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.

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?

17

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

have not yet expired; and by prorating the premium we find that at the end of the year there still remains to be earned $16,666.67 of the premiums collected in 1907 on these policies. As regards the business written in 1908, the company by the end of 1909 has only earned one half of the premiums from three-year policies and three tenths of the premiums from five-year policies, thus leaving $60,000 of premium income not yet earned. From its new business, yielding $450,000 of premiums, the company earns only $135,000 during the year in which the policies were written, and $315,000 must be assigned to the unearned premium fund. In all, therefore, the reserve at the end of the third year amounts to $391,666.67. If our illustration were extended to the fourth year, the reserve computation would be still more elaborate, because the company would then have to consider four classes of policies, viz., the three and five year policies of 1907, the three and five year policies of 1908, the one, three, and five year policies of 1909, and all the policies of 1910.

While the foregoing rule of equating the unearned premium is fairly safe for practical purposes, and meets the demands of the law, it should be remembered that it is only a system of averages, which does not always conform to real business conditions. Where a company's business is rapidly gaining, and more policies are written in the latter part of the year than in the early part, it is apparent that on the average the policies have not run for six months, and the reserve will, therefore, not be sufficiently high. Vice versa, if the company's business is declining, the reserve, if computed on the assumption that all policies written in the year have run six months, will be more than sufficient.

For this reason, if a large company wishes to know at any time exactly what its progress is, and whether its unearned premium liability is increasing or decreasing, it will be desirable to compute the unearned premium fund by

months instead of years. In fact, a few companies have adopted this method. Thus, in the case of one-year policies the assumption is made that as much business is done in one part of a given month as in another, and that consequently all policies written during a month may be assumed to have been in existence fifteen days. If the policy is written in January the company considers fifteen days, or one twenty-fourth of the premium earned at the end of the month, the remaining twenty-three twenty-fourths belonging to the reserve, while on the 31st of December twenty-three twentyfourths of the premium is earned, and one twenty-fourth unearned. If the policy was written in February, three twentyfourths of the premium will be unearned on December 31st. Similarly, as regards its three and five year business written in January, the company will consider fifteen days of premium as earned at the end of the month, while on December 31st the reserve on the three-year policies will be forty-nine seventy-secondths of the premium, and on the five-year policies ninety-seven one hundred and twentieths.