Chapter 5 Mean Reversion in Indian Commodities Market

Transcription

1 Chapter 5 Mean Reversion in Indian Commodities Market 5.1 Introduction Mean reversion is defined as the tendency for a stochastic process to remain near, or tend to return over time to a long-run average value. According to Vasicek (1977) short-term interest rate do not increase on average. This is close to the definition of mean reversion that time series will have constant growth rate if they mean revert. Mean-reversion model has more economic logic than the Geometric Brownian Motion(GBM). In the financial-economics literature one can notice several different ways to model the mean-reversion process. It is been observed that commodities prices mean revert or go back to their long term mean after hitting a minimum and a maximum. This phenomenon indicates that commodities prices have a tendency to revolve around their cost of production. A number of research papers provide empirical evidence of mean reversion in commodities market. (See Pindyck & Rotemberg (1990),Dare & Donaldson (1990), Kolb (1994), Irwin,Zulauf & Jackson (1996)). These evidences shows that mean reversion happens quite naturally in the commodities market. Random walk models were used to model commodity prices until mean reverting models started to prove their efficiency as a better commodity price forecaster. Laughton and Jacob (1993&1994), Ross (1995), Schwartz (1997), Cortazar & Schwartz (1997) came out with one factor mean reverting models. Geman and Nguyen (2002) used it for agricultural commodities and Pindyck (2000) for energy commodities. To study mean reversion in the Indian commodity market is a major objective of this study. Present study considers 9 commodities and their spot and future prices for studying mean reversion in the Indian commodity market. The next sections explain the methodology, data and results. 95

2 5.2 Methodology There are many way to study mean reversion. When we look at the formal mathematical definition of mean reversion in asset prices 61, generally it looks in to the presence of three patterns in the return series. Definition 1: An asset model is mean reverting if assets prices tend to fall (rise) after hitting a maximum (minimum) Definition 2: An asset model is mean reverting if returns are negatively autocorrelated Definition 3: An asset model is mean reverting if interest rates (and volatilities), yields or growth rates are stationary. One can verify empirically whether the return series follow these patterns or not and confirm the existence of mean reversion. Among the three definitions, the wider literature emphasize on the third definition for the presence of the mean reversion. Another way of testing for mean reversion can be done using Long Horizon Regression Test. Following Irwin, Zulauf & Jackson (1996), the Long Horizon Regression framework can be explained as follows A close look at mean reversion will compare the present spot/futures prices with its consequent long term mean value and should be able to predict the direction of the change in prices. Say if present price is above the mean value then the next change in price is expected to fall in order to reach equilibrium mean price or mean value. This is the basic idea behind using long horizon test for mean reversion and this proposition has been widely accepted in the literature. (See Fama and French; Cutler, Poteba and Summers (1990) Long horizon regression is the most widely used mean reversion test and according to Campbell (1991) long-horizon regressions have a superior power in predicting prices compared to alternative specifications. The regression equations can be specified as follows : The returns are with spot/ futures prices equation ΔP t = (ln P t ln P t ) 100 (1) 61 Jon Exley,Shyam Mehta,Andrew Smith (2004) Mean Reversion Presented to: Faculty & Institute of Actuaries Finance and Investment Conference Brussels, June

3 P t is the spot/futures price Where ΔP t percentage change in spot /futures prices ln is the natural log operator The mean deviation equation D t = (ln MP t - lnp t ) 100 (2) Where MP t is the estimated mean futures/spot price D t is the percentage deviation between MP t and P t The long horizon regression equation can be stated as ΔP t = α n + β n D t + e t (3) Where e t is the residual β n beta coefficient represent elasticity. If β n >0 evidence of mean reversion α n is the intercept coefficient represent the trend (drift) in spot/futures prices. One can say there is mean reversion if β n is significantly greater than zero. Beta is an important coefficient which is the percentage of deviation from the mean that is eliminated over the n-time horizon. The use of log price permits β n to be interpreted as elasticity. We considered both moving averages and trend estimation technique like filters for estimation of the long-run mean. When we use moving average, at least 3 years (156 weeks) of observations we lose; which is almost 40% of our time series data that is considered for study. Moving averages for shorter periods will not be able to reflect the properties of long run mean. Moreover estimations show filters give decent measure of long-run mean for this particular dataset (see the figures 5.1 to 5.18) both for spot and futures data series. By considering the data limitations, requirements and performance the present study use Hodrick-Prescott filter for estimating trend (long-run mean). A time series P t can be additively decomposed into a trend and business cycle component. Denote the trend component g t and the cycle component c t. Then P t = g t +c t. The HP filter finds a trend estimate, g t and a cycle estimate, c t by solving a penalized optimization problem. 97

4 5.2.1 Data This study looks at weekly prices of nine commodities spot and futures date for a time period of 2005 to Aluminum, copper, gold, led, crude, barley, jeera, guar and pepper are the 9 commodities. Period of data is the same as per table-4.1given in chapter Results Long horizon regression test results are given in the table.5.2. This research tested for mean reversion in nine commodities using both future and spot prices. The estimated β coefficients from all the 18 cases are significant at 1% level. As the coefficients are significantly different from zero the results indicates mean reversion in both spot and future prices of these nine commodities. The estimated α coefficient which indicates the existence of drift (trend) is significant only in three cases. The coefficients are significant for both future and spot prices of gold and spot pepper price. The computed F-statistics are significant at 1% level and it indicates that the regression estimates are valid. Test for stationality is done using Augmented Dickey Fuller (ADF) test which shows that the return series for nine commodities are stationary at 1% significance level see (table 5.1). It confirms that the series are mean reverting as per the definition (3). 5.3 Summary To verify the existence of mean reversion in the commodity prices in the Indian market is a major objective this study. We considered long horizon regression test and stationarity tests for analyzing the mean reverting behavior in the commodity prices. The long horizon regression test results for spot and future prices of 9 commodities shows existence of mean reversion in those price series. The estimated coefficients are highly significant in all the cases. Only in three cases drift coefficients are significant. The unit root test for stationarity using Augmented Dickey Fuller method confirms the mean reversion in the entire price series considered under study. 98

5 Test for stationary Augmented Dickey Fuller (ADF) test for unit roots Table 5. 1 Unit root Test Results for Return Series Unit root Test Results ADF Test statistics Commodities Spot Future Aluminum Lead Copper Gold Crude Barley Guar Jeera Pepper All are significant at 1% 99

6 Table 5. 2 Mean Reversion Long Horizon Regression Test Results for 9 Commodities Future Spot Commodity Coefficients Coefficients α β R 2 F α β R 2 F Lead t-value p-value Copper t-value p-value Aluminum t-value p-value Gold t-value p-value Crude t-value p-value Barley t-value p-value Pepper t-value p-value Guar t-value p-value Jeera t-value p-value

7 Figure 5.1 to Hodrick-Prescott filters for estimating trend in Spot Commodity series Figure 5. 1Aluminum Figure 5. 2 Copper Figure 5. 3 Lead Figure 5. 4.Crude

8 Figure 5. 5 Gold Figure 5. 6 Barley Figure 5. 7 Guar Figure 5. 8 Jeera Figure 5. 9 Pepper