The second term of the return equation is risk compensation. To know how much compensation bearing risk should earn, one must first know what risk looks like: it is not a number but a distribution of outcomes. This distribution has three stable features: the range of uncertainty widens with the square root of time, volatility clusters, and extreme outcomes are more frequent than the normal distribution predicts. Direction is hard to predict, but these features can be estimated to a considerable degree.
In 1900 the French mathematician Bachelier studied bond prices on the Paris Bourse in his doctoral thesis, Théorie de la spéculation. He proposed that price changes could be treated as random: at the next moment, a rise and a fall are about equally likely. He also derived a basic property of random change: the range of uncertainty widens with the square root of time, so when the time span grows fourfold, the range of possible outcomes only doubles. This description came five years before Einstein used the same mathematics to explain Brownian motion.
One point must be added to Bachelier's description: shares have a positive expected return over the long run, which is compensation for bearing risk. But this expected return is very small relative to volatility. With the illustrative parameters in the figure, the probability of a gain over a one-day holding is only slightly above one half; only over a ten-year holding is it clearly above one half. In the short run, direction is almost entirely drowned out by volatility.
Studying cotton prices in 1963, Mandelbrot observed that large changes tend to be followed by large changes, and calm by calm. The phenomenon was later confirmed in almost every financial market. The direction of moves remains unpredictable, but their size can be predicted to a considerable degree. The autoregressive conditional heteroscedasticity (ARCH) model proposed by Engle in 1982 was designed to capture exactly this property, and earned him the 2003 Nobel Memorial Prize in Economic Sciences; in 1986 Bollerslev extended it to the generalized autoregressive conditional heteroscedasticity (GARCH) model, which became the most widely used form.
Direction does not cluster; volatility does
Above: daily changes over 500 trading days. Below: the size of each daily change, ignoring direction. Switch to the comparison to see how a real market differs from one with constant volatility.
Mandelbrot also found that extreme changes occur far more often than the normal distribution predicts. Bachelier assumed that price changes are normally distributed, but in real price changes, large rises and falls far from the mean occur many times more often: the so-called "fat tails". On 19 October 1987 the Dow Jones Industrial Average fell about 22.6% in a single day. Under a normal distribution, measured against a long-run average daily volatility of about 1%, such a fall amounts to more than twenty standard deviations, with a probability so small as to be meaningless in practice.
Extreme moves are more frequent than the normal distribution predicts
The distribution of daily changes over 100,000 trading days (logarithmic vertical axis, so both tails are visible). The curve is a normal distribution with the same volatility. Choose a threshold and compare the number of days beyond it.
The shape of risk determines how a product should present outcomes, and also points to where models and AI are most useful.
Show the range of possible outcomes over different holding periods, rather than a single target price or an average return.
The probability of a gain differs greatly between a one-day and a ten-year holding; do not use long-run conclusions to steer short-term trading.
Volatility and risk exposure are more worth estimating than direction; update risk warnings promptly when volatility rises.
Show extreme episodes that have actually occurred, not only the range of volatility "under normal conditions".