## Abstract

Over the past two decades, commodity-linked products have become increasingly popular as an alternative asset class for many investors. This is mainly because long positions in commodities potentially offer diversification benefits in portfolios comprising other more traditional assets. Commodities also arguably provide equity-like returns and act as an inflation hedge. However, there is an ongoing debate concerning long-only positions in commodities as an adequate investment vehicle, especially given observed low or negative returns and increased correlation with other more common asset classes, as given periodically. The purpose of this article is to analyze the performance effects of including long/short commodity indices in more conventional stock–bond portfolios using out-of-sample tests over different periods and by employing different asset allocation strategies. Overall, the author identifies benefits of commodity inclusion from 2002 to 2011, but these benefits largely disappear from 2011 to 2015. An intriguing finding is that Henriksen observes a much lower risk–adjusted performance for the commodity indices after their launch date.

Recently, commodity investments have been the subject of heavy debate by both academics and financial industry practitioners. This article analyzes the performance effects from adding different long/short commodity indices to the more traditional stock and bond portfolio. We perform out-of-sample back tests under different market conditions and with different asset allocation techniques.

During the past two decades, there has been strong growth in commodity-linked derivative investments. In evidence, from the end of the 1990s to 2007, the number of open contracts on commodity exchanges almost doubled, resulting in volumes of exchange-traded derivatives of 20 to 30 times physical production for many commodities (Stoll and Whaley [2009]; Silvennoinen and Thorp [2013]). Furthermore, institutional holdings of commodity derivatives increased from $15 billion in 2003 to just over $200 billion in 2008 (Commodity Futures Trading Commission [2008]) and $435 billion in 2012 (Miffre [2014]).^{1}

This growing interest in commodity investments may result from the common perception that long positions in commodities have low correlations with other more traditional assets. This could relate to the drivers of commodity prices; that is, weather, politics, event risk, and interactions or constraints on the supply and demand, rather than discounted future cash flows (Geman [2005]). Furthermore, stocks and bonds tend to perform better when inflation is stable or slowing, while commodity prices tend to rise when inflation accelerates and therefore are seen as an inflation hedge (see e.g., Geman [2005]; Erb and Harvey [2006]; Gorton and Rouwenhorst [2006]; Fabozzi et al. [2008]).

The findings of several studies agree on the low correlation between commodities and traditional assets in a long-only framework. This suggests that long positions in commodities are potentially beneficial for diversification in a long-only portfolio of stocks and/or bonds (see e.g., Kaplan and Lummer [1998]; Georgiev [2001]; Erb and Harvey [2006]; Gorton and Rouwenhorst [2006]; Büyüks‚ahin et al. [2010]; Conover et al. [2010]; Bhardwaj et al. [2015]). However, correlation may vary across different phases of the business cycle. For example, at the peak of the business cycle, commodity prices are often high, given the increased demand for raw materials. Booming activity may also often account for an increase in interest rates and hence expectations for a decrease in growth, which again may cause financial assets to perform poorly (Geman [2005]; Fabozzi et al. [2008]). Together, these factors suggest that commodities do not provide equally good diversification benefits with other asset classes at all points in time (Gorton and Rouwenhorst [2006]; Kat [2006]; Bhardwaj et al. [2015]).

Conversely, other studies assert that the growing presence of commodity index funds in the market has increased the co-movement between commodities, stocks, and bonds, hence reducing the diversification benefits of adding commodities. This is commonly referred to as the *financialization* of commodities (see e.g., Domanski and Heath [2007]; Tang and Xiong [2012]; Silvennoinen and Thorp [2013]). Lombardi and Ravazzolo [2016] conclude that after the recent financial crisis, the correlation between commodities and the stock market increased. Lombardi and Ravazzolo [2016] also find that portfolios become substantially more volatile when commodities are included and thereby experience a decrease in the Sharpe ratio. Basak and Pavlova [2014] highlight that increased co-movements are also evident across different commodity sectors, although Steen and Gjølberg [2013] reject such herding behavior.

Importantly, low correlation alone is not sufficient for the diversification benefits of commodities; there should also be some return for the investor. Gorton and Rouwenhorst [2006] construct an equally-weighted index based on long positions of commodity futures and found that the portfolio provided equity-like returns. Erb and Harvey [2006] claim that this only occurred if there are positive spot returns in the future. Bhardwaj et al. [2015] maintain the conclusions of their 2006 study, while Sanders and Irwin [2012] provide evidence that long-only commodity futures markets produce no earnings and that the precise source of returns is unclear.

In a portfolio framework, several researchers observe a positive shift in the efficient frontier when adding commodity futures to the investment universe (see e.g., Ankrim and Hensel [1993]; Satyanarayan and Varangis [1996]; Anson [1999]; Abanomey and Mathur [1999]; Jensen et al. [2000]; Laws and Thompson [2007]; Idzorek [2007]). However, Belousova and Dorfleitner [2012] find that while commodities contribute to improving diversification, the diversification benefits vary considerably across different commodities. In contrast, Cao et al. [2010] report that the efficient frontier did not shift significantly when adding commodities in the period 2003 to 2010, which they argue might be a consequence of the increased co-movement across the asset classes.

In an out-of-sample analysis, You and Daigler [2013] suggest that commodities improve performance, while Daskalaki and Skiadopoulos [2011] find evidence of the opposite. Bessler and Wolff [2015] identify little or no improvement in performance by including agricultural and livestock commodities. However, they conclude that industrial metals and an aggregated commodity index generated improved performance. They also report that the portfolio benefits of commodities are time varying and that they vanished during the 2008 financial crisis.

More recently, there have been doubts raised in the literature regarding these positive effects in the investment environment, and several fund managers have moved away from commodity investment as a result. For instance, institutional investors had a net outfow of $48.3 billion from commodity strategies between 2013 and 2015 (Baker [2016]). Given the poor performance of long-only commodity investments, with a fall of 37% in the Bloomberg Commodity Index since 2011 (Meyer and Authers [2015]), a long-term return of zero (Sanders and Irwin [2012]), and the term structure of commodity futures (see e.g., Erb and Harvey [2006]), it may be natural to invest long/short in commodities, rather than long-only. See Miffre [2014] for additional arguments in support.

Some argue that more active strategies in commodities outperform long buy-and-hold strategies in commodities (see e.g., Akey [2005]; Erb and Harvey [2006]; Miffre and Rallis [2007]; Fuertes et al. [2010]; Miffre [2014]; Miffre [2016]). However, Marshall et al. [2008] find no evidence that these trading strategies outperform the market. Furthermore, Miffre and Fernandez-Perez [2015] find evidence that long/short commodity strategies based on momentum, term structure, and hedging pressure have higher returns, less volatility, and lower correlations with stocks and bonds than their long-only counterparts. This suggests that long/short commodity strategies may increase the performance or yield diversification benefits when added to a conventional stock and bond portfolio.

Unfortunately, research concerning the benefits of long/short commodity indices in stock and bond portfolios is limited. Both Yan and Garcia [2014] and Kremer [2015] find evidence of diversification benefits of third-generation commodity indices in an out-of-sample mean–variance framework, thereby indicating an increase in returns and a reduction in risk. Daskalaki et al. [2015] support this using a stochastic-dominance approach for portfolio optimization. The main drawback of the latter and Yan and Garcia [2014] is that they employ only momentum-based long/short commodity indices to test the diversification properties. Furthermore, they use only a single broad market-cap weighted index representing stocks and another for bonds.

In this article, we attempt to fill some of these gaps in existing research on the out-of-sample properties of long/short positions in commodities in a stock and bond portfolio framework. As previous studies employ the mean–variance and stochastic-dominance framework, we apply several different asset allocation techniques, including the maximum Sharpe ratio, minimum variance, equally-weighted, most-diversified, and three dissimilar risk-parity portfolios. Furthermore, we employ four different long/short indices—so-called “third-generation commodity indices”—as proxies for long/short positions in the commodity markets: (1) term structure; (2) market neutrality; (3) momentum; and (4) fundamentals/rules-based approaches, as the extant research is largely limited to momentum strategy. Furthermore, most existing studies specify at most two market-cap weighted indices and therefore may not well represent diversified portfolios. As an alternative, we create diversified portfolios composed of four stock and four bond indices.

We find from the full sample period (2002–2015) that the inclusion of the long/short commodity indices in general reduced the risk measures, and hence increase the risk-adjusted performance measures (RAPMs). However, the results from the subperiods vary. During the first (2002–2006) and second (2006–2011) subperiods, the RAPMs increased in most of the portfolios when adding the commodity indices. For the most recent subperiod (2011–2015), we observe completely different results compared to the two earlier subperiods. One intriguing observation is that the long/short commodity indices had a poorer performance after being launched compared to the results in “back-testing” period, that is, prior to their launching.

The remainder of the article is structured as follows. In the first section, we present and describe the data used. In the second section, we present some basic statistics of the indices used. In the third section, we provide our empirical results and in the fourth section, we discuss the recent poor performance. In the final section, we draw conclusions and suggest topics for further research.

## DATA

Our dataset consists of weekly Wednesday prices from December 27, 2000 to December 30, 2015, downloaded from Thomson Reuters DataStream and the SummerHaven homepage.^{2} If a given Wednesday is a non-business day with no trading, we use the previous trading day, or the nearest trading day available. The data cover 792 (791) observations for price levels (returns). All index values are total returns and denominated in U.S. dollars. Exhibit 1 details the indices used. We report our results based on weekly observations. The results do not change in any significant way using monthly or daily frequencies.^{3} Thus, the results are robust to different frequencies. We choose to report the weekly results since the historical tail-risk measures will be more robust when using weekly observations compared to lower frequencies.

We use the investment universe in Exhibit 1 to create diversified portfolios. Several previous studies only use two market cap-weighted indices, representing stocks and bonds as their investment universe (see e.g., Yan and Garcia [2014]; Kremer [2015]; Daskalaki et al. [2015]). When using a two-index approach, the results are not based on a well-diversified portfolio. We use three different stock indices which together represent investments at different continents, together with a real restate index to representing the international real estate markets. These potentially may improve risk-return profile of a mixed asset portfolio (NBIM [2015]). Including four bond indices likewise opens up the potential for increased diversification. Further, we use long-only indices for the stock and bonds since they generate a risk premium, and hence going short will yield a negative risk premium.

For the commodity market, we specify four long/short commodity indices for several reasons: (1) the recent poor performance of long-only commodity investments and the low long-term return, (2) the claimed diversification effects of such indices, and (3) long/short strategies are more intuitive in the commodity market because of their fundamentals, as previously discussed. We use the four basic types of third generation commodity indices using term structure, market neutrality, momentum, and fundamentals/rule based as the underlying strategies (see e.g., Kazemi et al. [2016]). Exhibit 2 describes the commodity indices used.

For the stock market, we specify three MSCI country indices for the US, Europe, and emerging markets, and another one for real estate. These indices provide a representation of the global stock market. These draw on the MSCI Global Investable Market Indexes Methodology, which aims to provide exhaustive coverage of the relevant investment opportunity set with a strong emphasis on index liquidity, investibility, and replicability. The indices are rebalanced semi-annually.

As for the bonds, we use four different indices; namely, the corporate, Treasury, and government inflation-linked bond indices provided by Bank of America Merrill Lynch (BofA ML). To represent emerging market bond markets, we employ the JP Morgan global diversified emerging market. For the risk-free rate, we use the 3-month US T-bill secondary market rate.

## SOME BASIC STATISTICS

In this section, we present some basic statistics and evaluate the performance of the indices over the full sample period (2001–2015). Further, we investigate the performance of the commodity indices before and after they were launched.

In Exhibit 3, we present cumulative returns. Regarding the commodity indices, we see that the CYDs (Long/Short and Market-Neutral) have relatively low volatility compared to the SummerHaven (Fundamental) index. We also see that the CYDs increase or are relatively flat during the financial crisis in 2008–2009. From the graphs, we can see that the SummerHaven and Morningstar (Momentum) indices display approximately the same pattern of cumulative returns as stocks and bonds.

Exhibit 4 presents descriptive statistics for the assets over the full sample. During this period, the SummerHaven commodity (SH) index displays the highest return^{4} of 8.96%, followed by the MSCI Emerging Market (EM) index with 8.6%, but the difference is not significant. In contrast, the CYD Long/Short (LS) index had the lowest return, followed by the MSCI Europe Index (EUR), with 3.13% and 4.43%, respectively, but they are not significantly different.

Looking at the risk of the indices in Exhibit 4, we see that the stock indices have higher risk measured in both annualized standard deviation and tail risk. As expected, the Treasury (TRE) index displays low values for standard deviation, ES and VaR. However, CYD Market-Neutral (MN) has lower standard deviation risk than the Treasury index.

All Sharpe ratios in Exhibit 4 are positive. The MN has the highest Sharpe ratio with a value of 1.37 due to its low standard deviation. The emerging market bond (EMB) has the second-highest Sharpe ratio with a value of 0.85. At the other end, we have EUR and LS with Sharpe ratios of 0.11 and 0.14, respectively.

Tests for normality suggest that TRE is the index closest to a normal distribution, followed by MOM. Returns for EMB and MN on the other hand, are clearly non-normally distributed. However, Jarque-Bera (JB) test rejects the hypothesis of normality in the return distributions for all indices, which emphasizes the importance of using alternative risk and performance measures in the portfolio optimization procedure. All indices exhibit negative skewness, except corporate (CORP) and inflation-linked (INFLNK) bonds. For investors, this means that there is a greater likelihood of high negative outcomes and that the standard deviation underestimates this risk. In addition, the indices display positive excess kurtosis, which implies heavier tails than under the normal distribution.

The profile of the commodity indices, especially compared with the equity indices, suggests that in this study, commodities shall be included in a portfolio framework when looking at the full period sample.

In Exhibit 5, we provide the correlation matrix over the full period. The correlations vary from ‒0.35 (US vs TRE) to 0.88 (CORP vs INFLNK). The CYD and MN and LS commodity indices have relatively low or negative correlations with the other indices. The SH index, on the other hand, has significantly positive correlations with all the other indices, except for TRE. TRE also has low or negative correlation with the other indices, except CORP. Furthermore, most of the long-term correlations between the commodity indices are low, but time varying. This also holds for the other statistics in Exhibit 4. So, we consider this in the portfolio construction procedure by using rolling windows in our calculations.

Since the commodity indices were launched in the period between 2006 and 2010, a part of their performance track record is based on back tests and does not represent the actual results for investors. Therefore, we will compare subsample performance of the commodity indices before and after they were launched.

The descriptive statistics and risk-adjusted performance (RAPM) for the commodity indices before and after their launching are shown in Exhibit 6 below. As can be seen, both the return and standard deviation decreased in the period after launching. The reduction in returns are so large compared to the reduction in standard deviation that all RAPMs decrease in the period after the launching of the indices. Skewness, on the other hand is less negative in the period after the launching, and the excess kurtosis in general is higher except for the MN index. There is no clear pattern in the changes of the downside risk measures when comparing the sample periods. However, the SH index has a reduction in both 5% VaR and 5% expected shortfall (ES).

There may be three reasons for the poor performance for the long/short commodity indices after their launching date: (1) It may be difficult to time the investments when the underlying commodity market turns from an increasing to a stable or decreasing trend, (2) The bid-ask spread may increase in a declining market resulting in a negative effect on the long/short trading strategies, and (3) It could be due to presence of capacity constraints in less liquid contracts, making it more difficult for funds to maintain the diversity of their trading books (see e.g., Naik et al. [2007]; Jones [2011]; Baltas and Kosowski [2013]; Van Hemert [2014]).

However, it is difficult to conclude whether the differences in performance are due to capacity constraints, due to the bid-ask spread, or whether they are due to changes in the underlying commodity market. A more solid conclusion on the reason of the recent poor performance needs further analysis, which is beyond the scope of this article.

## EMPIRICAL RESULTS

In this section, we present and evaluate the out-of-sample back tested performance of including commodity indices to the pre-specified allocation strategies consisting of stocks and bonds for the different asset allocation strategies. By out-of-sample back testing we mean that a portfolio is optimized for a given subperiod (“in-sample”), and the held for one subsequent period “out-of-sample”).

The portfolio weights are created using linear returns, while the back tested out-of-sample results employ compounded returns (see e.g., Meucci [2010]). We rebalance the portfolio weights annually with the re-estimated weights based on weekly observations from the previous year.

For robustness we use seven asset allocation techniques; (1) equally-weighted (EW), (2) naïve risk-parity (RP), (3) tail-risk-parity (RPES), (4) equal risk contribution (ERC), (5) most-diversified portfolio (MDP), (6) global minimum variance (GMV), and (7) max Sharpe (MS). Allocation techniques (2)-(5) are explained in Appendix II.

We exclude the possibility of short selling in the index allocation. We do this to obtain a better comparison of portfolio performance, as both the portfolios based on diversification and the fixed-weight portfolios exclude negative weights. Furthermore, the maximum Sharpe portfolio tends to incorporate extreme values in the asset positions when one allows short selling. We also preclude leverage when allocating capital to the indices used in the portfolios.

We start by presenting the most important metric for investors; namely, the investment return. We then analyze the portfolio risk using several risk measures for robustness. Finally, we analyze the risk-adjusted performance measures. We test these metrics for each strategy and the different investment universes: one consisting of stocks and bonds only, and the other consisting of stocks, bonds, and the commodity index.

### Performance of Portfolios

**Return and risk.** Looking at the returns of the different portfolios in Exhibit 7, we can see that there is no clear pattern when the commodity indices are included. In most of the portfolios during the different time-spans, there is more returns that are reduced than increased when the commodity indices are included in the portfolios. The reduction of the portfolio returns is most evidence in the full period and the most recent subperiod. However, during the full sample period and the two first subperiods, all the portfolios including the SummerHaven fundamental index (SH) have higher returns than the portfolios excluding commodities, and are most significant in the second subperiod.

It seems that taking positions solely based on the futures curve yields the poorest performance. These strategies are also the ones with the significantly highest reduction of returns in most of the portfolios where return is reduced. However, they increase the return significantly in some of the portfolios, as in the MDP and GMV in the first subperiod.

To obtain a sense of the amount of trading required to implement each portfolio strategy, we compute the portfolio turnover, defined as the average sum of the absolute value of the trades across the available assets (DeMiguel et al. [2009]). We can interpret the turnover quantity as the average percentage of wealth traded in each period, with the portfolio turnovers for our portfolios tabulated in Exhibit 8. As shown, the portfolio turnover generally increases when we include commodity indices in the portfolios. The only exceptions are the MS portfolio and MN in the EW and GMV portfolios.

We also see that the MS portfolios have the highest turnover, consistent with findings of Michaud [1989] on parameter instabilities of this portfolio.

Exhibit 9 presents the risk characteristics of the portfolios, as measured by standard deviation, the 5% VaR, and the 5% ES. These risk measures are the most commonly used, and capture both the volatility and the downside tail risk of the portfolios.

In general, we can see that when including the commodity indices, both the volatility and the tail risk are in general reduced, with a few exceptions. We can see that the two CYD indices (Long/Short and Market-Neutral) in almost every portfolio reduce the tail risk and the standard deviation, and are in most cases statistically significant.^{5} For the portfolios where the standard deviation and tail risk increase, the variance of the portfolios is not statistically different from the stock and bond portfolio. Furthermore, the SH index has the least impact on the reduction in risk, measured in both standard deviation and tail risk. For the MDP and GMV portfolios, the risk measures are relatively similar when including commodities, except for the MN index. This is because the optimization procedure provides relatively low weights to the MOM and SH indexes especially.

**Risk-adjusted performance.** The risk-adjusted performance measurements (RAPM) used in this analysis are the Sharpe, Sortino, and Omega ratios, which—given their differences—provide robustness of the results. For example, while the Sharpe ratio (Sharpe [1994]) is the excess return divided by the standard deviation, the Sortino ratio (Sortino [1994]) is the excess return divided by the downside deviations and thus only includes the downside risk in the risk adjustment. We also use the Omega ratio, as proposed by Keating and Shadwick in [2002], which is the ratio of average gains over average losses. The advantage of the Omega ratio is that it does not require any assumption concerning the distributional properties of the returns. We test the significance of the differences in the Sharpe ratios with and without commodities using the test in Memmel [2003], based Jobson and Korkie [1981]. These different measures are explained in Appendix I.

Exhibit 10 provides the RAPM results. For the full period, the RAPM generally increase, mainly because of the risk reduction properties of the commodity indices. For example, the LS index reduces the RAPM for the ERC, MDP, GMV, and MS portfolios, and the Sharpe ratios are significantly different from the stock and bond Sharpe ratio.

In the first subperiod, the inclusion of commodity indices also generally provides higher RAPM. The exception is MOM, as it only increases RAPM in the EW and MS portfolios, but we can observe a slight increase in Omega in all of the portfolios. The only portfolio with a significant difference in Sharpe is when we include MOM in the GMV.

During the second subperiod, we observe the same pattern as the full period. All of the RAPM increase when including the commodity indices, except the LS index. Including the LS index reduces all of the RAPM, and the differences in the Sharpe ratios are highly significant. For the MS portfolio, the Sharpe even has a negative value of ‒0.02 when the LS is included.

The third, and most recent, period provides a completely different interpretation for including commodity indices. The RAPM are less than the stock and bond portfolio for most of the portfolios, except for LS and MN in the EW portfolio, where the Sharpe and Omega ratios are identical. Furthermore, the inclusion of MN, MOM, and SH provides higher RAPM in the MS portfolio, except for the reduced Sortino including MN. The RAPM in the MS portfolio declines when the LS index is included, but the differences in the Sharpe ratios are not significant at any conventional level. We can also observe a relatively large decrease in the Sortino ratio when including LS in the MS portfolio.

The inclusion of the commodity indices appears to increase the RAPM in the period from 2002 to mid 2011, mainly because of the reduction in portfolio risk. However, from mid 2011 to the end of 2015, the inclusion of these commodity indices generally reduces the RAPM of the portfolios. This may relate to the findings in Dennin [2016] that the commodity indices are dependent on commodity markets. Therefore, when commodity spot or futures prices decrease as recently, long/short commodity indices also perform poorly, and their diversification benefits largely disappear.

## DISCUSSION AND CONCLUSION

Our objective was to analyze the possible out-of-sample benefits of adding long/short commodity indices, so-called third-generation indices, to diversified stock and bond portfolios. We did this by employing traditional allocation strategies such as the equally-weighted, minimum variance, and maximum Sharpe portfolios. Additionally, we implemented three approaches to risk parity, each based on different risk measures, comprising standard deviation, covariance, and expected tail loss, and the so-called most-diversified portfolio.

By analyzing the contribution of commodities under different asset allocation strategies, we performed a robust investigation of investor preferences for different portfolio choices. Under each portfolio strategy, we considered a stock and bond portfolio as the benchmark and four commodity indices based on different trading strategies; namely, (1) term structure, (2) market neutrality, (3) momentum, and (4) fundamentals/rule based, along with different stock and bond indices. We examined the period 2002–15 and further split this period into three equally long subperiods.

For the period 2002–15, the results showed that the inclusion of the commodity indices contributed to a decreased mean return, except when including the SH index. However, only 8 of the 28 mean returns were statistically different from the benchmark portfolio during this time. The risk of the portfolios decreased, in general, when the commodity indices were included. The reduction in standard deviation was significant compared with the stock and bond portfolios, especially when the LS and MN indices were included.

In addition, the expected-shortfall and VaR were less in most of the portfolios when we added the commodity indices. Moreover, commodities contributed to a significant increase in the Sharpe ratios when including commodity indices, except when adding the LS index to the ERC, MDP, GMV, and MS portfolios. The results for the Sortino and Omega ratios were mostly similar to those for the Sharpe ratio.

The subperiod results were ambiguous and did not confirm our findings from the full sample period. During the first and second subperiod, risk generally fell or remained unchanged when adding the commodity indices. Hence, the RAPM increased in most of the portfolios when adding the commodity indices, except for a significant decrease in the Sharpe ratio when adding MOM to the GMV.

For the most recent subperiod, we observe completely different results for returns and RAPM compared with the two earlier subperiods. However, risk is relatively similar. Most of the returns fell when adding the commodity indices, and 11 of the 28 returns were statistically different from those for the stock and bond portfolio. The mean returns also increased when adding MOM and SH to the MS portfolio, but they were not statistically different from the stock and bond portfolio. The RAPM in the final subperiod fell for all portfolios when including the commodity indices, except for MN, MOM, and SH.

Our results suggest that when long-only commodity futures increase in value in the second subperiod, the inclusion of long/short commodity strategies creates returns higher than or equal to the portfolio without commodities. However, during the most recent period when long-only commodity futures exhibited stable or negative price movements, the inclusion of the long/short commodity indices had negative or no impact on portfolio return. The reasons for this may be due to timing abilities of trading strategies, the bid-ask spread may increase in a declining market resulting in a negative effect on the long/short trading strategies, or be due to presence of capacity constraints.

In terms of future research on this topic, one possibility is to examine the benefits of adding the same type of commodity indices using other asset allocation strategies or optimization techniques. Additionally, one could examine the inclusion of third-generation commodity indices in portfolios comprising long/short stock and bond indices. In addition, a comprehensive study on the reason for the poor performance of the long/short indices after they were launched would be of great interest.

## APPENDIX I

### RISK-ADJUSTED PERFORMANCE MEASURES

## APPENDIX II

### ASSET ALLOCATION MODELS

#### Equal Risk Contribution

The ERC builds on the covariances between assets, and the derivation and optimization of this portfolio are given by:

1where *w* is the proportion invested in the given assets, σ_{i} is the standard deviation, and is the total risk contribution (TRC) of asset *i*.

For the optimization, we follow the objective function by Maillard et al. [2010] given by:

2where *w _{i}* is a unique solution and the condition

*f*(

*w*

_{i}) = 0 is ensured.

#### Naïve Risk-Parity

In the naïve risk-parity portfolio (RP), the optimal portfolio weights of the assets would be proportional to the inverse of its associated standard deviations, formally:

3#### Tail-Risk-Parity

Even though standard deviation is a commonly used risk measure, it is a limited estimate of the true risk of an asset when applied to financial analysis (see e.g., Inker [2011]). One is that two different assets with the same return and volatility might have different skewness and kurtosis. This highlights the importance of using alternative risk measures in this approach.

Popular approaches to measuring downside risk are Value-at-Risk (VaR) and the expected shortfall (ES). Many argue against VaR because it is not necessarily subadditive, which contradicts the principal of diversification. Without subadditivity the metric is unsuitable for risk budgeting (Alexander [2008], Roncalli [2013]). Since ES is subadditive, we will use this as the risk measure to construct the risk-parity portfolio based on tail risk.

To incorporate skewness and kurtosis and not make any assumptions about the parametric form of the return distributions, we use the historical method when calculating ES. However, this method assumes that we have experienced all possible future losses at some point in the past, which is an adverse assumption (see e.g., Alexander [2008]).

The risk-parity portfolio based on ES (RPES) has the same requirements as the risk-parity portfolio, meaning that we assemble that portfolio composition which achieves an equal risk contribution between assets and is given by:

4where the risk measure is 5% ES.

#### Most-Diversified Portfolio

The purpose of the most-diversified portfolio (MDP) proposed by Choueifaty and Coignard [2008] is to maximize the diversification ratio (DR). This is the weighted average of the standard deviation divided by the portfolio standard deviation, defined as follows:

5In terms of an equal risk contribution, this is equivalent to (see e.g., Roncalli [2013]):

6The inferior limit for the DR statistic, when long-only portfolios are considered, is one for a 100% weight in a single asset, so that values far from one indicate greater diversification (and hence, less concentration).

## ENDNOTES

I would like to thank my great students Nicoline Hansen-Tangen and Mats Overaae for writing on my idea on the inclusion of commodities in a portfolio framework during their master’s thesis. I would also thank Ole Gjølberg, Marie Steen, and Espen Haug for helpful comments. At last, I would like to thank Daniel Schild in Vescore for providing helpful information.

↵

^{1}Long commodity positions in the U.S. amounted to $200 billion as of February 2015. http://www.cftc.gov/idc/groups/public/@marketreports/documents/file/indexinvestment0215.pdf.↵

^{3}Results based on monthly and daily observations can be provided upon request.↵

^{4}Return refers to annualized logarithmic mean return.↵

^{5}The null hypothesis that there is a significant difference in variances is .

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