The type of contract which will be considered is a renewable oneyear term assurance on a group of lives which provides, in addition to a capital sum on the death of any life, that if at the end of the year the total claims in the year are less than some fraction, k, of the premiums paid, then a fraction, l, of the difference will be refunded.

Suppose that P is the net and P′ the office premium for n lives assured for a sum of 1 each and that q(r) is the probability that exactly r deaths will occur during the year, then

where c is the integral part of kP′. Since equals the expected deaths, μ say, the equation may be written

where and