the value of the term containing A 1 as calculated for one mortality table, might be used without material error in finding the values of annuities by other tables. The above examples show that the formula, as now completed, is capable of giving the values of annuities (and of course of
other quantities) with very great accuracy.So long as we consider the annuity to be payable yearly, no allowance being made for the time which elapses between the death of the nominee and the last previous payment of the annuity, it is, as we have seen, a very simple problem to calculate its value. But in practice annuities are generally payable by half-yearly instalments, and it is the custom to pay a proportionate part of the annuity for the odd time that elapses between the last half-yearly payment and the death of the nominee ; and the value found by the methods described above therefore require to be corrected before they are strictly applicable in practice. Approximate values of the necessary correc tions are very easily found ; but the strict investigation of their correct values is a problem requiring a considerable knowledge of the higher mathematics, and it would be quite beyond our present purpose to consider it.
When an annuity is payable half-yearly, the common rule for finding its value is to add -25, or a quarter of a year s purchase, to the value of the annuity payable yearly. When it is payable quarterly, 375 is added ; and when by instalments at n equal periods throughout the year (or by thly instalments), the addition is The values thus found are sufficiently correct for most purposes. More correct methods of finding the values of annuities payable half-yearly, quarterly, &c., are investigated in papers in the Assurance Magazine, by Woolhouse, xi. 327, and by Sprague, xiii. 188, 201, 305. Some authors have assumed that when an annuity is payable half-yearly, interest is also convertible half-yearly, overlooking the circumstance that the true rate of interest is thereby changed, as we have explained in the earlier part of this article. In fact, as we showed, 5 per cent, interest convertible half-yearly is equivalent to a true rate of interest, 5, Is. 3d. per cent. If, then, we have found the value of an annuity when payable yearly at 5 per cent, interest, and require, perhaps, in the course of the same investigation, the value of an annuity payable half-yearly, it is clear that that value should be computed, not at .5, Is. 3d. per cent, interest, but at 5 per cent. ; or if we prefer the rate 5, Is. 3d., then the value of the annuity payable yearly should also be calculated at that rate.
The approximate value of an annuity payable up to the day of the nominee s death, or of a " complete " annuity, as it is now usually called, is found in the case of annuities payable yearly by adding to the value of the ordinary annuity the value of i, payable at the instant of the nomi nee s death ; in the case of half-yearly annuities, by adding the value of \ ; and in the case of quarterly annuities, the value of
The previous remarks refer almost exclusively to annui ties which depend on the continuance of one life, or to " single life annuities," as they are commonly called. But an annuity may depend on the continuance of two or three or more lives. It may continue so long as both of two nominees are alive, in which case it is called an annuity on the joint lives ; or it may continue as long as either of them is alive, in which case it is called an annuity on the last survivor. Again, if it depends on the existence of three nominees, it may .either continue so long only as they are all three alive, when it is called an annuity on the joint lives ; or so long as any two of them continue alive, when it is called an annuity on the last two sur vivors ; or so long as any one of them is alive, when it is called an annuity on the last survivor. In addition to these, we have "reversionary" annuities, which are to commence on the failure of an assigned life, and continue payable for the life of a specified nominee ; or, more gene rally, to commence on the failure of a given status, or combination of lives, and continue payable during the existence of another status. There are also "contingent" annuities, which depend on the order in which the lives involved fail. Thus, we may have an annuity on the life of x, to commence on the death of ij, provided that take place during the life of z, and not otherwise, and to continue payable during the remainder of the life of x. Reversionary annuities are of considerable practical im portance, but contingent annuities are rarely met with. Lastly, we may mention annuities on successive lives, These are of importance in the calculation of the values of advowsons, and of fines on copyhold property. It does not fall within the scope of this article to treat at any length of annuities on more than one life, and we must refer the reader who wishes for further information with, regard to them to the works of Baily, Davies, and David Jones, already mentioned, and Milne s Treatise on the Valuation of Annuities and Assurances, 1815.
The student who wishes to pursue the subject more thoroughly, and to become acquainted with all the improvements in the theory of annuities that have been introduced of late years, should care fully study the various articles contributed to the Journal of the Institute of Actuaries, particularly those of Woolhouse and Make- ham. The Institute was founded in the year 1848, the first sessional meeting being held in January 1849. Its establishment has con tributed in various ways to promote the study of the theory of life contingencies. Among these may be specified the following : Before it was formed, students of the subject worked for the most part alone, and without any concert ; and when any person had made an improvement in the theory, it had little chance of becom ing publicly known unless he wrote a formal treatise on the wholo subject. But the formation of the Institute led to much greater interchange of opinion among actuaries, and afforded them a ready- means of making known to their professional associates any im provements, real or supposed, that they thought they had made. Again, the discussions which follow the reading of papers before the Institute have often served, first, to bring out into bold relief differences of opinion that were previously unsuspected, and after wards to soften down those differences, to correct extreme opinions in every direction, and to bring about a greater agreement of opinion on many important subjects. In no way, probably, have the objects of the Institute been so effectually advanced as by the publication of its Journal. The first number of this work, which was originally called the Assurance Magazine, appeared in September 1850, and it has been continued quarterly down to the present time. It was originated by the public spirit of two well-known actuaries (Mr Charles Jellicoe and Mr Samuel Brown), and was carried on by them for two years, we believe, at a considerable loss. It was adopted as the organ of the Institute of Actuaries in the year 18o2, and called the Assurance Magazine and Journal of the Institute of Actuaries, Mr Jellicoe continuing to be the editor, a post he held until the year 1867, when he was succeeded by Mr Sprague. Thf> name was again changed in 1866, the words Assurance Magazine" being dropped ; but in the following year it was considered desir able to resume these, for the purpose of showing the continuity cf the publication, and it is now called the Journal of the Institute of Actuaries and Assurance Magazine. This work contains not only the papers read before the Institute (to which have been appended of late years short abstracts of the discussions on them), and many original papers which were unsuitable for reading, together with correspondence, but also reprints of many papers published else where, which from various causes had become difficult of access to the ordinary reader, among which may be specified various papers- which originally appeared in the Philosophical Transactions, the Philosophical Magazine, the Mechanics Magazine, and the Com panion to the Almanac ; also translations of various papers from the French, German, and Danish. Among the useful objects which the- continuous publication of the Journal of the Institute has served, we may specify in particular two : that any supposed improvement in the theory was effectually submitted to the criticisms of the- whole actuarial profession, and its real value speedily discovered ;. and that any real improvement, whether great or small, being placet!, on record, successive writers have been able, one after the other, to
