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 cap ital 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 dis tant 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 = i ; this is equivalent to r= s; 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 correspond ing rate of premium on the present price (investment of cap ital) 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 indi vidual economy, showing itself in the whole state of improve ment 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 mod ern corporation bond is a promise to pay an annual sum, ex pressed as a percent age of the principal, and to repay the principal at the defi nite date of ma turity. 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 in terest. 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) ; usu ally 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 A note secured by real-estate mortgage is like the bond, but not so marketable, and is ordi • 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.
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.
narily 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 pay able 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 undis counted 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 mule comparable as to present worth with a uniform and perpetu4 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 for n, expected rents are made the basis for present capitalihstion. 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 sta tion. 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 pro gressive 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 sta tionary 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 depressive series of incomes.