Mathematics of Growth—I

Abstract

THE first three articles in this series discussed the squares method for fitting a straight line trend to a series of sales or earnings figures. It was suggested that the most useful method would be to fit a straight line trend to the logarithms of the individual items in each series. The antilog of this trendline would then give the annual percentage growth trend. While this annually compounded growth trend is a convenient measure for most applications, it does not lend itself as an eflicient measure when one wishes to measure several components of growth. Fortunately, annually compounded growth rates can be converted into a form which will enable the analyst to work such problems. In the last issue of this Journal, Associate Editor John Fountain's article Growth Figured the Simple Way presented a table of continuously compounded growth rates which are faster to use than a book of annually compounded interest rates (perhaps 15 seconds to locate the growth rate rather than 30 seconds with a book or a lot longer if you don't have the book and are working with logarithms). His table was a table of natural logarithms with different headings for the columns to make them easier to read. The article included a delightful exposition of the case for regarding continuous compounding as a more realistic concept of a company's growth pattern than annual compounding. Does a company really grow once a year in an annual jump? . . . No. A company is like a living thing that grows not in annual or monthly spurts, but continuously, every hour, every minute, every second. There is not a great deal of difference between annual growth rates and continuously compounded growth rates -at least not enough to cause one to undergo much of a change in his thought pattern. For example, if a savings and loan association paid 5% interest compounded only once a year, a $100 account would grow to $105.00 at the end of the year. If interest were calculated at the midyear as well as the close, the year-end value would be $105.06. If interest were compounded monthly, it would grow to $105.10 and if compounding was continuous, every second of every minute of every day in the year, the account would grow to $105.12 by the end of the year. Since this isn't very startling, the reader might be wondering what are the practical advantages, if any, of continuous compounding. The advantages come when one is analyzing the components of a company's growth. The reason is that continuous growth rates are additive in nature, while annually compounded growth rates are multiplicative in nature. One of the first practical applications of this difference was Daniel Murphy's reports when he was associated with Faulkner, Dawkins & Sullivan. To illustrate, let us use a hypothetical case in which a company has a 10% improvement in each of the three major factors of earnings growth-sales growth, pretax profit margins, and the tax rate. For this purpose it is more convenient to use the retention rate, the percent of pretax earnings carried down to net income. In Table 4.1 below, we have listed the annually compounded growth rates and also the continuously compounded growth rates: