The Appreciation of Equity Concept and Its Relationship to Multiple Rates of Return
Authors: Phillips Charles E. 
Published: December 2007

Abstract
In this paper, a method is developed and analyzed which resolves the multiple-rate-of-return dilemma, more accurately represents the earning power of invested capital than rate of return, and has the added bonus of being expressible in explicit form; i.e., no trial-and-error solutions are required, regardless of the shape of the income vs time curve. In essence, it assumes that earnings are continually or periodically reinvested at an average company appreciation rate, the total equity is calculated at the end of the project, thereby determining the annual appreciation rate, which indicates the relative attractiveness of the project. A multiple-rate-of-return project analyzed by the appreciation of equity method has a single appreciation rate, which accurately represents the earning power of the capital invested, taking into consideration the company‘s other operations. The fallacy of the high rates of return often found in multiple-rate-of-return projects is the tacit assumption that capital to be needed later in the project can be invested in other projects having that high rate of return until needed; whereas in actuality, the capital to be used later can only be invested in other of the company's projects at the average appreciation rate.
Introduction
The internal rate of return method has been discussed in the literature by a number of authors. It is the interest rate which results in zero profit when income and investment are discounted at that rate. It has a number of advantages over the methods used prior to it, such as payout time, ratio of ultimate profit to investment, or using a percentage of the present worth discounted at "bank interest rates". Various short-cut methods have been devised to calculate it for certain types of decline, such as exponential, harmonic and hyperbolic. However, unless the project follows one of these types of decline throughout its life, trial-and-error calculation must be used. Other disadvantages were recognized by Glanville in one of the original papers on the subject in the petroleum literature. One is that rate of return is only slightly affected by earnings discounted over a long period. This point is discussed later in the paper. Of more importance, however, is the confusion caused by certain projects having more than one rate of return, all apparently of equal economic significance.
Multiple Rates of Return
Fig. 1 is the present value profile of an "acceleration project", that is, one in which money is invested not necessarily to show a profit but to decrease the time required to obtain the income from a project. In fact, acceleration projects will generally result in a net loss. In the example, an investment of $630 results in an incremental income of $1,600 the first year and incremental income of minus $1,000 the second year, or a net loss of $30. This is the same as the present value profit discounted at zero per cent, as shown at the left of the curve. It is noted that both 11.1 and 42.9 per cent result in zero profit, and therefore meet the requirement of the definition of rate of return. Fig. 2 is an example of a project that requires not only an initial investment, but a substantial investment at a later date. This example has three rates of return: 11.1, 25 and 42.9 per cent.
JPT
P. 159ˆ