a given number of years, in which case it is called an annuity certain, or it may be dependent upon some particular circumstance, as the life of one or more individuals. The latter is called a contingent annuity. A perpetual annuity, is one which can only be terminated by the grantor, on the payment of a sum whose interest will be equivalent to the annuity. Of this character is the consolidated debt of England.

An annuity in possession, is one on which there is a present claim; an annuity in reversion, or deferred annuity, is one that does not commence until the lapse of a stated time, or the occurrence of some uncertain event, as the death of an individual.

The present worth of an annuity, is the sum which, at compound interest for the time of its duration, would amount to the sum of all the payments, each being placed at compound interest as it became due.

PROBLEM I.

To find the amount due on an annuity which has remained unpaid a given time.

As the payments are all at compound interest, this case evidently falls under Problem II., in Geometrical Progression.

RULE.

Find the sum of a Geometrical Progression, in which the first term is the annuity, the ratio is the amount of $1.00 for the time that should elapse from one payment to another, and the number of terms is the number of payments due.

1. If a person saves $250 per annum, and invests it at 7 per cent. compound interest, how much will he be worth at the end of 25 years?

2. What is the amount of an annuity of $500 payable semi-annually, forborn for 7 years, at 6 per cent. per annum? 3. What is the amount of a quarterly annuity of $400, in arrears for 5 years, at 5 per cent.?

4. What is the amount of an annual salary of $1000 for 10 years, at 6 per cent.?

5. What is the amount of a quarterly rent of $200 for 3 years, at 6 per cent.?

6. What is the amount of a pension of $400 a year, payable semi-annually, for 3 years and 6 months, at 7 per cent. per annum?

7. What is the amount of a rent of $1500, ten years forborn, at 5 per cent. per year?

8. An estate that yields an annual income of $2000, is offered for sale for the amount of 10 years income at 6 per cent. compound interest. What is the price of the estate?

9. What is the amount of a salary of $1300 for 16 years, at 5 per cent. compound interest?

PROBLEM II.

To find the present worth of an annuity certain. What is the present worth of a semi-annual rent of $500, for 6 years at 6 per cent?

By the preceding Problem, we find that the rent at the expiration of the 6 years, will have amounted to $7096.015. The question is, therefore, to find the present worth of $7096.015 due in 6 years, at a compound interest of 3 per cent. semi-annually, which is done by Problem I., in Geometrical Progression.

RULE.

Divide the amount of the annuity, (found by Problem I.) by the amount of $1.00 for the same time.

10. What is the present worth of an annual salary of $800 to continue 8 years, at 5 per cent. compound interest?

11. What is the present worth of an annual income of $500, to continue 11 years, at 6 per cent. compound interest?

12. What is the present worth of a quarterly rent of $200, to continue 5 years, at 7 per cent, compound interest?

13. What is the present worth of a semi-annual pension of $175, to continue 9 years, at 4 per cent. compound interest?

14. What is the present worth of an annuity of $3000 for 7 years, at 3 per cent, compound interest?

15. What sum invested at 6 per cent. compound interest, will yield me an income of $1600 per annum, for 25 years?

16. What sum at 5 per cent. will yield an annual income of $1200 for 15 years?

17. A gentleman wishes to present his estate to his children, reserving enough to yield $700 per annum for 15 years. How much must he reserve, allowing 5 per cent. compound interest?

PROBLEM III.

To find the present worth of a perpetual annuity.

The present worth of a perpetual annuity, is a sum which would yield an interest equivalent to the annuity. As the interest is found by multiplying the principal by the rate, the principal may be found by dividing the interest by the rate.

RULE.

Divide the annuity by the rate per cent.

18. For how much should an estate that rents for $175 per year, be sold, to allow the purchaser 6 per cent. interest on his investment?

19. What is the par value of an annual income of £500 in the 4 per cent. consols.*?

20. What sum of money must be laid out in the 3 per cent. consols., at 68 per cent. of the par value, to yield an income of £1000.

21. What sum of money at 4 per cent. will yield an annual interest of $620 ?

22. What sum, invested in an estate that rents for $400 per annum will yield an interest of 8 per cent.?

23. A farm rents for $750 per annum. For what price should it be sold, when money is worth 6 per cent. a year?

it

24. What sum will build a wall worth $1000, and renew every 15 years, at 5 per cent. compound interest?

* CONSOLS., is an abbreviation for the consolidated annuities of the British National Debt.

25. A railroad has been constructed through a farm, in consequence of which, the owner of the estate is obliged to expend $400 in fencing, that must be renewed at the expiration of every 12 years. What sum should he now receive, to compensate him for the required expenditure, money being worth 6 per cent. compound interest?

PROBLEM IV.

The present worth of a certain annuity being given, to find the annuity.

RULE.

Find the first term of a Geometrical Progression, in which the sum is the amount of the annuity for the whole time, the ratio is the amount of $1.00 for the time that elapses between two successive payments, and the number of terms is the number of payments.

26. The present debt of Pennsylvania (in 1844) amounts to about $40000000, which bears an interest of 5 per cent. per annum, payable semi-annually. What semi-annual appropriation will extinguish the debt in 40 years?

27. What sum of money must a man lay up annually, to amount to $10000 in 20 years, the investments being all made at 6 per cent. compound interest?

28. A builder takes a lease of a lot of ground for 25 years, and erects buildings on it which cost him $20000. Allowing money to be worth 6 per cent. compound interest, what clear* annual rent must he receive from the buildings to reimburse his expenditure, at the termination of the lease,—the rent commencing one year after the lease is given?

29. The executors of an estate wish to dispose of an unexpired lease that has 8 years to run, for a premium of $1500. What amount must be added to the annual rent, for that purpose?

* The clear annual rent, is the amount received after deducting ground-rent, taxes, and other expenses.

PROBLEM V.

To find the present worth of an annuity in reversion.

RULE.

Find the present worth of the annuity until the commencement of the reversion, and also the present worth until its termination. The difference of these two values will give the present worth of the reversion.

30. What is the present worth of an annuity of $1100, to commence in 3 years and continue for 8 years, interest at 6 per cent?

31. What is the present worth of a perpetual annuity of $300, to commence in 2 years, at 5 per cent. interest?

32. A father leaves an annual rent of $400 to his eldest child for 5 years, and the reversion of it for the 8 succeeding years to his youngest child. What is the present worth of each legacy, at 7 per cent.?

33. What sum must be paid, allowing 6 per cent. compound interest, to extend a lease 7 years,―the clear annual rent being $500, and the lease having 4 years to run?

34. What is the value of $3000 rail-road stock, that will yield no income for 4 years, but on which, after that time, there will be an annual dividend of 9 per cent. for 21 years?

35. What is the present worth of a reversion of $700 per annum, to commence in 20 years, and continue 40 years thereafter, allowing 6 per cent. compound interest?

The operations in annuities are generally so tedious, that tables similar to those given at the end of the book, are employed in their calculation.

The subject of LIFE ANNUITIES, involves the calculation of chances, as applied to the duration of life. Tables of the probable duration of life at each year, are prepared, and the number against each age is employed as a multiplier or divisor, for determining the present worth, or the annuity itself.