OF COUPLES AND COPULAS

李祥林与金融海啸 （上）
作者：英国《金融时报》萨姆•琼斯(Sam Jones)
约翰尼•凯什(Joh nny Cash)和琼•卡特(June Carter)在大奥普里(Grand Ole Opry)现场音乐秀的后台初次见面。他们的相识有点像乡村歌曲中的男女主角：他结过婚，她刚刚离婚，于是一段感情开始了；两位歌手都有小孩，前妻因他酗酒狂欢于1966年离开他之前，他有3个孩子。两年之后，他在舞台上向卡特求婚，尽管遭到卡特多次拒绝，但最终她还是答应了。他们各自又找到了自己的人生伴侣。
他们的人生结局也像是一首乡村歌曲。2003年，卡特心脏手术出现并发症，不幸逝世于纳什维尔，4个月后，凯什也与世长辞，和卡特在天堂相会。似乎，妻子因心脏并发症去世让他痛不欲生：“太痛苦了，”在最后一次音乐会上，他对观众这样说道。他一边为他的吉他调音，一边几乎流着泪表示，这种痛苦“让人难以接受，是最不能让人接受的。”
在刚刚丧失亲人的人群中，凯什只不过是一个突出的例子。约翰尼和琼相遇前，科学家们就已经注意到，这种“配偶相继去世的案例”其实并不是个案。到上世纪80年代，医学研究人员就开始著文提到“应激性心肌病”(stress cardiomyopathy)，又称“心尖球形综合征”(apical ballooning syndrome)，这个难懂的专业术语用来描述这样一种特殊情形：经历了一次非常严重的精神创伤之后，人体的大脑会神秘地向血液中释放出一些化学物质，而这些物质会削弱人体心脏机能——在某些情况下，会导致心脏完全停止工作。
医学界对此很有兴趣，因为研究人员可能有机会干涉并延长人的生命。另一个行业对这个现象也很感兴趣——但更多地是去理解它，而不是去阻止它。这些人就是寿险精算师。保险精算学是一个围绕生与死的统计学科——而心碎现象的统计也变得非常引人注目。一页又一页的死亡报告显示了相同的显著趋势：夫妻中，一方的去世会大大增加另一方死亡的几率。悲伤致死——从最一般意义上讲，不一定是死于应激性心肌病——并不是一种罕见现象，而多少是个统计概率事件。因此，寿险商为了经营期业务，需要把它融入统计模型中。在2008年3月做的一份研究中，卡斯商学院(Cass Business School)的亚普•施普鲁(Jaap Spreeuw)和王旭(Xu Wang，译音)发现，在爱人死后的一年里，女性可能死亡的概率是平常的2倍多，而男性的死亡概率则是6倍多。“这意味着……联合生存年金[继续支付相同金额，直至被保险人双双死亡]定价太低，而最后生存者年金[一方去世后，向另一方支付的年金增加]则定价太高，”这份研究的作者总结道。
然而，在卡斯商学院的权威研究之前，保险精算师就已开始把悲伤致死的趋势融入他们用来计算客户死亡概率的模型中。怎样才能可靠地抓住这种转瞬即逝的关系呢？当然，保险精算师们靠的是概率。虽然他们无法为一对具体的夫妇设计出一种能够预测心碎综合症致死的概率模型，但他们可以利用统计学，以一群人为样本，设计出一副相对精确的图景。
他们借用物理学，并以马尔可夫链(Markov chain)模型为基础设计出一个方程：一种表达一系列统计事件的方式，这些事件的结果互相独立。在物理学上，马尔可夫链过程基于我们对周围世界最基本的认识，从液体汽化的方式到一滴墨水在一杯水中逐渐扩散的方式。精算师们解释道，如果你把人视为原子，那么你就可以在人身上运用相同的原子数学了。
…
1987年秋天，一位男子从中国乘班机来到加拿大。这位当时有可能成为全球最具影响力精算师的男子叫李祥林(Xiang Lin Li)。在此之前，他和同行的许多年轻学者——全部来自南开大学——都未出过国门，但他们应中国和加拿大政府的邀请，来到加拿大做一些超乎寻常的事：研究资本主义。这一小群数学学者和统计学学者将攻读魁北克拉瓦尔大学(Quebec's Laval University)的商学学位。
对李祥林来说，去加拿大是他当时许多不可能的机会中最近的一个，而正是这些机会造就了他后来的人生。几十年前，在毛泽东的文化大革命行将结束之际，他的家庭受到了迫害：他的父亲——一位中层警官，恰好是一位低职位官僚，成了当时红卫兵再教育的对象，他们一家被赶往中国南部的一个小村庄。在乡下，年轻的李祥林上学的机会非常渺茫，更不用说上大学了。但他很有才能，再加上他有动力，最终他成功地走进了学堂，而且还进了南开大学——中国最具声望的大学之一。李祥林在南开大学学习经济学，而就在他通过硕士学位考试后，加拿大发出了邀请。为了成为被送往魁北克的一分子，他在四个月内学会了法语——学习语言的劲道似乎能和他捣弄数字时的劲道相提并论。
在国外，李祥林的干劲并没有减弱。去加拿大四年之后，他获得了工商管理学硕士学位；到了这个地步，他没有回国的念头了。在他在外的几年时间里，中国的微型公开性时期结束。共产党总书记胡耀邦和一些民主改革份子相继被革职，而当时的中国领导人邓小平对被释放出来的自由主义风气持非常谨慎的态度。1989年，中国的一些学生在天安门广场被射杀，全球的目光都聚焦到了中国。对于那些有志向的年轻学生来说——尤其是对刚在西方获得工商管理学硕士学位的学生来说，南开这样的大学并不是最安全的地方。
为了表明自己不再回国，他改名为大卫•李(David Li)。从拉瓦尔大学毕业后，他进入多伦多附近的滑铁卢大学学习保险精算学。而这并不是唯一的变化：从身处讲法语的蒙特利尔优雅社会到进入一个更世俗、更加以商业为中心的多伦多社会，这一变化意义深远，而且经过深思熟虑。据和他一同从中国移民出来的同伴、也是拉瓦尔大学的同学戴杰(Jie Dai，译音)称，“我清晰地记得[李]曾说过，如果你成了一位精算师，你会赚到很多钱。”
…
上世纪90年代，要赚大钱，当然不会是在滑铁卢大学，而是在硅谷、华尔街和伦敦金融城。对数学人才来说，硅谷是首选，但华尔街和金融城也在逐渐吸引李祥林这样的人才。1984年，罗伯特•鲁宾(Robert Rubin)——10年后他成了美国财长——为他当时的雇主高盛(Goldman Sachs)投行做出了一个大胆决定。鲁宾决定雇佣当时麻省理工大学斯隆管理学院的学者、经济学家费希尔•布莱克(Fischer Black)。1983年之前，有一些学者已开始研究经济学和市场学，但都是出于学术好奇心；布莱克个性独特——是一个严谨的学者，有成功的著述经验，并还有一个终身职位——他去了华尔街，并冒着象牙塔同事嘲笑的风险，将理论付诸实践。
鲁宾的豪赌为高盛带来的收入是布莱克薪水的好几倍。在高盛，这位教授开拓性地利用数学原理去追求利润。布莱克-斯科尔斯公式(Black-Scholes formula)的发明有一半是他的功劳，这个公式为市场风险制定一个合理的价格，给华尔街带来了革命性的变化。这个原则成为一个全新领域的奠基性教义，即定量金融学。定量金融学的倡导者试图用智慧来战胜市场，他们首先利用数学计算风险，利用数学排除这些风险。从事定量金融学的人数迅速增加。随着苏联的瓦解、军备竞赛的结束，以及1993年美国国会取消了超导超大型加速器（旨在成为当时世界上最大的物理实验仪器）的建造，粒子物理学家、量子力学专家，以及电脑工程师都无事可干了。对年轻一代的本科生和博士来说，把他们的知识应用到金融学，显然是防止他们领域工作流失很好的另一个选择。
粒子物理学家伊曼纽尔•德曼(Emanuel Derman)就做了这样的转变。他于1985年加入高盛集团，在在布莱克手下工作，最终接任导师的职位。他回忆道，当时大量涌入的“定量金融家”被称为是“POW”——即华尔街上的物理学家（POW原意为战俘）。另一位华尔街定量金融学家、现麻省理工大学讲师罗德城(Andrew Lo)引用的缩写词也相当精准。在最近一次讲座中，他说，现在华尔街真正需要的不是博士(PhD)，而是PSDs：“贫困、聪明，并且急切想致富”(poor, smart and with a deep desire to get rich)的人。在滑铁卢大学，李祥林非常适合这种描述。他在攻读保险精算博士学位，但没有人希望他今后走学术道路。于是，1997年获得博士学位后，他在加拿大最大的银行之一——加拿大帝国商业银行(CIBC)找了份工作。
对李祥林这样的毕业生来说，进入乱作一团的商界或许让人有些吃惊，即便拥有工商管理学硕士学位。像他这样的数学家至多也只不过是个定量金融家，多少被人看不起；如果他们幸运的话，交易员同事或许会拍拍他们后背，半拍马屁地说上一句“火箭科学家”。伊曼纽尔•德曼记得，有一次在高盛，他和另一个量子物理学家站在交易所内中央过道的两边，一位资深交易员从他们之间走过。这位交易员“赶紧跑开，极度痛苦地用双手抓着头，惊呼道：‘啊……！好强的力场！太强了！让我离开这里！'”
可是，李祥林1998年到纽约时，定量金融家统治了华尔街。那年夏天，长期资本管理公司LTCM)——由定量金融学中最好的专家管理的对冲基金——需要从联邦政府处获得大规模救助。但长期资本管理公司非但没有警示人们——数学模型可能会让投资者陷入严重的困境，它还把定量金融视为一个另类的后勤支持任务并将这一观点发扬光大了。这家基金公司倒闭前的巨大威力，以及倒闭有可能会在金融体系留下一个万亿美元的大洞这一事实，使人们对一个观念产生了怀疑，即交易员的直觉和经验要比数字情报重要得多。
萨姆•琼斯(Sam Jones)是《金融时报》驻阿尔法城记者。
译者/红岭
（待续）

李祥林与金融海啸（下）
作者：英国《金融时报》萨姆•琼斯(Sam Jones)
然而，定量金融家事实上并没有进入交易大厅。他们中的杰出人才仍旧在写论文，研究数字，把他们的理论知识运用到商业领域。李祥林来到纽约，在咨询公司RiskMetrics集团工作。该集团是从JP摩根(JP Morgan)独立出来的，但他仍然还是在考虑生、死和爱。2000年，他在著名的《固定收入期刊》(Journal of Fixed Income)上发表了一篇论文，引发了人们的强烈关注。在报告中，李祥林玩了一个非常优雅的把戏。借助于他在精算学和保险学以及对心碎症状的知识，他试图解决华尔街定量金融家最棘手的问题：违约相关性。
市场的功能和孤立的实验室不同，市场是相互联系、相互关联的。对于定量金融家来说，光试着去了解自己银行投资组合中的各个公司的破产概率还不够；他还要知道一家或多家公司的破产是如何增加（或减少）其他公司违约的可能性。例如，假设一家银行向两个企业——分别是乳牛场和乳品厂——提供贷款。根据评级机构，乳牛场破产的可能性为10%，而乳品厂破产的可能性为5%。但如果乳牛场真的破产了，而且这家乳牛场又是乳品厂的主要奶品供应商，那么乳品厂跟着乳牛场破产的可能性将迅速上升至5%以上。
事情就这样变得更加复杂了。爱尔兰乳牛场发行的债券和马来西亚软件公司发行的债券，它们之间的违约概率有何关系？或许你会这样认为，一点关系都没有：这些企业不仅提供的产品和服务完全不同，而且地理位置相距遥远。然而，假设两家公司都在从同一家陷入困境的银行贷款，而这家银行正要求收回贷款，那么情况又会怎样？
事实上，长期资本管理公司就是这样倒闭的。俄罗斯政府债券和墨西哥政府债券之间的相关程度如何呢？根据长期资本管理公司的模型（但应当指出的是，该模型使用的数据追溯到了100年前）显示，一点也不相关。但结果是，俄罗斯和墨西哥两个市场的主要投资者就是相同的那么几个人。1998年俄罗斯金融危机时，叶利钦政府债券违约，导致投资者因急于想降低其投资组合风险而恐慌性抛售墨西哥债券。
李祥林意识到，他的见解具有开创性。7年之后，他在接受《华尔街日报》采访时说：“突然，我觉得我[作为一个精算师]试图解决的问题就是那些人正在试图解决的。[贷款]拖欠就像是公司死亡一样。”而如果他能把痛苦致死的数学理论应用于破产公司中，那么他就有办法建立一个数学模型，用来评估一家公司的违约对其它公司出现违约可能性的影响。
…
当数学家和物理学家想描述事件发生的可能性时，他们通常会依靠一条叫联结(copula)的曲线。Copula是拉丁文中的一个名词词根，意思是一种“联系或关联”，当然，联结可以和许多变量相连，从中你可以看到它们之间的相互依赖性。在滑铁卢大学攻读博士学位及在加拿大帝国商业银行工作期间，李祥林的研究兴趣在于：如何利用联结曲线使当时的心碎综合症保险精算模型得以发展。依赖马尔可夫链的问题是，他们制作的人类寿命图景太过机械化、物理化，甚至是原子化了。李祥林推论道，利用能够更加合理地显示结果分布的联结曲线，可以设计出一张更精确、更综合的心碎综合症，或者说问题公司的图景。
他决定利用一条非常标准的曲线——高斯联结(Gaussian copula)曲线，更常见的说法是“钟形曲线”，或“正态分布曲线”——以绘制并决定任何给定资产组合的相关性。保险精算师能够在只知道凯什最近开始守寡，而不知道其它任何消息的情况下，告诉他们的雇主在琼•卡特去世后约翰尼•凯什去世的概率，同理，定量金融学家可以不用知道有关公司的任何消息，就能告诉他们的雇主一家公司违约有可能对另一家公司所产生的影响。从这个观点来看，这真的可能，或者是将成为一场数字计算游戏。
到2003年，李祥林的论文使他在华尔街一举成名。到现在为止，他担任过花旗集团(Citigroup)衍生品研究部总监和全球负责人,在11月一个阳光明媚的周二上午，他在年度定量金融大会(Quant Congress)上做了一个和他的研究相关的报告，如沐春风。在数百名定量金融同行面前（当时在场的一位回忆道：“这简直是一场科幻小说大会。”），他详细介绍了自己的模型——高斯联结违约函数。
报告中参杂着方程式、数学引理、拱形曲线和一系列矩阵。之后的提问对他充满敬意，非常专业。李祥林似乎发现了风险管理拼图的最后一块，而自定量金融人才引入华尔街以来，各大银行一直在努力把这块拼图的各部分拼接起来。
…
到2001年，相关性成了大事。一股新的热情在令控华尔街涌动——与导致上世纪80年代早期股票期货和衍生品爆发的布莱克-斯科尔斯(Black-Scholes)模型一样具有创新性。这就是结构化金融，使华尔街20年的定量金融发展达到高潮。基本原理很简单：银行不必再承担风险了。相反，银行可以使用复杂的数学原理并制定模型对风险进行定价，然后将这些风险打包，像交易其它普通有价证券一样进行交易。
抵押贷款就是个最好的例子。银行没有选择借出抵押贷款，然后在贷款期限内逐渐收取利息，相反，它们开始把这些贷款捆绑在一起，销售给那些专门设立的表外空壳公司。这些公司转而发行债券募集资金。而通过使用由定量金融专家制定的模型和数学原理，银行就可以调整抵押贷款组合的结构，确保针对投资者发行不同风险的债券。然而，问题就是相关性。任何表外证券化都无法正常把握的一样东西，就是他们拥有的上万种不同抵押贷款之间的相互关系。因此，上世界90年代期间，结构化金融始终是高度定制化的利基业务。
然而，2004年8月10日，评级机构穆迪(Moody's)把李祥林的高斯联结违约函数方程运用到自己的担保债务凭证(CDO)的评级方法中。CDO是一种结构性金融产品，最终证明是许多银行的噩梦。之前，穆迪公司主张CDO必须满足一个多样性分值——也就是说，每个CDO都应该包含不同种类的资产，比如商业抵押贷款、学生贷款和信用卡债务，还有很受欢迎的次级债。这确实是一个标准的投资妙计，它避免了把所有鸡蛋都放在一个篮子里，从而规避了风险。但李祥林的公式意味着，穆迪公司现在有一个能使公司判断风险相互关联性的模型——而上述提到的妙计或许可以扔出窗外了，因为风险可以用数学确定性来进行衡量。如果你知道你的篮子摔下的确切几率，你就没有必要用不同的篮子来装鸡蛋了。穆迪公司改变方法几周之后，世界另一个大型评估机构标准普尔(Standard & Poor's)也改变了自己的方法。
单单由次级抵押贷款组成的CDO风行一时。使用奇特的高斯联结相关性模型，以及一些聪明的表外架构，高风险抵押贷款被重新打包成具有3A评级的黄金投资产品。CDO市场迅速增长。2000年，CDO发行总量达数百亿美元。到2007年，发行的CDO债券总价值达2万亿美元。而随着越来越多的投资者希望将自己的资金投资于债券，使得借钱成本变得异常低，从而引发了房价大幅上涨，给世界各国经济注入了强劲动力。
…
现在我们已经知道，美国次级债住房市场开始出问题。2006年末，贷款违约率开始上升。银行起初并不担心。它们的模式假设，美国各地的小规模违约现象互不相关。但违约现象一直在出现。到2007年初，美国次级债市场明显出现了问题，到夏天，全美房产业主
开始拖欠抵押贷款。金融革命带来的便宜债务成本如此低，当初根本就不应该提供这种贷款。而相关性模型依旧像在上世纪90年代那样描述房产市场，并没有预测到它最终成为了“急剧膨胀的怪物”。具有讽刺意味的是，模型的发展改变了它自己建模的现实的本性。
银行开始承受持有CDO带来的损失，数目令人难以置信。随着各大机构对彼此的偿债能力变得担心起来，于是停止互相借贷。全球流动资金枯竭。问题从一个资产类别传染至另一个资产类别，银行的痛苦蔓延至整个实体经济。突然，每一件事都变得高度相关起来。
李祥林的方程为何没能预测到这个情况的发生呢？问题是，这个方程假设：事件是围绕着平均值——“正常”状态紧密联系在一起的。在保险精算学中，他的方程能充分抓住双重结果，如生或死，但在混乱的抵押贷款和经济学世界里，他的方程不再有效。在这里，可能出现的结果范围比面对保险公司客户时所要给出的那些结果范围来得更加复杂，无疑也更具一定的随机性。市场——尤其是抵押贷款市场——和那些保险公司相比，更倾向于极端相关性情况。心碎综合症死亡会引发富有诗意的联想，但预测心碎综合症死亡比起更加乏味的市场相关性预测来要容易得多，因为后者永远是那么不可知。
为什么没有人注意到这个方程的弱点呢？有些人注意到了。畅销书《黑天鹅》(The Black Swan)讲述了采用联结模型时考虑不相关事件的重要性，该书作者纳西姆•尼古拉•塔勒布(Nassim Nicholas Taleb)痛快淋漓地批评了定量金融学和李祥林的公式。“这东西就从来没有灵验过，”他说。“任何依赖相关性的行为都是江湖骗子做法。”
2007年，大卫•李（即李祥林）离开华尔街，回到中国。本文没有采访到他。但两年前，即在金融体系崩溃前，他做出过警告：“很少有人理解模型的本质。”统计学和精算学教授、李祥林在滑铁卢大学的导师哈里•潘尼尔(Harry Panjer)公平地看待了塔勒布的指控和李祥林的观点。今年早些时候，潘尼尔告诉《多伦多星报》(The Toronto Star)说：“我们统计学界有个说法，‘所有模型都是错的，但有些是有用的。'”而大卫•李的模型在一段时期内非常有用。

•琼斯(Sam Jones)是《金融时报》驻阿尔法城记者。
译者红岭

萨姆
未经英国《金融时报》书面许可，对于英国《金融时报》拥有版权和/或其他知识产权的任何内容，任何人不得复制、转载、摘编或在非FT中文网（或：英国《金融时报》中文网）所属的服务器上做镜像或以其他任何方式进行使用。已经英国《金融时报》授权使用作品的，应在授权范围内使用。


OF COUPLES AND COPULAS
By Sam Jones
Johnny Cash and June Carter met backstage at the Grand Ole Opry. It was a little like a country song: he was married, she recently divorced, and an affair ensued; both singers had young children, and Cash would have three more with his first wife before she left him in 1966, citing his drinking and carousing. Two years later, he proposed to Carter on stage and, despite having turned him down numerous times before, she accepted. They'd each found a life match.
It ended like a country song, too. In 2003, Carter died in Nashville of complications from heart surgery, and Cash followed her to the grave four months later. The heart complication for him, it seemed, was that it was broken: “It hurts so bad,” he told the audience at the last concert he would give. The pain, he said, tuning his guitar, close to tears, was “the big one. It's the biggest.”
Cash was speaking for many a bereaved partner – and well before Johnny ever met June, scientists had noticed that cases of spouses dying in rapid succession were not at all unusual. By the 1980s, medical researchers had started writing about “stress cardiomyopathy”, or “apical ballooning syndrome”, the ungainly name for the peculiar condition whereby an individual's brain, following an intense emotional trauma, would inexplicably release chemicals into the bloodstream that weakened the heart – in some cases, causing it quite literally to break.
The medical community was interested because it offered a chance, potentially, to intervene and prolong life. Another industry was interested in the phenomenon, too – but less to stop it and more to understand it. These were the actuaries working in life assurance. Actuarial science is the study of the statistics surrounding life and death – and the statistics surrounding the broken heart phenomenon were striking. Pages and pages of death records showed the same marked trend: that in human couples, the death of one partner significantly increases the chances of the death of the other. Dying of a broken heart – in the most general sense, not necessarily from stress cardiomyopathy – was not a rare occurrence, but something of a statistical probability. So much so that life assurers, in order to conduct their business, needed to incorporate it into their models. In a March 2008 study, Jaap Spreeuw and Xu Wang of the Cass Business School observed that in the year following a loved one's death, women were more than twice as likely to die than normal, and men more than six times as likely. “This implies … that joint life annuities [in which payments continue at the same price until both partners die] are underpriced while last survivor annuities [in which payments increase after one partner dies] are overpriced,” concluded the authors.
Even before the definitive Cass study, however, actuaries had begun to incorporate the broken heart trend into their mathematical models calculating the chances of clients dying. How could such an ephemeral relationship be reliably captured? The actuaries, of course, relied on probability. While they could not hope to devise a model that predicted the likelihood of death from a broken heart for a specific couple, they could use statistical science to devise a fairly accurate picture across a group of people.
They borrowed from physics and devised a formula based on something called a Markov chain: a way of expressing a series of statistical events whose outcomes are dependent upon one another. In physics, Markov chain processes underlie our most basic understanding of the world around us, from the way liquid turns to gas to the way a drop of vivid ink might slowly diffuse through a glass of water. If you treated people like atoms, the actuaries reasoned, you could apply the same maths.
. . .
In the autumn of 1987, the man who would become the world's most influential actuary landed in Canada on a flight from China. Neither Xiang Lin Li nor the handful of fellow junior academics with whom he was travelling – all from the University of Nankai – had ever been abroad before, yet they had come at the behest of the Chinese and Canadian governments to do something most unusual: study capitalism. The small band of mathematicians and statisticians would be taking business degrees at Quebec's Laval University.
For Li, going to Canada was just the latest in a series of unlikely opportunities that had shaped his life up to then. Decades earlier, his family had suffered at the blunt end of Mao's cultural revolution: his father, a mid-ranking police official, was precisely the kind of lowly bureaucrat that the red guard mob was intent on re-educating, and the family was uprooted to a small village in southern China. In the countryside, the chances of young Xiang Lin going to school – let alone university – were slim. But he was talented and driven, and made it not only into school, but on to Nankai, one of the country's most prestigious institutions. Li studied economics, and had just passed his master's examinations when the Canadians came calling. Determined to be among those sent to Quebec, he learnt French in four months – as much at home studying the language, it seemed, as he was crunching numbers.
Li's drive did not abate abroad. Four years after arriving in Canada, he'd earned his MBA; by this stage, he had no intention of returning home. In the few years he had been away, China's mini-glasnost period had withered. Hu Yaobang, general secretary of the Communist party and a pro-democratic reformist, had been ousted, and Chinese leader Deng Xiaoping was now wary of the liberalism genie that had been let out of the bottle. In 1989, the world had looked on as students were mowed down in Tiananmen Square. Universities such as Nankai were not exactly the safest places for ambitious young students – especially not ones returning from MBA courses in the west.
As if to make clear the break, Xiang Lin changed his name and became David Li. After graduation from Laval, he enrolled at a new university, Waterloo, near Toronto. He would now be studying actuarial science. And this wasn't the only change: the move from genteel, francophone Montreal to the more worldly and business-oriented Toronto was profound – and deliberate. According to Jie Dai, a fellow immigrant from China and a classmate at Laval, “I clearly remember [Li] mention that if you are an actuarial guy, you can earn a lot of money.”
. . .
The big money in the 1990s, of course, was not to be found at Waterloo but in Silicon Valley, on Wall Street and in the City of London. The first of these might have been an obvious destination for a talented mathematician, but the latter two were also becoming magnets for the likes of Li. In 1984, Robert Rubin, who a decade later would become US treasury secretary, made a bold decision for his employer at the time, the investment bank Goldman Sachs. Rubin decided to hire Fischer Black, an economist and academic at the Massachusetts Institute of Technology's Sloan School of Management. Prior to 1983, a few academics had toyed with economics and markets, but as intellectual curiosities; Black was the first of his kind – a serious academic, with publications under his belt and a tenured position to boot – to make the move to Wall Street, putting theory into practice and risking the scorn of his ivory tower colleagues.
Rubin's bet earned Goldman many multiples of Black's salary. At the bank, the professor pioneered the use of mathematics in pursuit of money. He was one half of the duo that came up with the Black-Scholes formula, which revolutionised Wall Street by promising to determine a rational price for market risks – a principle that would become the founding doctrine of a new field, quantitative finance. Quantitative finance's practitioners were trying to outwit the markets, using maths to eliminate risk by first using maths to calculate it. And the numbers of those practitioners grew quickly. With the collapse of the Soviet Union, the end of the arms race and, in 1993, the cancellation by the US congress of the superconducting super collider – intended to be the world's greatest physics experiment – particle physicists, experts in quantum mechanics and computing engineers were twiddling their thumbs. For the younger generation of newly qualified grads and PhDs, applying their expertise to finance was the obvious alternative to fighting it out for the dwindling number of jobs strictly in their fields.
Emanuel Derman – a particle physicist – was such a convert. He joined Goldman in 1985, working under Black before eventually taking over from his mentor, and recalls members of the “quant” influx being referred to as “POWs” – physicists on Wall Street. Equally accurate was the acronym used by Andrew Lo, another Wall Street quant and now a lecturer at MIT. What Wall Street was really after, said Lo in a recent lecture, was not PhDs, but PSDs: people who were “poor, smart and with a deep desire to get rich”. At Waterloo, Li fitted that description to a T. He was studying for his PhD in actuarial science, but no one expected him to go on to a career in academia. Instead, in 1997, after earning his doctorate, he took a job at one of Canada's largest banks, Canadian Imperial Bank of Commerce (CIBC).
For graduates such as Li, joining the rough and tumble world of business could be something of a shock, even when armed with MBAs. At best, the mathematicians were semi-disparagingly referred to as quants; if they were lucky, trader colleagues might slap them on the back and call them, half in flattery, “rocket scientists”. Emanuel Derman recalls a time at Goldman Sachs when he and another quant colleague were standing on the trading floor, on either side of a central aisle, and a senior trader passed between them. The trader “winced, clutching his head with both hands as though in excruciating pain, and exclaimed, ‘Aaarrggh-hhh! The force field! It's too intense! Let me out of the way!'”
And yet by the time Li got to New York, in 1998, the quants had taken over the asylum. In the summer of that year, Long Term Capital Management, a hedge fund run by the finest minds in quantitative finance, required a massive bailout from the federal government. But far from serving as a warning that mathematical models could get investors into serious trouble, LTCM exploded the notion of quantitative finance as a geeky, back-office support task. The fund's might before its fall – and the fact that its failure might have left a trillion-dollar hole in the financial system – discounted the notion that traders' instinct and experience counted more than numerical intelligence.
(To be continued)

The quants weren't exactly out on the trading floor, however. The best of them still spent their days writing papers, crunching numbers, applying their academic expertise to the world of business. Li had come to New York to work for a consultancy called the RiskMetrics Group, which had been spun out of JP Morgan, but he was still thinking about life, death and love. In 2000, he published a paper in the prestigious Journal of Fixed Income that gained some serious attention. In it, Li performed a most elegant trick. Borrowing from his work in actuarial science and insurance and his knowledge of the broken-heart syndrome, he attempted to solve one of Wall Street quants' most intractable problems: default correlation.
Markets do not function in laboratory-like isolation. They are linked, correlated. It isn't enough for any quant to try and know the probability of each individual company in his bank's portfolio going bust; he has to know how the bankruptcy of one company – or several – might increase (or decrease) the likelihood that other companies will default. Suppose, for example, that a bank loans money to two outfits – a dairy farm and a dairy. The farm, according to ratings agencies, has a 10 per cent chance of going bust and the dairy a 5 per cent chance. But if the farm does go under, the chances that the dairy will follow will rise above 5 per cent – quickly and steeply – if the farm was its main milk supplier.
And it gets more complicated from there. How correlated are the default probabilities on bonds issued by our Irish dairy farm and those issued by a software company in Malaysia? Not at all, you might think: the businesses not only provide totally different products and services, they're also geographically remote from each other. Suppose, though, that both companies have been lent money by the same troubled bank that is now calling in its loans.
In fact, this is exactly what sank LTCM. How correlated are Russian government bonds and those in Mexico? Not at all, according to LTCM's model, which, it should be noted, crunched data going back a hundred years. And yet it turned out for the hedge fund that both markets were dominated by the same few investors. The 1998 financial crisis in Russia, when Boris Yeltsin's government defaulted on its bonds, caused panic selling in Mexico as investors rushed to de-risk their portfolios.
Li realised that his insight was groundbreaking. Speaking to The Wall Street Journal seven years later, he said: “Suddenly I thought that the problem I was trying to solve [as an actuary] was exactly the problem these guys were trying to solve. Default [on a loan] is like the death of a company.” And if he could apply the broken hearts maths to broken companies, he'd have a way of mathematically modelling the effect that one company's default would have on the chance of default for others.
. . .
When mathematicians and physicists want to describe the chances of events occurring, they often rely on a curve called a copula. The Latin root is a noun meaning a “link or tie”, and indeed, copulas connect variables in such a way that their interdependence can be plotted. Throughout his PhD at Waterloo, and at CIBC, Li had been interested in how he could use copulas to develop existing actuarial models of the broken heart syndrome. The problem with relying on Markov chains was that they painted a far too mechanical, physical – atomic, even – picture of the human lifespan. Li reasoned that with a copula that showed a probable distribution of outcomes, a more accurate, encompassing picture of the broken heart or, for that matter, the broken company, could be devised.
He decided to use a very standard type of curve – the Gaussian copula, which is better known as a bell curve, or normal distribution – to map and determine the correlation on any given portfolio of assets. In the same way that actuaries could tell their employers the chances of Johnny Cash dying soon after June Carter without knowing anything about Cash other than the fact of his recent widowhood, so quants could tell their employers the effect one company defaulting might have on another doing the same – without knowing anything about the companies themselves. From this point on, it really could be, would be, a number-crunching game.
By 2003, Li's paper had made his name on Wall Street. By now he was director and global head of derivatives research at Citigroup, and on a bright Tuesday morning in November, he arrived at the annual Quant Congress to bask in the glory with a presentation about his work. In front of a room of hundreds of fellow quants (“not a million miles from some kind of science fiction convention”, one person who was there recalls) he ran through his model – the Gaussian copula function for default.
The presentation was a riot of equations, mathematical lemmas, arching curves and matrices of numbers. The questions afterwards were deferential, technical. Li, it seemed, had found the final piece of a risk-management jigsaw that banks had been slowly piecing together since quants arrived on Wall Street.
. . .
By 2001, correlation was a big deal. A new fervour was gripping Wall Street – one almost as revolutionary as that which had struck when the Black-Scholes model brought about the explosion in stock options and derivatives in the early 1980s. This was structured finance, the culmination of two decades of quants on Wall Street. The basic idea was simple: that banks no longer had to hold on to risks. Instead they could value them, using complex maths and modelling, then package and trade them like any other, ordinary security.
Mortgages were the prime example. Rather than make a mortgage loan and gradually collect interest over its lifespan, banks began to bundle the loans together and sell them into specially created off-balance-sheet shell companies. These companies in turn issued bonds to raise cash. And by using the modelling and maths being cranked out by quants, banks were able to tailor the structure of mortgage portfolios to ensure that bonds of varying risks could be issued to investors. The problem, however, was correlation. The one thing any off-balance-sheet securitisation could not properly capture was the interrelatedness of all the hundreds of thousands of different mortgage loans they owned. As a consequence, structured finance had remained a niche and highly bespoke practice throughout the 1990s.
On August 10 2004, however, the rating agency Moody's incorporated Li's Gaussian copula default function formula into its rating methodology for collateralised debt obligations, the structured finance instruments that subsequently proved the nemesis of so many banks. Previously, Moody's had insisted that CDOs meet a diversity score – that is, that each should contain different types of assets, such as commercial mortgages, student loans and credit card debts, as well as the popular subprime debt. This was standard investing good practice, where the best way to guard against risk is to avoid putting all your eggs in one basket. But Li's formula meant Moody's now had a model that enabled it to gauge the interrelatedness of risks – and that traditional good practice could be thrown out of the window, since risk could be measured with mathematical certainty. No need to spread your eggs across baskets if you knew the exact odds of your one basket being dropped. A week after Moody's, the world's other large rating agency, Standard & Poor's, changed its methodology, too.
CDOs built solely out of subprime mortgage debt became the rage. And using the magic of the Gaussian copula correlation model, and some clever off-balance-sheet architecture, high-risk mortgages were re-packaged into triple-A-rated investor gold. The CDO market exploded. In 2000, the total number of CDOs issued were worth somewhere in the tens of billions of dollars. By 2007, two trillion dollars of CDO bonds had been issued. And with so many investors looking to put their money in debt, that debt became incredibly cheap, fuelling a massive boom in house prices and turbo-charging the world's economies.
. . .
The unwinding started, as we all now know, in the US subprime housing market. Defaults started to increase in late 2006. The banks weren't worried at first. Their models assumed that the pinprick default points all over the US were not correlated. But the defaults kept coming. By early 2007, it was clear that the US subprime market had a problem and by that summer, homeowners all over the US were defaulting on their mortgages. The cheap debt made available by the finance revolution was so cheap, in fact, that the loans should never have been made. And the correlation model was still mapping the housing market as it had been in 1990s, not the grossly inflated monster it had become. The development of the model had, ironically, changed the nature of the reality it was modelling.
The losses the banks began to take against their holdings of CDOs were staggering. And as the institutions grew fearful about one another's solvency, they stopped lending to each other. Global liquidity dried up. The rot spread from asset class to asset class, and the banks' pain spread to the real economy. Suddenly, everything was highly correlated.
How had Li's formula failed to anticipate this? The problem was that it assumed events tended to cluster heavily around an average – a “normal” state. In actuarial science, Li's formula could adequately capture binary outcomes such as life or death, but in the messy world of mortgages and economics, it faltered. The range of possible outcomes here was more complicated, and indeed, random, than those facing an insurance company's clients. Markets – particularly the mortgage market – were far more prone to extreme correlation scenarios than were insurers. Death from a broken heart, for all its poetic associations, is far easier to predict than the more prosaic, but ultimately unknowable, interrelatedness of markets.
Why did no one notice the formula's Achilles heel? Some did. Nassim Nicholas Taleb, author of the bestselling The Black Swan – a book about the importance of considering outliers when looking at copulas – was a voluble critic of quantitative finance and Li's formula. “The thing never worked,” he says. “Anything that relies on correlation is charlatanism.”
In 2007, David Li left Wall Street and moved back to China. He could not be contacted for this story. But two years earlier, before the financial system blew up, he did warn: “Very few people understand the essence of the model.” Harry Panjer, a professor of statistics and actuarial science who was Li's mentor at Waterloo, strikes a balance between Taleb's accusations and the stance of Li. Earlier this year, Panjer told The Toronto Star newspaper: “We have a saying in statistics, ‘All models are wrong, but some are useful.'” And David Li's model was, for a period, profoundly useful.
Sam Jones is a reporter for FT Alphaville.
未经英国《金融时报》书面许可，对于英国《金融时报》拥有版权和/或其他知识产权的任何内容，任何人不得复制、转载、摘编或在非FT中文网（或：英国《金融时报》中文网）所属的服务器上做镜像或以其他任何方式进行使用。已经英国《金融时报》授权使用作品的，应在授权范围内使用。