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Breadth Momentum and Vigilant Asset Allocation (VAA);
Winning More by Losing Less
广度动量和警惕资产配置 (VAA);
通过减少损失来赢得更多

By Wouter J. Keller and Jan Willem Keuning
由沃特·J·凯勒和扬·威廉·克宁撰写
July 14, 2017, v0.99 2017 年 7 月 14 日,v0.99

Summary 摘要

VAA (Vigilant Asset Allocation) is a dual-momentum based investment strategy with a vigorous crash protection and a fast momentum filter. Dual momentum combines absolute (trendfollowing) and relative (strength) momentum. Compared to the traditional dual momentum approaches, we have replaced the usual crash protection through trendfollowing on the asset level by our breadth momentum on the universe level instead. As a result, the VAA strategy is on average often more than out of the market. We show, however, that the resulting momentum strategy is by no means sluggish. By using large and small universes with US and global ETF-like monthly data starting 1925 and 1969 respectively, we arrive out-of-sample at annual returns above with max drawdowns below 15% for each of these four universes.
VAA(警惕资产配置)是一种基于双动量的投资策略,具有强大的崩盘保护和快速的动量过滤。双动量结合了绝对(趋势跟随)和相对(强度)动量。与传统的双动量方法相比,我们用广度动量替代了通常在资产层面通过趋势跟随实现的崩盘保护。因此,VAA 策略平均而言通常超过市场 。然而,我们表明,所得到的动量策略绝不是迟缓的。通过使用自 1925 年和 1969 年起的美国和全球 ETF 类的月度数据,我们在样本外得到了四个宇宙的年回报率超过 ,最大回撤低于 15%。

1. Introduction 1. 引言

The VAA (Vigilant Asset Allocation) strategy described in this paper is a follow-up on our PAA strategy (Protective Asset Allocation, see Keller, 2016), targeting higher annual returns while at the same time offering stricter crash protection than PAA. Our target for VAA is offensive returns with defensive risks: winning more by losing less. To be more precise, with VAA we aim at moderate but offensive returns above but with defensive drawdowns of less than , preferably less than 15%. We will use monthly data starting Dec 1925 and Dec 1969 respectively for various US and global asset classes (as proxies for present day ETFs). More on our data later (see section 7).
本文所描述的 VAA(警惕资产配置)策略是我们 PAA 策略(保护性资产配置,见 Keller,2016)的后续,旨在实现更高的年回报,同时提供比 PAA 更严格的崩盘保护。我们的 VAA 目标是以防御性风险获取进攻性回报:通过减少损失来赢得更多。更准确地说,VAA 的目标是实现适度但进攻性的回报,超过 ,同时防御性回撤低于 ,最好低于 15%。我们将分别使用从 1925 年 12 月和 1969 年 12 月开始的月度数据,针对各种美国和全球资产类别(作为当今 ETF 的代理)。关于我们的数据将在后面详细说明(见第 7 节)。
VAA is part of the class of momentum based tactical asset allocations. Momentum (or "price persistence") can be applied to stocks (see eg. Jegadeesh, 1993) or asset-classes (see eg. Faber, 2007). We focus here on asset-classes (or assets, for short). Relative momentum (also called crosssectional or relative strength momentum, see eg. Faber 2010, Moskowitz 2011, Asness 2014 and Faber 2015) uses only the best (top T) performing assets within a universe (of size ) while absolute momentum (also called time-series momentum or trendfollowing, see eg. Moskowitz 2011, Antonacci 2013a, and Levine 2015) selects only the assets with positive momentum. The combination of absolute and relative momentum is often called dual momentum (Antonnacci, 2013b, 2014). For a historical overview, see also Faber (2013), Newfound (2015) and Antonacci (2014). As with most dual momentum models we will restrict ourselves to long-only trading strategies (no short-selling) with monthly portfolio reforms and rebalances.
VAA 是基于动量的战术资产配置的一部分。动量(或称“价格持续性”)可以应用于股票(参见例如 Jegadeesh, 1993)或资产类别(参见例如 Faber, 2007)。我们在这里关注资产类别(或简称资产)。相对动量(也称为横截面或相对强度动量,参见例如 Faber 2010, Moskowitz 2011, Asness 2014 和 Faber 2015)仅使用在一个宇宙(大小为 )中表现最好的(前 T)资产,而绝对动量(也称为时间序列动量或趋势跟随,参见例如 Moskowitz 2011, Antonacci 2013a 和 Levine 2015)仅选择具有正动量的资产。绝对动量和相对动量的结合通常称为双重动量(Antonnacci, 2013b, 2014)。有关历史概述,请参见 Faber (2013), Newfound (2015) 和 Antonacci (2014)。与大多数双重动量模型一样,我们将限制自己采用仅做多的交易策略(不做空),并进行每月的投资组合调整和再平衡。
To arrive at a vigilant model, we apply a "vigilant (fast and vigorous) crash protection (CP) strategy based on a "fast" filter for absolute momentum. We define a "bad" asset as an asset with nonpositive momentum. Our crash protection is based on the number of bad assets in the universe, instead of the replacement of individual bad assets, as with traditional absolute (and dual) momentum approaches. So, we use a kind of market breadth (in terms of our fast momentum filter) as crash indicator, as we did with PAA. Therefore, our breadth momentum extends the traditional absolute momentum when it comes to crash protection.
为了实现一个警惕的模型,我们应用了一种基于“快速”绝对动量的“警惕(快速而有力)崩溃保护(CP)策略”。我们将“坏”资产定义为动量为非正的资产。我们的崩溃保护基于整个市场中的坏资产数量,而不是像传统的绝对(和双重)动量方法那样替换个别坏资产。因此,我们使用一种市场广度(根据我们的快速动量过滤器)作为崩溃指标,正如我们在 PAA 中所做的那样。因此,我们的广度动量在崩溃保护方面扩展了传统的绝对动量。
Compared with PAA, however, we will use the number of bad assets relative to a protection threshold as a more granular crash indicator. As we will show, in-sample optimization of this threshold frequently results in an out-of-market allocation (i.e. cash), even when just one or only a limited number of the assets are bad. As a result, we will be in cash on average more than 50% over time in our four universes. When not fully in cash we will use the same (fast) filter for relative momentum to arrive at a more offensive strategy with only a limited number (top T) of best performing assets.
然而,与 PAA 相比,我们将使用相对于保护阈值的坏资产数量作为更细致的崩溃指标。正如我们将展示的,阈值的样本内优化经常导致市场外配置(即现金),即使只有一个或有限数量的资产是坏的。因此,在我们的四个领域中,我们平均会有超过 50%的时间处于现金状态。当不完全处于现金时,我们将使用相同的(快速)相对动量过滤器,以得出仅限于少量(前 T)表现最佳资产的更激进策略。
As we will see later, our "cash" concept is not limited to a risk-free asset but also includes various bonds in a separate "cash universe", besides the main universe of "risky" assets (which also might include bonds). Analogous to our relative momentum strategy for the risky universe, we will always choose the single best performing bond for cash, but without looking at the sign of the momentum of these bonds (ie. no absolute momentum for cash).
正如我们稍后将看到的,我们的“现金”概念不仅限于无风险资产,还包括在一个单独的“现金宇宙”中的各种债券,此外还有主要的“风险”资产宇宙(这也可能包括债券)。与我们针对风险宇宙的相对动量策略类似,我们将始终选择表现最佳的单一债券作为现金,但不考虑这些债券的动量符号(即现金没有绝对动量)。
As return measure, we will focus on CAGR (Compound Annual Growth Rate, so geometrical returns) while for risk we mainly take maximum drawdown in consideration, since we believe this is often felt as the more important risk indicator for investors than the traditional volatility measure. As our (insample) optimizing target we will therefore use a new return/risk measure: returns adjusted for drawdowns (RAD), besides the more familiar Sharpe (excess return/volatility) and MAR ratio (return/max drawdown). More on RAD later (see section 6).
作为回报衡量标准,我们将关注复合年增长率(CAGR,几何回报),而在风险方面,我们主要考虑最大回撤,因为我们认为这通常被投资者视为比传统波动性指标更重要的风险指标。因此,作为我们的(样本内)优化目标,我们将使用一种新的回报/风险衡量标准:调整回撤的回报(RAD),此外还有更为熟悉的夏普比率(超额回报/波动性)和 MAR 比率(回报/最大回撤)。关于 RAD 的更多信息请参见第 6 节。

2. Fast momentum 快速动量

For our momentum filter we will use a variant of the often used 13612 filter (see eg. Faber, 2007 and Keller, 2015), but now with an even faster response curve by using the average annual returns over the past and 12 months. We will denote this filter as 13612 W . The traditional 13612 filter uses the average total returns over the same four periods.
对于我们的动量过滤器,我们将使用一种常用的 13612 过滤器的变体(参见例如 Faber,2007 年和 Keller,2015 年),但现在通过使用过去 和 12 个月的平均年回报率,响应曲线更快。我们将这个过滤器称为 13612 W。传统的 13612 过滤器使用相同四个时期的平均总回报率。
In Fig. 1 we give the monthly return weights for various momentum filters (see also Beekhuizen, 2015, Zakamulin, 2015, and Keller, 2016, note 2), including our new 13612 W filter. Notice that our faster 13612 W filter gives a weight of (19/48) to the last month return as compared to and 18% for the simple 12-month return filter (RET12) as used eg. by Moskowitz (2011) and
在图 1 中,我们给出了各种动量过滤器的月度回报权重 (另见 Beekhuizen,2015 年,Zakamulin,2015 年和 Keller,2016 年,注 2),包括我们新的 13612 W 过滤器。请注意,我们更快的 13612 W 过滤器对上个月的回报赋予了 (19/48)的权重,而简单的 12 个月回报过滤器(RET12)则赋予了 和 18%的权重,例如 Moskowitz(2011 年)使用的。
Antonnacci (2013a), the SMA12-based filter as used by Keller (2016) for PAA, and the 13612 filter as used by Faber (2010), Hurst (2012), and Keller (2015), respectively .
Antonnacci(2013a),Keller(2016)用于 PAA 的基于 SMA12 的滤波器,以及 Faber(2010)、Hurst(2012)和 Keller(2015)分别使用的 13612 滤波器
Fig.1. Monthly return weights for RET12, SMA12, 13612, and our 13612W filter
图 1. RET12、SMA12、13612 和我们的 13612W 过滤器的月度回报权重
Because of the heavy weight to the last month in 13612W, we assume no 1-month reversal effects in our asset-class data. Although 1-month reversals are common for individual stocks, we found the opposite for most of our asset-class data. In-sample testing also shows better return/risk performance when using the 13612W filter instead of the 1-month filter or one of the other momentum filters (SMA12, RET12, and 13612).
由于 13612W 在上个月的重压,我们假设在我们的资产类别数据中没有 1 个月的反转效应。尽管个别股票的 1 个月反转很常见,但我们发现大多数资产类别数据则相反。样本内测试还显示,使用 13612W 过滤器而不是 1 个月过滤器或其他动量过滤器(SMA12、RET12 和 13612)时,回报/风险表现更好。
Now we can define our momentum measure of asset (class) for (the end of) each month as the average weighted lagged price according to the 13612W filter applied to the (dividend etc. adjusted ie. Total Return/TR) lagged monthly prices of the asset. Notice that we need 12 lagged prices for the 13612 W filter, so our backtests always start one year after the start of the data. We will use the same 13612 W filter for absolute and relative momentum in contrast to other authors (eg. Faber, 2007 using dual momentum).
现在我们可以为每个月(结束时)定义资产(类别) 的动量度量,即根据应用于资产的(经过股息等调整即总回报/TR)滞后月价格的 13612W 滤波器计算的加权滞后价格的平均值。请注意,我们需要 12 个滞后价格来使用 13612W 滤波器,因此我们的回测总是从数据开始一年后开始。我们将对绝对和相对动量使用相同的 13612W 滤波器,这与其他作者(例如,Faber,2007 年使用双重动量)形成对比。

3. Fast Crash Protection 3. 快速崩溃保护

For our crash protection (CP) we go along the lines of our PAA model where we used the number of so-called bad assets b (with non-positive momentum) in our universe to define the degree of cash. We will call this universe based approach breadth momentum, in contrast to traditional absolute momentum, which operates at the individual asset level by trendfollowing. Breadth momentum was
为了我们的崩溃保护(CP),我们遵循我们的 PAA 模型,在该模型中,我们使用了我们宇宙中所谓的坏资产 b(具有非正动量)来定义现金的程度。我们将这种基于宇宙的方法称为广度动量,与传统的绝对动量形成对比,后者通过趋势跟随在个别资产层面上运作。广度动量是
also used with our PAA strategy (Keller, 2016). However, the crash protection algorithm for VAA allows for much more granularity and aggressiveness than with PAA.
也用于我们的 PAA 策略(Keller,2016)。然而,VAA 的崩溃保护算法允许比 PAA 更高的细粒度和更强的攻击性。
We therefore define the breadth protection threshold B (or "breadth B", for short) as the minimal number of bad assets for which we go to cash, while we use the fraction (for ) as Cash Fraction, CF. In formula:
因此,我们将宽度保护阈值 B(简称“宽度 B”)定义为我们转为现金所需的最小不良资产数量 ,同时我们使用比例 (用于 )作为现金比例 CF。公式为:
if , and when (with , and )
如果 ,并且 (与 ,和
Notice that this is strikingly different from the traditional dual momentum approach (see Faber 2007, and others ) where CP is based on the number of bad assets in the relative momentum based best (top T) assets. So VAA allows us to go to cash more vigorously (ie. faster). When B<N/2, VAA is also faster than the most protective PAA variant, PAA2. In fact, the crash protection of PAA2 is equivalent to VAA with except for the momentum filter: SMA for PAA, 13612 W for VAA.
请注意,这与传统的双动量方法截然不同(参见 Faber 2007 及其他文献 ),其中 CP 是基于相对动量最佳(前 T)资产中的不良资产数量。因此,VAA 使我们能够更积极地转向现金(即更快)。当 B
Fig. 2 illustrates the vigorous crash protection (CP) of VAA. Here we show the cash fraction CF as a function of the number b of bad assets for three strategies (all with universe size and Top ):
图 2 展示了 VAA 的强大崩溃保护(CP)。在这里,我们将现金比例 CF 作为不良资产数量 b 的函数,针对三种策略(所有策略的宇宙大小为 ,前 ):
  • VAA with B=4 VAA 与 B=4
  • PAA2 (=VAA with  PAA2 (=VAA 与
  • Dual momentum, with
    双动量,带有
Fig. 2. for for VAA ( ), PAA2 (i.e. ) and Dual ( )
图 2. 对于 的 VAA ( ), PAA2 (即 ) 和 Dual ( )
Dual (Dual momentum) is defined here as the traditional dual momentum strategy where you select the best performing top assets with share (equal weight) while replacing the bad assets (ie.
双重(双重动量)在此定义为传统的双重动量策略,即选择表现最佳的前 个资产,权重为 (等权重),同时替换表现不佳的资产(即。
with non-positive absolute momentum) in this top by cash . As can easily be seen from the figure, for a given (and ) the cash fraction CF for VAA will always be more than that for Dual, ie. for . Only when (all assets bad) there holds for VAA, PAA and Dual.
在这个顶级 中,现金 的绝对动量为非正值。从图中可以很容易看出,对于给定的 (和 ),VAA 的现金比例 CF 总是会高于 Dual,即 对于 。只有当 (所有资产都不好)时,VAA、PAA 和 Dual 才满足
Notice again that for Dual the chosen relative momentum filter (for the top T selection) is sometimes different from the absolute momentum filter (see eg. Faber 2007). In contrast, for VAA our fast 13612 W filter is used for both types of momentum. Notice also that absolute momentum (the direction of the trend) in VAA works at the universe or market level while it works at the individual asset level in Dual approaches. In other words, in VAA absolute momentum only defines the number of bad assets (and therefore CF) in the universe and not the individual bad assets to be replaced by cash, as in Dual.
请注意,对于 Dual,选择的相对动量过滤器(用于顶部 T 选择)有时与绝对动量过滤器不同(例如,见 Faber 2007)。相比之下,对于 VAA,我们的快速 13612 W 过滤器用于两种类型的动量。还要注意,在 VAA 中,绝对动量(趋势的方向)在整体或市场层面上起作用,而在 Dual 方法中则在个别资产层面上起作用。换句话说,在 VAA 中,绝对动量仅定义了宇宙中不良资产的数量 (因此 CF),而不是要被现金替代的个别不良资产,如在 Dual 中所示。
In fact, one might recognize in the ratio (with being the number of good assets, ie. with positive momentum) the well-known market breadth, now based on our 13612 W momentum filter. This market breadth ratio equals one when all assets are good, minus one when all assets are bad and zero when the market is fifty/fifty. So, you might say that our cash fraction is a function of this market breath and the relative protection threshold . This also explains our term breadth momentum as the force behind our aggressive crash protection.
事实上,人们可能会在比率 中识别出著名的市场广度(其中 是良好资产的数量,即具有正动量),该比率现在基于我们的 13612 W 动量过滤器。当所有资产都是好的时,这个市场广度比率等于一;当所有资产都是坏的时,等于负一;当市场是五五开时,等于零。因此,可以说我们的现金比例是这个市场广度和相对保护阈值 的函数。这也解释了我们所说的广度动量作为我们积极崩盘保护背后的力量。

4. Easy Trading and the cash universe
4. 简易交易与现金宇宙

As is shown in Fig. 2, the CF fractions differ between VAA, PAA and Dual for . With dual momentum (Dual), the selected top asset fractions equal the cash fractions. If eg. all three selected assets have an equal share of , which is replaced by an equal share of cash in case of bad assets in the top T. This will make trading easy: you simply replace bad assets in the Top T by cash.
如图 2 所示,CF 分数在 VAA、PAA 和 Dual 之间存在差异,对于 。在双重动量(Dual)下,所选的前 个资产分数等于现金分数。例如,如果 所有三个选定资产的份额相等为 ,在前 T 中出现不良资产时,将其替换为相等的现金份额。这将使交易变得简单:您只需用现金替换前 T 中的不良资产。
Trading is, however, less easy with VAA. Take, for example, the case where and as displayed in Fig. 2. Starting with no bad assets at all in the universe of assets, so and for each of the top 3 assets, when next month (one bad asset in the universe) the cash fraction becomes . So, for each of the top3 assets and for cash (PAA style). As a result, we have to sell part of all three assets and buy cash to replace the remainder, which results in much more trading (and therefore possibly more slippage) than in the "easy trading" case of dual momentum where assets in the Top are sold and replaced by cash.
然而,使用 VAA 进行交易并不那么简单。以图 2 中显示的 为例。从 资产的宇宙中完全没有不良资产开始,因此每个前 3 个资产的 ,当下个月 (宇宙中有一个不良资产)时,现金比例变为 。因此,每个前 3 个资产和现金的 (PAA 风格)。结果,我们必须出售所有三个资产的一部分,并购买现金以替代其余部分,这导致比“双动量”的“简单交易”情况(在该情况下,前面的资产被出售并由现金替代)进行更多的交易(因此可能会有更多的滑点)。
To force "Easy Trading" (ET) in the case of VAA we map the fractions b/B to a multiple of the Top asset fractions , and remove the corresponding worst asset(s) from the Top . This is simply achieved by rounding down the raw fractions to multiples of . So, if eg. and with , no cash replacement is required and we keep the Top3 allocation since , for which rounding down results in . If eg. and with , the as the rounddown
为了在 VAA 的情况下强制执行“简单交易”(ET),我们将分数 b/B 映射到顶级资产分数 的倍数,并从顶级 中移除相应的最差资产。这可以通过将原始分数 向下舍入到 的倍数来简单实现。因此,例如 ,不需要现金替代,我们保持 Top3 的分配,因为 ,向下舍入的结果是 。如果例如 ,则 作为向下舍入的结果。
result of , consequently we replace the worst asset from the Top3 by cash. Explanding the same example to or which gives , or , we arrive at cash fraction of or 1 , respectively. The worst assets are the assets with the lowest 13612 W momentum in the top T . In general, the formula for CF with ET rounding becomes:
的结果,因此我们用现金替换 Top3 中的最差资产。将同样的例子扩展到 ,得到 ,我们分别得出现金比例为 或 1。最差资产是 Top T 中动量最低的 13612 W 资产。一般来说,带有 ET 舍入的 CF 公式变为:
By following this method, we always replace the worst asset(s) of the top by cash instead of sizing down all top T assets. This VAA-ET mechanism is also an essential difference between VAA on one hand and PAA at the other, since no ET was defined in case of PAA (see Keller, 2016).
通过这种方法,我们始终用现金替换前 的最差资产,而不是缩减所有前 T 个资产。这种 VAA-ET 机制也是 VAA 与 PAA 之间的一个重要区别,因为在 PAA 的情况下没有定义 ET(见 Keller,2016)。
When no Easy Trading rounding is necessary, as was the case with eg. the N12 universe in our PAA2 model (where ). When or the whole portfolio is fully invested in cash (when ) or fully invested in the top risky asset(s) (when ). Notice that when , the rounding of CF might give rise to less granular crash protection than with (see in Fig. 3), while when the crash protection becomes less granular than all possible multiples of (see in Fig. 3). For example, if and , only (for resp.) will be used, so we replace groups of 3 assets at a time by cash (and vice versa). In Fig. 3 we have depicted CF for various combinations of and .
不需要简单交易四舍五入时,就像我们 PAA2 模型中的 N12 宇宙(其中 )。当 时,整个投资组合完全投资于现金(当 )或完全投资于前 个高风险资产(当 )。请注意,当 时,CF 的四舍五入可能会导致比 更少的细粒度崩溃保护(见图 3 中的 ),而当 时,崩溃保护变得比所有可能的 倍数更少细粒度(见图 3 中的 )。例如,如果 ,则仅使用 (分别为 ),因此我们一次用现金替换 3 个资产组(反之亦然)。在图 3 中,我们描绘了不同组合的 的 CF。
Fig. 3. for VAA, PAA, Dual ( , for and )
图 3. 用于 VAA、PAA、Dual ( ,用于 )
Finally, some words about "cash". Traditionally, Faber (2007) uses (riskfree) 90-day TBill as cash by default, but also introduced alternative cash strategies with the Government bonds as "cash"proxy. We took cash one step further in our PAA paper by introducing a cash universe populated with SHY and IEF (short-term and intermediate-term US-treasury ETFs, respectively), picking each month the best of the two (in terms of momentum) as the cash asset. With VAA we extend the cash universe to three bond-like assets: SHY, IEF and LQD (or 30d TBill, IT Gov and LT Corp from Ibb for the longer backtests from 1927, see Section 7 below) and use our 13612W relative momentum filter to select the best bond. Notice that we use the same cash concept for the Dual strategy in our backtests too, where we replace the assets in the Top T with non-positive momentum by the best
最后,关于“现金”的一些话。传统上,Faber(2007)默认使用(无风险)90 天国库券作为现金,但也引入了以 政府债券作为“现金”代理的替代现金策略。在我们的 PAA 论文中,我们进一步发展了现金的概念,引入了一个由 SHY 和 IEF(分别是短期和中期美国国债 ETF)构成的现金宇宙,每个月根据动量选择其中表现最佳的作为现金资产。通过 VAA,我们将现金宇宙扩展到三种类债券资产:SHY、IEF 和 LQD(或 30 天国库券、IT 政府债券和 LT 公司债券,来自 Ibb,用于 1927 年以来的更长回测,见下文第 7 节),并使用我们的 13612W 相对动量过滤器选择最佳债券。请注意,我们在回测的双重策略中也使用相同的现金概念,在那里我们用最佳资产替换了 Top T 中动量为非正的资产。

cash asset out of our cash universe. We also use the same 13612W momentum filter for Dual as with VAA.
现金资产来自我们的现金宇宙。我们还使用与 VAA 相同的 13612W 动量过滤器用于 Dual。

5. Backtesting: In-sample optimization and out-of-sample validation
5. 回测:样本内优化和样本外验证

In each of our four VAA backtests below, we will split our backtest in two nearly equal parts: the first part being the in-sample "optimization" (or learning) period, followed by the second out-of-sample "test" (or validation) period. This way, "datasnooping" (a.k.a. overfitting or datamining, see e.g. Harvey 2013, 2014) for the optimal parameters is limited to the in-sample period. We also need the first year of the data for our momentum filter, hence each of the backtests requires a one year initialization period.
在下面的四个 VAA 回测中,我们将把回测分为两个几乎相等的部分:第一部分是样本内的“优化”(或学习)期,第二部分是样本外的“测试”(或验证)期。这样,“数据窥探”(即过拟合或数据挖掘,参见例如 Harvey 2013, 2014)对于最佳参数的限制仅限于样本内期。我们还需要数据的第一年作为动量过滤器,因此每个回测都需要一年的初始化期。
Therefore, in our short dataset from Dec 1969 - Dec 2016 ( 47 years) we will use Dec 1970 - Dec 1993 (23 years) as the In-Sample (IS) optimization period to determine the best parameters. Then, we will check these parameters in the Out-of-Sample (OS) test period of Dec 1993 - Dec 2016 (23 years). Besides IS and OS periods we will also define the Full Sample (FS) as the combination of IS+OS, so FS in our example equals Dec 1970 - Dec 2016 (46 years), and the Recent Sample (RS) as the last decade (Dec 2006 - Dec 2016, 10 years). Similarly, with the long dataset from 1925, IS and OS for the longer backtest equals Dec 1926 - Dec 1970 (44 years) and Dec 1970- Dec 2016 (46 years), respectively.
因此,在我们从 1969 年 12 月到 2016 年 12 月的短数据集中(47 年),我们将使用 1970 年 12 月到 1993 年 12 月(23 年)作为样本内(IS)优化期,以确定最佳参数。然后,我们将在 1993 年 12 月到 2016 年 12 月(23 年)的样本外(OS)测试期中检查这些参数。除了 IS 和 OS 期,我们还将定义全样本(FS)为 IS 和 OS 的组合,因此在我们的例子中,FS 等于 1970 年 12 月到 2016 年 12 月(46 年),最近样本(RS)为最后十年(2006 年 12 月到 2016 年 12 月,10 年)。同样,对于从 1925 年开始的长数据集,较长回测的 IS 和 OS 分别为 1926 年 12 月到 1970 年 12 月(44 年)和 1970 年 12 月到 2016 年 12 月(46 年)。
What parameters are optimized over IS in the VAA model and how? We will optimize both the breadth threshold (eg. when for ) and the "top" (eg. when we select the top 3 of best performing assets). Since we can't select more assets than the full universe population, T is constrained by . Similar, we have the feasible range (with we will always be in cash). As can be seen in Fig. 2 , in the case where , the (rounded) cash fraction CF is always higher with VAA than in the traditional Dual case. So no bad assets will show up in the remaining Top assets (after removing the CF fraction of worst assets for VAA with ET).
在 VAA 模型中,优化了哪些参数?我们将优化宽度阈值 (例如,当 时为 )和“顶级” (例如,当我们选择表现最佳的前 3 个资产时为 )。由于我们不能选择超过完整宇宙人口的资产,T 受到 的限制。同样,我们有可行范围 (如果 ,我们将始终处于现金状态)。如图 2 所示,在 的情况下,现金比例 CF 在 VAA 中始终高于传统的双重情况。因此,在剩余的顶级资产中不会出现不良资产(在使用 ET 的 VAA 中去除最差资产的 CF 比例后)。
To allow for vigorous crash protection for the three larger universes with , we will limit the breadth , while we also restrict diversification (the Top selection) by limiting . For the smallest universe with we simply take . So, the number of values (scenarios) to consider in-sample is for the larger universes while for the smallest universe the number is . Given the long in-sample periods ( 23 and 46 years, respectively) combined with only two parameters ( and ) and the limited number of scenarios (16-36), we expect limited datasnooping effects on eg. the estimated Sharpe ratios.
为了为三个较大宇宙提供强有力的崩溃保护,我们将限制广度,同时通过限制多样化(顶级选择)来进行限制。对于最小的宇宙,我们简单地取值。因此,考虑的样本内值(场景)数量对于较大宇宙是 ,而对于最小宇宙则是 。鉴于较长的样本内周期(分别为 23 年和 46 年)结合仅有的两个参数( )以及有限的场景数量(16-36),我们预计对例如估计的夏普比率的资料挖掘效应有限。
Notice that, although we limited the datasnooping bias by this in/out-of sample tests for both parameters and , there are always other choices which are not tested this way. In this respect, one might consider in particular our choice of the 13612 W momentum filter. Although there were some theoretical reasons for selecting this filter (above the usual 13612 filter), and although we have mainly focused on its in-sample performance when selecting it, we might not have chosen it with a bad out-of-sample performance. However, the fact that it works so well for all our 4 different universes and two different backtest periods, both in- and out-of-sample, might indicate some robustness and therefore limited datasnooping bias. Also, our choice of the 4 universes might be a source for datasnooping, although this choice was mainly determined by the available data and previous studies (PAA and GEM), and partly (VAA-U6) by some in-sample testing only. And finally, we like to focus on the new "breadth B" parameter, so that is the main object in our in-sample optimizations. But still, datasnooping is a serious risk when backtesting, so the best test is a live test (see, e.g., Jones 2017).
请注意,尽管我们通过对参数 进行的样本内/样本外测试限制了数据探测偏差,但总是存在其他未以这种方式测试的选择。在这方面,特别值得考虑的是我们选择的 13612 W 动量过滤器。尽管选择该过滤器(高于通常的 13612 过滤器)有一些理论依据,并且在选择时我们主要关注其样本内表现,但我们可能并没有选择一个样本外表现不佳的过滤器。然而,它在我们四个不同的宇宙和两个不同的回测周期中,无论是样本内还是样本外表现如此出色,可能表明了一定的稳健性,因此数据探测偏差有限。此外,我们选择的四个宇宙可能是数据探测的一个来源,尽管这个选择主要是由可用数据和之前的研究(PAA 和 GEM)决定的,部分(VAA-U6)仅通过一些样本内测试决定。最后,我们希望关注新的“广度 B”参数,因此这是我们样本内优化的主要对象。但仍然,数据探测在回测时是一个严重的风险,因此最好的测试是实时测试(例如,参见)。琼斯 2017).
The next question is: what performance indicator do we optimize in-sample?
下一个问题是:我们在样本内优化哪个性能指标?

6. Returns adjusted for drawdowns (RAD)
6. 调整后收益(RAD)

As said in the introduction, with our VAA strategy we aim at combining high (offensive) annual returns with low (defensive) drawdowns. Therefore, we will introduce a new return/risk measure of the resulting VAA equity line, called Returns Adjusted for Drawdowns (RAD). Its formula is
如引言中所述,我们的 VAA 策略旨在将高(进攻性)年回报与低(防御性)回撤相结合。因此,我们将引入一个新的回报/风险指标,称为调整回撤的回报(RAD)。其公式为
if and , and otherwise,
如果 ,否则
where and Max Drawdown ( , measured EOM in our case) of the VAA equity line over the chosen backtest period. Since RAD is an adjusted return, its interpretation is similar to any return (a simple percentage).
在所选的回测期间, 最大回撤( ,在我们的案例中以月末计算)是 VAA 股票线的表现。由于 RAD 是调整后的收益,其解释类似于任何收益(一个简单的百分比)。
This RAD measure is based on the observations that a max drawdown of often leads to the liquidation of a hedge fund. In this case our RAD , independent of R. We also recognize in RAD the term which is the necessary increase in price to recover to the previous top portfolio capital level after a drawdown of . When , this price increase equals , so RAD= , reflecting the difficulty of getting back to the previous top portfolio capital level after a severe drawdown.
该 RAD 指标基于以下观察:最大回撤 通常会导致对冲基金的清算。在这种情况下,我们的 RAD 与 R 无关。我们还在 RAD 中认识到术语 ,这是在经历 的回撤后,恢复到之前最高投资组合资本水平所需的价格上涨。当 时,这一价格上涨等于 ,因此 RAD= ,反映了在经历严重回撤后,恢复到之前最高投资组合资本水平的困难。
Why do we opt for this RAD measure instead of the usual ones like the Sharpe and MAR ratio? The Sharpe ratio is defined as the return (often in excess over a target return like the risk-free rate) divided by the (annual) volatility V of the returns. The MAR ratio (similar to the Calmar ratio) is
我们为什么选择这个 RAD 指标而不是通常的夏普比率和 MAR 比率呢?夏普比率定义为收益 (通常是超过目标收益,如无风险利率的部分)除以收益的(年)波动率 V。MAR 比率(类似于卡尔玛比率)是
simply return divided by max drawdown (with ). Both measures assume that you can apply leverage to arrive at higher and combinations with the same Sharpe and MAR ratio. But, as we know from leveraged ETFs, this only holds for constant growth (and lending rate equal to the riskfree rate), while in practice your Sharpe ratio will be much less after leverage. And not all investors have access to cheap leverage at the risk-free rate. So, when optimizing Sharpe or MAR ratios you might be stuck at relative low returns R with low risk, especially when we use a low or zero target (or risk-free) return for Sharpe.
简单地将 除以最大回撤 (与 一起)。这两种指标假设您可以使用杠杆来获得更高的 组合,同时保持相同的夏普比率和 MAR 比率。但正如我们从杠杆 ETF 中了解到的,这仅适用于恒定增长(且借贷利率等于无风险利率),而在实际操作中,您的夏普比率在使用杠杆后会大大降低。而且并非所有投资者都能以无风险利率获得廉价杠杆。因此,在优化夏普比率或 MAR 比率时,您可能会被困在相对低的回报 R 和低风险的情况下,特别是当我们使用低或零目标(或无风险)回报来计算夏普比率时。
This can be demonstrated eg. in Table 14, where EWC (the equal weight of our cash universe) has MAR= 0.66 with while VAA has MAR with . Also, assuming a zero-target return, EWC's Sharpe=2.9/2.1=1.4 is better than VAA's Sharpe=7.4/10.7=0.7. Still, VAA trumps EWC on RAD ( vs. for EWC) which seems appropriate for most non-defensive investors. This is also the reason why we take the (often higher than the risk-free rate) EWC return as target return in our Sharpe formula in all tables below, since a higher target return in Sharpe will give higher (more offensive) optimal returns.
这可以在表 14 中证明,其中 EWC(我们现金宇宙的等权重)具有 MAR=0.66 和 ,而 VAA 的 MAR 为 。此外,假设目标回报为零,EWC 的夏普比率=2.9/2.1=1.4 优于 VAA 的夏普比率=7.4/10.7=0.7。不过,VAA 在 RAD 上胜过 EWC( 对 EWC 的 ),这对于大多数非防御性投资者来说似乎是合适的。这也是我们在下面所有表格的夏普公式中将(通常高于无风险利率的)EWC 回报作为目标回报的原因,因为在夏普中更高的目标回报将带来更高(更激进)的最佳回报。
And while volatility V is statistically a much nicer risk measure than max drawdown D , because most stable stochastic processes have stable V but increasing D over time (see eg. Goldberg, 2016, for a review of the literature), most retail investors commonly identify true risk with D over V. For example, during the 2008 subprime crisis the SP500 (TR index) crashed over 50% in approximately 1.5 years from its late 2007 peak with 3 years to recover to break even, leaving B&H investors without any positive returns over nearly five years. The biggest drawback of the measure is that it is a "tail risk" which can only be meaningfully assessed in large backtests (preferably covering multiple decennia) with many drawdowns, since any backtest limited to only the last decennium (Dec 2006 - Dec 2016) will be based on a single event (ie. the 2007-2009 crash) and therefore is not very meaningful as some kind of average (as and are). Notice, that in this paper is measured monthly (EOM prices) so a flash crash might get unobserved.
尽管波动率 V 在统计上是比最大回撤 D 更好的风险度量,因为大多数稳定的随机过程具有稳定的 V 但随着时间的推移 D 却在增加(参见例如 Goldberg, 2016 的文献综述),大多数散户投资者通常将真正的风险与 D 而非 V 进行识别。例如,在 2008 年次贷危机期间,标准普尔 500 指数(总回报指数)从 2007 年底的高点下跌超过 50%,大约用了 1.5 年的时间,随后又花了 3 年才恢复到盈亏平衡,使得买入并持有的投资者在近五年内没有任何正回报。该度量的最大缺点是它是一个“尾部风险”,只能在大型回测中(最好覆盖多个十年)进行有意义的评估,并且需要有许多回撤,因为任何仅限于过去十年(2006 年 12 月 - 2016 年 12 月)的回测都将基于单一事件(即 2007-2009 年的崩盘),因此作为某种平均值(如 )并不太有意义。请注意,在本文中 是按月测量的(月末价格),因此闪电崩盘可能会被忽视。
To conclude, we will optimize RAD in-sample over many decades, but also give Sharpe and MAR ratios (besides and ) for both the in- and out-of-sample backtests. Since we use monthly data, and are also only measured using End-Of-Month (EOM) prices, so eg. the daily max drawdown might be greater.
总之,我们将在多个十年内优化样本内的 RAD,同时还会提供夏普比率和 MAR 比率(除了 )用于样本内和样本外的回测。由于我们使用的是月度数据, 也仅使用月末(EOM)价格进行测量,因此例如,日最大回撤可能会更大。

7. The Data 7. 数据

To test VAA, we will use two monthly datasets, one short global from 1969 and one long for the US from 1925
为了测试 VAA,我们将使用两个每月数据集,一个是 1969 年的短期全球数据,另一个是 1925 年的长期美国数据
The first (short, global) dataset is from ourselves (denoted KK, see also PAA, 2016, and TrendXplorer, 2017) and runs from (ult.) Dec 1969 to Dec 2016 (47 years). The second (long, US-only) dataset is combined from Ibbotson/Morningstar (denoted Ibb) and Fama-French (FF, see French, 2017) from (ult.) Dec 1925 until Dec 2016 ( 91 years). Since we need the first 12 months for our momentum filter
第一个(短期,全球)数据集来自我们自己(标记为 KK,另见 PAA,2016 年和 TrendXplorer,2017 年),时间范围从(最终)1969 年 12 月到 2016 年 12 月(47 年)。第二个(长期,仅限美国)数据集是由 Ibbotson/Morningstar (标记为 Ibb)和 Fama-French(FF,见 French,2017 年)结合而成,时间范围从(最终)1925 年 12 月到 2016 年 12 月(91 年)。由于我们需要前 12 个月的数据用于我们的动量过滤器。
\footnotetext{
The data has been generously provided to us by Morningstar.
这些数据是由晨星慷慨提供给我们的。

(see above) the actual data lengths used for backtesting are 90 years for Ibb/FF (from Dec 1926) and 46 years for KK (from Dec 1970). All sets contain monthly total return (TR) prices, so including dividends, etc.
用于回测的实际数据长度为 Ibb/FF 的 90 年(从 1926 年 12 月开始)和 KK 的 46 年(从 1970 年 12 月开始)。所有数据集均包含每月的总回报(TR)价格,因此包括股息等。
The shorter KK dataset contains (among others) the following global asset-classes (ETF proxies, see also Keller, 2016), for a total of 17 global assets from Dec 1969- Dec 2016: SPY, IWM, QQQ, VGK, EWJ, EEM, EFA, ACWX, IYR, GSG, GLD, SHY, IEF, TLT, LQD, HYG, AGG.
较短的 KK 数据集包含(其中包括)以下全球资产类别(ETF 代理,另见 Keller,2016),总共有 17 种全球资产,时间范围为 1969 年 12 月至 2016 年 12 月:SPY,IWM,QQQ,VGK,EWJ,EEM,EFA,ACWX,IYR,GSG,GLD,SHY,IEF,TLT,LQD,HYG,AGG。
The larger Ibb/FF dataset contains the following 21 US-only asset-classes (index series) from Dec 1925 - Dec 2016: SP500, Small Caps, 30d T-Bill, IT Gov, LT Gov, LT Corp, and High Yield from Ibbotson/Morningstar (lbb for short) and 10 US sectors and 4 US factors (Large/Small Cap x Growth/Value), from Fama French (FF). Here IT/LT stands for Intermediate/Long Term maturities, and Gov/Corp bonds for Government (Treasuries) and investment grade Corporate bonds.
更大的 Ibb/FF 数据集包含以下 21 个仅限美国的资产类别(指数系列),时间范围为 1925 年 12 月到 2016 年 12 月:SP500、小型股、30 天国库券、IT 政府债、LT 政府债、LT 公司债和高收益债券,来自 Ibbotson/Morningstar(简称 lbb),以及 10 个美国行业和 4 个美国因子(大/小盘 x 成长/价值),来自 Fama French(FF)。这里的 IT/LT 代表中期/长期到期,Gov/Corp 债券代表政府(国债)和投资级公司债券。
All early ETF data from 1969 are based on ETF proxies constructed by us (see Keller, 2016, and TrendXplorer, 2017 for details). By calibration, all our ETF proxies include ETF fees, etc. Notice that all Ibb and FF data represent TR index prices, so no corrections for ETF fees etc. were made in this case. Only the recent years of the KK database has observed and tradable ETF prices, all the other historical prices are non-tradable.
所有 1969 年的早期 ETF 数据均基于我们构建的 ETF 代理(详见 Keller,2016 年和 TrendXplorer,2017 年)。通过校准,我们所有的 ETF 代理都包括 ETF 费用等。请注意,所有 Ibb 和 FF 数据代表 TR 指数价格,因此在这种情况下没有对 ETF 费用等进行修正。KK 数据库的最近几年观察到了可交易的 ETF 价格,所有其他历史价格均为不可交易。
The in-sample out-of-sample (IS/OS) split will be half ways (see Section 5). Notice that in both IS periods (in particular in the 1971-1993 period) there are rising (and decreasing) bond yields, so we optimize ("train") our VAA strategy in a similar rate-environment as we might experience in the (near) future.
样本内样本外(IS/OS)划分将是中间的(见第 5 节)。请注意,在两个 IS 期间(特别是在 1971-1993 年期间),债券收益率有上升(和下降)的趋势,因此我们在与未来(近期)可能经历的类似利率环境中优化(“训练”)我们的 VAA 策略。

8. Four VAA universes 八个 VAA 宇宙

Using this data, we will consider four "market" universes. For each dataset we will choose a large (with size or assets) and a small (with size or ) universe to test VAA. In all our shorter backtests (from Dec 1970) we will us SHY, IEF and LQD as "cash universe" for the "out-ofthe-market" allocation. Compared to our choice of cash in our PAA paper we added LQD to get slightly more "pizazz" when out of the market. We will use a similar combination for the longer backtests from Dec 1926 with SHY replaced by 30day T-Bill, IEF by IT Gov, and LQD by LT Corp (all three from Ibb). Notice that the VAA model (like the PAA model) always chooses the single best cash asset in terms of our (13612W) momentum filter, irrespective of their sign (so no absolute momentum filter for the cash universe).
使用这些数据,我们将考虑四个“市场”宇宙。对于每个数据集,我们将选择一个大(规模为 的资产)和一个小(规模为 )的宇宙来测试 VAA。在我们所有较短的回测中(从 1970 年 12 月开始),我们将使用 SHY、IEF 和 LQD 作为“现金宇宙”进行“市场外”配置。与我们在 PAA 论文中选择的现金相比,我们添加了 LQD,以在市场外时获得稍微更多的“活力”。对于从 1926 年 12 月开始的较长回测,我们将使用类似的组合,SHY 替换为 30 天国库券,IEF 替换为 IT 国债,LQD 替换为长期公司债(这三者均来自 Ibb)。请注意,VAA 模型(如 PAA 模型)始终根据我们的(13612W)动量过滤器选择单一最佳现金资产,无论其符号如何(因此现金宇宙没有绝对动量过滤器)。
The first two universes are based on the global KK dataset from 1969 with backtests from Dec 1970. The large universe (N12) is the same as we use in our PAA paper:
前两个宇宙基于 1969 年的全球 KK 数据集,回测从 1970 年 12 月开始。大宇宙(N12)与我们在 PAA 论文中使用的相同:
  • VAA Global 12 (VAA-G12): SPY, IWM, QQQ, VGK, EWJ, EEM, IYR, GSG, GLD, TLT, LQD, HYG
The small universe (N4) is inspired by Antonnacci 's GEM (see Antonacci, 2014), where we replaced his ACWX (World ex US) by EFA (developed markets) and EEM (emerging markets) and added AGG to arrive at a N4+3 instead of his N2+1 universe:
小宇宙(N4)受到 Antonnacci 的 GEM(见 Antonacci,2014)的启发,我们将他的 ACWX(全球除美国)替换为 EFA(发达市场)和 EEM(新兴市场),并添加 AGG,从而形成 N4+3,而不是他的 N2+1 宇宙:
  • VAA Global 4 (VAA-G4): SPY, EFA, EEM, AGG
For cash we again use SHY, IEF and LQD (see above).
对于现金,我们再次使用 SHY、IEF 和 LQD(见上文)。
The last two universes are based on the Ibb/FF dataset for the US from Dec 1925 to Dec 2016 with backtests starting Dec 1926, where we also have chosen a large and a small universe. The large universe (N15) is based on FF's 10 US-sectors assets augmented with all the available five Ibb bonds (30d T-Bill, IT Gov, LT Gov, LT Corp, and High Yield):
最后两个宇宙基于 Ibb/FF 数据集,涵盖美国从 1925 年 12 月到 2016 年 12 月的数据,回测从 1926 年 12 月开始,我们还选择了一个大宇宙和一个小宇宙。大宇宙(N15)基于 FF 的 10 个美国行业资产,并增加了所有可用的五个 Ibb 债券(30 天国库券、短期政府债、长期政府债、长期公司债和高收益债)。
  • VAA US 15 (VAA-U15): FF 10 sectors, 30d T-Bill, IT Gov, LT Gov, LT Corp, High Yield
    VAA US 15 (VAA-U15): FF 10 个行业,30 天国库券,IT 政府,长期政府,长期公司,高收益
The small universe (N6) consists of the four US FF factors (Large/Small Cap Growth/Value) with two lbb. bonds added, so:
小宇宙(N6)由四个美国 FF 因素(大盘/小盘 成长/价值)和两个 lbb 债券组成,因此:
  • VAA US 6 (VAA-U6): FF 4 factors, T-Bill, LT Corp
    VAA 美国 6(VAA-U6):FF 4 因素,国库券,长期公司债
For cash we used 30-day T-Bills, IT Gov and LT Corp (all from Ibb, see above), equivalent to SHY, IEF, LQD in the shorter backtests.
对于现金,我们使用了 30 天的国库券、IT 政府债券和长期公司债券(均来自 Ibb,见上文),相当于短期回测中的 SHY、IEF 和 LQD。
The choice of Ibb's T-Bill (30d) and LT Corp for the smaller (risky) VAA-U6 universe was made by manual optimization on the in-sample period only. We opted for not too small (N4 global and N6 US) universes to have enough "breadth" for our crash-protection mechanism. Therefore, we preferred the 4-factors over Ibb's SP500/Small Cap combi for the small US universe.
选择 Ibb 的 T-Bill(30 天)和 LT Corp 用于较小(风险较高)的 VAA-U6 宇宙是通过仅在样本内期间进行手动优化得出的。我们选择了不太小的(N4 全球和 N6 美国)宇宙,以便为我们的崩溃保护机制提供足够的“广度”。因此,我们更倾向于使用 4 因子而不是 Ibb 的 SP500/小型股组合来应对小型美国宇宙。
For all universes, we used a (one way) transaction fee to account for brokers commission, slippage, etc. This is certainly too low for 1926 and probably too high for 2016, but it seems reasonable for ETFs over the last decade with EOM/EOD trading. We also show the yearly Total Transaction Costs (TTC) as turnover indicator, which equals per year ( ) when we have full monthly turnover.
对于所有宇宙,我们使用了一个 (单向)交易费用来计算经纪人佣金、滑点等。这对于 1926 年来说肯定太低,而对于 2016 年可能太高,但对于过去十年中进行 EOM/EOD 交易的 ETF 来说似乎是合理的。我们还显示了年度总交易成本(TTC)作为周转指标,当我们有完整的月度周转时,每年等于 )。
We also test our breadth momentum strategy for our each of the four VAA universes against the traditional Dual (or GTAA, see Faber 2007, 2013) momentum strategy. To make a fair comparison with VAA we will use the same data (incl. ETF fees) and cash universe, the same costs (TC=0.1%) and the same 13612 W momentum filter for Dual, while optimizing in-sample the top parameter T. We also tested the (somewhat special) GEM strategy from Antonacci, 2014 using his N2+1 universe and (excess RET12m) momentum filter against our VAA-G4.
我们还测试了我们的广度动量策略在四个 VAA 宇宙中的表现,与传统的双重(或 GTAA,见 Faber 2007, 2013)动量策略进行比较。为了与 VAA 进行公平比较,我们将使用相同的数据(包括 ETF 费用)和现金宇宙,相同的成本(TC=0.1%)以及相同的 13612 W 动量过滤器用于双重,同时在样本内优化顶层参数 T。我们还测试了 Antonacci 2014 年提出的(有些特殊的)GEM 策略,使用他的 N2+1 宇宙和(超额 RET12m)动量过滤器与我们的 VAA-G4 进行比较。

9. The VAA-G12 universe 9. VAA-G12 宇宙

The first universe is the same global universe as we used in our PAA paper, with 12 risky assets but now with three cash assets and some Vanguard data (SPY, IWM, QQQ, VGK, EWJ, VWO, VNQ, GSG, GLD, TLT, LQD, HYG with SHY, IEF, and LQD as cash). Our in-sample (IS) period is from (ult.) Dec 1970 - Dec 1993 (23 years), while our out-of-sample (OS) period is Dec 1993 - Dec 2016 (also 23 years). Data is from our own KK dataset, corrected for ETF fees etc. (see Keller 2016, and TrendXplorer 2017 for details).
第一个宇宙与我们在 PAA 论文中使用的全球宇宙相同,包含 12 种风险资产,但现在有三种现金资产和一些先锋数据(SPY、IWM、QQQ、VGK、EWJ、VWO、VNQ、GSG、GLD、TLT、LQD、HYG,现金为 SHY、IEF 和 LQD)。我们的样本内(IS)期间为(最终)1970 年 12 月 - 1993 年 12 月(23 年),而样本外(OS)期间为 1993 年 12 月 - 2016 年 12 月(同样为 23 年)。数据来自我们自己的 KK 数据集,已修正 ETF 费用等(详见 Keller 2016 和 TrendXplorer 2017)。

In-sample (IS) 样本内 (IS)

As said, we compute RAD over IS on top T=1.. 6 (selected number of Top assets) and breadth B=1..6 (protection threshold) for a total of values, using our 13612W momentum filter. Dual is the dual momentum strategy (called GTAA by Meb Faber) computed for various T=1..6 using our own momentum filter for both absolute and relative momentum.
如前所述,我们在 T=1..6(选定的顶级资产数量)和 B=1..6(保护阈值)上计算 RAD 相对于 IS,总共 个值,使用我们的 13612W 动量过滤器。Dual 是双动量策略(由 Meb Faber 称为 GTAA),使用我们自己的动量过滤器计算绝对和相对动量,适用于不同的 T=1..6。
From Table 1, the best VAA score at IS is RAD for and . For Dual optimal it is at . The best RAD values are clearly concentrated around this optimum and part of a "ridge" of high RAD values at We will use the optimal for VAA-G12 and T=3 for Dual in our backtests.
从表 1 可以看出,IS 的最佳 VAA 得分是 RAD ,适用于 。对于双重最优,它在 时为 。最佳 RAD 值明显集中在这个最优点周围,并且是 处高 RAD 值的“脊”的一部分。我们将在回测中使用 VAA-G12 的最优 和双重的 T=3。
RAD
1 2 3 4 5 6
0
1
2
3
4
5
6
Dual 双重
Table 1. RAD for VAA-G12 on IS (dec1970-dec1993) for T/B=1..6
表 1. VAA-G12 在 IS 上的 RAD(1970 年 12 月-1993 年 12 月),T/B=1..6
We can see more detail behind these T/B values when we draw the R/D points (like normally done for ) as function of and , as in Fig. 4. As with the frontier curve, points in the upper/left corner (with high return and low drawdown ) are preferable over points in the right/lower corner. Points in the left/upper corner will be called "efficient" when there are no other points with better R/D combinations (ie. no points in the upper/left quadrant), and "optimal" when they have the best and best values of all points.
我们可以在绘制 R/D 点(如通常对 所做的那样)作为 的函数时,更详细地看到这些 T/B 值背后的信息,如图 4 所示。与 边界曲线一样,左上角的点(具有高回报 和低回撤 )比右下角的点更可取。当没有其他点具有更好的 R/D 组合(即左上象限没有点)时,左上角的点将被称为“有效”,当它们在所有点中具有最佳 和最佳 值时,将被称为“最优”。
All VAA-T curves in Fig. 4 start with points with low (high protection) at the left/under (so low and ) and go to the right/upper (so high and ) for higher values (low protection). The VAA-B (and both Dual) curves in Fig. 5 (and Fig. 4, resp.) go from high T left/under (high diversification) to low T (less diversification) at the right/upper part. Notice that points left/under corresponds to more protection (lower B) and/or higher diversification (higher T ), and vice versa for points right/upper.
图 4 中的所有 VAA-T 曲线从左下角的低 (高保护)点开始(因此低 ),向右上方延伸(因此高 ),以获得更高的 值(低保护)。图 5(以及图 4)中的 VAA-B(和两个 Dual)曲线从左下方的高 T(高多样性)到右上方的低 T(低多样性)。请注意,左下方的点对应于更多的保护(较低的 B)和/或更高的多样性(较高的 T),右上方的点则相反。
From Fig. 4, it is clear that the point on the red VAA-T2 line (for ) is not only RAD optimal (in terms of return/risk) on IS (see Table 1) but also efficient. The point R/D=23/44% on the VAA-T1 (yellow line) wins in terms of only R but with a much larger .
从图 4 可以看出,红色 VAA-T2 线上的点 (对于 )不仅在 IS 上是 RAD 最优的(见表 1),而且是有效的。VAA-T1(黄色线)上的点 R/D=23/44%在仅 R 方面获胜,但 要大得多。
Notice that from Table 1 we see that is also high on RAD for other values of T. This is even more clear when we draw in Fig. 5 the R/D frontier for various B, which shows that the red VAA-B4 line (for various ) is efficient to all other values of . This also indicates some robustness (and possibly less datasnooping) for the choice. The first left/under points of all VAA-T lines in Fig. 4 corresponds to , the next to , etc. Similary, the right/upper points of all VAA-B in Fig. 5 (and Dual) lines corresponds to , the next to , etc. As we see from Fig. 5 (and Table 1), is RAD optimal for Dual (with R/D=22/21%), see the black line in Fig. 5. From Fig. 4 and 5 it is also clear how worse Dual (the black line) is wrt. VAA-T2 and VAA-B4, especially in terms of max drawdown D. In particular in Fig. 5, the VAA-B4 curve (for various T) dominates the Dual line (for various T) which resembles the VAA-B4 line shifted to the right side. Dual achieves nearly the same R but only at the cost of a higher D.
注意,从表 1 中我们可以看到, 在其他 T 值的 RAD 上也很高。当我们在图 5 中绘制不同 B 值的 R/D 边界时,这一点更加明显,红色的 VAA-B4 线(对于不同的 )对所有其他 值都是有效的。这也表明对于 选择的一些稳健性(可能还有较少的数据窥探)。图 4 中所有 VAA-T 线的第一个左下点对应于 ,下一个对应于 ,依此类推。同样,图 5(和 Dual)中所有 VAA-B 的右上点对应于 ,下一个对应于 ,依此类推。从图 5(和表 1)中我们可以看到, 对于 Dual 是 RAD 最优的(R/D=22/21%),见图 5 中的黑线。从图 4 和图 5 中也可以清楚地看出,Dual(黑线)相对于 VAA-T2 和 VAA-B4 是多么糟糕,特别是在最大回撤 D 方面。特别是在图 5 中,VAA-B4 曲线(对于不同的 T)主导了 Dual 线(对于不同的 T),这类似于向右移动的 VAA-B4 线。Dual 几乎达到了相同的 R,但代价是更高的 D。
Fig. 4. VAA-G12: R/D frontier on IS (dec1970-dec1993) for VAA-T and Dual
图 4. VAA-G12:VAA-T 和双重的 IS 上的 R/D 前沿(1970 年 12 月-1993 年 12 月)
Fig. 5. VAA-G12: R/D frontier on IS (dec1970-dec1993) for various VAA-B and Dual.
图 5. VAA-G12:不同 VAA-B 和双重的 IS 上的 R/D 前沿(1970 年 12 月-1993 年 12 月)。
Now we focus on the VAA-T2 curve, see Fig. 6 (red curve). This VAA-T2 curve manifests its typical "bent" appearance: starting at B=1 left under (with R/D=13/7%), we go upwards (higher R) through , to the optimal point or the "VAA knee" at (with ), all with nearly the same but with increasing for . The upper leg of the curve after the "knee" stretches nearly horizontally to the right (higher for nearly the same R) through , and 6 ( ). We see the same characteristic bent curve (but shifted to the right) for VAA-T1 (yellow curve). Surprisingly, when we do the same analysis for OS, we found a similar curve with the same "knee" at B=4 for VAA-G12 (but at T=1 instead of T=2). As we will see with other universes later, this "bent" R/D line for increasing B is typical for VAA. This could indicate that there might be some interesting timing mechanism at work here.
现在我们关注 VAA-T2 曲线,见图 6(红色曲线)。这条 VAA-T2 曲线表现出典型的“弯曲”外观:从 B=1 的左下方(R/D=13/7%)开始,我们向上(更高的 R)经过 ,到达最佳点或“VAA 膝部”在 (与 ),所有这些几乎具有相同的 ,但随着 的增加, 也在增加。曲线在“膝部”之后的上半部分几乎水平向右延伸(更高的 几乎对应相同的 R),经过 和 6( )。我们看到 VAA-T1(黄色曲线)也具有相同特征的弯曲曲线(但向右移动)。令人惊讶的是,当我们对 OS 进行相同分析时,我们发现 VAA-G12 在 B=4 处有类似的曲线,具有相同的“膝部”(但在 T=1 而不是 T=2)。正如我们稍后在其他宇宙中看到的,这条随着 B 增加而“弯曲”的 R/D 线对于 VAA 来说是典型的。这可能表明这里可能存在一些有趣的时序机制。
Fig. 6. VAA-G12: R/D frontier on IS (dec1970-dec1993) for VAA-T1, T2 and Dual.
图 6. VAA-G12:VAA-T1、T2 和双重的 IS 上的 R/D 前沿(1970 年 12 月-1993 年 12 月)。
From now one, we will refer to VAA-G12 as the optimal strategy (in terms of RAD) with T/B=2/4 for the N12+3 universe. For this optimal VAA-G12 strategy, we arrive at the following performance indicators on IS for VAA, EW (equal weight of the global risky universe, our benchmark), EWC (equal
从现在起,我们将把 VAA-G12 称为 N12+3 宇宙中 T/B=2/4 的最佳策略(就 RAD 而言)。对于这个最佳的 VAA-G12 策略,我们得出了以下关于 VAA、EW(全球风险宇宙的等权重,我们的基准)、EWC(等权重)的绩效指标。

weight of the Cash universe of assets SHY, IEF and LQD), the well-known 60/40 (SPY/IEF) portfolio, SPY, and Dual. See Table 2, where R, V, D equals return (CAGR), (annual) Volatility and the max Drawdown (EOM), and where the Sharpe ratio is computed with cash rate equal to the return of EWC, the MAD ratio equals R/D and RAD is the Return Adjusted for Drawdown.
现金资产组合的权重 SHY、IEF 和 LQD,以及众所周知的 60/40(SPY/IEF)投资组合、SPY 和 Dual。见表 2,其中 R、V、D 分别代表收益(CAGR)、(年)波动率和最大回撤(EOM),夏普比率的计算以现金利率等于 EWC 的收益为基础,MAD 比率等于 R/D,RAD 是调整回撤后的收益。
As can be seen from Table 2, the return of VAA on IS is a staggering R=21% with an unbelievable low drawdown (over 1970-1993) and a volatility , resulting in very high Sharpe=1.24, MAR=3.62 and RAD=19.7%. VAA's R is nearly twice that of EW, 60/40 and SPY while VAA's D is 5-7 times smaller. By definition, Sharpe EWC=0, while Sharpe VAA is more than 2, 5 and 6 times larger than EW, 60/40 (SPY/IEF) and SPY, while MAR is 7, 9, and 13 times larger, respectively. Dual (with optimal , see Table 1 ) is slightly better for vs. for VAA), but worse for vs. for VAA) , Sharpe, MAR, and RAD ( vs. for VAA). But remember, these results are the outcome of in-sample (IS) optimization, so possibly the result of datasnooping for parameters and .
从表 2 可以看出,VAA 在样本内的回报率为惊人的 R=21%,且回撤极低 (1970-1993 年期间),波动率 ,导致非常高的夏普比率=1.24,MAR=3.62 和 RAD=19.7%。VAA 的 R 几乎是 EW、60/40 和 SPY 的两倍,而 VAA 的 D 则小 5-7 倍。根据定义,夏普比率 EWC=0,而 VAA 的夏普比率则比 EW、60/40(SPY/IEF)和 SPY 分别大 2、5 和 6 倍,而 MAR 则分别大 7、9 和 13 倍。双重(在最佳 下,见表 1)在 VAA 的 方面稍好,但在 VAA 的 方面则较差,夏普比率、MAR 和 RAD( 在 VAA 中)。但请记住,这些结果是样本内(IS)优化的结果,因此可能是对参数 的数据挖掘的结果。
IS R V D Sharpe 夏普 MAR RAD
VAA-G12 21.0% 10.0% 5.8% 1.24 3.62 19.7%
EW 13.2% 10.4% 24.6% 0.45 0.54 8.9%
EWC 8.6% 5.6% 9.1% 0.00 0.95 7.7%
60/40 11.1% 10.6% 27.4% 0.24 0.41 6.9%
SPY 11.6% 15.4% 42.5% 0.20 0.27 3.0%
Dual3 双 3 21.5% 15.4% 21.3% 0.84 1.01 15.7%
Table 2. Performance indicators for VAA-G12 at IS (dec1970-dec1993)
表 2. VAA-G12 在 IS 的性能指标(1970 年 12 月-1993 年 12 月)

Out-of-sample 样本外

The out-of-sample (OS) results are displayed in Table 3, where we show all performance indicators for OS (Dec 1993 - Dec 2016) as well as some for the last 10 years, denoted Recent-Sample or RS (Dec 2006 - Dec 2016). As we see in Table 3, return R and max drawdown D over OS are "back to normal" (compared to IS), but are still within our preferred target range: and . This also holds for the last decade or RS (including the Global Financial Crisis), with an appealing , compared to (EW), (60/40), and (SPY), while return R (11%) and RAD (10%) of VAAG12 are also much better than the R and RAD readings for EW, 60/40, SPY, and Dual. This is all outof-sample (OS), so without datasnooping for parameters T and B.
表 3 显示了样本外(OS)结果,我们展示了 OS(1993 年 12 月 - 2016 年 12 月)的所有性能指标,以及最近 10 年的一些指标,称为最近样本(RS)(2006 年 12 月 - 2016 年 12 月)。如表 3 所示,OS 的收益 R 和最大回撤 D“回归正常”(与 IS 相比),但仍在我们首选的目标范围内: 。这对于最近十年或 RS(包括全球金融危机)也是如此,具有吸引人的 ,与 (EW)、 (60/40)和 (SPY)相比,VAAG12 的收益 R(11%)和 RAD(10%)也远好于 EW、60/40、SPY 和 Dual 的 R 和 RAD 读数。这一切都是样本外(OS),因此没有对参数 T 和 B 进行数据窥探。
The resulting Full-Sample (FS: Dec 1970 - Dec 2016, so 46 years) outcomes are displayed in Table 4. Return and max drawdown are well within our preferred VAA target range ( , ) resulting in , with corresponding high Sharpe and MAR ratios. Dual is slightly better at R ( vs. for VAA) but worse at V, Sharpe, MAR and especially D ( vs. for VAA) and RAD ( vs. for VAR).
结果的全样本(FS:1970 年 12 月 - 2016 年 12 月,共 46 年)结果显示在表 4 中。回报 和最大回撤 均在我们首选的 VAA 目标范围内( ),导致 ,并具有相应的高夏普比率和 MAR 比率。Dual 在 R 方面稍好( 对比 VAA 的 ),但在 V、夏普、MAR,尤其是在 D( 对比 VAA 的 )和 RAD( 对比 VAR 的 )方面表现较差。
OS R V D Sharpe 夏普 MAR RAD RS R V D RAD
VAA-G12 0.51 0.81
EW 0.19 0.19
EWC 0.00 1.11
0.34 0.28
SPY 0.26 0.18
Dual3 双 3 0.48 0.46
Table 3. Performance indicators for VAA-G12 at OS (dec1993-dec2016) and RS (dec2006-dec2016)
表 3. VAA-G12 在 OS(1993 年 12 月-2016 年 12 月)和 RS(2006 年 12 月-2016 年 12 月)的性能指标
Over FS the Total Transaction Costs per year are high (TTC=1.4%/y) indicating a monthly turnover of nearly (1.4/2.4=58%). The average Cash Fraction over FS equals , reflecting the vigorousness of the Crash Protection. Together with the resulting R and we think this underlines our VAA motto "Winning more by Losing Less". Notice that Dual has a much lower CF ( vs. for VAA), but a similar TTC as VAA ( vs. for VAA).
在 FS 上,每年的总交易成本很高(TTC=1.4%/年),这表明每月的周转率接近 (1.4/2.4=58%)。FS 上的平均现金比例为 ,反映了崩盘保护的强劲性。结合由此产生的 R ,我们认为这强调了我们的 VAA 座右铭“通过减少损失来赢得更多”。请注意,Dual 的现金比例要低得多( 对比 VAA 的 ),但与 VAA 的总交易成本相似( 对比 VAA 的 )。
FS R V D Sharpe 夏普 MAR RAD TTC CF
VAA-G12 0.85 1.20
EW 0.31 0.26
EWC 0.00 0.76
60/40 0.28 0.33
SPY 0.23 0.20
Dual3 双 3 0.66 0.65
Table 4. Performance indicators for VAA-G12 at FS (dec1970-dec2016)
表 4. VAA-G12 在 FS 的性能指标(1970 年 12 月-2016 年 12 月)
We also show in Fig. 7 the (log) equity line on FS, as well as the rolling 3 year annual returns (Fig. 8) for both VAA-G12 and its benchmark (EW). We also provide the relative price (green line) of VAA vs. EW in the equity line curve, which indicates for how far VAA is behind (sloping downwards) or ahead (upwards) of its benchmark. As Fig. 7 shows, from 2009-2016 the relative price is near horizontal, indicating similar price performance for the strategy as compared to the benchmark. It is also clear from Fig. 7 that recent returns of VAA above EW is made in the crash years like 2000/2002, 2008/9 (winning more by losing less) while in the years before 2000 VAA harvested risk premia during uptrending markets too.
我们在图 7 中还展示了 FS 上的(对数)股权线,以及 VAA-G12 及其基准(EW)的滚动 3 年年回报(图 8)。我们还提供了 VAA 与 EW 在股权线曲线中的相对价格(绿色线),这表明 VAA 相对于其基准的落后(向下倾斜)或领先(向上倾斜)程度。如图 7 所示,从 2009 年到 2016 年,相对价格接近水平,表明该策略与基准的价格表现相似。从图 7 中也可以清楚地看出,VAA 在 2000/2002 年、2008/9 年等崩盘年份的回报高于 EW(通过减少损失而赢得更多),而在 2000 年前的年份,VAA 在上涨市场中也获得了风险溢价。
The chart in Fig. 8 compares the rolling 3 year annual returns for VAA-G12 and its EW benchmark. The green curve depicts periods with relative outperformance of VAA over EW (curve above zero) and underperformance (curve below zero). Fig. 9 shows the drawdown profile of VAA against that of its EW benchmark. With only three outliers above the D=10% mark, VAA-G12 depicts a contained drawdown profile.
图 8 中的图表比较了 VAA-G12 及其等权基准的滚动三年年回报。绿色曲线描绘了 VAA 相对于等权基准的相对超额表现(曲线在零以上)和相对低于表现(曲线在零以下)的时期。图 9 显示了 VAA 与其等权基准的回撤情况。VAA-G12 仅有三个超出 D=10%标记的异常值,显示出其回撤情况相对可控。
Fig. 7. Equity line VAA-G12, EW and their relative price (log scale)
图 7. 股权线 VAA-G12、EW 及其相对价格(对数刻度)
Fig. 8. Rolling 3-year returns of VAA-G12, EW and their relative price
图 8. VAA-G12、EW 及其相对价格的滚动 3 年回报
Fig. 9 Drawdown of VAA-G12 and EW
图 9 VAA-G12 和 EW 的回撤

10. The VAA-G4 universe 10. VAA-G4 宇宙

The second universe is inspired by Antonacci's GEM (Antonnacci, 2014). He uses a risky universe of only two assets, SPY, and ACWX (Global Stocks ex US), and AGG (aggregate US bond) as cash. For VAA we will split ACWX into Vanguard's VEA (International Markets, including Europe, Japan and IM Pacific) and VWO (Emerging markets), and add one bond for some breadth to arrive at . Surprisingly, the best in-sample bond was BND (Vanguard's AGG), so we arrived at N=4 for our risky universe with SPY, VEA, VWO, and BND. Notice that adding a bond like BND to the risky universe has a somewhat similar effect as taking excess returns (as Antonacci did with his RET12m momentum) over that bond for the other (more risky) ETFs, since the best (Top1) asset should always be better than BND (or BND itself).
第二个宇宙受到安东纳奇的 GEM(Antonnacci, 2014)的启发。他使用了一个仅包含两个资产的高风险宇宙,SPY 和 ACWX(全球股票除美国外),以及 AGG(美国债券的综合)作为现金。对于 VAA,我们将 ACWX 拆分为先锋的 VEA(国际市场,包括欧洲、日本和 IM 太平洋)和 VWO(新兴市场),并添加一只债券以增加广度,得出 。令人惊讶的是,最佳的样本内债券是 BND(先锋的 AGG),因此我们得出了 N=4 的高风险宇宙,包括 SPY、VEA、VWO 和 BND。请注意,像 BND 这样的债券加入高风险宇宙的效果与对其他(更高风险)ETF 的超额收益(正如安东纳奇在他的 RET12m 动量中所做的)有些相似,因为最佳(Top1)资产应该始终优于 BND(或 BND 本身)。
As before, we will use SHY, IEF, and LQD as cash. We will denote our VAA strategy with this N=4 global universe as VAA-G4. Our in-sample (IS) period is (like VAA-G12) from (ult.) Dec 1970 - Dec 1993 (23 years), while our out-of-sample (OS) period is Dec 1993 - Dec 2016 (also 23 years). Data is from our own KK dataset, corrected for ETF fees etc.
如前所述,我们将使用 SHY、IEF 和 LQD 作为现金。我们将用这个 N=4 的全球范围表示我们的 VAA 策略,称为 VAA-G4。我们的样本内(IS)时期(与 VAA-G12 相同)为 1970 年 12 月到 1993 年 12 月(23 年),而我们的样本外(OS)时期为 1993 年 12 月到 2016 年 12 月(同样是 23 年)。数据来自我们自己的 KK 数据集,已修正 ETF 费用等。

In-sample (IS) 样本内 (IS)

Since N=4, we compute RAD over IS on top T=1..4 and breadth B=1..4 for a total of values (scenarios), using our 13612W momentum filter. See Table 5. Dual is the dual momentum strategy computed for various using our own momentum filter (13612W) for both absolute and relative momentum.
由于 N=4,我们在 T=1..4 和宽度 B=1..4 的基础上计算 RAD 相对于 IS,共有 个值(场景),使用我们的 13612W 动量过滤器。见表 5。Dual 是针对各种 计算的双动量策略,使用我们自己的动量过滤器(13612W)来处理绝对和相对动量。
RAD
T
1 2 3 4
0
1
2
3
4
Dual 双重
Table 5. RAD for VAA-G4 on IS (dec1970-dec1993) for T/B=1..4
表 5. VAA-G4 在 IS 上的 RAD(1970 年 12 月-1993 年 12 月),T/B=1..4
From Table 5, the best score for VAA is RAD=18.7% for and for Dual it is RAD=10.2% for . So we use T/B=1/1 and Dual4 in our backtests here. We can see more detail behind these T/B values when we draw the R/D frontier as function of and , as in Fig. 10 and 11. As with the R/V frontier curve, points in the upper/left corner (with high return and low drawdown ) are preferable over points in the right/lower corner. To show the "knee" in the VAA-T curves, we have added the B=0 ( cash) strategy with (independent of ), so now the range in eg. the VAA-T1 line in Fig. 10 is (5 points).
从表 5 可以看出,VAA 的最佳得分为 RAD=18.7%( ),而 Dual 的最佳得分为 RAD=10.2%( )。因此,我们在这里的回测中使用 T/B=1/1 和 Dual4。当我们绘制 R/D 边界作为 的函数时,可以看到这些 T/B 值背后的更多细节,如图 10 和 11 所示。与 R/V 边界曲线一样,左上角的点(高回报 和低回撤 )比右下角的点更可取。为了显示 VAA-T 曲线中的“膝部”,我们添加了 B=0( 现金)策略与 (独立于 ),因此现在例如图 10 中的 VAA-T1 线的范围是 (5 个点)。
From Fig. 10, it is clear that the point upper/left on the red VAA-T1 line (for ) is not only RAD optimal (in terms of return/risk, see Table 5) but is also optimal in terms of R. However, the point R/D=19/8% upper left on the yellow VAA-T2 line is only slightly worse on RAD (17.6 vs. 18.7%, see Table 5) but has a substantial smaller D ( instead of for T1).
从图 10 可以看出,红色 VAA-T1 线上的点 (对于 )不仅在回报/风险方面是 RAD 最优的(见表 5),而且在 R 方面也是最优的。然而,黄色 VAA-T2 线上的点 R/D=19/8%位于左上方,其 RAD 仅稍微逊色(17.6%对比 18.7%,见表 5),但 D 的值显著更小( 而不是 ,对于 T1)。
Fig. 10. VAA-G4: R/D frontier on IS (dec1970-dec1993) for VAA-T and Dual
图 10. VAA-G4:VAA-T 和双重的 IS 上的 R/D 前沿(1970 年 12 月-1993 年 12 月)
Fig. 11. VAA-G4: R/D frontier on IS (dec1970-dec1993) for VAA-B and Dual
图 11. VAA-G4:VAA-B 和双重的 IS 上的 R/D 边界(1970 年 12 月-1993 年 12 月)
Fig. 10 clearly shows the bend VAA curves as we had seen with the VAA-G12 strategy, starting at the point (giving a cash strategy with ) and bending at the "knee" at at and similar for all other VAA-T lines: they all have the knee at . From Fig. 11 (where we display the lines for various ) it is clear that the line VAA-B1 (for all ) is optimal both in terms of as in over nearly all other , including the Dual strategy, of which even the best solution at the left with is not near our red VAA-B1 solutions. This demonstrates (as with our VAA-G12 model) clearly the power (and robustness) of our new breadth parameter B for vigilant asset allocation. The same (and ) turns out also to be RAD optimal in our OS period (Dec 1993 - Dec 2016), showing remarkable stability, as with VAA-G12.
图 10 清楚地显示了 VAA 曲线的弯曲情况,正如我们在 VAA-G12 策略中看到的,从 点开始(给出 现金策略和 ),在 的“膝部”处弯曲,在 处,对于所有其他 VAA-T 线也是类似的:它们的膝部都在 。从图 11(我们展示了各种 的线)可以清楚地看出,VAA-B1 线(适用于所有 )在几乎所有其他 中都是最佳的,无论是 还是 ,包括双重策略,即使是左侧的最佳 解决方案在 处也远不及我们的红色 VAA-B1 解决方案。这清楚地展示了(与我们的 VAA-G12 模型一样)我们新的广度参数 B 在警惕资产配置中的力量(和稳健性)。相同的 (和 )在我们的 OS 期间(1993 年 12 月 - 2016 年 12 月)也被证明是 RAD 最优的,显示出显著的稳定性,正如 VAA-G12 一样。
From now on we will refer to VAA-G4 as the optimal strategy with T/B=1/1. In Table 6, we show the usual performance indicators for VAA-G4 (with optimal T/B=1/1) at IS (Dec 1970 - Dec 1993).
从现在起,我们将 VAA-G4 称为 T/B=1/1 的最佳策略。在表 6 中,我们展示了 VAA-G4(最佳 T/B=1/1)在 IS(1970 年 12 月 - 1993 年 12 月)期间的常用性能指标。
IS R V D Sharpe 夏普 MAR RAD
VAA-G4 22.0% 13.7% 13.0% 0.98 1.69 18.7%
EW 13.8% 12.9% 42.4% 0.41 0.33 3.6%
EWC 8.6% 5.6% 9.1% 0.00 0.95 7.7%
60/40 11.1% 10.6% 27.4% 0.24 0.41 6.9%
SPY 11.6% 15.4% 42.5% 0.20 0.27 3.0%
Dual4 双 4 14.2% 10.3% 21.9% 0.55 0.65 10.2%
Table 6. Performance indicators for VAA-G4 at IS (Dec 1970-Dec 1993)
表 6. VAA-G4 在 IS 的性能指标(1970 年 12 月-1993 年 12 月)
As we see in table 6, the return R is nearly double that of EW, 60/40 and SPY, while the max drawdown is at less than half (and of that of SPY). Sharpe and MAR for VAA-G4 are multiples of those of EW, 60/40 and SPY, as is RAD. Dual is worse for R ( vs. for VAA), as is D ( vs. for VAA), Sharpe, MAR and RAD ( vs. for VAA). This is all in-sample (IS), so possibly with datasnooping for parameters and .
如表 6 所示,回报 R 几乎是 EW、60/40 和 SPY 的两倍,而最大回撤 不到一半(且 是 SPY 的一半)。VAA-G4 的夏普比率和 MAR 是 EW、60/40 和 SPY 的倍数,RAD 也是如此。双重策略在 R 方面表现较差( 对比 VAA 的 ),D 也是如此( 对比 VAA 的 ),夏普比率、MAR 和 RAD( 对比 VAA 的 )。这一切都是样本内(IS),因此可能存在对参数 的数据挖掘。

Out-of-Sample (OS) 样本外 (OS)

The performance indicators for OS (Dec 1993 - Dec 2016) are shown in table 7. The return is again (as IS) roughly double that of EW, 60/40 and SPY, while the max drawdown D is less than (and of that of SPY). Sharpe and MAR for VAA-G4 are again multiples of those of EW, 60/40 and SPY, as is RAD. Dual is worse for vs. for VAA) , as is D, Sharpe, MAR and RAD ( vs. for VAA). Similar results (although with a smaller R and RAD) hold for the last decade (RS). This is all out-of-sample (OS), so without datasnooping for parameters and .
OS(1993 年 12 月 - 2016 年 12 月)的绩效指标如表 7 所示。回报 再次(如同 IS)大约是 EW、60/40 和 SPY 的两倍,而最大回撤 D 小于 (和 SPY 的 )。VAA-G4 的 Sharpe 和 MAR 再次是 EW、60/40 和 SPY 的倍数,RAD 也是如此。对于 VAA,Dual 在 方面表现更差,D、Sharpe、MAR 和 RAD 也是如此( 对于 VAA)。类似的结果(尽管 R 和 RAD 较小)在过去十年(RS)中也成立。这一切都是样本外(OS),因此没有针对参数 的数据挖掘。
OS R V D Sharpe 夏普 MAR RAD RS R V D RAD
VAA-G4 0.92 1.54
EW 0.09 0.14
EWC 0.00 1.11
60/40 0.34 0.28
SPY 0.26 0.18
Dual4 双 4 0.52 0.82
Table 7. Indicators for VAA-G4 at OS (Dec 1993 - Dec 2016) and RS (Dec 2006 - Dec 1993)
表 7. VAA-G4 在 OS(1993 年 12 月 - 2016 年 12 月)和 RS(2006 年 12 月 - 1993 年 12 月)的指标
In Table 8 we show the main statistics for the Full Sample (FS: Dec 1970 - Dec 2016, so 46 years), including the yearly Total Transaction costs (TC) and the average Cash Fraction (CF), both over FS. All the performance stats are impressive ( over nearly 50 years!) and similar to those at OS, showing limited datasnooping bias at IS. RAD is much better than EW, EWC, , SPY, Dual and GEM (with RAD=13.0% ). The TTC shows a heavy monthly turnover of ca. ) while average cash is , both high and similar to VAA-G12. Dual is roughly half of VAA on TTC and CF (with a very low TTC for GEM, see note 16).
在表 8 中,我们展示了完整样本(FS:1970 年 12 月 - 2016 年 12 月,历时 46 年)的主要统计数据,包括每年的总交易成本(TC)和平均现金比例(CF),这两者均为 FS 的统计数据。所有的表现统计数据都令人印象深刻(近 50 年内为 !),并且与 OS 的表现相似,显示出 IS 的数据挖掘偏差有限。RAD 明显优于 EW、EWC、 、SPY、Dual 和 GEM(RAD=13.0% )。TTC 显示出每月约为 的高换手率,而平均现金为 ,两者都很高且与 VAA-G12 相似。Dual 在 TTC 和 CF 上大约是 VAA 的一半(GEM 的 TTC 非常低,见注释 16)。
FS V D Sharpe 夏普 MAR RAD TTC CF
VAA-G4 0.94 1.46
EW 0.25 0.22
EWC 0.00 0.76
60/40 0.28 0.33
SPY 0.23 0.20
Dual4 双 4 0.53 0.54
Table 8. Performance indicators for VAA-G4 at FS (Dec 1970 - Dec 2016)
表 8. VAA-G4 在 FS 的性能指标(1970 年 12 月 - 2016 年 12 月)
Below (see Fig. 12, 13, and 14) we also give the (log) equity line, rolling annual returns and drawdowns for VAA-G4, EW and the relative performance of VAA/EW for the full sample (FS). Notice that the relative price in Fig. 12 is flat from 2009, indicating that returns (not risks) are similar to the benchmark (EW). The rolling 3y returns (Fig. 13) of VAA/EW are negative from 1977 to 1981, and positive most of the other years. Drawdowns (Fig. 14) for VAA-G4 are maximal at 13% in July 1990, and less than 10% in other years, except in 1974.
下面(见图 12、13 和 14),我们还给出了 VAA-G4、EW 的(对数)权益线、滚动 年收益率和回撤,以及 VAA/EW 的相对表现,涵盖完整样本(FS)。注意,图 12 中的相对价格自 2009 年起保持平稳,表明收益(而非风险)与基准(EW)相似。VAA/EW 的滚动 3 年收益率(图 13)在 1977 年至 1981 年间为负,而在其他大多数年份为正。VAA-G4 的回撤(图 14)在 1990 年 7 月达到最大值 13%,在其他年份则低于 10%,1974 年除外。
Fig. 12. Equity line for VAA-G4, EW and VAA/EW for FS (Dec 1970 - Dec 2016), log scale
图 12. VAA-G4、EW 和 VAA/EW 的权益线(1970 年 12 月 - 2016 年 12 月),对数刻度
Fig. 13. Rolling 3y Returns for VAA-G4, EW and VAA/EW for FS (Dec 1970 - Dec 2016)
图 13. VAA-G4、EW 和 VAA/EW 的滚动 3 年收益(1970 年 12 月 - 2016 年 12 月)
Fig. 14. Drawdowns (EOM) for VAA-G4, EW and VAA/EW for FS (Dec 1970 - Dec 2016)
图 14. VAA-G4、EW 和 VAA/EW 的回撤(EOM)对于 FS(1970 年 12 月 - 2016 年 12 月)

11. The VAA-U15 universe 11. VAA-U15 宇宙

The third universe is a long backtest (from Dec 1926) with a large universe, using the lbbotson (Ibb) and Fama French (FF) dataset. We will use a risky universe of 15 assets, consisting on the ten US sectors from FF and all the five US bonds (T-Bill, IT Gov, LT Gov, LT Corp, and HY) from Ibb, so N=15. We will use 30d T-Bill, IT Gov and LT corp from Ibb as Cash, similar to SHY, IEF and LQD for the shorter dataset (see Sections 9 and 10 above). So, all assets refer to the US. We will therefore denote our VAA strategy with this N15+3 US universe as VAA-U15.
第三个宇宙是一个长期回测(从 1926 年 12 月开始),使用了一个大范围的宇宙,采用 lbbotson(Ibb)和 Fama French(FF)数据集。我们将使用一个包含 15 个资产的风险宇宙,其中包括 FF 的十个美国行业和 Ibb 的五个美国债券(国库券、短期政府债、长期政府债、长期公司债和高收益债),因此 N=15。我们将使用 Ibb 的 30 天国库券、短期政府债和长期公司债作为现金,类似于 SHY、IEF 和 LQD 用于较短的数据集(见上面的第 9 和第 10 节)。因此,所有资产均指美国。因此,我们将用这个 N15+3 美国宇宙来表示我们的 VAA 策略,称为 VAA-U15。
Our in-sample (IS) period is from (ultimo) Dec 1926 - Dec 1970 (44 years), while our out-of-sample (OS) period is Dec 1970 - Dec 2016 (46 years). We use index data (EOM Total Returns) from Ibb/FF, so not corrected for ETF fees etc.
我们的样本内(IS)时期是从 1926 年 12 月到 1970 年 12 月(44 年),而样本外(OS)时期是 1970 年 12 月到 2016 年 12 月(46 年)。我们使用来自 Ibb/FF 的指数数据(EOM 总回报),因此未考虑 ETF 费用等。

In-sample (IS) 样本内 (IS)

Like with VAA-G12, we compute RAD on IS for this larger universe again over top T=1..6 and breadth for a total of values (scenarios), using our 13612 W momentum filter. The result is shown in Table 9.
与 VAA-G12 一样,我们在 T=1..6 和宽度 上再次计算这个更大宇宙的 IS 上的 RAD,总共得到 个值(场景),使用我们的 13612 W 动量过滤器。结果如表 9 所示。
RAD T
Table 9. RAD for VAA-U15 on IS (Dec 1926 - Dec 1970) for T/B=1.. 6
表 9. VAA-U15 在 IS 上的 RAD(1926 年 12 月 - 1970 年 12 月),T/B=1..6
Dual is the dual momentum strategy computed for various using our own momentum filter (13612W) for both absolute and relative momentum. From Table 9, the best score is RAD=7.4% for and , the best Dual strategy is Dual for , called Dual 6 . We can see more detail behind these T/B values when we draw the R/D frontier as function of and , as in Fig. 15 and 16.
双重是针对各种 计算的双重动量策略,使用我们自己的动量过滤器(13612W)来处理绝对和相对动量。从表 9 可以看出,最佳得分为 RAD=7.4%,适用于 ,最佳双重策略是针对 的双重策略,称为双重 6。当我们绘制 R/D 边界作为 的函数时,可以看到这些 T/B 值背后的更多细节,如图 15 和 16 所示。
As we see from Fig. 15, 16 (red lines) and Table 9 , the best point is with which is optimal for low in Fig. 15. The lines for and 6 are close to and therefore hidden to keep the picture clean. is also clearly optimal (and robust) for low D in Fig. 16 (where we hide VAA-B6: too close to VAA-B5). The VAA-T4 line in Fig. 15 has less clear "bend and knee" than in our global (G12/4) universes, but there is still a knee at (R/D=11/24%) in the red VAA-T4 line. The Dual line is clearly inferior in D to nearly all VAA-T (Fig. 15) and VAA-B (Fig. 16) lines, while optimal in R.
如图 15、16(红线)和表 9 所示,最佳点是 ,其 在图 15 中对低 是最优的。 和 6 的线接近 ,因此被隐藏以保持图像整洁。 在图 16 中对低 D 也显然是最优(且稳健)的(我们隐藏 VAA-B6:与 VAA-B5 太接近)。图 15 中的 VAA-T4 线的“弯曲和膝盖”不如我们全球(G12/4)宇宙中的明显,但在红色 VAA-T4 线的 (R/D=11/24%)处仍然有一个膝盖。双线在 D 方面明显劣于几乎所有 VAA-T(图 15)和 VAA-B(图 16)线,而在 R 方面是最优的。
Fig. 15. VAA-U15: R/D frontier on IS (Dec 1926 - Dec 1970) for VAA-T and Dual
图 15. VAA-U15:VAA-T 和双重的 IS 上的 R/D 边界(1926 年 12 月 - 1970 年 12 月)
Fig. 16. VAA-U15: R/D frontier on IS (Dec 1926 - Dec 1970) for VAA-B and Dual
图 16. VAA-U15:VAA-B 和双重的 IS 上的 R/D 前沿(1926 年 12 月 - 1970 年 12 月)
From now on we will refer to VAA-U15 as the optimal strategy with . In Table 10, we show the usual performance indicators for VAA-U15 (with optimal T/B=4/3) on IS (Dec 1926 - Dec 1970).
从现在起,我们将 VAA-U15 称为具有 的最佳策略。在表 10 中,我们展示了 VAA-U15(最佳 T/B=4/3)在 IS(1926 年 12 月 - 1970 年 12 月)上的常见性能指标。
As we see in Table 10, the return is larger than that of EW, 60/40 and SPY, while the max drawdown is less than half of EW ( ) and and less than of that of SPY ( ). Sharpe and in particular MAR for VAA-U15 are multiples of those of EW, 60/40 and SPY. Since max Drawdown D for EW, 60/40 and SPY are all larger than 50%, RAD=0%, in contrast to our VAA-U15 which is RAD=7.4%. Dual is better on R but worse on V, D ( vs. for VAA), Sharpe, MAR and RAD ( vs. on VAA). This is all in-sample (IS), so possibly with datasnooping for parameters T and B .
如表 10 所示,回报 大于 EW、60/40 和 SPY 的回报,而最大回撤 不到 EW( )和 的一半,并且小于 SPY 的 )。VAA-U15 的夏普比率,特别是 MAR,是 EW、60/40 和 SPY 的多倍。由于 EW、60/40 和 SPY 的最大回撤 D 都大于 50%,RAD=0%,而我们的 VAA-U15 则是 RAD=7.4%。Dual 在 R 上表现更好,但在 V、D(VAA 的 对比 )、夏普、MAR 和 RAD(VAA 的 对比 )上表现较差。这一切都是样本内(IS),因此可能存在对参数 T 和 B 的数据挖掘。
IS Sharpe 夏普 MAR RAD
VAA-U15 0.66 0.45
EW 0.36 0.12
EWC 0.00 0.66
60/40 0.35 0.12
SPY 0.30 0.11
Dual6 双 6 0.62 0.28
Table 10. Performance indicators for VAA-U15 at IS (Dec 1926-Dec 1970)
表 10. VAA-U15 在 IS 的性能指标(1926 年 12 月-1970 年 12 月)

Out-of-Sample (OS) 样本外 (OS)

The performance indicators for OS (Dec 1970 - Dec 2016) of VAA-U15 (with T/B=4/3) are shown in Table 11. As we can see in Table 11, the return R and max Drawdown D for VAA-U15 out-of-sample are both better than EW (the benchmark), 60/40 and SPY for R ( vs. 10% for EW and SPY, less for ) but much better for D ( vs. , and for EW, , and SPY, resp.). Sharpe is roughly double that of EW, 60/40 and SPY, while MAR is 3-4 times as good. RAD for VAA-U15 is also positive (RAD=10%) while RAD of EW, 60/40, and SPY at OS are all around or zero (SPY: D>50%). Dual is slightly better on R, but worse on V, D ( vs. for VAA), Sharpe, MAR and RAD ( vs. for VAA). This is all out-of-sample (OS), so without datasnooping for parameters and .
VAA-U15(T/B=4/3)的 OS(1970 年 12 月 - 2016 年 12 月)绩效指标如表 11 所示。从表 11 中可以看出,VAA-U15 的超出样本的回报 R 和最大回撤 D 均优于 EW(基准)、60/40 和 SPY(R 为 ,EW 和 SPY 为 10%, 更少),但 D 则更为显著( 对比 ,以及 EW 的 和 SPY 分别)。Sharpe 比 EW、60/40 和 SPY 大约高出一倍,而 MAR 则好 3-4 倍。VAA-U15 的 RAD 也是正值(RAD=10%),而 EW、60/40 和 SPY 的 OS RAD 均在 或零左右(SPY:D>50%)。Dual 在 R 上稍好,但在 V、D(VAA 的 对比 )、Sharpe、MAR 和 RAD(VAA 的 对比 )上表现较差。这一切都是超出样本(OS),因此没有对参数 进行数据窥探。
OS R V D Sharpe 夏普 MAR RAD RS R V D RAD
VAA-U15 11.7% 8.5% 12.1% 0.57 0.96 10.1% 8.6% 9.75% 10.46% 7.6%
EW 10.4% 10.6% 34.7% 0.34 0.30 4.9% 7.5% 10.7% 3.5%
EWC 6.8% 4.7% 8.3% 0.00 0.83 6.2% 3.9% 4.5% 5.0% 3.7%
60/40 9.4% 9.5% 29.9% 0.27 0.31 5.4% 6.0% 8.8% 29.9% 3.5%
SPY 10.4% 15.1% 51.0% 0.24 0.20 0.0% 7.0% 15.3% 51.0% 0.0%
Dual6 双 6 11.8% 11.1% 23.8% 0.45 0.50 8.1% 10.5% 10.3% 17.2% 8.3%
Table 11. Indicators for VAA-U15 at OS (Dec 1970-Dec 2016) and RS (Dec 2006- Dec 2016)
表 11. VAA-U15 在 OS(1970 年 12 月-2016 年 12 月)和 RS(2006 年 12 月-2016 年 12 月)的指标
In Table 12 we show the main statistics for the Full Sample (FS: Dec 1926 - Dec 2016, so 90 years), including the yearly Total Transaction costs (TTC) and the average Cash Fraction (CF), both over FS.
在表 12 中,我们展示了完整样本(FS:1926 年 12 月 - 2016 年 12 月,历时 90 年)的主要统计数据,包括每年的总交易成本(TTC)和平均现金比例(CF),这两者均基于完整样本。
All the performance stats are better than EW, 60/40, and SPY (R/D=11/24% over nearly 100 years) and return is similar to those at OS, showing limited datasnooping bias at IS. Notice the (on FS) in the early years (see Fig. 19), which is large for VAA but small compared to for EW, , and SPY. Dual is better on R, but worse on V, D ( vs. for VAA), Sharpe, MAR and RAD ( vs. for VAA). The TTC shows a monthly turnover of while average cash is , both high and similar to other VAA models.
所有的表现统计数据都优于 EW、60/40 和 SPY(近 100 年 R/D=11/24%),回报与 OS 相似,显示出 IS 的数据挖掘偏差有限。注意早期的 (在 FS 上),对于 VAA 来说很大,但与 EW、 和 SPY 相比则很小。双重策略在 R 上表现更好,但在 V、D 上表现较差(VAA 的 对比 ),在 Sharpe、MAR 和 RAD 上(VAA 的 对比 )。TTC 显示每月换手率为 ,而平均现金为 ,两者都很高且与其他 VAA 模型相似。
FS R V D Sharpe 夏普 MAR RAD TTC CF
VAA-U15 0.61 0.47
EW 0.35 0.14
EWC 0.00 0.59
60/40 0.31 0.14
SPY 0.27 0.12
Dual6 双 6 0.54 0.27
Table 12. Performance indicators for VAA-U15 at FS (Dec 1926-Dec 2016)
表 12. VAA-U15 在 FS 的性能指标(1926 年 12 月-2016 年 12 月)
Below (see Fig. 17, 18, and 19) we also give the (log) equity line, rolling 3y returns and drawdowns for VAA-U15, EW and the relative performance of VAA/EW for the full sample (FS). Notice that the relative price in Fig. 17 is roughly flat from 1926, 1939, 1974 and 2009, indicating that returns (not risks) in these three periods are similar to the benchmark (EW). The rolling 3y returns (Fig. 18) of VAA/EW are more than -10% in 1935, 1941 and from 2012, and around zero most of the other years. Max drawdowns (Fig. 19) for VAA-G4 are 24% before 1945, and 12% in all other years. So VAA-U15 shows outperformance in bear markets and similar performance in bull markets, as compared to EW.
下面(见图 17、18 和 19),我们还给出了 VAA-U15、EW 的(对数)权益线、滚动 3 年收益和回撤,以及 VAA/EW 的相对表现,涵盖完整样本(FS)。注意,图 17 中的相对价格在 1926 年、1939 年、1974 年和 2009 年大致保持平坦,表明这三个时期的收益(而非风险)与基准(EW)相似。VAA/EW 的滚动 3 年收益(图 18)在 1935 年、1941 年和 2012 年之后均超过-10%,而在其他大多数年份则接近于零。VAA-G4 的最大回撤(图 19)在 1945 年前为 24%,在其他所有年份为 12%。因此,与 EW 相比,VAA-U15 在熊市中表现优异,而在牛市中表现相似。
Fig. 17. Equity line for VAA-U15, EW and VAA/EW for FS (Dec 1926 - Dec 2016), log scale
图 17. VAA-U15、EW 和 VAA/EW 的股权线(1926 年 12 月 - 2016 年 12 月),对数刻度
Fig. 18. Rolling 3y Annual Returns for VAA-U15, EW and VAA/EW for FS (Dec 1926 - Dec 2016)
图 18. VAA-U15、EW 和 VAA/EW 的滚动 3 年年回报(1926 年 12 月 - 2016 年 12 月)
Fig. 19. Drawdowns for VAA-U15, EW and VAA/EW for FS (Dec 1926 - Dec 2016)
图 19. VAA-U15、EW 和 VAA/EW 的回撤(1926 年 12 月 - 2016 年 12 月)

12. The VAA-U6 universe 12. VAA-U6 宇宙

The fourth and last universe is a long backtest (from Dec 1926) with a small universe, using the Ibbotson (Ibb) and Fama French (FF) dataset. We will use a risky universe of 6 assets, consisting on the four US factors Size Value from FF plus two US bonds (30d T-Bill, and LT Corp) from Ibb, so N=6. As with VAA-U15 (see section 11), we will use 30d T-Bill, IT Gov and LT corp from Ibb as Cash, similar to SHY, IEF and LQD for the shorter dataset (see Sections 9 and 10 above). So, all assets refer to the US. We will therefore denote our VAA strategy with this N6+3 US universe as VAA-U6. Our in-sample (IS) period is from (ult.) Dec 1926 - Dec 1970 (44 years), while our out-of-sample (OS) period is Dec 1970 - Dec 2016 (46 years). We use index data (EOM Total Returns) from Ibb/FF, so not corrected for ETF fees etc.
第四个也是最后一个宇宙是一个较长的回测(从 1926 年 12 月开始),使用一个小的宇宙,采用 Ibbotson(Ibb)和 Fama French(FF)数据集。我们将使用一个包含 6 个资产的风险宇宙,包括 FF 的四个美国因子:规模 价值,加上 Ibb 的两种美国债券(30 天国库券和长期公司债),因此 N=6。与 VAA-U15(见第 11 节)一样,我们将使用 Ibb 的 30 天国库券、IT 政府债券和长期公司债券作为现金,类似于 SHY、IEF 和 LQD 用于较短的数据集(见上面的第 9 和第 10 节)。因此,所有资产均指美国。因此,我们将用这个 N6+3 的美国宇宙来表示我们的 VAA 策略为 VAA-U6。我们的样本内(IS)期间为(最终)1926 年 12 月到 1970 年 12 月(44 年),而我们的样本外(OS)期间为 1970 年 12 月到 2016 年 12 月(46 年)。我们使用来自 Ibb/FF 的指数数据(EOM 总回报),因此未考虑 ETF 费用等。

In-sample (IS) 样本内 (IS)

We compute RAD on IS over top T=1..6 and breadth B=1..6 for a total of values (scenarios), using our 13612 W momentum filter. Dual is the dual momentum strategy computed for various using our own momentum filter (13612W) for both absolute and relative momentum. From Table 13, the best VAA score is RAD=4.9% for and , and for Dual RAD=2.2% for . We can see more detail behind these T/B values when we draw the R/D frontier as function of and , as in Fig. 20 and 21. We include (as with VAA-G4) again the ( cash) point in Fig. 20 to show the knee more clearly and leave out some overlapping VAA-T and VAA-B lines.
我们计算了在顶级 T=1..6 和宽度 B=1..6 上的 IS 的 RAD,总共 个值(场景),使用我们的 13612 W 动量过滤器。Dual 是为各种 计算的双动量策略,使用我们自己的动量过滤器(13612W)来处理绝对和相对动量。从表 13 中可以看出,最佳 VAA 分数是 RAD=4.9% 对于 ,而对于 Dual RAD=2.2% 对于 。当我们绘制 R/D 边界作为 的函数时,可以看到这些 T/B 值的更多细节,如图 20 和 21 所示。我们再次在图 20 中包含了 ( 现金) 点 ,以更清晰地显示拐点,并省略了一些重叠的 VAA-T 和 VAA-B 线。
RAD 4.9% 1 2 3 4 T
5 6
0 2.3% 2.3% 2.3% 2.3% 2.3% 2.3%
1 2.8% 4.0% 4.7% 4.3% 4.7% 4.9%
2 0.0% 0.6% 0.0% 0.7% 1.6% 3.0%
3 0.0% 0.0% 0.0% 0.0% 0.1% 2.1%
4 0.0% 0.0% 0.0% 0.0% 0.0% 2.2%
5 0.0% 0.0% 0.0% 0.0% 0.0% 2.2%
B 6 0.0% 0.0% 0.0% 0.0% 0.0% 2.2%
Dual 双重 0.0% 0.0% 0.0% 0.0% 0.0% 2.2%
Table 13. RAD for VAA-U6 on IS (Dec 1926-Dec 1970) for T/B=1..6
表 13. VAA-U6 在 IS 上的 RAD(1926 年 12 月-1970 年 12 月),T/B=1..6
As we see from and Table 13 and Fig. 20, 21 (red line), the best point is . The VAA-T6 line (so with all 6 assets included when , and therefore no relative momentum) in Fig. 20 is less efficient than VAA-T2 and T4, except when where wins at . The red VAA-T6 line in Fig. 20 has less clear "bend and knee" than in our global (G12/4) universes, but there is still a knee at . The Dual line is less efficient in terms of return/risk to all VAA-T lines (Fig. 20) and to VAA-B1 (Fig. 21). From Fig. 21 we see that VAA-B1 for is also optimal in terms of return/risk to all other points with D<50%. The best Dual strategy (see also table 13) is T=6 (left point of the black Dual line).
从表 13 和图 20、21(红线)中可以看出,最佳点是 。在图 20 中,VAA-T6 线(因此在 时包含所有 6 个资产,因此没有相对动量)效率低于 VAA-T2 和 T4,除了在 时, 处获胜。图 20 中的红色 VAA-T6 线的“弯曲和膝盖”不如我们全球(G12/4)宇宙中的明显,但在 处仍然存在一个膝盖。双重线在回报/风险方面对所有 VAA-T 线(图 20)和 VAA-B1(图 21)效率较低。从图 21 中我们看到,VAA-B1 在 时在回报/风险方面对所有其他点(D<50%)也是最佳的。最佳双重策略(另见表 13)是 T=6(黑色双重线的左侧点)。
Fig. 20. VAA-U6: R/D frontier on IS (Dec 1926 - Dec 1970) for VAA-T
图 20. VAA-U6:VAA-T 的 IS 上的 R/D 前沿(1926 年 12 月 - 1970 年 12 月)
From now on we will refer to VAA-U6 as the optimal strategy with . Notice that relative momentum (eg. between the 4 factors) is disabled since . Only absolute momentum (at the universe level) is relevant in a very protective way ( , so when any of the 6 assets becomes bad), like with the small global strategy VAA-G4 (see Section 10).
从现在起,我们将把 VAA-U6 称为具有 的最佳策略。请注意,相对动量(例如,四个因素之间)自 以来已被禁用。只有绝对动量(在整体层面)以非常保护的方式相关( ,因此当任何六个资产变坏时 ),就像小型全球策略 VAA-G4 一样(见第 10 节)。
Fig. 21. VAA-U6: R/D frontier on IS (Dec 1926 - Dec 1970) for VAA-B and Dual
图 21. VAA-U6:VAA-B 和双重的 IS 上的 R/D 边界(1926 年 12 月 - 1970 年 12 月)
In Table 14, we show the familiar performance indicators for VAA-U6 (with optimal T/B=6/1) at IS (Dec 1926- Dec 1970). As we see in Table 14, the return is smaller than that of EW ( , 60/40 (7.6%), SPY (9.6%), and Dual6 (9.2%), while the max drawdown is less than Dual (43%) and less than half of EW (71%) and 60/40 (62%) and 1/3 of that of SPY ( ). Sharpe for VAA-U15 ( 0.42 ) is better than of those of EW and SPY , but not better than Dual (0.49). MAR of VAA-U6 ( 0.29 ) is nearly three times that of EW (0.13), 60/40 (0.12), and SPY (0.11), and better than Dual (0.21). Since max Drawdown D for EW, 60/40 and SPY are all larger than 50%,
在表 14 中,我们展示了 VAA-U6(最佳 T/B=6/1)在 IS(1926 年 12 月-1970 年 12 月)的熟悉绩效指标。如表 14 所示,回报 小于 EW( )、60/40(7.6%)、SPY(9.6%)和 Dual6(9.2%),而最大回撤 小于 Dual(43%),也不到 EW(71%)和 60/40(62%)的一半,仅为 SPY( )的三分之一。VAA-U15 的夏普比率(0.42)优于 EW 和 SPY ,但不如 Dual(0.49)。VAA-U6 的 MAR(0.29)几乎是 EW(0.13)、60/40(0.12)和 SPY(0.11)的三倍,并且优于 Dual(0.21)。由于 EW、60/40 和 SPY 的最大回撤 D 均大于 50%,
RAD , in contrast to our VAA-U6 which has RAD=4.9%, while for EWC and Dual RAD is and , resp. This is all in-sample (IS), so possibly with datasnooping for parameters and .
RAD ,与我们的 VAA-U6(RAD=4.9%)相比,EWC 和 Dual 的 RAD 分别为 。这都是样本内(IS),因此可能存在对参数 的数据窥探。
IS Sharpe 夏普 MAR RAD
VAA 0.42 0.29
EW 0.34 0.13
EWC 0.00 0.66
0.35 0.12
SPY 0.30 0.11
Dual6 双 6 0.49 0.21
Table 14. Performance indicators for VAA-U6 at IS (Dec 1926-Dec 1970)
表 14. VAA-U6 在 IS 的性能指标(1926 年 12 月-1970 年 12 月)

Out-of-Sample (OS) 样本外 (OS)

The performance indicators for OS (Dec 1970 - Dec 2016) of VAA-U6 (with T/B=6/1) are shown in Table 15. As we can see in Table 15, the return for VAA-U6 out-of-sample is slightly better than EW (10.6%, the benchmark), 60/40 (9.4%) and SPY (10.4%). D is much better ( vs. , and for EW, , SPY, resp. Sharpe ( 0.48 ) is nearly double that of EW, 60/40 and SPY, while MAR is three times as good. RAD=9.3% for VAA-U6, while RAD of EW, and 60/40 are roughly half of that, with RAD for SPY at OS, since . Dual is better than VAA on R ( vs for VAA) but worse on V, D ( vs. for VAA), Sharpe, MAR and RAD ( vs. for VAA). The picture for the last decade (RS: Dec 2006 - Dec 2016) is similar to that for OS, although with a lower return R. Dual is slightly better than VAA on R, V, D, and RAD ( vs. on VAA). This is all outof-sample (OS), so without datasnooping for parameters and .
VAA-U6(T/B=6/1)的表现指标(1970 年 12 月 - 2016 年 12 月)如表 15 所示。从表 15 中可以看出,VAA-U6 的超出样本回报率 略高于 EW(10.6%,基准)、60/40(9.4%)和 SPY(10.4%)。D 的表现更好( 对比 和 EW 的 、SPY 的情况)。夏普比率(0.48)几乎是 EW、60/40 和 SPY 的两倍,而 MAR 则好三倍。VAA-U6 的 RAD 为 9.3%,而 EW 和 60/40 的 RAD 大约是其一半,SPY 的 RAD 在超出样本时为 ,因为 。Dual 在 R 上优于 VAA( 对比 ),但在 V、D( 对比 )上表现较差,夏普比率、MAR 和 RAD( 对比 )也不如 VAA。过去十年(RS:2006 年 12 月 - 2016 年 12 月)的情况与超出样本相似,尽管回报率 R 较低。Dual 在 R、V、D 和 RAD 上略优于 VAA( 对比 )。这一切都是超出样本(OS),因此没有对参数 进行数据窥探。
OS R V D Sharpe 夏普 MAR RAD RS R V D RAD
VAA-U6 10.8% 8.1% 12.0% 0.48 0.90 9.3% 6.9% 9.5% 10.5% 6.1%
EW 10.6% 12.2% 39.4% 0.31 0.27 3.7% 6.6% 12.6% 39.4% 2.3%
EWC 6.8% 4.7% 8.3% 0.00 0.83 6.2% 3.9% 4.5% 5.0% 3.7%
60/40 9.4% 9.5% 29.9% 0.27 0.31 5.4% 6.0% 8.8% 29.9% 3.5%
SPY 10.4% 15.1% 51.0% 0.24 0.20 0.0% 7.0% 15.3% 51.0% 0.0%
Dual6 双 6 10.8% 9.1% 18.1% 0.44 0.60 8.4% 8.0% 9.2% 9.9% 7.1%
Table 15. Performance indicators for VAA-U6 at OS (Dec 1970-Dec 2016) and RS (Dec 2006-Dec 2016)
表 15. VAA-U6 在 OS(1970 年 12 月-2016 年 12 月)和 RS(2006 年 12 月-2016 年 12 月)的性能指标
In Table 16 we show the main statistics for the Full Sample (FS: Dec 1926 - Dec 2016, so 90 years), including the yearly Total Transaction costs (TTC) and the average Cash Fraction (CF), both over FS. While return R=9.1% are slightly less than EW (9.9%), SPY (10.0%), and Dual (10%), it is slightly better than 60/40 (8.5%). The max Drawdown ( ) is nearly three times smaller than that of EW (71%), 60/40 (62%), and SPY (83%). All the return/risk stats like Sharpe, MAR and RAD are better than EW, 60/40, and SPY, with RAD=6% while zero for EW, and SPY. Notice the (on FS) in the early years (see Fig. 24), which is large for VAA but small compared to D=62-83% for EW, 60/40, and SPY. Dual is slightly better at R and Sharpe, but clearly worse on D, MAR and RAD ( vs. for
在表 16 中,我们展示了完整样本(FS:1926 年 12 月 - 2016 年 12 月,历时 90 年)的主要统计数据,包括每年的总交易成本(TTC)和平均现金比例(CF),均为 FS 的统计数据。虽然回报率 R=9.1%略低于等权重(EW)的 9.9%、SPY 的 10.0%和双重策略的 10%,但略高于 60/40 的 8.5%。最大回撤( )几乎是等权重(71%)、60/40(62%)和 SPY(83%)的三分之一。所有的回报/风险统计数据,如夏普比率、MAR 和 RAD 均优于等权重、60/40 和 SPY,RAD 为 6%,而等权重、 和 SPY 为零。注意早期年份的 (在 FS 上)(见图 24),对于 VAA 来说很大,但与等权重、60/40 和 SPY 的 62-83%的 D 相比则较小。双重策略在回报率和夏普比率上略好,但在 D、MAR 和 RAD 上明显较差( 对比 )。
VAA). The TTC shows a monthly turnover of while average cash is , both high and similar to other VAA models. The TTC and average CF of Dual are roughly half of VAA.
VAA)。TTC 的月度周转为 ,而平均现金为 ,两者都很高且与其他 VAA 模型相似。Dual 的 TTC 和平均现金流大约是 VAA 的一半。
FS R V D Sharpe 夏普 MAR RAD TTC CF
VAA-U6 0.44 0.36
EW 0.32 0.14
EWC 0.00 0.59
60/40 0.31 0.14
SPY 0.27 0.12
Dual6 双 6 0.46 0.23
Table 16. Performance indicators for VAA-U6 at FS (Dec 1926 - Dec 2016)
表 16. VAA-U6 在 FS 的性能指标(1926 年 12 月 - 2016 年 12 月)
Below (see Fig. 22, 23, and 24 ) we also give the (log) equity line, rolling annual returns and drawdowns for VAA-U6, EW and the relative price of VAA/EW for the full sample (FS). Notice that the relative price in Fig. 22 is decreasing from 1933-1969, and mostly flat after that, indicating that returns (not risks) in the first period are less and in the last period similar to the benchmark (EW). The rolling 3y returns (Fig. 18) of VAA/EW are more than -20% in 1935, and 1941, and between and most of the other years with a score during the Great depression. Max drawdowns (Fig. 24) for VAA-U6 are 25% before WW2 and 12% in all other years.
下面(见图 22、23 和 24),我们还给出了 VAA-U6、EW 的(对数)权益线、滚动 年收益率和回撤,以及整个样本(FS)中 VAA/EW 的相对价格。注意,图 22 中的相对价格在 1933 年至 1969 年间下降,之后大致保持平稳,这表明第一阶段的收益(而非风险)低于基准(EW),而最后阶段则相似。VAA/EW 的滚动 3 年收益率(图 18)在 1935 年和 1941 年超过-20%,而在大萧条期间大多数其他年份则在 之间,得分为 。VAA-U6 的最大回撤(图 24)在二战前为 25%,在其他所有年份为 12%。
Fig. 22. Equity line for VAA-U6, EW and VAA/EW on FS (Dec 1926 - Dec 2016), log scale
图 22. VAA-U6、EW 和 VAA/EW 在 FS 上的权益线(1926 年 12 月 - 2016 年 12 月),对数刻度
Fig. 23. Rolling 3y returns for VAA-U6, EW and VAA/EW on FS (Dec 1926 - Dec 2016)
图 23. VAA-U6、EW 和 VAA/EW 在 FS 上的滚动 3 年回报(1926 年 12 月 - 2016 年 12 月)
Fig. 24. Drawdowns for VAA-U6, and EW on FS (Dec 1926 - Dec 2016)
图 24. VAA-U6 和 EW 在 FS 上的回撤(1926 年 12 月 - 2016 年 12 月)

13. Summary and concluding remarks
13. 总结与结论

In this paper, we have tested a dual momentum strategy (dual: with both absolute and relative momentum) with vigorous crash protection, called Vigilant Asset Allocation (VAA). However, instead of absolute momentum (trendfollowing) at the individual asset level, for VAA we use breadth momentum at the universe level for crash protection, as we did with PAA. Compared with PAA, however, we now use the number of bad assets (with non-positive momentum) in the universe (of size N ) relative to a breadth protection threshold B (or breadth B , for short, with ) as a more granular crash indicator. As we demonstrated, in-sample optimization of this breadth B often leads to crash protection when only one or a limited number of all assets in the universe are bad.
在本文中,我们测试了一种双动量策略(双:同时具有绝对动量和相对动量),称为警惕资产配置(VAA),并具有强有力的崩盘保护。然而,对于 VAA,我们在整体层面使用广度动量进行崩盘保护,而不是在单个资产层面使用绝对动量(趋势跟随),正如我们在 PAA 中所做的那样。与 PAA 相比,我们现在使用整体中坏资产(具有非正动量)的数量(大小为 N)相对于广度保护阈值 B(或简称广度 B, )作为更细致的崩盘指标。正如我们所展示的,广度 B 的样本内优化通常会导致 崩盘保护,当整体中只有一个或有限数量的资产是坏的时。
We have shown that with this aggressive crash protection and ditto momentum filter (13612W), we arrived out-of-sample (OS) at returns above with max drawdowns of less than , even over periods of more than 40 years (Dec 1970 - Dec 2016). In Table 17 we show the OS results for our four universes, with 12 and 4 global assets (VAA-G12 and VAA-G4, resp.) and with 15 and 6 US assets (VAA-U15 and VAA-U6, resp.).
我们已经证明,通过这种激进的崩溃保护和相应的动量过滤器(13612W),我们在样本外(OS)获得了超过 的回报,最大回撤低于 ,即使在超过 40 年的时间段内(1970 年 12 月 - 2016 年 12 月)。在表 17 中,我们展示了我们四个宇宙的 OS 结果,分别为 12 个和 4 个全球资产(VAA-G12 和 VAA-G4)以及 15 个和 6 个美国资产(VAA-U15 和 VAA-U6)。
OS
1993-2016 R V D Sharpe 夏普 MAR RAD N T B TTC CF
VAA-G12 0.51 0.81 12 2 4
VAA-G4 0.92 1.54 4 1 1
EWC 0.0 1.11 3 - -
60/40 0.34 0.28 2 - -
SPY 0.26 1 - -
OS
1970-2016 R V D Sharpe 夏普 MAR RAD N T B TTC CF
VAA-U15 0.96 15 4 3
VAA-U6 0.48 6 6 1
EWC 0.0 0.83 3 - -
60/40 0.27 0.31 1 - -
SPY 0.24 0.21 1 - -
Table 17. Out-of-sample results for two global (VAA-G) and two US (VAA-U) universes
表 17. 两个全球(VAA-G)和两个美国(VAA-U)宇宙的样本外结果
As above, Return (CAGR), Volatility and max Drawdown are all based on end-of-month (EOM) TR index prices (so including dividends etc.) with one-way transaction costs of and only corrected for ETF fees in the VAA-G universes. The Sharpe ratio is defined relative to the returns of the equal weight of the three-bond cash universe (EWC), MAR=R/D, and RAD is our own Return Adjusted for Drawdowns measure (see section 6).
如上所述,回报(CAGR)、波动率和最大回撤均基于月末(EOM)总回报指数价格(包括股息等),单向交易成本为 ,并且仅在 VAA-G 宇宙中修正了 ETF 费用。夏普比率是相对于三债现金宇宙(EWC)的等权重回报定义的,MAR=R/D,而 RAD 是我们自己调整回撤的回报指标(见第 6 节)。
Dual momentum is applied to the assets using the responsive 13612W (average annual returns over past 1,3,6 and 12 months) momentum filter for both absolute (trend following) and relative (cross sectional) momentum. Traditionally, absolute momentum implies trendfollowing where bad assets are replaced by cash as a form of crash protection. With VAA, we apply crash protection on the
双动量应用于资产,使用响应式 13612W(过去 1、3、6 和 12 个月的平均年回报)动量过滤器,适用于绝对(趋势跟随)和相对(横截面)动量。传统上,绝对动量意味着趋势跟随,其中不良资产被现金替代,以作为崩盘保护。通过 VAA,我们在此应用崩盘保护。

universe, using the number of bad assets b of the risky universe relative to breadth (so b/B) as the cash fraction (CF).
宇宙,使用风险宇宙中不良资产数量 b 相对于广度的比例 (即 b/B) 作为现金比例 (CF)。
With VAA, this breadth momentum driven crash protection might give rise to extensive trading since we do not necessarily replace individual assets by cash, as with traditional dual momentum. Therefore, we used for VAA a so-called "Easy Trading" (ET) approach where we limited cash to multiples of , where is the number of top assets used in relative momentum. The result is similar to traditional dual momentum, where we replaced risky assets by cash, but now much earlier, depending on the (rounded) ratio b/B.
通过 VAA,这种广泛的动量驱动的崩溃保护可能会导致广泛的交易,因为我们不一定像传统的双重动量那样用现金替代单个资产。因此,我们为 VAA 采用了一种所谓的“简单交易”(ET)方法,在这种方法中,我们将现金限制为 的倍数,其中 是相对动量中使用的顶级资产数量。结果类似于传统的双重动量,在这种情况下,我们用现金替代风险资产,但现在更早,具体取决于(四舍五入的)比例 b/B。
"Cash" itself is also based on a (simplified) momentum model, using the same 13612 W filter for relative momentum only for picking the best bond of our cash universe with the highest momentum, without regard to sign (so no absolute momentum). The cash universe for the shorter, global backtest from Dec 1970 (VAA-G) encompasses BIL, IEF, and LQD proxies, while the longer, US backtest uses similar indices (T-Bill, IT Gov, and LT Corp) from Ibbotson.
“现金”本身也是基于一个(简化的)动量模型,仅使用相同的 13612 W 过滤器来选择我们现金宇宙中相对动量最高的最佳债券,而不考虑符号(因此没有绝对动量)。从 1970 年 12 月开始的较短全球回测(VAA-G)涵盖了 BIL、IEF 和 LQD 代理,而较长的美国回测则使用来自 Ibbotson 的类似指数(T-Bill、IT Gov 和 LT Corp)。
Using ET, the cash fraction CF in the above four VAA models is based the two parameters and . The top number equals the maximum number of best risky assets allocated and is well known from dual momentum. The breadth parameter B is new for VAA. Both parameters and are estimated for each universe (by an in-sample optimization of RAD), see Table 17. Notice that the optimal parameters T (top) and B (breadth) increases with the universe size N , with T and B mostly of similar magnitude except for the universe (where and ), which seems an exception (due to too little diversification?).
使用 ET,上述四个 VAA 模型中的现金比例 CF 基于两个参数 。顶部数字 等于分配的最佳风险资产的最大数量,这在双动量中是众所周知的。宽度参数 B 是 VAA 的新参数。两个参数 是针对每个宇宙估计的(通过 RAD 的样本内优化),见表 17。请注意,最佳参数 T(顶部)和 B(宽度)随着宇宙大小 N 的增加而增加,T 和 B 的大小大致相似,除了 宇宙(其中 ),这似乎是一个例外(由于多样化不足?)。
Both small universes (VAA-G4 and VAA-U6) obtained optimal in-sample (IS) results with B=1 and therefore go completely to cash when only one of the (four or six) risky assets turns bad. For the VAA-G12 universe with 12 risky assets, and proved to be optimal resulting in a cash fraction when four or more of the twelve assets are bad, while for traditional dual this only happens when all risky assets are bad. Both examples demonstrate the responsiveness of our crash protection in all four VAA models compared to the traditional dual approach.
两个小宇宙(VAA-G4 和 VAA-U6)在 B=1 时获得了最佳的样本内(IS)结果,因此当四个或六个风险资产中只有一个表现不佳时,它们会完全转为现金。对于拥有 12 个风险资产的 VAA-G12 宇宙,当四个或更多的资产表现不佳时, 被证明是最佳选择,导致现金比例为 ,而对于传统的双重方法,这种情况仅在所有风险资产表现不佳时发生。这两个例子展示了我们在所有四个 VAA 模型中相较于传统双重方法的崩溃保护的响应能力。
Therefore, we were out of the market (in cash) in all VAA strategies for roughly of the time. However, we still arrived out-of-sample at nearly offensive ( ) returns with very defensive drawdowns ( . This demonstrates the responsive timing of breadth momentum in our VAA strategies and our motto "winning more by losing less". We also surpassed Antonacci's highly successful GEM model in return and return/risk with our VAA-G4 strategy but with more turnover and higher cash fraction. This is the price we pay for our aggressive strategy.
因此,我们在所有 VAA 策略中大约有 的时间处于市场外(持有现金)。然而,我们在样本外仍然实现了接近激进的( )回报,同时伴随非常防御性的回撤( )。这展示了我们 VAA 策略中广度动量的响应时机,以及我们的座右铭“通过减少损失来赢得更多”。我们的 VAA-G4 策略在回报和回报/风险方面超越了 Antonacci 的成功 GEM 模型,但伴随更高的周转率和更高的现金比例。这是我们为激进策略所付出的代价。
With such high cash fractions, the strategy of the cash universe becomes important, even more so with the current low and increasing yields of the coming years. However, similar rising yields were also present from 1950 and in the seventies (1970 - 1982) when eg. both VAA-G universes clearly outperformed EW and EWC (with double RAD's), while the same holds in the years after 1982 with the cash tailwind (decreasing yields).
随着现金比例如此之高,现金投资组合的策略变得重要,尤其是在未来几年低且不断上升的收益率背景下。然而,类似的收益率上升在 1950 年和 70 年代(1970 - 1982 年)也曾出现,例如,两个 VAA-G 投资组合明显优于 EW 和 EWC(双重 RAD),而在 1982 年之后的几年中,现金的顺风(收益率下降)同样适用。
Our four VAA strategies show a remarkable characteristic in our breadth momentum approach. This is the "knee" in the R/D space, going from low breadth B to higher values. Before the knee, return increases with B with nearly stable drawdown while after the knee, drawdown increases while returns are nearly stable. Also, the optimal breadth parameter B turns out to be the same for both VAA-G universes in both the in-sample and the out-of-sample periods, which demonstrates robustness. This all demonstrates some remarkable timing characteristic of our breath momentum approach for various regimes and universes. Against our expectations, it could be all just luck, but if not, future research should try to answer why. Why does this breath momentum work, what is the role of the universe used, is there a relationship with diversification and correlation between assets, and can we repeat the knee with other universes? Is breadth B constant or should we use some adaptive walk-forward scheme? Enough questions, enough to be done.
我们的四个 VAA 策略在我们的广度动量方法中显示出一个显著特征。这是 R/D 空间中的“膝部”,从低广度 B 到更高的值。在膝部之前,收益随着 B 的增加而增加,同时回撤几乎保持稳定;而在膝部之后,回撤增加而收益几乎保持稳定。此外,最佳广度参数 B 在两个 VAA-G 宇宙中,在样本内和样本外期间都是相同的,这表明了其稳健性。这一切都展示了我们广度动量方法在不同市场环境和宇宙中的一些显著时机特征。与我们的预期相反,这可能只是运气,但如果不是,未来的研究应该尝试回答为什么。为什么这种广度动量有效,所使用的宇宙有什么作用,资产之间的多样化和相关性是否存在关系,我们能否在其他宇宙中重复膝部?广度 B 是恒定的,还是我们应该使用某种自适应的前向行走方案?问题足够多,工作也足够多。

14. Literature 14. 文学

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  1. We thank Adam Butler, Walter Jones, Steve LeCompte, Bas Nagtzaam, Michael Roovers, and Valeriy Zakamouline for useful comments on earlier versions of this paper. All errors are ours.
    我们感谢亚当·巴特勒、沃尔特·琼斯、史蒂夫·勒孔普特、巴斯·纳赫特扎姆、迈克尔·鲁弗斯和瓦列里·扎卡穆林对本文早期版本的有益评论。所有错误均由我们负责。
  2. Monthly return weights are defined as the effect on the momentum filter of the return in a particular historical (lagged) month.
    月度回报权重被定义为特定历史(滞后)月份回报对动量过滤器的影响。
  3. The RET12 filter equals p0/p12 -1, the SMA12 filter equals p0/SMA12 -1, the 13612 filter equals , and the new 13612 W filter equals 19 , where pt equals price with lag .
    RET12 滤波器等于 p0/p12 -1,SMA12 滤波器等于 p0/SMA12 -1,13612 滤波器等于 ,新的 13612 W 滤波器等于 19,其中 pt 等于价格 ,滞后为
    With different filters for traditional absolute and relative momentum, it is possible that eg. in the Top T, the second best asset is replaced by cash but not the third best, which seems less logical. Therefore, we will always use the same filter for both relative and absolute momentum.
    通过对传统绝对动量和相对动量使用不同的过滤器,例如在 Top T 中,第二好的资产可能被现金替代,但第三好的资产却没有,这似乎不太合理。因此,我们将始终对相对动量和绝对动量使用相同的过滤器。
  4. Notice that Antonacci (2014) used a slightly different definition of Dual momentum for his specific GEM strategy, using only SPY for CP (see his p.98), in contrast to his informal (more usual dual) approach (in his flowchart on p. 101). Our definition of dual corresponds to Antonacci (2013b) and Faber's GTAA approach .
    注意到 Antonacci(2014)为其特定的 GEM 策略使用了稍微不同的双动量定义,仅使用 SPY 作为 CP(见其第 98 页),与他在第 101 页的流程图中所采用的非正式(更常见的双重)方法形成对比。我们对双动量的定义对应于 Antonacci(2013b)和 Faber 的 GTAA 方法。
  5. See also note 5 .
    另见注释 5。
    Apart from the required rebalancing of open positions to their prescribed weights .
    除了将未平仓头寸重新平衡到规定权重的要求外
  6. Harvey (2013) shows that to test the hypothesis Sharpe without datasnooping, one can use the t-test with sqrt( ) where is the sample size in years and is the annualized in-sample Sharpe ratio (over riskfree). When there is datasnooping with less than 100 scenarios (here ), a haircut of could be applied for around 0.5 , or if the is around 1 or higher. Recently, Paulsen (2016) proved that asymptotically (for large ) the haircut equals for an in-sample Sharpe ratio and for . For VAA, this results in a haircut of (for between ) when is the number of optimized parameters ( ) and year
    哈维 (2013) 表明,为了在没有数据挖掘的情况下检验 Sharpe 假设,可以使用 t 检验,公式为 sqrt( ),其中 是样本大小(以年为单位), 是年化的样本内 Sharpe 比率(相对于无风险收益)。当数据挖掘的场景少于 100 个时(这里 ),可以对 施加约 0.5 的削减,或者如果 在 1 或更高时,则为 。最近,保尔森 (2016) 证明,对于大样本量 ,削减等于 ,适用于样本内 Sharpe 比率 对于 。对于 VAA,这导致在 是优化参数数量( )和 年之间时,削减为
  7. for IS, and 5-18% when Y=46 years. Notice that these haircuts hold for (in-sample) Sharpe ratios, but not for MAR and RAD.
    对于 IS,当 Y=46 岁时为 5-18%。请注意,这些发型适用于(样本内)夏普比率,但不适用于 MAR 和 RAD。
    Using similar ideas, one might also construct a Return Adjusted for Volatility, e.g. RAV=R*(1-2V/(1-2V)) with a max Volatility of for which RAV .
    使用类似的思路,人们也可以构建一个调整波动率的收益,例如 RAV=R*(1-2V/(1-2V)),其最大波动率为 ,对于此 RAV
  8. We actually used (proxies for) Vanguard ETFs VEA, VWO, VNQ, and BND instead of the mentioned (and more common) iShares ETFs EFA, EEM, IYR, and AGG, respectively, in nearly all our backtest since these ETFs has lower fees and similar AUM's as the iShares ETFs. When we used Vanguard or iShares ETFs, our proxy ETFs from 1969 were also calibrated using data for recent Vanguard or iShares ETFs, respectively. See also note 16 for some difference in RAD etc. between using iShares and Vanguard ETFs.
    我们实际上在几乎所有的回测中使用了先锋 ETF VEA、VWO、VNQ 和 BND 的代理,而不是提到的(更常见的)iShares ETF EFA、EEM、IYR 和 AGG,因为这些先锋 ETF 的费用更低,且与 iShares ETF 的资产管理规模相似。当我们使用先锋或 iShares ETF 时,我们从 1969 年开始的代理 ETF 也使用了最近的先锋或 iShares ETF 的数据进行校准。有关使用 iShares 和先锋 ETF 之间在 RAD 等方面的一些差异,请参见注释 16。
  9. We used VWO and VNQ instead of EEM and IYR in our actual backtest. See also note 11.
    我们在实际回测中使用了 VWO 和 VNQ,而不是 EEM 和 IYR。另见注释 11。
    We used VEA, VWO and BND instead of EFA, EEM and AGG in our actual backtest. See also note 11.
    我们在实际回测中使用了 VEA、VWO 和 BND,而不是 EFA、EEM 和 AGG。另见注释 11。
  10. When using Sharpe or MAR instead of RAD we found a much more dispersed and irregular pattern, indicating less robust optimization results.
    当使用 Sharpe 或 MAR 而不是 RAD 时,我们发现了一个更加分散和不规则的模式,表明优化结果不够稳健。
  11. The setting resulted in an FS performance of and therefore RAD= over Dec 1970 - Dec 2016. We also used Meb Faber's Backtester (see Faber, 2017) with this T/B=2/1 setting for a preliminary test of this VAA variant from Dec 1927 - Dec 2016 and found an FS performance well within our target range of and over nearly 100 years.
    设置导致了 的 FS 表现,因此在 1970 年 12 月至 2016 年 12 月期间,RAD= 。我们还使用了 Meb Faber 的回测工具(见 Faber, 2017),在 1927 年 12 月至 2016 年 12 月期间以 T/B=2/1 设置对这个 VAA 变体进行了初步测试,发现 FS 表现完全在我们目标范围 之内,持续近 100 年。
  12. We have for the "official" GEM model (see Antonacci, 2014, and also TrendXplorer, 2016) but with our KK data (including ETF fees and a transaction cost), and Vanguard instead of iShares ETFs (SPY, VEU, and BND), the following performance statistics over Dec 1970 - Dec 2016: R/V/D=16.7/13.0/17.8%, Sharpe=0.70 (with EWC=AGG=7.6%), MAR= 0.93, and RAD=13.0%, with TTC=0.26% and . With Antonacci's original SPY, ACWX and AGG from iShare, we found R/V/D=16.0/12.9/19.4%, Sharpe=0.65 (with , , and , with and . When we used SPY, EFA, EEM, AGG for our VAA-G4 the FS results were R/V/D=18.8/13.1/16.4%, Sharpe=0.90 (with ), , and , with and . So, we conclude that both strategies (Antonacci's GEM and our VAA-G4) were dependent on the particular ETFs (iShares or Vanguard) chosen (thanks Walter). Notice that we also calibrated our proxy ETFs from 1969 using Vanguard or iShares ETFs, respectively; see also note 11 .
    我们有“官方”GEM 模型(见 Antonacci,2014 年,以及 TrendXplorer,2016 年),但使用我们的 KK 数据(包括 ETF 费用和 交易成本),以及 Vanguard 而不是 iShares ETF(SPY、VEU 和 BND),在 1970 年 12 月至 2016 年 12 月期间的以下表现统计:R/V/D=16.7/13.0/17.8%,Sharpe=0.70(EWC=AGG=7.6%),MAR=0.93,RAD=13.0%,TTC=0.26%和 。使用 Antonacci 的原始 SPY、ACWX 和来自 iShare 的 AGG,我们发现 R/V/D=16.0/12.9/19.4%,Sharpe=0.65(与 ,以及 )。当我们使用 SPY、EFA、EEM、AGG 进行我们的 VAA-G4 时,FS 结果为 R/V/D=18.8/13.1/16.4%,Sharpe=0.90(与 )、 ,以及 。因此,我们得出结论,两种策略(Antonacci 的 GEM 和我们的 VAA-G4)都依赖于所选择的特定 ETF(iShares 或 Vanguard)(感谢 Walter)。请注意,我们还从 1969 年开始使用 Vanguard 或 iShares ETF 校准了我们的代理 ETF;另见注释 11。
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