Читать книгу Financial Risk Management For Dummies - Aaron Brown - Страница 10

The mathematical study of risk began with casino gambling games. Later Bayesians did a reappraisal of that work using proposition bets in which you take one side or the other on a proposal (see the preceding section). Although you can get a great deal of insight from both approaches, they’re inadequate for risk management, even if you’re managing simple betting risk – the risk created for the purpose of making the bet.

Spinning the roulette wheel

You don’t need to understand the details of roulette for this discussion but I offer them here just in case you’re not familiar with the game. It may make some of the discussion clearer.

A roulette wheel consists of a wooden bowl with a spindle in the middle. The wheel head is a metal disk placed on the spindle, with the numbers 0 to 36 (plus 00 in American wheels) in slots around the outside. The slots are lined with a springy material – wood or plastic. The bowl has raised areas called deflectors that look like decorations but serve an important purpose.

The wheel head is spun in one direction, and a small, hard plastic ball is spun in the opposite direction against a lip at the outside of the bowl. As the ball slows, it drifts down from the lip, spiralling down toward the wheel head. Along the way, its direction gets changed when it hits deflectors. Eventually the ball touches the wheel head, which is still moving quickly enough to give the ball a hard bounce; also the springy wood or plastic in the number slots keeps the ball bouncing around for a while. But eventually the ball settles into a slot, and the wheel head slows to a stop.

Players make bets on the number the ball will land in, or its colour (18 of the numbers are red, 18 are black and the zeros are green) or other combinations. Most payouts are computed as the fair amount – the payout amount that would cause bettors to break even in the long run – ignoring the zeros. So if a player bets £1 on a single number, she wins £35 if she’s right and loses her £1 if she’s wrong. If you ignore the zeros, she wins one time for every 35 losses and breaks even. However, because she loses on zeros, she really has 36 (37 for an American wheel) £1 losses for each £35 gain.

Casinos allow bets for a brief period after the wheel is spun, which is what creates one of the main edges for advantage players (gamblers who play in way such that they, not the house, get the advantage). One reason is that a longer time for the bettors means more bets, which means more profit for the casino. But the other reason is that it makes the wheel seem more fair, because the croupier (the casino employee who spins the wheel) cannot try to bias the spin against your bet. Many players prefer to bet only on things that are in spin, meaning no human can influence the outcome any more.

Winners are paid, losing bets are collected and the next round begins.

Analysing roulette

Blaise Pascal, the 17th-century French mathematician who was one of the two founders of mathematical probability theory, actually invented the first roulette wheel in 1655 (about a decade before he got interested in probability) while trying to build a perpetual motion machine. But roulette didn’t get popular as a gambling game until nearly 150 years later. Almost immediately afterwards, however, gamblers got the idea that they could beat the game by exploiting biased wheels. That meant they observed results looking for numbers that came up more often than average, due to tilting or another defect in the wheel.

It took another 150 years for the next big insight by Ed Thorp, who realised that if the wheel was biased, of course you could beat it. But he appears to be the first person to also realise that if the wheel were not biased, it had to be machined so well that it would be predictable. This insight is a fundamental one about risk in general, not just roulette.

This point is obscured by the English language. When the average person says a roulette wheel is random, she means each number comes up with equal probability – there’s no advantage to betting one number over another. When a statistician says the roulette wheel is random, she means the numbers are unpredictable – not the same thing at all. If the roulette ball landed in the numbers 1, 2, 3 and so on in order up to 36 and then back to 0 (or 00 on American wheels), the outcome would be perfectly random in the first sense in that each number would come up the same number of times. But the outcomes would be completely non-random in the second sense in that the outcome was highly predictable. If the roulette wheel always came up 1 or 2, but mixed the two numbers perfectly, the wheel would be non-random to the average person, but perfectly random to the statistician.

You can easily build a roulette wheel that’s unpredictable – any kind of sloppy engineering will do. But you’re likely to find that this wheel is also biased and that some numbers come up more than others. If you machine the wheel so perfectly that the numbers all come up with exactly the same frequency, it’s likely that someone observing the spin can predict where the ball ends up – not necessarily perfectly every time but with enough success to win money in the long run. You’ll find it hard to build a wheel that’s both completely uniform and completely unpredictable. In fact, no one has ever managed to do it, with roulette or anything else. If no one can build one under controlled conditions, there’s no reason to expect events that are both completely uniform and completely unpredictable to occur naturally.

Beating roulette

What Thorp (and many others who have attacked this problem) discovered is that the roulette spin has two phases:

✔ In the first phase, the ball spins around the outer lip of the bowl. This action is highly predictable, you can easily compute when the ball will start spiralling down from the lip and what number will be underneath it when it does.

✔ The second phase starts when the ball begins spiralling downward. The path of the ball becomes hard to predict, due to deflectors built into the wheel and the violent bounces possible when the ball first makes contact with the wheel. But that unpredictability isn’t uniform and you can determine the segment of the wheel where the ball is most likely to end up.

Thus you have a period of predictability in which the result can be calculated, followed by a period of chaos, in which statistical patterns can be found. As you get deeper into this problem, you find phases within phases, but at each level you can segregate the phases into predictable elements to be computed and chaotic elements for which you compile statistics.

Although some people are quick to call something random, those with a practical interest in risk instead drill in to tease out aspects of a situation that can be calculated and aspects that can be analysed by frequency. Successful practical risk takers in almost any field ignore the obvious high-level prediction modelling or statistical analysis that occurs to a novice or is simple enough for a textbook approach. Risk takers end up obsessively measuring things other people think are irrelevant and compiling statistics about seemingly unrelated or trivial things while showing no interest at all in the things other people think matter.

For almost any risk of practical importance, a line risk taker, such as a portfolio manager, actuary and credit officer hired to choose which risks to take, automatically handles all the obviously predictable aspects and is well aware of the statistics about the range of outcomes. In order to add value, risk managers have to drill down to a deeper level to find the randomness within the predictability and the predictability within the randomness. It’s always there; you can always find deeper levels than those line risk managers use. Going one level deeper in your analysis is the trademark of a good risk manager.

Comparing to quantitative modellers

Most quantitative modelers model situations, including predictable aspects and random ones. But unless they have long experience managing real risk, they never have the obsessive drive to go deep enough in their risk analysis. Moreover, they’re constrained by needing to produce results that are explainable and statistically significant, and that can be achieved at reasonable cost through accepted methods. These handicaps usually make the results worthless for risk management.

The goal of most quantitative research is to model things at a level higher than the front-line risk takers. Research shows that if you model what an expert does, the model usually performs better than the expert. In other words, experts discover how to do something, but usually insist on adding intuitive judgement that actually makes things worse. If you just do what the expert says she does, you’re better off.

But risk management isn’t about making slightly better choices than the front-line risk taker. It’s about drilling down to a deeper level where the real uncertainty resides. The front-line risk taker already manages the risk at the level she understands, and all the preceding levels as well (or if she doesn’t, it’s easy to fix – you don’t need a risk manager to do it – fire her).

This is the big gulf between risk management and most conventional quantitative modelling. If you see people casually assuming something is random and compiling statistics or casually assuming something is predictable and making calculations, you’re not looking at risk managers. Risk managers are sure that they can exploit the wisps of pattern in other people’s randomness and the wisps of noise in other people’s signals. You see them obsessively cleaning data that everyone else thinks are both irrelevant and already clean enough for all practical purposes. At the same time, the risk managers are ignoring what everyone else thinks are the important data.