10 Mr. T. Weddle on Annuities and Assurances 



By (1.) we have a^^) = i;(l--rA^^'^), where zj=(l+r)-', the 

 present value of i6l, due in a year, hence (4.) becomes 



a„=w»(l-rA') . (1-rA") (1-rAW). . . . (5.) 



Problem II. To determine (ap) the present value of an an- 

 nuity of .£1 on thejoth life mentioned in Problem I. 



At the nomination of A^^'^ <£l will be due, and the annuity 

 on his life will be worth A^p^ ; hence the life will then be worth 

 I -\-A^P^; but £1 due at the nomination of A^^^ is now worth 

 tfp_ 1 ; hence the present value of the annuity on the joth life is 



a^_, .(1+A(/')), 



.-., (4..),a;=a'a"...a(/'-i).(l + A(^)), . ..... (6.) 



and this is the present value of £l per annum on the j9th life. 



The preceding (6.) is the form that will generally be found 



best adapted to computation ; but a^ may also be expressed 



in terms of a only, or of A only, as follows (see (1.) and (3.)) : — 



ap=:(l+^\a'ii"....a(P-^l-a(p^), .^ ..... (7.) 

 and 



«^=i;p-i(l-rA')(l-rA") .... (1 -?-A^^-'))(l +AW) . (s.) 



It may be observed too that (7.) is little, if any, inferior to 

 (6.) in point of easy application, as it requires a reference to 

 one table only, while (6.) requires a reference to two. 



Note. Unless the annuity be due, that is, unless a payment 

 is to be made immediately, none of the preceding formulas 

 will give ttp the value of the annuity on the first life, correctly. 

 If we suppose, as usual, that the first payment will be made 

 in a year, the true value of a^ will be 



And not 



fl,=A'=(l + i)(l-aO-l, (9.) 



z,= l+A'=(l+i)(l-a'). 



Problem III. To determine (An), the present value of an 

 annuity of jgl on the n successive lives A', A" . . . . A^"^ 

 We evidently have 



• A„=a,-f-flj+flrg , , , , -{-«^. 



But (7.) and (9.)— 



