Humble Pi

by Matt Parker

  Once the Making of a Fly market had corrected, Eisen’s colleague was able to buy a copy of the book for a normal price and the lab went back to trying to understand how genes work instead of reverse-engineering pricing algorithms. And I’m left with my copy of The Making of a Fly (which I bought second-hand; even ‘normal price’ US textbooks are beyond my budget). I even did my best to read it. I figured there must be some link between what happened to the book’s price and how genetic algorithms cause flies to grow. I could give the last word to the book itself. This is the best I could find:

  Studies of growth of this type give the impression of some mathematically precise control which operates independently in different body parts.

  – The Making of a Fly, Peter Lawrence (p. 50)

  I think we can all take something away from that. And it makes my purchase of the book now, technically, tax deductible. (Although probably not at the original price.)

  And they would have gotten away with it too, if it weren’t for the meddling laws of physics.

  In high-speed trading, data is king. If a trader has exclusive information about what the price of a commodity is likely to do next, they can place orders before the market has a chance to adjust. Or rather, that data can go straight into an algorithm that can make the order, placing decisions at incredible speeds. These times are measured in milliseconds. In 2015 Hibernia Networks spent $300 million laying a new fibre-optic cable between New York and London to try to reduce communication times by 6 milliseconds. A lot can happen in a thousandth of a second. Let alone six.

  For financial data, time is, literally, money. The University of Michigan publishes an Index of Consumer Sentiment, which is a measure of how Americans are feeling about the economy (produced after phoning roughly five hundred people and asking them questions), and this information can directly impact financial markets. So it was important how this data was released. Once the new figures were ready, Thomson Reuters would put them on its public website at 10 a.m. precisely so everyone could access them at once. In exchange for this exclusive deal to release the data for free, Thomson Reuters paid the University of Michigan over $1 million dollars.

  Why were they paying to give the data away for free? In the contract, Thomson Reuters was allowed to give the numbers out five minutes early to its subscribers. So anyone who paid to subscribe to Thomson Reuters could get the data five minutes before the rest of the market and start trading accordingly. And subscribers of their ‘ultra-low latency distribution platform’ received the data two seconds earlier, at 9:54:58 (plus or minus half a second), ready to be fed straight into trading algorithms. In the first half a second after this data is released, more than $40 million worth of trades can already have occurred in a single fund. The chumps who waited to get the data for free at 10 a.m. would find that the market had already adjusted.

  The ethics (and probably legality) are a bit blurry. Private institutions are able to release their own data however they want as long as they are transparent about it. And Thomson Reuters was able to point to a page on its website which outlined these times: the website equivalent to crossing your fingers behind your back. The practice only really came to public consciousness when CNBC (Consumer News and Business Channel) ran a story on it in 2013, and not long after that the practice came to an end.

  If this Thomson Reuters ad is a Venn diagram, it’s surprisingly honest.

  The release of government data is much more cut and dried: absolutely no one is allowed to trade on it before it is released to everyone simultaneously. When the US Federal Reserve is announcing something – for example, if it will continue with a bond-buying programme – that news can have a big impact on prices in the financial markets. If anyone knew the news in advance, they could start buying things that were destined to jump in value.

  So the Fed tightly controls the release of such information from within its headquarters in Washington DC. For instance, when an announcement was due to go out on 18 September 2013 at exactly 2 p.m. journalists had to go into a special room in the Fed’s building, which was locked at 1.45 p.m. Printed copies of the news were then handed out at 1.50 p.m. and people were given time to read them.

  At 1.58 p.m. TV journalists were allowed to go to a special balcony, where their cameras were set up. Moments before 2 p.m. print journalists could open a phone connection to their editors but not yet communicate with them. At exactly 2 p.m., as measured by an atomic clock, the information could be released. Financial traders all around the world want to be the first to get data like this. If one trader in Chicago can get the data even milliseconds before their competitors, that will give them an advantage. But how fast can the data travel?

  The two competing technologies are fibre-optic cables and microwave relays. Light going down a fibre-optic cable travels at about 69 per cent of the maximum speed of light in a vacuum, which is still blisteringly fast, covering around 200,000 kilometres every second. Microwaves move through the air at almost the full 299,792 kilometres per second maximum speed of light, but they have to be bounced from base station to base station to allow for the curvature of the Earth.

  There are also problems about where microwave base stations can be built and where fibre-optic cable can be laid. So the path from DC to Chicago taken by the data will not be the shortest possible route. But to get a lower limit, we can assume the data follows the shortest line between the Fed Building in DC to the Chicago Mercantile Exchange building (955.65 kilometres) at the full speed of light (and new hollow-core fibre-optic cables can reach 99.7 per cent of the speed of light) and calculate a time of 3.19 milliseconds. A similar calculation for the shorter DC to New York City journey gives 1.09 milliseconds.

  Those times assume the data is racing down a fibre-optic cable following the curvature of the Earth. Straight-line travel would be slightly faster. There are already ‘line of sight’ laser communication systems for financial data where the beginning and end points have nothing but air between them, for example, those that relay information between buildings in New York and buildings in New Jersey. To travel between Washington DC to Chicago, this would require going through the Earth.

  A laser ready to shoot financial data between cities. It holds the world record for the most boring laser ever.

  But this is not out of the question. Physicists have discovered exotic particles such as neutrinos which can move through normal matter almost unimpeded. Detecting them at the far end would be a major technological challenge, but such a system to fire data through the planet at the full speed of light is physically plausible. However, this shaves only about 3 microseconds off the DC to Chicago travel times (even less for New York). The fastest times possible for data to travel from the Fed building, as allowed by the laws of physics, are 3.18 milliseconds to Chicago and 1.09 milliseconds to New York.

  Which makes it mighty suspicious that trades happened in both Chicago and New York simultaneously at 2 p.m., when the Federal Reserve’s data was released on 18 September 2013. But if the data was coming from DC, then the New York markets should have flinched slightly before the Chicago markets. It looks as though people had been given the data early and tried to make it appear like they were merely trading at the first possible instant, except they had forgotten about the laws of physics. Fraud exposed because of the finite speed of light!

  Well, I say ‘fraud exposed’, but nothing has ever come of it. It was not discovered who was making these trades and who sent the data to them. And there was some unresolved confusion over whether moving data to an offsite computer but not releasing it until 2 p.m. was strictly disallowed by the Fed’s regulations. It seems the laws of finance are much more flexible than the rules of physics.

  Maths misunderstandings

  It would be remiss of me not to say something about the global financial crisis of 2007–8. It was kicked off with the subprime mortgage crisis in the US then rapidly spread to countries all around the world. And there are some interesting bits of mathematics which fed into
it. My personal favourites are collateralized debt obligation (CDO) financial products. A CDO groups a bunch of risky investments together on the assumption that they couldn’t possibly all go wrong.

  Spoiler: they all went wrong. Once CDOs could themselves contain other CDOs, a mathematical web was built which few people understood. I love my maths but, looking back at the global financial crisis as a whole, I don’t claim to understand what went wrong. If you want to look into it in greater depth, there are countless books out there dedicated to just this one topic. Or (if you’re old enough) watch the film The Big Short. I’m not going to say anything about it; instead I’ll talk about a more interesting and concrete example of people not understanding maths: company boards giving chief executive officers (CEOs), that is, the people in charge of companies, pay awards.

  A CEO in the US can today earn incredible amounts of money, sometimes tens of millions of dollars a year. Before the 1990s only CEOs who founded or owned a company would earn ‘super salaries’, but between 1992 and 2001 the median CEO pay for companies in the S&P 500 Index in the US rose from $2.9 million a year to $9.3 million (inflation-adjusted to 2011 dollars). A threefold real increase in a decade. Then the explosion stopped. A decade later, in 2011, the median CEO pay was still around $9 million.

  Some researchers at the University of Chicago and Dartmouth College noticed that, during the pay explosion, actual salaries and even the values of stocks given to CEOs did not increase similarly. The boom was coming from the remuneration paid to CEOs in one particular form: stock options.

  A stock option is a contract that allows someone to buy a certain stock in the future at a pre-agreed ‘strike’ price. So if you get a stock option to buy a certain company’s stock at $100 in a year’s time and the stock price goes up during that year to $120, it means you can now exercise your option and buy the stock for $100 and immediately sell it on the open market for $120. If the stock price goes down to $80, then you tear up your stock option and don’t buy anything. So a stock option has some value in itself: they can only make money (or break even). Which is why they cost money to buy in the first place and can then be traded.

  The calculation for the value of stock options is not straightforward and was developed only relatively recently, in 1973, with the Black–Scholes–Merton formula. Black passed away, but Scholes and Merton won the 1997 Nobel Prize for Economics for their formula. Pricing options involves factoring in things like estimating how likely it is that the value of the stock will change and how much interest could have been made with the money spent on the option. Which is all doable. It just ends up with a complicated-looking formula. And this is where company boards started to go wrong.

  It was not immediately obvious to all directors how the number of stock options directly related to the value being paid to the CEO. Look what happens when comparing other types of compensation to their true values:

  Value of salary = [number of dollars] × $1

  Value of stock = [number of shares] × [value per share]

  Value of options =

  [number of options] ×

  where

  S = current stock price

  T = time before option can be exercised

  r = risk-free interest rate

  N = cumulative standard normal distribution

  σ = volatility of returns on the stock (estimated with standard deviation)

  Even though it looks complex, the short story is that the S at the front means that the value of stock options scales with the current stock value. But while company boards would decrease the number of shares they gave a CEO as the value of those shares went up, the University of Chicago and Dartmouth College research showed that company boards experienced a kind of ‘number rigidity’ in granting stock options: the number they granted was surprisingly rigid. Even after a stock split, when CEOs were given twice as much stock to compensate for the stock now being worth half as much, the number of stock options would not change. Boards just kept giving the same number of stock options, seemingly ignoring their value. And during the 1990s and early 2000s that value went up a lot.

  Then, in 2006, a change in regulations meant that companies had to use the Black-Scholes-Merton formula to declare the value of the stock options they were paying their CEOs. Once the maths was compulsory and board members were forced to look at the actual value of stock options, the number rigidity went away and options were adjusted based on their value. The explosion in CEO pay stopped. This is not to say it decreased back to pre-explosion levels: once established, market forces would not let the level drop. The massive CEO pay packages still awarded today are a fossil of when company boards didn’t do the maths.

  NINE

  A Round-about Way

  In the 1992 Schleswig-Holstein election in Germany the Green Party won exactly 5 per cent of the votes. This was important, because any party getting less than 5 per cent of the total vote was not allowed any seats in parliament. With 5 per cent of the vote, the Green Party would have a member of parliament. There was much rejoicing.

  At least, everyone thought they had won 5 per cent of the vote, as was published at the time. In reality, they had won only 4.97 per cent of the vote. The system that presented the results had rounded all the percentages to one decimal place, turning 4.97 per cent into 5 per cent. The votes were analysed, the discrepancy was noticed and the Greens lost their seat. Because of this, the Social Democrats gained an extra seat, which gave them a majority. A single rounding changed the outcome of an election.

  Politics seems to contain all the motivating forces for people to try to bend numbers as far as they will go. And rounding is a great way to squeeze out a bit more give from an otherwise inflexible number. As a teacher, I used to give my Year 7 students questions like ‘If a plank is 3 metres (to the nearest metre) long, how long is it?’ Well, it could be anything from 2.5 metres to 3.49 metres (or maybe something like 2.500 metres to 3.499 metres, depending on rounding conventions). It seems some politicians are as smart as a kid in Year 7.

  In the first year of Donald Trump’s presidency, his White House was trying to repeal the Affordable Care Act (ACA), or Obamacare, as it had been branded. When doing this through legislation proved harder than they seem to have expected, they turned to rounding.

  For while the ACA laid down the official guidelines for the healthcare market, the Department of Health and Human Services was responsible for writing the regulations based on the ACA. In February 2017 the now-Trump-controlled Department of Health and Human Services wrote to the US Office of Management and Budget with proposed changes to the regulation. It seems that, if the Trump administration couldn’t change the ACA itself, it was going to try to change how it was interpreted. It’s like trying to adhere to the conditions of a court order by changing your dog’s name to Probation Officer.

  According to industry consultants and lobbyists who contacted the Huffington Post, one of those changes was to increase how much insurance companies could charge older customers. The ACA had laid down very clear guidelines stating that insurance companies could not charge older people premiums that are more than three times the premiums paid by young people. Healthcare itself should be a game of averages, with the goal being that everyone should share the burden equally. The ACA tried to limit how much insurance companies could stray from that ideal.

  It seems the Trump administration wanted to allow insurance companies to charge their older customers up to 3.49 times as much as younger people, using the argument that 3.49 rounds down to 3. I’m almost impressed at their mathematical audacity. But just because a number can be rounded to a new value does not make it the same number. They may as well have crossed out thirteen of the twenty-seven constitutional amendments and claimed nothing had changed, provided you rounded to the nearest whole constitution.

  This proposed change by the Trump administration was never adopted, but it does raise an interesting point. If the ACA had explicitly said ‘multiple of three, rounded to one significan
t figure’, they would have had an argument on their hands. There is an interesting interplay between the laws of mathematics and the actual law. I received a phone call from a lawyer a few years ago who wanted to talk about rounding and percentages. They were working on a case involving a patent for a product which used a substance with a concentration of 1 per cent. Someone else had started making a similar product but using a 0.77 per cent concentration of the substance instead. The original patent holder was taking them to court because they believed that the figure of 0.77 per cent rounds to 1 per cent and therefore violates their patent.

  I thought this was super-interesting because, if the rounding was indeed naively to the nearest percentage point then, yes, 1 per cent would include 0.77 per cent. Everything between 0.5 per cent and 1.5 per cent rounds to 1 per cent using that system. But given the scientific nature of the patent, I suspected that it was technically specified to one significant figure. Which is different: rounding to one significant figure sends everything between 0.95 per cent and 1.5 per cent to 1 per cent. By changing how the rounding is defined, the lower threshold is suddenly a lot closer and now excludes 0.77 per cent. And, sure enough, 0.77 per cent rounded to one significant figure is 0.8 per cent. It would not be in violation of the patent.