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The formula that changed the world
00:28:02
Rain and drought in numbers, the formula which changed Wall Street and then the world...
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This is more or less the statistical spine of the otherwise floppy media. The programme now airs year round on the BBC World Service, but this is a full length episode from Radio 4. Hello and welcome to More or Less, the programme that's the scourge of bad statistics, sloppy mathematics and pandas. This week we'll be telling the story of an equation that changed the world of finance forever. There will be more about nutrition and height. Forget about north and South Korea, what about Conservative and Labour MPs? But first, let's get a brief assessment of recent weather conditions from Carol Kirkwood in the BBC Weather Centre. It was a hard rain, a perpetual rain, a sweating and steaming rain. It was a mizzle, a downpour, a fountain, a whipping at the eyes and under, two at the ankles. It was a rain to drown all rains and the memory of rains. It shrank men's hands into the hands of wrinkled apes. It rained a solid, glassy rain and it never stopped. Carol Kirkwood, with a little help from the pen of Ray Bradbury, as the rain pours, keeping many more or less listeners inside. It seems some of you have taken the opportunity to write in and ask a number of rain related questions, like this one from loyal listener Jeff Wolff. I saw a factoid which claimed that domestic water use was only 8% of total water consumption, implying that hose pipe use is then only a fraction of that. Wesley Stevenson's here. What do you make of all this, Wes?
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Geoff, I don't know where you got that 8% figure from, but it's way out. Most of the water abstracted by the water abstracted? Yeah, it's kind of a word that water types use for getting water out of the ground. Most of the water abstracted by the water companies is used by domestic users and that amounts to almost half of all the water abstraction in the country.
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So not 8%. Do we know how much water is used in hose pipes?
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We don't know. And if the water companies do know, they're not telling us about it. But what we do know is roughly how much previous bans have saved. Jacob Tomkins is managing director of a charity called Waterwise, which is trying to get people to reduce the amount of water they use. And they made estimates last time around,
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the savings were between 5 and 10%. But the interesting thing was that it spread out effectively. The highest savings were in the middle of the area where there was a hose pipe ban. But then you got savings in areas like Severn Trent in the Midlands where there wasn't a hose pipe ban. You still Got savings of around 5%. There are several reasons for that. Obviously, hose pipe bans are quite high profile. They're covered in the media and what that means is you get national coverage and people hear there's a hose pipe ban, they're not exactly sure whether it's a hose pipe ban in their area. For instance, this time round, there's a hose pipe ban in Anglian regions area, but there isn't a hose pipe ban in Cambridge Waters area. And Cambridge Water is wholly surrounded by Anglian water. So all of the media that they're getting is, oh, there's a host pipe ban, but actually there's a little enclave there without one. So it's highly likely that Cambridge Water will also see a reduction in consumption in their area, despite not having a hose pipe ban.
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A
So last time around, we saved 5 to 10%. But 5 to 10% of what?
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B
Well, the industry estimate is that Everybody uses about 150 litres of water a day on average. So if we take a midpoint estimate and say that Everybody saves around 10 litres, there are 20 million people affected by the hosepipe ban. So that's a ballpark figure of 200 million litres of water a day, give or take.
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At 200 million litres doesn't seem a lot if you compare it to the 16,000 million litres of water abstracted by the water companies in England and Wales, or the 1200 million lost in leaks in those areas with bans. And I also have to make an obvious point. It's raining quite a lot right now. Yeah. What kind of person actually uses a hose pipe at the moment?
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B
Well, yes, no, that's a very good point. And it's one that Jacob Tompkins made to me and he says there's another big issue too.
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C
The other changes are we've got an increase in the level of metering and that reduces consumption by about 10%. But it also changes people's attitudes towards the way they can use water, so they're effectively paying for it. So there is more of a thought that, well, hang on a minute, I'm paying for this, therefore I can use as much as I like. So it'll be interesting to see whether people save as much water this time around as they did last time.
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So it's going to take more than a hose pipe ban to get us out of this. How long has the rainy weather got to go on before we're no longer in a drought?
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B
Well, unfortunately, this is the wrong type of rain. It's come at the wrong time of year. Jamie Hannaford is from the Centre for Ecology and Hydrology, and he says you need rain in the winter.
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D
It's in the winter season that the rainfall that falls can actually make its way down, soak down through the soil and go into our groundwater stores. Whereas in the summer season you've got much more evaporation. In the summer, plants are transpiring and so a lot of the rain that falls is actually lost and doesn't soak down to replenish those stores. This rainfall we've got at the moment in April is very useful, but doesn' change the overall water resources situation because any rain that falls from now on, really, particularly in the summer, is less likely to be useful.
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So we're getting drenched to the bone, thinking that at least this is doing something to ease the water shortage and it isn't?
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B
Well, it might be doing a little bit, but on the whole, no.
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Do we know how much rain we might need? Exactly.
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B
As Jamie Hannaford said, we need an exceptional amount of rainfall, but it's actually really hard to measure. Now, one man who's had a go at what is a ballpark figure, and this is a figure for those Dr. Drought areas, is John Rodden. He's a former Director of Hydrology and Water Resources with the World Meteorological Organization. So he sent me his assessment. He says in the 12 months to March, we'd recorded about 40% less rainfall than average, which means we're about 120 millimeters short. Now, usually over the six months from April to September, we actually fall further behind. John Rodder expects us to be about 30320 millimeters short by the end of September. So we need an extra 320 millimeters of rain this summer, which is about a foot, and that's roughly double what we'd normally expect. So I think it's probably safe to say that we'd notice if we got it. But he does add a caveat, which is that the rate of evaporation varies with vegetation, soil type and other factors. And in dry conditions, this can have a big effect. So measuring how much is really lost this way is very difficult. So what ultimately he's saying is you can't tell.
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Great. Well, thank you, Wes. Now, it's not just the weather, which is miserable at the moment. The economic climate is looking pretty inclement. The UK economy is once again officially in recession. Against the expectation of the city, the economy shrank during the first three months of the year by 0.2%. Not good news. But here on More or less, we feel a Bit of context is in order. Here's the good news. This is nothing like the recession of 2008 when economic output GDP dropped 7% over the course of 15 months. This so called recession has seen output fall by just half of 1% over the six months to March. And here's the bad news. Growth is very weak. If you look back to the peak of economic output, which was in 2008, we're still well below that peak. In a typical recession, we'd have recovered to the earlier peak after about three years. Even in the severe recessions of the early 1980s and the 1930s, output took about 44 years to recover to its previous peak. This ongoing slump has already lasted longer than that, and whether GDP growth over the past few months had been fractionally positive or fractionally negative wouldn't have changed that big picture in any important way. Oh, and one more thing. Although many people say that a recession is at least two consecutive quarters of negative growth, there is no official definition of a recession here in the uk. So the next time somebody tells you we're officially in recession, tell them they're officially talking bobbins. You're listening to More or Less with me, Tim Harford, in association with the Open University. In last week's program we looked at the fact that North Koreans are shorter than their South Korean counterparts, a difference which can almost certainly be attributed to malnutrition and poverty in the North. That item prompted several very interesting emails. Jean Eckersley wrote to there's no need to go as far as North Korea to see that food shortages stunt growth. My 1940s primary school photograph shows quite clearly which ones were the farmer's children. They were noticeably taller than the rest of us. Richard Bertol wrote, my late brother was born in 1937 and I followed in 1944. Both of us reached 5 foot 10 while our father was 6 foot 3. Our sons are both also 6 foot 3. Richard wonders whether the fact that he's shorter than both his father and his son can be blamed on the war years, rationing and austerity. And Ian Wise wrote, well, lets hear from the man himself. My mother was a Labour MP when she was elected in 1974 she said
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that all the Conservative MPs were much
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taller than the Labour ones. She speculated that that was stunting due
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to working class diet.
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By the time she died in 2000, she said that the new Labour MPs were as tall as the Conservatives and thought that this was due to the new intake being more middle class. Is there Any evidence that there used to be a difference in height between the parties, MPs? And has this got less under New Labour? Professor Sir Roderick Flood, the provost of Gresham College, has studied changes in human height for a long time.
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I don't know that Anybody's measured the MPs, but it is entirely plausible because in our society, and indeed in almost all societies that we know about, the upper classes are taller than the middle classes, who are taller than the working classes. So one would expect Tory MPs to be taller than Labour. And as the Labour Party becomes more middle class, you'd expect their average height to rise compared to what it was when they were predominantly working class people.
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A
So this is not a change in the diets of the working classes over the 20th century, or is that also going on?
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All of us have been growing taller since at least the end of the 18th century. There was probably a period in the middle of the 19th century with the growth of the cities, when living conditions deteriorated in the cities, that the average height diminished. But otherwise, basically our height's been going up at different rates, but going up for about the last two or three hundred years.
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And has the gap between the working classes, the height of the working classes and the height of the middle classes and the upper classes, has that gap changed? Is it narrowing?
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It has, it's significantly narrowed. We've got some very good evidence at the beginning of the 19th century for the heights of boys at Sandhurst compared with boys from the London slums. And that shows that almost all the boys from Sandhurst who were at that point from the middle and upper classes were taller than almost all the boys from the London slums. So it used to be literally true that the upper classes could look down on the working classes. That graph has diminished over time very significantly, but it still shows up in the surveys that have been done.
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Is it now still noticeable, I wonder? When I heard this story, maybe this is a little bit of myth making. I'm sure a Labour MP would like to see her colleagues as working class and as disadvantaged relative to the Tory toffs. But, I mean, it seems extraordinary that you'd actually notice the difference in height. But maybe we didn't.
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No. Well, I think we're very good at noticing differences in heights of different groups of people. I once attended a royal wedding because the bride in question was working with my wife. And my impression of that rather aristocratic group of people was that they were taller. Now, that's very unusual experience for me because I'm so six foot one. And a half or something like that. And therefore most groups that I meet are shorter than me. But that group of guests at a royal wedding were definitely on average taller than me, but probably only by half an inch or something like that. But actually, we're very good at visualising and seeing these differences.
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Do we know why this is? Is this to do with diet, Lifestyle? Is there a genetic component?
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The differences in heights between different groups, predominantly, almost entirely environmental. The effects of different living standards over the generations, in some cases between different social groups? It isn't just a question of differences between social groups. There are differences between different countries, different regions within countries and between different occupations. So the skilled working class, if you like, would be taller than the laboring classes. The Scots are shorter than people from the southeast of England, which was not true 200 years ago, when the Scots were probably taller than the people from the southeast of England.
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Professor Sir Roderick Flood, one of the authors of the Changing Body. We also spoke to another of the authors of that book, Professor Bernard Harris of Southampton University, who told us that listener Richard Bertell probably shouldn't blame the Second World War for the fact that both his father and his son outgrew him. Professor Harris says that in fact, overall diet improved during the war because rationing ensured every household received the minimum nutrition it needed and because there was a big increase in the provision of school meals. Perhaps, Mr. Bertle, you are instead a victim of regression to the mean. It's not every day that someone writes down an equation that ends up changing the world. But it does happen sometimes, and the world doesn't always change for the better. We're going to hear the story of the equation that transformed Wall street and the arguments over whether it made the world a better place or helped cause the financial mess we've all been dealing with for the past five years. It's called the Black Scholes formula. It was first written down in the early 1970s. But our story starts earlier than that, a lot earlier.
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The Black Scholes equation came from attempts to put a sensible price on a financial option. So financial options go back well over 100 years.
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This is Ian Stewart, professor of mathematics at Warwick University and The author of 17 Equations that Changed the World. Futures contracts were originally written for people who had rice to buy or to sell. In 17th century Japan, a simple futures contract says, I will agree to buy rice from you in one year's time at a price that we agree right now.
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They're almost like insurance policies. And once you can put a sensible price on such a contract, in the middle before it's matured, it becomes possible to buy and sell those contracts. Contracts. And so instead of buying and selling rice or wheat or gold, you buy and sell contracts for rice, wheat and gold.
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By the time we hit the 20th century, the Chicago Board of Trade is providing a marketplace for traders to deal not only in futures, but in options contracts. An example of an option is a contract where we agree that I can buy rice from you at any time over the next year at a price that we agree right now, but I don't have to if I don't want to. You can imagine why this kind of contract might be useful if I'm running a big chain of hamburger restaurants and I don't know how much beef I'll need to buy next year, and I'm nervous that the price of beef might rise. Well, all I need is to buy some options on beef. But that then leads to a very ticklish how much should I be paying for those beef options? What are they worth? And that's where this world changing equation, the Black Scholes formula, can help.
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The problem it's trying to solve is to define the value of the right, but not the obligation to buy a particular asset at a specified price within or at the end of a specified time period.
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That's Myron Scholes. He's a professor of finance at the Stanford University Graduate School of Business. And if you're wondering whether it's a coincidence that Professor Scholes is talking about something called the Black Scholes formula, it's not. The young Myron Scholes was fascinated by finance. As a teenager, he persuaded his mother to set up an account so that he could trade on the stock market. One of the amazing things about Scholes is that throughout his time as an undergraduate and then as a doctoral student, he was partially blind. And so he says, he got very good at listening and at thinking. When he was 26, an operation largely restored Scholes sight. The next year, he became an assistant professor at mit, and it was there that he stumbled upon the option pricing puzzle.
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F
I came upon the problem because my students at MIT had option data and they were writing a master's thesis and they tried to value the option. And when I looked at the way they were trying to value it, it became curious to me that they had to assume this idea of a constant discount rate to value the option when the risk of the option was changing. So you had this puzzle, and the puzzle bothered me a lot.
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Several factors contributed to the puzzle. One was this question of risk. The value of an option to buy beef at a price of, say, $2 a kilogram presumably depends on what the price of beef is and how the price of beef is moving around. But the connection between the price of beef and and the value of the beef option doesn't vary in a straightforward way. It depends on how likely the option is to actually be used. And that in turn depends on the option price and the beef price. All the variables seem to be tangled up in an impenetrable way. Myron Scholes worked on the problem with his colleague Fisher Black.
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F
We combined our thinking and we were able then to work out over a period of time the differential equation or describe, you know, the change in the option to various known parameters. But it took us a long time after that to actually achieve the simple Black Scholes formula.
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Black and Scholes figured out that rather than come at the option pricing formula directly, you could get at it indirectly. As Ian Stewart mentioned, these options work like insurance policies. It turns out that if I own just the right portfolio of beef plus options to buy and sell beef, and I have a delicious but totally risk free portfolio, since I already know the price of beef and the price of risk free assets, by looking at the difference between them, I can work out the price of these beef options. Well, that's the basic idea. The details took some time to work out.
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It might have taken us about a year and a year and a half to be able to solve and get the simple Black Scholes formula. But we had the actual underlying dynamics correct way before we were able to find the simple solution that has become the Black Scholes formula.
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The Black Scholes method turned out to be a way not only to calculate the value of options, but all kinds of other financial assets.
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We were like kids in a candy store in the sense that we described options everywhere and options were embedded in everything we did in life.
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But Black and Scholes weren't the only kids in the candy store.
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What the equation did was give everyone the confidence to trade options and very quickly, much more complicated financial instruments known as derivatives.
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Myron Scholes thought his equation was useful. He didn't see how it was going to transform the face of finance.
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About the time we had published this article, that's 1973, the Chicago board options Exchange started to trade options call options on 16 stocks.
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Scholes had just moved to the University of Chicago, and for several years he and his colleagues had been teaching the Black Scholes formula and methodology to their students.
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There were many young traders who either had taken courses at MIT or Chicago in using the option pricing technology. And as a result of that, many of the students picked up on the model and understood it. On the other hand, there was a group of traders who had only intuition. And in a very short period of time, the intuitive players were essentially eliminated by the more systematic players or the ones who had the option pricing technology.
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By 2007, the trade in derivatives worldwide was US$1,000,000,000,000. This is 10 times the total production of goods on the planet over its entire history. It shows how big this got. And okay, we're talking about the totals in a two way trade. People are buying, people are selling. You're adding it all up as if it doesn't cancel out. But it was a huge trade.
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The Black Scholes formula had passed the market test. But as banks and hedge funds relied more and more on this kind of equation, they became more and more vulnerable to mistakes or oversimplifications in the mathematics.
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The equation is based on the idea that big movements are actually very, very rare. There's a nice mathematical formula for exactly what the probability is. The problem is real markets have these big changes much more often than this model predicts. The other problem is everyone is following the same general mathematical principles, so they're all going to get the same answer.
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Now, these were known problems. What was not clear was that can't be clear until it's too late was whether the problems were small enough to ignore or well understood enough to fix. And then in the late 1990s, two remarkable things happened.
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The inventors got the Nobel Prize for Economics, and I would argue they thoroughly deserve to get it.
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Fisher Black died Young in 1995. When in 1997, Myron Scholes won the Nobel Prize, he shared it not with Black, but with Robert Merton, another option pricing expert. The Nobel Prize is a big deal in the life of most winners. But Myron Scholes had already made his mark. His work had inspired a generation of mathematical wizards on Wall Street. And both he and Robert Merton were players in the world of finance as partners of a hedge fund called Long Term Capital Management.
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The whole idea of this company was that it was going to base its trading on mathematical principles such as the Black Scholes equation. And it actually was amazingly successful. To begin with, it was outperforming the traditional companies quite noticeably, and everything looked great.
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You can guess how this ended badly. Long Term Capital Management ran into, among other things, the Russian financial crisis. The firm lost $4 billion in the course of six weeks. It was bailed out by a consortium of banks which had been assembled by the Federal Reserve. This was all happening in August and September of 1998, less than a year after Myron Scholes had been awarded his Nobel Prize.
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It showed the danger of this kind of algorithmically based trading. If you don't keep an eye on some of the indicators that the more conventional people would use, they were committed pretty much to just plowing ahead with the system they had. There wasn't a great deal else to do, and it went wrong.
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F
It has nothing to do with equations, it has nothing to do with models. It has to do with the idea that there's a group of traders in the firm. And I was not running the firm. Let me be very clear about that. Having been awarded the Nobel Prize, I was giving talks around the world, as most Nobel Prize winners do, and a group of traders were running the firm. I was not running the firm. So that's a wrong characterization. Second, wrong characterization was that it was models that actually had brought down the firm. There was not an ability to withstand the shock that occurred in the market later in summer and fall of 1998. It was just a matter of risk taking. It wasn't a matter of modeling.
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This is something people were still arguing about a decade later. Was the collapse of Long Term Capital Management an indictment of mathematical approaches to finance? Or as Myron Scholl said, was it just traders taking too much risk against the better judgment of the mathematical experts? Ten years after the Long Term Capital Management bailout, something else happened. Lehman Brothers collapsed. And the debate over Black Scholes and LTCM is now a broader debate over the role of mathematical equations in finance. Ian Stewart claims the Black Scholes equation changed the world. Does he really believe it caused the financial crisis?
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It was abuse of their equation that caused trouble. And I don't think you can blame the inventors of an equation if somebody else comes along and uses it badly. And it wasn't just that equation. In fact, it probably wasn't that equation as such at all. When we come to the real financial crisis, it was a whole generation of other mathematical models and all sorts of other techniques that followed on its heels.
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In other words, the Black Scholes formula didn't directly cause the crisis, but it set the stage for other mathematical innovations which were much more directly implicated. Black Scholes changed the culture of Wall street from a place where people traded based on common sense, experience and intuition to a place where the computer said yes or no.
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F
The Black Scholes methodology or technology has very specific rules and it has very specific requirements that technology attracted or caused investment banks to hire people who had mathematical skills and quantitative skills. I agree with that. They then develop products and develop technologies of their own that were not necessarily sorry. The technology itself or the applications of their technologies had flaws in them because they had assumptions that were wrong, or they had used data incorrectly to calibrate their models, or people who used their models didn't know how to use them.
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But Myron Scholes argues there's no going back. His equation is a kind of technological progress, and as with any technology, it needs to be used responsibly.
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F
The fundamental issue is that quantitative technologies in finance will survive and will grow. And that's what bothers me about the professor's claims. He looked and saw, okay, there was a blow up in certain parts of the market. And from that you make the conclusion that everything's gone amiss. It's the same way as saying because we had an explosion in a nuclear power plant, that all use of nuclear forever and a day is not valuable.
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D
It's very tempting to see the financial crisis and various things that led up to it as the sort of classic Greek tragedy of hubris begets nemesis. You try to fly, you fly too close to the sun, the wax holding your wings on melts, and you fall down to the ground. My personal view is it's not just tempting to do that, but actually there is a certain amount of truth in that way of thinking. I think the banker's hubris did indeed beget nemesis. The big problem is it wasn't the bankers on whom the nemesis descended, it
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was the rest of us, Professor Ian Stewart and Professor Myron Scholes. And that's all we've got time for this week in next week's programme. Well, who knows? If you think you know what should be in it, drop us a line More or less@BBC.co.uk. our website is BBC.co.uk more or less, where you can subscribe to our podcast. Until next week, stay dry and Goodbye More or Less was presented by me, Tim Harford, the undercover economist at the Financial Times. The producer was Richard Knight and the editor, Richard Varden. The program was produced in association with the Open University.