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

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%。

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。

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。

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 当 （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 ）：

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。

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 和 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 和 ，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 和 与 ，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 和 与 ，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 或 ，得到 或 ，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 .
当 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 或 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 和 ，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 和 的 CF。

Fig. 3. for VAA, PAA, Dual ( , for and )
图 3. 用于 VAA、PAA、Dual ( ，用于 和 )

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。

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 为 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（ 和 ）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 和 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？

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