385. To find the present worth of an annuity in perpetuity.

This is equivalent to finding a principal the interest of which is equal to the given annuity.

RULE.-Annex two ciphers to the annuity, and divide by

the rate.

EXAMPLE.-The annual income from an estate is $250. What is the present worth of the estate, allowing compound interest, at 6 per cent.?

Ans.

386. To find an annuity from its present value. RULE.-Divide its present value by the present value of an annuity of $1 for the given rate and time.

EXAMPLE. The present value of an annuity for 10 years, at 5 per cent. compound interest, is $3000. What is the annuity? Ans. $388.513.

387. To find an annuity from its amount.

RULE.-Divide the given amount by the final value of $1 for the given time and rate.

EXAMPLE. The final value of an annuity for 11 years, at 7 per cent. compound interest, amounts to $4735.08. What is the annuity? Ans. $300.

MISCELLANEOUS EXAMPLES.

1. What sum of money must a man invest annually at 6 per cent. compound interest, that he may have $5000 at the end of 10 years?

2. What is the present worth of $500 to be received annually for 6 years, allowing compound interest at 7 per cent.? 3. What sum invested at 6 per cent. compound interest will yield an income of $1800 per annum for 12 years?

4. The executors of an estate offer for sale an unoccupied lease that has 6 years to run, for a premium of $300. How much, added to the annual rent, will amount to the same sum ? 5. What is an interest of $250 annually in an estate for 10 years worth, allowing money to be worth 7 per cent. compound interest?

BUILDING AND LOAN ASSOCIATIONS.

388. BUILDING AND LOAN ASSOCIATIONS have for their object the accumulation of a fund from which the members can obtain the means to build or buy houses, purchase lands, or for similar purposes.

The shares are usually estimated at $200 each, and are paid for in monthly instalments, generally $1 per month for each share. When the accumulated payments reach a certain sum, the funds are offered at auction, and given to the shareholder paying the largest bonus or discount. Interest on the loan thus made is paid monthly, or at the same time as the periodical dues. The loans are generally secured by mortgage on real estate. To prevent delinquency, fines are imposed of 5 or 10 per cent. per month on all sums not paid when due. When the total amount received by the association is sufficient to give each shareholder the amount originally agreed upon, the association closes.

As promotive of habits of economy, and as affording means of profitable investment, these associations have been highly successful. The chief benefit, however, is derived from the increase in the value of the property purchased, and in the convenient form in which the payments are made. Practically, these associations have given homes to hundreds who would otherwise never have owned them.

389. To find the cost of a share at simple interest, when the monthly dues, time, and rate of interest are given.

RULE.-Multiply the interest on the monthly payment for one month by the number of months, less 1, that the association continues, and this product by one-half the number of

months the association continues. Then

Add the product of the monthly payment by the total number of months.

EXAMPLE.

What was the cost of a share for which $1 per month had been paid for 9 years, allowing 6% simple interest?

Total cost, allowing 6 per cent. interest, $136.890

390. To find the cost of a share at compound interest. RULE. Divide the monthly payment, after annexing two ciphers, by the given rate of interest, and find the compound interest of the quotient for the given rate and time.

391. To find the cost of a loan at simple interest.

RULE.-Add the present value of a share to the present value of the payment required for the loan.

As it is impossible before the association closes, owing to the variations of discounts, number of borrowers, etc., to know the exact time the association will continue, an approximate value is all that can be found. From eight to ten years is the usual time.

EXAMPLES.

1. What is the cost of a loan of $200, the association requiring $1 interest in addition to $1 as regular payment to be paid monthly for 6 years, the present value of a share being $50?

Interest on $2 for 1 month:

.01.

A bonus of 10% on $200 would leave $180.

of 20% would leave $160.

A bonus

2. Which would be preferable, the above loan and pay. ments, or to borrow $180 at 6 % compound interest?

392. To find the rate of interest paid for a loan.

RULE. From the cost of the loan subtract the amount received; then,

Find the rate it will require for the loan to gain this difference in the given number of years.

393. To find the amount of fines on dues remaining unpaid.

NOTE. The total amount of fines is equivalent to the sum of the compound interest of the dues for as many months as each payment remains unpaid; or to the final value of an annuity for the same time, with interest at the rate of the fine.

RULE.-Multiply the compound interest of $1, at the rate of the given fine, by the dues for one month, commencing with the interest of one month, and continuing for the whole number of months. The sum of all the products will be the total amount of fines. Or, by Annuity Tables,

From the final value of an annuity of $1 for one more year than the number of months that the dues remain unpaid, subtract one more dollar than there are such months; the remainder will be the fines on dues of $1 per month. This multiplied by the total dues per month, will give the total fines. Or,

Multiply the sum of the fines on $1, as given below, by the dues for 1 month.

EXAMPLE.

How much must a man, whose dues are $5 per month, pay for fines at 5% per month, on all sums remaining unpaid, after his dues remain unpaid 6 months?

LIFE INSURANCE.

394. LIFE INSURANCE companies base their premiums upon the number of years each person is expected to live after insuring, and the use of money for that time.

395. The EXPECTATION OF LIFE is the average number of years remaining to a person at a given age, and is deduced from tables of mortality, which have been prepared from various observations made in different places and periods, showing, out of a given number of persons, how many complete each subsequent year, and how many die in it, till the whole are extinct.

The Carlisle tables, formed by Mr. Milne according to the mortality observed at Carlisle (Eng.), the Northampton tables, formed by Dr. Price (Eng.), the Wigglesworth tables, prepared by Dr. Wigglesworth from data founded upon the mortality of this country, and others, are employed. The Carlisle tables are in general use in England, and to a considerable extent here. The Wigglesworth tables have been adopted by the Supreme Court of Massachusetts in estimating life estates: they show a smaller expectation of life than the Carlisle tables.

2894

56429

396. The PROBABILITY that a person of any designated age will attain any greater age is expressed by dividing the number of survivors at the greater age by the number that attain the given age. Thus, by the Carlisle tables, of 10,000 persons born together, 5642 attain to 30, and 2894 to 66 years of age. The probability that a person now 30 years will reach the age of 66 years is, therefore, about 1, or 1 chance in 2. The value of a sum of money, the receipt of which depends upon the person being alive at that time, will be reduced by that contingency one-half; so that if the sum to be received is $1000, its value is reduced to only $500. The present worth of $1000, due 36 years hence, interest at 6 per cent., is $122.74; but, depending upon the same contingency, it is worth only $61.37.