EXAMPLE 2 —What present sum is equivalent to a nett rent of 20/. per an nual for 69 years ? The value in the table againsf 69 years is 19,309;310, which multiplied by 20 gives the answer 3f-:6/. 38.11d If any of the annuities in the above ta ble, instead of being for an absolute term of years, had been subject to cease, if a given life should fail during the term, it is evident that the value would have been lessened in proportion to the probability of the life fading ; and that if, instead of being for a certain number of years, the annuity depended wholly on the uncer tain continuance of a given life or lives, its value must be ascertained by the pro bable duration of such life or lives. In order to compute the value of LIFE AN NUITIES, therefore it is necessary to have recourac to tAles that exhibit the number of persons, which, out of a cer tain number born, are found to be living at the end of every subsequent year of human life, which thus shew what are termed the probabilities of life.
Various tables of this kind have been formed by the different writers on this subject, as Dr. Halley, Mr. Thomas Simp son, M. Wersseboom, M. de I'arcieux, Dr. Price, Dr Haygarth, Mr. Wargentin, M. Susmilch, and others; and the true method of' computing the value of life annuities,according to the probabilities of any table of mortality, is laid down by Mr. liTilliam Morgan as follows: 44 Was it certain that a person of a given age would live to the end of a year, the value of an annuity of 1/. on such a life would be the present sum that would in crease in a year to the value of a life one y-ear olcler, together with the value of the single payment of 1/. to be made at the end of a year; that is, it vvould be 1/. to gether with the value of a life aged one year older than the given life, multiplied by the value of 1/. payable at the end of a year. Call the value of a life of one year older than the given life N, and the value of 1/. payable at the end of a year 1 — • then will the value of an annuity on r the given life, on the supposition of a cer 1 1tainty, be N x But the fact is, that it is uncertain whether the given life will exist to the end of the year or not; this last value, therefore, must be diminished in the proportion of this un certainty, that is, it must he multiplied by the probability that the given life will survive one year, or to ex press this probability, it will be X ar 1—N."
The values of annuities on the joint conti nuance of two lives are found by reason ing in a similar manner ; and such values, both for single and joint lives, are given in the following tables.
The values in this and the following tables suppose the payments to be made yearly, and to begin at the end of a year; but if all the payments are to he half yearly payments, and to be made at the end of every half year from the time of purchase, the value will be increased about one-fifth of a year's purchase.
The above table is formed from the pro babilities of life, as deduced from the re gister of mortality at Northampton for 46 years, from 1735 to 1780; and as it gives the mean values of lives between the highest and lowest, it is better adapted for general use than any other extant. It has of late years been generally adopted for calculating- the rates of assurance on lives, and is well suited to this purpose ; but it is by no means a proper table for individuals or societies to grant life annui ties from ; for having been formed from a register comprehending persons of all ages and conditions, it cannot give a cor rect representation of the duration and value of such lives as usually form a body of annuitants, such persons being gene rally a selection of the best lives from the common mass, the interest of every per son who purchases an annuity on any life requiring that he should take care that it is a good life. The best table for regula ting the grant of life annuities is that formed from the table of mortality pub lished by Mr. D. Parcicux, from the lists of the French tontines, but even this ta ble gives the values of the advanced ages considerably too low.