### The (Mis)Behavior of Markets

Saturday, December 6th, 2008The global financial system is a house built on sand. This is the main argument that the highly acclaimed mathematician Benoit Mandelbrot makes in his 2004 book “The Misbehavior of Markets”. And here we are, at the dusk of 2008, having seen global markets plummet an average 50% the last few months. Could he be on to something?

Mandelbrot wastes little time in the introduction and you get the main ideas simply by reading the first part of the book, titled: “The Old Way”. This is nothing less than a full-scale assault on the underlying assumptions of modern finance. He shows how the Capital Asset Pricing Model (CAPM), Markowitz Portfolio Theory (MPT) and the Black-Scholes options pricing method – the most important elements of orthodox financial theory – are all based on two shaky assumptions:

1. The magnitude of price moves are distributed according to a Bell curve, with small movements the most likely, and huge movements nearly impossible.

2. Price moves are temporally independent. At any given moment, all available information is priced into all securities, and thus the next price move is completely independent of the previous one. This is the heart of the Efficient Market Hypothesis.

Mandelbrot wastes no time shooting these assumptions down. The first assumption can be discarded fairly fast: Take the daily closing price of your favorite stock, currency or commodity and put them in a spreadsheet. Then take the log of each price to get relative price moves. Calculate all the daily differences and calculate a mean and standard deviation for the price moves. Now look at the largest moves: How many standard deviations are they? For instance, the 29% price move in the Dow Jones on October 19, 1987, has a probability of 1 in 10^50 based on standard financial reckoning. So what, you may say, we recognize these moves as “measurement errors” or “freak acts of God”. But these are the days when fortunes are made or lost. A model which simply discards the risk of such days, will effect a false sense of security.

The second assumption is harder to attack, but statistical tests show that the absolute size of price moves is correlated across days. If yesterday had a huge price move, you are more likely to see a huge price move today. Mandelbrot captures this in the concept of “trading time” – basically that time moves faster on volatile days and slower on dull days.

Of course, the author of this book should be known to anyone with even a slightest interest in popular mathematics, programming or computer graphics. As the founder of fractal theory – the theory of ruggedness – he has lent his name to the Mandelbrot fractal. Mandelbrot argues that price charts exhibit fractal behaviour: They are self-similar. If you remove the axis labels, you cannot tell if you’re looking on a 5-minute intraday chart or a long-time chart spanning decades. He also shows that simple fractal generators can create much more convincing fake charts than convential financial models can.

What are the implications of this? As I write this, my girlfriend is writing her Chartered Financial Analyst (CFA) exam. Her study material places great emphasis on the CAPM, MPT and Black-Scholes. And in the six volumes of study material, there is but one meager warning that these methods may not always work… Great, we have a horde of financial advisors and analysts out there who think they know how to diversify portfolios so that they give the optimal return for a given risk. But they are not even close to being able to quantify risk. Their alphas and betas are calculated with a mathematical distribution that does not fit the data. The map does not match the terrain, but our financial navigators are nevertheless taught that such discrepancies are small and that the current map is better than no map.

Is it better than no map? I would argue no. A man with no map is less likely to fall off a cliff than a man with a map that says there is no cliff.

In the context of banks, the picture gets particularly disturbing. A bank takes deposits and lends the money to others. In most countries there are reserve requirements, which dictates how much of the deposits should be available as cash or deposits in the central bank at all times. A common requirement is 10% (which is the US requirement). (As a side note, Canada has not had any reserve requirements since 1992. I have not been able to find any numbers for Norway, and Sweden also has no reserve requirement. Jordan is perhaps the safest place to have a bank account, with an 80% reserve requirement). The way risk is calculated on the bank’s assets forms the foundation for how much capital the bank will set aside to weather any financial storm. If risk is severly underestimated, the bank may end up having insufficient capital and go into bankruptcy.

Of course, if banks had completely transparent book-keeping and disclosed all their losses continuously, then the bank would declare bankruptcy just as its net value becomes 0 and the effect is that you have assets than can be liquidated to back all its liabilities. Of course, the banks’ book-keeping these days is extremely opaque, and their assets (loans, of which many might go bad) extremely illiquid. It doesn’t help that mark-to-market accounting has been suspended, so that the book values are far from what these assets would fetch in a forced liquidation. The end result is that when your bank finally declares bankruptcy, its assets will be inadequate to satisfy all the depositors demands.

But it says my deposits are insured! Indeed, sporadic bank failures are not a big problem because bank deposits are backed by government deposit guarantees up to a certain amount. However, the funds set aside for the deposit insurance schemes are based on similar risk calculations. The Federal Deposit Insurance Corporation (FDIC) does not set aside money equal to the full amount of deposits it insures. If even one big bank fails, the FDIC will have to draw on the goverment to do one of the following:

1. Spend more tax money to back the guarantees. You pay more taxes to pay yourself your money back. This is by far the most likely option, since most people will have trouble comprehending the previous sentence and are prone to accept this alternative.

2. Print money. The inflationary effect of this ensures that the money you receive is worth less than the money you had in the failed bank, and whatever money you earn every month is also going to be worth less. The net effect is likely to be that the bank deposit is lost. (This is also the kind of move that can lead to the collapse of currencies or even governments if it’s not carried out with extreme care.)

3. Don’t honor the deposit insurance. The deposit is lost. Prepare for riots in the streets!

Either way, your bank deposit is effectively lost. Anyone who tells you otherwise is saying you can have something for nothing. Depending on what option is chosen, you may or may not get to release your anger on bankers and politicians by going after them with torches and pitchforks. (You can tell that I’m favoring option #3)

The end result, guys, is that wherever risk in securities are measured, a new formula which can account for the sometimes violent swings in securities prices is needed. This is particularly important in banking where the public’s money and the stability of society is at risk. Stronger reserve requirements will follow, and bank failures can be a once every 10^47 years event, instead of once every 20 years.