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shares, value per share $25.27; third series, 400 shares, value per share 812.32; that a fourth series of 500 shares is then issued; the net profits for the fourth year are $3,000, and the total net profits for the four years are $5,325. Required: The value of a share of each series at the end of the fourth year.

The first series above alluded to has run four years, or forty-eight months. Forty-eight $1 payments have therefore been made on each share of stock. The first dollar paid has been invested forty-eight months; the second dollar paid, forty-seven months; the third dollar paid, forty-six months, etc., the last dollar of the forty-eight having been invested one month. The times of investment thus form a decreasing arithmetical series, with forty-eight for the first term, one for the last term, and forty-eight for the number of terms. The total investment is thus equal to $1 invested for 1,176 months (the sum of the series), equivalent to $48 invested for 244 months.

Treating the other series in the same way, we find that $35 paid per share in the second series has been invested for 18 months; $21 paid per share in the third series, for 12 months; and $12 paid per share in the fourth series, for 64 months; then

$18X500X244= $588,000, first series investment for one month.
$36 x 600 x 184 $399,600, second series' investment for one month.
$24 x 400X124= $120,000, third series' investment for one month.
$12X500 x 64 = $39,000, fourth series' investment for one month.

$1,146,600, total investment for one month. Hence the total net profits are divided as follows:

PAG or of the total profits belong to the first series.

YAM or 1997 of the total profits belong to the second series.
HIP or 94 of the total profits belong to the third series.

122898 or fit of the total profits belong to the fourth series. Total profits to be divided are $5,325.

10% of $5,325 == $2,730.77, first series' share of the profits.
1969 of $5,325 =$1,875.81, second series' share of the profits.
10% of $5,325 = $557.30, third series' share of the profits.
Thir of $5,325=$181.12, fourth serios' share of the profits.
$2,730.77 -- 500 = $5:46, profit of a share of the first series.
$1,855.81 - 600= $3.09, profit of a share of the seconıl series.
$557.30 - 400==$1.39, profit of a share of the third series.

$181.12 -- 500 ==$0.36, profit of a share of the fourth series. $18.00, dues paid, +- $5.46, profit, = $53.46, value of a share of the first series. $36.00, dues paid, + $3.09, profit, = $39.09, value of a share of the second series. $24.00, dues paid, + $1.39, profit, = $25.39, value of a share of the third series. $12.00, dnes paid, + $0:36, profit, =$12.36, value of a share of the fourte series.

In the above example, all the net profits made during the four years have been apportioned to the several series, but some associations apportion only each year's profits in this way. Other associations using this plan simplify the process, but obtain the same results, by dividing the total investment for one month (or for one week, as the case may be), into the profits to be apportioned, for the profit on $1 invested for

one month, which is then multiplied successively by the sum of the number of weeks, months, or other periods of time for which each dol lar of dues in each series has been invested. The products will be the amount of the profits belonging to a share in each series.

A few associations have been found that arrive at the same results by using the following method, which is known as Clark's plan:

1. Multiply the number of shares in force in each series by the quotient obtained by dividing the sum of the number of weeks, months, or other periods of time for which each dollar of dues in each series has been invested by the product obtained by multiplying the dues paid in on one share during the first year by the average time of investment, for the equalized results for each series.

2. Take the sum of these results and divide it into the total profits since the beginning of the association, for the rate per cent of profit.

3. Multiply the quotients already found by the rate per cent of profit, for the profit of a share in each series.

We have before seen that $48 dues per share in the first series have been invested for 244 months, which is equal to $1 invested for 1,176 months. In like manner $36 dues per share in the second series have been invested for 184 months, which is equal to $1 invested for 666 months; $24 dues per share in the third series, for 124 months, which is equal to $1 invested for 300 months; and $12 dues per share in the fourth series for 64 months, which is equal to $1 invested for 78 months. The average time of first year's payments is simply half the time of investment, which is 6 months. Twelve dollars invested for an average period of 6 months is equal to $1 invested for 72 months. Then:

1,176 -- 72 --- 16. 333.

666 = 72 9. 230.
300 —- 72 4. 166.

78 – 72= 1.083.
500 x 16. 333=8, 166. 50, equalized result for the first series.
600 x 9.250 =5,550.00, equalized result for the second series.
400 X 4. 166 =1, 666. 40, equalized result for the third series.
500 x 1.083 541.50, equalized result for the fourth series.

15,924. 40, equalized result for all series.
$5, 325, the total profits, = 15, 924. 10=33.4392, the rate per cent of profit.
$0.334392 X 16.333 = $5. 46, profit of a share of the first series.
$0.334392 x 9.250 =$3.09, profit of a share of the sccond series.
$0.334392 X 4.166 ==$1. 39, profit of a share of the third series.

$0.331392 x 1.083=$0.36, profit of a share of the fourth series. For the value of each share, add the dues as above.

There is a modification of plan 1, which follows the same general method as that shown in the first illustration, but differs in certain particulars and gives a different result. The modification is as follows: Instead of finding the exact equated time of investment, many associ

number of months a series has run. Using the same data as in the above illustration, we get 24, 18, 12, and 6 as the average number of months the series have run. It is this modification that is commonly, bat erroneously, called the partnership plan.

ILLUSTRATION.

$18 X 500 X 24 = $576, 000, first series' investment for one month.
$36 X 600 X 18 = $388, 800, second series’ investment for one month.
$24 X 400 X 12= $115, 200, third series' investment for one month.
$12 X 500 X 6 $36, 000, fourth series' investment for one month.

$1, 116, 000, total investment for one month. The total net profits are then divided in proportion to each series investment for one month, thus: 18 or of $5,325 = $2,748.39, first series' share of the profits.

0 or mats of $5,325 =$1,855.16, second series' share of the profits.
DYS% or less of $5,325 $549.68, third series' share of the profits.
rif:887 or is of $5,325 = $171.77, fourth series' share of the profits.

$2,748.39 --- 500 = $5.50, profit of a share of the first series.
$1,855. 16 = 600=$3.09, profit of a share of the second series.

$549.68 -- 400=$1.37, profit of a share of the third series.

$171.77 = 500 = $0.34, profit of a share of the fourth series. This modification of plan 1 has been simplified, the principle consisting in casting out common factors in the process of multiplication.

The first series has run 48 months; the second, 36 months; the third, 24 months; and the fourth, 12 months. The average time of investment, as we have before seen, is 24, 18, 12, and 6, respectively. Then we proceed thus:

48, age in months, x 24, average time, X 500 shares.
36, age in months, X 18, average time, x 600 shares.
24, age in months, X 12, average time, x 400 shares.

12, age in months, X 6, average time, X 500 shares. It will be readily seen that 12 is a factor coinmon to all the numbers of the first column, and that 6 is a factor common to all the numbers in the second column. Casting out these factors, we have

4 X 4 X 500= 8,000. Hences of the total profits belong to the first series.
3X3 X 600
5,400.

Mis of the total profits belong to the second series. 2 X 2 X 400 1, 600. Test of the total profits belong to the third series, 1X1 X 500 500.

TBg of the total profits belong to the fourth series. Total, 15, 500. Briefly put, then, the simplification is as follows: Multiply the number of shares in force in each series by the square of the time of investment expressed in terms or periods corresponding to the intervals between the series, and then divide the profits in proportion to these products.

The foregoing simplification has been still further simplified by finding the profit of a share in each series directly, instead of finding each series' share of the profit, as follows: 1. Multiply the number of shares in force in each series by the

H. Ex. 209-28

square of the time of investment expressed in terms or periods corresponding to the intervals between the series.

2. Divide the sum of these products into the results obtained by multiplying the total net profit by the square of the time of investment expressed as above.

The total of the products as in the last illustration is 15,500; then$5,325, total profits, x4x4-15,500 =$5.50, profit of a share of the first series. $5,325, total profits, x3x3--15,500= $3.09, profit of a share of the second series. $5,323, total profits, x2x2---15,500=$1.37, profit of a share of the third series. $5,323, total profits, x1x1--15,500 =$0.34, profit of a share of the fourth serios.

A share of the first series receives 16 times as much profit as a share of the fourth series; a share of the second series, 9 times as much; and a share of the third series, 4 times as much. This method, therefore, reveals the fact that, by multiplying the number of shares in force in each series by the square of the time each series has been invested, expressed in years, half years, quarter years, etc., corresponding to the intervals between the series, a correct basis of calculation is reached. These simplifications, however, are practicable only where series are issued at regular intervals, as fractions complicate the operation. This simplification is known as Rice's rule.

A few associations arrive at the same results by dividing the total investment for one month into the profits, for a rate per cent of profit, and then applying the rate to each series' investment for one month for each series' share of the profits. The process is also varied in the fol. lowing manner: Find what annual rate of interest the profits are equivalent to on the amount of dues paid for one-half the time that all the dues have been invested, and apply this rate on the dues paid per share for one-half the time of investment, for the profit of a share in any series.

Other variations are the following:

1. The profits are distributed on the amount of dues actually paid in on the shares in force in each series (not what the regular payments should have amounted to), multiplied by one-half the time of investment.

2. The profits are distribnted on the total amount of dues standing to the credit of the shareholders in the loan fund multiplied by onehalf the time of investment.

3. The series are not allowed to participate in the profits for the term in which they were issued.

4. The profits are distributed on the amount of dues actually paid in on all shares in force that are three months old or over, multiplied by one-half the time of investment, shares less than three months old not participating

5. The profits are distributed to the free shares only, dues on shares borrowed on being credited on loans.

6. Profits arising from withdrawals are divided equally among the

7. Profits arising from entrance fees are divided equally among the shares of the respective series in which the shares are taken.

8. A profit of $1 is given to all shares six months old or over. The remainder of the profits is distributed on the dues paid in on the shares in force six months old or over multiplied by one-half the time of investment.

9. Profits arising from premiums are divided equally among all the shares in force at the end of the period during which the loans were made. Profits from all other sources are distributed in accordance with the modified rule.

10. A fixed rate of interest is given on the total amount of dues paid on the shares in force at each apportionment. This interest is deducted from the profits for the term and the remainder distributed aecording to the modified rule.

11. A fixed rate of interest is given on the value of the shares in force as declared by the last report. This interest is deducted from the profits for the term and the remainder is distributed according to the modified rule.

12. A portion of the total amount of premiums received by and due the association is arbitrarily determined upon, and held in reserve to be applied in future dividends; the amount thus determined upon is deducted from the total profits, and the remainder of the profits is distributed as follows: The interest and dividends allowed on free shares withdrawn are added to the dues paid in on such shares, and the sum total of said interest and dividends is deducted from the amount of distributable profits; the balance is distributed among all the shares in accordance with the foregoing modified rule.

There is still another modification of plan 1, as follows: Multiply each series' investment (that is, the dues paid in on the shares in force) by one-half the number of months invested plus one and apportion the profits in proportion to these products.

Using the same data as before we proceed as follows:

ILLUSTRATION.

$48 x 500 x 25 =$600,000, first series' investment for one month.
$36 x 600 x 19=$410,400, second series' investment for one month.
$24 x 400 x 13=$124,800, third series' investment for one month.
$12 X 500 X 7= $42,000, fourth series' investment for one month.

$1,177,200, total investment for one month. Then, proceeding as before, we find that,

or gli of $5,325 = $2,714.07, the first series' share of the profits.

or 3 of $5,325 = $1,856.42, the second series' share of the profits. W8y or jøt of $5,325 = $564.53, the third series' share of the profits. 1499987 or A of $5,325 = $189.98, the fourth series' share of the profits. $48, dues, + ($2,714.07 - 500) = $53.43, value of a share of the first series. $36, dues, + ($1,856.42 - 600) =$39.09, value of a share of the second series. $21, dues, +($564.53 -- 400) = $25.41, value of a share of the third series. $12, dues, +($189.98 = 500)=$12.38, value of a share of the fourth series,

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