DISCOUNT.

Discount is the allowance made for the payment of money before it is due.

Present value is the present worth of money due at a certain time hence.

Suppose, for instance, I have a house for sale, and sell it to a man on the condition that he pay me £500 for it in six months' time, it is quite certain that if he were willing to pay me at once, that some abatement would be made on this £500; or if the money were paid now, that I should be willing to receive a few pounds less. In practice, the buyer would give me a bill promising to pay in six months, when I should either wait till the end of the time for my money, or, if in want of ready money, should take the bill to the banker, who, if he considered the man or myself solvent, would pay me the money, deducting simple interest for the unexpired part of the six months. This is called discounting a bill. The question naturally arises, If the banker takes off the simple interest, what is the difference between Discount and Simple Interest? The answer is, "practically, none,” ""theoretically, much." In fact, in the practical business of life, and in the rules laid down in arithmetic, there is on this point quite a divergence. The sooner the rule called Discount is banished from our arithmetic, and practice substituted for theory, the better for all parties.

When a banker takes the discount off a bill, say £200, for 12 months at 4 per cent., he would deduct £8, paying £192. Now, the person who has this money might say, If I put this £192 out at interest for 1 year at 4 per cent., it will only bring me in £7, 13s. 73d. I have the use of a sum of money for which I am charged £8, it ought to return me the same amount, £8, but it does not, and therefore the discounter has charged too much. So the matter remains thus: In Practice, discounting consists in taking off the simple interest, which is really too much. Arithmetic gives a rule by which the true quantity to be taken off can be found; must the arithmetic conform to the practice, or can we induce the bankers to give up their extra profit and become mathematically accurate, to the detriment of themselves and the advantage of their customers?

The discount on £105 for 1 year at 5 per cent. is £5 (5 × 1), because £100 put out to interest for 1 year at 5 per cent. would bring in £5, and so the person who holds the bill would have the exact value of his security.

So also the discount of £110 for 2 years at 5 per cent. is £10

(2 x 5), because £100 put out to interest for 2 years at 5 per cent. would yield £110.

Once more: the discount on £112 for 3 years at 4 per cent. is £12 (3 × 4), because £100 put out to interest for 3 years at 4 per cent. would yield £112.

Hence we see that if we multiply the number of years by the rate per cent., and add the product to £100, the discount of £100+that sum, is that sum for the given time.

It is required to find the discount of £240 for 2 years at 3 per cent.

In this case the interest on £100 for 2 years at 3 per cent. is 2×37.

Hence the discount on £107 for 2 years at 3 per cent. is £7. Now we have a simple Rule of Three problem. If the discount on £107 is £7, what is the discount on £240?

As £107: £240 :: £7: £15, 14s. 0247d. Ans.

Find the present value of £240 for 2 years at 3 per cent. By previous reasoning it is seen that if the discount on £107 is £7, the present value of £107 must be £100; so the question becomes this: If the present value of £107 is £100, find the present value of £240.

Present value.

As £107 £240 :: £100: £224, 5s. 11d. Ans.

It is a good method to treat all discount sums alike, i.e., consider that there is no distinction between the two cases in discount until you come to the final answer. To this end

understand that the discount is always found first, and then when the present value is required, you have only to subtract the answer from the principal, or the money, so as always to employ one kind of statement, and not two different statements. In the above case, the answer arises immediately from the first sum, thus:

1. Find the discount upon a bill of £239, 10s. 6d., payable 18 months hence, at 3 per cent. per annum.

2. Find the discount on £360, due 14 years hence, at £3, 10s. per annum.

3. What is meant by discount? Find the discount on £1807, 17s., due at the end of 2 years, at 4 per cent.

4. What ready money will discharge a debt of £1377, 13s. 4d., due 2 years, quarters, and 25 days hence, allowing discount at 43 per cent. per annum?

5. What is the discount on £325, 10s., due 146 days hence, at 3 per cent. per annum?

6. If I am liable for a bill of £380, due 3 months hence, and I propose to pay at once partly in cash and partly with a bill for £152, due 4 months hence, what sum must I pay down (interest reckoned at 4 per cent.)?

7. What is the present worth of £120, payable as follows, viz: £50 at 3 months, £50 at 5, and the remainder at 8 months, allowing discount at the rate of 5 per cent.?

8. Define discount. Find the discount at 5 per cent. per annum on £100, due one year hence.

9. It is required to find a sum of money, of which the discount for three years is £10 more at the rate of 7 than at 5 per cent. per

annum.

10. Calculate the discount on £2675, 13s. 4d. for 6 months at 2 per cent. per annum.

11. What is the discount on £257, 8s. 84d., paid 210 days before due, at 4 per cent.?

12. A bill of £760 is due 7 months hence; what is its present value at 5 per cent. per annum?

13. Find the present value of £496, 18s. 10d., due 2 years and 3 months hence, interest being calculated at £5, 10s. per cent.

14. A bill for £485 falls due; I pay £250; for the rest I give a 3 months' bill at 4 per cent.; for how much is the bill given, if its present value be the remainder of the sum owing?

15. A bill of £894 is drawn on February 16, 1860, at 7 months' date; what will be the immediate discount at 5 per cent.; and what will the discount be on the 1st of June 1860?

16. What sum of money paid down will discharge a debt of

£1000, due in two equal half-yearly instalments, interest being reckoned at 5 per cent. per annum?

17. Find the discount on a bill of £698, 17s. 8d., due in 1 year 7 months, at 4 per cent. per annum.

18. At what price must linen, which cost 2s. 74d. per yard, be marked, so as to gain 5 per cent., besides allowing a discount of 5 per cent. to the purchaser?

19. Find how much £240, 10s. is worth, paid 2 years before it is due, at 7 per cent.

20. What is the present value of £455, 12s. 6d., due at the end of 4 years, at 5 per cent.?

Discounting of Bills.-Sometimes merchants and others, who have stocks on hand, have not sufficient ready money to pay for their goods. They, under such circumstances, "give a bill," or a promise to pay, in 3, 4, or 6 months. The person to whom the bill is given takes it to a banker, who, if he considers the parties are solvent, advances the money, deducting the simple interest for the time unexpired. The person who takes the bill to the banker writes his name on the back of it, or "endorses it;" this, if the merchant or tradesman should fail, who gives the bill, renders the endorser liable to refund the money. Sometimes if the parties are strangers to the banker, or appear to have no assets, they have to obtain the signature of a known responsible person; then the banker will advance the money or pay the bill. Any person whose name is on a bill is liable to pay it.

"To give a bill" is to write an agreement on a stamped paper, by which the debtor agrees to pay the creditor by or on a certain day.

To meet a bill is to be ready with the money due to a creditor who was promised payment on a certain day.

While the bill is running means the time between giving a bill and the date at which it "falls due;" and at any time while the bill is running it may be discounted by a bill broker or by a banker. We must consider the banker as dealing in money: he, as it were, when he discounts a bill for £100, buys £100 for less than £100, which money is paid on a certain date; his profit is the discount minus his expenses of office, clerks, etc.

Bills are not payable, or, as it is termed, "presentable,” till three days after they are due. These three days are called "days of grace." Thus, for instance, if it is specified on the bill that it is to fall due on the eighteenth day of June, although nominally due on that day, it is not legally due till the 21st

of June; and if a bill broker were to discount such a bill on the 1st of June, he would consider how many days were unexpired till the 21st, and would take off the simple interest for 20 days. If the last of the days is a Sunday, the bill falls due on the Saturday preceding.

Discount is also a term applied to the deduction which many shopkeepers make on their marked prices. It will frequently happen that if a customer comes with ready money, or will take a large quantity of certain articles, that the seller can afford to let the purchaser have them cheaper, and will offer them with a certain allowance off; this is also termed discount. When a bill is settled, trifling allowances are often made: these all come under the general term, Discount.

A banker has a bill for £219 drawn on the 4th of June, at 4 months, presented to him for payment on the 14th of July; find how much he will pay for such a bill at 4 per cent.

The bill is drawn on the 4th of June for 4 months; it is due on the 7th of October, and is discounted on the 14th of July. Between the 14th of July and 7th of October are—

July. Aug. Sept. Oct.
17+31+30+7=85 days.

If £100 pays £41⁄2 for 365 days, how much will £219 pay for 85 days?

The usual way in practice, as before stated, is to find the simple interest on £219 for one year, or 365 days; and then by a simple Rule of Three sum find the interest for 85 days; but it will be seen that to make a Double Proportion sum of it is simpler and more sensible.

Find the discount on a bill for £219 drawn on the 4th of June at 4 months, and presented for payment on the 14th of July, discount being allowed at 4 per cent.