- Suggestions to Readers
- Part I.: Introduction
- Part I, Chapter I: Income and Capital 2
- Part I, Chapter II: Money Interest and Real Interest
- Part I, Chapter III: Some Common Pitfalls
- Part II.: The Theory In Words
- Part Ii, Chapter IV: Time Preference (human Impatience)
- Part Ii, Chapter V: First Approximation to the Theory of Interest Assuming Each Person's Income Stream Foreknown and Unchangeable Except By Loans
- Part Ii, Chapter VI: Second Approximation to the Theory of Interest Assuming Income Modifiable (1) By Loans and (2) By Other Means
- Part Ii, Chapter VII: The Investment Opportunity Principles
- Part Ii, Chapter VIII: Discussion of the Second Approximation
- Part Ii, Chapter IX: Third Approximation to the Theory of Interest Assuming Income Uncertain
- Part III.: The Theory In Mathematics
- Part Iii, Chapter X: First Approximation In Geometric Terms
- Part Iii, Chapter XI: Second Approximation In Geometric Terms
- Part Iii, Chapter XII: First Approximation In Terms of Formulas
- Part Iii, Chapter XIII: Second Approximation In Terms of Formulas
- Part Iii, Chapter XIV: The Third Approximation Unadapted to Mathematical Formulation
- Part IV.: Further Discussion
- Part Iv, Chapter XV: The Place of Interest In Economics
- Part Iv, Chapter XVI: Relation of Discovery and Invention to Interest Rates
- Part Iv, Chapter XVII: Personal and Business Loans
- Part Iv, Chapter XVIII: Some Illustrative Facts
- Part Iv, Chapter XIX: The Relation of Interest to Money and Prices
- Part Iv, Chapter XX: Objections Considered 73
- Part Iv, Chapter XXI: Summary
- Appendix to Chapter I
- Appendix to Chapter X
- Appendix to Chapter Xii
- Appendix to Chapter Xiii
- Appendix to Chapter Xix
- Appendix to Chapter Xx
PART II, CHAPTER IX
THIRD APPROXIMATION TO THE THEORY OF INTEREST
Assuming Income Uncertain
§1. More than One Rate of Interest
THE great shortcoming of the first and second approximations, from the standpoint of real life, is the complete ruling out of uncertainty. This exclusion of the risk element was made in order to make the exposition simpler and to focus the reader's attention on the factors most relevant to the theory of interest. But in real life the most conspicuous characteristic of the future is in its uncertainty. Consequently, the introduction of the element of chance, or risk, will at once endow our hypothetical picture with the aspect of reality. The foundation for our study of risk in relation to interest has already been laid in Chapter IV where the relation of risk to time preference was noted.
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 so-called interest rates. These rates vary because of risk, nature of security, services in addition to the loan itself, lack of free competition among lenders or borrowers, length of time the loan has to run, and other causes which most economists term economic friction. The very definition of loan interest as one implying no risk must now be modified so as to imply some risk that the loan may not be repaid in full according to the contract. Practically all of them are varieties of risk. Even in loans which theoretically are assumed to be riskless, there is always some risk. Modern corporate finance makes no pretence that risk is completely absent, but merely concerns itself with providing a more or less safe margin of protection varying with each specific case.
Furthermore, for our present purposes, contract or explicit interest is too narrow a concept. We now include not only the implicit interest realized by the investor who buys a bond, but the implicit interest realized by the investor who buys preferred stock. We may even include the rates realized on common stock, real estate, or anything else. Thus extended, the concept of interest becomes somewhat vague. And yet, if we exclude exceptional cases, there tend to emerge, at any time, several fairly definite market rates of interest according to the character of the security.
We find quoted rates on call loans, four months prime commercial paper, prime bankers' acceptances, first mortgages, second mortgages, as well as rates given by savings banks, rates allowed on active checking accounts, pawn shop rates, Morris Plan bank rates, rates realized on government bonds, railroad bonds, on other bonds, whether mortgage, debenture, or income bonds (all of which bond rates vary according to the character and credit of the issuer as well as in accordance with other circumstances), rates realized on preferred stock, and sometimes even the rates realized on common stocks. Wherever there is a sufficiently definite rate per cent per annum to be quoted as an expression of the current market, whether the quotation be in print or a verbal quotation in a broker's or banker's office, it seems proper to call it in a broad sense a rate of interest. Even when confined to such market rates, and excluding exceptional or individual rates, the rate of interest ceases to be the ideal, imaginary, single-valued magnitude hitherto assumed and takes on the myriad forms which we find in actual business transactions.
As indicated, this profuse variety is brought about chiefly by the introduction of risks of various sorts. The rate on call money is affected by the chance of the loan being "called" by the lender, or the sudden reduction of the total of such loans outstanding and the raising of the rate for those remaining. The rate on a loan-shark's loan is high because of the risk of non-payment by the borrower; the rate on a debenture and the rate on a second preferred stock of a moribund corporation are affected by the risk of inadequate corporate assets in event of liquidation; the rate on a government bond of a nation at war is affected by the chance of that nation's defeat; the rate of dividends on common stock is affected by the risks of the business; and so on in an infinite number of different cases.
§2. Relations Between the Various Rates
No very satisfactory theoretical treatment of the general relations between interest and risk has yet been worked out. But for practical purposes, a good usage is to limit the term "interest" to fairly safe loans and staple or standard market quotations and to designate by some other term, such as dividends or profits, the other less certain and less standardized rates. Another usage is to reckon as net profits or net losses the difference between these less standardized rates and a normal rate of interest so far as this can be expressed in figures. Thus a man who has invested $100,000 in common stock and is getting an income of $15,000 may think of $5,000 of this, or 5 per cent, as a fair interest on his investment and the remaining $10,000, or 10 per cent, as net profit. But we do not need here to enter into such discussions, especially in so far as they are only verbal. All that is here needed is to show briefly how risk modifies the theoretically perfect determination of the interest rate thus far made.
The rate in every loan contract is adjusted according to the degree of security given. Thus, security or guarantee may be furnished by a simple endorsement of reputable persons, in which case the degree of security will be the greater the larger the number of endorsers and the higher the credit which they possess, or it may be by the deposit of collateral securities. Thus the very name security has come to mean the properties themselves rather than their safety.
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 in general, 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 may sell only on a 6, or even on a 9 per cent basis.
The period of time a loan or bond runs is also an important factor as regards risk. There is a see-saw between the rates on short term and long term loans. That is, if the short term rate is greatly above the long term, it is likely to fall, or if greatly below, to rise. The long term rates thus set a rough norm for the short term rates, which are much more variable. When the future is regarded as safer than usual, loan contracts tend to be longer in time than otherwise. In a stable country like the United States, railway and government securities are thus often drawn for half a century or more. There is also a variability according to the degree of liquidity. A call loan which may be recalled on a few hours' notice has a very different relation to risk than does a mortgage, for instance. The call rate is usually lower than time rates because money on call is a little like money on deposit, or ready money. It is ready, or nearly ready, for use whenever occasion demands. This readiness or convenience takes the place of some of the interest. On the other hand, a sudden shortage of funds in the call loan market may send the call rates far above time rates and keep them there until the slow working forces release "time-money" and transfer it to the call loan market. Thus the call loan rates are very volatile and mobile in both directions.
The element of risk will affect also the value and basis of the collateral securities. Their availability for collateral will increase their salability and enhance their price. On the other hand, when, as in times of crisis, the collateral has to be sold, it often happens that for purposes of liquidation it is sold at a sacrifice.
§3. Limitations on Loans
The necessity of having to offer collateral will affect not only the rate which a man has to pay, but the amount he can borrow. It will limit therefore the extent to which he can modify his income stream by this means. Consequently it will not be possible, as assumed in the first two approximations, for a man to modify his income stream at will; its possible modification will be limited by the fear of the borrower that he may not be able to repay and the greater fear of the lender that he may not be repaid—because the borrower's credit may not prove good. In consequence of this limitation upon his borrowing power, the borrower may not succeed in modifying his income stream sufficiently to bring his rate of preference for present over future income down to agreement with the rate or rates of interest ruling in the market; and for like reasons he may not succeed in bringing the rate of return over cost into conformity with any rate of interest.
One feature of these limitations on borrowing may here be noted. The ability and willingness to borrow depend not only on the amount of capital which the would-be borrower possesses, but also on the form in which that capital happens to exist. Some securities are readily accepted as collateral, and accepted as collateral at a high percentage of their market value, whereas others will pass with difficulty and only at a low percentage of that value. The drift, especially during the last generation, toward the corporate form of business has had a striking effect in increasing the power and readiness to borrow. Whereas formerly many businesses were conducted 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. Similarly the small local companies, the stock of which was held almost exclusively by one family or group of friends, have been merged into large nationally known companies, the securities of which are widely marketable. This increase in size of business units, although it tends to decrease their number, has resulted in a rapid growth of the number of securities listed on the stock exchanges. The possessors of these securities have far wider opportunities to make use of collateral, and the tendency to borrow has received a decided impulse.
Where, on the other hand, the security needed is not available in the convenient form of engraved certificates, there is often considerable difficulty in negotiating a loan. A poor man may see what he believes to be an investment opportunity to make millions by exploiting an invention of his own, and he may be right. This option would have a much higher present value than the one he actually chooses, if only he could borrow the money needed to exploit it. But, being poor and hence without adequate collateral or other guarantees, he cannot get the loan. His choice of income stream, therefore, although the maximum open to him, is quite different from what it would be if he had that collateral or guarantee. If he goes into the enterprise at all he must choose a stopping point far short of what he would choose were he a large capitalist. This means that his marginal rate of return over cost will be higher than the market rate of interest, just as his rate of impatience will be higher than the market rate of interest. The last $100 he ventures to put in may promise a yield of 25 per cent as compared with a rate of interest of 5 per cent. Yet he does not go further into debt because he cannot. Supposing a definite limit of possible debt, all he can do toward further investing must be out of his own income, obtained by abstinence. But this possibility is also limited. He cannot cut his income down to zero, or he would starve. He can however cut it "to the quick", stopping at the point where his impatience has risen to meet the rate of return over cost which in turn will tend to fall with each additional dollar invested.
The story is told of the inventor of rubber making a last desperate—and, fortunately, successful—sacrifice in his experiments in which he resorted to burning up his furniture because he could not get funds with which to buy fuel.
Many such cases exist—cases of limitation on loans—which prevent a person's degree of impatience and his rate of return over cost from reaching the level of the rate of interest. But there is another part of the picture. The poor man who cannot borrow enough to exploit his invention can often find substitutes for loans and lenders. He may associate himself with others in a joint stock company and get the required capital partly from loans, by selling bonds secured by mortgage, partly by selling debentures on a higher interest basis, partly by selling preferred stock on a still higher basis, and partly by selling common stock. That is, the risks are recognized and pooled. One result may be to bring both his estimated rate of return over cost and his rate of impatience more nearly into harmony with these various rates of interest when due account has been taken of the various risks involved.
§4. Risk and Small Loans
Where the borrowing takes place in pawn shops, the rate of interest is usually very high, not so much because of the inadequacy of the security as because of its inconvenient form. The pawnbroker will need to charge a high rate of interest, if it is to be called 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, when not redeemed, he has to make an effort to find markets in which to sell them. He is, moreover, able to secure these high rates partly because pawnbroking is in bad odor, so that those who go into the business find a relative monopoly, and partly because of the fact that the customers usually have, either from poverty or from personal peculiarity, a relatively high preference for present over future income. While the effect of their accommodation at the pawn shop is to reduce their impatience to some extent, 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 except at high, or usurious, rates is often a great hardship, but it has one beneficent effect, that is, the discouraging of the improvident from getting unwisely into debt.
One of the very greatest needs has always been to sift out the relatively safe and sane from the relatively risky and reckless loans of the poor in order to encourage the one and discourage the other. When this has been more fully accomplished, the scandal of the loan shark will be largely a thing of the past. A loan which, to the shortsighted or weak willed borrower, seems to be a blessing, but which is really sure to prove a curse, ought certainly to be discouraged no matter what may be the rate of interest. The Russell Sage Foundation has studied the loan shark problem intensively and as a result has formulated a model small loans act which has been adopted by the legislatures in a large number of our States. This model act recognizes the greater risk and trouble involved in small loans for short periods, by permitting a maximum rate per month of 3½ per cent.
Only in the present generation has the age-long curse of the loan shark been met by constructive measures on a large scale. These are based on the simple principle that a man's friends and neighbors possess the necessary knowledge whereby to distinguish between a safe and unsafe extension of credit to him. The Morris Plan banks are founded on that principle. More effective are the Credit Unions founded by Edward Filene and others in America somewhat on the models of the Raiffeisen and other plans in Europe. Labor banks are rendering a similar service. These are enabling the poor to make effective use of personal character as a substitute for collateral security and are thereby greatly reducing the rate of interest on the loans of the poor. In 1928 one large bank in Wall Street instituted a similar system for loans without collateral to salaried employees. These devices and others are doing much to solve the problem of accommodating the reliable man of small means with loans at rates comparable with those ruling in the markets for the well-to-do.
§5. Salability as a Safeguard
When a security, because it is well known, or for any other reason, has a high degree of salability, that is, can be sold on short notice without risk of great sacrifice, its price will be higher than less favored securities, and the rate it yields will therefore be low. Salability is a safeguard against contingencies which may make quick selling advisable. In other words, in a world of chance and sudden changes, quick salability, or liquidity, is a great advantage. For this reason, the rate of interest on individual mortgages will be higher than the rate of interest on more marketable securities. It is, in general, advantageous to have stock listed on the stock exchange, for, being thus widely known, should the necessity to sell arise, such a stock will find a more ready market.
The most salable of all properties is, of course, money; and as Karl Menger pointed out, it is precisely this salability which makes it money. The convenience of surely being able, without any previous preparation, to dispose of it for any exchange, in other words, its liquidity, is itself a sufficient return upon the capital which a man seems to keep idle in money form. This liquidity of our cash balance takes the place of any rate of interest in the ordinary sense of the word. A man who keeps an average cash balance of $100, rather than put his money in a savings bank to yield him $5 a year, does so because of its liquidity. Its readiness for use at a moment's notice is, to him, worth at least $5 a year. There is a certain experienced buyer and seller of forests, in Michigan, who makes a practice of keeping a ready cash balance in banks of several million dollars in order better to be able to compete with other forest purchasers by having available spot cash to offer some forest owner who, becoming forest-poor, wishes to sell. Forests are extremely non-liquid while cash balances are extremely liquid.
§6. General Income Risks
Even when there is no risk (humanly speaking) in the loan itself, the rate realized on it is affected by risk in other connections. The uncertainty of life itself casts a shadow on every business transaction into which time enters. Uncertainty of human life increases the rate of preference for present over future income for many people, although for those with loved dependents it may decrease impatience. Consequently the rate of interest, even on the safest loans, will, in general, be raised by the existence of such life risks. The sailor or soldier who looks forward to a short or precarious existence will be less likely to make permanent investments, or, if he should make them, is 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 for long ahead.
When the risk relates, however, not to the individual's duration of life, but to his income stream, the effect upon the rate of interest will depend upon which portions of the income stream are most subject to risk. If the immediately ensuing income is insecure, whereas the remoter income is sure, the rate of preference for an additional sure dollar immediately over an additional sure dollar in the remoter period will, as was shown in Chapter IV, tend to be high, and consequently the effect of such a risk of immediate income upon the rate of interest will be to raise it. A risky immediate income acts on interest like a small immediate income.
But if, as is ordinarily the case, the risk applies more especially to the remoter income than to the immediate, the effect is the exact opposite, namely, to lower the rate of interest on a safe loan. The risky remote income acts as the equivalent of a small remote income. This example is, perhaps, the most usual case. If a man regards the income for the next few years as sure, but is in doubt as to its continuance into the more remote 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 with certainty the uncertain 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, making the future risky, we witness the anomalous combination of high rates where inadequate security is given coexistent with low rates on investments regarded as perfectly safe. When an investor cannot find many investments into which he may put his money without risk of losing it, he will pay a high price—i.e., accept a low rate of return—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 to hoard their capital in money form, even paying for storage charges, a payment which amounts to a negative rate of interest. During the World War some investors in the warring countries sought safekeeping for their funds in neutral countries.
§7. Securities Classified as to Risk
When risk thus operates to lower the rate of interest on safe investments and to raise the rate on unsafe investments, there immediately arises a tendency to differentiate 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. Some 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 tend to become a separate class. When enterprises came to be organized in corporate form, this classification of investors was recognized by dividing the securities into stocks and bonds, the stockholder being the person who assumes the risk and, theoretically at least, guaranteeing that the bondholder shall be free of all risk. Which person shall fall into the class of risk-takers and which not is determined by their relative coefficients of caution, as well as by the relative degree of risk which an enterprise would involve for the various individuals concerned. The same enterprise may be perilous for one and comparatively safe for another because of superior knowledge on the part of the latter, and the same degree of risk may repel one individual more than another, owing to differences in temperament or, most important of all, to differences in amount of available capital.
This shifting of risk from those on whom it bears heavily to those who can more easily assume it discloses another motive for borrowing and lending besides those which were discussed in a previous chapter. Borrowing or lending in corporate finance usually 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 ordinary stockholder and bondholder do not differ in this way so much as they do in respect to risk. They are both investors, and the positions, in which they stand as to the effect of their investment on the time shape of their income, are really very similar. But the stockholder has a risk attached to his income stream from which the bondholder seeks to be free. It is this difference in risk which is the primary reason for the distinction between stockholders and bondholders. The bondholder gives up his chance of a high income for the assurance, or imagined assurance, of a steady income. The stockholder gives up assurance for the chance of bigger gains.
The existence of this risk, tending as we have seen to raise the rate of interest on unsafe loans and lower that on safe loans, 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.
In the last few years, however, this disparity has been decreased from both ends. The public have come to believe that they have paid too dearly for the supposed safety of bonds and that stocks have been too cheap. Studies of various writers, especially Edgar Smith and Kenneth Van Strum have shown that in the long run stocks yield more than bonds. Economists have pointed out that the safety of bonds is largely illusory since every bondholder runs the risk of a fall in the purchasing power of money and this risk does not attach to the same degree to common stock, while the risks that do attach to them may be reduced, or insured against, by diversification. The principle of insurance of any kind is by pooling those risks virtually to reduce them. This raises the value of the aggregate capital subject in detail to these risks.
It is in this way that investment trusts and investment counsel tend to diminish the risk to the common stock investor. This new movement has created a new demand for such stocks and raised their prices; at the same time it has tended to decrease the demand for, and to lower the price of, bonds.
Again, speculation in grain, for example 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.
Risk is especially conspicuous in the financing of new inventions or discoveries where past experience is a poor guide. When new inventions are made, uncertainty is introduced, speculation follows, and then comes great wealth or great ruin according to the success or failure of the ventures. The history of gold and silver discoveries, of the invention of rubber, of steel, and of electrical appliances is filled with tales of wrecked fortunes, by the side of which stand the stories of the fortunes of those few who drew the lucky cards, and who are among today's multi-millionaires.
The rates of interest are always based upon expectation, however little this hope may later 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 an estimate which I saw in print of the value of a copper mine, allowance was made for future inventions which might reasonably be expected. In the same way, too, the buyer of machinery allows not simply for its depreciation through physical wear, but for its obsolescence. New investments in steam railroads are today made with due regard to the possibility that the road may, within a few years, be run by electricity, or that it will be injured by competition of bus lines or helped by terminal connections with them.
It may easily happen that, in a country consisting of overly sanguine persons, or during a boom period when business men are overhopeful, the rate of interest will be out of line with what actual events, as later developed, would justify. It seems likely that, in ordinary communities, realization justifies the average expectation. But in an individual case this is not always true; otherwise there would be no such thing as risk. Risk is synonymous with uncertainty—lack of knowledge. Our present behavior can only be affected by the expected future,—not the future as it will turn out but the future as it appears to us beforehand through the veil of the unknown.
§8. Effect of Risk on the Six Principles
We see, then, that the element of risk introduces disturbances into those determining conditions which were expressed in previous chapters as explaining the rate of interest. To summarize these disturbances, we may now apply the risk factor to each of the six conditions which were originally stated as determining interest. We shall find that its effects are as follows:
THE TWO PRINCIPLES AS TO INVESTMENT OPPORTUNITY
A. Empirical Principle
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 now include options with risk. That is to say, each individual finds open to his choice a given set of options (and opportunities to shift options, that is, opportunities to invest) which options differ in size, time shape, composition and risk.
B. Principle of Maximum Present Value
When risk was left out of account, it was stated that from among a number of different options the individual would select that one which has the maximum present value—in other words, that one which, compared with its nearest neighbors, possesses a rate of return over cost equal to the rate of time preference, and therefore to the rate of interest.
When the risk element is introduced, it may still be said that the maximum present value is selected, but in translating future uncertain income into present cash value, use must now be made of the probability and caution factors.
But when we try to express this principle of maximum present value in its alternative form in terms of the marginal rate of return over cost, we must qualify this expression to: the marginal rate of anticipated return over cost.
Three consequences follow. First, that the rate of return over cost which will actually be realized may turn out to be widely different from that originally anticipated. Second, there is in the market not simply one single anticipation; there are many, each with a different degree of risk allowed for in it. Third, the need of security may be such as to limit also the choice of options.
THE TWO PRINCIPLES AS TO IMPATIENCE
A. Empirical Principle
The rate of time preference depends upon the character of the income stream, but it must now take into account the fact that both the immediate and the future (especially the future) portions of that stream are subject to risk. There are many rates of time preference, or impatience, according to the risks involved. But they all depend on the character of the expected and possible income stream of each individual—its size, shape, composition and especially the degree of uncertainty attaching to various parts of it as well as the degree of uncertainty of life of the recipient.
B. The Principle of Maximum Desirability
It is still true, of course, that the individual decides on the option most wanted. But, in a world of uncertainty there are two features not present in a world of certainty. One (which however does not affect the formulation of the principle) is that what is now desired may prove disappointing. That is, though it seems, when chosen, the most desirable course, it may not prove the most desirable in the sense of deserving, in view of later developments, to be desired. The other new feature is that the most desired income stream is no longer necessarily synonymous with that which harmonizes the rate of preference with the rate of interest.
Rates of time preference in any one market tend toward equality by the practice of borrowing and lending, and more generally, that of buying and selling, but this equality is no longer, in all cases, attainable, because of limitations on the freedom to modify the income stream at will, limitations growing out of the existence of the element of risk and consequent limitations on the borrowing power.
THE TWO MARKET PRINCIPLES
A. Principle of Clearing the Market
In the first two approximations, where the element of risk was considered absent, it was shown that the aggregate modification of the income streams of all 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. The only difference is that, in a world of chance, the actual payments may be quite different from those originally anticipated and agreed upon, that is, will often be defaulted, in whole or in part.
B. Principle of Repayment
In the former approximations, the total present value of the prospective 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 caused by 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 fact of risk means that later there may be a wide discrepancy between the actual realization and the original expectation. 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. Only viewed in the present is the estimated value of the future return still the equivalent of the estimated cost.
We thus see that, instead of the series of simple equalities which we found to hold true in the vacuum case, so to speak, where risk was absent, we have only a tendency toward equalities, interfered with by the limitations of the loan market, and which, therefore, result in a series of inequalities. Rates of interest, rates of preference, and rates of return over cost are only ideally, not really, equal.
We conclude by summarizing in the accompanying table the interest-determining conditions not simply for the third approximation but for all three of our three successive approximations (distinguished by the numerals 1, 2, 3).
In this summary tabulation the "Principles as to Investment Opportunity" are formally inserted, under the first approximation, in order to complete the correspondence with the other two approximations, but of course they merely re-express the hypothesis under which the first approximation was made. They are therefore bracketed, since only the remaining four conditions are of real significance for the first approximation.
The first and second approximations were, of course, merely preparatory to the third, which alone corresponds to the actual world of facts. Yet the other two approximations are of even greater importance than the third from the point of view of theoretical analysis. They tell us what would happen under their respective hypotheses. Both these hypotheses are simpler than reality; hence they lend themselves better to formal analysis and mathematical expression.
Moreover, 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 projectile would follow a parabola if it were in a vacuum enables the student of practical gunnery better to understand the actual behavior of his bullets or shells. In fact, no scientific law is a perfect statement of what does happen, but only what would happen if certain conditions existed which never do actually exist. Science consists of the formulation of conditional truths, not of historical facts, though by successive approximations, the conditions assumed may be made nearly to coincide with reality.
The second approximation gives a clear cut theory applicable to the clear cut hypotheses on which it is based. The third approximation cannot avoid some degree of vagueness.
THE THEORY IN MATHEMATICS