【英语生活】风险能用数学模型吗

Amaranth, the hedge fund group destroyed by energy market losses, boasted of the sophistication of its risk models. One attempt to reproduce its risk metrics suggests that it lost more than half its capital from a nine sigma event – a probability so low that such events simply do not happen. It would be like being struck by lightening and attacked by a mad axeman, just at the moment you suffer a fatal heart attack.But similarly inconceivable contingencies do crop up. Long Term Capital Management was also the victim of a perfect storm. The one-day market fall in October 1987 was an event that would still have been unlikely even if the New York Stock Exchange had been open since the moment the earth was formed.

To understand the problem, it is better to look at much simpler models than the arcane ones needed to understand modern derivatives markets. Imagine yourself arriving at a bus stop, knowing the frequency of the buses – every 10 minutes – but not the exact arrival time. If buses keep exactly to schedule, the probability that one will arrive in the first minute is one in 10. If a bus does not arrive quickly, the likelihood of one appearing soon increases. After nine minutes, you can be certain that a bus will come in the next minute.

No one who has ever waited for a bus trusts this model. A better approach would make the frequency stochastic – buses arrive every 10 minutes on average, but with wide variability. This is the kind of model used in financial markets. It still predicts that the longer you have waited, the more likely it is a bus will arrive imminently.

No one who has waited for a bus or a friend or a pat on the back trusts that conclusion either. At first you have confidence in the model. The bus will adhere to its uncertain schedule. Both you and your friend have intended arrival times based on the agreed time of meeting. Your talents will eventually be recognised. But after an interval, you mistrust the original model. Perhaps there was an accident en route or a misunderstanding over the meeting place. Or perhaps your company does not value you as it should.

That is why your confidence in the rapid arrival of a bus rises at first but then falls. After a sufficiently long interval without a bus, no one remains at the bus stop. Ordinary people are too savvy for that. But there are always a few investors who go on asserting the rightness of their judgment in spite of the evidence before their eyes.

Any mathematical model of an uncertain environment must contemplate two types of risk. The first is incorporated within the structure of the model itself. If buses leave the garage at 10-minute intervals, what is the frequency distribution of their arrival time at a particular stop on the route? Such risks can be described using standard statistical distributions and historical data that reflect traffic conditions and the variable performance of drivers. Such techniques form the basis of the “value at risk” modelling employed in the financial community.

The second type of risk is uncertainty about whether the model you have developed describes the world accurately – either in the past, from which the data you employ are drawn, or in the future, in which the models you derive from them will be used. Such uncertainties necessarily exist and are unquantifiable.

A model of an imperfectly known environment must contemplate these two kinds of risk – one incorporated within the model and the other presented by failure of the model itself. People at the bus stop implicitly use this reasoning and more sophisticated commentators would do well to follow suit.

Attaching probabilities to forecasts is certainly better than clairvoyance. When someone does attach a probability to a forecast, they have – implicitly or explicitly – used a model of the problems. The model they have used accounts for in-model risk but ignores off-model risk. Their forecasts are therefore too confident and neither you nor they have much idea how over-confident they are. That is why mathematical modelling of risk can be an aid to sound judgment, but never a complete substitute

美国对冲基金集团Amaranth因能源市场巨亏而遭受毁灭性打击，但它曾对其风险模型的精准吹嘘不已。只要复制一下它的风险模型便可发现，它是在一个“9个标准差事件”（概率极低，以至于基本不会发生)中损失了一半多资本的。这就好比当你在致命的心脏病发作时，又被闪电劈到，还遭到了一个疯子用斧头攻击。然而，令人难以置信的类似偶然事件确实突然发生了。美国长期资本管理公司(LTCM)也是一场完美风暴的受害者。即使纽约证券交易所(NYSE)在地球形成时就已存在，1987年10月份那次持续一整天的市场下跌原本也不太可能出现。

为了弄清这个问题，让我们来看几个简单得多的模型，这比研究那些理解现代衍生产品市场所需的复杂模型更有帮助。设想你来到了一个公共汽车站，并且知道公共汽车的到站频率（每10分钟一班)，但不清楚准确的到站时间。如果公共汽车完全按预定时间行驶，那么一辆公共汽车在第一分钟到达的概率是10%。如果在短时间内没有车来，则一辆公共汽车很快出现的可能性就会上升。9分钟后，你便可以肯定下一分钟内会来一辆公共汽车。

所有等过公共汽车的人都不相信这个模型。一个更好的办法是把到站频率设为随机——平均每十分钟有一辆公共汽车到站，但到站时间有很大的可变性。金融市场上使用的就是这类模型。该模型的预测结论依然是：你等的时间越长，公共汽车马上到站的可能性就越大。

所有等过公共汽车、等过朋友或是等过一声赞扬的人也都不会相信这个结论。起初，你对这个模型颇有信心：公共汽车会按照不确定的时间表到来；你和你的朋友都根据约定的见面时间来计划到达时间；你的才能终将得到认可。但过了一段时间后，你对最初的模型产生了怀疑。也许公共汽车在途中发生了事故，或是双方对见面地点的理解有误，也可能是公司没有给予你应有的赏识。

这就是起初你对公共汽车很快会来的信心有所增强，后来却不断下降的原因所在。在隔了相当长的时间还没有来车之后，没人会留在车站。普通人都很清楚这个道理。然而，尽管证据就摆在眼前，却总还是有些投资者会坚持认为他们的判断是正确的。

任何针对不确定环境建立的数学模型都必须考虑到两类风险。第一类是包含在模型自身结构中的风险。如果公共汽车每隔10分钟离开车库，那么它们到达行车路线上某一特定车站的概率分布是怎样的呢？这种风险可以用标准概率分布和反映交通状况以及司机表现的历史数据来描述。这些技巧构成了金融界采用的“风险价值”(VaR)建模的基础。

第二类风险是一种不确定性：你所开发出的模型能否准确地反映出现实世界？过去是否曾准确（你使用的数据来自于过去)？将来（你从过去数据中推导出的模型要用于将来)能否做到？这种不确定性必定是存在的，而且是无法量化的。

一个针对不完全熟悉的环境建立的模型必须考虑到以下两种风险——一种是模型内部固有的风险，另一种是模型本身失败的风险。在公共汽车站等车的人无意中运用了这一推理方法，而考虑问题更为全面的评论员们也效仿了这一思路，并表现得非常出色。

利用概率进行预测当然要比凭空猜测好。当人们真的利用概率预测时，他们便已经——有意或无意地——使用了存在问题的模型。他们所使用的模型考虑了模型涵盖的风险，但忽略了模型未计入的风险。因此，他们的预测过于自信，无论是你，还是他们自己都不清楚他们过于自信到何种程度。这就是为什么对风险进行数学建模有助于做出合理决策，但绝不能完全替代决策的原因。



译者/朱冠华