§ 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 81/3 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: P:: 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 P= a/r when P is the present worth of all the incomes, a is the perpetual annuity, and r the rate per cent; e.g., 20,000= 1000/.05; this is equivalent to r = a/p; that is, the rate at which the future incomes are capitalized is the annuity divided by the capital sum; e.g., .05 = 1000/20,000.

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 = 1/1+r =1.00/1.05 . 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.