ANNUITIES.

ART. 148. An Annuity (L. annus, a year) is a fixed sum of money payable annually, or at the end of equal periods of time, to continue for a given number of years, for life, or for

ever.

A certain Annuity, or an Annuity certain, is one that is payable for a definite length of time.

A contingent Annuity is one that is payable for an uncertain length of time; as during the life of one or more persons. A perpetual Annuity, or an Annuity in perpetuity, is one that continues forever.

A deferred Annuity, or an Annuity in reversion (whether certain, contingent, or perpetual) is one that begins at a future time; as at the death of a certain person.

An immediate Annuity, or an Annuity in possession, is one that begins at once.

An Annuity forborne, or in arrears, is one whose payments have not been made when due.

The amount, or final value, of an annuity is the sum of the amounts of all its payments, at compound interest, from the time each is due, to the end of the annuity.

The present value of an annuity, at compound interest, is the sum of the present values of all its payments; or the present worth of its final value. The present value, put out at compound interest, will amount, at the time of the expiration of the annuity, to its final value.

The subject of annuities is one of great practical importance in the affairs of life. Its principal applications are leases, life-estates, rents, dowers, reversions, life-insurance, etc. The problems are readily solved by means of tables which give the present and final values of $1 for a given number of years at the ordinary rates of interest. A complete discussion of the principles upon which these tables are computed would require too much space.

Showing the present value, and also the amount, or final value, of an annuity of $1, for any number of years not exceeding fifty:

CACO2 Years.

Final value of an Annuity of $1.

4 per cent. 5 per cent. 6 per cent. 7 per cent.

1

41 19.993 052 17.294 368 15.138 016 13.894 120 41 99.826 536 127.889 763 165.047 684 214.609 570 49 20.185 627 17.423 208 15.224 543 13.452 449 42 104.819 598 185.281 751 175.950 645 230.632 240 43 20.370 795 17.545 912 15.806 173 13.506 962 43/110.012 882 142.993 839 187.507 577 247.776 496 44 20.548 841 17.662 773 15.383 182 13.557 908 44 115.412 877 151.143 006 199.758 032 266.120 851 45 20.720 040 17.774 070 15.455 882 13.605 522 45 121.029 392 159.700 156 212.743 514 285.749 811|

46 20 884 654 17.880 067 15.524 870 18.650 020 46 126.870 568 168.685 161 226.508 125 306.751 763 47 21.042 936 17.981 016 15.589 028 13.691 60S 47 182.945 890 178.119 422 241.098 612 329.224 886 48,21.195 131 18.077 158 15.650 027|13.780 474 48 189.263 206 188.025 893 256 564 529 353.270 093 49 21.841 472 18.168 722 15.707 572 18.766 799 49 145.833 734 198.426 663 272.958 401 878.999 000 50 21 482 185 18.255 925 15.761 861 18.800 746 50 152.667 084 209.847 976,290.835 905 406.528 929 50 152.667

81.969 202

85.719 252

39.992 727

44.865 177

84.247 970

88.505 214

43.392 290

49.005 789

86.617 859

41.430 475

46.995 828

53.436 141|

89.082 604

44.501 999

50.815 577

58.176 671

41.645 908

47.727 099

54.864 512

63.249 030

ART. 150. To find the amount, or final value, of an annuity certain, at compound interest, in arrears, or forborne.

Ex. 1. Suppose a rental of $500 a year to remain unpaid 8 years; what is the amount due, at 5 per cent. compound interest ? Ans. $47745.545.

Operation.

$9.549109, amount of $1 for 8 years. (See Table).

500

$47.745.545;

RULE.

-Multiply the amount, or final value, of an annuity of $1, for the given rate and time, by the given annuity.

Note. When the annuity draws simple interest, the amount is found as in annual interest.

Ex. 2. Find the final value of an annuity of $150, running 12 years at 4 per cent. compound interest.

Ans. $2253.87+. Ex. 3. An annuity of $200 has been in arrears 15 years; what is the amount due, at 6 per cent. compound interest? Ans. $4655.194.

ART. 151. To find the present value of an annuity certain. Ex. 1. What is the present value of an annuity of $120, to continue 25 years, at 6 per cent. ? Ans. $1534.

Operation.

$12.783356, present value of $1. (See Table.)

RULE.-Multiply the present value of $1,'as an annuity for the given rate and time, by the given annuity.

Note. Since the present value of an annuity is the present worth of its amount, or final value, the present value of an annuity may also be found by first finding the amount, and then the present worth of this amount.

Ex. 2. What is the present value of an annuity of $650, to continue 15 years, at 5 per cent. ? Ans. $6746.7777.

Ex. 3. What is the present worth of a leasehold of $1200, payable annually for 50 years, at 6 per cent. ?

Ans. $18914.23. Ex. 4. A widow is entitled to $140 a year, payable semiannually, for 18 years; what is the present value of her interest, at 10 per cent compound interest? Ans. $1158.80.

Ex. 5. I wish to purchase an annuity which shall secure to my ward, at 4 per cent. compound interest, $250 a year for 14 years. What must I deposit in the annuity office?

Ans. $2640.78. ART. 152. To find the present value of a perpetuity. Ex. 1. What is the present value of a perpetual leasehold of $1200 a year, at 5 per cent. ? Ans. $24000.

Operation.

$1200.00.05 $24000, present value.

--

Explanation. The present value must evidently be a principal which yields an annual interest of $1200, at 5 per cent. RULE.-Divide the given annuity by the interest of $1, for

one year.

Ex. 2. What is the present value of the perpetual lease of $4800 a year, at 8 per cent. interest? Ans. $60000.

Ex. 3. What is the present value of a perpetual leasehold of $1600 a year, payable semi-annually, at 6 per cent. interest per annum? Ans. $270663.

Suggestion. When an annuity is payable semi-annually, or quarterly, interest must be allowed on the half-yearly or quarterly payments to the close of the year. The annuity in the last example is $1624.

Ex. 4. The ground rent of an estate yields an annual income of $2400, payable quarterly, at 4 per cent. per annum. What is the value of the estate? Ans. $60900.

ART. 153. To find the present value of a certain annuity in reversion.

Ex. 1. What is the present value of an annuity of $250, deferred 12 years and to continue 10 years, allowing 6 per cent. compound interest? Ans. $914.43+.

Operation.

$12.041582-present worth of $1 for 22 yrs.

Explanation. The present worth of an annuity of $1 for 22 years must be equal to its present worth for 12 years, plus its present worth for the 10 succeeding years. Hence the present worth of an annuity of $1 for 10 years deferred 12 years must equal its present worth for 22 years, minus its present worth for 12 years. The present worth of $250 is evidently 250 times the present worth of $1.

RULE.-Find from the table the present value of an annuity of $1, commencing at once and continuing till the TERMINATION of the annuity, and also till the reversion COMMENCES. Multiply the differences of these present values by the given annuity.

Note.—If the annuity is perpetual, the present worth of $1, commencing at once, is found according to the last article. Ex. 2. What is the present value of a leasehold of $1800, deferred 10 years and to run 20 years, at 5 per cent. compound interest ? Ans. $13771.2888.

Ex. 3. A lease, whose rental is $1000 a year, is left to two sons. The elder is to receive the rent for 9 years and the youngest for the 12 years succeeding. What is the present value of each son's interest, allowing 6 per cent. compound interest? Ans. to last, $4962.385. Ex. 4. What is the present value of a perpetuity of $900, to commence in 30 years, allowing 4 per cent. compound interest ? Ans. $6937.17.

ART. 154. To find the annuity, the present or final value, time and rate being given.

Ex. 1. An annuity running 20 years, at 7 per cent. compound interest, is worth $15,000; what is the annuity?

Ans. $1415.89.