126 
LIFE ANNUITIES. 
the end of n years, multiplied by the chance of the life living n years, 
and the remaining part of the expression is the present value of an 
annuity on a life aged m+n years ; hence the following 
Rule. Find the value of an annuity on a life older by the number 
of years the annuity is to be deferred, than the present age ; multiply it 
by the present value of due at the end of that term, and by the 
chance of the life surviving that term. 
135. If the numerator and denominator of the expression be multi 
plied by r m , its value remains unaltered, and becomes 
k+n+I r m +”+ 1 + + / m+B+3 r m + f ‘+ 3 + lm+v+* r W+n+4 , + &c. 
a ( m hn~~ 
This formula, when we have tables calculated of the description men 
tioned in Art. 115, points out a very short method of calculating the 
values of deferred annuities ; for the number in column N, opposite the 
age (m+n) at which the annuity is to be entered upon, is the nume 
rator of the fraction, and the number in column D, opposite the present 
age (m) is the denominator of the fraction; the formula by Davies’s 
method is therefore 
136. Rule, Divide the number in column N, opposite the age at which 
the annuity will be entered upon, by the number in column D opposite 
the present age. 
When the annuity depends on the joint existence of any number of 
lives respectively aged rn, mi, , &c., the probability of their jointly 
surviving the term must evidently be substituted for the probability of 
one life surviving, i.e. 
lm l +n • &C. 
« ? ^ . £Lrt_|_ n , »¡¡+1!. m + n > 1 
TEMPORARY ANNUITIES. 
137. Let the present value of an annuity to continue the next n years 
provided any number of lives aged m, m t , w 2 , &c., continue jointly to 
exist during that term, be denoted by a Xmt mj> mi , &c>) . As the value 
of an annuity to continue for the next n years, together with the value 
of an annuity which is to be entered upon at the expiration of n years, 
and then continue during the remaining time of joint existence, is evi 
dently equal to the value of the annuity on the lives for the whole period 
of joint existence, to be entered on immediately, we have the equation ;