In this paper I discuss a simple application of option pricing theory to bonuses on with-profits life assurance policies. The approach is a new one and I have been able to make only an introductory exploration of its possible applications. Nevertheless it seems to me to give actuaries some sort of a handle for gripping the rather intractable problem of the relationship between reversionary bonus, terminal bonus and the proceeds of comparable unit-linked policies. I look at the problem very much from the policyholder's side, and I am not concerned here with aspects of valuation or solvency. These require further investigation.

1.1 My paper entitled ‘The Matching of Assets to Liabilities’ described a new study of the subject of matching. The paper outlined the mathematical framework, showed some calculations, and suggested various applications in the field of actuarial valuation.

1.2 Subsequently A. D. Wilkie published a note entitled ‘Portfolio Selection in the Presence of Fixed Liabilities: a Comment on “The Matching of Assets to Liabilities”’. Wilkie's note appears to represent a new extension of portfolio selection theory which enables him to encompass the conventional portfolio theory and to show where my approach to matching fits into the picture.

1.3 This paper is in the nature of a reply to that of Wilkie. My objective at the stage of the first draft was to re-examine the matching portfolio, as defined in relation to specified liabilities, in view of the wide range of alternative portfolios which Wilkie has described. The scope of this work progressed well beyond my original plans when I found a most interesting mathematical relationship between the ‘matching portfolios’ described in my earlier paper and the ‘efficient portfolios’ described in Wilkie's.