Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/184

 172 for m years that is, to be payable only if death should occur after that period is _5? ; which is equivalent to D, and hence to I, By subtraction, the single premium for a &quot; temporary&quot; assurance for m years on the same life is * T. X+M which is equivalent to and hence to A column R is sometimes inserted in commutation tables to facilitate calculations relating to &quot;increasing&quot; assurances. H x is the sum of the terms M*, M* T) . . . M. x + z ; so that 2 is the value of an assurance the amount of which shall be 1 if the life fails during the first year, 2 if during the second year, 3 if during the third year, and so on. Formu- When the value of any immediate annuity, calculated at a given Ife in rate of interest, is known, the value of a sum payable one year terms of after the last instalment of the annuity may be readily deduced annuity- from it. The value of any deferred payment is the difference values, between the sum to be ultimately paid and the discount for the period during which it is deferred. Let a be the value of an annuity of 1 at the rate of interest i, and let it be required to find the value of 1 due at the end of the year following the last pay ment of the annuity. The discount of 1 for one year at the rate of interest i is .__ l_/y : and the present value of such annual 1+t discount (payable in advance) for the whole period covered by the annuity and one year more is (1 -v) (1+a). Hence the value of the deferred payment of 1 is 1 - (1 -v) (1 +a). Putting a x for the value of an annuity on a life aged x, we have for the present value of a whole-term assurance on a life of that age 1 - (1 - 1?) (1 + a x ). The agreement of this result with those formerly deduced from the numbers dying in each year may be seen by substituting for d x, d x +i, &c., their equivalents (l x -l x +i (l x +i - lx+t), &c., when the foregoing expression becomes - l x + ) + v 2 ( lx = v(l + a x ) - a x ; as will be seen from the article ANNUITIES. By a simple transposition this expression takes the form v-(- v)a x ; which in its turn becomes 1 - (1 - v)(l + a x ). Annual Assurances, as formerly mentioned, are usually paid for &quot;by T&amp;gt;re- annual contributions or premiums, continuing either during the miums. whole subsistence of the assurance or during a limited period only. The annual premium for an assurance is deduced as follows. Since the present value of all the annual payments must be equal to the single premium, and since premiums are always payable in advance, we have (putting P for the annual premium required) P(l +a) = A ; whence P = . In this expression A may represent the single premium for any benefit whatsoever, whether depending on single or joint lives, or on any other description of status ; and (1 4- a) may represent the value, in any such case, of an annuity payable in advance during the period over which the payment of premiums is to extend. The annual premium, payable during the whole of life, for a whole-term assurance on a life aged x is l+a x a x ) - a x a x _ -- l+a x or it may be expressed in a variety of other ways by substituting different equivalents of the single premium and the annuity. When the premium is to be payable for m years only, its amount is expressed by _, whore the symbol |,_i^ represents the 1 + U . [LIFE. value of a temporary annuity for m - 1 years ; and 1 4 m _ a is there fore the value of an annuity for m years payable in advance. When the premium for the first m years is to b&amp;lt;; th of that for r the remainder of life, the ultimate annual payment is found by the expression where -i|a is the value of an -(1 +-!) +,-]] annuity deferred for m 1 years, and therefore of an annuity deferred for m years, but payable in advance. By the commutation method the annual whole-life premium is - -; -?- = _-- . The premium limited to m annual payments, for a whole-term assurance, is - I Na-1-Njc+..-i . The premium pay- able after m years, when the payment during that period is th of the ultimate annual payment, is -ICN^-x-N^. We do not propose to enter further on the investigation of for mula; for the calculation of premiums for the various descriptions of life assurances. These will be found in the works of Milne, Baily, Jones, and other authors who have treated of the subject of life contingencies. The student will find a very clear exposition of the nature and modes of calculation of the more ordinary kinds of premiums in a paper by Mr James Meikle, The Rationale of Life Assurance Premiums, reprinted by the Actuarial Society of Edin burgh in 1879. In the practical calculation of life assurance premiums various p r devices have been suggested for shortening labour and ensuring tic- accuracy. Mr Peter Gray s method of calculation, by means of me logarithmic tables on the plan originated by Gauss, may be specially the mentioned. His Tables ami Formula, in which this method is explained, is a work of great value to the student of life contin gencies. When the requisite annuity- values are available, the tables of assurance premiums constructed by Mr William Orchard afford great facilities, either in forming scales of premiums or in isolated calculations. The foregoing expressions for the single premium in terms of the corresponding values of annuities are of such a character as to be applicable to a great variety of cases to nearly every case, in fact, where the risk of the assurance is to be entered on imme diately, and the sum assured is to be payable at the r ml of the year following the last payment of the annuity embraced in the formula?. In like manner the formula; for the annual premium, - (1 - r)&amp;gt; and its equivalents are applicable in all such cases, but only when the premium is to be payable during the whole continuance of the assurance, so that in the expression Lll the annuity- 1 T&quot; ^ value a in the denominator corresponds with that in the nume rator. Mr Orchard has tabulated the values of v-(l-v}a and (1 - f) for all probable values of a, and for the several values 1 T&quot; Ct&amp;gt; of v corresponding to eight different rates of interest. By means of these tables, when the annuity -value corresponding to any required single or annual premium is known, the premium itself may be obtained by mere inspection. The tables may be employed with annuities derived from any table of mortality, and, as the various cases to which they apply are by far the most frequent in practice, they are found extremely useful by computers. We have throughout supposed that the payment of the sum assured is to be made at the end of the year in which death occurs. This supposition accords with the theory of annual mortality and annual conversion of interest into capital, upon which the usual system of calculation is based. It also agrees very nearly with fact when the sums assured are payable six months after death ; for, if it be supposed that the deaths occurring within each year of age take place at equal intervals of time, or that they occur in equal numbers in the first and second halves of each year respectively, the persons insured will, one with another, complete about half a year of age in the year when they die. When it is thought desirable to make allowance, in the calculation of premiums, for the circumstance of the sums assured being payable earlier than at the end of the year of death, that maybe done by a simple modification of the usual formulae. For example, A(l + i) is an approximation suffi ciently near for most purposes to the value of an assurance payable as soon as death occurs. The more scientific methods of calculation developed by Mr VV ool- house and others, and referred to in the article ANNUITIES, eliinin-