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are all of these events highly uncertain, but any model that we would develop to try to predict these events is likely to be highly subjective. There is no way of honestly estimating these probabilities. This type of probability is often referred to as uncertainty or Knightian uncertainty. We can say that these events are likely or unlikely or very unlikely, but not much more.

      In some rare and somewhat idealized cases, uncertainty is quantifiable. Some economists like to define risk as quantifiable uncertainty. A good example is the uncertainty involved in tossing an unbiased coin: will it come up heads or tails? The probability that an unbiased coin lands on heads is equal to the probability that it lands on tails, 1/2. Similarly, one can determine the probability for three successive heads followed by two successive tails to be (1/2)⁵, or 1/32. This is the frequentist definition of probability that defines the concept in terms of expected frequency of occurrences, in the limit, for an infinite number of tosses.

      You might argue that quantifiable uncertainty is unrealistic. On the one hand, a perfect coin is a Platonic ideal and no coin is perfectly fair. On the other hand, a coin toss is, in some theoretical sense, predictable. If we knew the velocity and angle of the flick, how the air was moving around the coin as it spun, and the irregularities of the floor upon which the coin was bouncing, we could predict the outcome of the coin flip with a high degree of accuracy. If we are willing to ignore quantum mechanics, we could argue that there are no truly random events, only pseudo-random ones. From a practical standpoint, though, outside of a laboratory, even without quantum mechanical effects, there are so many factors that might impact the result of a coin toss that we may as well consider it to be a random event.

      In human affairs, frequentist probabilities are rare. The world is constantly changing, and experiments with humans cannot easily be repeated with the same initial conditions. Importantly, human beings learn from experience. For example, credit markets after the great financial crisis won't behave like credit markets before the crisis, because we have all learned a lesson, at least temporarily.

      Put another way, human institutions display hysteresis: Their current state depends on their entire history. Though the history of the world doesn't affect a coin toss, the history of the world does have a bearing on the likelihood of a political revolution or the next change in a stock's price. The uncertainty in the behavior of a stock's price is qualitatively different from the uncertainty of a coin flip, because the behavior of people is very different from the behavior of coins. The likelihood of a stock market crash is not like the likelihood of throwing five tails in succession, because market crashes are societal events, and society remembers the last crash and fears the next. A coin doesn't fear a sequence of five tails, and isn't affected by the other coins in your pocket.

       The Behavior of a Share of Stock

      A company – take Apple Inc., for example – is a tremendously complex and structured endeavor. Apple has tens of thousands of employees, owns or leases buildings in many countries, designs products ranging from power plugs and cables to desktop and laptop computers through iPhones, iPads, and the Apple Watch. It manufactures some of them on its own, and outsources the manufacturing of others. It distributes its products through Apple's website and stores as well as via third parties, and sells music, videos, and books over the Internet. Apple advertises, provides product support, maintains websites, and carries out research and development.

      Amazingly, the entire economic value of this organization can, in theory, be summed up in just one number, the quoted price of a share of Apple's stock.2 The quoted stock price is the amount of money that was required to buy or sell just one incremental share of the company the last time the stock traded. Financial modeling is an attempt to project the value of the entire enterprise into that single number that symbolizes its value. It aims to tell you what you should pay today for a share of the company's future performance.

      The task of a would-be forecaster sounds impossibly difficult, and it gets worse. In order to predict the movement of stock prices, it is not enough to understand all of the complexities of a corporation and its place in the economy. In addition, we need to understand how all the other participants in the market view the company as well. Predicting the direction of stock prices, as Keynes wrote, is a lot like predicting the winner of a traditional beauty contest; you are not trying to figure out who is the most attractive, but who the judges think is the most attractive (Keynes 1936). In the long run, fundamentals, the state of the economy, and the state of the company count. Sentiment can maintain its influence for only so long. In the short run, though, people's opinions and passions count for a lot. But then again, the short run influences the long run. Short-term changes in the price of a company's stock will affect the behavior of the company, its customers, and its creditors; psychological reality and economic reality interact, and are in fact indivisible.3 When people thought Lehman Brothers might go bankrupt in late 2008, they wouldn't continue to lend it money, so it went bankrupt.

      The more you think about it, if you are honest and introspective, the more you realize that valuation is a vastly complex problem involving economics, politics, and psychology – the whole world, in fact – at both short- and long-term time scales. That the efficient market hypothesis is able to say anything universal about valuation is in fact quite remarkable. And it does it by ignoring as much of the particulars as possible.

       The Risk of Stocks

The most important feature of a stock is the uncertainty of its returns. One of the simplest models of uncertainty is the risk involved in flipping a coin. Figure 2.1 illustrates a similarly simple model, a binomial tree, for the evolution of the return on a stock with return volatility σ and expected return μ over a small instant of time Δt.4 The mean return during this time is μΔt, with a 50 % probability that the return will be higher, μΔt + σ

, and a 50 % probability that the return will be lower, μΔt − σ.

Figure 2.1 A Binomial Tree for the Future Returns of a Stock

      The volatility σ is a measure of the stock's risk. If σ is large, then the difference between an up-move and a down-move will be significant.

This simple model turns out to be extremely powerful. By adding more steps, as in Figure 2.2, and shrinking the size of Δt, we can mimic the more or less continuous motion of prices, much as movies produce the illusion of real motion by changing images at the rate of 24 frames per second. Assuming successive returns are uncorrelated with each other, in the limit as Δt → 0, the distribution of returns at time t becomes normally distributed with mean total return μt and standard deviation of returns σ

. Various normal distributions are portrayed in Figure 2.3.

Figure 2.2 Binomial Tree of Returns with Four Steps

Figure 2.3 Examples of Normal Distributions

      The key feature of this model of risky securities is that the entire behavior of the security is captured in just two numbers, the expected return μ and

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