It is a common experience in many fields of investigation to find that the standard methods which have been developed and used in a given field prove inadequate in a particular application. In these circumstances it is often found that no further progress can be made along lines hitherto considered conventional, because some fundamental question has not been explicitly posed and answered by the standard methods. In life assurance a feature of existing methods of approximate valuation, which has not received prominence, is their inability to measure satisfactorily the size of the possible error involved. A wide variety of approximate methods have been in use, and some have been found to give consistently ‘good’ results, but the question of what is meant by ‘good’ has been left unanswered. The background of this paper is that the authors, faced with this situation when tackling an essentially practical problem, were virtually forced to break away from traditional methods and seek a new tool.