XIII. Compound Interest and the Instantaneous Rate of Interest

This conclusion, however, applies o~ly to a particular interval of time. So far nothing has been said about the relationship between the r~tes of increase over different Rates of Inon ... for intervals of time, whether these intervals In ... tmenta for dU-are of the same length but beain and end fennt Intenals of e" time at different moments of time, or whether they are of different lengths. In so far as different but equidistant pairs of moments are concerned, one would be inclined to assume that the rate of increase would have to be the same. And under perfectly stationarycondi tions this would undoubtedly be true. But, as we have seen, a completely stationary state could be reached only gradually, and after a very long time; therefore as long as the relative values of the different commodities con170 CB.xm The 1 nBtanta'tIRffUS Rate of Interest 171 tinued to change, the rate at which any unit of investment in their production increased would necessarily change also. There would be no necessary relationship be tween the rates of increase over different periods in this case, except in so far as the periods overlapped, and then it would become a special instance of the problem of the rates which will rule for periods of different lengths.

This latter problem may be best considered by com paring the rates which will prevail during two or more very short periods of equal length immediately succeeding each other with the rate for the longer Intervals of cUfterent period to which they add up. If, for lengths example, we call the present moment 1 and consider two later moments which we will call 2 and 3, the question we have to answer is what will be the relationship of the rates of increase obtained by investing from 1 to 2 and from 2 to 3, to the rate of increase obtained by investing from 1 to 3. The answer will evidently depend on the conditions under which it will be impossible to increase the total product by investing more for the shorter periods and less for the longer period or vice versa. At first one might be inclined to assume that this con dition will be satisfied when the increase in the product obtained by investing a given quantity of input for the longer period is equal to the sum of the R"- f I .. a..,1 0 Dcrease DO.

increments of the product obtained if simply proportional din t ·t· f· t . to length of Interval correspon g quan lIes 0 mpu are Invested for each of the two shorter periods. But on closer examination this answer proves to be incorrect. It can easily be shown that equilibrium requires that the rate of increase over the longer period should be equal not to the sum of the (percentage) 1 rates of increase over the two shorter periods but to their product. If, for example, the rate of increase due to the investment over the shorter periods is the same for both periods and amounts in each 1 The essential point is that the rate is expressed as a ratio and not as a simple time rate in absolute terms. Cf. below, p. 177.

172 Investment in a Simple Economy PT. 11 case to an increase to 1·01 of the value of the input invested, then the rate of increase over the longer period will have to be such as to give a product not simply 1·02 times the value of the input invested but 1·01 x 1·01 or 1·0201 times that value. The proof is as follows. Let us suppose that at first the distribution of input between the longer and the two shorter investment periods was such that the rate of increase obtained over each of the two shorter periods was just half as great as the rate of increase obtained by investing for the longer period. If now some quantity of input which used to be invested for the longer period is invested only for the shorter period, i.e. from 1 to 2 instead of from 1 to 3, it will give an addition to the output at 2, which, compared with the original value of the input invested, will already show an increase by half the amount by which that input would have increased by the end of the longer period. And it will be possible, without changing the amount of output originally avail able at 2, to invest an amount of input equivalent to the output obtained at 2, from 2 to 3. This amount, which will already represent say 1·01 times the amount first invested, will then further increase to 1·01 times its present magnitude or, that is, to 1·0201 times its original magnitude. This means that if the rate of increase obtainable by investing for the longer period were only twice as large as the rate of increase over each of the two shorter periods, a greater return could be obtained by investing for the shorter period only in the first instance, and then reinvesting an amount equivalent to the resulting product for the second short period. It would be profit able to invest for the longer period only if the rate of increase were at least equal to the product of the rates of increase obtainable over the two shorter periods.

So long as we assume completely stationary conditions and, consequently, that the rates of increase for all periods of equal length will be the same, all that this result CH. XIII The Instantaneous Rate of Interest 173 means is that the rates of increase over periods of different lengths will have to correspond to the familiar law of com pound interest. The rate of increase over any long-period which is divisible into n shorter periods Theralerulinglorlhe of equal length will in general be equal longer Inlerval musl be equal 10 Ihe proto the nth power of the rate of increase duol 01 Ihe rales (or I al\ Ih. .horlor inlerapp ying to any of the shorter periods. vals Into which It can And, as will soon become apparent, it is be divided really compound interest which is the fundamental phenomenon: "simple interest" is only a simplifica tion which is convenient for practical purposes but is rather misleading if used in theoretical analysis. But for the present I want to emphasise the still more general concept of the rate of increase over any period being equal to the product of the rates of increase during all the shorter periods which it contains. These latter rates may, as we have seen, vary from one short interval to the next, but the rate will have to be uniform for all the input invested during anyone such interval.

The relationship which has to prevail between the rates of increase over shorter and longer periods must of course apply however short we make the shorter intervals of which we suppose the longer ones to be The InstantaneoUl composed. And by decreasing the length rale ollnleresl of these shorter periods further and further, until at the limit they approach mere moments of time, we finally arrive at a concept which will prove useful when we come to give a more exact formulation of the connection be tween the productivity of investment and interest. This concept is the instantaneous rate of interest or "rate of interest at a moment of time" (Wicksell's Verzinsungs energie: literally, "force of interest ").1 The meaning of this concept may be best explained by comparing it with the concept of the velocity at a 1 Cf. Wicksell, Lecture8, vol. i, p. 178, and 1. Fisher, The Nature of Capital and Income (1906), p. 359, where the same magnitude is described as the " rate of interest per annum computed continuously".

174 Investment in a Simple Economy PT. n moment of time of a body which is moving at a uniformly accelerating speed. At least for stationary conditions, where the force of interest would be the same at every moment, the case of a uniformly accelerating velocity provides a complete parallel. In this case the velocity of the body will change during any interval of time and, in consequence, the actual distance travelled during any interval will not give an exact expression of its speed at a moment of time. Similarly our rate of increase of the value of any unit of input invested will change during any interval, however short, and no actual increase during any such interval will give us an exact measure of the rate of increase at a moment of time. And because, since movement can be described only by stating the distance travelled during some finite period of time, the only way of stating the speed at a moment of time is to state the distance which would have been covered if the instant aneous speed had continued for a period of time, there fore, since growth in value can be described only by stating the amount of increase during some finite period of time, the only way of stating a rate of increase at a moment of time is to sta.te the amount of the increase that would have taken place if the instantaneous rate of increase had prevailed for a definite period of time. And just as we speak of a velocity of so-and-so many feet per second, although the velocity of a falling body never remains constant even for a second, so we speak of an instantaneous rate of interest of so many per cent per annum, although of course this rate does not actually continue throughout the year, but applies only to a particular moment.

In more concrete terms, an instantaneous rate of interest of 5 per cent per annum will, in consequence of the continuous compounding of interest accruing at every RelalloDllblpioelrect-moment, mean an effective increase by the Ive raie or Inie_i end of the year of 5·127 per cent, while in order to obtain an effective increase of only 5 per cent on the initial value by the end of the year, an instantaneous CR. XIII The Instantaneous Rate of Interest 175 rate of only 4·873 per cent per annum would be re quired. This relationship between the instantaneous rate (expressed per annum) and the resulting effective increase over the year if interest is compounded continuously, can best be shown by means of the familiar -Illustrated by com compound interest curve. The character-pound Interest eurve istic attribute of a compound interest curve (as of all " exponential" curves of which it is a particular example) v p o M N t FIG. 11 is that at every point on the curve the tangent is always in the same proportion to the corresponding value of the ordinate. If, for example,. the ordinate of the point P in the diagram (Fig. 11) is 2 and the slope of the tangent at this point is 2/5, then the tangent at the point P' with an 'ordinate of 3 will be 3/5, and so on. Now the slopeo( the tangent at any point divided by the ordinate repre sents the instantaneous rate of interest, or force of interest, at that point. Its immediate expression,a rate of increase divided by an absolute quantity, is, however, a pure number with no obvious meaning. It assumes concrete meaning only if we express it in terms of the proportional 176 Investment in a Simple Economy PT. II increase in the original quantity which it would have caused if it had continued to operate for a definite period, say one year. If in Fig. 11 the distance between the points M and N on the abscissa represents such an interval of one year, then the ratio of QR to RN will represent the force of interest expressed per annum.

It is at once apparent that the effective increase over the year is greater than this percentage. At this rate of instantaneous compound interest, the initial quantity MP will actually have increased by the end of the year to NP'. And the average rate of the actual increase during the year will be expressed by the slope of the chord P P' . This slope represents the effective per annum rate of interest in the usual sense. It will be seen that it must necessarily be greater than the instantaneous rate or force of interest. The situation becomes considerably more complicated, of course, as soon as we drop the assumption that the force of interest is the same at every moment, which will be true only under completely stationary conditions. In cases where the instantaneous rate is not the same at every moment, the effective rate of interest can, strictly speaking, be obtained only by integrating, over the rele vant interval, a function describing the absolute rate of interest at successive moments of time.

Before we proceed further it will be useful to consider Ambiguity of the in greater detail a distinction which is lerm .. rale" implicit in the discussion of the last sec tion, and which, if not clearly understood, is liable to cause considerable confusion. The source of this confusion is the ambiguity of mean ing, or perhaps merely the inexact use, in common parlance, of the term" rate" . The rate at which anything proceeds refers in the first instance to the absolute magnitude of the movement or other change during a unit of time. It is in this sense that we speak of movement at the rate of so many feet per second, of wage payments at the rate of CR. XIII The Instantaneous Rate of Interest 177 so many shillings per hour, etc. But when we are referring not merely to the rate of flow, but to the rate of continuous change in some magnitude, we can also express this rate as a ratio or proportion of that magnitude itself. This is, of course, what we do when we express the rate as a figure per cent. It is a time rate expressed as a ratio or pro portion.

So long as we think of interest merely as a flow of income which is drawn and consumed continuously as it matures, there is not much danger that this particular way of expressing it will mislead. The The Urate n of In .. difficulty arises only when there are periods terest a rate 01 growth d · h' h' t t . 11 d t expressed as a ra\lo urmg W lC In eres IS a owe 0 accumulate with the principal; this may be the case either between the dates at which interest is periodically paid, or over the longer period before a particular investment bears fruit. In such cases, where we have to deal not simply with a continuous flow but with a continuous growth of an initial magnitude at a given rate, the dis tinction becomes important. For a constant rate of growth in the absolute sense will be not a constant but a decreasing proportional rate (i.e. a rate expressed as a ratio of the magnitude which the quantity in question has reached at any moment), and a constant rate in the sense of a ratio will mean art increasing rate in the absolute sense of the term. 1 It is because the rate of interest is a time rate expressed as a ratio that, in order to obtain it, we have to divide the absolute rate of increase of the product due to a given extension of the investment period by the amount of the product. The difficulty which this seems occasion ally to cause is avoided if the difference between the 1 Cf. F. H. Knight. 1936, p. 444: "The ambiguity of the word • rate' is most unfortunate. In expressions such as the 'rate of interest' the word is inaccurately used as it combines a time rate of flow (correct meaning) with a ratio of this flow to a principal. And in addition there is really involved an instantaneous rate (ratio) of growth with reference to e. continuously changing base."

13 178 Investment in a Simple Economy FT. n, CR. xm concept of a rate in the absolute sense and the concept of a rate in the sense of a proportion or ratio (as used in the term" rate of interest") is always kept in mind. The rate of increase of the product is expressed as an absolute quantity per unit of time; it becomes a rate of interest if we express it as such-and-such a proportion (or per centage) of the total magnitude which is increasing. And the relevant total magnitude for this purpose is of course not that existing at the time the (pure) input was applied, but the magnitude to which it has grown by the point of time at which we wish to describe the rate of increase.