1.1. The concept of the matching of assets to liabilities is fundamental in matters of finance. In its broadest sense matching is relevant both to the investment of life office and pension funds and to actuarial calculations in relation to those funds.

1.2. Matching is inherently connected with the uncertainty of future conditions, as measured by such indicators as rates of interest, inflation and currency exchange. It is when conditions do not work out as hoped for or expected that the risks associated with mis-matching can materialize. Whether considering investment policy or an actuarial calculation it is important that the nature of a portfolio which minimizes these risks be appreciated, and yet it is often difficult to specify such a portfolio with precision. This may not be a source of difficulty in practice when the judgement of the actuary is applied to the problem in hand, but there is evident scope for refinement of the concepts.

0.1. Since the concept of immunization was introduced in Redington (1952) many new ideas have been advanced in connexion with the pricing of government securities. It is shown in Bierwag (1977) for example, that the applicability of an immunization strategy depends on the form of movements assumed possible in the yield curve.

0.2. Much of this earlier work has been framed in deterministic terms, where prices or yields are assumed to be known in advance, and uncertainty has been introduced via arbitrary instantaneous changes in those yields. It is easily shown that portfolios of securities with a given mean term or duration are immune (in the sense of yielding the same return at the required realization date) to certain forms of instantaneous change.

By the definition of generalized assurances and annuities, the relation 1 = Ās + δās is shown to be the simplest member of a family of relations in which the present value of the capital and interest repayments is equated to the original amount of a loan. When the force of interest is constant over time, the relation takes the simple form above, but when the force is a polynomial in time, the interest payments become a series of annuities, the payments on which increase over time at different rates.

The data presented in C.M.I.R. (1979) and (1984) on the Sickness Experience for Individual PHI Policies contain duplicate policies. In dealing with these data it is therefore necessary to be able to take account of the effect of duplicates on the variance of sickness claim inceptions and of weeks of sickness claims.

This paper deals with life assurance policies where the death benefit is the maximum of a fixed sum assured and a return of the gross premiums paid. The gross premium is then the solution of an implicit equation. The conditions for the uniqueness and existence of a solution are discussed.