Conditional higher order moments in metal asset returns

Abstract

This study examines the role of higher order moments in the returns of four important metals, aluminium, copper, gold and silver, using the asymmetric GARCH (AGARCH) model with a conditional skewed generalized-t (SGT) distribution. Implications of higher order moments in metal returns are evaluated by comparing the performances of conditional value-at-risk measures obtained from the AGARCH models with SGT distributions to those obtained from the AGARCH models with normal and student-t distributions. With the exception of gold, the AGARCH model with the SGT distribution appears to have the best fit for all metals examined.

Keywords:

• Metal returns volatility
• Kurtosis
• Skewness
• Value-at-risk

JEL Classification:

• G12

Notes

¹ Daily prices for aluminium (PALU), copper (PCOP), gold (PGOL) and silver (PSIL) are defined as daily closing spot prices. Metal prices are expressed in US dollars per unit/weight and are those reported on the NYMEX. Data on metal prices are obtained from www.Globalfinancialdata.com. Conditional volatility estimates are obtained from the empirical model discussed in this study.

² A detailed discussion of the impact of financialization on commodity prices can be found in Cochran et al. (2012).

³ Financial activity in the commodity futures market is large compared to the size of the physical production of such commodities. In 2005, the volume of exchange-traded derivatives for gold, copper and aluminium was approximately 30 times larger than physical production (see Domanski and Heath 2007).

⁴ See, for example, Hansen (1994), Theodossiou (1998), Harvey and Siddique (1999), Peirό (1999), Jondeau and Rockinger (2001), Giot and Laurent (2003), Brooks et al. (2005), Bali and Theodossiou (2007) and Bali et al. (2008).

⁵ For example, Harvey and Siddique (1999) show that inclusion of the conditional skewness eliminates the leverage effect observed in the conditional volatility of the stock index return. Brooks et al. (2005) found that the return dynamics of both stock and US Treasury bonds are in part influenced by the variance of their variance.

⁶ For a discussion see Brooks et al. (2005).

⁷ Alternatively, VaR can be defined as the minimum loss that can be expected in portfolio value over a given time period with a probability 1 − α.

⁸ See, e.g. Ciner (2001).

⁹ See CME publication http://www.cmegroup.com/trading/metals/files/MetalsRetailBrochure.pdf.

¹⁰ In the SGT density function, the shape parameters are in free form. However, the density function reduces to the distribution functions given above, when certain restrictions are imposed on the shape parameters. A detailed discussion of these restrictions is presented in subsequent sections of this paper.

¹¹ The first two non-centred moments M₁ = δ and M₂ = gθ² can be derived in a similar manner. Together, they can be used to calculate the mean and variance of z, which are and .

¹² The above model can accommodate several extensions. One obvious extension is to allow for asymmetric responses to standardized innovations (in the style of Glosten et al. (1993)) in the skewness and kurtosis equations.

¹³ In the literature, both spot and futures prices are used. In a survey article, Watkins and McAleer (2004) provide an extensive review of the economic analysis of pricing and return models applied to spot and futures markets for non-ferrous metals. The review suggests that futures prices are of particular interest to participants in industries reliant on the production and consumption of metals. Banks, investment funds and speculators are also participants in the futures market. As a result, studies that focus on such issues mainly use futures data, while studies examining the time series properties of metal prices mainly utilize spot price data. It is important to note that under no arbitrage conditions, current futures prices and current spot prices should be strongly correlated. The magnitude of the difference between current futures prices and current spot prices is, for the most part, a function of the convenience yield and storage cost. Finally, a number of studies have found that metal spot and futures prices are co-integrated (see, e.g. McMillian (2005) and Erbil and Roache (2010)). In an efficient market, co-integration is expected between spot and futures prices for the same underlying asset. This is because spot and futures prices do not drift apart in an efficient market. If spot and futures prices in a market do drift apart, it is likely that futures price may not be the best predictor of the next period spot price.

¹⁴ Additional diagnostic tests were performed to guard against the (remote) possibility that the empirical model is misspecified due to non-stationarity in the data. The test procedures include the augmented Dickey–Fuller (ADF) test (see Dickey and Fuller (1979, 1981)) and the Phillips–Perron test (see Phillips and Perron (1988) and Perron (1989)). The test results, reported also in table 1 (columns 11–14), indicate that all metal return series are stationary.

¹⁵ Results are not reported here, but are available from the authors upon request.

¹⁶ As a check on robustness, the GJR–GARCH model is estimated with normal and student-t distributions. The asymmetric volatility coefficients obtained from these models are also negative for all metals.

¹⁷ A long position in a futures contract must have a counterparty holding a short position in the same contract. In equity trades, a long position does not necessarily require a counterparty to hold a short position.

¹⁸ The critical value for rejection of the null using at the 5% level of significance is 3.841.

¹⁹ Updated requirements for capital reserves and market risk management are specified in the Revisions to the Basel II Market Risk Framework (2009).

²⁰ The (z) notations in S_t(z) and K_t(z)are dropped hereafter for convenience.

²¹ The VaR methodology employed here is similar to that in Bali and Theodossiou (2007) in the sense that coefficient estimates from the SGT–AGARCH model are obtained in a first step and the tail distributions using the Cornish–Fisher (1937) expansion are estimated in a second step.

²² Estimates of the coefficients from the AGARCH-n and AGARCH-t models are not presented, but are available from the authors upon request.

²³ For example, n₁₁ is the number of times a VaR violation is followed by another VaR violation.

²⁴ The critical value for rejection of the null using at the 5% level of significance is 3.841.

²⁵ The critical value for rejection of the null using at the 5% level of significance is 5.991.