PROBLEM 2. To form a table of loaded assurance premiums. The formation of w has been already exemplified. The present formation differs from that referred to in no other respect than that here the logarithm of (1 plus its loading) is included in the initial term. Thus, to form with a loading of 17 per-cent. :— PROBLEM 3. To form a table of the values of endowments payable at a specified age. The age at which the endowment becomes payable being +n, the value of the benefit is or, in logarithms, Dx+n log Da+n+colog DÅ. And the difference of this expression, in which + is constant, is A colog D, or colog vpn. Two examples follow, in which a+n takes the values 21 and 60, respectively. PROBLEM 4. To form a table of the assurances equivalent to a present value of a unit. The value of this assurance, when the age is x, is, The required table may be consequently constructed as The use of such a table as that of which the formation is here exemplified is to facilitate the conversion of a cash bonus into a reversionary bonus. Example. A policy on (15) has assigned to it a present bonus of 42 375. Find the equivalent reversionary bonus. 42 375×3 3742=142'982. This table could be very readily formed from the Assurances on p. 14 by means of Colonel Oakes's Table of Reciprocals. The method here employed possesses the advantage of giving also the logarithms of the values formed. PROBLEM 5. To form a complete table of the values of deferred and temporary annuities. By a complete table is to be understood a table, that shall comprise, in regard to the classes of annuities specified, all the cases that can present themselves in the use of the mortality table employed. The present age, (or, preferably, the age to which the value to be formed has reference,) being as usual a, denote that at which the deferred annuity is to be entered upon, and the corresponding temporary annuity to cease, by y. Then, the value of the deferred annuity being of which the logarithm is, N1 log N,+colog Dz we shall have to form the values of this expression for all the combinations of x and y, in which a does not exceed y. This can be very commodiously done in a series of paralel collumns. Either of the quantities x and y, as may be arranged, will vary in the columns; and the other will vary in passing from column to column, that is in the rows. If, as is most convenient, it is chosen to commence with the younger ages, the work will assume one or other of the following forms :-The first, if in the columns a be made to vary, and the second, if y be made to vary: The forms are theoretically equally eligible. The first, however, is to be preferred, because in it the addends will consist of fewer significant figures than in the second: ▲ colog D, contains generally only five significant figures, while A log N, contains six throughout. I The following is a specimen of the formation, in which the 24 1484 10 329276 213440 311094 20 4688 292735 19:6216 14971 2.8044 14971 23.8142 23.8142 219098 322279210029 23.6103 23.6103 13 373101 23 6103 354919 22 6422 336560 | 21*7050 14105 14 15 369024 233897 | 350665 | 22*4215 14039 0*9682 23.1581 364704 23 1582 The first six columns are here given complete. The initial terms are the successive values of log N,+colog D, when y is made to vary; and they are formed as follows, the addends being the terms of Alog N, -:^| |