Commodity futures and a wavelet-based risk assessment

Journal Pre-proof Commodity futures and a wavelet-based risk assessment Theo Berger, Robert L. Czudaj

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S0378-4371(20)30114-X https://doi.org/10.1016/j.physa.2020.124339 PHYSA 124339

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Physica A

Received date : 15 November 2019 Revised date : 10 January 2020 Please cite this article as: T. Berger and R.L. Czudaj, Commodity futures and a wavelet-based risk assessment, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2020.124339. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Commodity futures and a wavelet-based risk assessment∗ Theo Berger†

Robert L. Czudaj‡

January 10, 2020

Abstract

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This paper provides an in-depth assessment of commodity futures on applied risk measurement. We provide a thorough empirical study on deconstructed commodity futures returns and present a novel wavelet-based portfolio strategy. First, we examine the dependence structure between commodity futures and show that it is described by different dependence regimes in the short-run, mediumrun and long-run. Then, the out-of-sample portfolio study unveils that daily portfolio management is mostly driven by medium-run and long-run information. Furthermore, we also find that information inherent in long-run trends outperform the information included in short-run trends and underlines the usefulness of the wavelet approach for portfolio management.

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Keywords: Commodity futures, portfolio management, risk measurement, minimum variance, wavelet decomposition JEL Classification: C58, G17, Q47

∗

Thanks for valuable comments are due to two anonymous reviewers and the participants of the 7th International Symposium on Environment and Energy Finance Issues in Paris. † University of Bremen, Department of Economics and Business Administration, D-28359 Bremen, e-mail: [email protected]. ‡ Chemnitz University of Technology, Department of Economics and Business, Chair for Empirical Economics, D-09126 Chemnitz, e-mail: [email protected], phone: (0049)371-531-31323, fax: (0049)-371-531-831323 and FOM Hochschule f¨ ur Oekonomie & Management, University of Applied Sciences, Herkulesstr. 32, D-45127 Essen.

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Introduction

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After the turn of the Millennium the volatility of many commodity markets has increased substantially as shown in Figure 1 for crude oil, gold and wheat futures markets as three prime examples. In the academic literature the financialization of commodities, especially of crude oil, is provided as a major reason for the observed large swings in these markets (Hamilton and Wu, 2014, 2015). It is often argued that the emergence of futures speculators such as commodity index traders or hedge funds has intensified the financialization of commodities over the last two decades (Cheng et al., 2015; Henderson et al., 2015; Basak and Pavlova, 2016). Futures speculators are usually not interested in the commodities for production purposes but solely see them as financial assets for risk management and portfolio diversification (Czudaj, 2019). Previous literature has also

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documented that the financialization of commodities expressed by correlations between commodities and other financial assets has strengthened since the global financial crisis with the bankruptcy of Lehman Brothers in September 2008 as its starting point (Tang and Xiong, 2012; Wen et al., 2012; Silvennoinen and Thorp, 2013; B¨ uy¨ uksahin and Robe, 2014).

*** Insert Figure 1 about here ***

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Therefore, commodity futures today represent an important asset class relevant for

portfolio diversification (Cheng and Xiong, 2014). Following the early works by Johnson (1960) and Levy (1987), which stated that commodity futures trading is in principle not different from stock market trading, some researchers started applying portfolio theory to commodity futures markets (Satyanarayan and Varangis, 1996; Jensen et al., 2000; 1

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Miffre and Rallis, 2007; Fuertes et al., 2010; Cheung and Miu, 2010). You and Daigler (2013) and Daigler et al. (2017) showed the diversification benefits of using individual futures contracts instead of a commodity index relying on optimal Markowitz portfolios. In a recent paper, Yan and Garcia (2017) partly confirm these results but also argue that most commodity futures do little to improve the portfolio’s Sharpe ratio, especially in an out-of-sample context.

The ability to provide reliable return forecasts enhances portfolio risk diversification and wavelet techniques appear to be useful in this context. Such techniques to decompose financial returns into different components characterized as short-run, medium-run and long-run trends have already been shown to improve return forecasts for commodities and other financial assets (Berger, 2016; Faria and Verona, 2018; Risse, 2019). This decomposition mimics the heterogeneity of agents with respect to different con-

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sumption requirements, risk tolerance levels, assimilation of information, institutional constraints and heterogeneous beliefs (Chakrabarty et al., 2015; Czudaj, 2019). The same information might be interpreted differently by investors with different investment horizons. For example, negative information might provide a selling signal for shortterm investors but also offer a buying opportunity for long-term investors. Thus, the main advantage of the wavelet decomposition is that it allows to distinguish between different horizons and this seems crucial since short-run components might be related to speculative trading while long-run components might be attached to long-term supply

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and demand factors.

Therefore, the contribution of the present paper is to examine the benefit of a

wavelet decomposition of commodity futures for portfolio management. In doing so, we implement an out-of-sample approach to analyze the risk diversification potential of the individual frequency scale components. Previous literature has already shed light on the dependence structure across commodity futures and also between them and other 2

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financial assets relying on the wavelet approach (Berger and Uddin, 2016; Cai et al., 2019; Yahya et al., 2019). However, to the best of our knowledge this is the first study which goes one step further and examines the usefulness of the approach for portfolio management. This aspect is of great relevance from an investor’s perspective. We focus on a broad range of different commodity futures, including energy commodities, metals and agricultural commodities to study their relevance for portfolio management. Our findings highlight the usefulness of our wavelet approach for portfolio management and suggest that medium-run and long-run information are most relevant for daily commodity futures portfolio management.

The remainder of the paper is structured as follows. Section 2 presents the wavelet decomposition approach used for portfolio management and Section 3 describes our

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futures data set. Section 4 discusses our empirical results and Section 5 concludes.

Methodology

In this section, we present a methodological approach of wavelet filtering that enables us to deconstruct the underlying return series in order to exclusively focus on isolated short-, mid- and long-run components of the series. Also, we introduce the portfolio allocation algorithm and the relevant quality assessment.

2.1

Maximal Overlap Discrete Wavelet Transform

To deconstruct the futures return series into different short-, mid- and long-run sea-

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sonalities, we draw on Percival and Walden (2000) and apply the maximal overlap discrete wavelet transformation (MODWT) technique and refer to Crowley (2007) for an intuitive introduction to wavelets for economists. The MODWT technique is an expansion of the discrete wavelet transformation (DWT) and enables us to deconstruct return series without loosing the information of time, which is necessary in order to 3

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localize certain events in the deconstructed series.1 Moreover, as described by Gen¸cay et al. (2001), the MODWT technique allows us to achieve deconstructed return series components that are described by the same length as the original time series, which is crucial for our rolling-window out-of-sample assessment.2

Generally, the DWT provides the basis for the MODWT technique and it basically relies on the choice of the wavelet filter. Specifically, let hj,l be the DWT wavelet filter with l = 1, ..., L describing the length of the filter and j = 1, ..., J the level of deconstruction. Then the corresponding scaling filter is determined by gj,l and depends on hj,l by quadratic mirror filtering as given by gl = −1l hl .

Based on this introduced DWT setup, the MODWT approach expands the DWT ˜ j,l ) and scaling filter (˜ filtering framework and the respective MODWT wavelet (h gj,l )

and

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are directly obtained from DWT filters by

˜ j,l = hj,l /2j/2 , h

(1)

g˜j,l = gj,l /2j/2 .

(2)

Wavelet coefficients of level j are then achieved by the convolution of r and the MODWT filters. Let the underlying data be described by daily return series, r = {rt , t = 1, 2, ..., N }, then the deconstruction of the return series can be obtained as

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follows (see Percival and Walden, 2000):

and

fj,t = W Vej,t =

Lj −1

X

˜ j,l rt−l h

mod N ,

(3)

l=0

Lj −1

X

g˜j,l rt−l

mod N .

(4)

l=0

1 The localization of an event describes the main difference between MODWT and Fourier analysis, which is an alternative signal processing technique. 2 Due to boundary conditions, only the observations at the beginning of each series are reduced.

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with Lj = (2j − 1)(L − 1) + 1.

As we aim to utilize MODWT for an out-of-sample portfolio application, we are interested in the maximum number of boundary-free coefficients. Therefore, as described by Berger and Gen¸cay (2018), we use the Haar filter which has the smallest number of ˜ 1,0 = 1 , h ˜ 1,1 = − 1 and g˜1,0 = 1 , g˜1,1 = coefficients leading to h 2 2 2

1 2

for j = 1.

Then, the wavelet coefficients can be expressed in matrix notation at all scales as:

and

fj = ω W ˜j r

(5)

Vej = v˜j r.

(6)

Based on the MODWT specific concept of multi-resolution analysis (MRA), the underlying original futures return series can be described by the sum of all coefficients

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and the smoothed version of decomposition step J: r=

J X j=1

fj + v˜T VeJ = ω ˜ jT W J

J X j=1

e j + SeJ . D

(7)

ej = ω fj gives the detail coefficients and S˜J = v˜T VeJ the corresponding In this setup D ˜ jT W j

e j provides the local details of the smoothed version of the return series. Further, D trend at level j and captures short-term dynamics of the original return series by low

levels whereas long-term fluctuations are described by high levels. Consequently, S˜J is defined as the smoothed version of the time series given by a long-term weighted moving

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average. For instance, Figure 2 illustrates the time series of the individual components e j for WTI crude oil futures returns. D *** Insert Figure 2 about here ***

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In this context, Crowley (2007) and Berger and Gen¸cay (2018) provide an intuitive interpretation of individual scales of economic time series. We adapt this interpretation and define D1 and D2 as the scales that capture deviations related to short-run seasonalities, D3, D4 and D5 as deviations of medium-term seasonalities and D6, D7 and D8 as long-term trends. For a thorough discussion on the interpretation of individual deconstructed time series components of economic return series, we refer to Crowley (2007) and the references therein.

Based on the introduced setup, which allows for a decomposition and reconstruction of the underlying return series, we follow Berger and Gen¸cay (2020) and assess portfolio allocations on different scales. Hence, based on eight decomposition levels of the original futures return series, we achieve nine different portfolio allocations, one based on the

2.2

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original return series and eight competing allocations based on the eight detail levels.

Portfolio Allocation

As presented in Berger and Gen¸cay (2020), we apply the deconstructed return series as an input for the portfolio optimization problem. In order to study competing portfolio allocations, we apply Markowitz (1952) portfolio optimization to take the variance and covariance of every individual scale into account. Furthermore, we focus on the global minimum variance allocation and do not take short selling into account. The portfolio allocation algorithm is then given as follows:

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min wtT Ht wt

s.t. 1TN wt = 1.

wt

(8)

In this setup, portfolio weights wt are exclusively determined by the underlying co-

variance matrix Ht and therefore the differences in portfolio allocations can be described by the different information of the underlying deconstructed return series. Furthermore, we discuss the differences of portfolio allocations in terms of the respective out-of-sample 6

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performance by competing performance metrics. In this vein, we draw on De Miguel et al. (2009) and evaluate the out-of-sample returns by different performance metrics. Specifically, we study risk-adjusted out-ofsample returns and analyze the out-of-sample Sharpe ratio of strategy k: SRk =

µ ˆk . σˆk

(9)

Here, µ ˆk gives the out-of-sample mean return generated by strategy k divided by its sample standard deviation σ ˆk .

Due to potential drawbacks of the Sharpe ratio in the context of evaluating portfolio strategies (Marquering and Verbeek, 2004; Han, 2006), we also apply two measures which add to the information provided by the Sharpe ratio. As introduced by Sortino and van der Meer (1991), we assess the Sortino ratio to take account of the asymmetric

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pattern of financial volatility which can not be captured via Sharpe ratios: SoRk = q P T 1 n

µ ˆk

.

(10)

2 t=1 (min(rk , 0))

This ratio allows us to study the impact of wavelet decomposition on the left tail of the return distribution.

Additionally, as described by Shadwick and Keating (2002), we take into account the Ω-ratio of individual strategy k (ORk ), to study the information in the higher moments

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of wavelet out-of-sample return distributions: R∞ (1 − F (rk ))drk ORk = 0 R 0 . F (rk )drk ∞

(11)

As we deal with daily prices, we set the threshold to 0, which leads us to distinguish between the upside and the downside potential. Also, in order to quantify the differences between competing portfolio allocations,

we apply the information ratio for strategy k (IRk ) as applied by Grinold and Kahn 7

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IRk =

1 n

P

(rk − rb ) µ ˆT E = . σ ˆT E σ ˆT E

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(2000):

(12)

Here, strategy k describes portfolio allocations based on scale k and rk the returns of strategy k. We define portfolio allocations based on daily unfiltered return series as our benchmark b and the respective returns are described by rb . T E gives the tracking error, i.e. the difference between strategy k and the benchmark. µT E and σT E provide the corresponding mean and standard deviation. Hence, this ratio enables us to study the different performances of portfolio allocations based on covariance estimates of deconstructed returns in comparison to covariance estimates based on unfiltered returns.3

Data

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3

We use daily data on closing prices of first nearby futures contracts for 25 commodities traded either at the New York Mercantile Exchange (NYMEX), the Intercontinental Exchange (ICE), the Chicago Board of Trade (CBOT), the Chicago Mercantile Exchange (CME) or the Shanghai Futures Exchange (SHFE). These commodities can be classified into three groups: energy commodities (NYMEX WTI Crude Oil, ICE Brent Crude Oil, NYMEX Heating Oil, NYMEX Natural Gas (Henry Hub) and ICE Gasoil), metals (SHFE Aluminium, NYMEX Copper, NYMEX Gold, NYMEX Palladium, NYMEX Platinum and NYMEX Silver) and agricultural commodities (ICE

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Cocoa, ICE Coffee C, CBOT Corn, ICE Cotton, CME Lean Hogs, CME Live Cattle, CME Class III Milk, CBOT Oats, CBOT Rough Rice, CBOT Soybean Meal, CBOT Soybean Oil, CBOT Soybeans, ICE Sugar No. 11 and CBOT Wheat). The start of the sample period varies substantially across the 25 different futures. To synchronize the 3 For a thorough introduction to out-of-sample assessment of financial portfolio allocations, we refer to De Miguel et al. (2009) and the references therein.

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sample, it has been set to the period from August 4, 1999 to October 17, 2018, which is the longest period for which data for all 25 commodity futures is available. In addition, this enables us to start in the period that was associated with a large increase in financialization of commodities as illustrated in Figure 1 and discussed above. The whole data has been provided by Stevens Analytics via Quandl (https://www.quandl.com/) and continuous non-overlapping end-to-end concatenations of the nearby futures price series have been constructed by rolling over on the last trading day of the expiring or front contract. Daily futures returns have been computed as first difference of the natural logarithm ln(pt ) − ln(pt−1 ), where pt represents the futures price on day t.

Descriptive statistics for the futures returns of each commodity considered in the present study are reported in Table 1. The means of the returns are close to zero in all cases. All returns have heavy tails which is expressed by excess kurtosis (> 3) compared

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to the Gaussian. This feature is most pronounced for agricultural commodity futures returns, especially for lean hogs and rough rice. Most of the futures returns also show a substantially negative skewness. The latter indicates that downturns are often steeper than upturns. Therefore, the normality hypothesis can clearly be rejected for all futures markets according to the Jarque-Bera test.

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*** Insert Table 1 about here ***

4.1

Empirical Findings Dependence Structure

Before we turn to the assessment of our wavelet-based out-of-sample portfolio approach, we start with a descriptive analysis of the dependence structure across commodity fu9

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tures. Figure 3 provides a heatmap plot to visualize the correlation across the 25 commodity futures returns and shows that the correlation is relatively high within the three groups of commodities (i.e. energy, metals and agriculture) but relatively low across the three groups. The latter implies high diversification potential. The highest correlation is observed across energy commodity futures returns. Figure 4 also reports a heatmap plot to illustrate the correlation between the individual frequency scale components of the 25 commodity futures and basically confirms for each individual frequency scale that the correlation is relatively high within the three groups of commodities but relatively low across the three groups. The strongest correlation indicated by red color is observed across energy commodity futures for each frequency scale. Moreover, Figure 4 shows that correlation increases from short- to long-run trends (from D1 to D8) indicated by more warmer colors. The correlation across commodity futures for the

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short-run components (D1 and D2) is very similar to the correlations of the original return series illustrated in Figure 3 and therefore indicates that a portfolio approach based on these components will hardly outperform a portfolio based on the raw returns. The strongest correlation is observed for the eighth scale (D8). This indicates that the correlation regimes vary across scales and therefore across time horizons. In the following, we will exploit these different regimes varying across scales for portfolio diversification. It is also worth noting that commodity futures show different behavior than often observed for stock markets. For the latter the correlation decreases with the

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frequency scale. Different correlation regimes might incorporate different information for portfolio allocations, which could be beneficial for investors. Therefore, as a next step we assess the implications for portfolio management.

*** Insert Figures 3 and 4 about here *** 10

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Wavelet-based Portfolio Performance

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4.2

To assess the performance of our wavelet-based out-of-sample portfolio approach, Figure 5 first of all displays the rolling-window minimum variance portfolio returns based on the individual wavelet scales. Figure 5 basically shows that the variation of portfolio returns is the strongest during the period characterized by the global financial crisis between 2007 and 2009 and also indicates different performance for the individual wavelet scales. The highest spikes into both directions tend to be observed for scales higher than four, which are interpreted as medium-run and long-run components, displayed by red and especially dark red color (or blue and dark blue color, respectively). This is particularly observed for the financial crisis period.

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*** Insert Figure 5 about here ***

As a next step, Table 2 reports performance statistics for our wavelet-based portfolio management study. The table summarizes the performance of portfolios consisting of all 25 assets, for which the portfolio weights have been computed out-of-sample based on either the original returns or their individual wavelet components (i.e. D1, ..., D8). It becomes evident that the performance differs across the different wavelet components. This implies that the extracted trends comprise different information. Some portfolios

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constructed based on the wavelet components are able to outperform the portfolio based on the original return series. Especially, scales three, five and seven (D3, D5 and D7) exhibit a better performance than the raw returns when referring to the mean return and also the risk-adjusted mean return provided by the Sharpe ratio. According to the Sharpe ratio the medium-run component D5 gives the best performance. These 11

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results are also confirmed by asymmetric risk-adjusted mean measures such as the Omega or the Sortino ratio, the maximum drawdown (Max DD) and average tracking errors or information ratios, which compare the corresponding portfolio to a benchmark. As potential benchmarks we use the equal weights portfolio (eq) and the portfolio constructed based on the raw returns (mv). All these measures show that portfolios relying on medium-run or long-run information outperform portfolios based on shortrun information or based on all information inherent in the original returns. Therefore, excluding information related to short-run noise (D1 and D2) from the original time series is fruitful for applied portfolio management. This indicates that our approach is beneficial from an investor’s perspective and remarkably shows a clear difference of commodity futures markets compared to stock markets in portfolio management. For stock markets the short-run information provided by the first scales often shows a much

2020).

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better performance than medium-run or long-run information (Berger and Gen¸cay,

*** Insert Table 2 about here ***

To achieve robustness of our findings and to analyze commodity-specific patterns, we have also conducted the same analysis restricting the portfolio to the three sub-groups

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of commodities (i.e. energy, metals and agriculture). Tables 3 to 5 report the corresponding results. First, Table 3 reports performance statistics for energy commodity futures portfolios and also shows that the medium-run component D5 provides the best performance. However, for this sub-group of commodities the short-run component D1 also provides useful information. Second, Table 4 shows the summary statistics for metal futures portfolios and indicates that for this sub-group all portfolios constructed 12

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based on individual wavelet components are able to outperform the original returns portfolio except of the first two components D1 and D2. This implies that investors can benefit from removing the short-run trends from the returns before constructing the minimum variance portfolio. The best performance is provided by the seventh scale (D7). Finally, Table 5 gives the corresponding statistics for agricultural commodity futures and roughly confirms the findings that components D3, D5, D6 and D7 are able to outperform raw returns while the medium-run component D5 again shows the best performance.

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*** Insert Tables 3 to 5 about here ***

Overall, our main findings are as follows. The performance differs across the different wavelet components and this implies that the extracted trends comprise different information, which are useful for portfolio investors. In most cases the medium-run component D5 provides the best performance in terms of higher returns, which are also adjusted for risk according to different measures. But any other medium-run or long-run components are also able to outperform the raw returns. Therefore, investors can benefit from removing short-run trends from the returns before computing portfolio weights. This indicates that information inherent in medium-run and long-run trends

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outperform the information included in short-run trends and underlines the usefulness of the wavelet approach for portfolio management. Except for energy commodity futures the information inherent in short-run trends provides limited relevance for applied daily portfolio management. This provides a remarkable difference compared to the performance of wavelet-based portfolio management strategies on stock markets,

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and Gen¸cay, 2020).

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which usually show a much better performance based on short-run components (Berger

Our results show patterns, which vary across the three groups of commodity futures markets to some extend. However, all three groups have in common that the raw returns can be outperformed by any specific wavelet components. For energy and agricultural commodity futures markets the medium-run component D5 performs best and for metal futures the seventh scale D7 turns out to be the best component. Therefore, we conclude that the decomposition into individual scales is beneficial from a portfolio management perspective and that a portfolio manager can mostly benefit from information inherent in the medium-run and the long-run components D5 and D7, respectively.

5

Conclusion

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This study provides a thorough in-depth assessment of the dependence structure across commodity futures and shows that these are described by different dependence regimes in the short-run and in the long-run components. Therefore, we propose a waveletbased out-of-sample portfolio management approach, which unveils that the mediumrun trend is the most relevant for daily applied portfolio management. In addition, we show the usefulness of the wavelet decomposition of commodity futures returns for out-of-sample portfolio management. This has direct implications for portfolio investors, which can benefit from decomposing the futures returns into individual trends

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and extracting short-run trends before constructing portfolios based on the inherent information.

In this vein, our results provide empirical evidence that for commodity futures, the

information stored in the medium- and long-run covariance matrix leads to superior portfolio performance in terms of the applied portfolio metrics. Therefore, we conclude

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that excluding information related to short-run noise (D1 and D2) from the original time series is fruitful for applied portfolio management, which aims at minimizing overall portfolio risk. Interestingly, in comparison to stock prices, different information components are relevant. Berger and Gen¸cay (2020) provide evidence that for stock returns, it is the short-run information that comprises the relevant information for applied portfolio management. However, for commodity future prices our results indicate that this finding is not valid, which demonstrates a clear difference between these two asset classes. Here, our results suggest that for commodity futures, short-run information components seem to be of minor importance for applied risk management indicating that asset specific middle- and long-run seasonalities describe relevant information.

Our wavelet-based portfolio management approach offers the potential to be applied to many other assets of asset classes such as e.g. for trading cryptocurrencies. This

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provides further avenues for future research.

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Tables and Figures Table 1: Descriptive Statistics µ

σ

Max

WTI Brent Heating Oil Natural Gas Gasoil

0.00019133 0.00023283 0.00023718 5.55E-05 0.00024746

0.02367701 0.02188412 0.02256456 0.03375587 0.01989682

0.16409725 0.12706595 0.10536052 0.32407673 0.10731375

Aluminium Copper Gold Palladium Platinum Silver

-3.62E-05 0.00025852 0.00029741 0.00014507 0.0001221 0.00021772

0.00830832 0.01713655 0.01119313 0.02082224 0.01455242 0.01942556

0.04912579 0.11554965 0.08624971 0.15252981 0.1076167 0.12469481

Cocoa Coffee Corn Cotton Lean Hogs Live Cattle Milk Oats Rice Soybean Meal Soybean Oil Soybeans Sugar Wheat

0.00022292 1.98E-05 0.00011191 6.98E-05 -2.45E-05 0.00010206 9.44E-05 0.00020872 0.00014365 0.00014756 0.00013137 0.00011925 0.00018907 0.00014789

0.01953925 0.02121546 0.01813018 0.01914411 0.02162576 0.01114664 0.0168435 0.02220369 0.0172834 0.01815105 0.01482451 0.01572363 0.02208962 0.02009233

0.10269029 0.16392813 0.09801349 0.16710119 0.2505378 0.09123685 0.15921398 0.1157105 0.2671297 0.07528306 0.08716786 0.07542985 0.23547036 0.12929328

Min

Skewness

Kurtosis

JB-stat

LB-stat

LM-stat

-0.16544513 -0.14437161 -0.19633229 -0.1979969 -0.14157877

-0.12884692 -0.19561081 -0.41885291 0.52197554 -0.09317234

7.12694603 6.07258038 7.81191467 8.69647811 5.5718927

3339.11442 1873.59009 4658.93147 6550.02327 1298.56389

56.4507945 63.7909339 68.4542965 81.7671915 36.7711301

10.9020913 15.4404706 12.5121779 17.8381254 5.18368421

-0.05152724 -0.11709267 -0.09810481 -0.13201687 -0.09603311 -0.19497647

-0.31776376 -0.18146177 -0.27179219 -0.27444149 -0.4321132 -0.89891703

8.98123599 6.86891382 8.39693764 7.02682411 7.22305694 10.8307323

7065.47267 2948.94472 5745.95342 3225.5514 3628.73344 12606.58

137.290352 97.4376773 64.3275065 86.4698732 58.6863648 51.6168925

36.3602803 20.280669 7.54817995 29.7527647 11.8050942 6.93204337

-0.12513656 -0.13385103 -0.24528605 -0.271398 -0.26369797 -0.11937771 -0.15020026 -0.20017107 -0.21917529 -0.25400713 -0.0723905 -0.1408307 -0.18038169 -0.10016707

-0.2081147 0.28159743 -0.33792521 -0.481099 0.0378776 -0.49619629 -0.10499858 -0.44159535 0.6771862 -1.09640592 0.18754066 -0.65338297 -0.00757833 0.27991651

5.58369649 6.56539253 12.3951363 15.2989885 29.5977345 11.7197129 17.9045427 9.07031802 22.9806358 14.943618 5.23555904 8.18631191 9.11962739 5.2921818

1337.50077 2544.4966 17327.3235 29721.6232 138158.156 15041.0198 43391.756 7348.58847 78323.7013 28797.4035 1003.48869 5586.40962 7313.68642 1087.28839

44.1478058 60.9531636 55.336904 52.1487563 36.5989783 47.6756683 101.756818 62.6289194 59.2597659 63.3390163 37.06616 59.7715615 48.1693005 52.6927804

7.57026347 11.5617197 8.58542413 8.65968545 13.9536853 19.9708003 16.5093104 26.9892004 21.015675 11.1440145 2.96852774 9.83507752 13.3636969 5.67500836

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Note: This table presents the descriptive statistics of 25 commodity futures returns considered in the present study for a daily sample period running from August 4, 1999 to October 17, 2018. µ and σ denote the sample average and standard deviation for each market. Min and Max describe the minimum and maximum for each futures market. JB-stat provides the Jarque-Bera statistic for testing the null of normality. For the normal distribution, the Kurtosis is 3, the Skewness is 0 and the Jarque-Bera statistic (JB-stat) is 5.44. LB-stat gives the Ljung-Box test statistic for testing serial correlation up to a lag length of 10. LM-stat represents the Lagrange multiplier test statistic for serial correlation with respect to heteroscedasticity.

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Table 2: Wavelet-based portfolio management: Out-of-sample performance All assets

Original

D1

D2

D3

D4

D5

D6

D7

D8

Mean Min Max Var SR OR SoR Max DD IReq T Eeq IRmv T Emv

0.00015285 -0.03311159 0.03336567 2.64E-05 0.02973028 1.08635101 0.04317942 0.36708302 -0.00554487 0.00657409

0.00014072 -0.03552659 0.03435025 2.71E-05 0.02704669 1.07835538 0.03889863 0.38617224 -0.00744518 0.00652493 -0.01639455 0.00073968

0.00013844 -0.03234463 0.03471646 2.73E-05 0.02647592 1.07648381 0.03846782 0.3725853 -0.00780975 0.00651266 -0.02106205 0.00068416

0.00015833 -0.03415889 0.03565517 2.79E-05 0.02999108 1.08747611 0.04351207 0.37234512 -0.00465967 0.00664725 0.00534102 0.00102574

0.00014554 -0.0345054 0.03988241 2.93E-05 0.02688812 1.07778342 0.0391435 0.3715236 -0.00664994 0.00658057 -0.00509303 0.00143489

0.00020496 -0.03366595 0.04131771 3.10E-05 0.03681854 1.10600024 0.0538308 0.33616486 0.00248422 0.00630239 0.02480738 0.00210054

0.00015189 -0.03356836 0.0379981 3.39E-05 0.02608334 1.07360192 0.03735957 0.33885873 -0.00591358 0.00632574 -0.00035303 0.00270601

0.00022106 -0.03782458 0.04628407 4.02E-05 0.03485571 1.10094198 0.04993876 0.34417403 0.00511242 0.00621231 0.01802834 0.00378362

0.00012914 -0.03971387 0.04636184 4.50E-05 0.01924547 1.05410781 0.02732851 0.34129744 -0.0090322 0.00666132 -0.00558581 0.00424539

Note: This table presents the performance statistics of the portfolios consisting of 25 commodity futures returns considered in the present study for a daily sample period running from August 4, 1999 to October 17, 2018. The portfolios are constructed based on the original return series and the individual wavelet scales denoted by D1, ..., D8. Mean, Min, Max and Var denote the sample average, minimum, maximum and variance for each portfolio return. SR, OR and SoR represent the Sharpe ratio, the Omega ratio and the Sortino ratio described in Section 2.2. Max DD gives the Maximum Drawdown. Info and Tracking stand for the information ratio and the tracking error of the corresponding portfolio compared to a benchmark. As benchmarks we use two different portfolios: the equal weights portfolio (eq) and the Markowitz portfolio constructed based on the original return series (mv).

Table 3: Wavelet-based portfolio management: Out-of-sample performance for energy commodities Mean Min Max Var SR OR SoR Max DD IReq T Eeq IRmv T Emv

Original 0.0003 -0.0805 0.0817 0.0003 0.015 1.041 0.021 0.878 -0.004 0.007

D1

D2

D3

D4

D5

D6

D7

D8

0.0003 -0.0792 0.0814 0.0003 0.016 1.043 0.022 0.869 -0.002 0.007 0.013 0.001

0.0002 -0.0819 0.0861 0.0003 0.014 1.039 0.020 0.883 -0.005 0.007 -0.006 0.001

0.0002 -0.0841 0.0892 0.0003 0.014 1.038 0.020 0.893 -0.006 0.006 -0.008 0.002

0.0002 -0.0837 0.0811 0.0003 0.014 1.038 0.020 0.892 -0.006 0.007 -0.006 0.003

0.0003 -0.0868 0.0794 0.0003 0.016 1.044 0.023 0.883 0.001 0.007 0.006 0.005

0.0003 -0.1033 0.0961 0.0003 0.015 1.041 0.021 0.866 0.000 0.008 0.003 0.006

0.0002 -0.1045 0.1480 0.0004 0.010 1.028 0.014 0.895 -0.010 0.008 -0.006 0.009

0.0002 -0.1671 0.2687 0.0004 0.009 1.026 0.013 0.923 -0.007 0.012 -0.005 0.012

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Energy

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Note: This table presents the performance statistics of the portfolios consisting of energy commodity futures returns for a daily sample period running from August 4, 1999 to October 17, 2018. The portfolios are constructed based on the original return series and the individual wavelet scales denoted by D1, ..., D8. Mean, Min, Max and Var denote the sample average, minimum, maximum and variance for each portfolio return. SR, OR and SoR represent the Sharpe ratio, the Omega ratio and the Sortino ratio described in Section 2.2. Max DD gives the Maximum Drawdown. Info and Tracking stand for the information ratio and the tracking error of the corresponding portfolio compared to a benchmark. As benchmarks we use two different portfolios: the equal weights portfolio (eq) and the Markowitz portfolio constructed based on the original return series (mv).

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Metals Mean Min Max Var SR OR SoR Max DD IReq T Eeq IRmv T Emv

Original 0.0001 -0.0425 0.0415 0.0000 0.021 1.062 0.028 0.491 -0.010 0.008

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Table 4: Wavelet-based portfolio management: Out-of-sample performance for metals D1

D2

D3

D4

D5

D6

D7

D8

0.0001 -0.0452 0.0409 0.0000 0.019 1.056 0.026 0.516 -0.011 0.008 -0.011 0.001

0.0001 -0.0430 0.0400 0.0000 0.020 1.060 0.028 0.491 -0.010 0.008 -0.006 0.000

0.0002 -0.0453 0.0404 0.0000 0.023 1.068 0.031 0.496 -0.008 0.008 0.017 0.001

0.0002 -0.0430 0.0430 0.0000 0.022 1.065 0.030 0.492 -0.009 0.008 0.008 0.001

0.0002 -0.0441 0.0471 0.0000 0.022 1.065 0.031 0.484 -0.008 0.008 0.007 0.001

0.0002 -0.0490 0.0469 0.0000 0.023 1.068 0.031 0.441 -0.007 0.008 0.009 0.002

0.0002 -0.0664 0.0527 0.0001 0.029 1.086 0.040 0.427 -0.001 0.007 0.022 0.003

0.0002 -0.0599 0.0512 0.0001 0.025 1.074 0.034 0.476 -0.002 0.008 0.014 0.005

Note: This table presents the performance statistics of the portfolios consisting of metals commodity futures returns for a daily sample period running from August 4, 1999 to October 17, 2018. The portfolios are constructed based on the original return series and the individual wavelet scales denoted by D1, ..., D8. Mean, Min, Max and Var denote the sample average, minimum, maximum and variance for each portfolio return. SR, OR and SoR represent the Sharpe ratio, the Omega ratio and the Sortino ratio described in Section 2.2. Max DD gives the Maximum Drawdown. Info and Tracking stand for the information ratio and the tracking error of the corresponding portfolio compared to a benchmark. As benchmarks we use two different portfolios: the equal weights portfolio (eq) and the Markowitz portfolio constructed based on the original return series (mv).

Table 5: Wavelet-based portfolio management: Out-of-sample performance for agricultural commodities Mean Min Max Var SR OR SoR Max DD IReq T Eeq IRmv T Emv

Original 0.0001 -0.0342 0.0507 0.0000 0.020 1.054 0.029 0.346 -0.002 0.006

D1

D2

D3

D4

D5

D6

D7

D8

0.0001 -0.0347 0.0499 0.0000 0.018 1.050 0.026 0.354 -0.004 0.006 -0.014 0.001

0.0001 -0.0331 0.0529 0.0000 0.019 1.053 0.028 0.343 -0.002 0.006 -0.004 0.001

0.0001 -0.0351 0.0636 0.0000 0.021 1.059 0.031 0.353 0.001 0.006 0.010 0.001

0.0001 -0.0355 0.0606 0.0001 0.017 1.046 0.024 0.337 -0.005 0.006 -0.009 0.002

0.0002 -0.0348 0.0621 0.0001 0.025 1.071 0.037 0.347 0.006 0.006 0.020 0.002

0.0002 -0.0404 0.0566 0.0001 0.020 1.056 0.030 0.349 0.002 0.006 0.006 0.003

0.0002 -0.0492 0.0545 0.0001 0.024 1.066 0.034 0.388 0.008 0.006 0.013 0.004

0.0001 -0.0442 0.0517 0.0001 0.017 1.047 0.025 0.414 0.000 0.007 0.002 0.005

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Agriculture

Jou

Note: This table presents the performance statistics of the portfolios consisting of agricultural commodity futures returns for a daily sample period running from August 4, 1999 to October 17, 2018. The portfolios are constructed based on the original return series and the individual wavelet scales denoted by D1, ..., D8. Mean, Min, Max and Var denote the sample average, minimum, maximum and variance for each portfolio return. SR, OR and SoR represent the Sharpe ratio, the Omega ratio and the Sortino ratio described in Section 2.2. Max DD gives the Maximum Drawdown. Info and Tracking stand for the information ratio and the tracking error of the corresponding portfolio compared to a benchmark. As benchmarks we use two different portfolios: the equal weights portfolio (eq) and the Markowitz portfolio constructed based on the original return series (mv).

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Figure 1: Commodity futures prices

The plots show the futures prices for WTI crude oil (upper panel), gold (middle panel) and wheat (bottom panel) for a daily sample period running from March 30, 1983 to October 17, 2018.

Crude Oil Futures 150

100

50

1984

1987

1990

1993

1996

1999

2002

2005

rna lP

Gold Futures

2008

2011

2014

2017

2020

2008

2011

2014

2017

2020

1500

1000

500

1984

1987

1990

1993

1996

1999

2002

2005

Wheat Futures

1250

Jou

1000

750

500

250

1984

1987

1990

1993

1996

1999

22

2002

2005

2008

2011

2014

2017

2020

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Figure 2: Wavelet decomposition for WTI crude oil futures returns

The plots show the original return series for WTI crude oil futures (at the bottom) and the individual components of its wavelet

decomposition into eight scales denoted by D1, ..., D8 for a daily sample period running from August 4, 1999 to October 17, 2018.

D1

10 5 0 −5 −10

D2

5 0 −5

D3

3 0 −3 −6

D4

2 0 −2

D6 D7

2005

2010

Jou

2000

23

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D8

10 0 −10

D5

1 0 −1 1.5 1.0 0.5 0.0 −0.5 −1.0 1.0 0.5 0.0 −0.5 0.8 0.4 0.0 −0.4

2015

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Figure 3: Correlation between futures return series for all commodities

The plot visualizes the correlation matrix across the futures returns for all 25 commodities considered in this study for a daily sample period running from August 4, 1999 to October 17, 2018. The linear correlation coefficient ranges from -1 to 1 while the color scale

represents the absolute level of correlation from high (1) to low (0) displayed by a scaling from dark red to dark blue. The individual

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commodity futures are ordered in the same way as given in Table 1.

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Figure 4: Correlation between deconstructed futures return series

The plots visualize the correlation matrices across the individual wavelet scales of the futures returns for all 25 commodities considered in this study for a daily sample period running from August 4, 1999 to October 17, 2018. The linear correlation coefficient ranges from -1 to 1 while the color scale represents the absolute level of correlation from high (1) to low (0) displayed by a scaling from dark red to dark

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blue. The individual commodity futures are ordered in the same way as given in Table 1.

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Figure 5: Out-of-sample wavelet-based portfolio performance

The plot reports the portfolio returns of the minimum variance portfolio according to Markowitz (1952) containing of all 25 commodity futures considered in this study for a daily sample period running from August 4, 1999 to October 17, 2018. The portfolios are

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constructed out-of-sample based on a rolling-window approach relying on the individual wavelet scales.

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Research Highlights • We present a novel wavelet-based portfolio strategy for commodity futures. • Commodity futures are described by different dependence regimes.

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• Daily portfolio management is mostly driven by medium-run and long-run information.

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• News inherent in long-run trends outperform the news included in short-run trends.

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Declaration of interests

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Robert Czudaj and Theo Berger