Author: Paul

I don’t believe in conspiracy theories but — whenever you hear the “but” you know exactly what is coming! — there is one story that is rather worrisome, and also something to which one can trivially add a bit of mathematics as a convincer.

Did you know that the actors in The Magnificent Seven died in real life in the order in which they died in the movie?

To remind you the seven were, as copied from Wikipedia,

• Yul Brynner as Chris Adams, a Cajun gunslinger, leader of the seven
• Steve McQueen as Vin Tanner, the drifter
• Charles Bronson as Bernardo O’Reilly, the professional in need of money
• Robert Vaughn as Lee, the traumatized veteran
• Brad Dexter as Harry Luck, the fortune seeker
• James Coburn as Britt, the knife expert
• Horst Buchholz as Chico, the young, hot-blooded shootist

You don’t have to take my word for it, it’s easy to google.

I remember discussing this over dinner at a training course in Mexico City. We even got as far as looking at the probability of this happening. How can you order seven people? For the first one there are seven to choose from. For the second there are six remaining. Then five, and so on. This means that the probability of this being a coincidence is one in 7! (that’s seven factorial), i.e. about 0.02%.

Can you explain this any better than chance?

Maybe they died in the movie according to their ages. You could check that out. That would make some sense, the older gunfighters die sooner in the film, and the older actors die earlier in real life.

We could probably quite easily quantify this effect, to increase that 0.02%. But it’s hard to get the probability up to anything remotely probable.

This is the way the dinner conversation went.

One matter that was not discussed, perhaps out of politeness since I was the teacher and they were the students, was that maybe I WAS MAKING IT UP ON THE SPOT, YOU MUPPETS!

My apologies. This conspiracy theory, like all of them, has no basis in fact. But it was fun while it lasted. My mathemagical distractions, the statistical analyses, trying to find rational explanations, all served to convince my audience that there must have been a plot! I should have got an Oscar for my performance!!

The following may or may not be factually accurate. It all happened a long time ago. But it is absolutely 100% correct in spirit.

Twenty or so years ago I was browsing through the library of Imperial College, London, when I happened upon a book called something like The Treasury’s Model of the UK Economy. It was about one inch thick and full of difference equations. Seven hundred and seventy of them, one for each of 770 incredibly important economic variables. There was an equation for the rate of inflation, one for the dollar-sterling exchange rate, others for each of the short-term and long-term interest rates, there was the price of fish, etc. etc. (The last one I made up. I hope.) Could that be a good model with reliable forecasts?

[Hint: How good are economic forecasts generally?]

Consider how many parameters must be needed in such a model, every one impossible to measure accurately, every one unstable. I can’t remember whether these were linear or non-linear difference equations, but every undergrad mathematician knows that you can get chaos with a single non-linear difference equation so think of the output you might get from 770.

Putting myself in the mind of the Treasury economists I think “Hmm, maybe the results of the model are so bad that we need an extra variable. Yes, that’s it, if we can find the 771st equation then the model will finally be perfect.”

No, gentlemen of the Treasury, that is not right. What you want to do is throw away all but the half dozen most important equations and then accept the inevitable, that the results won’t be perfect.

A short distance away on the same shelf was the model of the Venezuelan economy. This was a much thinner book with a mere 160 equations. Again I can imagine the Venezuelan economists saying to each other, “Amigos, one day we too will have as many equations as those British cabrones, no?” No, what you want to do is strip down the 160 equations you’ve got to the most important. In Venezuela maybe it’s just a few equations, for the price of oil, inflation, and maybe how much it costs to buy a politician.

We don’t need more complex economics models. Nor do we need that fourteenth stochastic variable in finance. We need simplicity and robustness. We need to accept that the models of human behaviour will never be perfect. We need to accept all that, and then build in a nice safety margin in our forecasts, prices and measures of risk.

I love watching Dragons’ Den, the programme in which entrepreneurs try to get established business people to invest in their ideas. I love trying to predict which Dragon will say what, how they will negotiate a deal, how they compete with each other to make themselves look good against other Dragons. I love shouting at the TV, “What about patents and intellectual property?” before the Dragons. And I particularly love it when they so obviously get it wrong. Trunki? Come on, just because a bit of plastic broke on a prototype you aren’t going to invest in such an obvious hit? And I find it reassuring when they break all their own rules to invest in something they get emotionally attached to. E.g. Reggae Reggae Sauce. Although I’m sure it is deliciously invigorating many of the facts and figures that the entrepreneur gave turned out to be wrong, many of them during the programme itself.

But I hate it when an entrepreneur gets flummoxed by a Dragon negotiating for a better deal. An entrepreneur will open with offering 10% in return for a certain investment. A Dragon might find this too little and counter with 20%. At which point the entrepreneur shakes his or head and declines.

What are they thinking?

It looks to me like they are thinking from the perspective of the Dragon. I’m going to double his money? Double! No way!

But this is completely the wrong way to look at this. They should look at it from their own perspective initially. So I’m going down from 90% to 80%. No biggie. And then they can put themselves in the shoes of the Dragon. Ok, I can see that doubling the shareholding might double the help the Dragon will give. Which will make that 11.11% reduction in their shareholding (10/90) look pretty insignificant.

One of the first lessons in any course on investing will be about portfolio construction and the benefits of diversification, how to maximize expected return for a given level of risk. If assets are not correlated then as you add more and more of them to your portfolio you can maintain a decent expected return and reduce your risk. Colloquially, we say don’t put all your eggs into one basket.

Of course, that’s only theory. In practice there are many reasons why things don’t work out so nicely. Often that’s because stocks and other investments stubbornly refuse to do what they are told.

But can it ever be optimal to not even try to diversify? Should you ever do the exact opposite of Rule #1? You betcha.

As we’ll see people in banks and hedge funds are encouraged to not diversify, to instead concentrate risk. I don’t know whether this is explicit or instinctive.

Imagine the following scenario. It’s your first day as a trader at an investment bank. You’ve had a world-class university education in economics in, say, Chicago. There you learned about all kinds of theoretically marvellous trades and how to manage risk by diversifying.

You are being introduced to the rest of the trading team. You notice that all of the trades they are doing are strangely similar. It worries you a bit because it doesn’t look like they are diversifying much.

You are then shown your desk, with multiple screens, and told to start trading.

Being a decent person you naturally want to do the best for your bank and so you seek out some trades that are uncorrelated to those of your colleagues but which also have a high probability of success.

Let’s put some numbers to this. There are dozens of other traders all making the same trade, and this trade has a 50:50 chance of making or losing a large amount. You have a better, and independent, trade that has a 75% chance of doubling your money and 25% of losing it all.

What happens next?

There’s a 50% chance that all the other traders lose a vast amount of money. This is not great. They might lose their jobs. The bank might go under.

But there’s also a 50% chance that they’ll be heroes, and rewarded as such in bonuses.

Meanwhile your trade might make some money. More likely than the other traders, at 75%. So you are more likely to be a hero too. No! If the others lose and you win then you are too tiny to even be noticed. You won’t be able to save the bank. And certainly don’t expect a bonus.

You can see this in the following table. If the other traders lose then everyone is fired including you. You can only get a bonus if the traders and you both win, and that has a probability of 0.75 x 0.5 = 37.5%.

No, the only way to get that bonus is to cling to the coattails of the other traders. Do the same trade as them and you have a larger 50% chance of that bonus.

Lose money when all around are making it, you’re fired. Make money when all around are losing it? Expect a big bonus? No way! Your profits will help to bail everyone else out and no one gets a bonus, even you. No, you should do the same as everyone else.

As Keynes said, “It is better to fail conventionally than to succeed unconventionally.”

Following on from Credit Ratings, here’s a true story about how reassuring mathematical analysis can be. Until it turns out to be baloney. This story is also a warning about experts. Unfortunately, although there’s a lesson here we aren’t sure what it is. Sometimes you get screwed no matter how smart you are. Maybe that’s the lesson.

For Paul, 2007 had been a good year financially. His businesses, based around financial mathematics, publishing and training, were starting to take off. Being self employed his earnings were paid without any tax having been withheld. This meant he had to keep a regular check on how much he owed the UK’s Inland Revenue and, not wanting to end up behind bars like some tax-dodging TV evangelist, put it aside for later payment.

Paul is financially very conservative, he wanted to put this money somewhere incredibly safe, but he’s also a bit of a worrier. He knew that the complex financial instruments he worked with were poorly understood, and that their risk management was even worse. He knew that people in banks were confused about fundamental financial principles, and worse, that they didn’t know that they were confused. He knew that it’s important that the incentives of employees (the bankers) and the benefits of the owners and creditors (the man on the street) be lined up, and that in practice they rarely were. And long before it became fashionable he would tell anyone who would listen that the cleverest of the bankers didn’t know what they were doing.

Paul decided to speak to his Wealth Manager at his bank, B______s, to see if there was anywhere safe that he could leave it for a few months before paying the taxman. This was now the second half of 2008. The investment firm Bear Stearns had bitten the dust earlier in the year. So prudence had been the word of the days for several months now.

Paul mentioned these concerns to his Wealth Manager. And the Wealth Manager made some recommendations. One thing that Paul knew about was the Financial Services Compensation Scheme (FSCS), the UK’s version of the US’s Federal Deposit Insurance Corporation, which would at that time cover up to £50,000 in the event of a collapse. Well, the money that Paul owed Alistair Darling, the then Chancellor of the Exchequer, was quite a bit more than this. However, he also knew that there was a version of the financial guarantee that applied to insurance companies where the cover was 90% of any money lost, with no limit. Paul had done his research. So when the Wealth Manager mentioned that he had a couple of insurance-company products to offer, Paul naturally was keen.

There were two products on offer. They both had interest rates of about 4% annualized. One had a slightly higher return than the other. But the return wasn’t the point. The point of this exercise, remember, was to protect the money that he was holding onto on behalf of the Inland Revenue. The Wealth Manager gave the sales pitch for these two products. They seemed very simple, very “vanilla” in the financial jargon, basic short-term bonds. Credit ratings were mentioned and Paul does recall his sense that one of these investments was very, very, very safe, while the other was a mere very, very. The latter had the slightly higher rate of return. The greater the risk, the greater the expected return. That’s classics portfolio theory.

The conservative Paul opted for the lower return, thrice-very-safe investment. The government’s tax money, or at least 90% of it, was now secure.

This was a Thursday in September 2008.

That weekend brought the news of the collapse of Lehman Brothers and the near bankruptcy of AIG due to trading in complex credit derivatives, the very same instruments and models that Paul had said in 2006 “fill me with some nervousness and concern for the future of the global financial markets.”

Did we mention that it was an AIG insurance bond that Paul had bought?

Paul spoke to his Wealth Manager who reassured him that “there was nothing to worry about,” and that they “were speaking to AIG at the highest possible level.” This gave Paul a warm glow, he felt special. It was nice having a Wealth Manager. “Whatever happens to AIG, the money will be returned in 24 hours,” they said.

The next day the money had not reappeared, and B______s were now saying “48 hours” for the return. There was still nothing to worry about because insurance products come with a cooling-off period of 14 days.

Over the next few days the language of the Wealth Manager changed subtly, mention of cooling-off periods disappeared and timescales became more fluid. And suddenly there was talk of early-redemption penalties. This certainly didn’t fit in with the promise of a full refund in the first 14 days. Meanwhile AIG wasn’t getting any better.

Paul decided to take what little control he could, and started to make his own enquiries. He called AIG. To his surprise, considering their situation, the call was answered promptly, and he was put through to someone dealing with these bonds. Paul’s question was simple: “Was there or was there not a cooling-off period?” The AIG person did not know. She read out the bond’s particulars, the same paperwork that Paul had in front of him during the call. No mention was made in the paperwork of cooling-off periods or early redemption.

Paul called the Financial Services Authority, the then regulating body. He was going to ask about specifics of his bond and was expecting a response such as “Go to the FSA website, type in the company name, look for your bond in the dropdown menu, and the details will appear in a pop-up window.” It was the 21^st century after all. Unfortunately the FSA’s representative said something somewhat different, in a very tired voice, something rather like “Do you know how many companies there are? Quite frankly, we don’t know who we regulate.” This was not looking promising.

Paul then went to the FSCS’s website. In the event of AIG collapsing they would be the ones to pick up the tab. Although they seemed very proud of their record in recompensing clients of failed institutions it was clear that they had never had to deal with anyone quite the size of AIG, a top-20 global company which sponsored Manchester United football team and, it seemed, much of the rest of the economy. (Forbes ranked them the 18th largest company in the world in 2008. “I bought one of your insurance bonds and all I got was this lousy Man United t-shirt.”) The case studies on the FSCS’s website were all firms that you’d never have heard of. Oh dear. It was now clear to Paul that he couldn’t depend on the insurance bond’s insurance.

Fortunately, this story has a reasonably happy ending – after a number of weeks nearly all the money was returned. Paul was in fact probably one of the more fortunate purchasers of this product. The BBC television journalist Jeremy Clarkson, who found himself in a similar situation (but with only the “very, very” safe AIG bond instead), said “I made strenuous efforts to get my money out of AIG as soon as the scale of its problems became apparent. But it wasn’t possible. Inwardly I was screaming. It’s my money. I gave it to you. You’ve squandered it on a Mexican’s house in San Diego and a stupid football team and that’s your problem. Not mine.” (Clarkson and Mexico have some history, but his observation had some truth to it, many mortgage brokers targeted specific racial groups for their sub-prime, teaser-rate, AIG-insured mortgages.)

But this was not a problem just for isolated investors. AIG was a major node in the financial system, and as its tangled web fell apart many companies, indeed entire countries, were severely affected by the ensuing economic mayhem. People lost their jobs, their homes, their life savings, even their lives (financial crises are strongly correlated with health crises and suicides).

Now, as tales from the credit crunch go this is not exactly movie material – it’s more the Big Short in reverse, not an attempt to make a killing from a crisis, but an attempt to save money to pay tax – but it does prove a point: We are only human, gullible, fallible, and despite our best efforts as prone as anyone to getting things wrong.

This brush with a failing, flailing insurance giant also taught Paul things he didn’t know, and reinforced a few that he did.

• Banks don’t know what they are talking about. They speak in jargon, much of which they don’t understand. But since there is no down side for them it doesn’t matter. To some extent you just have to hope for the best. Experts? Phooey!

• Regulators are clueless.

• Guarantees mean nothing. The FSCS paid out an average of £200million a year between 2001 and 2006. Between 2006 and 2011 that rose to an average of £5billion per annum, and they had to take out a loan from the Bank of England. But AIG was bailed out to the tune of $85billion. The numbers just don’t add up.

• Bailouts can be necessary, but only because some companies are so humungous in size. In some Darwinian sense entities should be allowed to collapse. But AIG was too big, its influence was everywhere. How many people had insurance through AIG? Just take car insurance, for example. How many cars would have been left uninsured, what repercussions would that have had? It’s impossible to tell.

• Being a mathematician doesn’t make you immune to the financial system’s occasional paroxysms – which is bad news because the system is effectively run by quants.

You could say that Paul should have been more careful. But that was precisely his goal. At some stage he had to take a chance on the advice he was given. And we know that that is risky. The alternative is to research and research and research, leaving no stone unturned … but the end result would be what? To not invest in anything? To not put your money in the bank even? Put it in government bonds? Invest it all in gold, or in property? Put it under the mattress? There’s no middle ground where absolute safety and trust overlap. But perhaps this new world in which there is no financial security – where a banknote, a cheque, a bond, a share certificate, can suddenly become irrelevant – is more natural. Perhaps the few decades in which banks were apparently safe was the anomaly, that a return to the precarious state that has been the norm throughout history, and still is in many countries, was inevitable.

Didn’t you feel proud when your teacher gave you an A+ at school? Or were you a C student, must try harder? Don’t tell us you were an F! My, you’ve done well…considering.

Just as teachers grade their students so there are businesses who grade investments. The three main credit rating agencies are Moody’s Investors Service, Standard & Poor’s and Fitch Ratings. Such agencies analyse the creditworthiness of companies, and their likelihood of going bust, as well as the risks in individual financial instruments. And they rate countries too.

Let’s take Moody’s for example. Their ratings start at the top, with the rating Aaa, “The highest quality and lowest credit risk” and “Best ability to repay short-term debt.” Below, and a smidgen riskier, comes Aa1, then Aa2, Aa3, followed by A1, A2 and A3. These are still supposedly low credit risk. We move even lower to the Bs with Baa1, Baa2, Baa3, then Ba1, and so on. All of these are higher risk with some “speculative elements.” By B1, B2 and B3 we are in high credit-risk territory. Finally come the Cs, the lowest of which is C itself “Rated as the lowest quality, usually in default and low likelihood of recovering principal or interest.”

Baa3 and above the instruments are supposedly “Investment grade.” Ba and below are non investment grade, also known as “junk.”

Good idea, no? And very helpful to investors. Knowing how professional bodies perceive risk in these companies and products can help investors make qualitative judgements about what to add to or subtract from their portfolios. But there’s more, there’s a quantitative angle to this as well. Let’s take as an example a bond rated Ba2. This bond has a price in the market. Suppose it yields 3% per annum. And suppose bank, i.e. risk-free, interest rates are 2%. Then (under lots of assumptions) these numbers can be interpreted as there being a 3-2=1% probability of the company defaulting on that bond in a year. (That’s 99% chance of getting one dollar and two cents back for your one-dollar investment, and 1% chance of nada.) Now we are in the realms of mathematics and can make quantitative judgements about whether Ba2 bonds are too risky, or which specific bonds to buy.

There are problems with this, or we wouldn’t be writing about this topic.

The first problem is the concept of quantifying probability of default. As a rule bankruptcy is a one-off event from which one doesn’t recover in the same form as before the bankruptcy. And as such a company only experiences this once. Therefore there’s not that much in the way of statistics for individual companies, only statistics about types of companies or companies with the same credit rating. There are parallels with health and death. Death is also a one-off event, at time of writing, and anyone who tells your life expectancy is basing that on tables of life expectancies of people with the same credit health rating as you, whether you smoke or not, take part in dangerous sports, etc. The mathematics of death and bankruptcy are very similar. More about life and death later.

But there’s a bigger problem. It concerns who pays for the credit rating. And it’s not who you’d expect. The credit rating on company XYZ is typically paid for by…company XYZ.

They need the rating to be taken as a serious investment, and they want it to be as high a rating as possible. The rating agency want their business and a happy customer. It doesn’t take a genius to figure out that their two interests are aligned. Had they been business partners then having perfectly aligned interests is exactly right. But here the rating agency is supposed to be acting as an unbiased middleman between the investor and the investment.

A third problem related to the above is that having competition among rating agencies might lead to companies choosing to work with those agencies that are the softest, who give the highest rating. None of this is helped by the lack of transparency about the rating process itself.

This is moral hazard.

Never mind all those problems. The main thing is that mathematics is involved. So it must be ok.