概述 Abstract
This paper studies the relationship between idiosyncratic volatility and expected returns in
commodity futures markets. Measuring idiosyncratic volatility relative to traditional pricing
models that fail to account for backwardation and contango leads to the puzzling conclusion
that idiosyncratic volatility is negatively priced. In sharp contrast, idiosyncratic volatility is not
priced when the fundamental backwardation and contango cycle of commodity futures markets is
factored in an appropriate benchmark. Further evidence suggests that the idiosyncratic volatility
inferred from traditional benchmarks acts as proxy for the risk associated with contangoed
contracts.
Keywords: Commodity futures; Idiosyncratic volatility; Backwardation; Contango.
本文研究了商品期货市场特异波动率与预期收益之间的关系。如果使用传统定价模型对特异波动率进行定价，由于没有考虑到升水和贴水，将会导致特异波动率定价为负这一令人困惑的结论。与此形成鲜明对比的是，如果将商品期货市场基本的贴水与升水周期归因到一个合适的基准，那么特异波动率的定价应当为零。进一步的证据表明，从传统的基准导出的特异波动率代表了与升水合约相联系的风险。
关键词：商品期货、奇异波动率、升水、贴水
1，简介 Introduction
The link between idiosyncratic volatility and mean returns has been the subject of intense scrutiny
in the equity markets literature. Theoretical arguments rule out any such link since idiosyncratic
volatility can be diversified away and thus should not be priced (Sharpe, 1964) or maintain
that the link is positive since agents who hold poorly-diversified portfolios demand incremental
returns for bearing idiosyncratic volatility (Merton, 1987; Malkiel and Xu, 2002). The empirical
evidence is mixed. A number of studies support the contention that idiosyncratic volatility is not
priced (Fama and McBeth, 1973; Bali et al., 2005; Bali and Cakici, 2008; Fink et al., 2012; Huang
et al., 2010; Han and Lesmond, 2011) but others report evidence in favor of a positive (Malkiel and
Xu, 2002; Goyal and Santa-Clara, 2003; Fu, 2009; Garcia et al., 2011) or an anomalous negative
link (Guo and Savickas, 2008; Ang et al., 2006, 2009) between idiosyncratic volatility and average
returns in equity markets.
在股票市场相关的文献中，特异波动率和平均回报率之间的关联已经认真地研究过了。理论上的观点要么认为特异波动率和平均回报率之间不存在任何关联，因为特异波动率可以通过分散化消除掉（Sharpe，1964）。要么持有特异波动率定价为正的观点，因为持有分散化较差资产组合的客户将要求额外的风险补偿以承担特异波动率（Merton, 1987; Malkiel and Xu, 2002）。实证上的观点则比较混乱。许多的研究支持特异波动率定价为0的观点（(Fama and McBeth, 1973; Bali et al., 2005; Bali and Cakici, 2008; Fink et al., 2012; Huang et al., 2010; Han and Lesmond, 2011）。另外一些则提供证据支持特异波动率定价为正的观点（Malkiel and Xu, 2002; Goyal and Santa-Clara, 2003; Fu, 2009; Garcia et al., 2011）。还有一些很奇怪的观点认为在股票市场上特异波动率和平均回报率之间存在着一种负的关联(Guo and Savickas, 2008; Ang et al., 2006, 2009)。
This article studies the relation between idiosyncratic volatility and mean returns in commodity
futures markets. The role of idiosyncratic volatility as driver of commodity futures risk premia
was first conceptualized by Hirshleifer (1988) in a theoretical model that accounts for trading
costs and non-marketability of producers claims. The commodity futures risk premium can then
be decomposed into two components: the first one, in the spirit of the CAPM, depends on the
co-movement of the futures contract with a broad equity index; the second one depends on the
idiosyncratic volatility of the contract and net hedging which is positive when hedgers are net
short and negative when they are net long. Bessembinder (1992) validates the predictions of
Hirshleifer (1988) by showing that idiosyncratic volatility conditional on net hedging commands
a positive risk premium in agricultural commodity and foreign currency markets.
本文研究了商品期货市场中特异波动率和平均回报率的关系。特异波动率作为驱动商品期货风险溢价的角色这一概念最早是由 Hirshleifer 在1988年的一个理论模型中提出的，这个模型考虑了交易成本和生产者缺乏他们需要的销售市场的情况。商品期货的风险溢价于是可以分解成两个组成部分。按照CAPM的理论范式，第一个是取决于期货合约与其它一系列股票指数之间的共同运动情况，第二个取决于特异波动率和持有裸露头寸的情况。当持有多头头寸时，它是正的，当持有空头头寸时，它是负的。Bessembinder (1992)证实了Hirshleifer (1988) 的预测，他发现在农产品商品期货市场和外汇市场中，当持有裸露头寸时，特异波动率将导致正的风险补偿收益。
While studying the pricing of idiosyncratic volatility in commodity futures markets, the first
thorny issue we face relates to the choice of an appropriate asset pricing model or benchmark
upon which to model idiosyncratic volatility. Traditional risk factors from equity and fixed income
markets or emanating from the APT have been proven to be unsuccessful for pricing commodity
futures contracts (Dusak, 1973; Bodie and Rosansky, 1980; Kolb, 1996; Erb and Harvey, 2006; or,
more recently Daskalaki et al., 2012). Commodity-specific risk factors (Carter et al., 1983; Chang,
1985; Bessembinder, 1992; De Roon et al., 2000; Basu and Miffre, 2012; Daskalaki et al., 2012;
Gorton et al., 2012) have fared somewhat better, but not unanimously so, inasmuch as they
capture the fundamentals of backwardation and contango put forward in the theory of storage
of Kaldor (1939) and Working (1948, 1949) and in the hedging pressure hypothesis of Cootner
(1960) and Hirshleifer (1988). Given the lack of consensus on a suitable asset pricing model or
benchmark for commodity futures, the first objective of this paper is to provide evidence on
the ability of various risk factors to explain the cross-sectional variation in commodity futures
returns. The second objective is to test, through the various asset pricing models considered, the
nature of the relation between idiosyncratic volatility and mean returns both in a cross-sectional
framework and in a time-series portfolio formation setting.
我们研究商品期货市场中特异波动率定价问题时遇到的第一个棘手的问题，是如何选择一个合适的资产定价模型或者参照基准来衡量特异波动率。传统的权益类市场或固定收益市场中的风险因子，或者由APT定价模型产生的风险因子被证明对商品期货的定价是不成功的 (Dusak, 1973; Bodie and Rosansky, 1980; Kolb, 1996; Erb and Harvey, 2006; or,more recently Daskalaki et al., 2012)。由于考虑了Kaldor(1939) 和 Working(1948,1949)在贮藏理论，以及Cootner (1960) 和 Hirshleifer (1988)在套保压力假设理论中提出的升水和贴水的基础概念，一些商品期货独有的风险因子(Carter et al., 1983; Chang,1985; Bessembinder, 1992; De Roon et al., 2000; Basu and Miffre, 2012; Daskalaki et al., 2012; Gorton et al., 2012) 受到更好一些的评价，但并非毫无异议。尽管学术界对于商品期货定价的模型或参照基准缺少一致性，本文的第一个目标是证明多样化的风险因子能够解释不同商品期货在一个时间截面上的收益率的差异。本文的第二个目的，是为了用我们考虑的多样化资产定价模型来测试商品期货的特异波动率和平均收益率之间的关系，我们既基于时间截面框架来考虑这个问题，也基于组合信息的时间序列来研究这个问题。
The main contribution of our study is to demonstrate empirically that inferences on the relation
between lagged idiosyncratic volatility and expected returns in commodity futures markets hinge
on the choice of asset pricing model. When the pricing model fails to recognize the fundamentals
of backwardation and contango it is shown that: a) idiosyncratic volatility is negatively priced
cross-sectionally, and b) a mimicking portfolio that over time buys low idiosyncratic volatility
commodities and shorts high idiosyncratic volatility commodities offers sizeable alpha. These
results are aligned with those reported by Ang et al. (2006, 2009) in international equity markets.
In sharp contrast, if the natural propensity of commodity markets to be in backwardation/
contango is explicitly factored in the benchmark, idiosyncratic volatility is no longer priced crosssectionally
and the alpha of a portfolio that buys low idiosyncratic volatility commodities and
shorts high idiosyncratic volatility commodities becomes insignificant. This outcome agrees with the fundamental tenet of finance theory that idiosyncratic volatility can be diversified away and
hence, it is not priced. Further, we show that the seemingly negative premium associated with
idiosyncratic volatility when inappropriate asset pricing models are used reflects the pricing of
contangoed (rather than backwardated) portfolios, suggesting that idiosyncratic volatility acts,
in fact, as a proxy for the systematic risk associated with contangoed assets.
本文主要的贡献在于经验性地展示了商品期货市场延迟的特异波动率和预期的回报率会随着定价模型的选择而变化。如果定价模型没能识别期货贴水和升水的基础，那么 a) 特异波动率的定价是负的。b）一个反复地购买低波动率的商品期货并做空高波动率的商品期货的组合，将能够提供可观的alpha。 这些结论和Ang et al.（2006,2009）在全球股票市场得出的结论一致。与之形成鲜明对比的是，如果商品期货贴水或升水的自然倾向被明显地包含在基准中，特异波动率在时间截面上的将定价为0，买入低波动率的商品期货组合并卖空高波动率的商品期货组合所获得的alpha将不显著。这个结论和基础的金融学原则一致，特异波动率由于可以通过持有多样化组合的方法消除掉，所以其定价应当为0。进一步地，我们的研究显示，当采用不合适的资产定价模型来反映升水的（而非贴水）组合时，与特异波动率相关的看似为负的超额收益率实际上暗示我们，特异波动率其实代表着和溢价资产相关联的系统风险。
The paper is structured as follows. Section 2 introduces the commodity futures data. Section
3 motivates the pricing models considered both theoretically and empirically. Sections 4 and
5 provide cross-sectional and time-series evidence on the relationship between idiosyncratic
volatility and commodity futures returns. Section 6 concludes.
本文结构如下。第2部分介绍了商品期货的数据。第3部分阐述了从理论上和经验上考虑的定价模型。
第4部分和第5部分提供了时间截面和时间序列上的证据，展示特异波动率和商品期货收益率之间的关系。第6部分总结。
2，商品期货数据 Commodity Futures Data
The research is based on daily settlement prices and volume data for 27 commodity futures from
January 2, 1979 to August 31, 2011, from Datastream. The 27 commodities are: 12 agricultural
(cocoa, coffee C, corn, cotton n°2, frozen concentrated orange juice, rough rice, oats, soybean
meal, soybean oil, soybeans, sugar n° 11, wheat), 5 energy (electricity, gasoline, heating oil n° 2,
light sweet crude oil, natural gas), 4 livestock (feeder cattle, frozen pork bellies, lean hogs, live
cattle), 5 metal (copper, gold, palladium, platinum, silver), and random length lumber. We track
the futures prices on all nearest and second-nearest contracts in order to work with the most
traded contracts; the nearest contract is held up to one month before maturity when the position
is rolled over to the second-nearest contract. Excess returns (referred to as "returns" afterwards
for expositional ease) are measured as logarithmic price differences.
我们的研究基于27种商品期货从 1979年1月2日至2011年 8月31日每日的结算价格和交易量数据，数据来源是Datastream。这27种商品包括：12种农产品（cocoa, coffee C, corn, cotton n°2, frozen concentrated orange juice, rough rice, oats, soybean meal, soybean oil, soybeans, sugar n° 11, wheat），5种能源商品 (electricity, gasoline, heating oil n° 2,light sweet crude oil, natural gas)，4种牲畜（feeder cattle, frozen pork bellies, lean hogs, live cattle)，5种金属（copper, gold, palladium, platinum, silver）和 任意长度的木材。我们跟踪了这些商品期货中时间最邻近和第二邻近的合约的价格，以便找到主力合约：最邻近的合约在到期一个月之前被停止，并让位于第二邻近的合约。超额收益率（为叙述方便，下文将简称为"收益率"）通过使用对数价差方法来衡量。
Our empirical analysis also requires observations on the positions of large traders in commodity
futures markets, which are provided weekly by the Commodity Futures Trading Commission
(CFTC), from January 1986 to August 2011. Large traders have to report weekly to the CFTC
whether they are commercial (hedgers) or non commercial (speculators) and whether they are
long or short. Their declarations compiled in the Aggregated Commitment of Traders report
serve as inputs to calculate two hedging pressure measures. Speculators’ hedging pressure is
calculated as the number of their long positions (i.e., open interests or the amount of outstanding
contracts) divided by the total number of positions taken by non-commercial traders over the
week. Hedgers’ hedging pressure is defined in terms of their long positions as a fraction of the
total open interests associated with commercial traders over the week. For example, a hedging
pressure of 0.2 for hedgers means that 20% of hedgers were long and thus 80% were short
over the week, which is (as argued in detail in Section 3.1) a sign of backwardation. A hedging
pressure of 0.2 for speculators means that 20% of speculators were long and thus 80% were
short over the week, which is (as explained next) an indication of contango. The cross-section size
(N=27 commodities) of our sample is, in fact, dictated by data availability from the Aggregated
Commitment of Traders report.
我们的研究同时也需要对商品期货市场中大单交易的观察，这由商品期货交易委员会(CFTC)每周公布，时间从1986年1月至2011年8月。 大单交易必须每周向CFTC汇报交易情况：是套保交易还是投机交易，是多单还是空单。这些声明将被收集到交易商承诺汇总报告中，我们把它作为一个输入来计算两种对冲压力量。投机的对冲压力量用投机商一周内总的多头持仓（亦即未平仓合约）除以投资商一周内的总持仓来计算。套期保值的对冲压力量用一周内的多头套保持仓除以一周内的全部套保持仓来计算。例如，0.2 的套期保值对冲压力量意味着一周内有20%的套期保值持仓为多头，而另外的80%的套期保值持仓为空头，这是一种贴水的征兆（我们将在 Section 3.1详细讨论这一点）。0.2的投机压力量意味着一周内20%的投机仓位是多头而80%的投机仓位是空头，这是一种升水的征兆（我们将在后面解释这一点）。实际上，我们在一个时间截面上的合约数量（N = 27 种商品期货）正是由CFTC每周的交易商承诺汇总报告有效的数据量决定的。
3，时间截面上商品期货收益率的解释
Explaining the Cross-Section of Commodity Futures Returns
This section begins by presenting the theoretical motivation behind the pricing models employed
and continues by offering empirical evidence on their plausibility.
这个部分我们将首先展示我们应用的定价模型的理论动机然后为其合理性提供经验证据。
3.1 Theoretical Underpinning of Systematic Risk Factors
As there is no consensus on the systematic risk factors that should be adopted to model expected
commodity futures returns, we measure idiosyncratic volatility relative to several pricing models.
This section theoretically motivates the various risk factors. Some of them are borrowed from
the literature on the pricing of traditional assets (equities and bonds) under the Law of One Price
theory which contends that financial markets are fully integrated (Cochrane, 2005). All other
factors we consider are commodity-specific and are intended to account for the fundamental
backwardation/contango cycle of commodity futures.
3.1 系统性风险因素的理论基础
由于学术界对商品期货预期收益率的系统性风险因素定价应当采用的模型没有达成一致性，我们将用多个定价模型来衡量商品期货的特异波动率。这个部分我们将从理论上研究多种风险因子。它们中的一些借用自一些传统资产（股票和债券）的定价模型的一些文献，因为根据一价定律，各个不同的金融市场实际上是相连相通的一个集成体(Cochrane, 2005)。我们考虑的一些其它的因子是商品特有的，它们被用来将商品期货基本的贴水和升水周期纳入我们的理论体系。
We use as traditional risk factors: i) the three equity factors of Fama and French (1993), namely,
the equity market premium, the size premium (SMB), and the value premium (HML), obtained
from Kenneth French’s website, and ii) the excess returns of the Barclays US aggregate bond
index obtained from Bloomberg. The equity market excess return is used as substitute for the
true market portfolio assumed by the CAPM. SMB and HML are included as proxies for shifts in
the investment opportunity set of agents over time (Petkova, 2006); as such, they could be priced
in commodity futures markets too. The inclusion of a fixed income benchmark is motivated by
the theory of storage that explains the shape of the commodity futures curve via the cost of
borrowing.
The commodity-specific risk factors attempt mainly to capture the fundamentals of backwardation
and contango. Broadly speaking, backwardation means that the futures price of a commodity
is expected to appreciate as maturity approaches. Contango means the opposite. Figure 1
depicts the theoretical price evolution of two contangoed contracts (continuous lines) and two
backwardated contracts (dashed lines) with maturities n for the nearby contract and d for the
distant contract.
我们使用了以下一些传统的风险因子：i）Fama和French(1993)提出的三个股票因子，也就是股票市场超额收益率，市值超额收益率(SMB)，以及估值超额收益率(HML)，这些数据可以从 Kenneth French 的网站上获得。2）巴克莱美国综合债券指数的超额收益率，这个数据可以从Bloomberg获得。股票市场的超额收益率被用来代替CAPM模型中所假设的真实市场组合。SMB 和 HML 被包括进来作为客户投资机会随着时间变化的代表(Petkova, 2006);因此，在期货市场中，它们也是可以被定价的。 将固定收益基准包括进来是受到贮藏理论的启发，这个理论解释了商品期货曲线形状与借贷成本之间的关系。
我们采用商品期货特有的风险因子主要用来捕获贴水和升水的基础原理。一般来说，贴水意味着商品期货的价格预期将要上涨，随着交割日临近。升水则意味着相反。Figure 1 描述了两个升水合约(实线)和两个贴水合约(虚线)的理论上的价格演化走势，其中近期合约离交割日n天，远期合约离交割日d天。
Figure 1. Theoretical price evolution of commodity futures. The figure represents the evolution in the futures prices of two contracts
with maturity n and d (n < d) for a hypothetical commodity when the market is in backwardation (dashed lines) and when it is in
contango (continuous lines). Backwardated (contangoed) contracts are characterized by positive roll yield (negative roll yield), low (high)
hedgers' hedging pressure (HP), high (low) speculators' hedging pressure and are winners (losers) in a momentum portfolio. St denotes the
commodity spot price.
Figure 1. 理论上商品期货的价格演化。这个图片代表了两个期货合约价格的演化，它们的到期日分别是n和d天，我们假设市场是分别贴水的（虚线）和升水（实线）的。贴水(升水)合约的特征是有一个正的滚动收益（负的滚动收益），套期保值的对冲压力量较低（高），投机的对冲压力量较高（低），并且在一个动量组合 [momentum portfolio] 中为盈利方（亏损方）。St 表示商品现货价格。
We summon two theoretical rationales to motivate the price evolutions depicted in Figure 1. The
first relies on Kaldor’s (1939) and Working’s (1948, 1949) theory of storage which is empirically
supported by Gorton et al. (2012). This theory explains the shape of the commodity futures curve
by means of the incentive that inventory holders have in owning the commodity spot. Given
the difficulty of collecting reliable inventory data, we rely on the slope of the term structure
(hereafter TS) of commodity futures prices − instead of standardized inventory levels − as signal
for backwardation and contango (Erb and Harvey, 2006; Gorton and Rouwenhorst, 2006; Fuertes
et al., 2010; Daskalaki et al., 2012; Gorton et al., 2012). According to the storage theory, as
implicitly suggested in Figure 1, we extract the commodity futures risk premium by systematically
buying the backwardated contracts with the highest roll yields (or roll returns) and shorting the
contangoed contracts with the lowest roll yields.
我们总结了两个理论上的依据来解释 Figure 1中价格的演化。 第一个依据依赖于 Kaldor’s (1939) 和 Working’s (1948, 1949) 的贮藏理论，这个理论被 Gorton et al. (2012)从经验数据上给予了支持。这个理论用库存持有者持有现货的动机的方法来解释商品期货曲线的形状。考虑到收集可靠的存货数据是非常困难的，我们使用商品期货期限结构（此后用 TS 表示期限结构：term structure）的斜率，而不是标准得存货水平，来作为贴水和升水的征兆 (Erb and Harvey, 2006; Gorton and Rouwenhorst, 2006; Fuertes et al., 2010; Daskalaki et al., 2012; Gorton et al., 2012)。根据贮藏理论，如同 Figure 1所建议的那样，我们用如下方法提取商品期货的风险超额收益率，即系统性地买入拥有最高滚动收益的贴水合约，并做空拥有最低滚动回报的升水合约。
The second theoretical rationale for the backwardation/contango price evolution depicted in
Figure 1 comes from the hedging pressure (hereafter HP) hypothesis of Cootner (1960) which is
generalized in Hirshleifer (1988) and validated empirically in Bessembinder (1992) and Basu and
Miffre (2012). This theory relates backwardation and contango to the propensity of hedgers to
be net short or net long. As illustrated in Figure 1, the idea is to capture the risk premium of
commodity futures contracts by buying backwardated commodities for which hedgers are the
shortest and speculators the longest and by shorting contangoed commodities for which hedgers
are the longest and speculators the shortest. To put this differently, the HP mimicking portfolio
buys (sells) the commodities for which hedgers' hedging pressure is the lowest (highest) and
speculators' hedging pressure is the highest (lowest) following the double-sorting methodology
developed in Basu and Miffre (2012)
第二个针对Figure 1中 贴水和升水期货的价格演化的理论根据来自于 Cootner （1960）的对冲压力假设(此后用HP表示对冲压力因子，hedging pressure)，这个理论被 Hirshleifer（1988）所推广，并被 Bessembinder (1992) 和 Basu、Miffre (2012) 从经验证据上所证实。这个理论用套期保值交易者倾向于纯多头或纯空头来解释贴水和升水。如同 Figure 1所展示的那样，这个理论通过如下方法来捕获商品期货合约的风险超额收益，即买入套期保值交易商做空最多而投机商做多最多的贴水商品，同时卖空套期保值者做多最多而投机商做空最多的升水商品期货。换句话说，HP理论的模拟组合买入(卖空)套保对冲压力最低 (高) 而投机的对冲压力最高（低）的商品，这可以依据 Basu 和 Miffre（2012）年发展的双排序方法。
Our third risk factor is the commodity momentum portfolio (hereafter Mom) advocated by Erb
and Harvey (2006) and Miffre and Rallis (2007) that systematically longs commodities with the
best past performance and shorts commodities with the worst past performance. While it is
not directly intended to do so, this momentum portfolio can also capture to some extent the
backwardation/contango cycle. Indeed, Miffre and Rallis (2007) show that the winners (losers)
tend to have positive (negative) roll yields making them backwardated (contangoed). However,
as illustrated empirically in Fuertes et al. (2010), there is no full overlapping of Mom and TS
portfolio returns and therefore they can be used in conjunction to model commodity futures risk
premia.
我们的第三个风险因子来自于Erb、Harvey (2006) 和 Miffre、Rallis (2007) 所倡导的商品动量（此后用 Mom 表示动量组合因子，Momentum）组合，这种组合系统性地买入过去表现最好的商品并做空过去表现最差的商品。虽然并非出自本意，这种动量组合在某种程度上也能捕获贴水和升水的循环。实际上， Miffre 和 Rallis (2007) 显示 具有盈利（亏损）倾向的合约常常使得他们贴水（升水）。但是，就如同 Fuertes et al.(2010) 用经验数据论证的那样，Mom 和 TS 组合回报之间并没有完全重叠，因此它们可以联合起来使用，以便对商品期货的风险溢价建立模型。
Building on the Keynesian hypothesis that commodity futures markets are normally backwardated
(Keynes, 1930), we also include as potential risk factor the S&P-GSCI (Standard & Poor's Goldman
Sachs Commodity Index) as a long-only commodity market portfolio. Daily data for S&P-GSCI are
obtained from Bloomberg.
Finally, given that investors demand a premium for holding less liquid assets and in line with the
work of Han and Lesmond (2011) that relates the pricing of idiosyncratic volatility to liquidity
risk, all our benchmarks include as risk factor the liquidity risk premium (LRP) of Pastor and
Stambaugh (2003) that we apply to commodities.
In the remainder of the paper, we adopt the terminology fundamental commodity benchmarks
to refer to commodity pricing models that account for the fundamentals of backwardation
and contango (i.e., those including the TS, HP and/or Mom factors) and traditional benchmarks
to refer to models that fail to do so (i.e., those based on the Fama-French factors, bond risk
premium and S&P-GSCI). The liquidity risk premium is included consistently in all benchmarks of
either type.
根据凯恩斯假设，商品期货市场一般情况下是贴水的(Keynes, 1930)。作为一个仅可单向做多的商品市场组合的S&P-GCSI指数 (Standard & Poor's Goldman Sachs Commodity Index) 被我们作为一个潜在的风险因子包括进来了。S&P-GSCI 的每日数据从 Bloomberg获取。
最后，考虑到对于缺乏流动性的资产，投资者将要求额外的回报。与 Han 和 Lesmond (2011) 建立的特异波动率和流动性风险关联的工作一致，我们研究的所有商品期货的基准都包含了 Pastor 和 Stambaugh (2003) 提出的流动性风险溢价作为风险因子。
在我们论文的剩余部分，我们采用专有的商品期货基准来指代将贴水和升水原理考虑进来的商品定价模型（例如，那些包括了TS，HP 和/或 Mom 因子的模型）而用传统的基准来指代那些没能包括贴水和升水原理的模型（例如，那些建立在 Fama-French 因子，债券风险溢价，以及 S&P-GSCI上的模型）。无论对于哪种类型的基准，流动性风险溢价都被包括进来了。
3.2 Empirical Motivation for Fundamental Commodity Benchmarks
This section complements the preceding one by empirically providing some intuition for the
TS, HP and Mom factors as drivers of a systematic risk premium in commodity futures markets.
Figure 2 plots the evolution from January 1986 to August 2011 of end-of-month futures prices
for crude oil. Shaded areas signify backwardated months according to each of three different
signals: when the roll yield is positive (TS signal; Panel A), when speculators are net long at the
beginning and end of month (HP signal; Panel B), and when 12-month past returns are positive
(Mom signal; Panel C). Thus non-shaded areas indicate non-backwardated months when the
roll yield is negative (Panel A), speculators are net short at the beginning and end of the month
or changed positions within the month (Panel B), and when 12-month past performance was
negative (Panel C). Given that the shaded areas do not strictly coincide in the three panels, we
use the TS, HP and Mom factors conjunctly (in pairs or all three) in the benchmarks used to
extract and subsequently price idiosyncratic volatility.
3.2 基础性商品基准的经验性动机
在这个部分我们将用一些经验性的直观数据补充论证之前所说的 TS，HP，和 Mom 因子作为商品期货市场中系统性风险超额收益补偿的驱动的观点。Figure 2 画的是 1986年1月 至2011年 8月 原油商品期货月底价格的走势。阴影区域表示贴水月份，依据于以下三个不同的信号：当滚动收益率为正时（TS 信号； Panel A），当投机持仓在月初和月末为纯多头时（HP 信号； Panel B），当过去的12个月的收益为正时（Mom 信号；Panel C）。考虑到在这三个子图（Panel）里这些阴影区域不严格一致，我们在基准中使用TS，HP和Mom 因子联合起来（成对或三个一起）提取特异波动率并随后为之定价。
Figure 2. Historical crude oil futures prices. The figure plots monthly futures prices of crude oil alongside shaded areas which indicate
backwardated months when roll-returns are positive (Panel A), when speculators are net long at the beginning and end of month (Panel B)
and when 12-month past returns are positive (Panel C).
Figure 2. 原油期货的历史价格走势。这个图片画的是原油商品期货每月的价格，图片中的阴影区域表示贴水的月份，即当滚动收益率为正的时候(Panel A)，当投机持仓在月初和月末为纯多头的时候（Panel B）以及当过去12个月份收益率为正的时候（Panel C）.
Figure 2 中的三个子图支持了贴水（阴影区域）和商品期货价格上涨趋势是紧密相连的。与之相反，商品期货升水（非阴影区域）与价格下跌趋势紧密相连。为了量化在 Figure 2中的证据，我们创造了四个虚设的变量。一个是 D_R(t)，当商品期货每月回报为正时它取值为1，为负时它取值为0. 另外三个虚设变量是为了标注原油处于贴水状态：如果每月滚动收益为正，D_TS(t)取1否则取0；当投机持仓在月初和月末都为纯多头时D_HP(t)取1，否则取0；而 如果过去的12个月收益率平均值为正D_Mom(t)取1否则取0. D_R(t) 和其它三个虚设的贴水变量D_TS(t)，D_HP(t)，D_Mom(t)之间的相关系数(p-values) 分别是 30.18%（0.00）,13.68%(0.00) 和 11.17% （0.05）。这些显著为正的相关系数支持 TS，HP 和Mom 确实是非常合适的 捕获基础的 贴水和升水周期的信号，因此可以对商品期货市场内在的系统性风险溢价进行建模。
3.3 Pricing of Systematic Risk Factors
The following four principles are maintained in constructing the TS, HP and Mom mimicking
portfolios that act as proxies for a fundamental commodity risk factor related to the backwardation/
contango cycle. First, the long-short portfolios are formed by taking long positions in the 20%
most backwardated commodity contracts whose prices are expected to appreciate and short
positions in the 20% most contangoed commodity contracts whose prices are expected to
decline. Second, the ranking period over which the TS, HP or Mom signals are averaged out is
set to 12 months, and the holding period over which the long-short portfolios are being held
is set to 1 month.8 Third, in line with Erb and Harvey (2006) among others, the constituents of
the long and short portfolios are equally-weighted. Fourth, the long-short portfolios are fully
collateralized, meaning that 1/2 of the trading capital is invested in risk-free interest bearing
accounts for the longs and likewise for the shorts. Thus the return of the long-short portfolios
equals half the return of the long portfolios minus half the return of the short portfolios.
3.3 系统性风险因子的定价
在构建 TS，HP 和 Mom 模拟组合作为与贴水和升水周期相关联的基础性商品风险因子的代表时，我们始终遵循着下面四个原则。第一，我们构建的多空组合中，多头持仓选择 前20%贴水最多的商品合约，它们的价格预期将要上涨，空头持仓选择前20%升水最多的商品合约，它们的价格预期将要下跌。第二，计算TS，HP 和Mom信号平均数的排序期为12个月，持有多空组合的时间长度为1个月。第三，与 Erb 和 Harvey (2006) 以及其它研究者相一致，多空组合的各个组成成分是等权重分配的。第四，多空组合是完全对冲的，即有一半的资产被投资在与无风险收益率相关的多头头寸，另外一半类似地投资在空头头寸。因此，多空组合的收益一半取决于多头组合的收益，一半取决于空头组合的收益。
Our sample comprises daily observations on the three Fama-French risk factors, the commodity
futures returns and the (backfilled) S&P-GSCI returns for the period from January 2, 1979 to
August 31, 2011. Because the ranking window to construct the long-short TS and Mom mimicking
portfolios is set to 12 months, the first set of returns available for them corresponds to January
2, 1980. Since the ranking window for the liquidity risk mimicking portfolio is 60 months, as in
Pastor and Stambaugh (2003), the first set of returns for the liquidity risk factor corresponds to
January 2, 1985. On the other hand, the availability of hedging pressure data in the Aggregated
Commitment of Traders report forces us to start our analysis of the HP risk premium on January
2, 1987. Finally, as the returns on the Barclays bond index are available at a daily frequency from
January, 3 1989 onwards, the sample period that is common to all series runs from January 3,
1989 to August 31, 2011.
我们的样本包含了对1979年1月2日至2011年8月31日三个 Fama-French风险因子，商品期货回报以及S&P-GSCI 的每日观察。因为构建多空 TS和Mom模拟组合呢的排序时间窗口是12个月，因此，有效的第一个回报率数据是 1980年1月2日的。由于流动性风险的模拟组合的排序期是60 个月，如同 Pastor 和Stambaugh （2003）所提出的那样，第一个有效的流动性风险回报率数据在 1985年1月2日。
Table 1 presents pairwise correlations between the daily returns for all risk factors considered
over the longest period possible from January 3, 1989 to August 31, 2011. The average absolute
correlation is low at 0.08. Unsurprisingly, the highest pairwise correlation of 0.44 corresponds
to the two long-short commodity portfolios (Mom and TS), followed by a significant correlation
of 0.34 between Mom and HP. Relatively high pairwise correlations among TS, HP and Mom are
expected because, arguably, the three factors capture the fundamentals of backwardation and
contango. But the fact that these correlations statistically differ from 1 (according to t-tests
for Pearson correlation) motivates our decision to include two or three of these risk factors
simultaneously in the benchmarks; this was also borne out earlier by Figure 2. At the other
extreme, the correlation between the equity market factor and HML factor is small at -0.17, and
is followed closely by that between HML and SMB also very small at -0.12, as well as between
Barclays’ bond index and SMB at -0.12 too.
Table 1 展示了从1989年1月3日至2011年8月31日最长时期中每日收益率和所考虑的风险因子之间的相关系数。相关系数绝对值最低为 0.08。并不让人感到惊奇的是，最高的相关系数为0.44，这是两个多空商品组合之间的相关系数（Mom 和 TS），紧随其后的 Mom 和 HP之间0.34的相关系数。 TS，HP和Mom 之间相对较高的相关系数是预料之中的，因为按理说，这三个因子都可以捕获贴水和升水的基础原理。但是这些相关系数统计上地与1存在较大差别（根据 Pearson 相关系数的 t-检验）的事实，激发我们考虑将这些风险因子中的两个或者三个同时包括在我们的基准中，这一点在之前的 Figure 2 中已经得到了证实。在另一个极端，股票市场因子和 HML因子之间的相关系数 小至 -0.17，紧随其后的是 HML和 SMB之间 -0.12 的相关系数，同样地，巴克莱债券指数和 SMB 之间的相关系数也是 - 0.12.
Table 2 reports summary statistics for the daily (annualized or per annum, p.a.) excess returns
of the aforementioned risk factors over the period January 3, 1989 to August 31, 2011. Panel
A presents the results for the equity and fixed income motivated risk factors whereas Panel B
pertains to the commodity-specific risk factors.
Table 2 展示了前述风险因子在1989年1月3日至2011年8月31日期间的每日（年化）超额收益率的统计情况。Panel A 展示了股票和固定收益资产相关的传统风险因子，而 Panel B 是关于商品特有的风险因子。
In line with the theoretical argument that the fundamentals of backwardation and contango
matter to the pricing of commodity futures contracts, the statistics reported in Panel B provide
evidence that the Mom and HP factors, followed by the TS factor, offer the highest Sharpe ratios
and thus are more mean-variance efficient than the backwardated-only S&P-GSCI portfolio.
Aligned with the notion that long backwardated (contangoed) positions make (lose) money, more
detailed analysis reveals that the long TS, HP and Mom portfolios earn positive mean returns
of 4.69%, 4.09% and 7.32% p.a., respectively, while the short TS, HP and Mom portfolios earn
negative mean returns of -5.34%, -7.17% and -8.00% p.a., respectively.
与理论上贴水和升水将对商品期货合约的定价产生重要影响的观点一致，Panel B 的统计数据提供证据表明 Mom 和 HP因子，以及 TS因子将提供最高的Sharpe 比率。因此按照均值方差模型它们比 单纯贴水的 S&P-GSCI 组合更为有效。与上述观点一致，对贴水（升水）商品的多头持仓在盈利（亏损），更多的细节分析显示 多头的 TS，HP 和Mom组合平均每年分别赚得4.69%，4.09%，和7.32%的收益，而空头的 TS，HP 和Mom 组合平均每年分别赚得 -5.32%，-7.17%和 -8.00% 的收益。
An appropriate common risk factor should be able to explain the cross-sectional variation in
realized commodity futures returns. To gauge this, we follow the two-step approach of Fama
and MacBeth (1973) as deployed by Ang et al. (2009) to analyze the pricing of idiosyncratic
volatility for equities. First, we run time-series regressions for each of the 27 commodity futures,
to explain daily returns on the basis of M risk factors
一个恰当的共同风险因子应当能够解释真实商品期货在时间截面上收益的差别。为了评估这一点，我们仿照了 Fama 和 MacBeth (1973) 采用的量两步方法，这种方法被 Ang et al.(2009）扩展用来分析股票特异波动率的定价。首先，我们对27种商品期货跑一个时间序列回归，用以解释 M种风险因子的每日汇报。
其中 r_i,d 是第 i 个商品期货合约在给定月份中第d = 1,...,D 天的回报，其中D是那个月份的天数； f_ j,d 是第j个风险因子，j = 1,...,M; epsilon_i,d 是残差项，（beta_1,i,...,beta_M,i）'是我们关注的beta系数。因子的数量，M，取决于我们手上的基准。例如，我们的第一个传统的基准使用3个 Fama-French 因子，巴克莱债券指数，S&P-GSCI 和 LRP，因此一共 M= 6 个因子被使用。
第二步，我们用第一步得到的每月 beta系数跑一个时间截面上每月商品期货收益率的回归，可以表示成如下形式。
regression 2.jpg
其中 r_i,t+1 是第 i 个商品期货合约在 第 t+1月份的收益率； beta_ j, i, t+1是 equation (1) 中以t+1月份的每日收益率为基础使用 OLS方法估计出来的每月的beta （也就是说，估计beta依赖的变量和beta 是同一时期的）；（lambda_1,t+1,……，lambda_M,t+1）是 用equation(2) 用OLS方法估计出来的 第 t+1月份的 lambda 系数 或 风险因子定价。然后对 equation(1)的回归估计时间区间向前滚动一个月，然后得到的 beta 再带入到 equation (2) 来重新估计出一组新的 lambdas，如此滚动向前。 Shanken （1992）的修正的 t-检验方法被用于lambdas 的时间序列，并决定哪些因子对于时间截面上不同的商品期货具有定价功能。
Table 3 展示了平均的 lambdas , 显著性 t-检验 和调整的 R^2统计
Table 3 reports the averaged lambdas, significance t-tests and adjusted-R2 statistics.
Table 3.jpg
Notwithstanding the similar explanatory power for the cross-sectional variation in commodity
futures returns of all models, the analysis reveals that the HP factor stands out by being able to
price the cross-section of commodity futures returns. The largest lambdas in magnitude, strongly
significant at the 1% level, are those corresponding to the HP factor in the fundamental commodity
benchmarks. This evidence challenges the often-held view that commodities do not constitute
another asset class because they do not seem to have a common risk factor (Erb and Harvey,
2006; Daskalaki et al., 2012). The fact that there is at least one commodity-specific risk factor
that prices commodity futures cross-sectionally indicates that commodities instead constitute
a homogeneous alternative asset class. Finding that the sources of risk that explain the cross
sectional variations in stocks and bonds returns fail to command a risk premium in commodity
futures markets supports, however, the contention that financial markets are segmented.
尽管所有的模型对于时间截面上不同商品期货的收益率都具有类似的解释能力，但是我们的分析显示 HP 因子对于时间截面上不同商品期货的定价能力尤其突出。~~量级上最大的lambda, 在 1%的水平上具有非常强的显著性，是那些在基础商品期货基准中与HP因子对应的 lambda ~~。这个证据挑战了人们通常持有的观点——商品期货不构成另外一种类型的资产，因为它们并没有一个共同的风险因子（Erb 和 Harvey, 2006; Daskalaki et al., 2012）. 至少存在一个商品期货独有的风险因子可以对不同的商品期货在时间截面上定价的事实，标示着商品期货实际上构成了另外一种可供选择的资产类别。我们同时发现，用以解释股票和债券收益率在时间截面上差别根源的那些风险因子，并不能驱动商品期货市场的风险超额收益率。这个发现支持了金融市场实际上是隔离的观点。
