Читать книгу Financial Risk Management For Dummies - Aaron Brown - Страница 9

A famous scene in the film Zero Dark Thirty involves the director of the Central Intelligence Agency conferring with some subordinates about whether Osama bin Laden is in a house the agency has identified in Pakistan. “I’d say there’s a 60 per cent probability he’s there,” says the deputy director. What exactly does that mean?

The most common interpretation of probability statements among quantitative people is frequentism. In this view, the deputy means that given 100 potential missions with the same quality intelligence as is available for this one, he would guess about 60 of them would have the target’s location identified correctly.

The second favourite interpretation is that the deputy would bet $60 against $40 that Osama bin Laden is in the house. This goes by the name Bayesianism.

In neither interpretation are people talking about the actual risk of the mission. Frequentism talks about long-term average outcomes of long series of hypothetical future missions. Bayesianism talks about opinions of risk. That’s why we call them models. Models can be useful, but you have to be aware of the differences between model and reality.

Financial risk managers make use of both models, although Bayesianism is generally more useful than frequentism. But they use many other models as well. Most importantly, they pay careful attention to precisely which model is in use. Never make a probability statement without being sure about the model you’re using, and never fall in love with one particular model so that you ignore evidence from other approaches.

The next sections discuss these two risk models. Despite the deep philosophic gulf between the two camps, frequentist and Bayesian statisticians mostly use the same tools and mostly come to the same conclusions. When the data clearly indicate a conclusion, any reasonable method works. If a drug immediately cures 90 per cent of the people who take it, philosophic subtleties don’t matter. The drug works for frequentists and Bayesians and everyone else. On the other hand, if 51 out of 100 people survive after taking a drug, but 50 per cent survive untreated, and a few ambiguous cases come to light and some people experience serious side effects, statisticians cannot help. You need more data and doctors and other subject-matter experts to examine the details of the experiment – not a better analysis.

In cases where there’s moderate but not overwhelming evidence in favour of a proposition, statisticians have something to offer and may disagree. However, you don’t find that frequentists are more apt to agree with other frequentists, nor Bayesians more apt to agree with other Bayesians. Different conclusions depend on the models and forms of analysis and on adjustments to the data or assumptions, not on the fundamental approach to risk.

Counting frequency

Early risk theory was based on a limited idea of uncertainty. It models risk as a casino game that can be played over and over, with the range of outcomes and their probabilities known to all. The early view didn’t allow for the possibility of someone being able to get superior information about outcomes or to influence those outcomes. This type of risk simply doesn’t exist except in casinos and other gambling places where extreme care is taken to create it (and as I show in “Analysing Roulette” later in the chapter, it really doesn’t even exist there). The dice games and lotteries used in the early study of risk are poor models for the uncertainty that people face in real life.

Characterising risk

Although risk is one of the most important facts of life, language about it is extremely limited. It’s hard to discuss risk without expressing an opinion. You say someone was reckless, foolish or careless if she takes a risk that turns out badly. If things go well, that same person is judged as bold, innovative or a creative thinker. Even before you find out the result, if you like a risk you say the risk taker is shrewd, daring or taking a calculated risk. If you don’t like a risk, the risk taker is rolling the dice or speculating.

The difficulty of making neutral statements about risk isn’t a linguistic anomaly of English. It exists in many languages – perhaps all – and it reflects a deep misunderstanding of risk, which in turn leads to mismanagement of risk.

The first step in understanding risk is to accept that a bet is a bet whether you like the bet or not and whether it wins or loses. The same mathematical principles apply to all bets. Risk managers struggle constantly against entrenched superstition that ascribes outcomes to luck, fate, righteousness or other immeasurable moral forces – even from rational people who would never blame demons for fairies for a physical event.

Life is a casino, and everyone bets. The people who deny this fact make the biggest and least attractive bets.

The early model of risk is known as frequentism, which defines the probability of an individual event only in terms of the long-run frequency of a series of independent events.

But how does a frequentist answer a practical question about risk, such as, ‘What is the probability that it will rain tomorrow?’ You can’t repeat tomorrow 1,000 times (or even twice) to define the probability.

A frequentist cannot answer the rain question directly. She may build a model that estimates the probability of rain. Her model may say that there’s a 60 per cent chance of rain tomorrow. Running the model in the past, she finds that it rained on 52 of the last 100 days when the model said there was a 60 per cent chance of rain. The frequentist could construct a 99 per cent confidence interval for the probability of rain tomorrow that runs from 39 per cent to 65 per cent.

That sort of sounds like the frequentist is saying that the probability of rain is 99 per cent certain to be between 39 per cent and 65 per cent tomorrow. But she’s not saying even that. The frequentist can’t make any statement at all about tomorrow or about rain probability. The statement only concerns days in the past and refers to the probability of getting certain random samples in the past. Moreover, it’s not clear how you can put that prediction to use when deciding whether to carry an umbrella to work or to plan an outdoor wedding reception or to write a weather insurance policy.

To be fair to frequentists, they understand the problems and use techniques to give their statements practical meaning. For example, good frequentist statisticians insist on testing the weather model on the best available alternative models rather than arbitrary hypotheses, doing out-of-sample validation, and testing assumptions like constant rain probabilities. But most uses of frequentist statistics aren’t done with these safeguards, and that’s true of academic journals as well as popular media. And even with all the safeguards, the basic logical problems persist.

For most practical problems of risk management, frequentist statistical methods cause more misunderstanding and error than they provide solid guidance.

Betting with Bayes

An entirely different theoretical understanding of risk was developed in the 1930s by a brilliant Italian mathematician, Bruno de Finetti, and fully formalised in the 1950s by American Jimmie Savage. Like frequentist probabilities, Bayesian probabilities require events with a known range of outcomes that cannot be influenced by the individuals estimating the probabilities. However, Bayesians understand that different people can have different information and opinions about events, and that some events cannot be repeated.

Bayesians define probability as subjective belief, measured by how much you would bet on various outcomes. For example, if you’re willing to bet $40 against $60 that it will rain tomorrow; and equally willing to bet $60 against $40 that it won’t rain tomorrow; your subjective probability that it will rain tomorrow is 40 per cent. Of course, other people can have their own views, which may differ considerably from yours.

Bayesians choose their ideology and are often passionate about it. For many thinkers, Bayesianism is a reaction to the perceived failures of frequentism.

The great virtue of Bayesian methods is that they can give direct answers to questions such as, ‘What’s the probability that it will rain tomorrow?’ According to strict Bayesian theory, different individuals can have different probabilities for the same event, but one individual never has conflicting beliefs. That is, the Bayesian who thinks the probability that it will rain tomorrow is 40 per cent cannot also believe that the probability that more than a centimetre of rain will fall tomorrow is 50 per cent. The second statement must have the same or lower probability than the first.

Like frequentists, Bayesians turn all risk questions into gambling games. But frequentists use games like dice and coin flips, in which everyone can agree on the probabilities, and the games are fair; not necessarily fair in the sense of having equal odds of winning, but in the sense that no one can influence or predict the outcome.

Bayesians, by contrast, use what a gambler calls proposition bets, bets about the truth of some proposition, such as ‘It will rain tomorrow’ or ‘Germany will win the World Cup’ rather than bets on a mechanical device like dice or cards. Wagers on sporting outcomes are often proposition bets. De Finetti’s famous example is a bet that pays £1 if there was life on Mars a billion years ago. Assume that an expedition will settle this question tomorrow, and consider the price at which you would buy or sell the £1 claim. If you price the claim at £0.05 and are willing to buy it for that price or sell it to someone else at that price, then di Finetti says that the probability of life existing on Mars a billion years ago is 5 per cent … to you.

Notice that this bet isn’t fair in the frequentist sense. Both sides are expected to do their own research and have their own opinions about the outcome. In many such bets, both sides are expected to attempt to influence the outcome – the simplest example is two competitors betting on the outcome of a match they’re about to play.

In place of fairness, Bayesians prize consistency. It’s entirely possible for two equally good frequentist models based on the same data to give different answers to the probability of rain tomorrow. But for a Bayesian, any individual at any given time can give only one answer to that question. Moreover, frequentist methods can give inconsistent results – for example, a probability of rain tomorrow greater than the sum of the probability of rain before noon tomorrow plus the probability of rain after noon tomorrow. That cannot happen for a Bayesian.