NATIONAL ASSOCIATIONS OPERATING UNDER THE VARIOUS PLANS FOR THE DISTRIBUTION OF PROFITS. [There are no national associations operating under the omitted numbers.] The twenty-five principal plans and their modifications with illustrations showing the various methods of applying each are given in the pages immediately following: PLAN 1. This plan apportions the profits among series just as profits among partners are apportioned in a firm where the partners enter at different dates, each series representing a partner. 1. Multiply the dues paid in on the shares in force in each series by the equated time of investment. 2. Take the sum of these products and then find what fractional part each product is of the sum. 3. These fractions are the parts of the total net profits belonging to each series. To illustrate the rule, let us suppose that an association whose monthly dues are $1 per share had three series in force at the end of the third year, and that the number of shares in each series and their value per share were as follows: shares, value per share $25.27; third series, 400 shares, value per share $12.32; 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 mouths (the sum of the series), equivalent to $48 invested for 24 months. Treating the other series in the same way, we find that $35 paid per share in the second series has been invested for 183 months; $24 paid per share in the third series, for 12 months; and $12 paid per share in the fourth series, for 6 months; then $48×500×241- $24 × 400 × 124 --- $588,000, first series' investment for one month. $1,146,600, total investment for one month. Hence the total net profits are divided as follows: 588000 1146600 399600 or 980 666 200 of the total profits belong to the first series. 1146600 or 120000 39000 1148885 or Tir of the total profits belong to the fourth series. Total profits to be divided are $5,325. 960 19 of $5,325 = $2,730.77, first series' share of the profts. 666 200 of $5,325 1911 of $5,325 $1,875.81, second series' share of the profits. $557.30, third series' share of the profits. $181.12, fourth series' share of the profits. $5.46, profit of a share of the first series. = $3.09, profit of a share of the second series. 400-$1.39, profit of a share of the third series. $181.12 500 $0.36, profit of a share of the fourth series. $48.00, 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, dues paid, + $0.36, profit, = $12.36, value of a share of the fourth 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 dollar 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 cqualized 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 24 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 18 months, which is equal to $1 invested for 666 months; $24 dues per share in the third series, for 12 months, which is equal to $1 invested for 300 months; and $12 dues per share in the fourth series for 6 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. 500 × 16.333=8, 166. 50, equalized result for the first series. 15, 924. 40, equalized result for all series. $5, 325, the total profits, 15, 924. 4033. 4392, the rate per cent of profit. $0.334392 × 16. 333-$5. 46, profit of a share of the first series. $0.334392 × 9. 250-$3.09, profit of a share of the second series. $0.334392 × 4. 166 $1.39, profit of a share of the third series. $0.334392 × 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, but erroneously, called the partnership plan. $18 X 500 X 24 $12 × 500 × 6 = ILLUSTRATION. $576,000, first 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: if or 18% of $5,325 $2,748.39, first series' share of the profits. or of $5,325 = $1,855.16, second series' share of the profits. ARR or 18 of $5,325 15200 36000 $549.68, third series' share of the profits. r118880 or 15% 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, × 24, average time, × 500 shares. It will be readily seen that 12 is a factor common 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 8,000. Hence of the total profits belong to the first series. 5, 400. 500. Total, 15, 500. of the total profits belong to the second series. of the total profits belong to the third series. 183 of the total profits belong to the fourth series. 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 |