Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/94

84 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.

