### Are Copper Prices Mean Reverting? A Practitioner’s Point Of View

ESIC Business & Marketing School

June 17, 2012

**Abstract: **

It is difficult to decide which model to use to represent the price of copper in the projection of cash flows in capital budgeting analysis. In order to aid this decision, the following paper compares the methodologies most commonly used to detect random walk or mean reversion in a financial series of prices in order to prove whether or not the series is stationary.

The methods for analysing stationarity this research covers are classical Unit Root analysis and Variance Ratios. We also include two other methods, namely simulating the series using the information of the original one both with Geometric Brownian Motion and Mean Reversion and also Mean Reversion with Jumps. First, we compare the correlation of the series with the simulated one and then perform Principal Component Analysis in order to compare the similitude between the original series and the simulated ones.

There are some contradictions between the different methods, but important information is obtained for decision making in regard to the model to use in the projection.

### Are Copper Prices Mean Reverting? A Practitioner’s Point Of View – Introduction

The idea of the present analysis comes from the necessity of having a model of copper prices for the valuation of copper mines, both using simulated and Real Options. When such an analysis is performed, one of the main drivers of future cash flows are copper prices, because this is the most uncertain variable and it accounts for the highest value in cash flow per year.

Mining companies usually have a clear estimation of annual production and the total number of years of production, which can range from a few to as many as 30 or 40. There are models based on information from futures markets, such as the Schwartz (1997) model used in Moel and Tufano (1998), but this kind of model cannot be implemented in long-term projects because of the lack of futures contracts to match the project. What is more, even having this kind of long-term contract the lack of liquidity of the contracts do not incorporate all the information needed for the valuation.

The alternative to this kind of model is the family of Geometric Brownian Motion (GBM), the Mean Reversion (MR) and Mean Reversion with Jumps models, including all the different variations that have appeared over the time. One such example is the Heston (1993) model, which perhaps captures the price dynamics better. However, due to the difficulty involved in estimating the parameters and also the need for information on derivatives, we have decided to only use the basic models.

The next step is to decide whether copper price behaviour will be projected using a GBM or MR model, that is to say, the price has either no memory and is a martingale or some memory and corresponds to a stationary AR family model.

Intuitively, and looking at copper prices over the last few years, one might think behaviour is mean reverting and that prices fluctuate around a mean, as Graph 1 shows.

**Literature Review**

There is a huge body of literature in the field of random walk analysis of prices of different financial assets and markets, mainly focusing on detecting the fulfillment of the weak level of the Efficient Market Hypothesis (EMH).

Watkins and McAleer (2004)reviewed papers published on the spot and futures markets for non-ferrous metals from 1980 to 2002,while 13 out of 45 focused on the EMH (28%).

Poterba and Summers (1988) tested 20 stocks for randomness in the US and found evidence of mean reversion. Lo and MacKinlay (1988) used Variance Ratio tests to investigate the random walk hypothesis for many indices finding mixed results. Wahab (1995) used the ADF test and did not reject random walk for gold or silver futures. Barkoulasetal (1997) applied the Phillips-Perron(PP) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit root tests on 20 commodities and one composite index. The random walk hypothesis was rejected for copper by the PP test, but the stationarity hypothesis (KPSS) test was rejected for all commodities except copper, jute, wool and zinc.

Fraser and Andrew (1998) studied the behaviour of asset prices in terms of the EMH. They used the variance ratio test to investigate the departure from randomness in the form of mean-reversion (short memory) and persistence (long memory). Their conclusion was that aluminium, Brent, copper and gas oil prices were not random.

Engel and Valdes (2001) analysed copper prices from two perspectives, finding random walk behaviour in the short term (less than a year) and an AR(1) for the longer term. In the case of Watkins and McAleer (2003), the ADF test accepts the random walk for copper and the subseries created from the structural breaks.

Andersson (2007) compares the Lo-MacKinlay Variance Ratio, PP and KPSS using hedging errors modelling the series both as Mean Reversion and GBM. The mean reversion process provided the smallest mean absolute error for 125 out of 162 series, that is, 77%. The Lo-MacKinlay Variance Ratio rejects 117 series as random walk with respect to 280, PP only 39 out of 280 and KPSS rejects mean reversion in 247 out of 280 cases.

Borges (2010) reports the results of the random walk hypothesis test on European stock markets using the runs, different Variance Ratio, Lo and Mackinlay, Chow and Denning (1993) and Kim and Shamsuddin (2008) tests. In some periods, the random walk hypothesis is rejected while in others it is accepted.

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