4, 特異波動(dòng)率在時(shí)間截面上的定價(jià)
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.
在本節(jié)冒萄，我們將首先研究在時(shí)間截面上商品期貨收益率和它們過(guò)去的特異波動(dòng)率之間的關(guān)系。
4.1 在商品期貨市場(chǎng)中特異波動(dòng)率是否定價(jià)非零叠荠？
我們的研究方法參照了 Ang et al.’s (2009) 對(duì)股票期貨回報(bào)率和特異波動(dòng)率關(guān)系的研究。第一步敢会，對(duì) equation (1) 應(yīng)用時(shí)間序列回歸作出估計(jì)蔑穴，使用的每日數(shù)據(jù)的觀測(cè)時(shí)間窗口為 t-R 月份 至 t 月份鉴腻，其中 R = {1迷扇，3, 6,12} 。這使得我們可以估計(jì)第 i 種商品的時(shí)序特異波動(dòng)率爽哎， sigma^R_epsilon,i,t , 即用滾動(dòng)的時(shí)間窗口上回歸殘差的標(biāo)準(zhǔn)差來(lái)計(jì)算蜓席。
第二步，我們用 OLS方法來(lái)跑下面的每月份時(shí)間截面回歸方程
……（3）
其中 beta_ j,i, t+1 在前文中 equation (2) 后面已經(jīng)定義课锌，v_i,t+1 是一個(gè)隨機(jī)誤差項(xiàng)厨内。因此我們可以估計(jì) M+1個(gè)風(fēng)險(xiǎn)因子的定價(jià)，也就是 這些 lambdas( lambda_IV,t+1,lambda_1,t+1,...,lambda_M,t+1).第二個(gè)步驟要反復(fù)迭代計(jì)算直到樣本數(shù)據(jù)用完渺贤。 然后這些 lambdas的顯著性可以用 Shanken(1992)的修正 t-檢驗(yàn)方法來(lái)檢驗(yàn)雏胃。
Because of data availability constraints for Barclays’ bond index (only available from January
- 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 巴克萊債券指數(shù)數(shù)據(jù)獲取的限制（只有1989年1月以后的數(shù)據(jù)），以及我們?cè)趀quation(1)中選取的用來(lái)獲取特異波動(dòng)率的最大的排序周期是12個(gè)月志鞍，表格中總結(jié)的每月的lambdas (風(fēng)險(xiǎn)的定價(jià))因此是從1990年1月開始的瞭亮。 Table 4 展示了不同時(shí)間窗口（即 R = {1,3,6,12}）回歸后 lambdas 的平均值，以及回歸后調(diào)整R^2系數(shù)的平均值. 由于在傳統(tǒng)的基準(zhǔn)中述雾，市值因子（SMB）在 1% 的置信水平上顯著定價(jià)不為零街州，如 Table 4 中 展示的那樣（也就是兼丰，模型 A和 B）玻孟， 我們考慮了如同初始時(shí)候定義的那種基礎(chǔ)性的商品基準(zhǔn)（也就是，模型 D 到 G）以及擴(kuò)張的 包括 SMB因子的版本（也就是鳍征，模型 H到 K）黍翎。
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.
在傳統(tǒng)的基準(zhǔn)，也就是模型 A到 C艳丛，與基礎(chǔ)性的商品基準(zhǔn)匣掸，也就是模型 D到K，的結(jié)果之間存在著尖銳的對(duì)比氮双。與 Ang et al. (2009) 年提出的有關(guān)股票的證據(jù)一致碰酝，當(dāng)以傳統(tǒng)的基準(zhǔn)作為參照時(shí)，商品期貨的特異波動(dòng)率的定價(jià)顯著地不為0戴差，并導(dǎo)致負(fù)的風(fēng)險(xiǎn)超額收益率送爸。然而，如果我們用的是基礎(chǔ)性的商品基準(zhǔn)，特異波動(dòng)率則定價(jià)為0. 由一個(gè)達(dá)到 - 3.90 的 t 統(tǒng)計(jì)量可知袭厂，用傳統(tǒng)基準(zhǔn)建模得出的特異波動(dòng)率的定價(jià)平均來(lái)看顯著低于用基礎(chǔ)性商品基準(zhǔn)建模得到的結(jié)果墨吓。 這個(gè)發(fā)現(xiàn)表明，雖然預(yù)期回報(bào)率反映了貼水和升水的原理纹磺，但特異波動(dòng)率的定價(jià)實(shí)際上為0帖烘，這與特異波動(dòng)率可以被分散化的金融學(xué)信條相一致。
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
- 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.
作為穩(wěn)健性測(cè)試橄杨，我們估計(jì)了其它三種規(guī)格的傳統(tǒng)基準(zhǔn)秘症，即在 Table 4的 模型 A 和 C 中替代 S&P-GSCI 組合為 Thomson Reuters/Fefferies CRB 指數(shù)， Dow Jones-UBSCI 指數(shù)或 對(duì)這27中商品構(gòu)建的多頭等權(quán)重每月平衡倉(cāng)位的組合式矫。我們同樣估計(jì)了 三種不同規(guī)格的商品基準(zhǔn)历极，即在 Table 4的 模型 D至G中，我們考慮讓基礎(chǔ)性風(fēng)險(xiǎn)因子(TS衷佃，HP 或 Mom) 單獨(dú)起作用趟卸，而不是同時(shí)考慮讓它們成對(duì)或三個(gè)一起作用。最后氏义，由于在 Table 4 的基準(zhǔn)B中锄列，股票風(fēng)險(xiǎn)的超額收益率定價(jià)為負(fù)，我們最后估計(jì)了一組基礎(chǔ)性的商品基準(zhǔn)惯悠，這個(gè)基準(zhǔn)同時(shí)包括了股票的風(fēng)險(xiǎn)溢價(jià)以及 模型 H至K的SMB邻邮。這些穩(wěn)健性測(cè)試沒有能夠挑戰(zhàn)我們之前在 Table 4 中提供的證據(jù)。我們的一些沒有發(fā)表的結(jié)果(如果需要可以向我們索取)實(shí)際上表明如果以傳統(tǒng)的基準(zhǔn)為背景克婶，特異波動(dòng)率和平均回報(bào)率之間存在著顯著的負(fù)相關(guān)性筒严，而如果以基礎(chǔ)性的商品基準(zhǔn)作為背景，特異波動(dòng)率和平均回報(bào)率之間的關(guān)聯(lián)則不顯著情萤。
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.
作為進(jìn)一步的穩(wěn)健性檢驗(yàn)鸭蛙，我們對(duì) equation (3) 進(jìn)行了包括額外的一個(gè)月延遲回報(bào) r_ i,t 的回歸，最后的結(jié)果展示在 Table 5 中筋岛。
此處分析的動(dòng)機(jī)來(lái)源于 Huang et al.(2010) 的工作娶视，他們主張說(shuō) Ang et al.(2006,2009)報(bào)道的股票市場(chǎng)的特異波動(dòng)率的定價(jià)為負(fù)的事實(shí)實(shí)際上是由一個(gè)被忽略的收益反轉(zhuǎn)因子所引起的。這里發(fā)現(xiàn)的結(jié)果表明考慮延遲的回報(bào)后睁宰，當(dāng)以傳統(tǒng)的基準(zhǔn)為背景時(shí)肪获，特異波動(dòng)率的定價(jià)依然為負(fù)，而以基礎(chǔ)性的商品基準(zhǔn)為背景時(shí)柒傻，特異波動(dòng)率的定價(jià)依然為0孝赫，這與我們之前的發(fā)現(xiàn)是一致的。
Bessembinder（1992）報(bào)告說(shuō)以純對(duì)沖壓力為條件的特異波動(dòng)率在農(nóng)產(chǎn)品和外匯期貨市場(chǎng)的定價(jià)為正红符，這個(gè)結(jié)果與 Hirshleifer (1988)年建立的理論模型是一致的青柄。根據(jù)他們的引導(dǎo)劫映，我們用我們的數(shù)據(jù)，以及A到 K的定價(jià)模型刹前，也研究了一下以對(duì)沖壓力為條件的特異波動(dòng)率是否驅(qū)動(dòng)正的風(fēng)險(xiǎn)超額收益率這個(gè)問(wèn)題泳赋。
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.
我們通過(guò)將我們的特異波動(dòng)率和一個(gè)虛設(shè)的變量相互作用來(lái)完成這個(gè)工作。這個(gè)虛設(shè)變量的取值是這樣的：如果投機(jī)持倉(cāng)在月初和月末都為凈多頭（貼水）喇喉，虛設(shè)變量取值為1祖今；如果投機(jī)持倉(cāng)在月初和月末都為凈空頭（升水），虛設(shè)變量取值為-1拣技；如果投機(jī)持倉(cāng)在當(dāng)月改變了持倉(cāng)方向千诬，虛設(shè)變量取值為0. 和采用的基準(zhǔn)無(wú)關(guān)，我們證實(shí)了Hirshleifer（1988）的預(yù)測(cè)和 Bessembinder (1992)的論證膏斤，確實(shí)在以對(duì)沖壓力為條件的特異波動(dòng)率和平均回報(bào)之間存在著一個(gè)正的關(guān)聯(lián)徐绑。換句話說(shuō)，在一個(gè)貼水的市場(chǎng)（投機(jī)持倉(cāng)為凈多頭）特異波動(dòng)率驅(qū)動(dòng)著一個(gè)正的風(fēng)險(xiǎn)回報(bào)而在一個(gè)升水的市場(chǎng)（投機(jī)持倉(cāng)為凈空頭）特異波動(dòng)率驅(qū)動(dòng)著一個(gè)負(fù)的風(fēng)險(xiǎn)回報(bào)莫辨。詳細(xì)的結(jié)果可以向我們索取傲茄。
4.2 特異波動(dòng)率令人困惑的負(fù)定價(jià)現(xiàn)象來(lái)自于貼水還是升水呢？
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.
目前為止我們主要的推理路線是認(rèn)為令人困惑的特異波動(dòng)率定價(jià)為負(fù)的現(xiàn)象其實(shí)是由于認(rèn)為忽略了商品期貨市場(chǎng)基礎(chǔ)性的 貼水/升水 周期沮榜。如同 Table 4 所清楚展示的那樣盘榨，特異波動(dòng)率顯著為負(fù)的定價(jià)現(xiàn)象在我們采用了考慮了貼水和升水的基礎(chǔ)性商品基準(zhǔn)時(shí)消失了。商品期貨市場(chǎng)可能逃脫貼水和升水這兩種自然狀態(tài)蟆融，因此或許是它們當(dāng)中的一個(gè)在解釋商品期貨特異波動(dòng)率負(fù)的定價(jià)方面扮演了更強(qiáng)作用的角色草巡。為了評(píng)估這種推測(cè)，我們重新對(duì)基礎(chǔ)性的商品指數(shù)采用 Ang et al (2009) 的兩步分析法型酥，不過(guò)這次我們只單獨(dú)考慮多頭（貼水）的TS山憨，HP 和 Mom 組合，在另一方面弥喉，只單獨(dú)考慮空頭（升水）的TS郁竟，HP 和 Mom 組合。結(jié)果呈現(xiàn)在 Table 6 中档桃。
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.
正如同貯藏理論和對(duì)沖壓力假說(shuō)所預(yù)測(cè)的那樣枪孩，與多頭貼水的 TS, HP 和 Mom 組合相關(guān)聯(lián)的風(fēng)險(xiǎn)定價(jià)通常是正的，在1%的置信水平上具有顯著性藻肄，而與空頭升水的TS,HP 和 Mom 組合相關(guān)聯(lián)的風(fēng)險(xiǎn)定價(jià)傾向是負(fù)的，對(duì)大部分?jǐn)?shù)據(jù)都具有顯著性拒担。這意味著投資者需要一個(gè)正的超額收益（也就是更大的回報(bào)）來(lái)持有一個(gè)貼水組合的多頭或者是一個(gè)升水組合的空頭嘹屯，這和 Figure 1 所展示的價(jià)格演化情況一致。
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.
關(guān)于特異波動(dòng)率因子从撼，多頭貼水組合的結(jié)果和之前展示的傳統(tǒng)基準(zhǔn)的結(jié)果非常相似州弟，超額收益率都顯著為負(fù)钧栖。例如，在Table 4中婆翔，參照傳統(tǒng)的基準(zhǔn)A拯杠，B，C的定價(jià)模型給特異波動(dòng)率的平均定價(jià)為 -0.4916啃奴，在Table 6 中潭陪，給多頭貼水的 TS，HP 和 Mom組合的特異波動(dòng)率的平均定價(jià)為 -0.3546最蕾。與此形成鮮明對(duì)比的是依溯，以空頭升水的 TS，HP和Mom為基準(zhǔn)對(duì)特異波動(dòng)率的定價(jià)只有 0.0470瘟则，這個(gè)結(jié)果無(wú)論在經(jīng)濟(jì)學(xué)意義上還是統(tǒng)計(jì)學(xué)意義上和 0 都區(qū)別不大黎炉。
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.
從我們的分析中可以得到的最重要的一個(gè)信息是令人困惑的特異波動(dòng)率的負(fù)定價(jià)現(xiàn)象實(shí)際上是一個(gè)人為現(xiàn)象，原因是定價(jià)模型中沒有包括升水風(fēng)險(xiǎn)組合因子醋拧。當(dāng)把后者包括進(jìn)去時(shí)慷嗜，特異波動(dòng)率將變得不再重要。因此之前的那些發(fā)現(xiàn)丹壕，即以傳統(tǒng)的商品基準(zhǔn)為基礎(chǔ)時(shí)洪添，持有特異波動(dòng)率較高資產(chǎn)的投資者將獲得一個(gè)負(fù)的超額收益率的觀點(diǎn)是虛假的。因?yàn)檫@僅僅反映了與持有多頭的升水組合相關(guān)的負(fù)的超額收益率雀费。這實(shí)際上相當(dāng)符合直覺干奢，通過(guò)持有多頭組合，投資者實(shí)際上扮演了套期保值者的角色盏袄，因此他們需要付給做空的投機(jī)商一些保險(xiǎn)費(fèi)用忿峻。換句話說(shuō)，在升水的市場(chǎng)中辕羽，通過(guò)持有多頭（相反情況下是空頭）頭寸逛尚，投機(jī)商放棄了他們通常能夠獲得的保險(xiǎn)費(fèi)用。
5刁愿，特異波動(dòng)率的時(shí)間序列定價(jià)
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.
在本節(jié)我們將通過(guò)采用風(fēng)險(xiǎn)模擬組合的方法研究特異波動(dòng)率和商品期貨市場(chǎng)回報(bào)之間的時(shí)間序列關(guān)系绰寞。
5.1 特異波動(dòng)率模擬組合
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)對(duì)股票市場(chǎng)的分析方法，我們根據(jù)特異波動(dòng)率對(duì)商品排序并劃分成五等分铣口，這是通過(guò)使用過(guò)去 R = {1,3,6,12}月份的每日收益率計(jì)算得到的滤钱，采用的基準(zhǔn)是傳統(tǒng)的基準(zhǔn)或者基礎(chǔ)性的商品基準(zhǔn)，如同 equation(1) 那樣脑题。然后我們構(gòu)建了一個(gè)模擬組合件缸，這個(gè)組合買入特異波動(dòng)率最低的1/5商品期貨的同時(shí)做空特異波動(dòng)率最高的1/5商品期貨。我們持有這個(gè)多空組合1個(gè)月叔遂，然后按照相同的交易過(guò)程他炊，一個(gè)新的多空特異波動(dòng)率組合將被創(chuàng)建争剿。
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.
為了與基礎(chǔ)性的風(fēng)險(xiǎn)因子(TS，HP 和 Mom)相一致痊末，特異波動(dòng)率組合是完全對(duì)沖的( fully-collateralized）蚕苇，在每個(gè)月末重新平衡持倉(cāng)，使頂部和底部的五分之一的構(gòu)成成分為等權(quán)重凿叠。等權(quán)重的計(jì)劃可以方便地避免組合集中到一個(gè)特定的商品因此能夠確保更好地分散化涩笤。過(guò)分集中可能導(dǎo)致流動(dòng)性問(wèn)題，讓投資者開倉(cāng)或空倉(cāng)出現(xiàn)潛在的困難幔嫂。我們通過(guò)在所有的基準(zhǔn)中包含Pastor 和 Stambaugh (2003)提出的流動(dòng)性風(fēng)險(xiǎn)溢價(jià)來(lái)處理這個(gè)問(wèn)題辆它。
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 展示了多空特異波動(dòng)率組合表現(xiàn)的統(tǒng)計(jì)數(shù)據(jù)，其中在左側(cè)部分特異波動(dòng)率是用傳統(tǒng)基準(zhǔn)衡量的履恩，也就是模型 A到C锰茉，而右側(cè)的特異波動(dòng)率是用基礎(chǔ)性的商品基準(zhǔn)衡量的，也就是模型D到K切心。 一個(gè)由12個(gè)參照傳統(tǒng)基準(zhǔn)的特異波動(dòng)率的多空組合等權(quán)重分配的策略每年可以盈利4.94%飒筑，在 5%置信水平上具有顯著性。12個(gè)策略中的10個(gè)在10%的置信水平上具有顯著超額正收益（12個(gè)策略中的7個(gè)在5%或者更好的置信水平上具有顯著超額收益）绽昏。與之形成鮮明對(duì)照的是协屡，一個(gè)在32個(gè)參照基礎(chǔ)性的商品基準(zhǔn)的特異波動(dòng)率多空策略等權(quán)重分配的組合只能取得上述收益率的一半，只有2.53%每年全谤，并且在統(tǒng)計(jì)上不具有顯著性肤晓；這32種策略中沒有任何一個(gè)能夠取得顯著的超額正收益。用夏普比率可以得到類似的推論认然，對(duì)于參照傳統(tǒng)基準(zhǔn)的超額收益率的多空組合补憾，其夏普比率（平均為 0.4543）遠(yuǎn)大于參照基礎(chǔ)性商品基準(zhǔn)的多空組合（只有約一半，平均為 0.2361）卷员；Opdyke t-檢驗(yàn)結(jié)果為 2.30盈匾，可以確認(rèn)前者的夏普比率顯著比后者更大。
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).
如同在時(shí)間截面分析框架中所做的那樣毕骡，我們測(cè)試了我們時(shí)間序列分析對(duì)于特定傳統(tǒng)的或基礎(chǔ)性商品基準(zhǔn)的穩(wěn)健性削饵。這通過(guò)如下方法實(shí)施：i)將 模型 A和 C 中的S&P-GSCI 指數(shù)分別更換為 Thomson Reuters/Jefferies CRB 指數(shù)， Dow Jones-UBSCI 或27中商品期貨的多頭等權(quán)重組合未巫；ii）考慮在模型D至G中讓基礎(chǔ)性風(fēng)險(xiǎn)因子（TS窿撬，HP，或 Mom）單獨(dú)作用而不是讓它們成對(duì)或三個(gè)聯(lián)合起作用橱赠；iii) 在模型 H至 K中同時(shí)包括股票市場(chǎng)風(fēng)險(xiǎn)收益率和 SMB 市值風(fēng)險(xiǎn)收益率尤仍。此處未呈現(xiàn)的結(jié)果表明Table 7中得到的結(jié)果在這些不同的規(guī)格條件下也是成立的：參照傳統(tǒng)基準(zhǔn)構(gòu)造的特異波動(dòng)率組合的夏普比率的平均值（0.3895）遠(yuǎn)遠(yuǎn)超過(guò)參照基礎(chǔ)性商品基準(zhǔn)構(gòu)造的特異波動(dòng)率組合的夏普比率（0.2497），顯著性O(shè)pdyke t-統(tǒng)計(jì)量達(dá)到了2.59狭姨。
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 , 這是用每月特異波動(dòng)率多空組合的收益率和多種風(fēng)險(xiǎn)因子做回歸后的截距來(lái)衡量的宰啦，Table 8 同時(shí)還呈現(xiàn)了根據(jù)自回歸和異方差性的魯棒性的 Newey and West (1987) 標(biāo)準(zhǔn)誤差的顯著性 t-統(tǒng)計(jì)量。 Table 8 同時(shí)在最后一列展示了兩種 alpha之間差別的顯著性 t-檢驗(yàn)結(jié)果饼拍。一種是等權(quán)重的12個(gè)基于傳統(tǒng)基準(zhǔn)的特異波動(dòng)率策略的alpha赡模，另外一種是等權(quán)重的32個(gè)基于基礎(chǔ)性商品基準(zhǔn)的特異波動(dòng)率的alpha。
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.
上述分析的結(jié)論如下师抄，如果采用傳統(tǒng)的基準(zhǔn)漓柑，特異波動(dòng)率組合產(chǎn)生alpha的能力將大為“樂觀”。事實(shí)上叨吮，參照基礎(chǔ)性的商品基準(zhǔn)（也就是模型 D至K）辆布，alphas 統(tǒng)計(jì)值實(shí)際為0，而參照傳統(tǒng)的基準(zhǔn)（也就是模型A到C）alpha 顯著為正茶鉴，大概是每年 5.21%锋玲。經(jīng)驗(yàn)數(shù)據(jù)上賣出高波動(dòng)率的組合的同時(shí)買入低波動(dòng)率的組合可以獲取反常收益的證據(jù)實(shí)際上是兩個(gè)模型問(wèn)題的人為產(chǎn)物：a）從傳統(tǒng)的基準(zhǔn)中導(dǎo)出的波動(dòng)率信號(hào)實(shí)際上并不是真的特異波動(dòng)率，因?yàn)樗艘粋€(gè)被忽略的基礎(chǔ)性貼水和升水周期的系統(tǒng)性風(fēng)險(xiǎn)成分 b) 這個(gè) alpha 實(shí)際上是用一個(gè)不合適的基準(zhǔn)測(cè)量的涵叮。
5.2 高特異波動(dòng)率組合和低特異波動(dòng)率組合
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.
在研究股票時(shí)惭蹂，Ang et al. (2006, 2009) 撰文指出特異波動(dòng)率多空組合的收益率表現(xiàn)更多地受益于較高特異波動(dòng)率的股票表現(xiàn)顯著地弱于市場(chǎng)平均而不是由于較低特異波動(dòng)率股票的表現(xiàn)強(qiáng)于市場(chǎng)平均。為了研究這一點(diǎn)是否也適用于商品期貨市場(chǎng)割粮，我們分別衡量了多頭低特異波動(dòng)率組合和空頭高特異波動(dòng)率組合的alphas盾碗。其中alphas 分別基于前面定義的傳統(tǒng)的或基礎(chǔ)性的商品基準(zhǔn)計(jì)算得到。Table 9 展示了這個(gè)結(jié)果舀瓢。
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.
使用傳統(tǒng)基準(zhǔn)的結(jié)果與Ang et al.(2006,2009)的結(jié)論一致：低特異波動(dòng)率組合似乎略微有正的超額收益（平均的alpha 為每年 1.81%）廷雅，而高特異波動(dòng)率組合則有非常低的年化 alpha(平均每年為 -5.37%)。有趣的是京髓，如果基礎(chǔ)性的貼水/升水周期被合理地歸因到基準(zhǔn)中航缀，并用來(lái)對(duì)特異波動(dòng)率建模和對(duì)反常收益率進(jìn)行衡量，多頭和空頭組合的alphas的絕對(duì)值都會(huì)減小朵锣。Table 9 中最后一行圓括號(hào)中的是根據(jù)這兩種不同基準(zhǔn)計(jì)算出來(lái)的平均alphas(依次多頭和空頭)的差值是否顯著的 t-檢驗(yàn) 統(tǒng)計(jì)量谬盐。t-統(tǒng)計(jì)量 為 2.99（-4.43），依照傳統(tǒng)基準(zhǔn)的多頭(空頭)組合的alphas诚些，標(biāo)記為 A 至C 飞傀，統(tǒng)計(jì)上比 依照基礎(chǔ)性商品基準(zhǔn)的多頭（空頭）組合，標(biāo)記為D至K诬烹，的alphas 更大（性曳场）。這進(jìn)一步支持了我們的觀點(diǎn)：依靠傳統(tǒng)的基準(zhǔn)構(gòu)建的特異波動(dòng)率信號(hào)實(shí)際上部分是系統(tǒng)性的绞吁，因?yàn)樗从沉撕唾N水和升水周期相聯(lián)系的系統(tǒng)性風(fēng)險(xiǎn)幢痘。一旦把貼水和升水周期考慮進(jìn)來(lái)，多頭和空頭的特異波動(dòng)率模擬組合的alphas將變成可忽略不計(jì)的家破。
6颜说，結(jié)論 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.
這篇文章研究了商品期貨特異波動(dòng)率和預(yù)期收益率之間的關(guān)系购岗。本文寫作的動(dòng)機(jī)是由Ang et al. (2009) 提出的在全球股票市場(chǎng)上存在著特異波動(dòng)率定價(jià)為負(fù)的經(jīng)驗(yàn)性證據(jù)所引起。把他們的分析擴(kuò)展到商品期貨市場(chǎng)上市非常有趣的门粪，不僅是因?yàn)檫@或許可以證明這種令人困惑的負(fù)相關(guān)關(guān)系是否具有普遍性喊积，還因?yàn)檫@可以顯示為什么這些數(shù)據(jù)為什么會(huì)包含這種反常的效應(yīng)。這里展示的經(jīng)驗(yàn)證據(jù)支持這樣的推論玄妈，特異波動(dòng)率和商品期貨的預(yù)期收益率取決于選取的定價(jià)模型乾吻，或者說(shuō)是用于提取特異波動(dòng)率的基準(zhǔn)信號(hào)。當(dāng)選用的定價(jià)模型不能夠識(shí)別商品期貨市場(chǎng)不可阻擋的貼水和升水周期循環(huán)時(shí)拟蜻，特異波動(dòng)率似乎將引起一個(gè)令人困惑的負(fù)的超額收益率绎签。于是，構(gòu)建一種買入低波動(dòng)率的商品期貨資產(chǎn)并賣出高波動(dòng)率的商品期貨資產(chǎn)的模擬組合將可以獲得可觀的平均收益酝锅。表面上看這是把 Ang et al. (2009) 針對(duì)全球股票市場(chǎng)的文章擴(kuò)展到商品期貨市場(chǎng)诡必。
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，由于商品期貨在定價(jià)時(shí)應(yīng)當(dāng)考慮升水和貼水的基礎(chǔ)事實(shí)屈张，構(gòu)建特異波動(dòng)率的多空組合所獲取的超額收益率將消失擒权，特異波動(dòng)率將不再產(chǎn)生一個(gè)負(fù)的超額收益率。這個(gè)結(jié)論和基礎(chǔ)金融學(xué)的信條一致：特異波動(dòng)率可以通過(guò)分散化消除掉因此定價(jià)為0阁谆。進(jìn)一步的證據(jù)表明當(dāng)采用傳統(tǒng)的基準(zhǔn)時(shí)碳抄，看起來(lái)與特異波動(dòng)率相聯(lián)系的負(fù)的超額收益率是升水組合而不是貼水組合的定價(jià)現(xiàn)象。特異波動(dòng)率代表著和升水合約相聯(lián)系的風(fēng)險(xiǎn)场绿。
進(jìn)一步的研究可以將這里的分析擴(kuò)展到FOREX期貨市場(chǎng)剖效，在FOREX期貨市場(chǎng)上，升水和貼水現(xiàn)象已經(jīng)被顯示是有重要影響的(Bessembinder, 1992)焰盗。
