ABSTRACT
I n his book A Mathematician Plays the Stock Market, JohnAllen Paulos describes a scenario that occurred during the wildtimes when dotcom companies were going public on a daily basis. A certain investor is offered the following opportunity: Every Monday for a period of 52 weeks the investor may invest funds in the stock of one dotcom company. On the ensuing Friday, the investor sells. The following Monday, he purchases new stock in another dotcom company. Each week, the value of the stock purchased has
a probability of 12 of increasing by 80%, and a probability of 1 2 of
decreasing by 60%, depending on market conditions in the previous weeks. This means that on average, the increase in value of the purchased stock is equal to 0.8 × 12 − 0.6 × 12 = 0.1, giving a return of 10% per week. The investor, who has a starting bankroll of ten thousand dollars to invest over a period of the coming 52 weeks, doesn’t hesitate for a moment; he decides to invest the full amount, every week, in the stock of a dotcom company. After 52 weeks, it appears that our investor only has 2 dollars left of his initial ten-thousand-dollar bankroll. He is, quite literally, at a loss to figure it all out. But in fact, this investment result is not very surprising when you consider how dangerous it is to rely on averages in situations involving uncertainty. A person can drown, after all, in a lake that has an average depth of 25 cm. For situations involving uncertainty factors, you should never work with averages, but rather with probabilities! It is easily explained that the probability of nearly depleting the bankroll is large if the investor invests his whole bankroll in each transaction. The most likely path to develop over the course of 52 weeks is one in which the stock increases in value 50% of the time, and decreases in value 50% of the time. This path results in a bankroll of
1.826 × 0.426 × $10,000 = $1.95
after 52 weeks. Running one hundred thousand simulations of these investments over 52 weeks renders a probability of about 50% that the investor’s final bankroll will not exceed $1.95, and a meagre probability of 5.8% that the investor’s final bankroll will be greater than his starting bankroll of ten thousand dollars.
