_{1}

^{*}

The study was intended to reveal the relationship among the spot and future price of crude oil, which in turn will help in determining the prices of crude oil. While structuring a portfolio, high correlation among assets alone cannot be taken as a satisfactory measure for long run diversification paybacks. There is a crucial need to enhance the traditional risk-return modeling methodologies by giving due consideration to common long term trends among the asset prices. Considering this pressing need, the present paper attempts to explore the long run and short run relationship between spot and future prices of crude oil using time series data. To estimate the long and short run dynamics of crude oil prices, the present study applies the Johansen cointegration, and vector error correction modelling to time series analysis.

Asset allocation is perceived as one of the most significant and strategic decision to be taken by an investor or a fund manager. Asset allocation is about the proportion of investment in a class of asset in the portfolio. While deciding the portfolio, the fund manager has to decide on the types of asset to be included in the portfolio [

There is growing interest in finding out the relationship between the spot and future prices of oil and other commodities because of the long term implications resulting from commodity price movements. The commodity market for oil has experienced higher levels of volatility ever since the first oil crises reported in the 1970s. Last few years witnessed record oil prices and climate-change-related interest in biofuels, which in turn have resulted in search for explanations in this area. High commodity prices, whether or not related to oil prices, have obvious effects on purchasing power and economic growth [

The structure of this paper is as follows. Section 2 addresses the relevant literature and the objective of the study. Section 3 shows the model and empirical evidence drawn from the regression, cointegration tests, Granger causality tests and VECM estimations. Conclusions are provided in Section 4.

A good number of empirical studies confirm the low correlation among commodity futures and other asset classes over certain periods of time [

Stelios & Cees [

Maslyuk & Smyth [

Huang, Yang & Hwang [

Ghaith and Awad [

The study attempts to provide evidences on the extent to which the spot and future price of crude oil move together, so that the investor is able to take better investment decisions. The study is conducted, by giving due consideration to the perspective of an investor while analyzing the relationship between spot and future prices. Modern portfolio theory suggests that the relevant information matrix for such an investor includes the expected asset returns, the variability of these returns, as well as cross-asset correlations. Additionally, leads or lags in the time series make correlations almost useless. For example, if the data is lagged by one or two days some of the daily time series, the effect on the correlation between the series will be significant, the correlation might even turn from positive to negative. On the other side, the effect on the common long term relationship between the series will be minimal. Cointegration allows for short term divergence between two different time series, meaning that in a day to day basis, the series does not necessarily have to go up or down at the same time, one might go up while the other goes down, thus there is no need for the two series to move in daily synchrony at all. In the long run, however the two price series cannot wander off in opposite directions for very long without coming back to their long term equilibrium. The distinction between stationary and non-stationary time series is extremely important because stationarity is a precondition to make statistical inferences. If the mean or variance of time series change with time, then it is impossible to generalize results from regressions made for a specific period of time into a different period of time. So, it is necessary to identify whether the time series is stationary or not before making any statistical inference. If one performs regression analysis on time series where the dependent, independent, or both variables have a unit root process, then the results will have no economic significance, in particular, the estimates will be biased and the results of hypothesis tests will be invalid. This is the problem of spurious regression which was first reported back in 1926 by Yule. In order to confirm stationarity of the series, Augmented Dickey Fuller test was conducted. To analyze long-term cointegration, the study made use of the daily closing prices for both spot and future. The study was based on the methods of Johansen’s cointegration analysis. The idea for the analysis is that if two series each follow upward trend, then, in general, they will diverge in the long run. The data analysis comprises of four parts: 1) testing for a unit root; 2) testing for the number of cointegrating vectors in the systems of asset prices, provided the null hypothesis of a unit root for every series is not rejected; 3) testing the vector autoregression between the assets; and 4) testing the causality effect among the two assets.

To test for a unit root in each series, the researcher employed the Augmented Dickey-Fuller (ADF) [

Δ Y t = α 0 + Z t + α 1 Y t − 1 + ∑ α t Δ Y t − 1 + ε t (1)

where α_{0} is constant, t is a deterministic trend, and enough lagged differences (p) are included to ensure that the error term becomes white noise. If the autoregressive representation of Y_{t} contains a unit root, the t-ratio for α_{1} should be consistent with the hypothesis, α_{1} = 0. The results of unit root test are presented in

Reading from

To investigate the existence of a long-term relationship between spot and future prices of crude oil, cointegration analysis was performed. If the spot and future prices are cointegrated with one another, then this will provide statistical evidence for the existence of a long-run relationship. Though, a set of economic series are not stationary, there may exist some linear combination of the variable

S/N | Variables | ADF Fisher Chi-square | Order of Integration | |
---|---|---|---|---|

Critical Values | Probability Values | |||

1 | Crude Oil Spot | −4.6448 | I(1) | |

2 | Crude Oil Spot | −5.1771 | I(1) |

Sources: Unit Root Test, Eviews 8.

which exhibit a dynamic equilibrium in the long run [_{t} is a vector of n stochastic variables, then there exists a p-lag vector auto regression with Gaussian errors of the following form:

Δ Y t = k + γ 1 Δ Y t − 1 + ⋯ + γ ρ − 1 Δ Y t − ρ + 1 + Π Y t − 1 + z t (2)

where γ 1 , ⋯ , γ p − 1 and Π are coefficient matrices, z_{t} is a vector of white noise process and k contains all deterministic elements.

The focal point of conducting Johansen’s cointegration tests is to determine the rank (r) of matrix γ_{k}. In the present application, there are three possible outcomes. First, it can be of full rank, (r = n), which would imply that the variables are stationary processes, which would contradict the earlier finding of nonstationarity. Second, the rank of k can be zero (r = 0), indicating that there is no long-run relationship among the variables. In instances when γ_{k} is of either full rank or zero rank, it will be appropriate to estimate the model in either levels or first differences, respectively.

Finally, in the intermediate case when there are at most r cointegrating vectors 0 ≤ r ≤ n (i.e., reduced rank), it suggests that there are (n − r) common stochastic trends. The number of lags used in the vector Error Correction model is chosen based on the evidence provided by Akaike’s Information Criterion. The cointegration procedure yields two likelihood ratio test statistics, referred to as the maximum eigenvalue (λ-max) test and the trace test, which will help in determining which of the possibilities is supported by the data. Unrestricted Cointegration Trace statistic and Max-Eigen statistic are established to determine whether cointegration exists. The results are given in

Johansen Cointegration results can be studied either on the basis of Trace value or Max Eigen value. From the above table, trace value indicates that there is no cointegration at level as p-value of 0.0000 is less than 0.05 and critical value (15.49471) is less than the trace statistic (63.22908), therefore the study fails to accept the null hypothesis that there is no cointegration equation among the variables. On the similar lines, Max-eigen value also indicates existence of cointegration by rejecting the null hypothesis that there is zero cointegration equations among the variables, with p-value 0.0000 less than 0.05 and critical value (14.36460) is less than the max eigen statistics (38.18780). Therefore, both the tests indicate that there exists cointegration among spot and future prices of crude oil.

If the variables are found to be cointegrated in long run, then the next step is to employ vector error correction model followed by the granger causality. The vector error correction (VECM) is commonly used for forecasting systems of interrelated time series and for analyzing the dynamic impact of random disturbances on the system of variables. The optimum lag length is identified using Akaike Information Criteria (AIC). The VECM approach sidesteps the need for structural

Hypothesized No. of CE(s) | Trace Statistic | 0.05 Critical Value | Probability Value |
---|---|---|---|

None | 63.22908 | 15.49471 | 0.0000 |

At most 1 | 24.45403 | 9.841466 | 0.0000 |

At most 2 | 14.2237 | 7.75366 | 0.0000 |

At most 3 | 8.2251 | 4.81889 | 0.0000 |

At most 4 | 2.91598 | 3.85613 | 0.0033 |

At most 5 | 0.032623 | 2.79707 | 0.0566 |

Source: Johansen Cointegration Test, Eviews 8.

Hypothesized No. of CE(s) | Max-Eigen Statistic | 0.05 Critical Value | Probability Value |
---|---|---|---|

None | 38.18780 | 14.36261 | 0.0000 |

At most 1 | 24.18396 | 3.23142 | 0.0000 |

At most 2 | 13.99862 | 2.07757 | 0.0000 |

At most 3 | 11.30912 | 1.87687 | 0.0000 |

At most 4 | 9.58975 | 1.58434 | 0.0273 |

At most 5 | 0.00109 | 1.13162 | 0.1236 |

Source: Johansen Cointegration Test, Eviews 8.

modelling by treating every endogenous variable in the system as a function of the lagged values of all of the endogenous variables in the system.

Consider two time-series variables, yt and xt. Generalizing the discussion about dynamic relationships to these two interrelated variables yields a system

y t = β 10 + β 11 y t − 1 + β 12 x t − 1 + v t y (3)

x t = β 2 0 + β 21 y t − 1 + β 22 x t − 1 + v t x . (4)

The equations describe a system in which each variable is a function of its own lag, and the lag of the other variable in the system. The results VECM are shown in

No coefficient in the table has probability value of 0.05 or less, this implies there is no short-run relationship among the variables used in the model.

The results of regression model estimated shows that there exist a positive relationship between the spot price and future price of crude oil. Future price of crude oil would be 11.89638 when the spot price of crude oil is assumed to be zero. When spot price increases by 1 percent, Future price of crude oil increases approximately by 88 percent. The model shows that the explanatory variable captured in the model explained 97 percent of the variations as R^{2} is 0.9715. The result of OLS regression model is presented below (

CRUDE _ FUTURE = β 0 + β 1 CRUDE _ SPOT + μ i t (5)

Error Correction | D(CRUDE_SPOT) | D(CRUDE_FUTURE) |
---|---|---|

CointEq1 | −0.736710 | −0.670553 |

(0.30153) | (0.33631) | |

[−2.44321] | [−1.99385] | |

D(CRUDE_SPOT (−1)) | 0.341176 | 0.271840 |

(0.52064) | (0.58068) | |

[0.65531] | [0.46814] | |

D(CRUDE_SPOT (−2)) | 0.825303 | 1.134791 |

(0.51736) | (0.57703) | |

[1.59522] | [1.96661] | |

D(CRUDE_FUTURE (−1)) | −0.138169 | −0.115614 |

(0.46906) | (0.52316) | |

[−0.29457] | [−0.22099] | |

D(CRUDE_FUTURE (−2)) | −0.804160 | −1.113752 |

(0.46236) | (0.51569) | |

[−1.73924] | [−2.15973] |

Sources: Vector Error Correction Model, E Views 8.

Dependent Variable: FUTURE_PRICE | ||||
---|---|---|---|---|

Method: Least Squares | ||||

Variable | Coefficient | Std. Error | t-Statistic | Prob. |

C | 11.89638 | 1.783911 | 6.668704 | 0.0000 |

SPOT_PRICE | 0.879634 | 0.019619 | 44.83672 | 0.0000 |

R-squared | 0.971488 | Mean dependent var | 91.27033 | |

Adjusted R-squared | 0.971005 | S.D. dependent var | 10.09249 | |

S.E. of regression | 1.718537 | Akaike info criterion | 3.953061 | |

Sum squared resid | 174.2487 | Schwarz criterion | 4.022270 | |

Log likelihood | −118.5684 | Hannan-Quinn criter. | 3.980184 | |

F-statistic | 2010.332 | Durbin-Watson stat | 0.504490 | |

Prob(F-statistic) | 0.000000 |

Sources: Regression Analysis, E Views 8.

CRUDE _ FUTURE = 11 . 89638 + 0. 879634CRUDE _ SPOT + μ i t . (6)

The regression coefficient is statistically significant, therefore, it can be concluded that spot price causes future price of crude oil in long run. These results reconfirm the results of cointegration analysis.

The study examined both the short run and long run relationship between spot and future prices of crude oil, so that crude oil futures could be considered as a diversification tool for investors to earn an extra return by using the data across 2009-2014. The analysis shows that there is a strong correlation among the two variables and the two variables are also found to be cointegrated, resulting in evidences for long term relation between the two variables meaning that the two series share a common stochastic move. If a passive investor includes crude oil futures to the traditional crude oil spot, he/she would be able to earn high return in lieu of low risk. The present study supports the diversifying properties of commodity futures. Future research can be conducted in other commodities, so that the results can be more generalized.

Minimol, M.C. (2018) Relationship between Spot and Future Prices of Crude Oil: A Cointegration Analysis. Theoretical Economics Letters, 8, 330-339. https://doi.org/10.4236/tel.2018.83023