13. When the annual dividend of railroad stock is 15%, and the interest of money is 10%, at what premium ought the railroad stock to sell? Ans. 50%.

14. At what per cent. discount must I buy bank stock, paying 6%, that the investment may pay 9%. Ans. 331%.

15. If the C. & E. RR. Co. declare a dividend of 15% per annum, what is the value of its stock, money being worth 8% ?

16. The free banking law of New York requires that the stocks deposited with the superintendent, as security for banknote circulation, shall be made equal to stock producing an interest of 6% per annum. What amount of circulating notes could a bank receive on a five per cent. stock?

Ans. 83% of the par value of the stock. What on a 7% stock? Ans. 116%. 17. In January, 1848, the total amount of British consols was £378,019,855. What was the amount of interest paid on them semi-annually? Ans. £5,670,297.

18. The debt of Great Britain and Ireland, in round numbers, is £780,000,000, and the annual revenue £56,000,000. Supposing the annual interest to average 31%, what per cent. of the revenue is needed to pay the interest on the debt?

19. In July, 1859, forty-five New York Fire Insurance Companies (out of fifty), on a capital of $8,712,000, divided among the stockholders, as a semi-annual dividend, $679,950. Compared with railroad stock paying 5% semi-annually, which would yield the greater income, railroad stock bought at 65% or insurance stock at par?

20. A man subscribed $20,000 stock in a mining company, the capital stock of which is $500,000, but only 20% paid in. A cash dividend of 2% on the par value is declared and a dividend of 10% to be credited to the stockholders as an installment on their unpaid stock. What is the amount of cash he receives, and what is the balance due on his subscription ?

21. I buy 100 shares of $100 each in a railroad company, the capital stock being $3,000,000. The first year they de clare a cash dividend of 10%. The second year they increase their stock by declaring a stock dividend of 10%. The third

year they divide among their stocknclders the same amount as in the first year. What would be the per cent. of the last dividend? Ans. 9 per cent. How much more would they need to declare a dividend of 10%, the same as in the first year? Ans. $30,000. 22. If the paid up stock in a railroad company be worth 100%, and a stock dividend of 10% be made to the stockholders, what would be the value of the stock after the dividend ? Ans. 90 per cent.

23. If the net earnings of a bank with $200,000 capital be sufficient to pay an annual dividend of 10%, and reserve $4000 as a surplus to provide for future losses, and it pay 6% on its net earnings to the State in lieu of taxes, what would be the rate of taxation on its capital?

Ans. per cent.

NEW RULE FOR FINDING THE VALUE OF A BOND.

ART. 138. Most of the problems respecting stocks and bonds, and brokerage in money and exchange, can be solved. by the application of the ordinary principles of percentage, without special rules. One problem, however, not unfrequently arises, more complicated, to the solution of which the attention of the student is now directed.

To find the present value of a bond having several years to run, with interest payable semi-annually, in order to realize from the dividends and final payment an equivalent to a given rate per cent. per annum on the investment, use the following

RULE.-1st. Taking a single dividend or semi-annual interest on the bond for a principal, compute the simple interest on it at the proposed rate, for one-fourth as many years as would be the product of the number of semi-annual dividends into the number less one. To this interest add the sum of the several amounts of semi-annual interest, and the face of the bond, setting this sum down for a DIVIDEND.

2d. Suppose another bond, differing from the given bond only in its rate of interest being the same as the proposed rate for investment. Proceed with this as with the other, and use the result for a DIVISOR.

The quotient, after division, will express, decimally, the rate per cent. of the par value equal to the present value.

Ex. 1. What should I pay for a bond for $1000 due in 10 years, with interest at.5%, payable semi-annually, in order to make it a 10% investment ?

Solution.

Interest on $25 at 10%, for 20x12 years,

Total amount of semi-annual dividends=$25×20=

Face of the bond,

Dividend,

Interest on $50 at 10% for 20x12 years,

$237.50

500

1000

1737.50

475

Total amount of semi-annual dividends =$50×20= 1000

Face of the bond,

Divisor,

$1737.50 $2475.70202.

1000

2475

$1000 ×.70202=$702.02, the present value of the bond. REMARK.-A strictly accurate solution of the above problem requires the aid of logarithms, and the operation is tedious. The above rule is simple and brief, and gives a result sufficiently approximate for all practical purposes. The question involves compound interest, the interest on the investment being supposed to be compounded annually, while the interest on the dividends is compounded at the proposed rate at the end of each year. Though annual interest gives a result somewhat less than compound interest, yet if two problems be wrought, first by annual interest and then by compound, the ratio between the results by the first operation will not differ essentially from the ratio by the second. This principle forms the basis of the rule given above. The work of computing the 'annual interest," or rather semi-annual interest, is much shortened by incorporating in the rule an expression for the sum of the arithmetical series of years, during which a single dividend would draw interest. The approximation to strict

accuracy is furthermore increased by treating the supposed bond or investment the same as the one given, so far as that its interest should be payable semi-annually instead of annually, as proposed in the conditions of the problem.

The answer given to the above example in PRICE'S STOCK TABLES, computed by logarithms, is 70% instead of 70%, as given by the above rule.

If the rate per cent. to be realized be the same as the rate of interest on the bond, the present value, by the above rule, would be the par value. By Price's Stock Tables it would be at a premium; if 7%, and running 50 years, the premium would be 1% per cent.

Ex. 2. Money being worth 10% per annum, what is the present value of a 7% bond, interest payable semi-annually, running 20 years? Ans. 76 per cent. " 75,9%

06

66

By Price's Stock Tables. Note. As the ratio only is sought, any convenient amount may be assumed for the face of the bond.

Ex. 3. In 1813 the United States government borrowed $16,000,000, selling their bonds to run 12 years, at 6% interest, payable semi-annually, at 12% discount. At what discount should the purchasers have taken them, to realize on their investment an average annual interest of 8% ?

Ans. 14,8%. REMARK.—It is manifest that, if a corporation sells in New York its bonds, drawing 7% interest, for less than par value, it is borrowing money at a higher rate of interest than the legal rate, and the contract under the general law of that State, regulating interest, becomes tainted with usury. But for the accommodation of corporations, and the security of capitalists investing in such bonds, it was enacted by the Legislature of New York, in 1850, that "no corporation shall hereafter interpose the defense of usury in any action." With this restriction upon them, corporations can negotiate their bonds more readily and at better rates than without such restriction. A large class of individual borrowers desire a similar legal prohibition for a like accommodation,

EQUATION OF PAYMENTS.

ART. 139. Equation of payments is the process of finding the mean or average time for the payment of several sums of money due at different dates. The mean or average time sought is called the equated time.

The common methods of finding the equated time are based upon the principle that money kept after it is due is counterbalanced by an equal sum of money paid the same length of time before it is due.

This principle obviously depends upon another which may be expressed as follows: The payment of $100 down, and $100 in two years without interest is equivalent to the payment of $212 in two years, without interest, the rate of interest being 6 per cent.; or to express the same abstractly, the use of any sum of money is worth its interest for the time it is used.

ART. 140. To find the equated time for the payment of several sums of money with different terms of credit.

Ex. 1. A owes B $1200, of which $300 is due in 4 months, $400 in 6 months and $500 in 12 months. What is the equated time for the payment of the whole sum?

300 × 4-1200 400 × 6=2400

500 × 12-6000

1200

)9600

FIRST METHOD.

Explanation.-Suppose the sums

to be paid respectively, 4 months, 6 months, and 12 months before due. The amount to be paid will be

8 mos. Ans. $300-its discount for 4 months; $400-its discount for 6 months; and $500-its discount for 12 months. The interest or discount of $300 for 4 months equals the discount of $1 for 1200 months; the discount of $400 for 6 months equals the dis