The amounts apportioned under the various modes in whic might be received were as follows: £57,113 in Reversionary Bonus. 1,958 in Reduction of Premiums. 830 in converting Whole Life Assurances into Assurances, payable at a fixed age, or at £59,901 Consolidated Revenue Account of the Crown Life Assurance for Five Years, commencing 25th March, 1865, a 25th March, 1870. The average Rate of Interest on the Investments of the Life A Fand of the Company was, on 25th March, 1866, £4 9s. 6d. £4 10s. 7d.; 1868, £4 11s. 2d.; 1869, £4 12s.; and 1870, £4 per Cent. At the respective dates mentioned portions of the F not invested; and if it be intended by the legislature that the rate calculated on the Gross Fund, including Agents' Balances, Out Premiums and Interest, and Interest accrued, &c., then the Intere Invested portion of the Fund was equal to the following rates Gross Fund, namely:-1866, £4 6s. 10d.; 1867, £4 5s. 5d. £4 2s.; 1869, £4 6s. 5d.; and 1870, £4 7s. 5d. JOURNAL OF THE INSTITUTE OF ACTUARIES AND ASSURANCE MAGAZINE. On "Extra Premium." By JAMES R. MACFADYEN, of the Legal and General Life Assurance Society, Fellow of the Faculty of Actuaries (Scotland). [Read before the Institute, 25 March 1872.] A PAPER on Extra Premium may treat of the subject in one or other of two ways. It may be, on the one hand, an attempt to deduce tables of mortality for the various classes of underaverage risks occurring in a Life Office's transactions. Or, on the other hand, this part of the subject may be ignored and the paper be simply an enquiry into the effects resulting from the various methods employed in charging extra—an analysis of the actuarial points raised in consequence of a Society admitting not merely minimum risk lives, but others not eligible on similar terms. In short, the subject of Extra Premium may be considered either in respect to the amount to be imposed or in regard to the manner and effect of its imposition. The first question must be determined mainly by actual mortality observations-the second does not require in the same degree for its examination the light of experiIt is this latter question which I have taken up in the following paper; but as both branches of the subject are intimately connected, it may be well to make a few general remarks on the former also. ence. VOL. XVII. F ance practice to a set of rates founded on observations of lif similar circumstances. We can thus only adminis of rough justice-carry out the cardinal principle of that the many must bear the burden of the place every life that is insurable at all under such scale as shall most nearly represent its condition. the same thing is done in cases taken as minin and is the very essence of life insurance. The whol then is Where shall the line be drawn? How man mortality shall be used? Were the selection of the assu the Office done away with, one might suffice; and ad the force of this selection, will be the number of varyin rates required. Anything tending to diminish this forc to simplify the classification of assured lives. As an illu this, let me take the following: It is at present the practice in many Offices to policyholders free residence in any part of the world, pr privilege be not exercised during a probationary period; lation, together with a sufficiently broad foundation mortality, being considered adequate to check the force against the Society. Similarly, any method acting as g regulator to the flow of assurances to an Office will necessity of a great variety of life premiums whether for for climate, and will thus narrow the field of requisite to the rates to be charged in under-average cases. V remarks I shall leave the first branch of my subject and remainder of this paper to the second form in which th presents itself, namely, the actuarial results of the variou now in use for charging extra premium. Circumstances having an unfavourable tendency on th of human life may be divided into two great classes-th taining those cases in which the age of the person affec more important bearing on the prejudicial circumstance other, those cases in which less weight is attached to th life. I use the words " more and "less" advisedly whatever be the nature of the circumstance the age r some weight; but there are obviously certain kinds o which this is more important than in others. Following mode of classification-in the former section we see case diseases, and also what may be called the "special circumstance" extras; in the latter, climate and occupation risks and a few maladies, such as hernia and gout, find a place. Let us now proceed to examine the first of these two sections. The ordinary manner in which extra is charged in such cases is as follows: The Medical Examiner. of the Office finding the proposer ineligible at the minimum rate, though not altogether uninsurable, recommends the acceptance of the risk with a certain number of years addition to the age; that, in fact, if the person be r years old, he be taken at the premium rate for, say age (+t). Let us assume that the life is accepted on these terms, and that Pr+t correctly represents the premium required to provide the benefit granted. Does it follow that on future occasions of dealing with the policy, the Office should treat it as if it were an ordinary assurance of entry age (x+t)? I believe that this is the method generally adopted. Let us now consider how far it is justifiable. If we have any number of quantities a, b, c, d, e, &c., such that a : b: :c: d and e: b :: f: d and g: b::h: d, then we know that a+e+g:b::c+f+h:d; but the converse proposition is not true —that if a+e+g: b::c+f+h:d it necessarily follows that a:b::c:d e:b::f: d g:b::h:d -the reason evidently being that we can completely vary the values of a, e, and g, and also c, f, and h, without affecting their sum, so that like ratios between the individual quantities do not nccessarily follow, though they may possibly do so. To apply this to my subject. A person aged a opening a policy of assurance by annual premiums has from some cause to pay extra to such an extent that he is charged the premium which would be the normal rate at age (x+t). It does not follow that because his premium is the same as that of age (x+t) his chance of death in any individual year will be the same as that of a person of real age (x+1). Let dashes over the following symbols denote that mortality in accordance with extra is meant. Then (disregarding loading) since Pr+t=P', we have 1 1+ax+t -d= But as shown above, though the sum of the numera side of this equation is to its common denominator as the numerators of the other side is to its denominator, follow that each of the terms in one of these numerato denominator as the corresponding term of the other n to its denominator; and thus Divide these ratios of possible inequality by &c., respectively, and we have nor x+1 not necessarily equal to and generally, it does not follow that l'z+n-l'z Ꮖ So that if any life be charged an extra such that the paid is equivalent to that paid by a life t years older n to such extra, it does not follow that the chances of individual years after entry are the same in each case. individual parts into which the numerators containing t of v may be split are quite arbitrary; and therefore, alth premiums to ages ≈ and (~+t) are the same, the ex of life at entry as well as afterwards in each case may be |