Random rates of growth and return: introducing the expo‐normal distribution

Expected values and standard deviations of the geometric means of independent positive random variables are useful indicators of the long‐term profitability of an investment, or survival of a biological population. Often these quantities cannot be evaluated in a closed form, or even if they can, there may be a choice between several probability models for the ‘annual’ growth factors. This paper formulates approximations for geometric means and standard deviations. It evaluates their performance and compares the best of them with the exact values for selected probability models of the annual factors. Among these is a new model for annual log‐returns, called the expo‐normal law. This is the law of log X, where X has a normal law conditioned on X>0. Its properties are developed in some detail. It is found that for the ranges of annual means and standard deviations typically encountered in financial applications, the longer horizon values depend little on the choice of probability model, and that, where possible, exact evaluation is computationally simpler than using approximations. Copyright © 2002 John Wiley & Sons, Ltd.