key: cord-0870674-fqhpaec1 authors: Gagnon, Marie-Hélène; Manseau, Guillaume; Power, Gabriel J. title: They're back! Post-financialization diversification benefits of commodities date: 2020-05-15 journal: nan DOI: 10.1016/j.irfa.2020.101515 sha: 8b131e698d65510860ad8fe50c7f1d6bc2fc3b09 doc_id: 870674 cord_uid: fqhpaec1 Abstract Do alternative assets such as commodities improve portfolio diversification? The empirical evidence is generally positive but mixed, and almost exclusively focuses on U.S. data. Using several distinct commodity indexes over the period 1993–2019, we investigate the case of an investor in Canada, a commodity-currency country where equities are already exposed to commodity beta. We use spanning tests and several out-of-sample performance measures for both risk-averse and disappointment-averse investors. Overall, we find that while the diversification potential of commodities was limited in Canada before and during financialization, the post-financialization period offers new opportunities. The evidence suggests that portfolio performance is significantly improved using some, but not all, commodity indexes. Thus, the choice of a relevant commodity index matters as a vehicle for diversification. Finally, compounding an international component to the sectorial diversification of the portfolio can significantly improve its performance. J o u r n a l P r e -p r o o f (i.e., the commodity boom and financialization period), research finds that diversification benefits are amplified during volatile periods (Gorton and Rouwenhorst, 2006) . Although most of the literature concerns the U.S., optimal portfolios for U.S. investors may not be the best guide for investors in other countries, especially commodity-currency economies. Research on European representative investors has found that adding commodities lowers portfolio risk and can increase returns . However, for UK investors this may be true for only certain weighted portfolios or indexes of commodities (Giamouridis, Sakkas and Tessaromatis, 2014) . Commodities seem to help European investors even during bear markets (Delatte and Lopez (2013) , in particular gold and silver (Sarafrazi, Hammoudeh and Araújo Santos (2014) . On the other hand, research on commodity-currency countries is very limited. Cheung and Miu (2010) study commodity diversification for U.S. and Canadian portfolios over 1970-2005. They find that commodities help, but only during bull markets, and that benefits are greatest when risk-free rates are low. The literature has expanded to look at emerging markets, in particular China. This research finds that commodities are beneficial except during crisis periods, when they do not in general provide a safe haven, whether for BRICS investors (Berkiros et al. (2017) or China . In China, only soybeans and soybean meal futures seem to be consistently useful (Liu, Tse and Zhang, 2018) while gold provides a safe haven for energy stocks (Wen and Nguyen 2017) . Commodities also help investors in Asia by indirectly gaining international risk exposure (Batten, Szilagyi and Wagner, 2015) . Gaining commodity exposure can involve positions in specific commodity indexes, or in individual commodity futures or portfolios of futures. In general, research finds that individual commodities are highly correlated within groups, so J o u r n a l P r e -p r o o f it is sufficient to examine one representative commodity per group (see e.g., Batten, Szilagyi and Wagner, 2015; Gao and Liu, 2014; and Silvennoinen and Thorp, 2013) . In some cases, however, specific commodities perform better than others, which justifies reviewing such evidence. Overall, the recent literature finds that for U.S. equity investors, newer, thirdgeneration indexes perform better than early-generation indexes over different sample periods (Daigler, Dupoyet and You, 2017; Henriksen et al., 2019; Miffre, 2012; Yan and Garcia, 2017) . Daskalaki, Skiadopoulos and Topaloglou (2017) confirm this finding for both second-and third-generation commodity indexes. Fethke and Prokopczuk (2018) agree that third-generation commodity indexes perform better, but find that their performance is not consistent. Henriksen (2018) concludes that diversifying using older commodity indexes was only beneficial prior to 2011. Some of the literature explicitly considers dynamic models, assuming that investors can generate and exploit short-term forecasts. The findings are consistent with the rest of the literature. Commodities are generally beneficial, but help less during crisis periods (e.g., Fousekis and Grigoriadis, 2019; Öztek and Öcal, 2017) . While dynamic models using forecasts increase portfolio returns, economic gains for investors may be limited due to higher volatility and turnover costs (e.g., Bernardi, Leippold and Lohre, 2018; Lombardi and Ravazzolo, 2016; Pouliasis and Papapostolou, 2018) . Lastly, benefits vary by investor risk aversion (Cotter, Eyiah-Donkor and Potì, 2017; Lin and Zhang, 2019) and investment horizon (Cai et al., 2020) . Research on the benefits of individual commodities is not conclusive, although the value of adding precious metals such as gold seems to be a consistent finding (e.g., Baur and Lucey, 2010) . Demiralay, Bayraci and Gencer (2017) find that lean hogs, feeder cattle, natural gas, orange juice, and gold are beneficial to add to a portfolio, while Bekiros et al. (2016) find that gold, platinum and heating oil are most helpful. Adding energy commodities may lower risk for a commodity J o u r n a l P r e -p r o o f portfolio in the post-financial crisis period (Rehman et al., 2019) . However, indexes that are energy-light or focused on metals perform better for U.S. investors over 1983-2013 than do indexes that load more on energy or agricultural commodities (Bessler and Wolff, 2014) . While precious metals reduce downside risk, they may also lower the portfolio's Sharpe ratios (Bredin, Conlon and Poti, 2017) . Gold is usually considered the best safe haven commodity, but other metals like palladium may perform as well (Agyei-Ampomah, Gounopoulos and Mazouz, 2014) . We use a portfolio optimization methodology based on the approach of Skiadopoulos (2011) (thereafter, DS (2011) ). The methodology allows us to move beyond the mean-variance framework and include higher-order moments for more general investor preferences. In this section, we present empirical evidence on the diversification potential of commodities, using both spanning tests and out-of-sample tests. Regression-based spanning techniques work by adding a risky asset to the initial MV efficient frontier used as a reference point (DeRoon and Nijman, 2001) . This spanning test has the following null hypothesis: the mean-variance frontier of the reference assets increased by the test asset coincides with the reference asset boundary only. Then, no mean-variance investor will benefit from adding the new asset to their optimal portfolio constructed only from reference assets. The technique used here is based on the stochastic discount factor (SDF) framework (DeRoon et al., 1996 (DeRoon et al., , 2003 ). An SDF is such that where is the vector of gross returns of all k assets forming the traditional portfolio (equities, bonds, and risk-free assets), denotes the information available at time t and is a unit vector of dimension k. The SDF is derived from first-order conditions of a J o u r n a l P r e -p r o o f portfolio optimization problem where the investor maximizes the expected utility of his or her terminal wealth (DeRoon and Nijman, 2001) . In this case, the SDF is proportional to the first derivative of the utility function of wealth, given the investor's optimal portfolio choice ( ) where λ is a constant corresponding to the subjective discount factor specific to the individual and is the optimal portfolio weighting vector. M is a set of SDFs allowing the recovery of the price of the k assets in the reference portfolio. It shows that returns of the tested asset are M-spanned by the returns of the reference assets, if and only if where * +, so that W includes the portfolio choices that are valid for an investor, and where the asset portfolio weights sum to 1. This setup assumes that the new asset is M-spanned by the benchmark assets if and only if the return of the benchmark asset can be written as the return of a portfolio containing the benchmark assets with a zero mean error term . where is orthogonal to the set of SDFs M considered. Hansen and Jagannathan (1991) prove that SDFs associated with MV optimization behavior have the lowest variance among all those that are admissible and linear in asset returns. Therefore, equation (3) can be estimated by the following linear regression: The null hypothesis for the spanning test is J o u r n a l P r e -p r o o f Since the model uses excess returns, the slope coefficients of risky assets do not need to add up to 1. The remaining allocation is satisfied by loading on the risk-free asset. These linear restrictions are easy to test using a Wald test. If the set of returns also spans for investors with a non-mean-variance utility function ( ), then the error term in (5) must be orthogonal to the derivative of the marginal utility ( ). Therefore, the set M considered includes MV linear SDFs as well as non-MV utility function SDFs that correspond to different degrees of risk aversion. Equation (2) implies that any value given for the risk aversion coefficient imposes a different SDF that should be included in the set M. Therefore, the spanning test must be performed by examining whether the relative restrictions are valid for each specific risk aversion coefficient value. Following DeRoon et al. (1996 DeRoon et al. ( , 2003 , DS (2011) estimate equation (3) by projecting the returns of the tested asset on the set M of SDFs: where i corresponds to the i-th risk aversion value. The null hypothesis that the test asset is M-spanned by the reference assets is now equivalent to (8) To account for the presence of autocorrelation and heteroskedasticity in the residual term, we use Newey-West (1987) standard errors. Moreover, before running the regression for equation (7), the unobserved regressors (i.e., marginal utilities) must be estimated. To this end, and to obtain the optimal portfolio weights, we assume the investor's preferences are described by a power utility function. This function implies decreasing absolute and constant relative risk aversions, which are desirable properties. The power utility function is defined as: where γ is the risk aversion coefficient (RRA). The optimal portfolio weights are estimated using the generalized method of moments (GMM) (see e.g., Cochrane, 2005) . To implement this approach, the moment conditions generated by our SDFs must be defined. Given the non-MV utility function, equations (1) and (2) imply that the returns of the K reference assets must satisfy the following conditions: where i corresponds to the i th risk aversion value. ( ) corresponds to the SDF which prices each return . The GMM approach estimates the parameters by making sample averages as close as possible to one another. For a sample of size T, the moment conditions ( ) are defined as the sample mean of the errors By definition, for each risk aversion value i, the SDF should fix the price of the three reference assets. Therefore, this approach provides three moment conditions for estimating . Finally, we obtain the GMM estimator of , that is, ( ), minimizing the following quadratic function: where W is a positive definite weighting matrix. Since the number of unknowns is equal to the number of moment conditions, W is equal to the identity matrix. In this section, we describe how we perform out-of-sample portfolio optimization by directly maximizing the expected utility of investors. We assume that investors' preferences are described by the power utility function (eq. 9). To ensure the robustness of our results, we consider several levels of relative risk aversion (RRA = 2, 4, 6, 8, 10) . 3 In addition, to account for more general behavioral characteristics in investor preferences we use the disappointment aversion framework (DA) introduced by Gul (1991) . This J o u r n a l P r e -p r o o f model allows for investors to be loss-averse in addition to risk-averse. It means that at a certain reference point, investors are more sensitive to wealth losses than to gains. That is, they react asymmetrically. Our DA value function is based on the power utility function as in DS (2011), that is: Optimization by direct maximization of the utility function involves solving the following problem: J o u r n a l P r e -p r o o f where ( ) is the joint allocation function of N returns at time t + 1, is the investor's wealth level at time t, is the return of asset i during the period and is the weighting invested in the asset i. The joint allocation function must be estimated and requires assumptions about the estimators or the parametric form of the distribution. This optimization is therefore subject to estimation errors. To circumvent this problem, we use the full-scale optimization (FSO) method proposed by Carmer, Kritzman and Page (2005) and Adler and Kritzman (2007) , as in DS (2011). With this method, each of the periodic T sets of N empirical returns is treated as a future scenario with probability . Utility is computed for each possible vector as well as each scenario in the sample. Vector , having the highest average utility among all the scenarios, is the combination of optimal allocations . Full-scale optimization therefore involves solving the following problem: where Ω contains a budget constraint ( , where ι is a unit vector) and a constraint on weights. These constraints ensure that all funds have been invested and allow shortsales up to a certain level. The formation of the optimal portfolio is dynamic with a rolling-window approach of K months. At each point t in time, the last K observations are used to estimate the asset allocation that maximizes the expected utility function. The weights estimated at time t are used to calculate the out-of-sample return achieved during the period [ ]. This process is repeated until the end of the sample is reached, and a series of T-K monthly out-of-sample returns is obtained. The series of time series returns obtained is then used to compute the out-of-sample performance of the two optimal portfolios. As a robustness exercise, we also perform additional out-of-sample analysis accounting for higher order J o u r n a l P r e -p r o o f moments by estimating the expected utility function using a Taylor second-order series expansion. 6 We use three commonly used performance measures: Sharpe ratio (SR), portfolio turnover, and a return/loss measure of risk-adjusted returns that are net of transaction costs [see e.g., DeMiguel et al. (2009) , Kostakis et al. (2011) , and Daskalaki and Skiadopoulos (2011) ]. The first measure compares the risk-adjusted performance of our investment alternatives. A Memmel (2003) statistic is used to test whether the Sharpe ratios (SR) of the two strategies are statistically different. The estimator ̂ of strategy c is defined as the average of excess out-of-sample returns, ̂ , divided by the standard deviation ̂ : Second, we compute the portfolio turnover rate to quantify the degree of rebalancing required to implement each of the two strategies. The portfolio turnover rate of strategy c is defined as the average absolute change in the weights on the T-K rebalancing periods and on all N available assets: where is the optimal weighting of the portfolio in asset j under strategy j at time t, is the weighting before rebalancing at time t + 1, and is the desired weighting after rebalancing at time t + 1. The turnover rate of the portfolio can be interpreted as the average percentage of the portfolio value that is redistributed over the period. Third, we use a measure of risk-adjusted returns that is net of transaction costs [DeMiguel et al. (2009) ]. Consistent with prior research, we set the transaction cost per share to 50 basis points for equities and bonds, 35 basis points for commodities J o u r n a l P r e -p r o o f futures indices, and 0 for risk-free assets. The net-of-transaction-costs wealth for strategy c is given by: where is the portfolio return achieved prior to rebalancing. Therefore, the net-oftransaction-costs return is defined by (20) Therefore, using this metric of returns that are net of transaction costs, the return-loss measure is interpreted as the additional return required for the traditional investment strategy to perform as well as the commodity-augmented strategy, in terms of the Sharpe ratio. Let , be the monthly average of the RNTC sample of the strategy with the expanded and restricted investment opportunity, respectively, and , be the corresponding standard deviations. Then, the return-loss measure is given by: This is also true for the Commodity Currency Index. The evidence we report is therefore consistent with previous studies, and confirms that commodity indices taken alone underperform other asset classes (e.g., Jensen, Johnson, and Mercer, 2000) . With the exception of the Bloomberg Commodity Index and the Bank of Canada Commodity Ex Energy Index, we reject at the 5% level the null hypothesis that returns are normally distributed (using the Jarque-Bera test). Comparing the U.S. and Canadian trends in indexes underscores the differences between the two markets and further justifies our analysis. Figure 2 further shows, from a Canadian and U.S. perspective, the evolution of the main indices during two sub-periods of interest. Indeed, for the period from January 2005 to June 2008, Daskalaki and Skiadopoulos (2015) conclude that adding commodities to a U.S. investor's portfolio is profitable. This result is not surprising since this period is the longest and the strongest commodity bull cycle since the Second World War (Conceição and Marone, 2008) . This is clearly seen in Panel B. Turning to the Canadian market, the S&P TSX has generally outperformed commodity indices. But given the significant presence of the commodities sector in Canada, the impact of a commodities boom is not the same for the Canadian and U.S. economies. As the Canadian dollar appreciates, this mitigates the impact of the boom. Indeed, prices in the domestic currency increase less than do world prices, the profitability of the export sector increases less than expected, and thus domestic consumers benefit from the appreciation in the form of cheap imports. This section presents and discusses the results of spanning tests where a commodity index is added to the conventional mix of equities, bonds, and risk-free assets. If the spanning condition holds, the asset being tested (here, commodities) does not provide any significant improvement to the tangency portfolio or global minimum-variance portfolio. Thus, it does not improve upon the efficient frontier. The MV spanning hypothesis (eq. 5) and the non-MV spanning hypothesis (eq. 7) are tested separately with a power utility function. The non-MV spanning tests use risk aversion coefficients of different levels (RRA = 2, 4, 6, 8, 10) and a subjective annualized discount factor of 0.95. We consider the full sample (1993-2019) and four sub-periods, namely before, during, and after the financialization period as well as during the global financial crisis. The results are summarized in Table 2 , which reports Wald test statistics, with pvalues in parentheses. This test has the null hypothesis that there is spanning, meaning that the portfolio frontier augmented with the tested asset (i.e., commodities) coincides with the frontier from the reference portfolio. First, we consider the full sample period. We fail to reject the null hypothesis of MV spanning at the 5% level for all indexes considered except the Morningstar commodity currency index. In the non-MV case, we fail to reject the null for all indexes except Morningstar CC (p < .10). However, we find J o u r n a l P r e -p r o o f important differences when we look at each sub-period. In period 1 (before financialization), we generally fail to reject the spanning null hypothesis. However, we reject the null with the BCOMXE in period 2 (during financialization) and in all cases in period 3 (after financialization) for power utility (at the 1% level). These results suggest that period 1 results are driving the full sample findings. For the global financial crisis period, we fail to reject the null hypothesis of MV spanning at the 5% level for all indexes considered, while we reject the null of non-MV at 5% for every index except the MSCC. Thus, the benefits of commodities vary by investor type and by commodity index type, and in addition are time-varying. Note that the full-sample performance of the Morningstar CC index suggests that investors should consider combining international diversification with commodity diversification. During the third sub-period (where commodities perform best), institutional investors retreated from commodity markets, as prices-which partly recovered after the 2008-09 financial crisis and recession-dropped even further in 2014. For example, WTI crude oil prices fell by about 73% between July 2014 and February 2016. 8 As institutional investors usually take long-only positions, falling commodity prices encouraged an outflow of funds. Moreover, commodity index investment returns historically benefited from positive roll yield returns, as commodity futures curves were usually in backwardation. Over the period 2005-2015, however, curves were more often in contango, implying a negative futures roll return (Erb and Harvey, 2016) . Given the outflows of institutional investor positions in commodities, the evidence suggests that correlations with financial instruments have decreased, and therefore that benefits from commodity diversification have increased. The finding that we reject more often for power utility than MV is consistent with DS (2011), and suggests that gains from commodity diversification are greater when we consider the impact of higher moments on expected utility (which by definition MV ignores). This result also corroborates the broader literature's emphasis on risk measures related to skewness and kurtosis (crash risk, asymmetry, jump risk, tail risk), whose importance for investors is now well established. However, the results in period 1 suggest J o u r n a l P r e -p r o o f that adding commodities to the investment portfolio may not always help against crash and tail risk. The reason for our findings differ somewhat from DS (2011) is likely the nature of the Canadian stock market index. This index loads more heavily on commodityrelated industries and firms, which are exposed to commodity beta (e.g, Boons, Roon and Szymanowska, 2014) . This result suggests that the investor should carefully choose the commodity index to yield benefits, particularly if the equity market is already exposed to commodity beta. Utility This section presents results for the out-of-sample (OOS) performance of traditional portfolios and those augmented by a commodity index. All optimal portfolios are constructed to directly maximize investor expected utility (for a range of risk aversion and disappointment aversion specifications). Results are presented in Tables 3 to 5 and include the Memmel test on Sharpe ratios (SRs). The null hypothesis is equality of the SRs for traditional investment opportunities and the commodity-increased portfolio. First, we discuss the results regarding the Bank of Canada (BCIP) indexes (with and without energy), and present these in table 3. Optimal portfolios formed using the BCIP index generate Sharpe ratios that are lower than those excluding commodities. Nearly all differences are statistically significant at the 5% level using the Memmel test. This finding is confirmed in sub-periods. The results improve if we use the index excluding energy (which is a particularly important industry in Canada, especially since the shale gas boom). Overall, however, the investor's portfolio performance is not improved by adding the BCIP index. Next, we consider the Bloomberg commodity index in Table 4 . Sharpe ratios increase and the improvements are generally significant based on the Memmel test (at least at the 10% level and often 5%). In the full sample, improvements are significant for the 72month window, both for the risk-averse and disappointment-averse investor. However, there are important differences across sub-periods, consistent with the evidence from J o u r n a l P r e -p r o o f spanning tests. Performance is significantly better in sub-period 3 for the 36-and 72month windows, and to a lesser extent also in the 60-mo window. However, performance in periods 1 and 2 is either similar or worse (in the 60-month window). The gains from the Bloomberg index are consistent with the spanning tests in terms of sub-period analysis, as they are clustered in the full period and the last sub-period of the sample. Excluding energy has a marginal impact here, as the results are similar. However, the comparison between diversification opportunities with and without energy in the index presented in Table 6 shows the possibility of some gains over the BCOM when including only the energy sector for 60-and 72-month windows for the full sample. During the global financial crisis, excluding the energy sector results in significantly smaller Sharpe ratios than the full index. The main point we make is that there are economically and statistically significant improvements in OOS performance (across measures and utility specifications) at the 60and 72-month horizons when investors use the Bloomberg index. However, performance is usually worse when the Bank of Canada commodity index is used. Thus, consistent with our evidence from spanning tests, we find that commodity diversification tends to work best when combined with international diversification. Indeed while the Bloomberg index is foreign (since U.S.-based), the BoC index is domestic. Indeed, the results suggest that it is better to turn to international diversification at the same time as adding commodities, and that is what the Bloomberg (BCOM) index offers. This finding is related to evidence from Carrieri, Errunza and Sarkissian (2004) who find that investors should use international and sector diversification to improve portfolio performance. Next, we discuss the commodity related foreign exchange indexes. First, we consider the return-loss measure, where a positive value implies that despite a portfolio turnover that is greater than that of a traditional investment portfolio, investors can still earn a higher return by adding commodities to his or her portfolio. We find an improvement for the Bloomberg index for the full sample (36 and 72 mo) and for period 3 (all windows), but performance is worse in periods 1 and 2. The global financial crisis period also shows positive return-loss measures, but gains desappear with disappointment aversion A=0.6. These findings lend support to the main results, as investors benefit more from commodities in the most recent sub-period, post financiarisation (2008-2019P), and for longer windows. Using the Bank of Canada index yields consistently worse performance, except during the global financial crisis for the index excluding the energy sector. This is another indication that excluding energy in the case context can lead to interesting diversification. The commodity-currency index shows mixed but typically weakly negative results, irrespective of the value for disappointment aversion. Finally, portfolio turnover is similar across commodity indices. It is always higher when we add a commodity index, and it is nearly always decreasing in risk aversion and decreasing in the length of the window. However, this relationship is less clearly monotonic under disappointment aversion, as turnover is sometimes first decreasing and then increasing in risk aversion or window length or both. Overall, the results for alternative performance measures are robust and lend strong support to our main findings using Sharpe ratios for risk-averse or disappointment-averse investors. J o u r n a l P r e -p r o o f This paper investigates the diversification potential of several commodity indexes over a long sample period (1993-2019) that includes the pre-, during, and post-financialization phases in commodity markets as well as the global financial crisis. We present evidence using spanning tests and out-of-sample performance measures. Our study focuses on Canada, a commodity-currency market the most opened country in the G-7 for this period and an ideal benchmark for this question because of its closeness with the USA, for which evidence is widely studied in the literature. Thus, this analysis examines the diversification potential of commodities for countries whose economies and currencies are strongly affected by commodities. Our results highlight three main findings. First, while the potential for diversification using commodities had diminished prior to and during financialization, it is now back. We argue that the retreat of institutional investors from commodity markets market-associated with prices falling and volatility increasing--is one of the reasons why correlations are now lower than before. This translates into better diversification potential in the third sub-period we study (2008) (2009) (2010) (2011) (2012) (2013) (2014) (2015) (2016) (2017) (2018) (2019) . This improved diversification in the post financialization sup-period also translates into a reasonable diversification potential using commodities during the global financial crisis. Therefore, not only are commodities back in style, but our evidence suggest that they may offer some diversification against economic turmoil, suggesting a safe-haven characteristic sometimes associated with commodities (see e.g., Miffre and Fernandez-Perez, 2015) . Our results also suggest that, in general, commodity-linked improvements in terms of Sharpe ratios decrease with the investor's level of risk aversion. This finding is in line with recent evidence presented in Lin and Zhang (2019) and Cai et al. (2020) . Thus, our results are most relevant for investors with a lower relative risk aversion, who have the most to gain. Second, we find that the choice of a particular diversification vehicle matters, as the indexes we consider do not contribute the same diversification profile. How much an index loads on the energy sector and how liquid it is appear to be important factors. Our results suggest that the Bloomberg index is superior to the others in terms of diversification for investors in commodity-currency economies. In terms of country J o u r n a l P r e -p r o o f diversification, our results suggest that a lighter loading in the energy sector (which tends to be pro-cyclical) provides some diversification during the global financial crisis. Third and last, our paper provides a strong case for coupling international diversification along with sectorial (commodity) diversification. In our results, we find that the Morningstar commodity currency index improves on the efficient frontier, while regular commodity indexes do not. Furthermore, the U.S. commodity index has better out-of-sample performance than the Canadian commodity index. These results are especially relevant for more sophisticated investors, who are not constrained by the same investment set as institutional investors, and who can profit from the current diversification potential of commodities despite their volatility and strong cycles. Table 2 presents the Wald test statistics and respective values for the null hypothesis: the mean-variance frontier of the reference assets increased by the test asset coincides with the reference asset boundary only. The results are based on monthly observations from February 1993 to April 2019. In each period, the MV spanning hypothesis (eq. 5) and the non-MV spanning hypothesis (eq. 7) are tested separately. The non-MV spanning test is done with a power utility function and use different levels of risk aversion coefficients (RRA = 2, 4, 6, 8, 10) and a subjective annualized discount factor of 0.95. The reference assets consist of the S&P/TSX Composite Index, the S&P Canada Aggregate Bond Total Return Index and the Bank of Canada's 1-month commercial paper rate. All test statistics are based on a Newey-West covariance matrix with five offsets. One, two and three stars indicate significance at the level of 10%, 5% or 1% respectively. J o u r n a l P r e -p r o o f Table 4 presents performance measures (annualized Sharpe Ratio (SR), portfolio turnover, and annualized return-loss) in the case where the expected utility is maximized as part of a power utility function (A= 1) and with a disappointment utility A= 0.6. The values of the Memmel's SR test (2003) -0.6% -0.3% -0.7% 0.0% 0.2% -1.3% -0.6% -0.1% Table 5 presents performance measures (annualized Sharpe Ratio (SR), portfolio turnover, and annualized return-loss) in the case where the expected utility is maximized as part of a power utility function (A= 1) and with a disappointment utility A= 0.6. The values of the Memmel's SR test (2003) are indicated in parentheses below the test. The null hypothesis is that the SR obtained from the set of traditional investment opportunities is equal to that derived from the augmented set of commodities. The results are reported for different rolling window sizes (K = 36, 60, 72 observations) and different relative risk aversion degrees (RRA = 2, 6, 10). Investors have access to commodity investments through the Morningstar Commodity Currency Index TR (MCC). Results are based on monthly observations from February 1993 to April 2019 and prices are in Canadian dollars. One, two and three stars indicate that annualized SR is statistically significantly different at the level of 10%, 5% or 1% respectively. Results are in bold when the commodity-augmented portfolio improves performance compared to the traditional portfolio. J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f Ret.-loss -9.6% -7.4% -5.1% -5.0% -6.9% -2.8% -5.3% -6.9% -4.0% (continued on next page) J o u r n a l P r e -p r o o f Ret.-loss -7.7% -5.5% -3.3% -7.1% -2.5% -1.8% -8.7% -3.5% -2.5% Table 6 presents performance measures (annualized Sharpe Ratio (SR), portfolio turnover, and annualized return-loss) in the case where the expected utility is maximized as part of a power utility function (A= 1) and with a disappointment utility A= 0.6. The values of the Memmel's SR test (2003) are indicated in parentheses below the test. The null hypothesis is that the SR obtained from the augmented set of Bloomberg Commodity Total Return Index (BCOM) is equal to that derived from the augmented set of sub-indices commodities (the Bloomberg Energy Subindex Total Return (BCOMEN) or the Bloomberg ExEnergy Subindex Total Return (BCOMEX)). The results are reported for different rolling window sizes (K = 36, 60, 72 observations) and different relative risk aversion degrees (RRA = 2, 6, 10). Results are based on monthly observations from February 1993 to April 2019 and prices are in Canadian dollars. One, two and three stars indicate that annualized SR is statistically significantly different at the level of 10%, 5% or 1% respectively. Results are in bold when the commodity-augmented portfolio improves performance compared to the traditional portfolio. 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An analysis of stocks, bonds and gold Black swan events and safe havens: The role of gold in globally integrated emerging markets On the time scale behavior of equitycommodity links: implications for portfolio management On the diversification benefits of commodities from the perspective of euro investors On the diversification benefits of commodities from the perspective of Euro investors Maximum diversification strategies along commodity risk factors Do commodities add value in multi-asset-portfolios? an out-of-sample analysis for different investment strategies Investor interest and the returns to commodity investing Boom or Gloom? Examining the Dutch Disease in Two-speed Economies Risk and return in commodity futures The price of commodity risk in stock and futures markets The price of shelter-Downside risk reduction with precious metals Are the gains from international portfolio diversification exaggerated? The influence of downside risk in bear markets Commodities and equities: ever a -market of one‖ Co-movements in commodity markets and implications in diversification benefits Industry risk and market integration Commodity currencies Can exchange rates forecast commodity prices? Diversification benefits of commodity futures Commodity Currencies and Currency Commodities Is now the time to add commodities to your portfolio Predictability and diversification benefits of investing in commodity and currency futures Optimal hedge fund allocations Spicing up a Portfolio with Commodity Futures: Still a Good Recipe Should investors include commodities in their portfolios after all? New evidence Are there common factors in individual commodity futures returns? Diversification benefits of commodities: A stochastic dominance efficiency approach Commodity and equity markets: Some stylized facts from a copula approach Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Time-varying diversification benefits of commodity futures Testing for mean-variance spanning: A survey Hedging pressure effects in futures markets Currency hedging for international stock portfolios: The usefulness of mean-variance analysis Testing for spanning with futures contracts and nontraded assets: A general approach An empirical portfolio perspective on option pricing anomalies The tactical and strategic value of commodity futures Conquering misperceptions about commodity futures investing Can oil prices forecast exchange rates? An empirical analysis of the relationship between commodity prices and exchange rates Is commodity index investing profitable Investment potential of agricultural futures contracts How well can investors diversify with commodities? Evidence from a flexible copula approach The volatility behavior and dependence structure of commodity futures and stocks Benefits of commodity investment The role of commodities in strategic asset allocation A theory of disappointment aversion Identification and estimation of Gaussian affine term structure models Effects of index-fund investing on commodity futures prices Dependence of stock and commodity futures markets in China: Implications for portfolio investment Implications of security market data for models of dynamic economies New evidence on the financialization of commodity markets Properties of long/short commodity indices in stock and bond portfolios Can commodities dominate stock and bond portfolios? World Economic Outlook -Growth Resuming Stock market predictability and industrial metal returns Predictability and under-reaction in industry-level returns: Evidence from commodity markets Efficient use of commodity futures in diversified portfolios Performance hypothesis testing with the Sharpe and Treynor measures Can commodity returns forecast Canadian sector stock returns? What every investor should know about commodities, part II: Multivariate return analysis Determinants of the real exchange rate in a small open economy: Evidence from Canada Transmission of volatility between stock markets Market timing with option-implied distributions : a forward-looking approach Including commodity futures in asset allocation in China On the correlation between commodity and equity returns: implications for portfolio allocation Portfolio management with heuristic optimization Index tracking with constrained portfolios. Intelligent Systems in Accounting Global optimization of higher order moments in portfolio selection Performance hypothesis testing with the Sharpe ratio Long-short commodity investing: A review of the literature The case for long-short commodity investing Financial crises and the nature of correlation between commodity and stock markets Linearly constrained global optimization and stochastic differential equations Skewness in financial returns Volatility and correlation timing: The role of commodities Commodity trade and the carry trade: A tale of two countries Generalized disappointment aversion and asset prices Commodity investing Downside risk, portfolio diversification and the financial crisis in the euro-zone Diversification benefits of commodity assets in global portfolio What is the opportunity cost of mean-variance investment strategies? Management Science Investor flows and the 2008 boom/bust in oil prices Advances in the commodity futures literature: a review Sampling uniformly from the unit simplex Commodity index investing and commodity futures prices Commodity index investing: Speculation or diversification Differential evolution --a simple and efficient heuristic for global optimization over continuous spaces Index investment and financialization of commodities Can investors of Chinese energy stocks benefit from diversification into commodity futures? Economic Modelling Diversification return, portfolio rebalancing, and the commodity return puzzle Can the VAR model outperform MRS model for asset allocation in commodity market under different risk preferences of investors?