A NNTITY. 



known by th* name of life annuities, fall undrr the latter Urm. Closely 

 oanneetsd with this put of the subject in oorntouM (which Me), in 

 which an esUte u held during oerUin lives, but in which there U a 

 power of renewing any life when it drop*, that U, aubiUtuting another 

 life in place of the former, on payment of a fine REVFIUIOXI, or the 

 interest which the next proprietor ha* in any eetate, Ac,, after the death 

 of the present and Uft-im*nuu* [IxiURAsac], in which the queetion 

 u, what annuity muit A. pay to B. during hu life, in order that B. may 

 pay a given nun to A.'s exeouton at hu death. 



If money ouuld not be improved at interest, the value uf an annuity 

 certain would simply t the yearly mini multiplied by the number of 

 yean it it to continue to be paid. Thus a lease {or three yean of a 

 bouse which U worth NXV. a year, might either be bought ny paying 

 the rent yearly, ur by paying SOOT. t once. A life annuity, in such a 

 cue, will be worth an annuity certain, continued for the average num- 

 ber of yean lived by individual! of the same age ai the one to whom 

 the annuity U granted. But if compound interact be supposed, which 

 u always the case in real transactions of this kind, the landlord, in t In- 

 case of the annuity certain just alluded to, must only receive such a 

 sum as, when put out to intercut, with lOiV. subtracted every year for 

 rent, will just be exhausted at the end of three years. To exemplify 

 this, let us suppose that money can be improved at 4 per cent. In 

 Table I., in the column headed 4 p. c. (4 per cent.) we find 2775 oppo- 

 riU to 3 in the first column, by which is meant that the present value 

 of an annuity of out pound to but three yean is 2775/., or '2j,\i/. The 

 present value of an annuity of 1001. under the some circumstances is, 

 therefore, 277'5i, or 277'. 10*. Thin is the value of a lease for three 

 yean corresponding to a yearly rent of 100/. The landlord who re- 

 ceives this, and puts it out at 4 per cent., will, at the end of one year, 

 have 3981. 12*. From this he subtracts 100/. for the rent which haa 

 become due, and puts out the remainder 188/. 12. again at 4 per cent. 

 At the end of a year this has increased to 196/. 2*. 10}</., from which 

 100/. is again subtracted for rent. The remainder, 961. it. 10j</., again 

 put out at interest, becomes at the end of the year 99/. 19*. &<!., within 

 three pence of the last year's rent This little difference arises from 

 the imperfection of the Table, which extends to three decimal places 

 only. 



TABLE I. PansucT VALI-X op AX ASNITTY OF OSF POI-MP. 



For I 

 ever ] 



33-333 



11-000 



10-tOO 



To find the present value of an annuity of II. per annum continued 

 for 10 yean, interest being at 5 per cent, look in the column headed 

 5 p. c., and there, opposite to 10 in the first column, will be found the 

 value of 7722A, or 71. 14*. 6W. This would be commonly said to be 

 7*722 ytari pun-hate of the annuity. Decimals of a pound may be 

 reduced to shillings and pence, and the convene, by the following 

 Table : 



TAL It. AXD III. FOR RIDVCIXU Demurs or A POUND TO SBILLIKOH 



AXU PCXCE, AMD THE COXTXXU. 



ANNUITY. 

 For example, what is -66&V. in shillings and pence f 



TABLE II. is ii 



Ofl i 



665 13 3| 



Again, what is 1 ?<. 10}</. in decimals of a pound f 



TABLE III. in o u -5 



7 u -85 



10 .. -iHJ 



i -MO.-J 



.1" 17 13 



895 



These conversion* are not made with .. ;L , only 



three decimal places are taken. Tin- error will m-ver U- more tlmii 

 one farthing. 



To use Table I. where the number of yean u not in the table, but U 

 intermediate between two of those in tin- table. MU-II a mean must be 

 token between the annuities belonging to the nearest vean atx>\ 

 In-low the given year, as the given year is between those two yean. 

 This will give the reniilt with Kiitlic-ient nearness. We muHt observe, 

 that no tables which we have room to give art- - r nnm- th m 



a first guess, so to speak, at the value required, such oa may enabli- any 

 one who is master of cominon arithmetie, not to form a decisive opinion 

 on the case before him, but to judge whether it i- worth hi* uhil, i , 

 make a more exact emjuiry, either by taking professional adv. 

 consulting larger tables. As an example of the case mentioned, suppose 

 we ask for the value of oil annuity of I/, continued for 12 yean, ii. 

 being at 4 per cent. We Cud in Table 1., column -1 



For 10 yean 8'111 



15 ., 1M18 



i>iir. i 



8-007 



Since 5 yean adds 8'007 to the value of the annul' ii- will 



add about one-fifth part of this, or -tidl, aiul two years will add about 

 1-202. This, added to S'lll, gives 9'313. The real \ e near 



to 9'385, and the error of our table ia '07 out of S-313, or about the 

 133rd part of the whole. The higher we go in th.- uUe. tin- 1, 

 portion of the whole will this error be. 



The last line in Table I. gives the value of the annuity of II. con- 

 tinued for ever : for example, at 5 per cent, th- 

 or, as it is called, a jierpetuily of If., is 20/. This is the sum 1 

 5 per cent, yields II. a year in interest only, without diminut 

 the ]>riucii>al. We see that au annuity for a long term of yean differ* 

 very little in present value from the game continued f<'. 

 example, II. continued for 70 years at 4 per cent. J-395/., 



while the perpetuity at the same rate is worth only ~2,'il. Hence the 

 present value of on annuity which U not to begin to be paid 

 yean have elapsed, but is afterwards to In 1-005 



at 4 per cent: which sum improved during the 7U years, would yield 

 the 25/.ueces3ary to pay the annuity for all yean succeeding. 



TABLE IT. Axotnrr of AS ASXVITY OF OXE POI-NII. 



Iii this Table we see what would be possessed by the receiver of ;m 

 annuity at the end of his term, if he put each year's annuity out at. 

 interest as soon as he received it. For example, an annuity of I/., in 

 40 yean, at 5 per cent, amounts to 120-8/., which includes 401. r. 

 altogether at the IT different yean, and 80'8/., the compound 



interest arising from the first year's annuity, which has been 39 yean 

 at interest, the second year's annuity, which has been 38 yean at'int 

 rest, and so on, down to the last year's annuity, which Las only ju - 

 been received. When the annuity is payable half yearly, or quarterly, 

 its present value is somewhat greater than that given in the preceding 

 Table. For the aunuiu ng certain portions of his annuity 



sooner than in the cane < yinenta, gains an additional portion 



of interest. Since 4 per cent, is 'i per cent half-yearly and 1 per cent. 

 quarterly, and since every term contains twice as many half-yean as 



