“DON’T put all your eggs into one basket” is the layman’s expression encouraging diversification. This was put into mathematics in a financial context by Harry Markowitz in the 1950s, in his Modern Portfolio Theory. He showed how to construct a portfolio of financial assets so as to maximize expected return for any given level of risk. This idea has inspired much theory and practice of investing, at all levels.
Only last month my bank manager had me complete a questionnaire so he could figure out how much risk I was comfortable with, and then he gave me the computer-generated result that told me how much of my money should be in cash, how much in bonds and how much in stocks. So with your grandmother telling you what not to do with all your eggs, your bank manager giving similar advice, and Professor Markowitz, a Nobel laureate, backing this up with mathematics, you’d expect that traders in banks would have got the message. Yet as the following simple thought experiment shows, it is not quite so straightforward.
It is your first day in your first job out of business school. You are going to be a trader in an investment bank! You will be rich! You will retire by the age of 30 and spend the rest of your days doing charitable works (when not on your yacht, of course). You are shown to your desk and introduced to your fellow traders. You notice something very strange, that they are all making similar trades using similar financial instruments. That’s odd, you think, there doesn’t seem to be much diversification going on.
Never mind, you are going to put into practice everything you’ve learned in school, and that includes diversification, so your trades will be safely diversified from those of your colleagues. Now to see if that makes any sense, we’ll put some numbers to this, and imagine what could happen to your plans for buying that yacht. Does diversifying improve your chances of getting a big bonus?
Suppose that you have 100 colleagues, each trading with $10 million. Bearing in mind Einstein’s advice, we are going to keep things simple, so as to make the mathematics as transparent as possible, and assume that they are betting on a coin toss. And, crucially, they are all betting on heads on the same toss of the same unbiased coin — it doesn’t get more undiversified than that.
It’s 50-50 whether they win or lose. If the single toss comes up heads then they all win, and the bank makes 100 times $10 million, of which each trader perhaps gets a tidy $2 million bonus. That’s their down payment on a decent yacht. Everyone’s happy: traders, management, shareholders and depositors. But if it comes up tails, they lose, and the bank goes bust. But while the traders and management only have to find new jobs, the shareholders and the depositors potentially face losing their life savings.
You come along, and, thanks to your college education, you have found a much better trade than your colleagues. Let’s say that you are betting on another, independent coin — but one that is biased. This coin has a 75 percent chance of heads. And you’ve also got $10 million to invest.
Let’s look at two possibilities: first, that you do the responsible thing of betting the good odds on the biased coin, and second, that you bet on the heads on the 50-50 toss just like your colleagues. I say that the first case is “responsible” for two reasons: one because it’s a better bet than that of your colleagues and so will increase the bank’s expected return; and two because it also helps the bank diversify. That’s classic Modern Portfolio Theory, and is also common sense.
O.K., so you bet $10 million on your coin. What is the probability of your getting your $2 million bonus? Easy, it’s just the probability of getting heads, 75 percent, isn’t it? Well, no, it’s not. Yes, there’s a 75 percent chance of your making money for your bank, but if your colleagues have meanwhile tossed a tail, your bank is broke and no one’s getting a bonus, even you. They’ve cost the bank a billion dollars, and you’ve made it a mere $10 million. But what if you toss tails on the biased coin when the others toss heads? The others get their bonus, but you’ve just lost $10 million, what a terrible trader you must be, and are shown the door.
No, the only way to get that bonus is if both you and the others make winning trades — that is, if both coins land heads up. And the probability of that is 50 percent times 75 percent — that’s 37.5 percent. So, even though you have a biased coin working in your favor, the chance of you getting a bonus is still substantially less than half.
By now you can probably see where I’m going with this. Suppose that instead of betting on the biased coin you join in with all your colleagues and bet on the same toss of the first coin. Now you all win or lose together, the odds are even and the probability of getting your bonus is 50 percent. This is significantly higher than if you’d done the “responsible” thing of helping your bank to increase its expected return and decrease its risk.
This example makes it clear that your interests and those of the shareholders and depositors can be complete opposites. They probably didn’t teach you that at business school. And the plan Treasury Secretary Tim Geithner unveiled yesterday didn’t seem to take it much into account. But it’s a lesson we’re all learning to our detriment in the current economy.
Paul Wilmott is the founder of Wilmott, a journal of quantitative finance.迄今为止，华尔街经纪们多年来是如何赚到钵满盘满的？著名英国学者、数量金融工程学家威尔莫特（Paul Wilmott）道出了其中的秘密。
他在2月10日的《纽约时报》发表题为“红利宝宝”（Bonus Babies）的文章说，“不要把鸡蛋放在一个篮子里”，是凡夫俗子们对投资理财多样化的理解，也是1950年代诺贝尔奖获得者马科维茨（ Harry Markowitz）的“现代证券投资组合理论”之一。他的理论告诉大家如何通过证券投资组合，对金融资产进行风险管理，以便获得最大的期望回报。他的这一理念，成为后来许多投资理论和操作方法的灵感。
大家当然希望自己的银行投资经理也照此理念办事。然而事实却与此大相径庭。如果你是投资银行的经理人，从你离开商学院，从事投资工作的第一天开始，就会注意到某些非常奇怪的事情：大家都在按照类似的金融信条，进行类的交易，并没有什么多样化可言。
如果你照足商学院学到的那一套操作，你的交易可能因贯彻了多样化原则，比你的同事们的交易安全稳妥。但你想买游艇的计划可能就泡汤了。假设你有 100名同事，每人都有权处理1,000万美元的投资事宜。他们采用共同投掷1枚硬币的方法来投资，并且都以投中硬币的头像一面为全赢，反面为全输。于是大家输赢的机会是一半对一半。
如果掷中头像一面，那么大家全赢。银行赚进10亿美元，即1,000万美元的100倍。而每名经理人都获得200万美元的红利——大家买游艇的钱都到手了。于是不论经理人、投资管理人还是银行股东、银行存款户，人人皆大欢喜。
但如果掷中硬币的反面，上述人等全盘皆输，银行破产倒闭。不过经理人、投资管理人只是丢了饭碗，他们还可以另觅工作。而股东和存款人则可能失去毕生积蓄。
如果你是异数，认为从自己的商学院教育中，可以找到比那些同事更好的交易方式。那么假定你单独使用1枚硬币，而且这枚硬币有75%的机会投中头像一面！而你的投资额也是1,000万美元。
于是出现两种可能性：一种是你负责任地善用75%的机率，投中头像一面；第二种是按照其他同事的机率行事，只给自己一半的机会投中头像那一面。我说 “负责任地”使用第一种机会，是基于两条理由：一是它比其他同事的机会好得多，可以增加银行盼望的回报；二是也有助于银行投资多样化，既符合典型的“现代证券投资组合理论”，也符合常理。
但结果是什么？你可能得到200万美元的红利吗？当然可能，因为你的硬币投中头像的几率是75%！然而你错了。你只想到自己有75%的机会投中头像，却没有想想其他同事的那枚硬币，如果同时投中了反面，那么你的银行将破产，包括你在内无人能获得红利。
很简单，因为你为银行赚到了1,000万美元，但那100名同事则使银行损失了10亿美元。但如果你的硬币投中了反面，而其他同事那一枚则投中了头像，大家都得到200万美元红利，只有你一人分文不获。
所以，惟一能使你和其他100名同事双赢的方法就是，你的硬币和他们的硬币同时都投中头像——但对你来说，出现那种情况的机率是75%的50%，即37.5%。所以即使是你和大家双赢，你获得红利的机会还不到50%，比其他100人都要低。
现在你可能慢慢明白了，你如果“负责任地”办事，想为你的银行增加投资回报，减少投资风险，最后你将一无所获。但如果你放弃自己的硬币，和大家一起投掷同一枚硬币，输赢共进退，你获得红利的机会将显著提高到50%。
这个例子清楚地揭示出：华尔街金融经纪人的利益，和银行股东、存款人的利益是完全对立的。但这些秘密是不会告诉商学院学生的。奥巴马政府财政部长盖特纳日前的挽救经济计划，似乎不想对此着墨太多。但我们每一个人都应当从中吸取教训。
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