We will now state, with the necessary technical detail, our reasons for the opinion we have already expressed, that it is not permissible to add to the assets, or otherwise bring into the valuation as a credit, the present value of the extra charge made upon half-yearly and quarterly premiums in excess of the aliquot part of an annual premium. This extra charge is made for three different purposes, (1) to cover the cost and trouble of collecting premiums at frequent intervals; (2) to make good the loss of interest on the portion of the premium which is not paid at the beginning of the year; and (3) when the half-yearly or quarterly payments are really premiums, and not instalments of yearly premiums, to compensate for the loss of premium which will occur in some cases through the life assured dying before the full year's premium for the current policy-year has been paid. The proper amount of the extra to be charged for (2) and (3) is easily determined. Since the value of an annuity payable in advance is 1 + a, ¾ + a, or ⅝ + a, according as it is payable yearly, half-yearly, or quarterly, the total yearly net premium for the age x is

according as it is payable yearly, half-yearly, or quarterly.