but that portion of the public securities, whose dividend is paid out of the Public Funds, is specially termed the Funded Debt, and whenever temporary loans have been raised by government, by the issue of securities called Exchequer Bills, they constitute the Unfunded Debt.

When a person becomes proprietor of stock, the transaction is recorded in the account books of that stock, kept by government, in the Bank of England.

295. The same view is taken of the capital of joint-stock associations; the money is raised by shares, among the partners. The shares are fixed at a certain sum: £10 each, £50 each, or any other sum. Now, should the undertaking prove lucrative, and yield a percentage larger than was expected, the shares find a ready market, and rise in price: a £50 share may bring £64, in which case the shares are said to be £14 above par, &c. Ex. 1. What is the purchase of £816. 15s. 3d., in the 3 per cent. Annuities, at 744 per cent.?

£100 stock cost £74 sterling,

..£816. 15s. 3d. cost

sterling.

£816. 15s. 3d. x £744

100

or £608. 9s. 9d.

Ex. 2. What stock can be bought for £2418, in the 3 per cent., at 93 ?

£93 sterling buy £100 stock,

.. £2418 sterling buy

2418 × £100

or £2600 stock.

93

Ex. 3. How much per cent. will be obtained by purchasing 34 per cent. stock, at 92?

£92 sterling gain £3,

..£100 sterling gain

100 × £31/

92

or £347.

Ex. 4. If £1170 be invested in the 3 per cent, at 78, what annual income arises?

...£78 gain £34,

Ex. 5. What sum must be invested in the 3 per cent. stock,

at 91, to produce an income of £72?

Ex. 6. How much stock, in the 3 per cent. Consols, can be bought for £2600, when the price is 97%, and the commission to the stock broker per cent.?

..the commission is for £100 stock,

..£973+, or £97, is the cost of £100 stock;

..£2600 is the cost of

260000
97

or £2666. 13s. 4d.

Ex. 7. How much stock can be purchased by the transfer of £1000 stock from the 3 per cent. at 72 to the 4 per cent. at 90, and what would be the difference of income?

Income on £1000 stock, at 3 per cent.=10 × £3, or £30.
And 100 is bought by 72,

=

.. 1000 is bought by 10 x £72 £720 sterling=money invested.

And income on £800 stock, at 4 per cent. = 8 × £4, or £32. Difference of income £32-£30=£2 gain per year.

The second income might have been found without determining the second stock.

Thus, . 90 brings 4 income,

720 brings 8x £4, or £32 income.

Ex. 8. In which is it the most advantageous to invest, the 3 per cent. at 96 or the 3 per cent at 881?.

Hence it is most advantageous to invest in the 3 per cent., at 961.

296. EXERCISES.

1. What is the purchase of £3260 Bank Stock, at 156 per

cent.?

2. What quantity of stock, in the 3 per cent., at 634, can be purchased for £780?

3. At what price must a purchase be effected in the 3 per cent., to secure 5 per cent. interest?

4. What sum must be invested in the 3 per cent. stock, at 881, to produce an income of £420 ?

5. If I buy £840 stock in the 3 per cent., at 803, and pay per cent. brokerage, what does it cost me?

6. What must be paid for the purchase of £12000 stock, at 105, and the commission per cent.?

7. What would be the difference in the incomes of two persons, one of whom invests £5650 in the 4 and the other the same sum in the 3

per cent. stock, at 1023, per cent., at 93 ?

8. A person invests £6800 in the 3 per cent. stock, at 93%, and soon after sells out, the stock having risen 2 per cent.; in both buying and selling he pays brokerage at per cent. His gain is required.

9. Bought, some £25 railway shares, when they were at £134, and sold them again at £33, and thus made a profit of £760. How much was invested, and what was the gain per cent. ?

10. The £100 share in a railway pays a dividend of £5. What must be given per share to receive 7 per cent. for my money?

11. Bought £1242 stock, in the 4 per cent., for £1471. 16s. 8d. What is the price of stock, and what is my income?

12. What is the value of £1000 India stock, at 1153 per cent. ? 13. What must be the price of 3 per cent. stock to enable me to receive 4 per cent. for my money ?

14. Having invested £840 in the 3 per cent. stock, at 85, I want to know how much I must invest in the 8 per cent., at 151, to secure an income of £540?

15. By selling out of the 3 per cent., at 84, a person obtains £1784, of which he invests £840 in the 3 per cent., at 86, and the remainder in the 7 per cent., at 162. Find the difference in his income.

16. What is the rate per cent. obtained for money in the 8 per cent., at 124 ?

17. What would be the difference of income made by the transfer of £10000 stock from the 3 per cent., at 80, to the 4, at 98?

18. In which is it the most advantageous to invest money, the 5 per cent. stock, at 1214, or the 3 per cent., at 84?

19. When the 3 per cent. stock is at 94, what is the corresponding rate in the 5 per cent., and what is the price of £3000 stock of each ?

20. If the 3 per cent. stock falls 2, what must be the corresponding fall in the 3 per cent.?

ANNUITIES.

297. When a sum of money is to be paid yearly for a certain number of years it is called an annuity.

There are two kinds of annuities, those that are for a given number of years, called annuities certain, and those that are to be paid so long as one or more individuals shall live, and are, therefore, called contingent or life annuities.

Questions relating to annuities involve some principles of interest, but they require the assistance of algebraical formulæ, which it would be out of place to explain here.

But, nevertheless, we shall give a few instances, which can be solved without a direct reference to these formulæ.

Ex. 1. What is the amount of an annuity of £60, to continue 5 years, allowing 5 per cent. compound interest ?

Suppose the last payment to be £1, the 4th would be £1.05, that is, £1+its interest for 1 year; then the 3rd would be (1.05)x(1.05) or (1.05); the 2nd, (£1.05)x(1.05)x(1.05) or (£1.05); and the 1st, (£1.05)1. Therefore, the amount of an annuity of £1, for 5 years, is £1+£1.05+(£1.05)2+(£1.05)3+ (£1.05)1 = £5.52563125.

And hence the amount of an annuity of £60 is 60 × £5.52563125, or £331. 10s. 9d.

Ex. 2. What is the present worth of an annuity of £40, to continue 5 years, compound interest, at 6 per cent. per annum? £1.06 in 1 year is at present worth £1,

..£40 in 1 year are at present worth

..£40 in 2 years are at present worth

£35.599857.

..£40 in 3 years are at present worth £33.584771.

..£40 in 4 years are at present worth £31.683746.

=

..£40 in 5 years are at present worth

(1.06)5

£40

1.3382255776

= £29.910327. ..£168.51455 or £168. 10s. 34d. = the present worth of the annuity.

Ex. 3. A freehold estate (or an annuity to continue for ever, or a perpetuity) brings in yearly £75. What would it sell for, allowing the purchaser 5 per cent., compound interest, for his money?

By the question, we have to find a sum which, if put to interest, will bring the same as the yearly rent.

Ex. 4. A person will buy a freehold farm, provided that he can get 63 per cent. for his money. How many years' purchase should he offer?

Since "How many years' purchase" signifies the same as how many years' rent, we must then find in how many years the rent must be paid to make £100.

63 are made in 1 year,

100

..100 are made in or 15 years' purchase.

Ex. 5. I bought a freehold estate at 16 years' purchase. What rate per cent. have I for my money?

By the last example, the answer is evidently 100, or 61 per

cent.

Ex. 6. What is the present worth of an annuity of £75, to commence 10 years hence and to continue 7 years after, at 6 per cent. ?

This is an example of an annuity in reversion, or an annuity which does not come into possession till some time has elapsed.