Transactions of the Actuarial Society of Edinburgh Volume 1

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1886 Excerpt: ... unit paid down is the value of an annuity of i for n years (produced by the investment of the unit for the term), and of the original unit to be returned at the end of the term: that is (1--if1) is the value of an annuity of i for n years, and consequently the value of an annuity of 1 is l t 18. Also, by means of perpetuities, we can obtain the same result A unit invested will yield an annuity of i for ever. Therefore the value of a perpetuity of 1 is-4 » Now, an annuity for n years is a perpetuity entered on at once, less a perpetuity deferred n years, and its value is ax (1--ti") or 1--A" i 19. If the annuity be payable p times in a year, and interest be convertible q times, we have 20. In all cases the value of an annuity is the total discount on 1 during the whole currency of the annuity, divided by the product of the number of instalments per annum and the interest on 1 during the interval between two instalments of the annuity. 21. If the intervals between the payments of the annuity become indefinitely small, so that the payments are made momently, the annuity is said to be continuous. 22. To find the amount and the present value of a continuous annuity, interest convertible momently, we must in formulas (9) and (13) make q infinite, when i becomes S. We then have, agreeably with Chapter I. Art. 13, «S)=--s--(u) (15) 23. If, however, the annuity be not continuous, but be payable p times in a year, while interest is convertible momently, we must in formulas (8) and (12) make q infinite, when i becomes 5. We then obtain.-I?=i (16) 1 i_ e-f «=--T (17) When the annuity is payable yearly these become en3_l (18) (19) l_e-«a a=J= 24. Again, if the annuity be continuous while interest is convertible q times a year, p becomes in...