cent of the premiums received on all unexpired risks that have less than one year to run, and a pro rata on all premiums received on risks that have more than one year to run."

This rule is only approximately correct in its application to actual conditions, since it is based on the assumption that the volume of the company's business is uniform throughout the year, i.e., that as many policies of a given term are written on the first day of the year as on the last, and that as many are written on June 30 as on July 1. If this assumption is granted, it follows that the average life of all policies written in a given year is six months, and that consequently six months of the premium is earned, while the balance is still unearned. If all the policies written by a company in a given year are one-year policies, our rule thus provides, since all these policies are assumed to have been in force six months, that the company can consider one half of the total premium income from these policies as earned, and that the other half still remains to be earned. This unearned half of the total premiums, however, which constitutes the reserve for that year on one-year policies, will be earned in the following year.

If policies are written for longer terms, such as two, three, four, and five years, the same principle is applied. Thus in the case of two-year policies the term under consideration extends over twenty-four months. It is assumed that in a given year as many two-year policies are written at the beginning of the year as at the end of the year. Consequently, all two-year policies written in that year are assumed to have been in force six months, and during the year in which the policies were written the company earns the premium in the proportion that six months bears to the total term of twentyfour months or one fourth. One fourth of the premium is, therefore, considered earned during the year in which the two-year policies were written and three fourths is still un

earned, or in the reserve. At the end of the second year the policy is assumed to have been in existence eighteen months (six months during the first year and twelve months during the second year), and the company is now entitled to the premium in the proportion that eighteen months bears to the full term of twenty-four months, or three fourths. One fourth of the premium, however (the balance for the remaining six months of the term), is still in the reserve, and will be considered as earned in the third year.

In the case of three-year policies the term covers thirtysix months, and all such policies are again assumed to be in force for six months during the year in which they are written. Applying the same method used in the above illustration, the company earns during the year in which these policies are written, that portion of the total premium represented by the ratio of six months to the term of thirty-six months, or one sixth, while five sixths still remains to be earned. At the end of the second year the company earns another twelve months of the premium or one third of the total, and the premium is now one half earned and one half unearned. At the end of the third year the earned portion of the premium amounts to five sixths and the reserve to one sixth, and this remaining one sixth is considered earned in the fourth year. In the case of four-year policies the company earns during the year in which the policies are written one eighth of the total premium (six months out of forty-eight months) and seven eighths is in the reserve. At the end of the second year the earned premium and the reserve amount respectively to three eighths and five eighths; at the end of the third year to five eighths and three eighths; at the end of the fourth year to seven eighths and one eighth; while during the fifth year the remaining one eighth of the premium is considered earned. Similarly, in the case of five-year policies, one tenth of the premium is earned during the first year and nine tenths is in the reserve. In each succeeding year

the company earns another one fifth of the premium and the reserve decreases correspondingly, until in the sixth year the premium becomes fully earned and the reserve exhausted.

PORTION OF PREMIUM EARNED AND UNEARNED DURING VARIOUS YEARS

Term of Policy.

First Second Third Fourth Fifth Sixth
Year. Year. Year. Year. Year. Year.

Earned.

In applying the foregoing method of computing the reserve, let us assume that an insurance company begins business in the year 1907, and during the first three years receives the following premium income: During the first year $50,000 of premiums from one-year policies, $25,000 from threeyear policies, and $25,000 from five-year policies; during the second year $100,000 from one-year policies, $50,000 from three-year policies, and $50,000 from five-year policies; and during the third year $200,000 from one-year policies, $150,000 from three-year policies, and $100,000 from five-year policies. Assuming that all these policies continue in force and that there are no cancellations, what should be the reinsurance reserve of this company at the end of each year?

During the first year of its history this company, according to the rule adopted for reserve computations, earned one half of its $50,000 of premium income from one-year policies written during the year, one sixth of its $25,000 of in

come from three-year policies, and one tenth of its $25,000 of income from five-year policies, or a total of $31,666.67. The reserve for the three types of policies amounted respectively to one half, five sixths, and nine tenths of the premiums received, or a total of $68,333.33.

In the year 1908 this company earns the remaining one half ($25,000) of the premiums received on the one-year policies written in 1907. It also earns two sixths of the premiums received in 1907 from the three-year policies, and two tenths of the premiums received in 1907 from the fiveyear policies, or a total of $38.333.33. But the company also wrote new business during 1908, receiving $100,000 from one-year policies, $50,000 from three-year policies, and $50,000 from five-year policies. Of these new premiums the company is again entitled to one half as regards one-year policies ($50,000), one sixth as regards three-year policies ($8,333.34), and one tenth as regards five-year policies ($5,000), or a total of $63,333.34. In all, the company earned during 1908 on its new business of that year and on its policies of 1907, which were still in force, a total of $101,666.67. As regards its three-year policies written in 1907, however, there remains unearned at the end of 1908 three sixths of the premium ($12,500), and as regards fiveyear policies seven tenths of the premium ($17,500), or a total of $30,000. By applying the proper percentages to the 1908 business, it is found that the company must keep in the reserve $136,666.66, or, in other words, the difference between the $63,333.34 earned on the 1908 business and the total premium income of $200,000 received. At the end of the second year, therefore, the company has earned a total on all the policies in force of $101,666.67, and must have in the reserve $167,666.66.

In the third year of its business (1909) our hypothetical company must make a reserve allowance for three classes of policies. Its three and five year policies written in 1907