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 5l. 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 1001. 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 20., 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, be 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 to 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 2007., and 2. 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, 2107. 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 34 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. 8. 210 13 4 1,00) 8,42 12 ( 8,52 6,24 96 principal. rate per cent. 25 5 61 1 year's interest. 3 years' interest. 4 4 3 4a year's interest. L. 29 9 9 3 years' interest. 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 a 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 cent. Interest for 4 years = L. 37-8000 840 25 days = -6472 L. 43-9597 L. 43 19. 24d. 6472; that is, it is equal to the interest for a year multiplied by the fraction The interest for 25 days is Division by 100 is performed by cutting off two figures to the right. 25 385 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. 44d. 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 36,500 (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; 2 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, 107 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 fusure date, at simple interest, see DISCOUNT. Table for ascertaining the Number of Days from any one Day in the Year to any other Day. 98 128 159 189 220 251 281 3.2 342 21 109 139 170 20 231 262 262 323 353 79 110 140 171201 232 263 293 324 354 90 111 141 172 202 233 264 224 325355 8112 142 173 203 234 265 295 326 356 82 113 143 174 204 235 266 296 327 357 55 83 114 144 175 2.5 236 267 297 328 358 26 24 115 145 176 206 237 268 298 329 359 85 116, 146 177 237 238 269 299 330 360 5 36 6 37 65 96 126 157 39 66 97 127 158 188 219 250 280 311 341 23 54 39 9 40 10 41 69 100 130 161 191 222 253 283 314 344 26 57 103 133 164 194 225 256 256 317 347 29 88 119 149 180 210 241 272 302 333 363 42 70 101 131 162 192 223 254 294 315 345 27 58 86 117 147 178 208 239 270 300 331 361 In counting-honses, 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 1. accumulating at compound interest, at 3, 31, 4, 41, 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 5001. will amount to in 7 years at 4 per cent. In the column marked 4 per cent. and opposite to 7 years, we find 1315,932, which shows that 1. 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 1:315,932/. or 657·9661.; that is, 6571. 19s. 4d. For the same purpose of facilitating calculation, the present value of 11. due any number of years hence, not exceeding 70, at 3, 3, 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 5007. due 7 years hence, reckoning compound interest at 4 per cent. Opposite to 7 years, and under 4 per cent., we find 75291,731, the present worth of 11. due at the end of 7 years; and multiplying this sum by 5001., the product, being 379-95897., 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, 31, 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 507. 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 1. for the given time at the given rate per cent.; and this multiplied by 50 gives 1181 8756195, or 1.1841. 17s. 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 253443, or 6081. 5s. 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 upecified 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 2791, &c., by 523, and the quotient will be 20789, &c., which under 5 per cent. in Table 1. 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 17. 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. 5s. 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 1. for the given time and rate; and dividing 1057-2791, &c., by this sum, the quotient 50-387, &c., or 201. 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 501. 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 9-62002,528, which multiplied by 50 gives 481., the answer required. Supposing the annuity, instead of being for 25 years, had been a perpetuity, it would have been worth 1,250, 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 17., 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, 2; and the present value of 17., 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. 4.000 2,994 X 0-6755642. 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 tality, which he deduced from observations made at Breslaw, in Silesia. In 1724, M. De Moivre published the first edition of his tract on Annuities on Lives. In order to facilitate the calculation of their values, M. De Moivre assumed the annual decrements of life to be equal; that is, he supposed that out of 86 (the utmost limit of life on his hypothesis) persons born together, one would die every year till the whole were extinct. This assumption agreed pretty well with the true values between 30 and 70 years of age, as given in Dr. Halley's Table; but was very remote from the truth in the earlier and later periods. Mr. Thomas Simpson, in his work on Annuities and Reversions, originally published in 1742, gave a Table of mortality deduced from the London bills, and Tables founded upon it of the values of annuities. But at the period when this Table was calculated, the mortality in London was so much higher than in the rest of the country, that the values of the annuities given in it were far too small for general use. In 1746, M. Deparcieux published, in his Essai sur les Probabilités de la Durée de la Vie Humaine--a work distinguished by its perspicuity and neatness,-Tables of mortality deduced from observations made on the mortuary registers of several religious houses, and on the list of the nominees in several tontines. In this work, separate Tables were first constructed for males and females, and the greater longevity of the latter rendered apparent. M. Deparcieux's Tables were a very great acquisition to the science; and are decidedly superior to some that are still extensively used. Dr. Price's famous work on Annuities, the first edition of which was published in 1770, contributed powerfully to direct the public attention to inquiries of this sort; and was, in this respect, of very great utility. Of the more recent works, the best are those of Mr. Baily and Mr. Milne, which indeed, are both excellent. The latter, besides all that was previously known as to the history, theory, or practice of the science, contains much new and valuable matter; and to it we beg to refer such of our readers as wish to enter fully into the subject. The Table on which Dr. Price laid the greatest stress, was calculated from the burial registers kept in the parish of All Saints in Northampton, containing little more than half the population of the town. There can be no doubt, however, as well from original defects in the construction of the Table, as from the improvement that has since taken place in the healthiness of the public, that the mortality represented in the Northampton Table is, and has long been decidedly above the average rate of mortality in England. Mr. Morgan, indeed, the late learned actuary of the Equitable Society, contended that this is not the case, and that the Society's experience shows that the Northampton Table is still remarkably accurate. But the facts Mr. Morgan disclosed in his View of the Rise and Progress of the Equitable Society (p. 42.), published in 1828, are quite at variance with this opinion: for he there states, that the deaths of persons insured in the Equitable Society, from 50 to 6C years of age, during the 12 years previously to 1828, were 339; whereas, according to the Northampton Table, they should have been 545! And Mr. Milne has endeavoured to show (Art. Annuities, new ed. of Ency. Brit.) that the discrepancy is really much greater. The only other Table used to any extent in England for the calculation of life annuities, Is that framed by Mr. Milne from observations made by Dr. Heysham on the rate of mortality at Carlisle. It gives a decidedly lower rate of mortality than the Northampton Table; and there are good grounds for thinking that the mortality which it represents is not very different from the actual rate throughout most parts of England; though it cannot be supposed that a Table founded on so narrow a basis should give a perfectly fair view of the average mortality of the entire kingdom. In life insurance, the first annual premium is always paid at the commencement of the assurance, and the others at the termination of each year so long as the party assured survives. Hence, at the beginning of the assurance, the whole of the annual premiums payable for it exceed the value of an equal annuity on the life by one year's purchase. And, therefore, when the value of an assurance in present money is given, to find the equivalent annual premium during the life, the whole present value must be divided by the number of years' purchase an annuity on the life is worth, increased by 1. Thus, for an assurance of 100%. on a life 40 years of age, an office, calculating by the Carlisle Table of mortality, and at 4 per cent. interest, requires 53-4467. in present money. Now according to that Table and rate of interest, an annuity on a life just 40 years of age is worth 15-074 years' purchase, so that the equivalent annual premium is 15-04X13325, or 3l. 6s. 8d. The annual premium may, however, be derived directly from the value of an annuity on the life, without first calculating the total present value of the assurance.— -(See Mr. Milne's Treatise on Annuities, or the art. Annuities in the new edition of the Ency. Britannica.) 53.4461. In order to exhibit the foundations on which Tables of life annuities and insurance have been founded in this and other countries, we have given, in No. V. of the following Tables, the rate of mortality that has been observed to take place among 1,000 children born together, or the numbers alive at the end of each year, till the whole become extinct, in England, France, Sweden, &c., according to the most celebrated authorities. The rate of mortality *The greater part of this Table was originally published by Dr. Hutton in his Mathematical Dictionary, art. Life Annuities. Mr. Baily inserted it with additions in his work on Annuities; and it |