Randomly Walking Through Wall Street

The daily closing values of the S&P 500 Index from January 1, 1926 through June 11, 1993, a total of 17,610 values, were entered into Mathematica, and the day-to-day percent changes were calculated. Using the Standard Mathematica Package Statistics 'Continuous Distributions' and the built-in function NonLinearFit, procedures were developed to find the probability distribution that best models these daily changes. Although the log-normal distribution has been used traditionally, we found that a logistic distribution provides the best model, having a coefficient of determination 0.998. Using this model and Mathematica to simulate stock market performance we have found that, although the short-term changes in the stock market can often be explained by world events, longer-term behavior of the market can be modeled with accuracy. Simulations for time periods between 6 months and 10 years show that, although dollar-cost average investing has less volatility, the long-term investor can expect a higher return from a lump-sum investment.