Coin Toss Banking

Op-Ed Contributor
Bonus Babies
new_york_times:http://www.nytimes.com/2009/02/11/opinion/11wilmott.html

By PAUL WILMOTT
Published: February 10, 2009
London

“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.