The following are the premiums demanded by the Equitable Society for insuring 1001., or an equivalent annuity on the contingency of one life's surviving the other :

It is stated by Mr. Morgan, in his Account of the Equitable Society already referred to, that the number of insurances in that institution for terms of years does not much exceed one hundredth part of those for the whole period of life; and that the business of the office at present is almost wholly confined to the assurance of persons on their own lives-those on the lives of others, whether for termis or for continuance, being, in consequence of the commission money allowed to agents and attorneys, engrossed by the new offices.-(Account of the Equitable Society, p. 53.)

[The reader is referred to Kent's Commentaries on American Law, Lecture 48th, for information concerning the law of marine insurance in the United States,-and to the 50th Lecture of the same work for the law relating to life and fire insurances.

It is to be regretted that life insurance is so little practised in the United States. There is no country to which its benefits are more important. That country in which enterprise and activity is most rapidly developed and becomes characteristic of a people is precisely the country where the practice of life insurance accomplishes the most in alleviating calamity and in securing social comfort.

The Massachusetts Hospital Life Insurance Company was incorporated in 1818; since which time the privilege of effecting insurances upon lives has been conferred on a number of other companies. Of these the principal are the Baltimore Life Insurance Company, the Pennsylvania Company and the Girard Life Insurance and Trust Company in Philadelphia, the New York Life Insurance and Trust Company and the Farmers' Loan and Trust Company in the city of New York. It may be stated that the Girard Life Insurance and Trust Company, which commenced business in 1836, is the only one in the United States that has offered, to those who make insurance for the whole of life, a bonus, or addition to the value of their policy, after the expiration of a term of years. This practice has, in several of the London offices, contributed greatly to the benefit of both the insurers and the insured. Calculations of the earnings upon life insurance are usually made after a lapse of seven years, and a proportion of the amount is added to the policies for the whole of life.

No tables of mortality of a general nature have been constructed in the United States. It is, however, believed by those who have directed their attention to the subject, that the duration of life in the northern and middle states is equal to its duration in England and Scotland. Hence all insurances for lives have in this country been computed from the English tables, founded for the most part upon the Carlisle rates of mortality.

The American policies of insurance, when they have reference to the lives of persons in the northern states, stipulate that they shall be void if the insured enter into the military or naval service, or in the event of his dying by suicide, in a duel, or by the hands of justice. They are also declared to be void if the insured should die on the high seas, or the great lakes; or if he pass beyond the settled limits of the United States, or of the British pro

vinces to the north of them, or south of the states of Virginia and Kentucky, without the consent of the insurers being first obtained.—Am. Ed.]

INTEREST AND ANNUITIES. Interest is the sum paid by the borrower of a sum of money, or of any sort of valuable produce, to the lender, for its use.

The rate of interest, supposing the security for and facility of re-possessing the principal, or sum lent, to be equal, must obviously depend on what may be made by the employment of capital in industrious undertakings, or on the rate of profit. Where profits are high, as in the United States, interest is also high; and where they are comparatively low, as in Holland and England, interest is proportionally low. In fact, the rate of interest is nothing more than the nett profit on capital: whatever returns are obtained by the borrower, beyond the interest he has agreed to pay, really accrue to him on account of risk, trouble, or skill, or of advantages of situation and connection.

But besides fluctuations in the rate of interest caused by the varying productiveness of industry, the rate of interest on each particular loan must, of course, vary according to the supposed solvency of the borrowers, or the degree of risk supposed to be incurred by the lender, of either not recovering payment at all, or not recovering it at the stipulated term. No person of sound mind would lend on the personal security of an individual of doubtful character and solvency, and on mortgage over a valuable estate, at the same rate of interest. Wherever there is risk, it must be compensated to the lender by a higher premium or interest.

And yet, obvious as this principle may appear, all governments have interfered with the adjustment of the terms of loans; some to prohibit interest altogether, and others to fix certain rates which it should be deemed legal to charge, and illegal to exceed. The prejudice against taking interest seems to have principally originated in a mistaken view of some enactments of the Mosaical law-(see Michaelis on the Laws of Moses, vol. ii. pp. 327— 353 Eng. ed.), and, a statement of Aristotle, to the effect that, as money did not produce money, no return could be equitably claimed by the lender! But whatever may have been the origin of this prejudice, it was formerly universal in Christendom; and is still supported by law in all Mohammedan countries. The famous reformer, Calvin, was one of the first who saw and exposed the absurdity of such notions-(see an extract from one of his epistles in MCulloch's Political Economy, 2d ed. p. 510.); and the abuses caused by the prohibition, and the growing conviction of its impolicy, soon after led to its relaxation. In 1554, a statute was passed, authorising lenders to charge 10 per cent. interest. In 1624, the legal rate was reduced to 8 per cent.; and in the reign of Queen Anne it was further reduced to 5 per cent., at which it still continues. It is enacted, by the statute (12 Ann. c. 16.) making this reduction, that "all persons who shall receive, by means of any corrupt bargain, loan, exchange, chevizance, or interest of any wares, merchandise, or other thing whatever, or by any deceitful way or means, or by any covin, engine, or deceitful conveyance for the forbearing or giving day of payment, for one whole year for their money or other thing, above the sum of 51. for 100l. for a year, shall forfeit for every such offence, the treble value of the monies, or other things, so lent, bargained," &c.

It is needless to waste the reader's time by entering into any lengthened arguments to show the inexpediency and mischievous effect of such interferences. This has been done over and over again. It is plainly in no respect more desirable to limit the rate of interest, than it would be to limit the rate of insurance, or the prices of commodities. And though it were desirable, it cannot be accomplished. The real effect of all legislative enactments having such an object in view, is to increase, not diminish, the rate of interest. When the rate fixed by law is less than the market or customary rate, lenders and borrowers are obliged to resort to circuitous devices to evade the law; and as these devices are always attended with more or less trouble and risk, the rate of interest is proportionally enhanced. During the late war it was not uncommon for a person to be paying 10 or 12 per cent for a loan, which, had there been no usury laws, he might have got for 6 or 7 per cent. Neither is it by any means uncommon, when the rate fixed by law is more than the market rate, for borrowers to be obliged to pay more than they really stipulated for. It is singular than an enactment which contradicts the most obvious principles, and has been repeatedly condemned by committees of the legislature, should still be allowed to preserve a place in the statute book. •

Distinction of Simple and Compound Interest.-When a loan is made, it is usual to stipulate that the interest upon it should be regularly paid at the end of every year, half year, &c. A loan of this sort - is said to be at simple interest. It is of the essence of such loan, that no part of the interest accruing upon it should be added to the principal to form a new principal; and though payment of the interest were not made when it becomes due, the lender would not be entitled to charge interest upon such unpaid interest. Thus, suppose 100l. were lent at simple interest at 5 per cent., payable at the end of each year; the lender would, at the end of 3 or 4 years, supposing him to have received no previous payments, be entitled to 151. or 201., and no more.

Sometimes, however, money or capital is invested so that the interest is not paid at the periods when it becomes due, but is progressively added to the principal; so that at every term a new principal is formed, consisting of the original principal, and the successive accumulations of interest upon interest. Money invested in this way is said to be placed at compound interest.

It appears not unreasonable, that when a borrower does not pay the interest he has contracted for at the period when it is due, he should pay interest upon such interest. This, however, is not allowed by the law of England; nor is it allowed to make a loan at compound interest. But this rule is often evaded, by taking a new obligation for the principal with the interest included, when the latter becomes due. Investments at compound interest are also very frequent. Thus, if an individual buy into the funds, and regularly buy fresh stock with the dividends, the capital will increase at compound interest; and so in any similar case.

Calculation of Interest.-Interest is estimated at so much per cent. per annum, or by dividing the principal into 100 equal parts, and specifying how many of these parts are paid yearly for its use. Thus 5 per cent., or 5 parts out of 100, means that 51. are paid for the use of 1001. for a year, 101. for the use of 2001., and 21. 10s. for the use of 501. for the same period, and so on.

Suppose, now, that it is required to find the interest of 2101. 13s. for 34 years at 4 per cent. simple interest. In this case we must first divide the principal, 210. 138. into 100 parts, 4 of which will be the interest for 1 year; and this being multiplied by 3 will give the interest for 3 years. But instead of first dividing by 100, and then multiplying by 4, the result will be the same, and the process more expeditious, if we first multiply by 4, and then divide by 100. Thus,

L. 5.

210 13 principal.

4

1,00) 8,42 12 (
20

8,52
12

6,24

rate per cept.

L. s. d.,

8 8 6 1 year's interest.

L. 29 9 93 34 years' interest.

4 96

It is almost superfluous to observe, that the same result would have been obtained by multiplying the product of the principal and rate by the number of years, and then dividing by 100.

Hence, to find the interest of any sum at any rate per cent. for a year, multiply the sum by the rate per cent., and divide the product by 100.

To find the interest of any sum for a number of years, multiply its interest for one year by the number of years; or, without calculating its interest for one year, multiply the principal by the rate per cent. and that product by the number of years, and divide the last product by 100.

When the interest of any sum is required for a number of days, they must be treated as fractional parts of a year; that is, we must multiply the interest of á year by them, and divide by 365. Suppose that it is required to find the interest of 2101. for 4 years 7 months and 25 days, at 4 per

9.45 X 25
365

The interest for 25 days is
=-6472; that is, it is equal to the interest for a year multiplied by the fraction 25
Division by 100 is performed by cutting off two figures to the right.

365

Many attempts have been made to contrive more expeditious processes than the above for calculating interest. The following is the best :

Suppose it were required to find the interest upon 1721. for 107 days at 5 per cent.

This forms what is called in arithmetical books a double rule of three question, and would be stated as follows:

£ Days. £ £ Days.

100 X 365 5 :: 172 X 107: 21. 10s. 43d. the interest required.

Hence, to find the interest of any sum for any number of days at any rate per cent., multiply the sum by the number of days, and the product by the rate, and divide by 38,500 (365 X 100); the quotient is the interest required.

When the rate is 5 per cent., or 1-20th of the principal, all that is required is to divide the product of the sum multiplied by the days by 7,300 (365, the days in a year, multiplied by 20).

Five per cent. interest being found by this extremely simple process, it is usual in practice to calculate 4 per cent. interest by deducting 1-5th; 3 per cent. by deducting 2-5ths; 24 per cent. by dividing by 2; 2 per cent. by taking the half of 4, and so on.

In calculating interest upon accounts current, it is requisite to state the number of days between each receipt, or payment, and the date (commonly the 31st of December) to which the account current is made up. Thus, 1721. paid on the 15th of September, bearing interest to the 31st of December, 167 days. The amount of such interest may, then, be calculated as now explained, .or by the aid of Tables. The reader will find, in the article BOOKKEEPING (p. 161.) an example of interest on an account current computed as above, without referring to Tables.

The 30th of June is, after the 31st of December, the most usual date to which accounts current are made up, and interest calculated. In West India houses, the 30th of April is the common date, because at that season the old crop of produce is generally sold off, and the new begins to arrive.

It is of great importance, in calculating interest on accounts current, to be able readily to find the number of days from any day in any one month to any day in any other month. This may be done with the utmost ease by means of the Table on the following page.

By this Table may be readily ascertained the number of days from any given day in the year to another. For instance, from the 1st of January to the 14th of August (first and last days included), there are 226 days. To find the number, look down the column headed January, to Number 14, and then look along in a parallel line to the column headed August, you find 226, the number required.

To find the number of days between any other two given days, when they are both after the 1st of January, the number opposite the 1st day must, of course, be deducted from that opposite to the second. Thus, to find the number of days between the 13th of March and the 19th of August, deduct from 231, the number in the Table opposite to 19 and under August, 72, the number opposite to 13 and under March, and the remainder, 159, is the number required, last day included.

In leap years, one must be added to the number after the 28th of February.

For the mode of calculating discount, or of finding the present values of sums due at some future date, at simple interest, see DisCOUNT.

VOL. II.-I

13

Table for ascertaining the Number of Days from any one Day in the Year to any other Day.

In counting-houses, Interest Tables are very frequently made use of. Such publications have, in consequence, become very numerous. Most of them have some peculiar recommendation; and are selected according to the object in view.

When interest, instead of being simple, is compound, the first year's or term's interest must be found, and being added to the original principal, makes the principal upon which interest is to be calculated for the second year or term; and the second year's or term's interest being added to this last principal, makes that upon which interest is to be calculated for the third year or term; and so on for any number of years.

But when the number of years is considerable, this process becomes exceedingly cumbersome and tedious, and to facilitate it Tables have been constructed, which are subjoined to this article.

The first of these Tables (No. I.) represents the amount of 17. accumulating at compound interest, at 3, 3, 4, 4, and five per cent. every year, from 1 year to 70 years, in pounds and decimals of a pound. Now, suppose that we wish to know how much 500l. will amount to in 7 years at 4 per cent. In the column marked 4 per cent. and opposite to 7 years, we find 1-315,9321., which shows that 11. will, if invested at 4 per cent. compound interest amount to 1:315,932 in 7 years; and consequently, 5001. will, in the same time and at the same rate, amount to 500 X 1-315,9321. or 657-9667.; that is, 6577.

For the same purpose of facilitating calculation, the present value of 11. due any number of years hence, not exceeding 70, at 3, 31, 4, 41, and 5 per cent. compound interest, is given in the subjoined Table No. II. The use of this Table is precisely similar to the foregoing. Let it, for example, be required to find the present worth of 5001. due 7 years hence, reckoning compound interest at 4 per cent. Opposite to 7 years, and under 4 per cent., we find 75291,7811., the present worth of 11. due at the end of 7 years; and multiplying this sum by 500l., the product, being 379-95891., or 3791. 19s. 2d., is the answer required.

ANNUITIES.

1. Annuities certain.-When a sum of money is to be paid yearly for a certain number of years, it is called an annuity. The annuities usually met with are either for a given number of years, which are called annuities certain; or they are to be paid so long as one or more individuals shall live, and are thence called contingent annuities.

By the amount of an annuity at any given time, is meant the sum to which it will then amount, supposing it to have been regularly improved at compound interest during the intervening period.

The present value of an annuity for any given period, is the sum of the present value of all the payments of that annuity.

Numbers III. and IV. of the subjoined Tables represent the amount and present value of an annuity of 1., reckoning compound interest at 24, 3, 3, 4, 4, 5, and 6 per cent., from 1 year to 70. They, as well as Nos. I. and II., are taken from "Tables of Interest, Discount, and Annuities, by John Smart, Gent. 4to. London, 1726." They are carried to 8 decimal places, and enjoy the highest character both here and on the Continent, for accuracy and completeness. The original work is now become very

scarce.

The uses of these Tables are numerous; and they are easily applied. Suppose, for example, it were required to tell the amount of an annuity of 501. a year for 17 years at 4 per cent. compound interest.

Opposite to 17 (Table III.) in the column of years, and under 4 per cent., is 23-69751,239, being the amount of an annuity of 11. for the given time at the given rate per cent.; and this multiplied by 50 gives 1184 8756195, or 1,1847. 178. 6d., the amount required.

Suppose now that it is required what sum one must pay down to receive an annuity of 501. to continue for 17 years, compound interest at 4 per cent. ?

Opposite to 17 years (Table IV.) and under 4 per cent. is 12-16566,886, the present value of an annuity of 11. for the given time and at the given rate per cent.; and this multiplied by, 50 gives 602 283443, or 6081. 58. 8d., the present value required.

When it is required to find the time which must elapse, in order that a given sum improved at a specified rate of compound interest may increase to some other given sum, divide the latter sum by the former, and look for the quotient, or the number nearest to it, in Table No. I. under the given rate per cent., and the years opposite to it are the answer. Thus,

In what time will 5231. amount to 1,0871. 5s. 7d. at 5 per cent. compound interest?

Divide 1087 2794, &c., by 523, and the quotient will be 2.0789, &c., which under 5 per cent. in Table I. Is opposite to 15 years, the time required.

If it had been required to find the time in which a given annuity, improved at a certain rate of compound interest, would have increased to some given sum, the question would have been answered by dividing, as above, the given sum by the annuity; and looking for the quotient (not in Table No. I., but) in Table No. III., under the given rate per cent., it would be found on a line with the time required. Thus,

A. owes 1,000l. and resolves to appropriate 101. a year of his income to its discharge: in what time will the debt be extinguished, reckoning compound interest at 4 per cent.?

1,000 divided by 10 gives 100, the number in Table No. III. under 4 per cent., and nearest to this quotient is 99-8265, &c. opposite to 41 years, the required time. Had the rate of interest been 5 per cent., the debt would have been discharged in somewhat less than 37 years. This example is given by Dr. Price (Annuities, 6th ed. vol. ii. p. 289.); and on this principle the whole fabric of the sinking fund was constructed. Of the abstract truth of the principle there cannot, indeed, be a doubt. But every thing depends on the increasing sums annually produced being immediately invested on the same terms; and this, when the sum is large, and the period long, is altogether impracticable.

Let it next be required to find an annuity which, being increased at a given rate of compound interest during a given time, will amount to a specified sum: in this case we divide the specified sum by the amount of 11, for the time and rate given, as found in Table III., and the quotient is the answer. Thus,

What annuity will amount to 1,0871. 58. 7d. in 15 years at 5 per cent. compound interest?

Opposite to 15 years in Table III., and under 5 per cent., is 21-5785, &c., the amount of 11. for the given time and rate; and dividing 1087-2794, &c., by this sum, the quotient 50 387, &c., or 501. 7s. 9d., is the annuity required.

Deferred Annuities are those which do not commence till after a certain number of years; and reversionary annuities, such as depend upon the concurrence of some uncertain event, as the death of an individual, &c.

The present value of a deferred annuity is found by deducting, from the value of an annuity for the whole period, the value of an annuity to the term at which the reversionary annuity is to commence. -Thus,

What is the present value of an annuity of 50l. to continue for 25 years, commencing at 7 years from the present time, interest at 4 per cent. ?

According to Table No. IV., the value of an annuity of 11. for 25 years at 4 per cent. is 15-62207,995, and that of 11. for 7 years is 6:00205,467, which being deducted from the other, leaves 962002,528, which multiplied by 50 gives 4811., the answer required.

Supposing the annuity, instead of being for 25 years, had been a perpetuity, it would have been worth 1,2501., from which deducting 3001. 2s., the value of an annuity for 7 years at 4 per cent., there remains 9491. 18s., the value of the reversion.

For a selection of problems that may be solved by Tables of annuities certain, see Smart's Tables, pp. 92-100.

2. Life Annuities.-After what has been stated in the article on INSURANCE (GENERAL PRINCIPLES OF), respecting Tables of mortality, it will be easy to see how the value of a life annuity is calculated. Supposing, to revert to the example given before (p. 73.),that it were required to find the present value of 1., the receipt of which is dependent on the contingency of a person, now 56 years of age, being alive 10 years hence, taking the Carlisle Table of mortality, and interest at 4 per cent.: Now, according to that Table, of 10,000 persons born together, 4,000 attain to 56, and 2,894 to 66 years of age. The probability that a person, now 56 years, will be alive 10 years hence, is consequently, 4,000 and the present value of 1., to be received certain 10 years hence being 0.6755647., it follows, that if its receipt be made to depend on a life 56 years of age, attaining to 66, its value will be reduced by that contingency to ·0-488771., or 9s. 94d. If, then, we had to find the present value of an annuity of 17. secured on the life of a person now 56, we should calculate in this way the present value of each of the 48 payments, which, according to the Carlisle Table, he might receive, and their sum would, of course, be the present value of the annuity.

2,894

;

2,894 X 0-6755641.
4.000

This statement is enough to show the principle on which all calculations of annuities depend; and this also was, in fact, the method according to which they were calculated, till Mr. Simpson and M. Euler invented a shorter and easier process, deriving from the value of an annuity at any age, that of an annuity at the next younger age. There is a considerable discrepancy in the sums at which different authors, and different insurance offices, estimate the present value of life annuities payable to persons of the same age. This does not arise from any difference in the mode of calculating the annuities, but from differences in the Tables of mortality employed. These can only be accurate when they are deduced from multiplied and careful observations made, during a long series of years, on a large body of persons; or when the average numbers of the whole population, and of the deaths at every age, for a lengthened period, have been determined with the necessary care. It is to be regretted, that governments, who alone have the means of ascertaining the rate of mortality by observations made on a sufficiently large scale, have been singularly inattentive to their duty in this respect. And until a very few years since, when Mr. Finlaison was employed to calculate Tables of the value of annuities from the ages of the nominees in public tontines, and of individuals on whose lives government had granted annuities, all that had been done in this country to lay a solid foundation on which to construct the vast fabric of life insurance had been the work of a few private persons, who had, of course, but a limited number of observations to work upon.

The celebrated mathematician, Dr. Halley, was the first who calculated a Table of mor