The value of an annuity which is not to be payable till either one or other of two persons is dead, and which is to con tinue during the life of the survivor, is found as in the last case, only subtracting twice the value of the joint annuity, in stead of that value itself. In the preced ing case it is 8-31. For this case only differs from the preceding, in that the annuity is not payable while both are alive, that is, during the joint lives. Con sequently the value in this case is less than that in the last, by the value of an annuity on the joint lives.
The value of an annuity to be paid to A from and after the death of B, if the latter should happen to die first, is the value of an annuity on the life of A, diminished by the value of an annuity on the joint lives of A and B. For the situation is exactly the same as if the dice granted an annuity to A, to be re turned as long as both should live. The ages and Table being as before, and the life on whose survivorship the annuity depends being that aged 25, we have:— whence the value of the required annuity of I/. is 7.31.
The following Table, extracted with abridgment from Morgan on Insurances, deduced from the Northampton Table, with interest at 4 per cent., gives the average sum to which the savings of an individual may be expected to amount at the end of his life, improved at com pound interest from the time he begins to lay by :— That is to say, according to the North ampton Tables, if a person were, at the age of 26 (that is, a year after 25), to begin laying by 1001. a year at interest, he might expect the amount at the end of his life to be 79.21. for each pound laid by yearly ; or 7920/. Or, to speak
more strictly, if 100 persons were to do this, they might expect that the average amount of their savings, reckoning the accumulations at their deaths, would be 79201. each. As we have already ob served, the mortality of the Northampton Table is greater than the fact, and the average accumulations would be greater, from young ages considerably greater, than those shown in the preceding table.
We have seen that the security of the method for estimating the value of life annuities depends upon the presump tion that the average mortality of the buyers is known. This average cannot be expected to hold good, unless a large number of lives be taken. Therefore, the granting of a single annuity, or of a few annuities, as a commercial speculation, would deserve no other name than gam bling, even though the price demanded should be as high as that given in any tables whatsoever.
In the preceding tables, we would again remark, that our object has been simply to furnish the means of giving a mode rately near determination of a few of the most simple cases. We should strongly recommend every one not to venture on important transactions without profes sional or other advice on which he can depend, unless he himself fully under stands the principles on which tables are constructed. The liability to error, even in using the most simple table, is very great, without considerable knowledge of the subject ; and most cases which arise in practice contain some circumstances peculiar to themselves, which have not and could not have been provided for in the general rules.
The following references to works on this subject may be found useful :—