ASSURANCES ON LIVES. 
186. When an engagement is entered into to secure the payment of 
a sum on the death of an individual, in consideration of a stipulated 
single or annual payment, such transaction is denominated an Assurance 
on the life of that individual. 
The object of the present part of this Treatise is to investigate rules 
for determining the proper premiums, single or annual, that should be 
charged in different cases of Life Assurances. 
In the valuation of annuities the money was supposed payable at the 
end of any year, in the event of the life being in existence at the end 
of the year; so, in determining the values of assurances, it is assumed 
in making the calculations, that the money is payable, not at the exact 
time of the failure of existence, but at the end of the year in which the 
failure of the particular life or lives shall take place. 
The formula will also assume the sum assured to be £l, from which 
the value of an assurance of any other sum may evidently be found by 
multiplying by that sum. 
The same letter of the alphabet will be used as the characteristic for 
denoting the present value of an assurance as was used to denote the 
present value of an annuity; the italic capital (A) representing the 
assurance, and the small italic (a) the annuity. 
187. To determine A 
(m, m\, m2, &c.) 5 
the present value of an assur 
ance on the failure of the joint existence of the last v survivors of any 
number of lives aged m, m lf m 2 , &c. respectively. 
The present value of the expectation of receiving the sum at the end 
of the nth year is found by multiplying the probability of the event 
taking place in the nth year by the present value of £l due at the end 
of n years, which gives (Art. 177) 
r i 
V (m, my, m2, &c.), n—I - mi, m2, &c.) 
and if there be found the present value of the expectation of receiving 
the sum at the end of each particular year during the whole time which 
they may exist, the sum of these values will be the present value of ¿£l 
to be received at the end of the year in which the failure of the joint 
existence shall take place, whenever that event may happen. 
(m, mj, m 2 , &c.)^ ”b®), (m, mj, mg, P(m, my, mg, &c.),n ^ * 
but 
m, mu »¡a, 
(m, i»i, mg, Sc.).