesis on select commodity futures. Testing the Samuelson hypothesis through the β term of the GARCH (1,1) yields meaningful results, as the GARCH (1,1) assumes that the returns are uncorrelated, with zero mean. Moreover, in the GARCH (1,1) process, the present volatility does not depend on past returns, and thereby makes it a suitable methodology to test the Samuelson hypothesis. In this respect, the present analysis aims at filling a gap in the existing literature. Finally, in India, while the volatility issues related to dominant financial assets, such as company shares, have been well researched and documented, only a few studies on commodity futures have been carried out. More specifically, the trend and pattern of the volatility in the daily returns from commodities have been largely ignored in the existing literature. The remainder of this paper is organized as follows. The second section presents the literature. The third section deals with the methodology used in this paper and describes the relevant data sources. The result and discussion of the analysis are carried out in the fourth section. Finally, the fifth section provides our concluding remarks.

Literature review

Many researchers, such as W. R. Anderson (1985), examined the Samuelson hypothesis using selected agricultural futures contracts and found support for wheat, oat, soybeans, and soybeans meal futures. Bessembinder et al. (1996) provided a new framework for the maturity effect, the ‘BCSS hypothesis’ (based on Bessembinder, Coughenour, Seguin and Smoller). This hypothesis is an extension of the Samuelson hypothesis. The authors found that the Samuelson hypothesis is more likely to hold for those commodities whose price changes can be reversed in future. Black and Tonks (2000) investigated the pattern of the volatility of commodity futures prices over time and revealed the conditions which support the Samuelson hypothesis. Allen and Cruickshank (2000) analyzed the Samuelson hypothesis for selected commodity futures of three different futures markets in three different countries. They performed a regression analysis complemented by ARCH models, and the result suggests that the Samuelson hypothesis holds in the case of maximum selected contracts. Floros and Vougas (2006) investigated the Samuelson hypothesis in the context of the Greek stock index futures market and examined the maturity effect through linear regressions and GARCH models. The result of the study suggests that volatility depends on time to maturity and gives a stronger support to the Samuelson hypothesis compared to linear regressions. Duong and Kalev (2008) examined the Samuelson hypothesis for 336 selected commodities from five futures exchanges observed between 1996 and 2003. Using the Jonckheere-Terpstra Test, OLS regressions with realized volatility, and various GARCH models, the authors find mixed evidence concerning the support for the Samuelson hypothesis. Even though many studies investigated the Samuelson hypothesis, very few contributions analyzed it in the context of the Indian commodity futures market.

Notable exceptions are Verma and Kumar (2010), who examined the application of the Samuelson hypothesis and BCSS hypothesis in the Indian commodity futures market. Gupta and Rajib (2012) also examined this issue for eight commodities, and they concluded that the Samuelson hypothesis does not hold for the majority of the considered commodity contracts.

Numerous studies investigate the volatility of futures prices worldwide.

Locke and Sarkar (1996) examined the changes in market liquidity following changes in price volatility. The results of the study suggest that market makers are most hurt by volatility in the case of inactive contracts. Richter and Sorensen (2002) analyzed a volatility model for soybean futures and options using panel data. The study suggests the existence of a seasonal pattern in convenience yields and volatility, in line with the storage theory. Chang et al. (2012) examined a long memory volatility model for 16 agricultural commodity futures. The empirical results are obtained using unit root tests, GARCH, EGARCH, APARCH, FIGARCH, FIEGARCH, and FIAPARCH model. Manera et al. (2013) examined the effect of different types of speculation on the volatility of commodity futures prices. The authors selected four energy and seven non-energy commodity futures observed over the period 1986–2010. Using GARCH models, the study suggests that speculation affects the volatility of returns, and long-term speculation has a negative impact, whereas short term speculation has a positive effect. Christoffersen et al. (2014) analyzed the stylized facts of volatility in the post-financialization period using data of 750 million futures observed between 2004 and 2013.

Two strands in the existing literature focused on volatility in the Indian commodity futures market. First, the literature is largely dominated by spot price volatility and its spillover effect on future price volatility, that is, the price discovery mechanism of the futures market. Brajesh and Kumar (2009 ) examined the relationship between future trading activity and spot price volatility for different commodity groups, such as agricultural, metal, precious metal, and energy commodities in the perspective of the Indian commodity derivatives market. P. Srinivasan (2012) examined the price discovery process and volatility spillovers in Indian spot-futures commodity markets and the result points to dominant volatility spillovers from spot to futures market. Sehgal et al. (2012) examined the futures trading activity on spot price volatility of seven agricultural commodities and found that unexpected futures trading has strong correlation on spot volatility. Chauhan et al. (2013) analyzed the market efficiency of the Indian commodity market. They found that for guar seed, the volatility in futures prices influences the volatility in spot prices and the opposite result holds for chana. The work by Chakrabarti and Rajvanshi (2013) also explored the determinants of return volatility of select commodity futures in the Indian context. Sendhil et al. (2013) examined the efficiency of commodity futures through price discovery, transmission, and the extent of volatility in four agricultural commodities and found persistence volatility in spot market. Kumar et al. (2014) examined the price discovery and volatility spillovers in the Indian spot-futures commodity market. Gupta and Varma (2015) reviewed the impact of futures trading on spot markets of rubber in India and observed bidirectional flow of volatility between spot and futures market. Vivek Rajvanshi (2015) presented a comparative study on the performance of range and return-based volatility estimators for crude oil commodity futures. Malhotra and Sharma (2016) investigated the information transmission process between the spot and futures market and found that bidirectional volatility spillovers exists between the spot and futures market.

Second, a few studies specifically focus on the volatility of commodity futures. Kumar and Singh (2008) examined the volatility clustering and asymmetric nature of Indian commodity and stock market using S&P CNX Nifty for the stock market, and gold and soybean for the commodity futures market. Kumar and Pandey (2010) examined the relationship between volatility and trading activity for different categories of Indian commodity derivatives. They find a positive and significant correlation between volatility and trading volume for all commodities, no significant relationship between volatility and open interest, and an asymmetric relationship between trading volume and open interest. Kumar and Pandey (2011) examined the cross market linkages of Indian commodity futures with futures markets outside India. However, all these studies focus on the price volatility of commodity futures. In contrast with the above-mentioned studies on the Indian commodity futures market, the present study attempts to examine the return volatility of select commodity futures as financial assets.3

Methods

The data on commodity futures are obtained from the official website of Multi Commodity Exchange (MCX), Mumbai, and cover the period from 2004 to 2012. We selected four commodities (potato, crude oil, gold, and mentha oil) from four different categories of commodity futures: Agricultural Commodity Futures, Energy, Bullions and Oil, and Oil Related Products, respectively. This choice satisfies two basic criteria: (i) the high frequency of future contracts; (ii) the large volume/value of such futures within the study period. Table 1 justifies the choice of the commodity futures.
Table 1

List of traded contracts (in volume and value) of commodity futures in MCX, India

  

Bullions

Energy

Agricultural products

Oil & oil related products

Year

 

Golda

Silver

Platinum

Crude Oila

Natural Gas

Gasoline

Potatoa

Kapas

Pepper

Mentha Oila

Mustard seed

RBD

Palmolien

2004

Traded Contracts(in lots)

632,843

138,977

5

NA

NA

NA

NA

NA

1715

8

6406

NA

678

59

Total Value(in lakhs)

394,070

4.99

449,984

5.34

NA

NA

NA

NA

NA

1197

5.84

4750.

96

NA

1268.4

9

25.43

2005

Traded Contracts(in lots)

260,040

7

584,476

5

NA

515,781

1

NA

NA

NA

1144

81

1027

08

87,369

3

3826

16,945

Total Value(in lakhs)

175,513

30.18

196,148

49.81

NA

137,708

85.61

NA

NA

NA

7778

8.06

6997

9.86

18,607

59.84

6708.5

6

23,099.3

5

2006

Traded Contracts(in lots)

995,735

1

949,854

4

NA

446,653

8

19,537

56

NA

3979

08

3815

73

4431

7

24,282

30

17,073

3759

Total Value(in lakhs)

760,489

1

506,073

78.39

NA

130,325

62.32

32,625

19.14

NA

7299

85.37

2781

05.65

4612

0.65

55,089

50.19

27,798.

17

13,706.0

5

2007

Traded Contracts(in lots)

140,242

17

918,327

3

NA

139,388

13

17,327

59

NA

4227

23

2112

87

1924

81

81,674

9

0

NA

Total Value(in lakhs)

171,474

191.96

515,680

68.06

NA

421,132

66.31

25,869

80.10

NA

7654

16.31

1801

29.77

2509

52.27

16,257

74.77

0.00

NA

2008

Traded Contracts(in lots)

140,242

17

109,726

76

3790

205,070

01

74,750

6

NA

3900

9

7162

3

2430

42,502

3

NA

NA

Total Value(in lakhs)

171,474

191.96

704,073

59.66

5133

4.27

859,472

48.64

30,021

86.10

NA

6356

6.98

6958

7.30

3566.

32

84,761

1.43

NA

NA

2009

Traded Contracts(in lots)

121,449

67

115,555

01

1291

1

410,928

21

11,124

491

5494

2

8996

9

6544

7

4180

3

50,604

2

NA

NA

Total Value(in lakhs)

184,997

191.41

828,910

95.67

6110

9.94

121,020

964.66

27,497

924.35

2059

35.14

2428

96.24

8029

2.97

4859

4.66

10,114

10.54

NA

NA

2010

Traded Contracts(in lots)

120,522

25

164,405

33

221

415,370

53

11,176

937

842

4918

52

8759

6

NA

15,710

93

NA

NA

Total Value(in lakhs)

219,874

783.77

159,664

842.35

1302

.21

150,743

390.24

27,919

327.69

3511.

97

7572

78.05

1168

88.73

NA

50,139

65.14

NA

NA

2011

Traded Contracts(in lots)

126,557

60

244,345

44

210

547,536

58

98,821

19

20

4671

50

1941

38

NA

15,689

17

NA

NA

Total Value(in lakhs)

314,713

353.71

408,239

010.89

1325

.36

242,044

737.34

23,293

743.77

107.5

9

8096

64.01

3507

01.82

NA

66,592

52.09

NA

NA

2012

Traded Contracts(in lots)

102,876

09

172,845

29

21

577,902

29

27,886

670

20

2740

47

3033

23

NA

22,891

39

NA

NA

Total Value(in lakhs)

305,672

442.56

297,774

497.73

139.

66

289,229

240.48

54,440

421.01

123.6

7

8034

93.25

5644

01.58

NA

12,470

449.09

NA

NA

Source: MCX, India and authors’ own calculation

Note: NA denotes data not available

Here we choose three active commodity futures from each group for the entire study period to show the relative prominence of particular commodity futures(based on volume and value of trade) within the group

a denotes the selection of commodity futures based on volume and value of trade, with the only exception for bullions futures for the year 2011 where the silver futures dominates

In the commodity futures exchanges, trading takes place for 1-month, 2-month, and 3-month contract expiry cycles. However, in India, the 4-month, 5-month, and up to 1-year contract expiry cycles exist, in some cases, and we treat them as unusual exceptions. We only focus on the 1-month (near), 2-month (next-near), and 3-month (far) expiry cycles for futures. All futures contracts expire on the last Thursday of the month.

Hereafter, we provide a hypothetical example to demonstrate the steps involved in calculating the return in the logarithm form. We introduce a case based on crude oil.
  • The contract starts on July 30, 2010, and expires on October 20, 2010.

  • Nominal return for 1-month contract = ln(closing price on October 20)-ln(opening price on October 1); (October 1 is the Friday following the last Thursday of September, with 1 month to expiry, approximately.).

  • Nominal return for 2-month contract = ln(closing price on October 20)-ln(opening price on August 27); (August 27 is the Friday following the last Thursday of August, with 2 months to expiry, approximately).

  • Nominal return for 3-month contract = ln(closing price on October 20) -ln(opening price on July 30).

Here, the daily return on futures is calculated as the value of the continuously compounded rate of the return multiplied by 100. As such, the Log return of the price series = ln(Ft /Ft-1) *100, where Ft and Ft-1 are the closing prices on day t and (t-1) of a futures contract. The standard deviation of the daily return is also calculated for all the three categories of contract cycles.

We use the conventional standard deviation approach as the measure of the volatility of daily returns. A hypothetical example is as follows (Table 2).
Table 2

Returns of near, next near and far month contracts of mentha oil maturing on 31st December, 2010

Col.1

Col.2

Col.3

Col.4

Col.5

Col.6

Col.7

Contract/expiry month

Near month

Returns

Next near month

Returns

Far month

Returns

31-Dec-10

1-Dec-10

0.018

1-Nov-10

0.039

1-Oct-10

0.012

31-Dec-10

2-Dec-10

0.017

2-Nov-10

0.039

4-Oct-10

−0.012

31-Dec-10

3-Dec-10

−0.009

3-Nov-10

0.003

5-Oct-10

0.016

31-Dec-10

4-Dec-10

0.005

4-Nov-10

−0.023

6-Oct-10

0.015

31-Dec-10

6-Dec-10

−0.030

5-Nov-10

0.018

7-Oct-10

0.039

31-Dec-10

7-Dec-10

−0.012

6-Nov-10

0.032

8-Oct-10

0.035

31-Dec-10

8-Dec-10

−0.019

8-Nov-10

−0.018

9-Oct-10

−0.004

31-Dec-10

9-Dec-10

−0.041

9-Nov-10

−0.029

11-Oct-10

−0.013

31-Dec-10

10-Dec-10

−0.028

10-Nov-10

−0.041

12-Oct-10

0.012

31-Dec-10

11-Dec-10

−0.041

11-Nov-10

−0.024

13-Oct-10

0.039

31-Dec-10

13-Dec-10

0.005

12-Nov-10

0.003

14-Oct-10

−0.017

31-Dec-10

14-Dec-10

0.032

13-Nov-10

0.022

15-Oct-10

−0.029

31-Dec-10

15-Dec-10

0.012

15-Nov-10

0.007

16-Oct-10

0.036

31-Dec-10

16-Dec-10

−0.006

16-Nov-10

0.011

18-Oct-10

−0.007

31-Dec-10

17-Dec-10

−0.035

17-Nov-10

0.000

19-Oct-10

−0.008

31-Dec-10

18-Dec-10

0.002

18-Nov-10

−0.004

20-Oct-10

−0.004

31-Dec-10

20-Dec-10

0.035

19-Nov-10

−0.030

21-Oct-10

0.038

31-Dec-10

21-Dec-10

−0.012

20-Nov-10

0.023

22-Oct-10

0.039

31-Dec-10

22-Dec-10

−0.011

22-Nov-10

0.015

23-Oct-10

−0.006

31-Dec-10

23-Dec-10

0.023

23-Nov-10

−0.033

25-Oct-10

0.039

31-Dec-10

24-Dec-10

0.000

24-Nov-10

0.001

26-Oct-10

0.039

31-Dec-10

27-Dec-10

0.013

25-Nov-10

0.011

27-Oct-10

0.011

31-Dec-10

28-Dec-10

0.008

26-Nov-10

−0.005

28-Oct-10

−0.004

31-Dec-10

29-Dec-10

0.009

27-Nov-10

0.014

29-Oct-10

0.006

31-Dec-10

30-Dec-10

0.024

29-Nov-10

0.011

30-Oct-10

0.011

31-Dec-10

31-Dec-10

0.100

30-Nov-10

−0.005

 

Std. dev

0.029

 

0.022

 

0.021

Source: MCX database and authors’ own calculation

We also introduce the concept of 25-day moving standard deviation (also known as the rolling standard deviation) as a measure of the trend in the volatility of the daily returns.

The method for calculating the rolling standard deviation is explained with the help of a hypothetical example based on crude oil futures.
  • The contract starts on July 30, 2010, and expires on October 20, 2010.

  • We consider the first 25 days starting from July 30, 2010 to calculate the standard deviation.

  • For the next period, the initial day (July 30, 2010) is left out and 1 day is added to the end of the period (August 24, 2010) so that the 25 days begin from July 31, 2010, and end on August 24, 2010. The standard deviation is calculated for these 25 days.

  • The above process is repeated for the entire length of the contract cycles to obtain the rolling standard deviation for the concerned futures.

  • In this example, 25-days are considered as the average number of trading days per month (leaving aside Sundays and other holidays). Therefore, the total annual trading days for commodity futures is 305 days.

We then proceed to plot graphically the daily returns series over time so that volatility clustering can be verified.

Descriptive statistics

To analyze the characteristics of the daily return series of the commodity futures market during the study period, the descriptive statistics show the mean (X), standard deviation (σ), Skewness (S), Kurtosis (K), and Jarque-Bera statistics results.

We calculated the coefficients of Skewness and Kurtosis to verify whether the return series is skewed or leptokurtic. To test the null hypothesis of normality, the Jarque-Bera statistic (JB) has been applied, as follows:
$$ JB=\frac{N-k}{6}\left[{S}^2+\frac{1}{4}{\left(K-3\right)}^2\right], $$
(1.1)
where N is the number of observations, S is the coefficient of Skewness, K is the coefficient of Kurtosis, k is the number of estimated coefficients used to create the series, and JB follows a Chi-square distribution with 2 degrees of freedom (d. f.). We perform a joint test of normality where the joint hypothesis of s = 0 and k = 3 is tested. If the JB statistic is greater than the table value of chi-square with 2 d. f., the null hypothesis of a normal distribution of residuals is rejected.

Test for stationarity

For testing whether the data are stationary or not, we performed the Augmented Dickey-Fuller (Dickey and Fuller 1979) and Philips-Perron Test (PP) (Phillips and Perron 1988). The stationarity of the return series has been checked by ADF test by fitting a regression equation based on a random walk with an intercept, or drift term (φ), as follows:
$$ \varDelta {y}_t=\varphi +\partial {y}_{t-1}+\sum {\theta}_j\varDelta {y}_{t-j}+{\mu}_{t,}, $$
(1.2)
where μ t is a disturbance term with white noise. Here the null hypothesis is H 0 : ∂ = 0 (with alternative hypothesis H 1 : ∂ < 0). If this hypothesis is accepted, there is a unit root in the yt sequence, and the time series is non-stationary. If the magnitude of the ADF test statistic exceeds the magnitude of Mackinnon critical value, the null hypothesis is rejected, and there is no unit root in the daily return series.
Phillips and Perron (1988) suggested an alternative (non-parametric) method to control for serial correlation when testing for the presence of a unit root. The PP method estimates the non- augmented DF test equation, and it can be seen as a generalization of the ADF test procedure, which allows for fairly mild assumptions concerning the distribution of errors. The PP regression equation is as follows:
$$ \varDelta {y}_{t-1}=\varphi +\partial {y}_{y-1}+{\mu}_t, $$
(1.3)
where the ADF test corrects for higher order serial correlation by adding lagged differenced terms on the right-hand side, while the PP test corrects the t statistic of the coefficient ∂ obtained from the AR(1) regression to account for the serial correlation μt. The null hypothesis is H 0 : ∂ = 0 (with alternative hypothesis H 1 : ∂ < 0).

Test for heteroskedasticity

The presence of heteroskedasticity in asset returns has been well documented in the existing literature. If the error variance is not constant (heteroskedastic), then, the OLS estimation is inefficient. The tendency of volatility clustering in financial data can be well captured by a Generalized Autoregressive Conditional Heteroskedastic (GARCH) model. Therefore, we modeled the time-varying conditional variance in our study as a GARCH process.

To identify the type of GARCH model that is more appropriate for our data, we performed the ARCH LM test (Engle 1982). This is a Lagrange Multiplier test for the presence of an ARCH effect in the residuals. We first regressed the return series on their one-period lagged return series and obtained the residuals \( \left({\widehat{\varepsilon}}^2\right) \). Then, the residuals have been squared and regressed on their own lags of order one to four to test for the ARCH effect. The estimated equation is:
$$ {\widehat{\varepsilon}}_t^2={\vartheta}_0+\sum_{i=1}^4{\vartheta}_i{\widehat{\varepsilon}}_{t-1}^2+{K}_t, $$
(1.4)
where K t is the error term. We, then, obtained the coefficient of determination (R2). The null hypothesis is the absence of ARCH error, H 0 : ϑ i  = 0, against the alternative hypothesis H 1 : ϑ i  ≠ 0. Under the null hypothesis, the ARCH LM statistic is defined as TR2, where T represents the number of observations. The LM statistic converges to a χ 2 distribution. Hence, we use the Lagrange Multiplier (LM) test for Autoregressive Conditional Heteroskedasticity (ARCH) to verify the presence of heteroskedasticity in the residuals of the daily return series for all commodity futures. If the ARCH LM statistic is significant, we confirm the presence of an ARCH effect.

The ARCH model as developed by Engle (1982) is an extensively used time-series models in the finance-related research. The ARCH model suggests that the variance of residuals depends on the squared error terms from the past periods. The residual terms are conditionally normally distributed and serially uncorrelated. A generalization of this model is the GARCH specification. Bollerslev (1986) extended the ARCH model based on the assumption that forecasts of the time-varying variance depend on the lagged variance of the variable under consideration. The GARCH specification is consistent with the return distribution of most financial assets, which is leptokurtic and it allows long memory in the variance of the conditional return distribution.

The Generalized Arch Model (GARCH)

The GARCH model (Bollerslev 1986) assumes that the volatility at time t is not only affected by q past squared returns but also by p lags of past estimated volatility. The specification of a GARCH (1, 1) is given as:

mean equation:
$$ {r}_t=\mu +{\varepsilon}_t, $$
(1.5)
variance equation:
$$ {\sigma}_{t-1}^2=\omega +{\alpha \varepsilon}_{t-1}^2+{\beta \sigma}_{t-1}^2, $$
(1.6)
where ω > 0, α ≥ 0, β ≥ 0, and r t. is the return of the asset at time t, μ is the average return, and ε t is the residual return. The parameters α and β capture the ARCH effect and GARCH effect, respectively, and they determine the short-run dynamics of the resulting time series. If the value of the GARCH term β is sufficiently large, the volatility is persistent. On the other hand, a large value of α indicates an insensitive reaction of the volatility to market movements. If the sum of the coefficients is close to one, then, any shock will lead to a permanent change in all future values. Hence, the shock is persistent in the conditional variance, implying a long memory.

Wald test

The Wald test estimates the test statistic by computing the unrestricted regression equation, without imposing any coefficient restrictions, as specified by the null hypothesis. The Wald statistic (under the null hypothesis) measures how the unrestricted estimates satisfy the restrictions. If the restrictions are valid, then, the unrestricted estimates should fulfill the restrictions.

We consider a general nonlinear regression model:
$$ y=x\left(\beta \right)+\epsilon, $$
(1.7)
where β is a k vector of parameters to estimate. Any restrictions on the parameters can be written as:
$$ {\mathrm{H}}_{\circ }:\mathrm{g}\ \left(\upbeta \right)=0 $$
where g is a smooth q dimensional vector imposing q restrictions on β.

Under the null hypothesis H˳, the Wald statistic has an asymptotic χ 2 (q) distribution, where q is the number of restrictions.

The result of the above tests is derived using Eviews 7.

Result and discussion

The Samuelson hypothesis is tested by the daily returns for the select commodity futures, and the results are reported in Table 3. There is no clear trend and pattern in the percentage of the standard deviation among the selected commodities, except the gold futures, for which the Samuelson hypothesis holds. For other commodity futures (potato, crude oil, and mentha oil) this assumption is not confirmed.4 For crude oil and mentha oil, the volatility of daily returns is greater for the 3-month (far) contract, followed by the 1-month (near) contract and the 2-months (next near) contract. The only exception is observed for potato futures, for which the volatility of daily returns for the 2-month (next near) contract is greater than that for the 1-month (near) contract. This phenomenon may be attributed to two possible reasons: (1) the underdeveloped and/or developing futures market in India, which acts as a barrier to the fulfillment of the Samuelson hypothesis; (2) since the volatility of daily returns for the 3-month (far) contract is greater for the selected three commodity futures (potato, crude oil, and mentha oil), the trend may be attributed to the initial euphoric behavior in the futures market, resulting from the initiation of a future contract.
Table 3

Volatility of daily returns

Commodity futures

1 month contract

2 month contract

3 month contract

 

S.D(%)

Rolling s.d

S.D(%)

Rolling s.d

S.D(%)

Rolling s.d

Potato

1.96

Decreasing

2.19

Decreasing

3.79

Decreasing

Crude oil

1.86

Constant

1.71

Marginal increase

2.4

Decreasing

Mentha oil

2.56

Constant

2.16

Decreasing

4.65

Constant

Gold

1.06

Marginal increase

0.97

Increasing

1.01

Increasing

Table 3 also presents the rolling standard deviation of the four commodity futures for all the three types of contract cycles.

To explore the trend in the volatility of daily returns for the selected commodity futures, we used the methodology known as 25-days rolling standard deviation. Figures 1, 2 and 3 depict the trends of the volatility for each commodity futures, where the x-axis measures the number of contracts traded and the y-axis measures the standard deviation in percentage (%) terms.
Fig. 1

Trends based on rolling standard deviation for 1 (Near) month contract

Fig. 2

Trends based on rolling standard deviation for 2 (next near) month contract

Fig. 3

Trends based on rolling standard deviation for 3 (far) month contract

For potato futures, there is a decreasing trend in volatility for near, next near, and far month contracts, with near contract exhibiting the least declining trend in volatility, and far month contract showing the maximum declining trend in volatility.

For crude oil and mentha oil futures, the near month volatility trend of daily returns is almost constant, and the magnitude of rolling standard deviation (volatility trend) is the highest for the far month contract.

For gold futures, the trend in volatility is increasing for all types of contract (1-month, 2-month, and 3-month). Moreover, this rise in the trend in volatility is greater for the 1-month contract, suggesting that the gold futures trend is more volatile as the contract approaches the maturity date.

The descriptive statistics for daily return series of the select commodity futures are summarized in Table 4.
Table 4

Simple Statistics for all three types of contracts for four commodities

  

Mean

Median

Maximum

Minimum

Std. Dev.

Skewness

Kurtosis

Jarque-Bera

Probability

Contracts

 Potato

Near

0.000978

0.002409

0.285775

−0.406582

0.019603

−3.660024

181.8045

1,802,719

0.000000

Next Near

0.00158

0.002358

0.264099

−0.437677

0.021901

−4.900318

182.2342

1,440,545

0.000000

Far

0.000579

0.002266

0.47847

−0.557147

0.037906

−2.459749

121.0494

449,624.1

0.000000

 Mentha oil

Near

0.00275

0.000226

0.150534

−0.094422

0.025606

1.582898

11.14921

6391.611

0.000000

Next Near

−3.34E-05

0.000000

0.101813

−0.410464

0.021692

−4.171503

77.02822

461,557

0.000000

Far

−0.002403

0.000000

0.669725

−0.591267

0.046594

−3.343164

83.00441

525,838.3

0.000000

 Crude oil

Near

−0.000249

0.000621

0.088606

−0.094389

0.018635

−0.228756

5.63786

628.0626

0.000000

Next Near

−0.000128

0.000763

0.084266

−0.09662

0.017191

−0.276003

5.628188

634.6635

0.000000

Far

−0.000574

0.000362

0.33352

−0.242462

0.024035

−0.12481

48.82034

180,300.2

0.000000

 Gold

Near

0.000515

0.000677

0.081194

−0.064016

0.010629

−0.235882

9.932995

2690.098

0.000000

Next Near

0.000429

0.000797

0.051936

−0.065173

0.009777

−0.649765

8.615752

1582.362

0.000000

Far

0.000434

0.000511

0.08112

−0.08509

0.010107

−1.026436

17.93436

12,290.4

0.000000

The average daily returns for all commodity futures are either close to zero or negative throughout the study period. The descriptive statistics show that the returns are negatively skewed. Since the estimated coefficients for the Skewness of the return series are different from zero, the underlying return distributions are not symmetric. The estimated coefficients for the Kurtosis of the daily return series are relatively high, suggesting that the underlying distributions are leptokurtic or heavily tailed and sharply peaked toward the mean compared to a normal distribution. The observed Skewness and Kurtosis indicate that the distribution of daily return series is non-normal. The Jarque-Bera normality test also shows the non-normality of the return distributions, as the estimated values of the Jarque-Bera statistic of all the return series are statistically significant at the 1% level (Figs. 4, 5, 6 and 7).
Fig. 4

Daily return series graph of Potato futures

Fig. 5

Daily return series graph of mentha oil futures

Fig. 6

Daily return series graph of crude oil futures

Fig. 7

Daily return series graph of gold futures

The correlogram test is conducted to address the presence of serial correlation in the residuals. We observe no serial correlation in the residuals up to 24 lags for the gold and crude oil futures in all types of contract cycles. This result holds for the 3-month (far) mentha oil contracts and potato near and next near contracts, as reported in Table 5.
Table 5

Correlogram test (upto 24 lags)

Commodity

futures →

Potato

Mentha oil

Crude oil

Gold

Contract types →

Near

Next

Near

Far

Near

Next

Near

Far

Near

Next

Near

Far

Near

Next

Near

Far

Residuals are serially correlated

No

No

Yes

Yes

Yes

No

No

No

No

No

No

No

The ADF and PP tests are performed to verify the stationarity of the daily return series, and the statistics are presented in Table 6. The p values of the ADF and PP tests are <0.05, which leads to conclude that the data used for the entire study period are stationary.
Table 6

Result of unit root test

Commodity futures →

Potato

Mentha oil

Crude oil

Gold

Contract types →

Near

Next

Near

Far

Near

Next

Near

Far

Near

Next

Near

Far

Near

Next

Near

Far

ADF Test Statistic

−28.242

−31.121

−27.875

−41.075

−45.046

−5.682

−43.527

−43.770

−43.860

−37.804

−32.710

−34.323

 Prob.

0.000

0.000

0.000

0.000

0.0001

0.000

0.000

0.0001

0.0001

0.000

0.000

0.000

Philips Perron

Test Statistic

−36.475

−31.120

−27.875

−41.179

−44.800

−44.315

−43.525

−43.781

−43.860

−37.821

−32.692

−34.331

 Prob.

0.000

0.000

0.000

0.000

0.0001

0.0001

0.000

0.0001

0.0001

0.000

0.000

0.000

Test critical value

 1%

−3.434

−3.436

−3.438

−3.433

−3.433

−3.433

−3.433

−3.433

−3.433

−3.435

−3.435

−3.435

 5%

−2.863

−2.864

−2.865

−2.862

−2.862

−2.862

−2.862

−2.862

−2.862

−2.863

−2.863

−2.863

 10%

−2.567

−2.568

−2.568

−2.567

−2.567

−2.567

−2.567

−2.567

−2.567

−2.567

−2.568

−2.567

Note: ADF Test Statistic is estimated by fitting the equation of the form: Δy t = φ + ∂y t−1 + ∑θ j Δy t − j + μ t and PP test statistic is estimated by the equation: Δy t−1 = φ + ∂y t−1 + μ t

Both the test statistics reported in Table 6 reject the null hypothesis at the 1% significance level, with the critical value of −3.43 for both the ADF and PP tests. These results confirm that the series are stationary.

The graphs of daily returns confirm the absence of a clustering effect for potato futures and menthe oil futures. Only the 3 month contracts for menthe oil futures exhibits a small clustering effect for some periods. The graphs of crude oil and gold futures for all types of contracts show that the daily return series exhibits a clustering effect or volatility.

Table 7 presents the result of the ARCH-LM test (Engle 1982) of heteroskedasticity. This test detects the presence of the ARCH effect in the residuals of the daily return series. The ARCH-LM test statistic is significant for all types of contract cycles of gold commodity futures and the near and next near month contract of crude oil commodity futures, as well as for the mentha oil next near5 contracts. The result confirms the presence of ARCH effects in the residuals as the test statistics are significant at 1% level. Hence, the results confirm the need for the analysis of the GARCH effect. The ARCH-LM statistic is not statistically significant for all types of potato contracts and mentha oil near contracts. Moreover, in the case of far month contracts of crude oil and mentha oil futures, we find no evidence of ARCH effect in the residuals. These findings are in line with the negligible amount of volatility clustering exhibited by the daily returns’ volatility graph. Hence, the results seem to confirm the need for the analysis of the GARCH effect.6
Table 7

Result of ARCH-LM test for residuals

 

Potato

Mentha oil

Crude oil

Gold

 

Near

Next

Near

Far

Near

Next

Near

Far

Near

Next

Near

Far

Near

Next

Near

Far

Obs R-squared

0.604

0.046

0.010

4.45

8

30.52

2

0.037

78.95

5

64.85

4

0.099

48.72

8

26.39

4

142.6

915

Prob. Chi-Square

0.437

0.831

0.919

0.03

5

0.000

0.847

0.000

0.000

0.752

0.000

0.000

0.000

Note: ARCH- LM Statistic (at lag-1) is the Lagrange Multiplier test statistic to examine the presence of ARCH effect in the residuals of the estimated model. If the value of ARCH LM Statistic is greater than the critical value from the Chi-square distribution, the null hypothesis of no heteroskedasticity is rejected

The GARCH model is used for modeling the volatility of daily return series for the three types of contracts (near, next near, and far contracts) for crude oil and gold commodity futures and only for next near and far month contracts for mentha oil futures. The result of the GARCH (1,1) model is shown in Table 8. All the parameters of the GARCH analysis are statistically significant.
Table 8

Estimated result of GARCH (1,1) Model

Commodity futures→

Mentha oil

Crude Oil

Gold

Contract types →

Next

Near

Far

Near

Next

Near

Far

Near

Next

Near

Far

Co-efficients↓

Mean

μ (constant)

−4.30E − 05c

−0.002299b

8.02E-05c

0.000215c

-0.00044c

0.000251c

0.000436c

−5.84E − 05b

Variance

 ω (constant)

5.01E − 06b

0.000315b

2.34E-06b

1.72E-06b

5.14E-05b

1.60E-06b

1.35E-06b

4.82E-06b

 α (arch effect)

0.099648b

-0.003595b

0.032133b

0.030668b

-0.00364b

0.071647b

0.084121b

0.217631b

 β (garch effect)

0.891336b

0.856938b

0.960178b

0.962725b

0.9111b

0.917732b

0.903237b

0.775297b

 α + β

0.990984

0.853343

0.992311

0.993393

0.90746

0.989379

0.987358

0.992928

 Log likelihood

5376.339

3241.84

5582.369

5772.936

4778.326

4315.175

3862.321

4295.068

 Akaike info. Criterion (AIC)

−5.384801

−3.307962

−5.30673

−5.464648

−4.6343

−6.45236

−6.75537

−6.61537

 Schwarz info. Criterion (SIC)

−5.370770

−3.293707

−5.29329

−5.451254

−4.62063

−6.43291

−6.73330

−6.59545

Residual Diagnostics for GARCH (1, 1):ARCH-LM (1) test for heteroskedasticity

 Obsa R-squared

1.204135

0.025106

0.264945

0.057879

0.003309

2.001935

0.12552

0.617633

 Prob. Chi-Square(1)

0.2725

0.8741

0.6067

0.8099

0.9541

0.1571

0.7231

0.4319

Wald Test

 F-statistic

1433.702

1191.443

1497.929

1524.253

2935.317

10.09822

9.188416

3.26E-34a

 Probability

0.000

0.000

0.000

0.000

0.000

0.0015

0.0025

1.000

Note: For Wald test the null hypothesis is α + β = 1

aSignificant at 1% level, b Significant at 5% level, c Significant at 10% level

The constant (ω), ARCH term (α), and GARCH term (β) are statistically significant at the 1% level. In the variance equation, the estimated β coefficient is considerably greater than the α coefficient, which implies that the volatility is more sensitive to its lagged values. The result suggests that the volatility is persistent. Moreover, the β term is greater for the near month contract cycles for gold futures, which confirms the validity of the Samuelson hypothesis. The sum of these coefficients (α and β) is close to unity, which indicates that a shock will persist for many future periods, suggesting the prevalence of long memory. However, the Wald test indicates the acceptance of the null hypothesis that α + β = 1 for far month contract cycles of gold futures only.

To check the robustness of the GARCH (1,1) model, we employed the ARCH-LM test (Engle 1982) to verify the presence of any further ARCH effect. As shown in the Table 7, the ARCH- LM test statistic for the GARCH (1,1) model does not show any additional ARCH effect in the residuals of the model, which implies that the variance equation is well specified for the select commodity futures.

As a result, we can conclude that, among the select commodity futures, the clustering effect is present in the volatility of daily returns for crude oil and gold commodity futures in all contract cycles. Mentha oil futures also present a clustering effect in far month contracts.

Conclusions

This paper addresses the volatility of four select commodity futures: potato, mentha oil, crude oil, and gold. All the three types of contract cycles (near month, next near month, and far month) are considered for volatility analysis. The conventional approach based on standard deviation as a measure of volatility is considered to test the Samuelson hypothesis. To further corroborate the findings, the β-term of the GARCH (1,1) is also used to verify the Samuelson hypothesis. The results suggest that the Samuelson hypothesis does not hold for the select commodity futures in the Indian context, except for the gold futures. These results are in line with the findings of Gupta and Rajib (2012) and suggest that the Indian gold futures market is as developed as in the advanced countries.

The trend in the volatility of daily returns is captured by the concept of rolling standard deviation. The volatility trends in crude oil and mentha oil futures highlight the significance of the available information as the far month volatility is higher than the near month volatility. The fluctuations in the world markets for oil commodities have a lagged impact on the domestic market. Finally, the objective of futures market in terms of price discovery and hedging against future risks seems to be satisfied for potato futures. To test the presence of a unit root in the daily return series, we performed the ADF and PP tests. The results confirmed the stationarity of the daily return series for all the commodity futures.

For volatility modeling, we first considered the graphical representation of volatility clustering along with the descriptive statistics for all contract cycles of each commodity future. We, then, introduced a correlogram to check for serial correlation in the residuals, and, finally, the ARCH-LM test was conducted to check for the presence of an ARCH effect. All contract cycles of potato futures did not show any volatility clustering, and the result of the ARCH-LM test ruled out any ARCH effects in the daily return series. However, for all types of contract cycles of gold futures, we found unambiguous volatility clustering, and the ARCH-LM test results also suggested the presence of an ARCH effect. These results are in line with the findings of Kumar and Singh (2008) for gold futures.

For mentha oil and crude oil futures, the result obtained from the volatility clustering and ARCH- LM test was ambiguous for different contract cycles. Although the result of the ARCH-LM test implied no ARCH effect for the far month of mentha oil and crude oil futures, a trace of volatility clustering was observed in the daily return graph. Hence, we considered the far month contracts of mentha oil and crude oil futures for the GARCH analysis.

Furthermore, the result of the GARCH (1,1) model shows that three parameters, the constant(ω), ARCH (α) term, and GARCH (β) term, are significant at the 1% level. In the variance equation, the estimated β coefficient is greater than the α coefficient, which implies that the volatility is more sensitive to its lagged values. Hence, the volatility is persistent. The sum of these coefficients (α and β) are close to the unit, which suggests that a shock will persist for many future periods. This is particularly true for gold futures of far month contract, in line with the findings of Kumar and Singh (2008).

The volatility clustering effect shows that the crude oil and gold futures markets are rather similar. The crude oil futures market is largely dependent on the global market conditions, which are highly volatile. The spillover effect of global volatility has an impact on the Indian crude oil futures market. Other significant macroeconomic variables (such as the interest rate, exchange rate, and so on, which are fluctuating in nature) have a significant impact on gold futures market in India. Thus, after examining the Samuelson hypothesis and volatility features, we concluded that, out of the selected commodity futures, gold futures are well developed and organized in the Indian market.

Footnotes
1

The aim of this paper is to portrait the simplest form of return volatility of the select commodity futures. Therefore, advanced volatility models (like EGARCH, TGARCH, PGARCH) are not considered, although the inclusion of such models would definitely enrich the present study.

 
2

Data source: www.fmc.gov.in

 
3

The factors affecting the return volatility of commodity futures (like trading volume and open interest) are not under the purview of the present study as that would unnecessarily complicate and shift the focus out of the presented issue.

 
4

Identical results hold for gold futures, for which we test the Samuelson hypothesis using the β term of GARCH (1, 1) model as a measure of volatility, as reported in Table 8.

 
5

The graph for the next near month contract of menthe oil shows volatility clustering although the Jarque-Bera value suggests that the residuals are not normally distributed. In addition, the correlogram shows that the residuals are serially correlated. Therefore we perform the ARCH-LM test and we observe the presence of ARCH effect.

 
6

Although the result of the ARCH-LM test implies no ARCH effect for the far month contract of mentha oil and crude oil futures, a trace of volatility clustering is observed in the daily return graph. Hence, we also consider the far month contracts of mentha oil and crude oil futures for the GARCH analysis.

 

Declarations

Acknowledgements

The authors are indebted to three anonymous referee of this journal for their constructive comments of on the earlier draft of the manuscript. However, the usual disclaimer applies.

Funding

There is no financial assistance received in carrying out this particular research activity.

Availability of data and materials

The dataset is obtained from the publicly available repository, MCX, India website.

Authors’ contributions

BG initiated the thematic concept of the current research while IM carried out the exercise using statistical tools and techniques with the help of EViews 7. Both authors read and approved the final manuscript.

Ethics approval and consent to participate

Not Applicable.

Consent for publication

Not Applicable.

Competing interests

The authors declare that they have no competing interests.

Publisher’s Note

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Economics, The University of Burdwan

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© The Author(s). 2017

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