In case of withdrawal, a non-borrower receives the dues paid in, and an equitable part of the accrued profits. By the Installment Plan, a borrower pays the difference between the withdrawal value of the shares and the gross amount of the loan. By the Net or Gross Plans, a borrower pays the difference between the sum of the withdrawal value of the shares, increased by the premium for the unexpired years of the series, and the gross amount of the loan.

The profits of an association accrue from interest and premiums. The True Profit at any date of a series is the legal interest on the payments, plus that part of the profit on premiums which the present value of a share is of the par value, $200. The Withdrawal Profit is the True Profit less a Withdrawal Discount fixed by the Association By-Laws. NOTE.-Building Associations are not, as often supposed, builders of houses. They are corporations organized to enable their members to build houses, or buy them in their individual capacity, and might perhaps as appropriately be called Savings Fund and Loan Associations.

CASE I.

940. To find the actual cost of any amount of stock. 1. What would be the annual aggregate dues on 50 shares of stock at $1 a month per share?

SOLUTION. Since the dues on 1 share for 1 month are $1, on 50 shares for 1 month, they will be $50, and for 1 year 12 times $50, or $600.

OPERATION.

$1x50×12= $600

Rule. Multiply the periodical dues by the number of periods, and to this product add the sum of the fines, if any have been levied.

2. I buy 10 shares in 1st series, 8 in 2d, and 16 in 3d of Investment Building Association; if these series run out in 8, 8, and 9 years respectively, how much money in monthly dues will then have been paid in on the three series when closed out? Ans. $3488.

3. Bought 45 shares of Franklin Association, 5 months after the date of issue, for $252, but was unable to pay up dues for 3 months after purchase; how much did I pay for the year, including fines of 10% on unpaid dues?

Ans. $594.

4. At the end of the 4th year of a series, John Doe borrows of the Decatur Building Association, on 20 shares at $40 premium, Gross Plan, but in two years he becomes insolvent and ceases to pay his dues; if the fines are 5% on unpaid dues, what would be due the association at the end of the seventh year? Ans. $636.

CASE II.

941. To find the true premium charged on loans, and refunded on their payment.

1. Mr. Lee bought a loan on 6 shares of stock 1 year and 1 month old, premium $72; what is the true premium, if the last annual report gives the value of a share as $15.50?

SOLUTION.-Since the dues for 1 month have been paid since the report, the accumulated value is $15.50+$1

= =

OPERATION.

$15.50+$1=$16.50 $200-$16.50 $183.50 $16.50, and the unaccumulated value $72×183.50÷200=$66.06 $183.50. Since

$200

183.50

$16.50

of the par value is unaccumulated,

183.50
200

of the premium bid 200 must be the premium due, or the true premium; 183:50 of $72=$72 X 183.50-200= $66.06.

200

Rule. Multiply the premium bid by the unaccumulated value divided by 200 (or by one-half of the unaccumulated value regarded as a rate per cent.), and the result will be the True Premium.

NOTES.-1. This method is more accurate and equitable than that of deducting 10 per cent. for each year, or the "stated premium" method. A money basis, and not a time basis, is the only one that will permit exact results.

2. In finding the Net Loan, deduct the True Premium; and in finding the Total Cost of a loan, multiply the monthly payments by the time in months. 3. Upon Payment of Loans, the True Premium at the date of payment should be refunded. The Pennsylvania law requires one-eighth of the premium to be refunded for every year unexpired of eight.

2. In June, 1876, I buy a loan on 5 shares of stock, issued April, 1873, at a premium of $65; what amount of cash do I receive, if by the last annual Report a share of this series is worth $51.80 ? Ans. $764.05.

3. If the first series of the Centennial Building Association, issued July, 1876, should be published in the sixth annual Report as worth $140.50, what would be the net amount of a loan made in December, 1882, on 20 shares of this series at a premium of $95? Ans. $3491.75.

4. If this loan should be returned May, 1884, and the Report of '83 gave the shares at $165.80, what would be the true premium due the borrower on a share, and the balance due on the loan? Ans. $11.02; $3779.60. 5. The Provident Building Association sold money to Mr.

Collins on 23 shares for 679 a month premium per share ; what was the amount of his loan and what did his monthly payments aggregate for 9 years? Ans. $4600; $6632.28.

CASE III.

942. To find the amount and the actual cost of a loan to a borrower.

1. I bought a loan on 12 shares in a new series of Benefit Building Association, Gross plan, at $11 and "stated premium;" if the series runs out in 8 years, what will be the actual cost of my loan?

months, and so on; hence the interest of a payment of $1 for the different periods equals the interest of $1 for a number of months represented by an arithmetical series whose first term is 1, last term, 104, and number of terms 104, or (Art. 848) 1⁄2 of (104+1)×104. The interest of $1 for 1 month is 1, and for the aggregate months, of 105× 104×3o= 105×104×1 $27.30; and on $24 it is $27.30×24=$655.20. The sum of the payments equals $24 × 104, or $2496; and the cost of the loan equals $2496+$655.20, or $3151.20.

Rule.-I. Multiply the number of months increased by 1, by the number of months, and divide by 4, to find the interest at 6% on the aggregate monthly payments of $1.

II. Multiply the interest on the aggregate payments of $1, by the monthly payment, to find the interest on the payments. Find the sum of the payments, and to this sum add the interest; the result will be the cost of the loan.

NOTES.-1. It is here assumed that the payments draw simple interest from their payment to the close of the series. It would be more correct, perhaps, to reckon annual interest, or even compound interest; but the method given is more convenient.

2. The loan in the Installment plan, is the value of the shares on which it is made, the premium being a part of the monthly payment; in the other plans, the loan equals the value of the share minus the premium.

2. Mr. Smith and Mr. Jones each buy a house for $4000, and Mr. S. gives a 6% bond and mortgage due in 10 years

for the full amount; Mr. J. gives a bond and mortgage to the Investment Building Association for a loan on 20 shares at 75 cents a month premium; if the shares are cancelled in 10 years, who pays the most cash for his house, and how much? Ans. Mr. Jones; $200.

3. Mr. Henry and Mr. Williams buy houses for $2460, each paying $500 cash. Mr. H. gives a 6% bond and mortgage for the balance, int. payable annually, which he pays off at the end of 9 years; Mr. W. borrows of a building association a net loan of $1960 at $60 premium, Net plan, in a new series, which “ runs out" in 9 years; which house cost the more, reckoning interest on payments? Ans. 1st, $1.58%.

4. Mr. Brown rents a house for $16 a month for 10 years, and then buys it for $2000; Mr. White buys a house for the same price, and to pay for it obtains a loan from a Building Association, at 65 cents a month premium, on 10 shares of a series which runs out in 10 years; how much less does Mr. B. pay for his house than Mr. W., who pays annually 2% for taxes, and 1% for repairs, interest reckoned in both cases on monthly payments? Ans. $530.15.

CASE IV.

943. To find the rate of interest received by a nonborrower.

1. What rate of interest do I receive on 8 shares of building association, dues $1 per share, if the series runs out in 8 years?

OPERATION.

$200-$102
103 × 102

$98

24

$98

equated time.

103×102

= 22.39%

24

SOLUTION. The installments paid on 1 share for 8 years, or 102 months, is $102, and the difference between $200, the final value, and $102, the amount paid, equals $98, which is the gain, or interest on the investment. $1, the first payment, is on interest 102 months, the second payment is on interest 101 months, etc.; hence the interest on the payments for the different periods is equivalent to the interest on $1 for a number of months represented by the sum of an arithmetical series whose first term is 102, last term 1, and number of terms 102, or of (102+1)× 102, months = 2+ of (102+1)× 102, years; hence the interest on $1 for 1 103×102 year, or the rate, is $98÷$.2239+, or 22.39% 24

Rule.-I. Subtract the sum of the installments paid on one share from the value of the share, and the difference will be the interest on the investment.

II. Multiply the number of payments by the number of payments increased by one and divide by 24, to find the equated time, or the number of years in which $1 will produce the same interest as the installments.

III. Divide the interest on the investment by the equated time; the quotient will be the equated rate per cent.

2. What equated rate % of profit has been made by the fourth series of the Schuyler Building Association, if at the end of 23 months it is worth $33.26 a share? Ans. 44.6%. 3. What rate of interest will a building association pay that runs out in 8 yr. ? 9 yr.? Ans. 26.8%; 18.76%.

4. If I buy 25 shares of Penn Building Association (new series) paying $1 dues each month and 75 a year for contingent expenses, and the series runs out in 10 years, what will be the equated rate of interest on the investment for the given time? Ans. 11.21%.

5. Mr. Black buys from a friend 24 shares of Decatur Building Association, during the 8th month of the 4th year of the series, paying $45.50 (estimated value by last Report) and the dues paid since the Report; what rate of interest does he receive on his investment, if the series runs out in 8 years? Ans 20%+.

6. Buy 10 shares second series paying $17.58 (estimated value at last report) and dues for 9 months; if at the end of the year the series is valued at $38.30 per share, what is the amount of profit, and what the equated rate of interest? Ans. $87.20; 1225%.

CASE V.

944. To find the rate of interest paid by a bor

rower.

1. I bought a loan of the Penn Building Association on 10 shares, new issue, at $95 premium, Net plan; what rate % of interest will I have paid if the series expires in 83 yr.?