The (Mis)Behavior of Markets

by Benoît B. Mandelbrot

Brownian motion is a bank economist’s best friend. When asked by his boss to predict the dollar-sterling rate a year from now, he can smartly sidestep the question. Working from today’s rate of $1.65 to the pound, he gives, not a specific forecast of $1.70, but a vague Brownian range: “The pound will trade between $1.55 and $1.75, and there is a good case to make for it trending up within that range—if the U.S. economy stutters, if inflation in Britain rises moderately, if ...” Of course, he is only staking his job on the vague range, not on the what-ifs—so he survives to forecast another year. But what happens if the exchange rate wanders farther than the square-root-of-time law forecasts? Trouble for the economist, obviously. How could that happen? Easily, if exchange rates exhibit long dependence. A rate move in one direction will tend to continue on the next day, and a few days later. The rate will still bounce around from day to day. But in the long run it will drift farther and farther from its starting point. The rates are no longer fluctuating by the blindest form of chance. Now the game is rigged, just like the Nile floods running in sequence.