XI. Third Approximation

CHAPTER XI THffiD APPROXIMATION TO THE THEORY OF INTEREST (ASSUMING INCOME UNCERTAIN) § 1 Up to this point we have ignored the element of chance, by assuming. that the entire. future income~stream, or at any rate, such portions of it as need to influencepresent choice, are foreknown and mapped out .in advance. In the preceding chapter, we have assumed inventions to be ~prisesc-sudden enlargements of knowledge coming upon us without previous anticipation. In other words, we have assumed that men disregard future inventions and act as though their knowledge of the future were per fect. This assumption, like the assumption that bodies fall in vacuo, in the ordinary presentation of the theory of gravitation, has enabled us to complete our formal statement of the theory more easily, although at the expense of exact conformity to actual historical fact; for, · world, the most conspicuous characteristic of the future is its uncertaIn y.. onsequen y t e intro duction of the element of risk will give, as by magic, the aspect of reality. The general principles which have been stated, however, will still hold true when we assume uncertainty instead of certainty; they merely require to be supplemented· by other principles.

One consequence of changing our assumption as to the certainty of future events is to compel the abandonment of the idea of a single rate of interest. Instead of a single rate of interest representing the rate of exchange between this year and next year, we now find a great variety of rates according to the risk involved. The rate in every loan 207 208 THE RATE OF INTEREST [CHAP. XI contract is adjusted on the basis of the degree of security given. TIlus, security may be furnished by simple indorse ment of reputable persons, in which case the degree of se curity will be the greater the larger the number of indorsers and the higher tIle credit which they possess; or it may be by the deposit of collateral securities. The necessity of the latter operation will affect a mall's ability to borrow, and limit the extent to which he may modify his income-stream by this means. It will not be possible, as hitherto assumed, for a man to modify his income-stream at will, but only up to the limit of his credit. In conse quence of this limitation upon his borrowing power, he may not succeed in modifying his income-stream sufficiently to bring the rate of preference between present and future income down to the rate of interest ruling in the mar ket; and for like reasons, he may not succeed in bringing the rate of return on sacrifice into conformity with the rate of interest.

One feature of this limitation is the fact that the ability to borrow depends, not so much on the amount of capital which the wouldbe borrower possesses, as on the form in which that capital happens to be. Some secllrities are readily accepted as collateral, and accepted for a high per centage of their face value, whereas others will pass with difficulty and only fQr a low percentage. The recent ten dency to change the organization of business to the corpo rate form has had a striking effect in increasing the power to borrow. Whereas formerly many businesses were con ducted as partnerships and on a small scale, numerous stocks and bonds have now been substituted for the old rights of partnership and other less negotiable forms of security; hence the possessors of these securities have wider opportunities to deposit collateral, and the tendency to borrow has received a decided impulse. This explains to a large extent the investing and speculative mania which followed the recent widespread consolidation and formation of trusts.

SEC. 1] THIRD APPROXIltlATION 209 Where the security does not exist in the' convenient form of written certificates, there is often considerable difficulty in negotiating a loan. If commodities are used for security, they must ordinarily be themselves deposited with the lender, - in other words, tPU~ in pawn. Where the bor rowing takes place in E!!wn sh~ps, it is not because of the inadequacy of the security, but because of its incQnvenient t form, that the rate of interest is usually very high. The f pawnbroker will need -to charge a high rate of interest, partly because he needs storage room for the security he accepts, partly because he needs special clerks and experts to appraise the articles deposited, and partly because, in many cases, he needs to find a market in which to sell them when not redeemed. He is, moreover, able to secure these //'" ~_ high rates partly because ~wnbroking is in bad odo~, and ~O\-t/ lC·L·'\.....those who go into it therefore find a relative monopoly; and partly because of the fact that the customers usually have, either from poverty or pe~~~-p-ecul.iarity, a reia. tively high valuation of presentOverJllt1lre income. The effect of their flocking to the pawn shop is to reduce this high valuation; but it will not reduce it to the general level in the community, because these persons do not have access to the loan market in which the ordinary business man deals. To them, undoubtedly, the fact that they cannot borrow ~xcept at high or usurious rates is often a great hardship; but it has, as one beneficent effect, the discouragement of getting unwisely into debt. Those who patronize pawn shops to a large extent do so because they possess little foresight and selfcontrol, and the im pediment which they find in the shape of a high interest rate in a measure takes the place of the selfcontrol and foresight which they should possess. Were it possible for this class to borrow at lower rates, many of them would sink even more deeply into debt than they actually do. Thus, if slave-rJ were legalized ,and it were possible for a man t(; .:.mortgage the, income from his own labor, it isunfortu: natel~e that many WOllld avail themselves of ,this privi p 210 THE RATE OF INTEREST [CHAP. XI lege, and would drop to the lowest place in the economic scale,- slavery. The fact that such contracts are illegal fixes a limit below which the ordinary ne'er-do-weel can not fall. At this point his rate of preference for present over future is not in harmony with the rate of interest in the community. When the market rate of interest is 5 per cent., he may feel a rate of preference of 25 per cent.

§ 2 We find, therefore, that the introduction of the element of chance, and the necessity of overcoming it by the giving of security, has as one of its effects the splitting up of the market into a number of sub-markets. Instead of one huge market in which there is a single rate of interest, to which every individual conforms his own rate of preference, we now find a number of separate markets, a number of different rates of interest, and a very imperfect adjustment of the individual rates of preference to those rates. We need here to emphasize the distinction 'between a commercial rate of interest which includes risk and a pure or riskless rate of interest. 1 The commercial rates vary widely, although the range of variation for rates on loans easily negotiable is relatively small. In ordinary real estate mortgages in the same market the range of variation is seldom over 1 per cent. If we pass from explicit interest, or the rate of interest involved in a loan contract, to implicit interest or the rate involved in purchases and sales of property of various kinds, we see again that the greater the risk the higher the "basis"

on which a security will sell. A" gilt-edge" security may sell on a 3 per cent. basis, when a less known or less salable security will be selling on a 6 per cent. basis. The element of risk will affect also the value of the collateral securities. Their availability for this purpose will increase their sala1 See Glossary under HBasis"; also The Nature of Capital and Income, Chap. XVI.

SEC. 3] THIRD APPROXIMATION 211 bility and enchancetheir price. On the.other hand, when,as in times of crises,the collateral is imperatively demanded, it often happens that, for purposes of liquidation, it is sold at a sacrifice. In the same way that risk causes the rates of explicit interest in a community to diverge from each other, or causes rates of preference to separate from rates of interest, it will cause the rates of implicit interest to diverge. The same individual who would borrow, if he couId, at 25 per cent., but who lacks the necessary security, must devote his energies instead .to acquiring or producing instru ments which will have a return on sacrifice at the 25 per cent. level. Although it would be more economical, if he could only borrow the money, to build durable houses, he will build inferior ones. Hence the anomaly, that even in countries where the rate of interest is low, there will be primitive communities in which the instruments possessed, in the form of dwellings, tools, implements, etc., are far less substantial than is compatible with the low rate of int.erest.

§3 Among other pheuQmenawhich followfrom the existence of risk are the variations· in the d.!lrationof loans. Where the future is regarded as safe, loan contracts tend to be longer in time than otherwise. Railway and government securities are thus often drawn for half a century or more. On the other hand, to provide' for the uncertainty of the immediate future, the "call loan" is devised. This is a loan which has no specifieddue date, but can be.demanded at the option of the lender whenever some circumstance makes this course advisable. A loan contract of this kind brings the burden of risk on the borrower and relatively relieves the lender, and in consequenge, under such con ditions, the rate on·call loans will usually tend to be low. The same principles will explain the low rate of implicit interest in many cases. Where a security, because it is well 212 THE RATE OF INTEREST [CHAP. XI I I known, or for any other reason, has a high degree of sala bility, that is, can be sold on short notice without great sacrifice, the price will be higher than otherwise, and the rate of interest it yields will therefore be low. Hence it is that the rate of interest on individual mortgages will be higher than the rate of interest on more marketable se curities. It is usually considered an advantage to any stock to be listed on the stock exchange; for, being thus widely known, should the.necessity arise to sell it, there will be found a more ready market.

The most salable of all properties is, of course, money; and as Karl Menger has pointed out, it is precisely this salability which makes it money. The convenienceof being able, without any previous preparation, to dispose of it for any exchange, is itself a sufficient return upon the capital which a man keeps in this form, and takes the place of any rate of interest in the ordinary sense of a money payment;. § 4 A further consequence of the introduction of the element of ~is the wide divergence between the actual rate of return realized by an investor and the expected rate. When risk was regarded "as absent, it was assumed that the ex pected always happened; but in the actual world this is far from true. Those who invested in some of the mining "bonanzas" many years ago have received a rate of return of many hundred per cent.; and far in excess of the rate of interest which they would have been willing to take for a loan. Reversely, those who invested in the South Sea Bubble found an opposite disparity between their expecta tions and their realizations. Risk is especially conspicu ous at the time of new inventions or discoveries. Almost , all prediction is based on a belief in the repetition of p~st experience; but at these times, past experience is a poor guide. When new inventions are made, uncertainty is introduced, speculation follows, and after that, great SEC. 5] THIRD APPROXIMATION 213 wealth or ruin. The history of gold and silver ~iscoveries and of the invention of rubber, steel, and electrical appli ances, is filled with tales of thousands of wrecked fortunes, by the side of which tower the fortunes of to-day's nou veaux riches.

The rate of interest is always. based upon expectation, however little this may be justified by realization. ..Man / makes his guess of the future and stakes his action upon it. In his guess he discounts everything he can foresee or estimate, even future inventions and their effects. In a recent estimate of the value Qf a copper mine, allowance-;asH j ~ made for future economies from inventions which might reasonably be expected. So, also, the buyer of machinery allows not simply for its depreciation through physical wear, but for its being possibly superseded. New investments in steam railroads are to-day made with due regard to the possibility that the road may within a few years be run by electricity. It may easily happen that in a country consisting of oversanguine persons, or during a period when business men are overhopeful, the rate of interest will be higher than the event justifies. It is probable that, in ordinary communities, realization justifies the average expectation; but in the· individual case this is not always true, otherwise there would be no risk. Risk f J is due to partial knowledge. Our present acts must be controlled by the future, not as it actually is, but as it looks to us through the veil of chance.

§ 5 In the preceding section we discussed the effect of risk on the pseudo-or impure rate of interest; ~ that is, the rate on unsafe investments. But even the pure rate of interest; or the rate on safe investments, is affected by risk. The effect is different according to the various conditions which may influence the rate of preference for 1 See The Nature 01 Capital and Income, Chap. XVI, § 8.

214 THE RATE OF INTEREST [CHAP. XI present over future income. Where the risk relates to human life, the rate of preference for present over future income is increased.! Consequently the rate of interest, even on safe loans, will be raised by the existence of such risk. The man who looks forward to a short or precarious existence will be less likely to make permanent investments, or, if he makes them, less likely to pay a high price for them. Only a low price, that is, a high rate of interest, will induce him to invest. When the risk relates, however, not to the duration of life, but to the incomej stream, the effect upon the rate of interest will depend [ upon which portion of the income-stream is subject to risk. If the immediately ensuing income is insecure, whereas the remoter income is sure, the rate of preference for immediate as compared with remote income will, as was shown, be high, and consequently the effect of such a risk upon the rate of interest will be to raise it. But if, as is ordinarily the case, the risk applies more especially to the remoter income than to immediate, the effect is the exact opposite; namely, to lower the rate of interest on a safe loan. This is, perhaps, the typical case. If a man regards the income for the next few years as sure, but is in doubt as to its continuance into the remoter future, he will be more keenly alive to the needs of that future, and will consequently have a less keen preference for the present. He will then be willing, even at a very low rate of interest, to invest, out of his present assured income, something to eke out the less sure income of the future. The effect of risk in this case, therefore, is to lower the rate of interest on safe loans, though at the same time, as already explained, it will raise the rate of interest on unsafe loans. Consequently, in times of great social unrest and danger, we witness the anomalous combination of high rates where inadequate security is given, coexistent with low rates on investments regarded as perfectly safe. In commercial language, when an investor cannot find many investments into which he may 1 Cf. Carver's The Distribution 01 Wealth, p. 256.

S:mc.6] THIRD APPROXIMATION 215 put his money without risk of losing it, he will pay a high price for the few which are open to him. It has been noted in times of revolution that some capitalists have preferred to forego the chance of all interest and merely hoard their capital in money form, even paying for storage charges, which amounts to a negative rate of interest. § 6 When risk thus operates to lower the rate of interest on safe investments and to raise the rate on unsafe in vestments, there immediately arises a tendency to dif ferentiate two classes of securities and two classes of investors, -precarious securities and adventurous investors on the one hand, and safe securities and conservative investors on the other. Risk is inevitable in every business, but is regarded by most people as a burden; hence the few who are able and willing to assume this burden become a separate class. To-day, when any enterprise is organized in corporate form, it is usual to recognize this tendency by dividing the securities into stocks and bonds, the stock holder being the person who assumes the risk and, theoreti cally at least, guarantees that the bondholder shall be free of all risk. Which persons shall fall into the class of risk takers and which not, is determined by their relative co~cients of caution, l as wel.. l as by the relative degree of 1\ ~...a.n enterprise would involve for the various in- , ~. The same enterprise may be perilous to oIlel ,,,,:, and comparatively safe to another, because of superior ::7 .,~ knowledge or other conditions; and the same degree of risk may repel one individual more than another, owing to differences in temperament, or, most important of all, di~erences in amount of capitaL2 This shifting of risk from those on whom it bears heavily to those who can best assume it, discloses another motive 1 See The Nature 0/ Capital and Income, Chap. XVI, § 6.

2 Ibid., Appendix to Chap. XVI, p. 409.

216 THE RATE OF INTEREST [CHAP. XI for borrowing and lending besides those which were discussed in a previous chapter. Lending, in modern finance, often indicates not simply a difference in time-shape as between two income-streams, but also a difference of risk. The object of lending which was emphasized in earlier chapters, before the risk element was introduced into the discussion, was to alter the time-shape of the income-stream, the borrower desiring to increase his present income and decrease his future, and the lender desiring, on the contrary, to decrease his present income and add to his future. But the stock holder and bondholder do not differ in this way so much as in respect to risk. They are both investors, and stand in a very similar position as to the effect of their investment on the time-shape of their income. For the stockholder, however, there is a risk attached to his income stream from which the bondholder is relatively free. It is this difference in risk which is the-primary reason for the distinction between stockholders and bondholders.

The bondholder tt commutes" 1 his chance of a high income for the certainty of a steady income. ~. The existence of this risk, tending, as we have seen, to raise the rate of impure interest and lower that of pure interest, has as its effect the lowering of the price of stocks and the raising of the price of bonds from what would have been their respective prices had the risk in question been absent. On the other hand, the separation of the investors into stockholders and bondholders reacts upon the prices of stocks and bonds and tends to lessen the disparity between them. Were there no bonds, but only stocks, the price of the stock would have to be lower than it now is, in order to induce the timid investor to buy. In other words, the effect of the separation between risky and safe investments is at once to moderate the lowness of the low 1 Cf. Hadley, Economics, p. 270. The rate of interest, however, is not, properly speaking, the rate of commutation; for the rate of commutation would be the ratio between the average earnings of the stockholder and the.. average earnings of the bondholder, whereas the rate of interest is the ratio between income and capital.

Slilc.7] THIRD APPROXIMATION 217 rate on safe investments and the height of the high rate on the unsafe investments. The same tendency, to reduce the disparity between the rates of interest on safe and unsafe investments, grows out of the practice of insurance. One effect of insurance 1 is to raise the value of capital subject to risk of fire or other in surable risks, by consolidating those risks and thus virtually reducing them. But this rise in the value of capital implies a reduction in the rate of·interest which it yields. Again, the effect of speculation, by setting aside a certain class of persons to assume the risks of trade, has the effect of reducing these risks by putting them in the hands of those who have most knowledge; for, as we have seen, risk varies inversely with knowledge. In this way the whole plane of business is put more nearly on a uniform basis so far 'as the rate of interest is concerned.

§ 7 We see, then, that the element of risk introduces dis turbances into those determining conditions which were expressed in previous chapters as explaining the rate of interest. To summarize these disturbances, we may apply the risk factor to each of the six conditions which were originallystated as determining interest. We shall find that its effects are as follows:1. The condition that each individual has a given range of choice still holds true, but these choices are no longer confined to absolutely certain optional income-.streams, but include options with risk. That is, each individual finds open to his choice a given set of options which differ in size, time-shape, composition, and risk. 2. Rates of preference for present over future goods are of two kinds, according as the goods are comparativelycer tain or uncertain. The marginal rate of preference fora certain present over a certainfuture good, or thep'U'f8rate 1 See The Nature o/Capital and Income, Chap. XVI.

218 THE RATE OF INTEREST [CHAP. IX of timepreference, depends upon the character of the total income..stream, - not only its size, shape, and com position, but also and particularly upon the degree of cer tainty attaching to various parts of it and the degree of certainty of life of the recipient. Again, the preference rate for present certainincome as compared with future un certain income, or the impure rate of timepreference, will, in normal individuals, be greater than if both in comes were certain, and will be the greater, the higher the risk and the higher the caution in assuming it. 3. Pure rates of timepreference (as among certaingoods) in different individuals tend toward equality by the prac tice of borrowing and lending, and more generally, buying and selling; but this equality is not in all cases attain able, because of limitations on the freedom to modify the income-stream at will. These limitations grow out of the existence of the element of risk. There are various means of reducing or avoiding risk,-in particular, by the devices of collateral security, indorsement, under writing, etc., but all of these processes have more or less definite limitations. In consequence, it is not always possible to provide security for as large a loan as would be necessary to change the income-stream enough to reduce the rate of preference of the borrower to the rate of interest. If the security is adequate, the rate of preference will be equalized with the rate of interest; if not, it will remain above it. Where the security introduces impedi ments which affect the lender as well as the borrower, it will also happen that the rate of interest will be raised, as in the case of pawn shops which hold in pledge their motley assortment of cumbersome merchandise. Thus, instead of one rate for several loans, there will be a number of separate rates and a number of separate markets, accord ing to the nature of the security asked and given.

At the same time there will be a tendency to ask and receive loans with inadequate security. This introduces a pseudo-or impure rate of interest which will be above SBc.7] THIRD APPROXIMATION .219 the pure rate 'by a margin differing according to risk and caution. 4. When risk was left out of account, it was stated that from among a number of different alternatives the indi vidual would select that one which had the maximum pres ent value, - in other words, that one which, compared with its nearest neighbors, possessed a rate of return on sacrifice equal to the rate of preference, and therefore to the rate of interest. When the risk element is introduced, it will still be true that the maximum present value is selected; but in translating future uncertain income to present cash value, use must now be made of the probability and caution factors. One consequence is that when we express this principle of maximum present value in its alternative form in terms of the "marginal rate of return on sacrifice,"

we must qualify this expression as the "marginal rate of anticipated return on sacrifice." The rate of return on sacrifice' which will be actually realized may turn out to be widely different from that originally anticipated. 5. In the former approximations, where the element of risk was considered absent, .it was shown that the aggre gate modification of the income-streams of individuals for ·every period of time was zero. What was borrowed equaled what was lent, or what was added by sale was equal to what was subtracted by purchase. The same principle still applies; for what one person pays, another person must receive. 6. In the former approximations, the total present value of the projected modifications of one's income-stream was zero; that is, the present value of the loans equaled the present value of the borrowings; or the present value of the additions and subtractions due to buying and selling balanced each other. In our present discussion, in which future income is recognized as uncertain, this principle still holds true, but only in the sense that the present market values balance at the moment when the future loans or other modifications are planned and decided upon. The 220 THE RATE OF INTEREST [CHAP. XI fact that risk is present may lead to a wide discrepancy between the original expectation and the actual realiza tion. In liquidation there may be default or bankruptcy.

When the case is not one of a loan contract, but relates merely to the difference in income-streams of two kinds of property bought and sold, the discrepancy between what was expected and what is actually realized may be still wider. But, viewed in the present, the estimated value of the future return is still the equivalent of the sacrifice. The present value of a future uncertain event is equal to its mathematical value multiplied by a caution factor,! and the mathematical value is equal to the expected value multiplied by a probability factor and discounted according to the rate of preference for present over future income. § 8 We thus see that instead of the series of simple equalities which we found to hold true in the vacuum where risk was absent, we have only a tendencytoward equalities, interfered with by the limitations of the loan market, and therefore resulting in a series of inequalities. Rates of interest, rates of preference, and rates of return on sacrifice are only ideally, not really, equal.

We conclude by summarizing in the following table the interestdetermining conditions for our three successive approximations: 1 See The Nature 0/ Capital and Income, Chap. XVI, § 6.

tv l\j ...... rn t!rJ ? 00 ~ THIRD APPROXIMATION 1. He has the choice of anyone from among a specific list of eligible uncertain income-streams (some of which may depend on the rate of interest). 2. His preference rate depends on his uncertain income-stream as (1) selected and (2) modified by ex change. 3. Preference rates of different individuals tend toward a common ~ rate of interest. t-4 4. The individual selects, .from ~ the eligible list, the income-stream > which has in present estimation, ~ whether truly or falsely, the maxi-. ;g mum present value. If the alter-0 natives are numerous and vary ~ continuously from each other, this t( condition is equivalent to the con- > dition that the marginal rate of ~ estimated return on sacrifice shall ~ equal the rate of interest. 5. The modifications, through ex change, of the income-streams at the same time for different indi viduals mutually cancel one an other for each separate time in terval.

6. The modifications, through ex cha~ge, of the income-stream of the same individual through different time periods mutually cancel one another in e8timatedpresent value. SECOND APPROXIMATION 5. The modifications, through exchange, of the income-streams at the same time for different indi viduals mutually cancel one an other for each separate time in terval. 6. The modifications, through exchange, of the income-stream of the same individual through differ ent time periods mutually cancel one another in present value. 1. He has the choice of anyone from among a specific list of eligible certain income-streams (some of which may depend on the rate of interest). 2. His preference rate depends on his certain income-stream as (1) selected and (2) modified by exchange. . 3. Preference rates' of different individuals become equal to the common rate of interest. 4. The individual selects, from the eligible list, the income-stream which possesses the maximum present value. If the alternatives are numerous and vary continu ously from each other, this condi tion is equivalent to the condition that the marginal rate oj return on 8acrifice shall equal the rate of interest.

FIRST APPROXIMATION 5. The modifications, through ex change, of the income-streams at the same time for different individuals mutually cancel one another for each 8eparate time interval. 6. The modifications, through ex change, of the income-stream of the 3ameindividual through different time periods mutually cancel one another in present value. [1. Each individual has a rigid and certain income-stream unalterable except by exchange.] 2. His preference rate depends on that income-stream as modified by exchange. 3. Preference rates of different in dividuals become equal to the com mon rate of interest. [4. The individual, having no choice, accepts the income-stream given.] 222 THE RATE OF INTEREST [CHAP. XI In the first approximation, conditions 1 and 4 are in serted to complete the correspondence with the other two approximations; but they are both really the same con dition, and merely reexpress the hypothesis under which the first approximation was made. It is the remaining four conditions which are of real significance.

The first two approximations were, of course, merely pre paratory to the third, which alone corresponds to the actual world of facts. Yet the other two approximations are of equal importance with the third from the point of view of analysis. They tell us what would happen if future income were (1) fixed and certain, and (2) flexible but certain; and to know what would happen under these hypothetical conditions enables us better to understand what does happen under actual conditions, just as the knowledge that a pro jectile would follow a parabola if it were in a vacuum, and that it would follow a certain other cunre if in a still at mosphere of given density, enables the student of practical gunnery better to understand the actual behavior of his cannon balls. In fact, no scientific law is a perfect state ment of what does happen, but of what would happen if certain conditions existed which do not actually exist.!

Science consists of the formulation of hypothetical se quences, not of historical facts; though by successive approximations the hypotheses may be made nearly to coincide with reality.2 1 See the writer's llEconomics as a Science," Science, August 31, 1906. J See Appendix to Chap. XI, § 1.