商品期货收益率和特异波动率（下）

4, 特异波动率在时间截面上的定价

Cross-sectional Pricing of Idiosyncratic Volatility

In this section, we begin by investigating the relationship between commodity futures returns
and their past idiosyncratic volatility in a cross-sectional setting.

在本节，我们将首先研究在时间截面上商品期货收益率和它们过去的特异波动率之间的关系。

4.1 在商品期货市场中特异波动率是否定价非零？

我们的研究方法参照了 Ang et al.’s (2009) 对股票期货回报率和特异波动率关系的研究。第一步，对 equation (1) 应用时间序列回归作出估计，使用的每日数据的观测时间窗口为 t-R 月份 至 t 月份，其中 R = {1，3, 6,12} 。这使得我们可以估计第 i 种商品的时序特异波动率， sigma^R_epsilon,i,t , 即用滚动的时间窗口上回归残差的标准差来计算。

第二步，我们用 OLS方法来跑下面的每月份时间截面回归方程

……（3）

其中 beta_ j,i, t+1 在前文中 equation (2) 后面已经定义，v_i,t+1 是一个随机误差项。因此我们可以估计 M+1个风险因子的定价，也就是 这些 lambdas( lambda_IV,t+1,lambda_1,t+1,...,lambda_M,t+1).第二个步骤要反复迭代计算直到样本数据用完。 然后这些 lambdas的显著性可以用 Shanken(1992)的修正 t-检验方法来检验。

Because of data availability constraints for Barclays’ bond index (only available from January

1989. and given our choice of largest ranking period R at 12 months to obtain the idiosyncratic
      volatility in equation (1), the monthly lambdas (prices of risk) summarised in the tables thus far
      are for months from January 1990 onward. Table 4 reports averages of the lambdas pooled across
      the different window sizes considered R = {1, 3, 6, 12} months, and the adjusted-R2 averaged
      across regressions. Given that the size factor (SMB) appears significantly priced at the 1% level in
      the traditional benchmarks shown in Table 4 (i.e., models A and B), we consider as fundamental
      commodity benchmarks those as initially defined (i.e., models D to G) and augmented versions
      thereof that include the SMB factor (i.e., models H to K).

由于 巴克莱债券指数数据获取的限制（只有1989年1月以后的数据），以及我们在equation(1)中选取的用来获取特异波动率的最大的排序周期是12个月，表格中总结的每月的lambdas (风险的定价)因此是从1990年1月开始的。 Table 4 展示了不同时间窗口（即 R = {1,3,6,12}）回归后 lambdas 的平均值，以及回归后调整R^2系数的平均值. 由于在传统的基准中，市值因子（SMB）在 1% 的置信水平上显著定价不为零，如 Table 4 中 展示的那样（也就是，模型 A和 B）， 我们考虑了如同初始时候定义的那种基础性的商品基准（也就是，模型 D 到 G）以及扩张的 包括 SMB因子的版本（也就是，模型 H到 K）。

There is a striking contrast between the results for the traditional benchmarks, models A to C, and
those for the fundamental commodity benchmarks, models D to K. Consistent with the evidence in
Ang et al. (2009) for equities, idiosyncratic volatility of commodity futures is significantly priced
and commands a negative risk premium when modelled relative to traditional benchmarks. It is
not priced, however, when measured using fundamental commodity benchmarks.As suggested
by a large t-statistic equal to -3.90, the prices of idiosyncratic volatility modelled using traditional
benchmarks are on average significantly lower than those suggested by fundamental benchmarks.
These findings suggest that, as long as expected returns accurately reflect the fundamentals of
backwardation and contango, idiosyncratic volatility does not appear to be priced which is in line
with the finance tenet that idiosyncratic volatility can be diversified away.

在传统的基准，也就是模型 A到 C，与基础性的商品基准，也就是模型 D到K，的结果之间存在着尖锐的对比。与 Ang et al. (2009) 年提出的有关股票的证据一致，当以传统的基准作为参照时，商品期货的特异波动率的定价显著地不为0，并导致负的风险超额收益率。然而，如果我们用的是基础性的商品基准，特异波动率则定价为0. 由一个达到 - 3.90 的 t 统计量可知，用传统基准建模得出的特异波动率的定价平均来看显著低于用基础性商品基准建模得到的结果。 这个发现表明，虽然预期回报率反映了贴水和升水的原理，但特异波动率的定价实际上为0，这与特异波动率可以被分散化的金融学信条相一致。

Table 4.jpg

As robustness tests, we estimate three other specifications of the traditional benchmarks that
replace in models A and C of Table 4 the S&P-GSCI portfolio by either the Thomson Reuters/
Jefferies CRB Index, the Dow Jones-UBSCI or a long-only equally-weighted monthly-rebalanced
portfolio of all 27 commodities. We also estimate three other re-specifications of the fundamental
commodity benchmarks labelled D to G in Table 4 that consider either one of the fundamental
risk factors (TS, HP or Mom) in isolation instead of considering them as pairs or triplets. Finally,
as the equity risk premium is negatively priced in benchmark B of Table 4, we estimate a final set
of fundamental commodity benchmarks that include both the equity risk premium and SMB in
models H to K. None of these robustness checks challenges the earlier evidence presented in Table

4. Unreported results (available upon request) indeed show a negative and significant relationship
   between idiosyncratic volatility and mean returns in the context of traditional benchmarks and
   an insignificant relationship between idiosyncratic volatility and mean returns in the context of
   fundamental benchmarks.

作为稳健性测试，我们估计了其它三种规格的传统基准，即在 Table 4的 模型 A 和 C 中替代 S&P-GSCI 组合为 Thomson Reuters/Fefferies CRB 指数， Dow Jones-UBSCI 指数或 对这27中商品构建的多头等权重每月平衡仓位的组合。我们同样估计了 三种不同规格的商品基准，即在 Table 4的 模型 D至G中，我们考虑让基础性风险因子(TS，HP 或 Mom) 单独起作用，而不是同时考虑让它们成对或三个一起作用。最后，由于在 Table 4 的基准B中，股票风险的超额收益率定价为负，我们最后估计了一组基础性的商品基准，这个基准同时包括了股票的风险溢价以及 模型 H至K的SMB。这些稳健性测试没有能够挑战我们之前在 Table 4 中提供的证据。我们的一些没有发表的结果(如果需要可以向我们索取)实际上表明如果以传统的基准为背景，特异波动率和平均回报率之间存在着显著的负相关性，而如果以基础性的商品基准作为背景，特异波动率和平均回报率之间的关联则不显著。

As a further robustness check, we estimate equation (3) including the one-month lagged returns
ri,t as an additional regressor and the results are shown in Table 5.

The motivation for this analysis stems from Huang et al. (2010) who argue that the negative
idiosyncratic volatility premium for equities documented by Ang et al. (2006, 2009) is induced
by a return-reversal omitted variable. The findings suggest that idiosyncratic volatility is still
negatively priced for commodities in the context of traditional benchmarks when the lagged
return is factored in,and it is still not priced with fundamental commodity benchmarks, consistent
with our earlier findings.

Bessembinder (1992) shows that idiosyncratic volatility conditional on net hedging pressure is
positively priced in agricultural and foreign exchange futures markets, a result consistent with
the theoretical model formulated in Hirshleifer (1988). Following their lead, using our dataset
and pricing models (A to K) we investigate whether hedging-conditioned idiosyncratic volatility
commands a positive risk premium too.

作为进一步的稳健性检验，我们对 equation (3) 进行了包括额外的一个月延迟回报 r_ i,t 的回归，最后的结果展示在 Table 5 中。

此处分析的动机来源于 Huang et al.(2010) 的工作，他们主张说 Ang et al.(2006,2009)报道的股票市场的特异波动率的定价为负的事实实际上是由一个被忽略的收益反转因子所引起的。这里发现的结果表明考虑延迟的回报后，当以传统的基准为背景时，特异波动率的定价依然为负，而以基础性的商品基准为背景时，特异波动率的定价依然为0，这与我们之前的发现是一致的。

Bessembinder（1992）报告说以纯对冲压力为条件的特异波动率在农产品和外汇期货市场的定价为正，这个结果与 Hirshleifer (1988)年建立的理论模型是一致的。根据他们的引导，我们用我们的数据，以及A到 K的定价模型，也研究了一下以对冲压力为条件的特异波动率是否驱动正的风险超额收益率这个问题。

Table 5.jpg

This is done by interacting our idiosyncratic volatility measure with a net hedging pressure
dummy that takes value: 1 if speculators are net long at the beginning and end of month
(backwardation), -1 if speculators are net short at the beginning and end of month (contango)
and 0 if speculators change positions within month. Irrespective of the benchmarks used, we
confirm the predictions of Hirshleifer (1988) and the evidence of Bessembinder (1992) of a positive
relationship between hedging-conditioned idiosyncratic volatility and mean futures returns. In
other words, idiosyncratic volatility commands a positive risk premium in backwardated markets
(when speculators are net long) and a negative risk premium in contangoed markets (when
speculators are net short). Detailed results are available upon request.

我们通过将我们的特异波动率和一个虚设的变量相互作用来完成这个工作。这个虚设变量的取值是这样的：如果投机持仓在月初和月末都为净多头（贴水），虚设变量取值为1；如果投机持仓在月初和月末都为净空头（升水），虚设变量取值为-1；如果投机持仓在当月改变了持仓方向，虚设变量取值为0. 和采用的基准无关，我们证实了Hirshleifer（1988）的预测和 Bessembinder (1992)的论证，确实在以对冲压力为条件的特异波动率和平均回报之间存在着一个正的关联。换句话说，在一个贴水的市场（投机持仓为净多头）特异波动率驱动着一个正的风险回报而在一个升水的市场（投机持仓为净空头）特异波动率驱动着一个负的风险回报。详细的结果可以向我们索取。

4.2 特异波动率令人困惑的负定价现象来自于贴水还是升水呢？

Is the Idiosyncratic Volatility Puzzle Driven by Backwardation or Contango?

Our main line of reasoning thus far is that the puzzling negative idiosyncratic volatility premium
is an artefact of neglecting the fundamental backwardation/contango cycle of commodity futures
markets. As Table 4 clearly illustrates, the significantly negative price of idiosyncratic volatility
vanishes once backwardation and contango are taken into account through the fundamental
commodity benchmarks. Out of the two natural states a commodity futures market can be in, it is
possible that one of them plays a stronger role to explain the idiosyncratic volatility phenomenon.
To gauge this conjecture, we reconduct Ang et al.’s (2009) two-stage analysis for the fundamental
commodity benchmarks but considering this time around the long (backwardated) TS, HP and
Mom portfolios, on the one hand, and the short (contangoed) TS, HP and Mom portfolios, on the
other. The results are set out in Table 6.

目前为止我们主要的推理路线是认为令人困惑的特异波动率定价为负的现象其实是由于认为忽略了商品期货市场基础性的 贴水/升水 周期。如同 Table 4 所清楚展示的那样，特异波动率显著为负的定价现象在我们采用了考虑了贴水和升水的基础性商品基准时消失了。商品期货市场可能逃脱贴水和升水这两种自然状态，因此或许是它们当中的一个在解释商品期货特异波动率负的定价方面扮演了更强作用的角色。为了评估这种推测，我们重新对基础性的商品指数采用 Ang et al (2009) 的两步分析法，不过这次我们只单独考虑多头（贴水）的TS，HP 和 Mom 组合，在另一方面，只单独考虑空头（升水）的TS，HP 和 Mom 组合。结果呈现在 Table 6 中。

As predicted by the storage theory and the hedging pressure hypothesis, the prices of risk associated
with the long backwardated TS, HP and Mom portfolios are generally positive and significant
at the 1% level and the prices of risk associated with the short contangoed TS, HP and Mom
portfolios tend to be negative and significant for the most part. This means that investors demand
a positive premium (i.e., larger returns) for taking long positions in backwardated assets and short
positions in contangoed assets, which is consistent with the price evolution depicted in Figure 1.

正如同贮藏理论和对冲压力假说所预测的那样，与多头贴水的 TS, HP 和 Mom 组合相关联的风险定价通常是正的，在1%的置信水平上具有显著性，而与空头升水的TS,HP 和 Mom 组合相关联的风险定价倾向是负的，对大部分数据都具有显著性。这意味着投资者需要一个正的超额收益（也就是更大的回报）来持有一个贴水组合的多头或者是一个升水组合的空头，这和 Figure 1 所展示的价格演化情况一致。

Table 6.jpg

With respect to the idiosyncratic volatility factor, the results for the long backwardated portfolios
are very similar to those reported earlier for the traditional benchmarks since the premium is
significantly negative. For example, while the prices of idiosyncratic volatility inferred from the
asset pricing models referred to as traditional benchmarks A, B and C in Table 4 stand at -0.4916
on average, the prices of idiosyncratic volatility associated with the long backwardated TS, HP
and Mom portfolios in Table 6 average out at -0.3546. In sharp contrast, with an average of only
0.0470, the idiosyncratic volatility premia inferred on the basis of the short contangoed TS, HP and
Mom portfolios are economically and statistically undistinguishable from zero.

关于特异波动率因子，多头贴水组合的结果和之前展示的传统基准的结果非常相似，超额收益率都显著为负。例如，在Table 4中，参照传统的基准A，B，C的定价模型给特异波动率的平均定价为 -0.4916，在Table 6 中，给多头贴水的 TS，HP 和 Mom组合的特异波动率的平均定价为 -0.3546。与此形成鲜明对比的是，以空头升水的 TS，HP和Mom为基准对特异波动率的定价只有 0.0470，这个结果无论在经济学意义上还是统计学意义上和 0 都区别不大。

The most important message stemming from this analysis is that the puzzling negative premium (or
discount) that idiosyncratic volatility attracts is an artefact from using an asset pricing model that
fails to factor in the risk of contangoed portfolios; once the latter is accounted for, idiosyncratic
volatility no longer matters. Thus the earlier finding, on the basis of traditional commodity
benchmarks, that investors earn a negative premium (i.e., are penalized with a discount) for taking
idiosyncratic volatility is spurious since it merely reflects the negative premium (i.e., discount)
associated with long positions in contangoed markets. This is quite intuitive because, by being
long, investors play the role of hedgers (see Figure 2) and thus pay an insurance premium to short
speculators. In other words, by taking long (as opposed to short) positions in contangoed markets,
speculators lose the insurance premium they would typically earn.

从我们的分析中可以得到的最重要的一个信息是令人困惑的特异波动率的负定价现象实际上是一个人为现象，原因是定价模型中没有包括升水风险组合因子。当把后者包括进去时，特异波动率将变得不再重要。因此之前的那些发现，即以传统的商品基准为基础时，持有特异波动率较高资产的投资者将获得一个负的超额收益率的观点是虚假的。因为这仅仅反映了与持有多头的升水组合相关的负的超额收益率。这实际上相当符合直觉，通过持有多头组合，投资者实际上扮演了套期保值者的角色，因此他们需要付给做空的投机商一些保险费用。换句话说，在升水的市场中，通过持有多头（相反情况下是空头）头寸，投机商放弃了他们通常能够获得的保险费用。

5，特异波动率的时间序列定价

Time-Series Pricing of Idiosyncratic Volatility

This section studies the time-series relation between idiosyncratic volatility and returns in
commodity futures markets using a risk-mimicking portfolio construction method.

在本节我们将通过采用风险模拟组合的方法研究特异波动率和商品期货市场回报之间的时间序列关系。

5.1 特异波动率模拟组合

Idiosyncratic Volatility Mimicking Portfolios

Following the analysis in Ang et al. (2006, 2009) for equities, we sort commodities into quintiles
based on their idiosyncratic volatility , which is computed using daily returns over the past
R={1,3,6,12} months relative to either traditional benchmarks or fundamental commodity
benchmarks, as in equation (1). We then construct a factor mimicking portfolio that buys the
quintile with the lowest idiosyncratic volatility and shorts the quintile with the highest idiosyncratic
volatility. We hold the long-short portfolio for one month, at which time the same trading process
is repeated to obtain a new long-short idiosyncratic volatility portfolio.

仿照Ang et al.(2006,2009)对股票市场的分析方法，我们根据特异波动率对商品排序并划分成五等分，这是通过使用过去 R = {1,3,6,12}月份的每日收益率计算得到的，采用的基准是传统的基准或者基础性的商品基准，如同 equation(1) 那样。然后我们构建了一个模拟组合，这个组合买入特异波动率最低的1/5商品期货的同时做空特异波动率最高的1/5商品期货。我们持有这个多空组合1个月，然后按照相同的交易过程，一个新的多空特异波动率组合将被创建。

For the sake of consistency with the fundamental risk factors (TS, HP and Mom), the idiosyncratic
volatility portfolios are fully-collateralized, rebalanced at the end of each month, and based on
equal weights for the constituents of the top and bottom quintiles. While an equal-weighting
scheme conveniently avoids portfolio concentration on specific commodities and thus ensures
better diversification, it can also cause illiquidity problems, making it potentially difficult for
investors to open or close positions. We tackle this issue by including the liquidity risk premium of
Pastor and Stambaugh (2003) in all benchmarks.

为了与基础性的风险因子(TS，HP 和 Mom)相一致，特异波动率组合是完全对冲的( fully-collateralized），在每个月末重新平衡持仓，使顶部和底部的五分之一的构成成分为等权重。等权重的计划可以方便地避免组合集中到一个特定的商品因此能够确保更好地分散化。过分集中可能导致流动性问题，让投资者开仓或空仓出现潜在的困难。我们通过在所有的基准中包含Pastor 和 Stambaugh (2003)提出的流动性风险溢价来处理这个问题。

Table 7 presents summary statistics for the performance of the long-short idiosyncratic volatility
portfolios, where idiosyncratic volatility is measured relative to traditional benchmarks on the
left-hand side, i.e. models A to C, and relative to fundamental commodity benchmarks on the
right-hand side, i.e. models D to K. An equally-weighted portfolio of all 12 long-short idiosyncratic
volatility strategies built upon the traditional benchmarks earns 4.94% a year, significant at the
5% level; 10 out of those 12 strategies earn significantly positive mean excess returns at the
10% level (7 out of those 12 mean excess returns are significant at the 5% level or better).
In sharp contrast, an equally-weighted portfolio of all 32 long-short idiosyncratic volatility
strategies built upon the fundamental commodity benchmarks earns about half the above returns,
only 2.53% a year, which is statistically insignificant; none of these 32 strategies earn significantly
positive mean excess returns. Similar inferences are drawn based on Sharpe ratios, which tend to be
much larger (at 0.4543 on average) for traditional benchmarks than for fundamental commodity
benchmarks (at about half, 0.2361 on average); the Opdyke t-test statistic equal to 2.30 confirms
that the Sharpe ratio of the former is significantly larger than that of the latter.

Table 7 展示了多空特异波动率组合表现的统计数据，其中在左侧部分特异波动率是用传统基准衡量的，也就是模型 A到C，而右侧的特异波动率是用基础性的商品基准衡量的，也就是模型D到K。 一个由12个参照传统基准的特异波动率的多空组合等权重分配的策略每年可以盈利4.94%，在 5%置信水平上具有显著性。12个策略中的10个在10%的置信水平上具有显著超额正收益（12个策略中的7个在5%或者更好的置信水平上具有显著超额收益）。与之形成鲜明对照的是，一个在32个参照基础性的商品基准的特异波动率多空策略等权重分配的组合只能取得上述收益率的一半，只有2.53%每年，并且在统计上不具有显著性；这32种策略中没有任何一个能够取得显著的超额正收益。用夏普比率可以得到类似的推论，对于参照传统基准的超额收益率的多空组合，其夏普比率（平均为 0.4543）远大于参照基础性商品基准的多空组合（只有约一半，平均为 0.2361）；Opdyke t-检验结果为 2.30，可以确认前者的夏普比率显著比后者更大。

table 7.jpg

As in the cross-sectional framework earlier, we test the robustness of our time-series analysis to
the specification of the traditional and fundamental commodity benchmarks. This is carried out
as follows: i) replacing the S&P-GSCI in models A and C by either the Thomson Reuters/Jefferies
CRB Index, the Dow Jones-UBSCI or a long-only equally-weighted portfolio of the 27 commodity
futures, ii) considering either one of the fundamental risk factors (TS, HP or Mom) in isolation
instead of treating them as pairs or triplets in models D to G, and iii) including both the equity
risk premium and SMB in models H to K. Unreported results indicate that the conclusions of
Table 7 hold for these specifications too: as borne out by a significant Opdyke t-statistic of 2.59,
the average Sharpe ratio of idiosyncratic volatility portfolios based on traditional benchmarks
(at 0.3895) exceeds that of idiosyncratic volatility portfolios based on fundamental commodity
benchmarks (at 0.2497).

如同在时间截面分析框架中所做的那样，我们测试了我们时间序列分析对于特定传统的或基础性商品基准的稳健性。这通过如下方法实施：i)将 模型 A和 C 中的S&P-GSCI 指数分别更换为 Thomson Reuters/Jefferies CRB 指数， Dow Jones-UBSCI 或27中商品期货的多头等权重组合；ii）考虑在模型D至G中让基础性风险因子（TS，HP，或 Mom）单独作用而不是让它们成对或三个联合起作用；iii) 在模型 H至 K中同时包括股票市场风险收益率和 SMB 市值风险收益率。此处未呈现的结果表明Table 7中得到的结果在这些不同的规格条件下也是成立的：参照传统基准构造的特异波动率组合的夏普比率的平均值（0.3895）远远超过参照基础性商品基准构造的特异波动率组合的夏普比率（0.2497），显著性Opdyke t-统计量达到了2.59。

Table 8 reports annualized abnormal returns or alphas measured as the intercept of a regression
of monthly idiosyncratic volatility portfolio returns on various risk factors, alongside significance
t-statistics based on autocorrelation and heteroskedasticity robust Newey and West (1987)
standard errors. Table 8 also presents in the last row a t-test for the significance of the difference
between two alphas: one corresponding to an equally-weighted portfolio of all 12 idiosyncratic
volatility strategies based on traditional benchmarks, the other corresponding to an equallyweighted
portfolio of all 32 idiosyncratic volatility strategies based on fundamental commodity
benchmarks.

Table 8 展示了年化的反常收益率或者叫做alphas , 这是用每月特异波动率多空组合的收益率和多种风险因子做回归后的截距来衡量的，Table 8 同时还呈现了根据自回归和异方差性的鲁棒性的 Newey and West (1987) 标准误差的显著性 t-统计量。 Table 8 同时在最后一列展示了两种 alpha之间差别的显著性 t-检验结果。一种是等权重的12个基于传统基准的特异波动率策略的alpha，另外一种是等权重的32个基于基础性商品基准的特异波动率的alpha。

Table 8.jpg

The upshot of this analysis is that the inferences on the alpha generation ability of the idiosyncratic
volatility portfolios are far more “optimistic” when traditional benchmarks are used. In fact, the
alphas inferred from fundamental commodity benchmarks (i.e., models D to K) are zero statistically
whereas those inferred from traditional benchmarks (i.e., models A to C) are positive and often
significant, averaging 5.21% a year. The empirical evidence that abnormal profits can be made
by selling high idiosyncratic volatility portfolios and buying low idiosyncratic volatility portfolios
appears to be an artifact of two modeling issues: a) the volatility signal derived from traditional
benchmarks is not truly idiosyncratic because it contains a systematic risk component related
to the neglected fundamental backwardation/ contango cycle and b) the alpha is subsequently
gauged using an improper benchmark.

上述分析的结论如下，如果采用传统的基准，特异波动率组合产生alpha的能力将大为“乐观”。事实上，参照基础性的商品基准（也就是模型 D至K），alphas 统计值实际为0，而参照传统的基准（也就是模型A到C）alpha 显著为正，大概是每年 5.21%。经验数据上卖出高波动率的组合的同时买入低波动率的组合可以获取反常收益的证据实际上是两个模型问题的人为产物：a）从传统的基准中导出的波动率信号实际上并不是真的特异波动率，因为它包含了一个被忽略的基础性贴水和升水周期的系统性风险成分 b) 这个 alpha 实际上是用一个不合适的基准测量的。

5.2 高特异波动率组合和低特异波动率组合

High versus Low Idiosyncratic Volatility Portfolios

In the context of equities, Ang et al. (2006, 2009) document that the performance of long-short
idiosyncratic volatility portfolios is more strongly driven by the underperformance of stocks with
high idiosyncratic volatility than by the outperformance of stocks with low idiosyncratic volatility.
To investigate whether the same applies to commodity futures, we measure the alphas of the long
idiosyncratic volatility and short idiosyncratic volatility portfolios, separately, where the alphas
are calculated relative to either the traditional or the fundamental commodity benchmarks as
defined earlier. Table 9 shows the results.

在研究股票时，Ang et al. (2006, 2009) 撰文指出特异波动率多空组合的收益率表现更多地受益于较高特异波动率的股票表现显著地弱于市场平均而不是由于较低特异波动率股票的表现强于市场平均。为了研究这一点是否也适用于商品期货市场，我们分别衡量了多头低特异波动率组合和空头高特异波动率组合的alphas。其中alphas 分别基于前面定义的传统的或基础性的商品基准计算得到。Table 9 展示了这个结果。

Table 9.jpg

The results using traditional benchmarks are in line with Ang et al. (2006, 2009): while the
portfolios with low idiosyncratic volatility tend to marginally outperform (with an average alpha
at 1.81% a year) the portfolios with high idiosyncratic volatility have very low annualized alphas
(at -5.37% on average). Interestingly, when the fundamental backwardation/contango cycle is
properly factored in the benchmarks, which are used both to model idiosyncratic volatility and
measure abnormal performance, the alphas of the long and short portfolios decrease in absolute
value. The last row of Table 9 reports in parenthesis t-tests for the significance of the difference in
the average alphas (longs and shorts, in turn) for the two types of benchmarks. With t-statistics at
2.99 (-4.43), the alphas of the long (short) portfolios for the traditional benchmarks denoted A to
C are found to be statistically larger (smaller) than the alphas of the long (short) portfolios for the
fundamental commodity benchmarks denoted D to K. This further substantiates our conjecture
that idiosyncratic volatility signals built upon traditional benchmarks are partly systematic
because they reflect the risk associated with the backwardation/ contango cycle. Once the latter
is taken into account, the alphas of the long and short idiosyncratic volatility mimicking portfolios
become negligible.

使用传统基准的结果与Ang et al.(2006,2009)的结论一致：低特异波动率组合似乎略微有正的超额收益（平均的alpha 为每年 1.81%），而高特异波动率组合则有非常低的年化 alpha(平均每年为 -5.37%)。有趣的是，如果基础性的贴水/升水周期被合理地归因到基准中，并用来对特异波动率建模和对反常收益率进行衡量，多头和空头组合的alphas的绝对值都会减小。Table 9 中最后一行圆括号中的是根据这两种不同基准计算出来的平均alphas(依次多头和空头)的差值是否显著的 t-检验 统计量。t-统计量 为 2.99（-4.43），依照传统基准的多头(空头)组合的alphas，标记为 A 至C ，统计上比 依照基础性商品基准的多头（空头）组合，标记为D至K，的alphas 更大（小）。这进一步支持了我们的观点：依靠传统的基准构建的特异波动率信号实际上部分是系统性的，因为它反映了和贴水和升水周期相联系的系统性风险。一旦把贴水和升水周期考虑进来，多头和空头的特异波动率模拟组合的alphas将变成可忽略不计的。

6，结论 Conclusions

This paper investigates the relation between idiosyncratic volatility and expected returns in
commodity futures markets. The analysis is motivated by the puzzling empirical evidence for
international equity markets provided by Ang et al. (2009) suggesting that idiosyncratic volatility
is negatively priced. Extending their analysis to commodity futures markets is of interest not only
because it could prove that the puzzling negative relationship is a pervasive phenomenon, but also
because it could shed light on the reasons as to why this anomalous effect is revealed by the data.
The empirical evidence presented suggests that inferences on the relation between idiosyncratic
volatility and expected commodity futures returns depend on the choice of asset pricing model
or benchmark used to extract the idiosyncratic volatility signal. When the asset pricing model
fails to recognize the inexorable backwardation/contango cycle of commodity futures markets,
idiosyncratic volatility seemingly commands a puzzling negative risk premium and, relatedly,
mimicking portfolios that systematically buy low idiosyncratic volatility commodities and short
high idiosyncratic volatility commodities seem to offer sizeable mean returns. Prima facie these
results extend those of Ang et al. (2009) from international equity markets to commodity futures
markets.

这篇文章研究了商品期货特异波动率和预期收益率之间的关系。本文写作的动机是由Ang et al. (2009) 提出的在全球股票市场上存在着特异波动率定价为负的经验性证据所引起。把他们的分析扩展到商品期货市场上市非常有趣的，不仅是因为这或许可以证明这种令人困惑的负相关关系是否具有普遍性，还因为这可以显示为什么这些数据为什么会包含这种反常的效应。这里展示的经验证据支持这样的推论，特异波动率和商品期货的预期收益率取决于选取的定价模型，或者说是用于提取特异波动率的基准信号。当选用的定价模型不能够识别商品期货市场不可阻挡的贴水和升水周期循环时，特异波动率似乎将引起一个令人困惑的负的超额收益率。于是，构建一种买入低波动率的商品期货资产并卖出高波动率的商品期货资产的模拟组合将可以获得可观的平均收益。表面上看这是把 Ang et al. (2009) 针对全球股票市场的文章扩展到商品期货市场。

However, should commodity futures be priced instead with reference to the fundamentals of
backwardation and contango, the abnormal performance of long-short idiosyncratic volatility
mimicking portfolios fades away and idiosyncratic volatility no longer commands a negative
risk premium. These results align well with the fundamental tenet that idiosyncratic volatility
should be diversified away and thus it is not priced. Further evidence shows that the seemingly
negative premium associated with idiosyncratic volatility when traditional benchmarks are used is
a manifestation of the pricing of contangoed, rather than backwardated, portfolios: idiosyncratic
volatility acts as proxy for the risk associated with contangoed contracts.

Further research could extend this analysis to the FOREX futures market where backwardation and
contango have been shown to matter (Bessembinder, 1992).

然而，由于商品期货在定价时应当考虑升水和贴水的基础事实，构建特异波动率的多空组合所获取的超额收益率将消失，特异波动率将不再产生一个负的超额收益率。这个结论和基础金融学的信条一致：特异波动率可以通过分散化消除掉因此定价为0。进一步的证据表明当采用传统的基准时，看起来与特异波动率相联系的负的超额收益率是升水组合而不是贴水组合的定价现象。特异波动率代表着和升水合约相联系的风险。

进一步的研究可以将这里的分析扩展到FOREX期货市场，在FOREX期货市场上，升水和贴水现象已经被显示是有重要影响的(Bessembinder, 1992)。