Front Page Titles (by Subject) CHAPTER 23: CAPITALIZATION OF MONETARY INCOMES - Economics, vol. 1: Economic Principles
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CHAPTER 23: CAPITALIZATION OF MONETARY INCOMES - Frank A. Fetter, Economics, vol. 1: Economic Principles 
Economics, vol. 1: Economic Principles, (New York: The Century Co., 1915).
Part of: Economics, 2 vols.
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CAPITALIZATION OF MONETARY INCOMES
§ 1. Buying with money. § 2. Capitalization of agricultural land-incomes. § 3. Years’ purchase and rate of income on capital. § 4. Price and rate of income. § 5. Bonds and mortgages as saleable incomes. § 6. Price of variable terminable incomes. § 7. Depreciation funds. § 8. Corporate securities. § 9. Capital value of public franchises. § 10. Incomes sold in perpetuity. § 11. Bonds with fixed maturity.
§ 1. Buying with money. Where money is used the usual case is that of the sale of one good for money, which is then spent for another good. In all these trades time-preference is only one factor helping to fix the price, but the important thing to note is that it always is a factor and is logically and practically a matter for separate consideration. Wherever, to-day, there is a business income that has a market-value, that may be bought and sold, it may be capitalized. Men compete in the purchase of income-yielding agents. There is a continual contest in judgment among investors to secure the largest return for the smallest outlay. On the other hand, the owners of any income strive to secure the largest capitalization for it that they can. Buying as cheaply as they can the present goods they need, and selling as dearly as they can the future goods they offer, each man fits his valuation to the market. In any market the individual finds an established price (Chapter 7, section 6) and all he can do is to buy or to refuse to buy, sell or refuse to sell, at that price. A trader’s valuation may be such that he is an included buyer at one price, and at another price he ceases to buy and begins to sell.
§ 2. Capitalization of agricultural land-incomes. An interesting example and one of great historical importance, showing the capitalization of a series of incomes looked upon as perpetual and uniform, is agricultural land of western Europe since the latter part of the Middle Ages when money had come into more general use. Suppose the annual net income is $1000 (after deducting from rents all repairs, taxes, and other costs) and every one believes that it will continue at that amount indefinitely. The ownership of the estate represents the right to this annuity, and whatever price is paid for the ownership is the price of the whole series of incomes. As the series of incomes is looked upon as perpetual, if the future rents were to be counted as if they were already present, with no discount on their future value, the capital sum would be infinite. On the other hand, if the ownership is worth nothing just after a rent-payment when no more rents are due for a year, the discount on the future rents would be 100 per cent. Evidently either extreme is impossible, and as a fact of observation, just such purchases are made every day at a finite price bearing a pretty regular relation to the amount of the annual income. The practice is plainly indicated by the phrase in which the price for land is spoken of still in England and the continental countries—a phrase unfamiliar to American ears—as a certain number of “years’ purchase.” If an estate is sold for twenty or thirty times the annual net rental, it is said to be sold at twenty or thirty “years’ purchase,” as the case may be. This does not mean that the rental for twenty years only is sold, but that the rental in perpetuity is sold for twenty times the annual rent; that is, the land is sold outright for the amount of twenty years’ rent paid at once. The estate is looked upon primarily as providing a fixed income; the value of the permanent possession of the estate is thought of as a certain number of times the value of the income secured. “Years’ purchase” means, therefore, the length of time required for the incomes to amount to the purchasing price.
§ 3. Years’ purchase and rate of income on capital. Now at ten years’ purchase every piece of property yields 10 per cent on the capital invested (purchase price $10,000, annual income $1000); at twelve years’ purchase 8⅓ per cent; at twenty years’ purchase 5 per cent; at twenty-five years’ purchase 4 per cent, etc. Increase in the number of years’ purchase involves a reciprocal decrease in the rate of return which the original investment of capital will yield; that is, one divided by the years gives always the rate per cent of income, .10, .083, .05, .04, etc. The arithmetic process is the simple one of aliquot parts. The number of years’ purchase expresses a ratio of capitalization, thus: a : S :: 1 : 10, ten years’ purchase being a low ratio and 40 years’ purchase a high ratio. Corresponding with this is a rate of annual premium at which the price a year distant will appear in comparison with present price, the difference being a net income; thus, present principal is to future principal plus a year’s income as 100 is to 105, the rate being .05.1
Whatever the rate is, it is an arithmetical fact, entirely independent of any calculation by purchaser or lender, but necessarily resulting whenever the property changes hands at any price. Another arithmetical fact is that this rate of yield is that at which the annual income of a perpetual uniform series must be compound-discounted to produce the capital sum; that is, a perpetual series of $1000 discounted at 10 per cent gives a present worth of $10,000; or ten years’ purchase, a perpetual series of $1000 discounted at 5 per cent gives a present worth of $20,000, or 20 years’ purchase. The rate at which a perpetual series is compound-discounted to purchase a capital sum is always the rate of simple interest the investment will yield, and vice versa. The present income is worth most, next year’s less, and so on in a decreasing series. Whatever the rate prevailing, incomes infinitely distant became infinitesimally small when compound-discounted. The formula is S = when S is the present worth of all the incomes, a is the perpetual annuity, and r the rate per cent; e.g., 20,000 = ; this is equivalent to r = ; that is, the rate at which the future incomes are capitalized is the annuity divided by the capital sum; e.g., .05 =
§ 4. Price and rate of income. It may be shown by a price diagram how every price arithmetically involves a corresponding rate of premium on the present price (investment of capital) which will be unfolded as an income to the investor. Take the case of a house affording a net rent to the owner of $100 a year (after allowing for taxes, costs, depreciation). The price of the series of incomes is the amount at which the bids are brought to equilibrium, the marginal bidders being those just ready to drop out of the market if a slight change is made. This reflects the rate of time-preference in the individual economy, showing itself in the whole state of improvement and depreciation of agents in the possession of each man. B will prefer to rent so long as the house is priced at $2000 (involving a rate of 5 per cent) but prefers to buy when it is priced at $1800, a discount of 5.55 per cent. The expression of the price of time as a percentage is merely a convenience.
§ 5. Bonds and mortgages as saleable incomes. The modern corporation bond is a promise to pay an annual sum, expressed as a percentage of the principal, and to repay the principal at the definite date of maturity. A twenty year 5 per cent bond for $1000 thus is a promise to pay 19 annual incomes of $50 at the end of each year (but see note below) and one payment of $1050 at the end of the twentieth year. It could as well be termed a 20 payment $50 a year bond with a maturing value of $1000, without mentioning a rate of interest. The rate it truly yields the investor depends on the price he pays, which is fixed by market conditions. Such a bond does not necessarily sell at par (its denomination); usually it sells at a premium or at a discount. The investors treat a bond as so many incomes distributed at certain points of time, and bidding in the market fixes the market-price for future incomes of that type.2 A note secured by real-estate mortgage is like the bond, but not so marketable, and is ordinarily held by the same investor until maturity. It usually (but not always) is bought at its face value and the holder looks upon it as capital to that amount. But as it is not payable until the date named as maturity, he could, if he wished, convert it into ready funds before it is due, selling it at the best price he can get, which may be above or below par. Thus a ten year mortgage for $5000, bearing 5 per cent interest, may be looked upon as containing nine annual payments of $250 each and a tenth payment of $5250. The total undiscounted sum of all the payments is $7500; and if the mortgage is bought at par it yields an annual net income of 5 per cent on the investment; if bought above par it yields an income of less than 5 per cent (e.g., bought at $5406 it yields 4 per cent).
This is a good illustration of what is meant by capital as contrasted with wealth. If the mortgaged house will bring a price of $10,000, that is the price of the wealth; but the owner has a capital of $5000, and the holder of the mortgage has a capital of $5000—which together are the total value of the wealth.
§ 6. Price of variable and terminable incomes. Cases of entirely uniform and perpetual incomes (even in expectation) are very rare. Most incomes are variable and terminable. These are capitalized and made comparable as to present worth with a uniform and perpetual series. Incomes may change in an upward direction, more or less regularly from year to year; they may decrease or they may remain the same for a series of years and then terminate abruptly; or they may vary by any combination of these changes.
Especially in modern times real-estate rentals, formerly the type of stability, have been rapidly altered by social changes, and so far as these changes are foreseen, expected rents are made the basis for present capitalization. Investors and owners alike may foresee that a piece of land used only for agriculture will, within a few years, be taken up for city lots, or will be needed for a factory or as the site of a railroad station. A vacant lot may be held for a number of years at a good price while yielding nothing; in this case the incomes are all future, and the capitalization must be based upon the progressive series expected, beginning at zero. In some cases the physical output of any agent may decline while the price of the product increases, the resultant being perhaps a stationary yield or an increasing one. When foresters foresee that the selling price of the timber will be greater twenty-five years later than it is to-day, and they estimate the future yield of the forest on this basis, they advise expenditure that would be unwise if present prices were to continue. Again when the expected series of incomes is declining, as the royalties secured from mines, being certain to disappear at some more or less calculable date, the capital value of the mine is the present worth of a limited and degressive series of incomes.
The value of a short series may be calculated by simple arithmetical methods, and more easily by the aid of a table of present worth, when any rate of premium on the present is assumed. Suppose the rate to be .05 and the incomes expected are as follows: at the end of the 1st year, 100, of the 2nd year 75, of the 3d year 50, then ceasing. The present worth is the sum, = 95.238 + 68.027 + 43.192 = $206.46. Again it must be observed (see Chapter 22, sections 8, 9) that if the price is $206.46 it mathematically involves the rate of 5 per cent, quite independent of any thought; the calculation merely reveals and expresses exactly a rate inherent in the transaction.
§ 7. Depreciation funds. As a matter of practice the more difficult calculations of variable and terminable annuities are avoided by investors by taking the perpetual uniform series of incomes as the standard with which all the other series may be made comparable. They treat the original capital invested as a sum to be kept intact to be reinvested. The payments at the end of each year are treated as gross income to be divided between a depreciation fund sufficient to maintain the capital unimpaired at the end of the period, and net income which may either be spent or be saved (reinvested as an additional capital). In all bonds bought above par the amount to be treated as a depreciation fund is larger in proportion to the nearness of the maturity of the bond, and in turn, the more distant the date of maturity the higher the present price of a bond bought or sold above par.3 This method of calculating the capital investment as equivalent to a fixed sum of money is convenient, especially in distinguishing between income and principal. When losses are great they fall upon capital, and the income is a negative quantity. When prices of bonds rise, the income is larger than was expected, and (unless taken out in some way) is by a mere bookkeeping process counted as added to the capital, and the rate of income thereafter is reckoned on a higher basis of annual investment.4
§ 8. Corporate securities. Corporations are business enterprises which issue stock, or certificates of a share in their ownership and income. Doubtless the convenience of the sale and transfer of invested capital by the use of stock has been one of several reasons for the large increase of this form of organization since the beginning of the nineteenth century. Originally the stock of a company taken collectively represented all the capital invested, and each share entitled the owner to a given portion (called a dividend) of the total income earned. The shares are issued in regular denominations in terms of money; this amount expressed on the face of the share is the so-called nominal value, or par value, or face value.5 But as a business proves more or less profitable, the value of a share of its income rises and falls regardless of the nominal amount of stock issued. At once there is a divergence between the denomination and the market-price (often called value) of the stock. The nominal amount of stock is relatively permanent, the same year after year; it may be increased by further issues, or it may be decreased by cancelation after purchase with funds in the corporation’s treasury. But when stock is the only form of claim on the earnings that is issued, the fluctuations of the market-price of the stock record the market’s judgment of the business; that is, its expected dividends capitalized at a market-rate for investments of that grade of safety.
But in present practice there are several forms (of which stock is but one) in which corporate incomes are marketed and in which an investor may buy a share in the earnings of the business. Bonds, representing money loaned to a company and entitling their holder to regular interest payments, hold legal priority to the claims of any variety of stock. They usually do not give their owner a vote in the management or make him in the technical sense a part owner in the business. Next stands preferred stock, which entitles the owner to share first in the dividends, if there are any; and finally the common stock, which gets a share only when the other claims are satisfied. All of these are called corporate “securities,” tho they are in many cases far from “secure,” in the sense of being free from risk. By the multiplication and further variation of these readily saleable claims on industrial incomes, the investors’ desires are met more fully and with greater precision, and correspondingly the corporations more conveniently obtain the funds needed.
§ 9. Capital value of public franchises. Franchises for the use of public highways by railroad, gas, water, and lighting companies enable their owners to get incomes which can be capitalized. If a company is given the exclusive right to operate in a locality (with use of streets, alleys, public roads, and the right of eminent domain to condemn private property) any income above an average rate of return on the investment is capitalized either in the higher price of the stock or in additional stock issued without additional outlay upon the plant. If the franchise is unlimited, the income may be capitalized as practically perpetual; if the franchise is limited, and is to expire in thirty or forty years, only the limited series of privileged incomes can ordinarily be capitalized. When, however, the managers are able to exert influence enough to have the franchise extended, and the investors believe in the skill of the managers to influence the legislators by fair means or foul, the value of the stock continues higher than it could usually be under a limited franchise. Such circumstances becloud the question whether the exceptional income arising under the franchise should go to the public or to the company. The important question, however, is whether the company is entitled to the income, for if so, the capitalizing of the income somehow, as is done in every other business, is inevitable.
§ 10. Incomes sold in perpetuity. In creating any income-yielding debt, public or private, the seller is capitalizing the promise of regular incomes to be paid to the buyer. It is not essential that a debt agreement should call for the repayment of the capital sum advanced by the lender. Many if not all of the early “public stocks” were in form promises to pay an annual income perpetually (without specified date of maturity) as the “British consols” (national bonds) are to-day. They sold for whatever they would bring in the market as a means of borrowing money. They could be redeemed and canceled only by purchasing them at their price in the open market.
The sale, in the Middle Ages, of a “charge” on the rents of a landed estate, called a “rent charge,” was in very similar form.6 A landowner, wishing money to go on a crusade or to improve his estate or to invest in some other business, sold a rent-paper entitling the purchaser to receive permanently a given sum, to be paid out of the rent of the estate. The debt was a “charge” upon the rent, until it was paid. The seller gave up the right to retain that amount of rent as it came in year by year, and received a capital sum in hand. Generally he had the right to repay the sum whenever he wished and thus extinguish the rent-charge. Logically viewed, the purchaser, in buying an equitable part of the income, bought an equitable part of the rent-bearing wealth. In effect it was just like a loan except that the purchaser of the rent-charge could not demand the repayment of his money. He could, however, sell the rent-charge when he wished to get his capital out. Gradually it became usual to sell and transfer rent-papers just as is done to-day with mortgages and bonds. Rent-papers thus came in the fifteenth century to be negotiable paper in somewhat general use. There was a rise and fall of the value of the rent-paper with changes in the demand for investment in rent-charges or with changes in the security.
§ 11. Bonds with fixed maturity. The modern public and corporate bonds issued by nations, states, municipalities, and corporations, for war, public buildings, public works, such as wharves, canals, water supply, etc., are looked upon by both the borrower and the lender much in the same way as were the old annuities. The main difference is that the modern obligations promise to repay a stated capital after a stated number of years. If the income of a $1000 4 per cent bond (interest payable semi-annually) running for 20 years, is $40, and the bond sells for $900, it will yield an income of $40 each year for 19 years and an income of $1040 the 25th year, of which $900 equals the original capital invested, and $100 is the increment of value distributed over the whole period. Such a transaction would be said to net the investor 4.78 per cent. The incomes received from public bonds are paid from the proceeds of taxation and are a charge on the rents and incomes of all taxable property; and the incomes received from bonds of industrial and public service corporations are a charge on the earnings of the enterprises.
Even a deposit evidenced by an interest bearing “certificate of deposit” in a commercial bank or in a savings bank may be seen to have this same character. The bank is the borrower, exchanging the promise to pay each year, or half year, or quarter, an amount of interest proportioned to the amount deposited, and to repay the capital sum on certain agreed conditions. The life annuities sold by insurance companies preserve very closely the character of the old annuities and rent-charges, tho the annuity that can be bought for a present payment is proportionately larger than a perpetual annuity, because the number of payments is limited to the lifetime of the purchaser.
[1 ]The rate of premium is reckoned on the basis of present worth as 100. This rate is ordinarily used to discount the future by dividing the future income by the rate plus 1. P = To express the true rate of discount on the future, however, the future value must be taken as a basis of 100; discount is proportional to premium, as present worth is to future worth; thus p : f :: 100 : 105 :: 95.238 : 100 :: 4.762 (the rate per cent of discount) : 5 (the rate per cent of premium). As a matter of convenience in business practice, the rate of premium (which becomes an interest rate) is generally employed in all banking, insurance, annuity, forestry, and other problems, and when used as a divisor, in the manner just explained, is for convenience spoken of as the rate at which the sums are “discounted”; e.g., in the next paragraph of the text. In the well-known method of bank discount, however, the premium (interest) rate is used as a multiplier, the interest being simply taken out in advance and the borrower receiving a smaller sum than that nominated in the note. This is equivalent to charging him interest at a somewhat higher rate, as interest ordinarily is payable at the end of the year. See ch. 25, note 2.
[2 ]In most cases the interest on bonds is payable semi-annually (at the end of each six months) and the bond tables showing the “rate of interest realized if purchased at prices named and held to maturity,” otherwise known as the investment yield, are usually prepared with this condition in mind. This is equivalent to a slightly higher rate of interest.
[* ]This shows graphically that, the net yield of a durative agent being given, every possible price (capitalization) arithmetically corresponds with and involves a rate, which evolves as a rate of income on the investment.
[3 ]E.g., in a circular issued in 1908 prices were quoted: “$100,000 City of Boston registered 4 per cent bonds, due June 1, 1928, price 106⅜, yielding 3.55 per cent; same kind due June 1, 1938, price 108¼ yielding 3.55 per cent; same kind due June 1, 1948, price 109½, yielding 3.55 per cent.” The gross yearly income is $4000, of which but $3550 would be treated as net income, and $450 would have to be reinvested each year to leave the capital undiminished at the end of the period for which the bonds run. The opposite is necessarily the case with bonds bought at a discount, the rate of net income on the investment being somewhat larger than the percentage that the annual payment is of the par value.
[4 ]The practice of counting the capital as equal to a fixed sum of money has suggested to some the idea that capital is, in its very nature, a permanent quantity of value. But this is an illusion, for the “sum of value” may be both increased and diminished, and may be utterly swept away. Capital is no more than the bookkeeping expression of the present worth of a person’s control over income, definite at the moment that the first investment has been made, and thereafter merely an estimate, until the property (right) is again sold at a price.
[5 ]This is an unfortunate use of the term value. It would be better to speak of the amount of a single share as the denomination, and of the product of the denomination by the number of shares outstanding as the nominal amount of stock.
[6 ]See ch. 14, sec. 4.