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The Wall Street Math Hustle
金融业如何拥抱坏数学是BS。
华尔街擅长卖东西 - 但学术实业资金复合人对复杂数学的声誉是一个神话。
金融业以抵御任何其他科学领域的方式使用数学。学术文学和行业研究具有伪数学废话。您不必望而解见到动机所在的位置：许多作者都是由丰富的投资管理和咨询公司雇用或保留的。面对诸如算法供电的投资策略的投资者兴趣，习惯 - 确实是requirement— today is for firms to use scientific language and notation to nourish the idea that they’ve proved mathematically that there’s a way to systematically beat the market.
They haven’t.
It’s not surprising that more is claimed by their suggestive language than they actually “prove” because the field suffers from a subtle corruption. There is a pattern developing of publishing a semi-quantitative paper and using it as the basis to establish an investment advisory firm, become the seller of an investment product “relying only on math,” and go off on a globe-circling marketing bender. The fastest-growing asset management firms are purveyors of investment products that draw upon mathematical finance research, so theories ridden with poorly specified mathematics and wildly exaggerated results abound.
如果公式可以杀死
有一种说法是，数学没有谎言，但在2015年，着名诺贝尔奖获奖经济学家保罗·罗默发表了一篇标题为“数学”的论文，以“真实的”，由Satirist Stephen Colbert创造出来。数学不是真正的数学，你需要设计飞机或建造桥或土地的那种。真正的数学是精确的。（看电影Hidden Figures如果您还没有知道。）投资金融数学是销售数学。你不需要它并不重要，或者每个人都以自己的方式解释它。您正在使用它来留下深刻印象的客户。
多年来，阿尔法是投资的圣杯，几乎所有的钱经理都声称他们可以生产它。然后，在JUPITERS大小的证据表明，没有投资经理 - 不是共同基金，而不是对冲基金 - 可以可靠地生产阿尔法，投资经理停止奠定了直接要求。
现在，对于许多人来说，有一个新的圣杯：多样化。
不能作为一个小组表明他们可以获得退货，行业通过转移投资的焦点来改变受试者。虽然追求多样化永无止境，但目前尚不清楚符合多元化的承诺：风险高效，最大多样化，最低方差，相等风险贡献，反向波动率加权，多样性加权，无限制。
Bedazzled by such superficial complexity of the math, risk-based portfolio construction products are all the rage.
Don’t let them pull the wool over your eyes.
特别容易想出的策略that are more “diversified” than the market, beat the market in hindsight, and require no math. City University of London's Cass Business School created one of the more innovative weighting schemes when it introduced a portfolio based on the popular board game Scrabble using each company’s ticker symbol to calculate the Scrabble weight for each stock.
Xerox Corp. (XRX), for example, has nearly three times the weight of Apple (AAPL). Pulled from the largest 500 stocks in the University of Chicago's Center for Research in Security Prices (CRSP) database each December, it’s a better “diversified” portfolio and returned almost double the cap-weighted index.
如果所有股票具有相同的资本化，则会实现最大多样性 - 该称为相同的加权，这是多元化策略的最基本。它购买了一点一切，如果你思考它，那些有500种股票的拼写组合只是在平等加权组合的边缘上调。
What no one is telling you is that the historical performance and risk statistics of easier-to-understand, equal-weighted portfolios are very close to — and often dominate — these new, mathematically inspired portfolios. It is a supreme irony that almost anything that’s not of the capitalization kind will work — or not — as well as these new formulas.
误差
重新平衡在替代形式的股权索引方面发挥着巨大的争论。支持者声称重新平衡提供了一些返回奖金 - 利用波动性作为回报来源 - 但是你不能说，“重新平衡的溢价总是积极的，如同随机组合理论所证明的。”不可能区分重新平衡回报 - 这是对重新平衡的重新平衡行为的特定，可以通过重新平衡和不平衡的战略来获得。
而所谓的重新平衡证明是janky。一个声称有重新平衡的奖金，但它是基于计算的average annualizedreturn — a meaningless number — instead of年化平均值return. You can’t earn that number in real life; it’s a math error. Others rely on the provably false assumption that assets always mean-revert, when, in reality, all markets have portions that are trending and de-trending. Markets experienceboth平均逆转和动量;它们不是永久性的负面自相关的状态。在任何随机时间段，您可以获得势头或均值逆转。但他们的“证据”不可能认为，重新平衡的那一刻也是换气发生的那一刻。鬼使神差。
It’s obviously not true; it’s easy to find an unlimited number of cases in which rebalancing doesn’t outperform buy and hold. If a lot of stocks persistently display strong trending behavior (up or down), this will harm a rebalancing strategy, as you will be constantly trimming the winners and adding to the losers. When assets have unequal and skewed-to-the-right returns — as in real life — a buy-and-hold strategy will produce a higher expected return as assets with higher returns drift upward and become larger components of the portfolio. You don’t need to invoke unnecessarily complicated stochastic calculus to understand that.
This rebalancing issue may explain why most active managers underperform: According to Hendrik Bessembinder, a finance professor at Arizona State University, the entire gain in the U.S. stock market since 1926 is attributable to the best-performing 4 percent of listed companies, and the cap-weighted indexes captured all of it because they don’t rebalance.
炒作没有界限
The promotion behind formula-based approaches is shameless.
例如，最近的一篇文章标题为“Mathematician认为他有魔术公式生成财富：基于巴黎的投资组合经理创造了一个”反基准“策略，他说仍在保持增长的同时降低风险。”
Magic formula? Seriously?
当然，重要的是你在投资建议中定位所以故事的不同之处拉什蒙方式 - 但是当你看在引擎盖下时，这些多样化方法的许多方式都是如此相似，他们可能是另一个母亲的兄弟。
理论只是一个理论。这不是一个事实。根据定义，具有多样化的组件的投资组合将具有低于其内部股票的加权平均值的投资组合方差。但是挂在一分钟：在这种差异很大的程度上，投资者汇总了彼此内在不同的证券 - 但它并不一定意味着它更多的多样化比市场上限加权组合。在轮盘赌的同时投注红色和黑色可提供猛犸象的多样化率。这也是愚蠢的。
最大多样化有一个名称和故事，如此脂肪，就像大喊大叫“嘿kool-aid！”投资者撞到了墙壁。他们离开销售促销的是，在风险贡献方面，最大多样化组合总是相对于市场上加权组合更集中。
多样化通常意味着拥有许多不同的投资，但最大多样化和最小方差方法通常采用相对较小的可投资集合;他们只使用1000个股票中只有100个百分比并不少见 - 这一百分比减少较大的可投资资产集。我们来自哪里，那是集中风险，而这一整个过程开始因为我们担心这一指数太浓，只有几个资产施加了太多重量。新方法对浓度进行了偏见的争论，但随后在不同方向上集中了投资组合。
真相只是简单地，在实践中没有这样的事情作为最大多样化。Assuming it is possible to create a portfolio more diversified than one that includes every company, the apostles of maximum diversification would have you believe they capture more independent sources of risk, but they’re erroneously calling random stock jiggles “more bets” when they're really evanescent uncorrelated sources of randomness associated with no risk premium.
“最大多样化”是一个提取的呼号，但假设正在推广为真理而不是披露。这些天是流行的，声称战略不依赖预测;这些风险方法索赔是不可知的，没有偏见，没有预测或赌注的制造。但假设所有资产都具有相同的锐利比率在功能上相当于反向工程预期回报，因此他们实际上并不避免这种问题 - 他们只是使用代数作为解决它的后门方式。历史上，历史上，预计返回与测试版线性的回报是一个糟糕的赌注。
The advertising says diversification is the firms' only bet, so maximum diversification is commonly misunderstood to have even exposure to all risk drivers, but as Apollon Fragkiskos, director of research at Markov Processes International, notes, “They claim to be unbiased, but it’s a rather technical definition on their side that isn’t what many practitioners consider it to be because their portfolios have biases and sector concentrations. You cannot have it both ways; you either diversify against stocks or factors.”
最大多样化的赌注使得不完全符合多元化;这是市场是错误的。无论在市场测试版中反映出什么新的影响力，策略都会偏离它。在市场测试版的范围内反映了越来越多的影响 - 互联网商务，无人驾驶汽车，人工智能或加密货币 - 这种策略将从它倾斜。但市场并不总是永远的错误。让Amazon.com，字母表和Netflix复合 - 而不是购买Sears Holdings Corp.是因为它越来越无关 - 正是使索引难以击败的原因。
幻灯片规则品种的事实
The investment world is characterized by one overarching feature — covariance — but everything has been utterly miscalculated, right from the start.
最典型的“反基准”用于最大多样化的公式，正在寻找相对于剩余宇宙的最奇怪的回归历史的股票 - 这是一个目标，这些客观落入了比大多数市场更不相关的股票。
But since when is de-correlation equivalent to diversification?
The central idea behind all this is safety. Yet the risk-mitigating capability of anti-correlation is somewhat suspect. The looming spoilsport observation is this: De-correlation won’t protect investors because two assets can be negatively correlated and crash simultaneously. Even if all of your investments are完美negatively correlated, you should still be nervous as a cat:
低相关不会减少危机Beta;当下一个无拘无束的股票清算正在进行时，这将变得明确。那是你在冰箱里找到宝宝的时候！
自最大的公司基本上是该指数,成为不相关的转化为biasing away from the mega-cap stocks — but then so do all of the new strategies.
他们都设计持有less资本较大的股票。这是一个新的故事，但具有相同的小写和价值答案，我们已经听到了几十年。和普遍存在的：价值和小帽暴露是自然的特征除非与CAP加权基准一样，投资组合旨在与公司估值和体重之间具有积极的关系 - 而且没有人，但没有帽加权分度人员。最新的策略正在转向平等甚至inverseweighting. Since small stocks typically have larger market betas than large stocks, part of the size premium may simply be the equity market risk premium in disguise. If that’s the case, these new, risk-based approaches will work no better than a prayer for a plague to pass you by.
（最后评论中有一个极低的开玩笑。）
Old Wine in New Bottles
But still. If the end result of all these fancy ideas is to tilt toward small caps and value, why not just do that?
数学是一个可以相对容易地完成的烟幕。英语灯主任嘲笑这是多么明显。您不需要详细的模型或必须优化任何内容。
The most sellable products have hitched onto the intuitively plausible — but false — premise that cap-weighted indexes overweight overpriced stocks and underweight underpriced ones.
Naturally, there’s a “proof” that any non-cap weighted portfolio will beat the cap-weighted market — but this is logically impossible because for every non-cap weighted portfolio there’s a complementary non-cap weighted portfolio, together with which they are the market. They can’t both beat it.
它据说很容易击败指数：只需避免泡沫股票，所有你需要做的就是踢出 - 或体重减轻 - 指数中最大的股票。
对不起，但学校仍在这个问题上仍在会议上。
If it were that easy, active managers would be beating their indexes year after year because by now we’re sure they’ve gotten the memo about Rolf Banz’s 30-year-old paper “proving” that small-cap stocks beat large caps, which, by the way, isn’t true: He forgot to transform monthly holding-period returns to log returns before running his regression.
'So . . . Are We Alone?'
当我们为自己的正确偏见而自豪时，我们将与您分享正确的解释。
建立一个索引并跳过或超重最大的控股不是天才。确实，最昂贵的股票 - 像亚马逊和字母表 - 占该指数的最大比例，但高帽公司可以轻松低估，低帽子被高估。它不是一个统计财产，即市场上的加权让您过度高估的公司，并且曝光不足，因为最大的股票不低于索引。
这一切都取决于你如何铺设事实。根据Bessembinder的90年的研究，当购买和持有的股票投资组合按开发市场上限进行排序并持续十年来，大写组合回报率高（153％）比小帽的平均回报更高（153％）投资组合（97％）。
最大的股票也不表现出罗素1000年内的罗素。
All that conference rigamarole sounds good — but do you know how the song goes?
We don’t want to disturb the sure-thing delusion, but this might curb your enthusiasm: Whether you look at the number of constituents or compound growth of the group overall, the largest 200 stocks do not underperform small stocks — not since Russell/FTSE has been keeping track.
等等，但为什么？
有趣的是，相信最昂贵的股票有趣的是，同时我们知道数字启示录的五骑兵 - Facebook，Apple，Amazon，Netflix和字母表 - 在过去的五个中绝对粉碎它年。
2017年Dow中超过50％的集会降至仅为五股;超过80％的进步以十个名称为中心。有趣的是什么？像小丑？
It’s a highly successful act of hypnosis, but there are lateral truths.
这可能是最大的股票often卑鄙 - 恢复 - 但这并不意味着您可以将其转化为策略。Blockbuster是其日期的Netflix，如果历史是任何指南，大约50％的标准普尔500指数将在未来十年内更换。这也是如此mostfirms fail. Disappearance or decline was almost three times as likely as growth, and corporate mortality rates are rising. And guess what? Larger companies are less likely to fail than smaller companies; older companies are less likely to fail than younger companies. Despite the existence of a small-cap premium, just 37 percent of small stocks have holding-period returns that exceed those of the one-month Treasury bill — versus 69 percent of stocks in the largest decile.
一旦实施了一个获胜的赌注，就在其最大的投注中进一步集中了一个加权的基准，但这并不意味着它已经完成了村庄白痴，并在最糟糕的时间最大化其分配。
避免市场最极端的赌注听起来像一个聪明的想法，因为CAP加权致力于最多的货架空间，不仅是最大的股票，也是最昂贵的股票。但 - 在这里密切关注 - 昂贵与高估的东西也不相同。最近的研究表明，很少有公司 - 麦当劳，约翰逊和约翰逊，字母表等 - 对大多数CAP加权指数的收益负责。它们也是所有的大型昂贵的公司，不断变大。今天，丰富的越来越富裕：超过1％的全球GDP通过亚马逊/阿里巴巴生态系统。对抗流行股票和产品的交易必须意味着您不相信数字平台？我们的猜测是没有。但也许你是对的;这只是一个fad。喜欢电脑。 And the internet.
如果较大的公司是 - 而且他们无疑是 - 比较小公司更多样化？那么如果你想多样化，那么你不应该拥有更多的较大公司，而不是较小的公司吗？事故确实发生在20世纪30年代的彻头彻尾的恐怖损失，因此我们不会考虑驾驶规模较小，以成为一个很大的风险控制策略。
Avoiding bubbles is one kind of risk-control strategy; surviving macro-economic crashes is another.
刻录令人叙述和数学 - 但到目前为止，没有送货
We’ve disbelieved plenty in our long lives, but only because so many of the facts don’t tie out.
这些策略载有科学认证的印章，但华尔街的研究人员就像所有科学家一样，具有道德责任，使这些据说系统地证明模型的局限性传达给令人兴奋但不充分持怀疑态度的公众。
他们的大部分世界都是胡说八道。是的，也许我们有点枯萎了。但我们不希望他们将您销售更多的内容。
理查德威根works as a strategist at a large corporate pension fund. He authored this article independently of his employer.
Michael Edesess是一位数学家和经济学家。他目前是Edhec-Risk Institute的研究助理。