Unit-Linked life insurance can be simply defined as the application of unit trust principles to the ‘savings element’ of each premium paid and the application of insurance principles to the remaining ‘insurance element’ only. It has been appearing in various forms for some time now. In North America, it has been mainly of the variable annuity variety, commencing with CREF in approximately 1952. In the U.K. it has been mainly of the endowment assurance variety, commencing in approximately 1957 but not really making a big impact until 1962–63 when the unit trust groups first came on the scene. In Australia it has so far been limited to superannuation schemes with the so-called (by one life office) ‘E.F.G.’ approach the prime example, commencing in approximately 1965. In each country the position is the same—a remarkably favourable response from the public, particularly in the last few years, causing an urgent reassessment of previously held views, sometimes, unfortunately, most widely and eloquently recorded! There have also been legislative difficulties of varying degrees, most pronounced in North America, least pronounced in the U.K., with the position in Australia not clear.

All business is, to a greater or lesser extent, concerned with the taking of risk. This paper attempts to explain some of the main approaches that have been adopted in pursuing the concept of risk from a probabilistic viewpoint. The word ‘risk’ itself has been used fairly indiscriminately by writers, and thus gets given a number of different senses, but it has attained a clear and precise definition at various times and in different types of literature. Indeed we can distinguish three main avenues of development. The first is the pure economic concept, the second the actuarial approach, and the third the more recent linkage of risk analysis with the general approach to decision-making under uncertainty.

In my earlier paper I expressed disappointment at failure to find b.i.b.d.s for the case ν= 4p, k = 4, b = ½ν(ν−1), λ = 6 (p is an odd prime). The failure was not complete because I had given a solution for ν = p + 1, k = 4, λ = 6 which covered the cases of ν = 4p where ν−1 was also a prime. In what follows I shall complete my earlier note and also give simple formulae for ν = p + 1, k = 4, λ = 3.

Consider a very large number of persons, and probability distributions for the age at death of the last survivor, next to last survivor, etc. First, suppose that the persons are statistically independent with the same probability distribution for age at death (random sample case). Then, some approximations to distributions of extremes are often usable. These approximations are completely specified except for at most three parameters. This simplifies distribution estimation to estimation of the parameters. Moreover, previous large samples (possibly different sizes) from the same population of persons, and much of their data on extremes, can be used for estimation. Also, the sample results of ten remain applicable for the more general case of independence (or mild dependence) but possibly different distributions for the ages at death. Here, the average of these distributions is ‘sampled’. Very recent results show that distributions of extremes are of the sample type for any joint distribution of the ages at death. However, the distribution ‘sampled’ can be greatly different for the last survivor, next to last survivor, etc. This effectively limits estimation of parameters to previous groups having very nearly the same joint distribution and use of one observed extreme per group.