CHAPTER ,X BUILDING AND LOAN ASSOCIATIONS 60. Definitions and first principles. Abuilding and loan asso- ciation is an association of persons formed for the purpose of enabling its members to accumulate money by periodical pay- ments into its treasury, to. be invested from time to time in loans to those of its members who wish to build homes. The membership of such associations usually consists of two classes: shareholders who are investors only, and shareholders who are at the same time investors and borrowers. There are many plans for building and loan associations, but most of them require a small monthly or weekly payment, which in the case of borrowers provides for the payment of interest and for the establishment of a sinking fund which will extin- guish the indebtedness when the stock matures, and in the case of investors constitutes a savings account which is augmented from time to time by the dividends which may be declared out of profits earned by the association. The sources of profits are interest earned on loans made to borrowing members; gains due to the difference between the book vahte and the withdrawal valve of stock belonging to members who retire before their stock has matured; fines assessed upon members who may have been delin- quent in the payment either of dues or of interest; fees charged on new business; and, finally, any undivided surplus that may have remained over at .the last distribution.* The net profits are the profits remaining after all the expenses of conducting the business have been paid. The net profits art » One other source of profit frequently utilized by loan associations is the differ- ence between the interest received on money borrowed by the association at a lower rate than that charged to borrowing shareholders, and the interest paid on loans thus made. The amount of such loans is usually limited by law. For example, in Wis- consin the limit is 20% of the assets of the association. 164 BUILDING AND LOAN ASSOCIATIONS 165 sometimes distributed on the basis of the book values imme- diately after the last distribution of profits, and sometimes the basis of distribution is found by adding to the book values, immediately after the last distribution, a portion of the amount paid in dues since the last distribution. The latter plan is more equitable, since each shareholder's profits are more nearly pro- portional to the amount of money he has invested. The former plan will give a slightly higher rate of profit. PROBLEM. To determine the rate of profit for a period, The rate of profit for a period cannot be determined in ad- vance, since it depends upon elements that cannot be known until the end of the period. For example, loans in force at the beginning of the period may be repaid within the period, or a considerable amount of stock may be withdrawn, or part of the funds of the association may remain uninvested. At the end of the period when the elements of the problem are all known, it reduces to a simple problem in percentage. Let I denote the total interest received during the period, / the amount received from fees and fines, and e the total expenses and losses. Further, let B be the book value immediately after the last distribution of all stock remaining in force to the end of the period, and d the amount added to the book values to enable the amount paid in as dues during the period to partici- pate equitably in the profits. The net profits N will then be y-^i+f-e, (1) and the rate of profit r will be ^^ (2) if the distribution is made on the basis of book values at the beginning of the period, or r^-^^+f-6 (3) £+d B-\-d ' ' if dues paid in during the period are allowed to participate in the profits. 166 MATHEMATICAL THEORY OF INVESTMENT To find d, one has, in effect, a problem in the equation of payments for every shareholder. If, for example, a shareholder pays $1 a month per share, and pays promptly on the first of the month for six months, each share of his stock will contrib- ute $3.50 toward d. The stock of a shareholder delinquent in all dues for the period would contribute nothing to d. The stock of a member who-paid $1 the first month, $1 the second month, nothing the third and fourth months, $3 the fifth month, and $1 the sixth month, would contribute $3 to d, since, by the approxi- mate rule for equation of accounts, $1 for six months, $1 for five months, $3 for two months, and $1 for one month would be equivalent to $3 for six months. It would be impossible to give even a concrete example of the determination of the rate of profit without taking into account the receipts and disbursements, and the status of every "share- holder's stock for an entire period. The rate of profit once determined, the apportionment of the net profits to each shareholder's stock is a simple matter.
EXAMPLES 1. B holds 10 shares in the Provident Building and Loan Association, which, immediately after the distribution of profits on January 1, 1910, had a book value of $72.98 per share. During the following half-year period he paid as follows: January 1, $10; February 1, $10 ; Slay 1, $30; June 1, $10. On July 1 the rate of profits was .0254. What was his share of the profits? Solution. To find the amount of the dues participating in the profits, we note that B had invested $10 for 6 months, $10 for 6 months, $30 for 2 months, and $10 for one month. These sums would be equivalent to a sum x for 6 months, where x is determined from the equation 6x = 10 x 6 + 10 x 5 + 30 x 2 + 10 x 1 = 180. Therefore x = 30. The profits on B's stock will then be computed on $729.80 + $30, or $759.80. Consequently, B's share of the profits will be $769.80 x .0254 = $19.30.