In bringing these Notes under the consideration of the members I feel some apology is needed for the reason that the subject may he said to be somewhat outside the scope of the ordinary proceedings of the Institute.

In addition, however, to its legal aspects, which will doubtless appeal to a profession which demands from its members a not inconsiderable knowledge of law, I venture to think there are other grounds on which the subject may be considered suitable for discussion.

The Public Trustee Act brings into being a Department, which, from its intimate connection with the affairs of the people and the large financial interests involved, may in course of time rank only second to some of the great Departments of the State, and, from this point of view alone, the subject is of interest.

Moreover, it has a close connection with the work of many Actuaries owing to the large investments made by Life Offices in reversionary securities, this kind of investment being peculiarly affected by questions concerning the trusteeship.

When, further, it is remembered that Insurance Companies form one of the classes of Corporate Bodies which may be authorized by the Treasury and Public Trustee to undertake the duties of Custodian Trustees, I think it will be admitted that the subject is one with which we are practically concerned.

In many staff pension funds there is a provision that if a pensioner dies before his pension payments amount to the oaccumulation of his contributions (with or without compound interest) to the date of retirement, the difference shall be paid to his representatives.

Although Mr. H. W. Manly and Mr. George King have dealt with the problèm of valuing this benefit, and Mr. Manly has shown how to calculate the rate of contribution to provide for it, the formulas presently to be deduced may be thought of sufficient interest to warrant their publication.

In considering the benefit mentioned above, it will be seen that if—

(i) the contributions amount to n times the yearly pension,

(ii) the annuity used in valuing the pension be continuous,

(iii) the benefit be valued on the assumption that the funds earn no interest—

the value of the benefit at the date of retirement, for each unit of annual pension, will be as follows, assuming the deaths to be equally distributed over the year:

Of lx persons retiring at age x, dx will die in the year of age x to x + 1, and as they will, on the average, die in the middle of the year, there will be payable in respect of them (n − ½)dx, being the amount of their contributions less ½dx which they will have received as pension.