A Parsimonious Risk Factor Model for Global Commodity Future Market

Transcription

1 A Parsimonious Risk Factor Model for Global Commodity Future Market Abstract Using 10-year option and future data of global market, the risk-neutral skewness, estimated following the method from Bakshi et al. [5] has been found with the ability of pricing the average cross-sectional return in the global commodity future market, generating extra 8.3% return annually. The higher (lower) current risk-neutral skewness, the lower (higher) the subsequent return. The two-step regression based approach following Fama and MacBeth [29], showing that risk-neutral skewness has negative relation on the future return. This result is robust to scenarios in which traditional risk factors and recent new risk factors like total skewness, idiosyncratic skewness and volatility of return have been considered. Bootstrap simulation is applied to confirm that no single product has extra significant alpha when then new parsimonious risk factor is concerned. Moreover, risk-neutral skewness is found with the predictability regarding the future macroeconomics factors like bond return, default risk as well as TED spread. Keywords: Asset pricing, Risk-Neutral, Skewness, Factors, Bootstrap 1

2 2 Contents 1 Introduction 5 2 Related Literature Review Theory of Risk Premium Theory of Storage Theory of Normal Backwardation Risk Factor Portfolio Approach Skewness Preference Data and Methodology Trading Data Hedging Position Database Futures Data Options Data Underlying Marco-Economic Data Risk-Neutral Skewness Estimation Risk-Neutral Skewness Commodity Variables - Traditional Risk Factors Risk Factor Portfolio Construction Rationale for OLS Analysis Summary Statistics Analysis Empirical Results Time Series Analysis Cross Sectional Analysis Robustness Check Risk Factor Based Approach Bootstrap Analysis Economics Rationale for A Risk Factor Conclusion 46 6 Appendix 56

3 3 List of Tables 1 Statistics for Risk Factors Summary Statistics for Return Series on Individual Assets Level Summary Statistics for Risk-Neutral Skewness Factor on Individual Assets Level Time Series Analysis Portfolio Property based on Risk-Neutral Skewness Cross Sectional Analysis Fama Macbeth Regression Principal Component Analysis for Individual Assets Time Series Analysis for Equally Weighted Factor Time Series Analysis for New Risk Factors Cross Sectional Two-Step Regression Individual Regression for Alpha Coefficient and Statistics Risk Factor Prediction Analysis U.S. Bond Return Average Level Risk Factor Forecasting Analysis - Change of Money Stock and TED Spread Default Risk Forecasting Testing Risk Factor Exposure Testing

4 4 List of Figures 1 Time Series Variation of Cross-Sectional Risk-Neutral Skewness Factor Portfolios Performance Comparison Overlapping Analysis Between TS and RN.SKEW Overlapping Analysis Between MOM and RN.SKEW Overlapping Analysis Between HP and RN.SKEW Principal component analysis on the Assets Return Bootstrap Simulation Analysis

5 5 1 Introduction Recently, commodity futures have gained much more attention than it was in the past years. One idea could be the popularity transfer from stock to other derivatives. Also, it is attractive for researchers to do study on the comparison between stock market and future market. One of highly arguable fact was that whether future market is integral with stock market or shows strong segmentation? The second incentive for studies on commodity futures is caused by the considerably compound return from investing in commodity futures, with 9.85% and 9.81% for commodity futures and common stocks from 1950 to 1976, and 12.2% annual return of Goldman Sachs Commodity Index (GSCI) since 1969 comparing 11.2% annual return of S&P 500, there is an obvious advantage for considering studying and investing futures is that return from trading future contract is less or even negatively correlated with the stock return (Bodie and Rosansky [12], Erb and Harvey [25] and Gorton and Rouwenhorst [33]). This means investors can diversify their risk by investing in combination of both stocks and futures rather than merely bear the risk from stock market. Far more than this, futures can be also used for some real industry companies to hedge the demand and supply side risk like suddenly capacity shortage or unexpected industry recession. It offers somehow like an insurance for companies which can reduce cost fluctuation and investors who can make profits by getting involved in. In the modern finance field, given the success of mean-variance portfolio selection theory (Markowitz [50]) and then initialization of CAPM system (Sharpe [60] and Mossin [53]), return of investment in the financial products is able to be explained by this standard and simple model. Inheriting the fundamental spirit from CAPM, return in financial asset is only caused by the variation in market portfolio, which is constructed by assuming all rational investors will hold such portfolio (combination of risky and risk-free asset). In the earlier study, market portfolio is treated as an efficient portfolio. This is well known as market efficiency, stating that there is no one who can get extra return as the total variations in assets return are from market portfolio. However, this assumption has been challenged for a long time as several studies found that market portfolio is not able to ex-

6 6 plain all return variations, one famous among all of those is by Fama and French [28] who introduced the well-known three-factor model (Market portfolio, HML and SMB). After this stage, factor model become more and more popular and lots of researchers seek new factors which can explain the return in market while has not been found before. Following the study framework by stocks, risk factors consideration in commodity futures is pursued and tested. Consistent with law of one price theory, factor that can explain cross sectional average return in equity should be able to explain average return of commodity futures as well. However, by applying the CAPM model, extended CAPM model and traditional equity motivated approach, no significance has been found, which states the heterogeneity of commodity futures return (Jagannathan [39] and Erb and Harvey [25]). This documents the segmentation between equity and futures, enforcing researchers find out new heterogeneous or idiosyncratic factors for commodity futures return explanation. Now, it turns our to ask whether there are factors that can explain the crosssectional average return in commodity futures market. Moreover, if there are, whether we can find out a parsimonious representative factor model for both timeseries and cross-sectional variation in commodity futures? Due to the lack of huge number of commodity futures products, can we find the spectacular significance from the individual perspective rather than portfolio way? Are those factors able to provide some forecasting views on the real economy, like bonds, default rate, equity and money stock and liquidity? In this paper, the aim is to figure out those factors which can answer all the questions mentioned above and give more precise evidence about which factors should be taken into account for academically asset pricing theory in global commodity futures and for the physical investors when they are doing investment in the global commodity future market. The current proved and popular risk factors for commodity futures are focused on term structure and momentum factors. Accompanying with this popular ones, average and hedging pressure factors are also tested and used with relatively high frequency compared with others. This paper is to extend the current literature by proposing a new risk factor. Unlike the no significance of factors in the com-

7 7 modity futures proposed by Daskalaki et al. [19] and differ from the findings of three-factor model (average factor, term structure factor and momentum factor) by Bakshi et al. [6], this paper introduces a new three-factor model (term structure factor, momentum factor and risk-neutral skewness factor). To be specific, using a large cross-sectional of commodities option dataset from the DataStream, a new risk factor is estimated, called risk-neutral skewness. Afterwards its pricing ability is analysed by using two-step regression method in the spirit of Fama and Mac- Beth [29]. Not exactly following their methods, full sample is applied in this paper to extract more information, which is theoretically and practically approachable (Cochrane [17]). Different from the previous commodity feature based risk factors which are estimated based the futures price or return series, the new risk factor is estimated from the commodities out-of-money call and put option data simultaneously. As for the property of new factor, it shows strong negative relation with subsequent commodities future return and its predictability still exists even the traditional risk factors have been taken into account. Time-series analysis is also tested, showing that risk-neutral skewness sorted portfolio has relatively higher annual return and sharp ratio and smaller volatility and maximum draw-down. In addition, this new factor sorted portfolio can generate extra alpha (abnormal return) when considering the traditional risk factors like equally weighted portfolio, term structure, momentum and hedging pressure. Consistent with the finding in (Fernandez-Perez et al. [30]), even though the difference in construction of skewness factor, risk-neutral skewness is also showing asymmetric property for the portfolio construction. That is, the mean return is monotonically decreasing from P1 (lowest skewness sorted portfolio) to P5 (highest skewness sorted portfolio). However, the absolute mean return from P1 is considerably smaller than that of return from P5, which is also stated by the extra significant alpha from time-series analysis with tables in appendix. With respect to the stability of the test, certain adjustments have been applied: (i) Shanken-correction is calculated following (Shanken [58]), which is referred as the standard way of dealing with error-in-variables (EIV) problem. The corrected standard error can reduce the upward basis of the t statistics compared with using the general ols method (ii) Newey-West adjustment with lag 12 is considered fol-

8 8 lowing (Newey and West [54]), which is used for reduce the upward basis especially when the autocorrelation is strong enough (iii) bootstrap re-sampling method following the idea from (Kosowski et al. [45]) has been implemented for time-series analysis in individual products level in terms of 1000 re-sampling times, which is regarded as a reliable way of eliminating the lucky reason for alpha to be significant. Far more than this, robustness check is done from a lot of angles: (i) Equally weighted portfolio is argued by Bakshi et al. [6] as they insist that it is an important factor which should be considered. Therefore, principal component analysis is applied to extract the first main component, which is confirmed to be an equally weighted strategy as it gives nearly equal weight on each commodity future product, explaining the main variation of commodity futures return. Following their suggestion and PCA analysis, I add equally weighted portfolio as an extra factor in the Fama Macbeth regression to see whether the skewness factor is still valid (ii) in addition to the conventional risk factors, recent new factors like idiosyncratic volatility, total skewness and idiosyncratic skewness are argued by some researchers that they can price the commodity futures return (Fernandez-Perez et al. [30] and Fuertes et al. [32]). So, they are also added in the two-step regression as control variables. After controlling all of these literature proposed risk factors, risk-neutral skewness is still significant at 99% level. Same to the result in non-robustness test results, term structure and momentum factors are significant at least 10% level, which is consistent with the literature. Going further, several macroeconomics related proxy variables are chosen as the dependent variable to test whether the new risk factor can predict their movement in the future. Following Warren Bailey [64] and Bakshi et al. [6], money stock change (M2), TED spread (spread between 3-month LIBOR rate and 3-month Treasury bill), 10-year bond return, 30-year bond return, corporate default spread (Moody s Seasoned Baa Corporate Bond Yield relative to is the Moody s Seasoned Aaa Corporate Bond Yield) and default premium (Moody s Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year Treasury Constant Maturity) are

9 9 used in this analysis. Compared with other factors, skewness factor can predict middle and long-run bond return, showing positive correlation. The same correlation can be found in TED and corporate default rate spread and default premium forecasting as well. These all relation signs imply that skewness factor is regarded as a gauge of macroeconomics prosperity. The higher the skewness factor, the higher the probability of being involved in a higher default state and the higher premium can be obtained from less qualified corporate bond. Similarly, this higher skewness will reduce the money liquidity in the market and increase the premium generated from these bonds guaranteed by government. As for the contribution, to my best knowledge, there is no research doing asset pricing on global commodity futures with respect to the risk-neutral third moment risk factor both cross-sectional and time-series testing in the literature. Consequently, the contribution of this paper is to fill some gaps in the following aspects: (i) based on model free method, risk-neutral skewness is estimated from the OTM option data and is shown to be a necessary risk factor when considering the asset pricing theory in global commodity futures (ii) risk factor testing is far more than Fama Macbeth two-step regression, maceconomics variable forecasting and exposure is also provide to make the risk factor more solid than previous research. The sections are organized as follows: section 2 will give the review of related literature and explain the way that traditional risk factors and recent new factors price the commodity futures; section 3 will go through data and methodologies used in the skewness estimation procedure and regression analysis; section 4 is mainly about the results analysis; section 5 gives the conclusion and section 6 shows appendix with tables and figures.

10 10 2 Related Literature Review In this section, comprehensive literature but not complete will be reviewed. The structure for this part is as follows: first, this part goes to the theory for risk premium and risk factors, then their corresponding empirical studies are analysed and discussed. 2.1 Theory of Risk Premium Theory of Storage In the existing literature, there are two branches of theory to explain the risk premium embedded in the commodity futures return. The first study about the futures are by Working [65] and Kaldor [41] who propose the theory of storage. Convenience yield is proposed implied from holding the physical commodity products. This is defined as a potential benefit especially under the scenario where the spot market is getting involved in the situation of scarcity of specific inventory. Therefore, the holder of physical products can sell product with a higher price than that in normal period to obtain the positive difference as profits. They argue that convenience yield is a negative function of inventory level. When inventory level is going up, the convenience yield is decreasing. However, it is hard to measure the convenience yield as the inventory data is not available in the earlier study. One straightforward way of identifying this inventory level is by substitution of adjusted future basis. According to the definition of future basis, it is defined as the differential of spot price and future price. For simplicity, thinking of one period future price pricing equation, mathematically, convenience yield can be derived from the non-arbitrage pricing equation as follows: F i,t = S i,t (1 + r t δ i,t + C i,t ) (1) Where, F i,t and S i,t are the future and spot price level for product i at time t. δ i,t is the convenience yield for product i at time t. C i,t is the cost of carrying goods for product i at time t. r t is the risk free rate. As we can see from the equation, δ is regarded as profit that is subtracted from the

11 11 equation on the right hand side. Following this non-arbitrage equation, interest adjusted future basis (1 + r) S i,t F i,t, applied by Fama and French [27] and Ng and Pirrong [55] who, among others, find the evidence for negative relation between inventory level and convenience yield. More precisely, Fama and French [27] use metal data from London Metals Exchange (LME) from 1986 to 1992, and document that spot price is more volatile than future price especially in the period of low inventory level, which is high level of convenience yield. On the contrary, these two price levels are nearly similar when the inventory level is high. These two findings support the theory of storage, which is confirmed by the study from Ng and Pirrong [55] who use same data same time period. In their studies, more precise estimation (error-correction and GARCH) is used to care more about reduction of the endogenous bias error. Their results add more measures of these relation: correlation coefficient, volatility and elasticity measurements shows consistency with the theory of storage. However, their estimation is not exactly accurate as they use estimated inventory data rather the real one. Real inventory data have been applied by several researchers to figure out the relation between convenience yield and inventory level recently. Standing on the side of production firm, Pindyck [56] decomposes total cost into production cost and cost of holding inventory. He documents the convenience yield as a convex function of inventory level, even though his intertemporal optimization model is not approachable for the underlying chosen. His failure might due to the rational expectation theory which are not able to describe or the absence of consideration in external factors that can affect production function to some extent. Based on reliable inventory data (copper, natural gas, crude oil and gold) from administrative channel although not too much database, Dincerler et al. [22] do the regression based approach with inclusion of squared term of convenience yield as well as inventory level and evidence that convenience yield is not linear to stocks scarcity but quadratic. They also confirm that the previous documented relation sign is right, implying that lower inventory level will cause the increase in the convenience yield as well as the spot price. But more importantly, they find that the relation is not that simple as a linear function by plotting convenience yield against spot price. Continuing in this fashion, Gorton et al. [34]

12 12 directly write the future basis as a non-linear function of inventory to work out the data fitting. Far more than previous study, they provide a view of comprehensive dataset about inventory, covering 31 commodity products, which is a largest one so far. By changing the normalized inventory data level from 1 to 0.75, they find that the coefficient of inventory level to basis in spline regression analysis is deepen sharply over this range, showing the necessity of non-linear curve to fit the data. Such relation is robust to seasonality effect as authors have already put dummy to exclude this potential effect Theory of Normal Backwardation The second branch theoretical set-up is the normal backwardation theory introduced by Keynes [42] who gives a new perspective of how and why it is possible to hold the future contract? The intuitive idea behind is the demand and supply relation in which hedgers (either individuals or firms) need to sell (supply) future contract to reduce their risk of unanticipated drop in underlying products in the future. For those who hold (demand) the contract, by return, this future contract should be discounted with price level smaller than expected future spot price, which is reasonable as investors only buy (hold) this undervalued contract for premium purpose when price converge upward in the future. Mover forward, theoretical model is established by (Stoll [61] and Hirshleifer [38]). In terms of the idea from Stoll [61], he use inter-temporal portfolio optimization way and end up with the pricing relation equation showing two possibilities: contango and backwardation. Backwardation refers to the situation where spot price is larger than future price, while contango refers to the opposite direction. By assuming the variation in commodity price is larger than that of storage cost, if the correlation between stock return and commodity return is negative, backwardation will be persistent, while contango will be persistent if the correlation is positive. Some empirical studies have been done by (Bodie and Rosansky [12] and Gorton and 1 future basis based study about the spot price and inventory level can also be found in stochastic analysis studies for linking stochastic convenience yield and price level, see (Schwartz [57] and Casassus and Collin D. [15]).

13 13 Rouwenhorst [33]) recently. They prove that the correlation is negative and even more negative if time horizon is long enough. Built on this result, the model from Stoll [61] mainly indicate the normal backwardation theory. Similarly, Hirshleifer [38] gives the explanation of why backwardation premium exists? The informational barriers (participation cost) in the commodity futures will make speculators outside the market, which in return force the bias of risk premium generated by hedgers consistent in the market as not enough speculators can join the market to push market back to perfect situation. These two explanations theoretically demonstrate why investors are willing to hold contract with price lower than expected price. No matter what views above two theories stand for, the purpose of these is to explain the existence of risk premium in the commodity futures. The following literature are those in which researchers have done a lot on risk premium searching. The first empirical study starts from Dusak [23], who adopts value-weighted S&P 500 index as market portfolio and point out that commodities like wheat, corn and soybean do not generate risk premium based on CAPM asset pricing model (same results followed by Kolb [44] and Bessembinder [10]). His conclusion is that the results can not be related to Keynesian theory and the principal reason might be the uncorrelatedness between the data he use and index (given few information from commodity agriculture are embedded in). If the commodity he use is copper that is more related to the industrial process (or the underlying economy development), the beta could be significant to some extent. Instead of applying one factor model, Ehrhardt et al. [24] propose two-factor model to justify the risk premium within APT framework and fail to find the result. However, strong evidence of normal backwardation has been documented by Carter et al. [14], who use the same model like Dusak [23] but give two more extension: allowing speculators to be net long or net short, and inclusion of commodity index as weighted market portfolio. Different from previous studies in commodity futures risk premium findings above, Baxter et al. [9] argue that the market index chosen by both of them is overweighted for commodities as S&P 500 index has already included certain percentage of commodities (similar argument inspired by Black [11]). After constructing a new index with weight construction method consistent

14 14 with theory suggested by Marcus [49], they confirm that wheat, corn and soybean do not expose to systematic risk and are in absence of risk premium. By applying test on more commodity categories, Bodie and Rosansky [12] show that the beta coefficient is roughly one if looking at more commodity products, even though their first trail is also consistent with the result from Dusak [23] with no significance on specific products. What they suggest is that results are more precise considering more products as well as longer testing periods. Moreover, Fama and French [26] separate the return and get the estimation of return premium, which is the basis, denoted as the differential of future price and spot price. They find the time-varying property and statistically significant when regressing excess return on it in most commodities out of 21 commodities. So far, it is still hard to argue whether market portfolio is able to explain the risk premium in the commodity future as the results in the literature are mixed 2.2 Risk Factor Portfolio Approach From now on, this is more about current research on risk premium chasing with risk factor portfolio approach instead of merely focusing on correlation with market portfolio return. This means the factor is constructed by the portfolio sorted method which is widely used in stock market pricing model (eg. the famous threefactor model given by Fama and French [28]). In spirit of the relation between future basis and price variation, term structure factor has been proposed and tested in the literature (Koijen et al. [43], Erb and Harvey [25], Szymanowska et al. [62] and Fuertes et al. [32]). Term structure is designed following risk factor portfolio way by basis sorted two extreme quantile portfolio return differential. Erb and Harvey [25] apply term structure and long only strategies on commodities and its index. The results imply that strategy with long backwardated contracts and shorting contango contracts outperforms the strategy with long only no matter in individual commodities or index from 1992 to Fuertes et al. [32] and Koijen et al. [43] also do the strategy testing based on large sample of commodity futures in the global market. Both of them

15 15 define the same future basis (except that Koijen et al. [43] give their own name carry factor ). The first study pays more attention on the multi-factor sorting portfolio while the second one emphasizes on generalizing the factor far more than commodity futures. The results they show are consistent as term structure factor can give reasonable prediction on the future return for most products. Different from above study, Szymanowska et al. [62] propose the idea of self-explanation mechanism in the commodity futures. In another word, futures return are totally explained by spot premia (holding return on future contract) and term premia (spread return among future price with different time to maturities). High minus low (HML) risk factors are constructed by these two premium, which to some extent eliminate the alpha with price error equal to zero. Although there is no theory supporting the existence of momentum factors, empirical study on the factor searching has confirmed that momentum is common factor for illustrating the cross-sectional average return in commodity futures. Among researchers, Asness et al. [4] identify two factors (One of them is momentum factor) existing in the global market, which is not only in the future groups. Their conclusion is solid to some extent as the proposal is passing two-step regression with risk premium value significant deviating from zero. However, they did not separate commodity future as a single market, it is hard to figure our how much the future return is exposed to their momentum factor in commodity category. Extending the method with only one momentum strategy in study of Erb and Harvey [25], Miffre and Rallis [51] use 56 momentum strategies in terms of different holding and ranking periods, showing that the average return from momentum strategies is to maximum around 9% among 31 commodity futures. More importantly, their research on momentum factor give the intuition why and how it provides risk premium in commodity future market? The momentum sorted strategy actually choose contracts which are in backwardation to long and contracts which are in contango to short. Their conclusion is converging to the backwardation theory by Keynesian. Under the framework of hedgers hedging and speculators speculation, hedging pressure is proposed and measured as an efficient indicator for investors to predict

16 16 future return in the market. The mathematical calculation for hedging pressure is the net short position of hedgers divided by total open interest in the market for specific contract. If the hedging pressure is larger, hedgers short position is larger than long position, there is positive risk premium likely to be exist in the market and speculators can hold the future contract for being compensated. Bessembinder [10] find that future return is positively related to the hedging pressure. Following the literature, De Roon et al. [20] document that risk premium generated by hedging pressure can be regarded as a non-systematic risk premium after controlling the market factor. More than this, cross sectional hedging pressure is also found among 16 commodity futures by applying different hedging factors, except for financial index. More recently, Hedging pressure has also been confirmed with the capacity of pricing future contracts return cross sectionally (Basu and Miffre [8]). They argue that their results are consistent with Keynesian s theory with high hedging pressure for speculators indicating backwardated contract, which will yield higher subsequent expected return by holding them. Above all three risk factors are currently treated as the common risk factors which can explain the variation in futures return. However, the literature for this conclusion is still mixed as some of them are arguing the reality of risk factors. Daskalaki et al. [19], among others, offer a comprehensive comparison study for the risk factor model. More than market portfolio proxy, CAPM extended series model (CCAPM,MACPM,MCCAPM...), macro-economics factors (consumption growth, money growth, FX factor...), equity-motivated factors (FF, Carhart, LFF and LCarhart) and even the commodity-specified risk factors mentioned above are all tested with no significance. Their conclusion is that commodity futures show segmentation property and not able to be illustrated by anything currently popular in the literature. Even though they nearly reject all factors in commodity market, they haven t considered risk-neutral high moments effect.

17 Skewness Preference Within the case of asset pricing model, mean-variance model, used as describing the behaviour of investors in the whole financial market, is not accurate or to some extent not valid. The argument for this is due to the doubt that the underlying asset is not distributed normally, which results in the imprecise modelling of investors decision based on only expected mean and volatility. Talking more about the investors behaviour, it is hard to simply narrow the definition of investors behaviour to be rational by always avoiding risk and pursuing return. Somehow, investors have been witnessed by taking risky activities as they want to obtain extra benefit. This is documented as risk-lottery like behaviour like investing in negative skewed asset. In summary, their risks are, no matter how, will be compensated by providing them abnormal return. In this paper, the key point is focusing on skewness, also called the third moment of return series. It sheds some lights on the current two-dimension pricing model, both in equity market and future market. Following the Sharper-Lintner asset pricing model, demonstrated by (Sharpe [60], Mossin [53] and Lintner [48]), Arditti and Levy [2] extend thie two-factor model to be three-factor model with introduction the concern on assets skewness. Given the theoretical proof, they find that skewness is positively related to both mean return and volatility, which leads to a new efficient frontier on which portfolios are strictly dominating others points out of the frontiers with these three concerns. One year later, Kraus and Litzenberger [46] show that after considering the utility function with concave property and positive skewness chasing, systematic skewness is necessary for the pricing model, which states the necessity of skewness in asset pricing. Empirical test is also supporting their proposal with the coefficient for systematic skewness is positive and significant. This indicates a new fact that previous miss-pricing in equity market is not attributed to borrowing constraints or lending rates. Instead, it is due to the lack of the role in systematic skewness in asset pricing model. This is supported by Lim [47] who use GMM instead of Fama Macbeth approach to identify the fact that skewness does matter.

18 18 Besides the equity market, skewness searching has also been done in future market. Unlike the result in equity market, there is no strong evidence to support the skewness will affect the return in future market especially when systematic risk is controlled (Junkus [40]). However, the reasons for them failing to test the significance may be attributed to the wrong selection of market portfolio: S&P 500 and BLS wholesale price index. More than the study by Junkus [40], Christie- David and Chaudhry [16] adopt nine market indexes as the market portfolios and use co-skewness and co-kurtosis as the risk factor in terms of the method from Fama Macbeth, showing that systematic skewness has strong relation with return generated in the future period. R square is increased when adding co-skewness and co-kurtosis in asset pricing model, which is also robust regardless of the chosen of market index. Rather than concentrating on mixed result for the success of co-movement in asset pricing model, more recently, the popular risk factor is more related to idiosyncratic property of return distribution, like volatility and skewness. Earlier study about individuals skewness is more likely to be found in the equity market. Conditional skewness is tested for equity market among stock pool with different sorting criteria and sub-sample analysis, they find that conditional skewness can price to some extent but not general for all assets (Harvey and Siddique [37]). One reason for their failure might be the imperfection of skewness as it is not the ex ante measure of skewness based approach. As for the attractive attribute of ex ante measure, Boyer et al. [13] find that idiosyncratic volatility can be a good proxy variable to estimate expected idiosyncratic skewness. Based on this regression approach, the expected idiosyncratic skewness plays an excellent role generating 1% abnormal return monthly. This is recorded with a negative relation between skewness and subsequent return indeed after controlling Fama French three-factor model. Their result is also consistent with Amaya et al. [1] who use new estimation method with intra-day (high frequency) data to measure realized skewness and document that it has a significant negative relation on subsequent returns. The success for their findings is more related to the high frequency data as this will increase the accuracy of estimation compared with usage of low frequency data. Unlike previous research, real ex ante skewness is obtained by two different ways, implemented by Bali and Murray [7] and Conrad et al. [18], both of them

19 19 find the same relation for risk-neutral skewness and expected return. However, they empirical results are more focused on stock market rather than on the future market, which point out how we find the gap in the literature. Lots of recent empirical studies have studied the effect of skewness in future market, but none of them are same to this paper. Inspired from the equity research, Fuertes et al. [32] extract the residual from regression of asset return on MOM and TS and use them to estimate the second moment called idiosyncratic volatility (IDIO VOL). Based on factor sorted approach, triple-sorted portfolio (MOM, TS and IDIO VOL) can offer more smoothing return and lower drawbacks. Crosssectional regression with dummy included also confirms its pricing ability as coefficient is relatively larger than other two factors among sub-samples. Skewness, calculated in third central moment way, in idiosyncratic way and in expected way, shows non-trivial effect on asset return generating process (Fernandez-Perez et al. [30]). Their empirical results demonstrate that after controlling the traditional risk factors, extra 8% average annual return can be obtained from buying lower skewness and selling high skewness commodity futures contracts. This means, a strong negative relation is documented in their studies, which is consistent with the relevant studies applied in the equity market. In conclusion, regarding the literature of asset pricing in global future market, new risk factors exploring are more emphasized on heterogeneous part instead of the co-movement with market portfolio. Idiosyncratic factors of the future products have not only explained the variation from traditional risk factor, but also contributed to the variation excluded from the traditional part. However, there is no one using risk-neutral skewness as pricing factor in global future market. The most closest research that has been done by using the same estimation method is by Triantafyllou et al. [63]. However, the key point they specify is the differential of realized volatility and risk-neutral volatility with the later part estimated by the same method. Therefore, by going through the literature, using new risk factor estimated by well-proved model free approach in future market is the contribution in this paper, which is unique as far as I know.

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21 21 3 Data and Methodology 3.1 Trading Data Hedging Position Database Consistent with the literature, the data for hedging pressure are collected from the Commodity Futures Trading Commission (CFTC) website with weekly frequency. According to the rule conducted by CFTC, all qualified large participators in the future contracts trading should identify their trading purpose (for hedging or speculating reason) to CFTC. Every Tuesday, the data for trading positions are gathered and calculated based on some weighting approximations, and finally released on Friday in the same week. With respect to the details of position data, the original definition of hedgers in data is belonging to trading for commercial purpose, while that of speculators is belonging to trading for non-commercial purpose. For each definition, data are separated into short and long position. Since the hedging pressure data are fixed with its publication frequency, the following data frequency are chosen correspondingly Futures Data As to the commodity futures price data, they are collected from the DataStream with time span from 2007/10/10 to 2016/03/01, daily frequency. Consistent with the previous literature, there is a necessity to construct the continuous price time series data for one specific product based on discrete and different maturities future contracts. The underlying intuition for researchers doing future contract in this way is to make future return comparable to other financial products. Another reason is due to the inconsistency of future contracts. It is impossible to buy and hold the contract like the general way treated in stock market as contract will expire for future certain period. Therefore, investors need to roll over contracts to avoid physical transaction as well as the lack of liquidity. The approach is done in this paper that continuous time series price data are constructed by choosing the nearest-maturity contract and rolling to the next nearest-maturity contract until

22 22 when there is less than one month maturity date left for the nearest-maturity contract. After the price data selection, weekly frequency is applied based on data publication day introduced by hedging position data. Return then is logarithm form of price difference based on the continuous time series price data constructed before, which emphasises the fact the return is not calculated from different contracts. Null value will be fitted in to return series if there is a gap caused by rolling and different maturities when applying the weekly frequency selection. This means, there is no way to calculate the return in terms of two different maturities future contract Options Data Options price data are obtained with the same source and time length as future data. Generally, this period contains a relatively comprehensive investing cycle in the global market as there is bear market after the 2008 financial crisis and bull market from This is also the maximum length of data available from the DataStream, which limits the time length of future data and hedging position data as these two are longer than option data somehow. Like the future price data, option data have different maturities dates, which implies the short live property of option data. However, different from other types of data, options have their specialities as two more parameters are added in to assure one. One specific parameter is strike price by which one option can be categorized into call or put. The second parameter is indicated by the premium (how much investors have to pay if they want to hold the option contract). Therefore, for each commodity product, all available call and put contracts with parameters maturity, premium and strike price are collected. To give explicitly idea of what the data is like to be, by expressing in numbers, on the mean level, for one year, one future product has roughly 2500 call and put option contracts receptively with different maturity dates. This totally accounts for 2375 daily observations with around 5000 call and put option contracts including strike price, option value, maturity data columns. In matrix expression, it is a 2375 times 5000 times 3 matrix for one product. Data

23 23 for options are also daily frequency, this will be finally converted to weekly frequency based on the criterion given in hedging potion data when we estimate new factor, which exactly follows literature from Conrad et al. [18]. 3.2 Underlying Marco-Economic Data In this part, some macroeconomic related data and trading specific data are collected with data source following the literature mentioned above. For the trading specific data, classical equity motivated three factors are downloaded from Fama French data library website with weekly frequency 2. Based on the assumption of market homogeneity, the factors pricing stock market have relation with factors pricing the commodity futures to some extent. As to the macroeconomic data, 10-Year Bond, 30-Year Bond, change of money stock, TED spread, Moody s Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year Treasury Constant Maturity and Moody s Seasoned BAA Corporate Bond Yield relative to the Moody s Seasoned AAA Corporate Bond Yield data are all from the Federal Reserve Economic Data St. Louis database with weekly frequency 3. The first two are bond related components, which can be taken as an indicator for the market fear propensity. The less confidence for investors in other financial market, the higher the yield from the bond holding. In another word, investors are more likely to pursue the less risky asset (bond), which will in return pushes up the bond yield. The change of money stock and TED spread are regarded as liquidity pressure when market is experiencing a downward trend. Lower change of money stock and higher TED spread imply less liquidity in the market, pointing out the less money inflow on risky asset and driving down the price of risky assets. The last two components introduced here to be the proxy variables for the probability of corporates default. Quantitatively, the higher the differential, the higher the probability for occurrence of corporate and bond defaults, which results in the potential drop in the financial system especially when some large financial service companies are included in. This is given the huge business scale for these companies with credit 2 library.html 3

24 24 based financial assets. Once there is high probability of default rate, chain effect on default pay-off will lead to vicious circle which contaminates the whole financial sector. In conclusion, risk factor who has the ability of pricing the commodity futures, should has the potential capability of explaining the change for these proxy variables for macroeconomics activities. 3.3 Risk-Neutral Skewness Estimation Risk-Neutral Skewness Risk-neutral skewness is renamed in the spirit of the implied volatility (also know as risk-neutral volatility). The reason for it is to be risk-neutral is due to the risk-neutral probability measurement which assumes there is no arbitrage under this measurement. Implied volatility has been found with the ability of reflecting the market expectation about the future market change degree. Based on the same idea, risk-neutral skewness should possess similar property. Risk-neutral estimator based on the idea of future expectation has a lot of advantages than the ex post one. From the intuition behind, the incentive for investors to hold the future contract is due to the future potential outcome, here is profitability. This means that investors are making decision based on the future expectation rather than the past performance. In this paper, the estimation method of the futures risk-neutral skewness is based on Bakshi et al. [5], who gives specific equations to be followed when estimation is applied. Unlike the method mentioned by Boyer et al. [13] who uses regression approach including past skewness and volatility to estimate the future skewness, the model-free method used in this paper does not require any contribution from the past moments estimator. Meanwhile, an obvious merit is worthy mentioning that for each product, only one single daily future and corresponding option price are needed. Technically, this has greatly reduced the work to estimate time series skewness. When estimating the realized skewness and volatility, it is a trade-off to select estimation time length. Technically, this accuracy of estimation result is not consistent as different time length can affect both estimation bias and trading time left after estimation period. Generally, the longer the data, the more accurate the results will be. At the same time, this is

25 25 also a dilemma for time series estimation as the more information you include, the less sample you can use. In this paper, such trade-off can be offset when using the model-free method. Generally, Bakshi et al. [5] shows that pay-off of securities can be priced by using a set of option data with different strike prices on that securities. Firstly, the quadratic, cubic and quartic pay-off need to be constructed in terms of different maturities strike and option prices. Then, the risk-neutral moments is calculated by some combinations of these pay-off. Mathematically, the equation for estimating the skewness is as follows: RN.SKEW.F Q i,t (τ) = erτ W i,t (τ) 3µ i,t (τ)e rτ V i,t (τ) + 2µ i,t (τ) 3 [(e rτ V i,t (τ) µ i,t (τ)) 2 ] 3 2 (2) Where, RN.SKEW.F Q i,t (τ) is the risk-neutral Q measure of skewness of underlying i at time t with τ maturity. V i,t (τ), W i,t (τ) and X i,t (τ) are the time t prices of τ maturity quadratic, cubic and quartic contracts, respectively. r is the risk-free rate, which is three-month treasury bill. V i,t (τ) = S i,t K i 2(1 ln( S i,t )) C i,t (τ, K i )dk i K 2 i (3) Si,t 2(1 + ln( S i,t K + i )) P i,t (τ, K i )dk i 0 K 2 i W i,t (τ) = S i,t K i 6(ln( S i,t ) 3(ln( K i S i,t )) 2 C i,t (τ, K i )dk i K 2 i (4) Si,t (6ln( S i,t K + i ) + 3(ln( S i,t K i ))) 2 P i,t (τ, K i )dk i 0 K 2 i X i,t (τ) = S i,t K i 12(ln(( S i,t ) 2 4(ln( K i S i,t )) 3 ) C i,t (τ, K i )dk i K 2 i (5) Si,t 12(ln(( S i,t K + i ) 2 + 4(ln( S i,t K i )) 3 ) P i,t (τ, K i )dk i 0 K 2 i

26 26 µ i,t (τ) = e rτ 1 e rτ V i,t (τ)/2 e rτ W i,t (τ)/6 e rτ X i,t (τ)/24 (6) Where, C i,t (τ, K i ) and P i,t (τ, K i ) are the time t prices of European out-of-money calls and puts written on the underlying product with strike price K and expiration τ periods from time t, S i,t is the i th underlying security s price, in the commodity future market, standardized nearest to maturity contract price is a proxy variable. As for the specific operation, since the practical analysis is to deal with discrete data, so trapezoidal approximation is implemented to estimate the integral for equations (3)-(5). In order to make the result precise, certain filter conditions are applied during the estimation procedure. The minimum number of call and put OTM option price for the calculation is at least 4. At the same time, the number of call and put options should be equal in order to estimate. Then it is necessary to exclude the deep OTM option data as they are more likely to be less traded due to its non-profitability. Including this kind of options will increase the estimation bias to some extent. Therefore, to be specific, moneyness is defined for OTM call and put option with different calculation formula K i S i,t and S i,t K i. Trapezoidal rule only deal with the data with moneyness varying within the range of (1,1.5]. Finally, I exclude those options which have only 4 days left to maturities as the trading behaviour on these options will distort the fair value of option value themselves to some extent. So it is unwise to include these contracts in estimation process. After the estimation procedure with daily option price data, the estimated riskneutral skewness factors are obtained with different maturities. These different maturities imply that different variations in skewness for the future corresponding time maturities. If there is a risk-neutral skewness with 30 days left to maturity, this means the expected future 30 days skewness variation at current time period. In order to analyse the effect from this factor, following literature, an interpolation method is applied by constructing a constant time maturity series risk-neutral skewness for each underlying product. Consequently, 30 days constant maturity

27 27 skewness is obtained by the following interpolation formula: RN.SKEW.F 30 = (T 1 RN.SKEW.F 1 T 2 T 30 T 2 T 1 + T 2 RN.SKEW.F 2 T 30 T 1 T 2 T 1 ) T 365 T 30 (7) Where, RN.SKEW.F 30 is risk-neutral skewness with maturity date equal to 30. RN.SKEW.F 1 is selected risk-neutral skewness with maturity date less than 30 days and RN.SKEW.F 2 is selected risk-neutral skewness with maturity date larger than 30 days. T 1 and T 2 are corresponding maturity date. T 30 and T 365 are exactly equal to 30 and 365. After interpolation, there will be a single time series RN.SKEW.F 30 for each underlying product. If there is 30 days risk-neutral skewness estimated already, it then will be used without interpolation step. If either RN.SKEW.F 1 or RN.SKEW.F 2 is missing, the value for RN.SKEW.F 30 at this point will be replaced by null value. This process is applied to all underlying products. As a result, the outcomes for all global future products are risk-neutral skewness matrix with constant maturity day Commodity Variables - Traditional Risk Factors The future basis, traditional defined as the price differential between spot price and future contract price. However, in the real trading, spot price is not as liquid as future contract price. Consequently, following the literature, basis is the log difference between the second-nearest to maturity contract price and the current nearest to maturity contract price. Consistent with Koijen et al. [43], the traditional future basis is scaled by their maturity differential (T 2 T 1 ), which transfers the log price difference into same unit. This is a necessary operation for the comparison in the cross sectional ranking procedure. Mathematically, the future basis 4 The idea why we focus on one month constant maturity is due to skewness estimation purpose. By moving maturity to 60 and 120 days, the number of interpolated constant risk neutral skewness shows decreasing effect. Therefore, to avoid data inconsistency and bias in the future analysis, we only focus on 30 day maturity

28 28 factor is as follows: Basis.F i,t = log(p ricet 2 i,t /(P ricet 1 i,t )) T 2 T 1 (8) Where, Basis.F i,t is the future basis factor for contract i at time t; log(p rice T 2 i,t ) is log price of the second-nearest to maturity contract i at time t; log(p rice T 1 i,t ) is log price of the nearest to maturity contract i at time t; T 2 and T 1 are the corresponding time to maturity, measured by days. Intuitively, it shows the relation between two prices with different maturities. If the basis is positive, the future contract price with long maturity date is larger than the one with short maturity date. In order to satisfy the non-arbitrage condition, they will converge to the expected spot price, which will push up short maturity price and drive down longer maturity price to some extent. Then, it generates profits when investors either but the nearby contract or sell the future contract. Consistent with the literature and to avoid the liquidity trap, the nearby contract is the most liquid one that can be traded in portfolio construction. In conclusion, this implies a positive relationship between future price and basis. For the hedging pressure specific factor construction, it measures the degree to which hedgers and speculators participate in specific direction. Formally, construction of the hedging pressure for speculators is the net non-commercial short position divided by total non-commercial open interest. Mathematically, for each contract i at time t, the hedging pressure factor is as follows: HP.F i,t = NCS i,t NCL i,t NCS i,t + NCL i,t (9) Where, HP.F i,t is the hedging pressure factor for contract i at time t; NCS i,t is number of commercial short position in contract i at time t; NCL i,t is number of commercial long position in contract i at time t. If Hedging Pressure value is positive and large, this demonstrates the increasing short positions hold by the non-commercial investors in the nearest to maturity

29 29 future contract. In terms of the fundamental idea, future contract is offered when commercial traders want to hedge their risks in spot market, therefore high Hedging Pressure value in this definition indicates the opposite side for investors to short the contract. In another word, the future contract holding premium shows when Hedging Pressure Factor value is high and investors short the future contract. There signs a negative relation between HP value and future price. Following the proposal from Asness et al. [3] and Miffre and Rallis [51], the momentum factor, another non-trivial return driven factor, is defined as the simple moving average of the past 12-month return. The window length chosen is random in the general case, but here is strictly due to the fact that they prove the best profitability when using past 12-month as the window calculation length. Mathematically, this factor is constructed as follows: MOM.F i,t = t=k 1 t=k 52 Return i,t 52 Where, MOM.F i,t is the momentum factor for contract i at time t, Return i,t is the return series for contract i at time t, 52 stands for weeks data used in calculation. In this paper, return data are converted to weekly frequency, so the moving average for momentum factor is calculated by using past 52 weeks return series for each underlying. (10) The rationality for MOM.F factor working properly is the belief that past performance will continue in the future. Past positive average return will continue to exist in the next period. So, this concludes a positive relationship between MOM.F factor and future price Risk Factor Portfolio Construction After getting the estimation of all risk factors mentioned above, based on the risk factor estimation process populated in stock market, the standard high minus low risk factor portfolio return is constructed. To be specific, TS, HP, MOM

30 30 and RN.SKEW are one dimensional time series long and short portfolio return in terms of each risk factor. For example, the average value of paste 12-month future Basis.F are cross-sectional ranked ascending and grouped into 5 subgroups. In the next stage, the equally weight future contracts return in the top group asset minus the equally weight future contracts return in the bottom group asset is denoted as the high minus low portfolio return, which is represented by TS. This means the total portfolio return that investors can get if they buy top sorted group asset and short bottom sorted group. In this paper, according to the sign between factor and return, high minus low approach can be reversed to low minus low in order to make factor based strategy profitable (positive average return). Specifically, TS and MOM are following high minus low portfolio, while HP and RN.SKEW are low minus high portfolio. This difference inherits from risk factors attribute based on behind intuition. All portfolios are sorted on 12-month (52 weeks in weekly frequency setting) ranking period, and the holding period is one week. Portfolios are rebalanced weekly when new observation is collected. The 52 weeks ranking period are for comparison convenience among different strategies and this also reduce the seasonality effect by making the factors smoothing enough as commodity futures are strongly related to the demand and supply for underlying products Rationale for OLS Analysis In asset pricing world, both generalized moment (GMM) and ordinary least square (OLS) methods are widely used in literature. OLS estimation is popular among nearly all empirical studies as it is easier and more straightforward for purpose of understanding and implementation. Fama and MacBeth [29] are the most famous among researcher applying OLS estimation, which is followed by studies from (Bakshi et al. [6],Basu and Miffre [8],Fuertes et al. [32],Daskalaki et al. [19] and Fernandez-Perez et al. [30]) on global commodity futures pricing. However, OLS estimation has been argued with lack of ability in handling with heterogeneity in error term. In terms of this argument, Newey and West [54] propose unbiased covariance variance matrix estimation based on kernels, lags and bandwidths selec-

31 31 tion. By estimating for certain lags chosen, asset pricing OLS regression error due to large portfolios or assets data generating process correlation can be corrected. GMM estimation introduced by Hansen [36], focusing on matching theoretical moments and empirical moments in terms of robust penalized matrix, can also provide stable estimation in regardless of concerning on heterogeneous effect inherited from asset pricing theory. Recently, comparison study on estimation methods in simulation analysis shows the better performance for OLS rather than GMM (Shanken and Zhou [59]). Sampling from multivariate normal and t distribution shows small pricing error for OLS estimation given fact that the number of asset is relatively small. Based on these analysis, I will apply only OLS estimation via time series and cross section dimension to analyse the pricing ability in new risk factor Summary Statistics Analysis In the table 1, panel A shows the correlations among different risk factors portfolios return. It is clear that all risk factors are partial correlated with correlation coefficient less than 0.5. It is also worthy mentioning that MOM is the one with small correlation value when computed with other factors, which illustrates that MOM contains relatively much more different information than others. The riskneutral skewness is mainly related to TS and HP factors with coefficient over 0.4, shedding some lights on the common information they may overlap with each other. Although the correlation coefficient is not small, uncorrelated part is still important, which should be studied to some extent. Panel B in the table 1 gives the summary statistics of risk factors portfolios. In spite of the fact that riskneutral skewness sorted portfolio is not the least volatile one, it has the largest annualized mean return and sharp ratio, 10% and 1.2 respectively. At the same time, from the view of portfolio return distribution, risk-neutral skewness sorted portfolio are more close to the normal distribution, showing nearly no potential extreme negative or positive value along the trading period. This can be also confirmed by the 99% VAR and positive return month, which proves that the new risk factor outperforms other traditional risk factors sorted portfolios quantitatively and qualitatively.

32 32 The summary statistics of return series for all commodity products are listed in the table 2. Average mean for this sampling period is generally negative implying the loss when simply holding contracts in the commodity futures. High moments, skewness and kurtosis show that all the return distribution is not standard normal, with either negative or positive skewed and fat tail in the distribution. In order to compare with the traditional factor, here future basis is taken as an example as it is widely stated in the literature. Obviously, column 7 is the correlation between return series and future basis, positive correlation implies fact that term structure is indeed implies the truth of backward theory. Higher basis will give the incentive for nearest to maturity contract converging upward to the future contract with higher price, then higher return. Compared with the future basis, column 8 shows that the correlation coefficient between risk-neutral skewness and future contract return is negative. Moreover, this is matched with the correlation coefficient in column 7, which supports the different portfolio construction perspective. From column 6, autocorrelation is small enough to support the independence attribute of future contract return series mentioned in the literature. For the details of the new risk factors, statistics information is listed in table 3. The first four columns are the general descriptive statistics like mean, standard deviation, skewness and kurtosis of risk-neutral skewness factor. With respect to the mean value, the majority is negative, pointing out that the risk level for most commodity products is under low level averagely. This is consistent with our data time span as the only extreme time period is 2008 financial crisis, while other left period shows strong incentive for investing in commodity future market. The second-last column is about risk factor one period lag autocorrelation, which gives the idea that skewness is to some extent correlated with themselves. With respect to the last column, it is the correlation coefficient with the traditional future basis. On the average, the apparent negative correlation support the portfolio building theory, while the relatively small absolute value for coefficients represent that new risk factors deliver more extra information after consideration of the traditional future basis effect.

33 33 4 Empirical Results In this part, estimated risk-neutral skewness will be analysed from different perspectives. Continuing the criteria of identifying and testing risk factor mentioned in the literature, time-series and cross-sectional regression based approach are adopted, which can explain both how much impact risk factors have on return from assets and whether the risk factor is a satisfied and necessary factor. Robustness check is included to make sure that results are general and solid when other potential risk factors are taken in account in the literature. Bootstrap simulation is applied on individual level to illustrate the fact of no abnormal return generated when our selected risk factors are used. In addition to these steps, prediction and exposure regarding the macroeconomics factors will be concerned in order to see the property and usage scale of risk factor. Before everything on procedure, some apparent ways of showing the pattern or non-strict relation for cross-sectional level and risk-neutral skewness is reported via figure. In the figure 1, it shows the time-varying relation between price level and new risk factor. Price level and risk factor are calculated by averaging cross all available products at each time period. The red line, denoted as price level, hits the lowest point after the black line (risk-neutral skewness level) hits the peak. After this peak, skewness gradually declines accompanied by the ascending witnessed from price level. Another obvious time point is after year 2014 when price level suddenly jumps a lot as the skewness value shifts to lowest point measured over this whole sample period. Therefore, combined with the real word analysis, it is reasonable to argue that there should be some significant negative relation between these two variables, especially risk-neutral skewness reflect fast and quick than the price level. In another word, skewness factor can be regarded as a potential useful factor with strong predictability with respect to the return from commodity futures.

34 Time Series Analysis In this part, time series analysis is first introduced and the aim of this is to investigate whether the new risk factor based approach can generate extra information when all traditional risk factors are taken into account. All available product contracts are sorted ascending by risk-neutral skewness factor based on the estimation procedure mentioned in the methodology part. In total, there are five groups for products to be grouped in, which leads to the number of products in each group at least larger than three. For each group, portfolio return is calculated by averaging all products return on the same cross-sectional time date. In another word, portfolio is constructed by equal weight, which is more meaningful when different factors are compared. Firstly, new risk factor sorted approach performance will be argued among groups. For this purpose, portfolio related statistics are calculated in a standard way, including annual return, standard deviation, sharp ratio, adjusted value at risk and etc. Then, among each group, statistical analysis from ordinary least square (OLS) estimation is implemented for each group portfolio return against traditional risk factors introduced in the literature and methodology. Intuitively, the performance will shed some lights on the relation between factor and asset return in terms of strategy performance. OLS regression is aimed at exploring the extra information new risk factor can generate, which is reflected by Alpha (well known as the abnormal return by excluding the unneeded information). The whole process is following the equation as follows: P R i,t = α i + β 1,i T S t + β 2,i HP t + β 3,i MOM t + ɛ i,t (11) Where, P R i,t is the sorted portfolio return at time t, for instance P R 1,t is matched for P1 in this paper at time t. α i is the abnormal return in i th portfolio regression analysis. ɛ i,t is the i th portfolio regressed error term at time t. i is correlated to the which portfolio specify from P1 to P5. TS, HP and MOM are buying and selling portfolio return, which are constant for each asset time series regression. From the table 4, panel A, all available commodities have been assigned into 5 quantile portfolio with ascending order based on risk-neutral skewness factor. For the annualized mean, it is obvious that there is a general down-sloping trend for

35 35 portfolio return from P1 to P5, with annual return from 5% to -15%, except slightly increase in portfolio 4. This demonstrates the general idea that risk-neutral skewness factor is negatively related to return. Regarding to the last column, the low minus high portfolio (P1-P5), generates stable annual return 10% with volatility is only 9% annually. From the view of return distribution, low minus high portfolio is distributional close to normal with skewness close to zero and little flat tail. Compared with single buying portfolio, buying and selling portfolio is more reliable when applied in the real market with less risk to some extent. Meanwhile, the corresponding VAR, widely used as a measure of potential risk, is the least for the low minus high portfolio with only 4%. Compared with other portfolio combination, buying and selling strategy has 1% probability with loss over 4% of total capital. Sharp ratio is large on both two extreme quantile, but showing strong asymmetric property combining the result from annual return. This implies that the high skewness sorted portfolio implies strong sensitivity during this selection period. After including selling availability, the low minus high portfolio delivers 1.2 sharp ratio, which is significant higher than long only strategies here and even other strategies mentioned in the literature considering the global future market so far (0.47 term structure strategy in Erb and Harvey [25], 0.75 hedging pressure strategy in Basu and Miffre [8], 0.67 carry strategy in Koijen et al. [43], value and momentum factors less than 1 in normal case in Asness et al. [4] and 1.1 time series momentum strategy with all available future products in global market in Moskowitz et al. [52]). From the panel B, it is clear that all quantile portfolios are generally highly affected by the traditional risk factor, especially the momentum factor (MOM). It is worth mentioning that only P5 and P1-P5 portfolios can generate extra alpha, which is statistically significant. Combined with the result from panel A, asymmetric effect from risk-neutral skewness is still existed with high skewness contributing huge significant abnormal return while low skewness does not. Since the data is based on weekly frequency, the extra annual return from regression approach is 8.3% for buying and selling strategy. Also on the last column, it shows that risk-neutral

36 36 skewness sorted portfolio does not only contain the information from backwardation and momentum theory but also provide extra idiosyncratic information.

37 Cross Sectional Analysis In this section, two dimensional regression approach is applied following Fama and MacBeth [29] two-step regression. In the first step, time series OLS regression is used to get the beta coefficient for each asset. Then, assuming all betas are constant for the whole portfolio construction period, the second step is to run cross-sectional regression at each time period to get the risk premium. Specifically, cross-sectional asset returns are regressed against all constant betas, yielding the risk premium called gamma. It is worth mentioning that gammas are time series estimators for betas. Finally, gammas are averaged out to get the estimated risk premiums and their t-statistics can be calculated as well. Intuitively, the idea held here is to explore the average risk premium that can be obtained by assuming the beta coefficients in portfolios are constant. After this analysis, explicit explanation can be given to quantify how much average risk premium will change by one unit change in the underlying risk exposures, betas. In our analysis, single factor and factors combination are tested separately, which is for controlling comparison. Mathematically, in the first step, it follows as: R i,t = α i + β 1,i T S t + β 2,i HP t + β 3,i MOM t + β4, i RN.SKEW + ɛ i,t (12) Where, i is standing for each asset instead of each portfolio in time series regression. R i is the i th asset return in time series dimension. β i is the i th asset risk exposure to risk factors. In the second step: R i,t = α i,t + γ 1,t β 1,i + γ 2,t β 2,i + γ 3,t β 3,i + γ 4,t β 4,i + µ t (13) In the table 5, TS, HP, MOM, and RN.SKEW are sorted buying and selling strategy return based on following smoothing factors: paste 12-month future basis, paste 12-month hedging pressure, paste 12-month average return and paste 12- month risk-neutral skewness. From column 1 to 4, the separate risk factor is tested, showing that except for the hedging pressure all other factors are statisti-

38 38 cally significant.this might be the idea of sampling problem given few researchers testing the same sample as we do. From the column 4 to 6, RN.SKEW is primarily tested by controlling each traditional risk factor receptively, aiming at seeing whether our new risk factor still matters when previous proved information added in. The testing result indicates that the new risk factor, RN.SKEW, is still valid by providing extra information. Obviously, coefficient for RN.SKEW in column five is reduced, which is consistent with the result from correlation analysis in table 1 as common information has been filtered out via regression idea. Momentum factor inclusion, on the contrary, pushing up both coefficient in column 7 as their correlation is relatively small. The column 8 is to follow the literature to see whether those traditional popular risk factor is significant as they are in the previous way. Insignificance has been found for hedging pressure (HP) factor, while significances still exist for term structure (TS) and momentum (MOM) factors. Again, this contradiction may due to financial crisis effect given strong momentum effect given the convergent expectation on future volatility. The last column is the most important one as the new risk factor is tested by controlling all other factors. The significance of in column 9 proves the fact that new factor include different information than the traditional factors have. Meanwhile, the last row, adjusted R square is increasing from 21% to 27% by adding the new risk factor, RN.SKEW. That also confirms the pricing ability of our new risk factor as it implies the necessity of including RN.SKEW when investors want to pricing the commodity futures return in the global market. Moreover, in the regression with only RN.SKEW factor, the adjusted R square is nearly 5%, while the overall R square increase when considering all risk factors is more than 6%. This is a clue for the fact that the new proposed risk factor has some information relating to traditional ones, therefore, the inclusion of them will make new risk factor outstanding in regression performance. All of these evidences suggest that it is possible to regard this new risk factor as a common pricing factor in global future market pricing model. Meanwhile, new risk factors performs better than traditional one given the cumulative return evidenced in figure 2. More evidence on factor properties can be found in overlapping analysis (illustrate the idea that to what extent do two risk factor based portfolios share common

39 39 information) in figure 3, 4 and 5. For each ranking period, assets name sorted into bottom and top quantile will be extracted out separately for each portfolio. Depending on different signs (theory suggested properties), two quantiles will also be compared to our interest RN.SKEW in order to find out the number of common assets sharing. Specifically, for TS and HP factor, top-bottom (top quantile assets should be compared with bottom quantile suggested from RN.SKEW), while MOM is consistent with top-top and bottom-bottom. After comparing this two quantiles, summing up the number of common asset from two comparisons will give the information on to what extent (measure by number of asset) do these two strategies share common interest. Overall speaking, the overlapping number is all below 5, which confirms the risk factor correlation matrix shown in the table 1. To be specific, in figure 3, it shows the number of overlapping asset between TS and RN.SKEW are more likely to shift around 1, 2 and 3, implying there difference asset selection base. Similarly, MOM, in figure 4, also shifts a lot to show different selection ability. However, compared with HP in figure 5, they are both losing connect during the end of 2013 and the beginning of 2014 when HP are gradually showing matching ability with RN.SKEW. One possible idea to identify is the economic announcement from OPEC stating the disagreement on oil production cutting, which drive down the crude oil price dramatically. Such high correlation among asset selection during this period and even afterwards can be due to heterogeneous belief effect embedded in RN.SKEW (Han [35], Diavatopoulos et al. [21] and Friesen et al. [31]).

40 Robustness Check Risk Factor Based Approach Built on the result in the empirical analysis, there is still something arguable as recent literature propose more common risk factor needed to be considered in the global future market. Therefore, in this section, robustness analysis is introduced from different perspectives to justify the stable influence from risk-neutral skewness. In the literature, some researchers argue that the equally weight portfolio (average the return for all available contracts return) is an important risk factor when doing commodity future pricing. Instead of directly using equally weighted portfolio without any evidence, the principal component analysis (PCA) is implemented on all commodity futures returns. According to results in table 6, the component 1 does explain the most variation of future return and the factor loadings for each products are nearly equal. This can be seen from the figure 2 with first 5 main components are plotted. The blue line is the first component line for all 25 commodity products with the weight is nearly close to each others. Consequently, equally weighted portfolio is constructed by averaging all available contracts return at each time period, denoted as EW 5. Then, time series analysis is first adopted to justify the influence when considering EW in table 7. The result shows strongly significant positive effect between EW and single buying risk-neutral sorted portfolio. In the last column, buying and selling portfolio return has a significant negative exposure on equally weighted portfolio, suggesting the loss of new strategy is inherited from the average moving on all assets return. Rather than simply using equally weighted portfolio as a risk factor, some researchers propose other new risk factors: total skewness (estimated based on daily return series), idiosyncratic skewness (estimated by running regression of single asset return on traditional risk factors and calculating the third moment of residual 5 Principal component (PC) based factor for the fist component is also constructed by multiple between single return series and corresponding first component loading to test cross sectional pricing ability, showing no significance, which is not reported in this appendix.

41 41 term) and idiosyncratic volatility (similar to idiosyncratic skewness but calculating the standard deviation of residual term). Robustness check in time series analysis is applied by taking above risk factors into consideration. In table 8, asymmetric property is still apparent with high skewness contributing only slightly more on abnormal return when more variables added in. Among all of these variables, idiosyncratic volatility shows strong relation in most scenarios including extreme tail and buying-selling strategy. The coefficient for it is positive, which is consistent with the literature as investors buy low volatility and sell high volatility. This can be clearly supported by the study from Boyer et al. [13] who use lagged idiosyncratic volatility to predict the value of expected skewness and show the correlation between expected skewness and asset return is positive and strongly significant. Estimated expected skewness also shows good property in asset pricing test by passing two-step regression test. Moreover, total skewness based portfolio return also accounts for the variation from risk-neutral skewness sorted portfolio. The same sign direction for both idiosyncratic volatility and total skewness illustrates the same information contained, but risk-neutral skewness can provide more extra different information when looking at the alpha term at the top of table. In addition to time series analysis, robustness check about cross-sectional regression is reported in table 9 with all proposed variables included. Based on results, it is hard to conclude that most proposed risk factors have pricing ability when they are tested alone except for the TOT.SKEW which is significant in single testing in column 3. However, this effect vanishes clearly when previous traditional risk factors are added in regression. Their exclusion implies that the same information has been dominated by those traditional factors. Considering other columns with different combinations, RN.SKEW coefficient is time-varying but still show statistical significance. The Most comprehensive regression is on the last column in table 9, all proposed risk factors are put together with results concluding that RN.SKEW, TS and MOM are the most efficient risk factors needed to be concerned in the global future market.

42 Bootstrap Analysis Inspired by Kosowski et al. [45], bootstrap simulation analysis is used to demonstrate the no abnormal return from individual asset level. The intuition for doing this is due to the unbalanced panel data for future contracts. For data structure, time series data length inconsistency may not explicitly test the asset pricing error in a joint way. Initially, individual assets are regressed on TS, MOM and RN.SKEW in time series dimension. Asset.Return i,t = α i + β 1,i T S t + β 2,i MOM t + β 3,i RN.SKEW t + ɛ i,t (14) Where, ( ˆα i, ˆ β1,i, ˆ β2,i, β3,i ˆ and ɛˆ i,t for i = 1,2,3,... and for t = 1,2,3,...) is obtained. The corresponding t-statistics for alpha are recoded to show assets significance on abnormal return when all risk factors are considered. After getting the regression estimation, in the next step, a new pseudo time-series residual data is generated by sampling from ˆɛ with replacement for each product times re-sampling process have been repeated, which generates 1000 pseudo residuals columns. Then, new estimated asset return is calculated by using real data for risk factors, estimated betas and pseudo residuals for 1000 times. Asset.Return i,t = ˆ β 1,i T S t + ˆ β 2,i MOM t + ˆ β 3,i RN.SKEW t + ɛ i,t (15) Where, ɛ i,t is pseudo time-series residuals for 1000 columns. Asset.Return i,t is estimated asset return without alpha concern theoretically. It is necessary to mention that for estimated asset return series, estimated alpha is not included, which will give the pseudo asset returns without abnormal information. Finally, time series regression is applied again by regressing pseudo asset returns on risk factors. Asset.Return i,t = αi + β1,i T S t + β2,i MOM t + β3,i RN.SKEW t + ɛ i,t (16) Where, αi for i = 1,2,3... are estimated with its t-statistics documented correspondingly. In this bootstrap analysis, table 10, the value of α and t-statistics t α are shown

43 43 for each asset. Given the better property of t-statistics rather than coefficient (volatility standardized for the former one), we choose t-statistics for result interpretation. The underlying idea behind this method is to find out the whether real alpha t-statistics is out side the threshold of re-sampling alpha t-statistics distribution. If empirical value is larger than the re-sampling distribution threshold, then it is possible to conclude that this asset does generate non-negligible extra information (abnormal return) when risk factors are taken into account. In the individual regression analysis, soybean meal shows strong significant alpha with t-statistics larger than 5% significance level, while others do not. Since the t- statistics of alpha for soybean meal among all products is the maximum, bootstrap is implemented by choosing maximum t-statistics from each re-sampling regression cross over all future products. As a result, there will be 1000 times bootstrap alpha t-statistics, which is plotted by kernel density function in figure7. The intuition behind this is that if only a few value larger than real alpha t-statistics, it is impossible to state non-existence of abnormal return. Based on result, from figure 7, it is obvious to see that empirical value (vertical blue line) is nearly in the middle of sampling distribution density, stating that the significance of alpha in the real analysis is mainly due to sample variation instead of the extra information this asset presents.

44 Economics Rationale for A Risk Factor According to Cochrane [17], the necessary condition for a factor being common factor is that the factor should have capacity of predicting the macroeconomic investing proxy variables. Also inspired by the analysis of economic exposure from Moskowitz et al. [52], some economy relations should be studied to provide more information on risk factor suggesting strategy. Therefore, risk factors are used to test to what extent they are exposed to and predict the macro-economy related proxies. The first category of macroeconomics proxy variables I choose are US treasury bill return, 10-year and 30-year return series. Different forecasting periods are selected and taken as the dependent variables in OLS model. The second category of proxy variables will be the average level of money stock change and TED spread. In another word, moving average is applied by choosing varying window lengths on data smoothing. The third category is about the opportunity for default and credit risk, which I use two representatives following the literature: DFS (Moody s Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year Treasury Constant Maturity), CRS (Moody s Seasoned Baa Corporate Bond Yield relative to is the Moody s Seasoned Aaa Corporate Bond Yield). From table 11, the new factor is statistically significant when it comes to predict the middle and long time relation for 10-Year Bond by up to 12 month ahead. While for 30-Year bond, RN.SKEW can only account for predictability in 6 month ahead. The positive sign for RN.SKEW in time series forecasting regression suggests that higher skewness (also related to higher risk) will result in the seeking of risk protected asset like government bond. This turns out that the return from bond will increase as it becomes the popular asset in the market with huge cash inflow. Momentum factor presents negative significant effect on 30-Year bond s return prediction. When there is market trend existing, momentum behaviour investors will catch the trend with large cash outflow from the bond investment. This risk compensation from trading in momentum behaviour becomes more attractive than holding in less risky bond, impairing the return for government bond. As for the average level prediction on money change and TED spread, from the

45 45 table 11, there is no evidence that any factors can predict the change in the money stock in any time period. While, RN.SKEW is the only factor which shows the predictability in TED spread from more than 3-Month period. The most strong evidence is the middle-term (6-Month) with coefficient 0.02 and 0.01 for 9 and 12 months. The positive relation implies that higher RN.SKEW will amplify the difference between interbank rate and Libor rate. As a result of increasing TED spread, liquidity is restricted among banks, indicating the liquidity shortage in the financial trading market. Consequently, funding and financing activities for investors are limited, which lead to the unwinding of investors positions in margin requirement transactions. Such unwinding of positions will exaggerate the risk exposure in the market, creating the market sentiment for escaping away from trading in financial market. Finally, prices for assets will go down by stairs and involve in an cycle for driving assets value down further. Therefore, past RN.SKEW can give a clear view of future market cash flow and market sentiment direction prediction. Avoiding trading in high level of RN.SKEW can reduce investors loss to some extent. In terms of the default and credit risk, the result from table 13 confirms the positive relation with RN.SKEW and future DFS and CRS. The increase in RN.SKEW will increase the rate yielded by the high risky company rated in Moody s benchmark. Following the modern asset pricing theory, the intuition for explaining the increase of DFS and CRS is due to the idea of risk compensation. It is the reason of bad state suggested by RN.SKEW with an increasing level of value, holding risky or less credit guaranteed asset should provide extra risk premium for investors. The significant positive sign confirms the explanation. Like previous two aspect, the main prediction period is focusing on middle-term for both DFS and CRS. Finally,in the table 14, risk factors are taken as the dependent variable with respect to some underlying factors like Fama French three factors (MR, SMB and HML), cross-sectional volatility (CROSS VOL), DFS, CRS and TED. In the column 2 and 3, term structure and momentum factors are exposed to SMB factor, indicating the fact that backwardation theory and trend following idea is similar to the intuition of buying stocks which is in small size group. In another word, this suggests that

46 46 the return is the compensation for the risk investors take. For momentum factor, the cross-sectional volatility is also a good way to explain the return losing from this strategy as volatile market can not be clearly marked with entry and exist. During this period, momentum strategy loss its advantage to some extent, with one unit increase in CROSS VOL. Different from all these, RN.SKEW is not explained by any underlying as the construction of this factor is about future expectation not on the past belief 6. Even though RN.SKEW share some information with TS ans MOM in time series regression, they are not presented here based on above listed underlying proxies. 5 Conclusion Global future market return is tested based on Fama Macbeth two step regression, showing the sufficient and necessary risk factors for pricing future return. 10 years option data, future data and trading position data are used in empirical analysis, which suggests that the parsimonious risk factors for return pricing are term structure, momentum and risk-neutral skewness. Several new proposed common factors in the literature have been considered in regression analysis, shedding some lights on robustness of risk-neutral skewness. That is, RN.SKEW is strong valid factor for global commodity future return generating process even though some current popular factors are considered. Moreover, bootstrap analysis is adopted on individual asset level to demonstrate the pricing abilities from these three factors due to unbalanced panel and non-normal distribution. Results confirm the nonexistence of pricing error in terms of the empirical t-statistics is less than sampling t-statistics distribution based on 1000 sampling times. In another perspective, predictability of RN.SKEW and its determinants are studied with outcome suggesting that RN.SKEW has strong capacity for predicting Marco-economy related variables like bond return, TED spread and corporate default rate. Meanwhile, no underlying variables are related to the data generating process of RN.SKEW. 6 Results for prediction on equity based dependent variable are not shown here given no significance found, suggesting the heterogeneity for futures and stocks, which is consistent with literature.

47 47 This implies the fundamental of RN.SKEW: the expectation from investors about future risk movement. Therefore, it is not explained by underlying property as past information is now unrelated. In conclusion, all of these point out the necessary and sufficient condition for taking RN.SKEW into account when investors are pricing global future return. In another word, RN.SKEW is a common factor for pricing global future return.

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56 56 6 Appendix Table 1: Statistics for Risk Factors Table 1 gives a general introduction on some property of risk factors (TS,MOM,HP and RN.SKEW). Pair correlation for all risk factors mentioned in literature is in panel A. Corresponding statistics are listed in panel B with each column standing for each portfolio statistics sorted on risk factor. Pabek A: Correlation Matrix TS MOM HP RN.SKEW TS MOM HP RN.SKEW Panel B: Summary statisitcs Mean StDev Sharp ratio Skewness Kurtosis % VAR(Cornish-Fisher) % of positive months

57 57 Table 2: Summary Statistics for Return Series on Individual Assets Level Mean Volatility Skewness Kurtosis Autocorrelation Ln(Basis) RN.SKEW Brent.Oil Cocoa Coffee Corn Cotton Feeder.Cattle Gold Heating.Oil Lean.Hogs LS.Crude.Oil Live.Cattle Oats Orange Palladium Platinum RBOB.Gasoline Rough.Rice Silver Soy.Meal Soybean Soy.Oil Sugar Wheat WTI.Crude.Oil Natural.Gas

58 58 Table 3: Summary Statistics for Risk-Neutral Skewness Factor on Individual Assets Level Mean Volatility Skewness Kurtosis Autocorrelation Ln(Basis) Cocoa Coffee Corn Cotton Soy.Meal Soybean Soy.Oil Sugar Wheat Rough.Rice Lean.Hogs Feeder.Cattle Live.Cattle Oats Orange Natural.Gas RBOB.Gasoline Brent.Crude.Oil Heating Oil WTI.Crude.Oil LS.Crude.Oil Gold Silver Palladium Platinum

59 59 Table 4: Time Series Analysis Portfolio Property based on Risk-Neutral Skewness This table is for reporting about risk-neutral skewness based portfolio properties. In panel A, assets are sorted based on estimated risk-neutral skewness, with the lowest five commodities (P1) and the highest five commodities (P5). The first column is for statistics and the last column is for low minus high portfolio with equal weight. In panel B, time series linear regression is for testing the factor exposure as well as abnormal alpha. Simple standard error (Newey-West standard error) is reported in bracket (square bracket). Portfolios P1 P2 P3 P4 P5 P1-P5 Panel A: Summary statisitcs Mean StDev Sharp ratio Skewness Kurtosis % VAR(Cornish-Fisher) % of positive months Panel B: Regression analysis Alpha (0.0012) (0.0014) (0.0015) (0.0014) (0.0006) [0.0012] [0.0014] [0.0015] [0.0015] [0.0006] TS (0.0974) (0.1115) (0.1173) (0.1228) (0.1161) (0.0506) [0.1210] [0.1272] [0.1349] [0.1412] [0.1397] [0.0531] HP (0.0939) (0.1075) (0.1131) (0.1184) (0.1119) (0.0487) [0.1117] [0.1267] [0.1159] [0.1385] [0.1331] [0.0524] MOM (0.0707) (0.0809) (0.0851) (0.0891) (0.0842) (0.0367) [0.0825] [0.1352] [0.1232] [0.1010] [0.1014] [0.0424] Adj.R.square p < 0.01, p < 0.05, p < 0.1

60 60 Table 5: Cross Sectional Analysis Fama Macbeth Regression Fama Macbeth two step regression is adopted for factor alone and combined factors. Standard error in square bracket is adjusted by autocorrelation and heterogeneity with lag 12 based on(newey and West, 1987). Standard error in the curly bracket is adjusted by error in variable (EIV) problem proposed by (shanken,1992). The last row is the report for the adjusted R square for each regression. (1) (2) (3) (4) (5) (6) (7) (8) (9) Alpha [0.0011] [0.0011] [0.0010] [0.0011] [0.0011] [0.0009] [0.0010] [0.0010] [0.0008] {0.0017} {0.0016} {0.0014} {0.0017} {0.0012} {0.0014} {0.0012} {0.0009} TS [0.0010] [0.0010] [0.0010] [0.0011] {0.0012} {0.0013} {0.0011} {0.0012} HP [0.0008] [0.0008] [0.0008] [0.0008] {0.0009} {0.0009} {0.0009} {0.0009} MOM [0.0014] [0.0014] [0.0014] [0.0014] {0.0020} {0.0019} {0.0019} {0.0019} RN.SKEW [0.0010] [0.0010] [0.0010] [0.0009] [0.0010] {0.0010} {0.0011} {0.0010} {0.0010} {0.0010} Adj.R.square p < 0.01, p < 0.05, p < 0.1

61 61 Table 6: Principal Component Analysis for Individual Assets Principal component analysis is applied on 25 future products, giving the result of nine major components from second column to the tenth column. The table shows the factor loading on each individual asset with respect to each component. Factor loading can be interpreted as the weight for each asset. Component one gives nearly equally factor loading on each asset, implying the equally weighted portfolio can be delegated by the first principal component. Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9 Brent.Oil Cocoa Coffee Corn Cotton Feeder.Cattle Gold Heating.Oil Lean.Hogs LS.Crude.Oil Live.Cattle Oats Orange Palladium Platinum RBOB Rough.Rice Silver Soy.Meal Soybean Soy.Oil Sugar Wheat WTI.Oil Natural.Gas

62 62 Table 7: Time Series Analysis for Equally Weighted Factor According to the suggestion of PCA analysis and findings in (Bakshi et al. [6]), equally weighted portfolio is formed by longing all available commodity futures at each time period and then used as a controlling risk factor for robustness check: whether risk-neutral skewness risk factor still price the cross-sectional average return after considering the main component. The first row is the portfolio sorted number with P1 (the lowest risk-neutral skewness) and P5 (the highest risk-neutral skewness) and P1-P5 (low minus high). P1 P2 P3 P4 P5 P1-P5 Alpha (0.0008) (0.0008) (0.0009) (0.0009) (0.0008) (0.0006) [0.0008] [0.0008] [0.0009] [0.0010] [0.0008] [0.0006] EW (0.0370) (0.0404) (0.0444) (0.0468) (0.0390) (0.0292) [0.0416] [0.0537] [0.0651] [0.0540] [0.0542] [0.0361] TS (0.0630) (0.0686) (0.0754) (0.0795) (0.0663) (0.0497) [0.0653] [0.0756] [0.0867] [0.0907] [0.0668] [0.0518] HP (0.0618) (0.0674) (0.0740) (0.0780) (0.0651) (0.0488) [0.0654] [0.0861] [0.0976] [0.0920] [0.0720] [0.0514] MOM (0.0476) (0.0519) (0.0570) (0.0601) (0.0501) (0.0376) [0.0546] [0.0649] [0.0712] [0.0914] [0.0663] [0.0426] Adj.R.square p < 0.01, p < 0.05, p < 0.1

63 63 Table 8: Time Series Analysis for New Risk Factors Controlling the effect from other factors suggested from the literature, time series regression is applied by regressing risk-neutral skewness sorted portfolio in different quantile groups on idiosyncratic skewness (IDIO.SKEW), idiosyncratic volatility (IDIO.VOL) and total skewness (TOT.SKEW). Simple standard error (Newey-West standard error) is reported in bracket (square bracket). P1 P2 P3 P4 P5 P1-P5 Alpha (0.0007) (0.0008) (0.0009) (0.0010) (0.0008) (0.0006) [0.0008] [0.0009] [0.0009] [0.0010] [0.0008] [0.0006] EW (0.0419) (0.0464) (0.0505) (0.0540) (0.0448) (0.0330) [0.0464] [0.0618] [0.0862] [0.0623] [0.0643] [0.0422] TS (0.0628) (0.0696) (0.0758) (0.0811) (0.0672) (0.0495) [0.0674] [0.0796] [0.0886] [0.0916] [0.0736] [0.0569] HP (0.0689) (0.0763) (0.0831) (0.0889) (0.0736) (0.0543) [0.0747] [0.0904] [0.1054] [0.1102] [0.0877] [0.0636] MOM (0.0487) (0.0540) (0.0588) (0.0629) (0.0521) (0.0384) [0.0556] [0.0723] [0.0765] [0.0919] [0.0786] [0.0513] IDIDO.SKEW (0.0715) (0.0793) (0.0863) (0.0923) (0.0765) (0.0564) [0.0842] [0.0954] [0.1043] [0.0909] [0.0850] [0.0653] IDIO.VOL (0.0585) (0.0649) (0.0706) (0.0756) (0.0626) (0.0461) [0.0655] [0.0852] [0.1078] [0.0995] [0.0736] [0.0556] TOT.SKEW (0.0585) (0.0649) (0.0706) (0.0756) (0.0626) (0.0461) [0.0587] [0.1016] [0.0910] [0.1176] [0.0898] [0.0579] Adj.R.square p < 0.01, p < 0.05, p < 0.1

64 Table 9: Cross Sectional Two-Step Regression EW is the equally weight portfolio for all commodity futures which includes all available futures during each time period. TOT.SKEW is the long minus shot portfolio sorted on third moment of return series. IDIO.SKEW (IDIO.VOL) is the long minus short portfolio sorted on third moment (second moment) of residual that is estimated from regression of asset return on traditional risk factors (TS, MOM, HP).Standard error in square bracket is adjusted by autocorrelation and heterogeneity with lag 12 based on(newey and West, 1987). Standard error in the curly bracket is adjusted by error in variable (EIV) problem proposed by (shanken,1992). The last row is the report for the adjusted R square for each regression. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) Alpha [0.0010] [0.0009] [0.0017] [0.0009] [0.0010] [0.0016] [0.0009] [0.0009] [0.0010] [0.0012] [0.0008] [0.0009] [0.0009] [0.0013] RN.SKEW [0.0014] [0.0011] [0.0014] [0.0011] [0.0013] [0.0014] [0.0015] [0.0019] [0.0018] [0.0019] {0.0010} {0.0010} {0.0010} {0.0010} {0.0009} {0.0010} {0.0012} {0.0012} {0.0012} {0.0011} EW [0.0016] [0.0017] [0.0017] [0.0017] [0.0019] {0.0021} {0.0018} {0.0018} {0.0017} {0.0015} TS [0.0010] [0.0010] [0.0010] [0.0010] [0.0010] [0.0010] [0.0011] [0.0011] [0.0011] [0.0011] {0.0011} {0.0011} {0.0011} {0.0012} {0.0011} {0.0011} {0.0011} {0.0012} {0.0012} {0.0012} HP [0.0008] [0.0008] [0.0009] [0.0008] [0.0008] [0.0009] [0.0008] [0.0009] [0.0009] [0.0009] {0.0009} {0.0009} {0.0009} {0.0009} {0.0009} {0.0009} {0.0009} {0.0009} {0.0009} {0.0010} MOM [0.0017] [0.0018] [0.0019] [0.0015] [0.0021] [0.0018] [0.0017] [0.0025] [0.0020] [0.0024] {0.0020} {0.0022} {0.0022} {0.0018} {0.0020} {0.0019} {0.0021} {0.0022} {0.0020} {0.0022} TOT.SKEW [0.0015] [0.0031] [0.0030] [0.0032] [0.0031] [0.0032] {0.0016} {0.0028} {0.0028} {0.0027} {0.0027} {0.0027} IDIO.SKEW [0.0014] [0.0016] [0.0017] [0.0016] [0.0019] [0.0019] [0.0020] {0.0016} {0.0013} {0.0012} {0.0013} {0.0015} {0.0015} {0.0016} IDIO.VOL [0.0010] [0.0013] [0.0016] [0.0015] [0.0018] {0.0011} {0.0011} {0.0015} {0.0013} {0.0014} Adj.R.square p < 0.01, p < 0.05, p <

65 65 Table 10: Individual Regression for Alpha Coefficient and Statistics Individual asset return is regressed on risk factors (TS, MOM, RS.N), yielding the regression coefficient on alpha as well as t-statistics. Results in this table have been divided into 5 sections, with each section showing asset name in the first row, alpha coefficient in the second row and alpha t-value in the third row. Asset.Ri,t = αi + β1,i T St + β1,i MOMT + β3,i RN.SKEWt + ɛi,t Asset Brent.Crude.Oil Cocoa Coffee Corn Cotton Alpha T-Statistics Asset Feeder.Cattle Gold Heating.Oil Lean.Hogs Light.Sweet.Crude.Oil Alpha T-Statistics Asset Live.Cattle Oats Orange Palladium. Platinum Alpha T-Statistics Asset RBOB.Gasoline Rough.Rice Silver Soybean.Meal Soybean Alpha T-Statistics Asset Soybean.Oil Sugar Wheat WTI.Crude.Oil Natural.Gas Alpha T-Statistics p < 0.01, p < 0.05, p < 0.1

66 66 Table 11: Risk Factor Prediction Analysis U.S. Bond Return Time series OLS regression is applied with dependent variables 10-Year and 30-Year bond return with different forecasting period. For each of them, return is calculated by log differential between two prices with T period gap (T=3,6,9,12). Independent variables are listed on the left column of table, standing for risk factors. The coefficient in the table shows the forecasting coefficient in each single time series regression. It means how much the bond return will reflect by observing one unit change in risk factors. 10-Year-Bond 30-Year-Bond 3-M 6-M 9-M 12-M 3-M 6-M 9-M 12-M Alpha (0.00) (0.00) (0.00) (0.00) (0.00) (0.01) (0.02) (0.03) TS (0.05) (0.10) (0.15) (0.20) (0.39) (0.97) (1.68) (2.78) MOM (0.04) (0.08) (0.11) (0.14) (0.29) (0.71) (1.24) (2.04) RN.SKEW (0.05) (0.10) (0.14) (0.19) (0.38) (0.95) (1.65) (2.73) Adj. R RMSE p < 0.01, p < 0.05, p < 0.1

67 67 Table 12: Average Level Risk Factor Forecasting Analysis - Change of Money Stock and TED Spread Time series OLS regression is applied with dependent variables change of money stock and TED spread with different forecasting period. For each of them, average level is calculated by taking the mean of variable for data with T period length(t=3,6,9,12). Independent variables are listed on the left column of table, standing for risk factors. The coefficient in the table shows the forecasting coefficient in each single time series regression. It means how much the average money stock and average TED change will reflect by observing one unit change in risk factors. M2 TED 3-M 6-M 9-M 12-M 3-M 6-M 9-M 12-M Alpha (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) TS (0.00) (0.00) (0.00) (0.00) (0.01) (0.01) (0.01) (0.00) MOM (0.00) (0.00) (0.00) (0.00) (0.01) (0.00) (0.00) (0.00) RN.SKEW (0.00) (0.00) (0.00) (0.00) (0.01) (0.01) (0.01) (0.00) Adj. R RMSE p < 0.01, p < 0.05, p < 0.1

68 68 Table 13: Default Risk Forecasting Testing Four different time forecasting periods are used in regression, marked in the table from 3-month up tp 12-month for both DFS and CRS. The dependent variables are calculated by averaging future corresponding time period DFS and CRS value. The coefficient in the table shows the average value change in the future time period responding to one unit change in the risk factors listed in the left column of table. DFS CRS 3-M 6-M 9-M 12-M 3-M 6-M 9-M 12-M Alpha (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) TS (0.03) (0.04) (0.04) (0.04) (0.02) (0.02) (0.02) (0.02) MOM (0.02) (0.03) (0.03) (0.03) (0.02) (0.01) (0.01) (0.01) RN.SKEW (0.03) (0.04) (0.04) (0.04) (0.02) (0.02) (0.02) (0.02) Adj. R RMSE p < 0.01, p < 0.05, p < 0.1

69 69 Table 14: Risk Factor Exposure Testing Three risk factors exposure to underlying factors have been tested with HP factor is excluded as it shows no significance during the previous analysis. MR, SMB and HML are standard Fama French three factors (data obtained from Fama French Library), CROSS VOL is the cross sectional volatility from all commodity futures cross-sectional estimation. DFS is the Moody s Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year Treasury Constant Maturity. CRS is the Moody s Seasoned Baa Corporate Bond Yield relative to is the Moody s Seasoned Aaa Corporate Bond Yield. TED is the spread between 3-month LIBOR rate and 3-month treasury bill. RN.SKEW TS MOM Alpha (0.01) (0.01) (0.01) MR (0.07) (0.07) (0.09) SMB (0.13) (0.13) (0.16) HML (0.13) (0.13) (0.16) CROSS VOL (0.25) (0.26) (0.32) DFS (0.32) (0.33) (0.41) CRS (0.52) (0.52) (0.65) TED (0.59) (0.59) (0.73) R Adj. R RMSE p < 0.01, p < 0.05, p < 0.1

70 Figure 1: Time Series Variation of Cross-Sectional Risk-Neutral Skewness Factor 70

71 Figure 2: Portfolios Performance Comparison 71

72 Figure 3: Overlapping Analysis Between TS and RN.SKEW 72

73 Figure 4: Overlapping Analysis Between MOM and RN.SKEW 73