140 .LIFE ANNUITIES.
160. If the annuity cease at the expiration of t years from the present
time the present value will be
®(m) a Cn>,m,)
n n •
161. To find the value of the annuity payable during the joint lives of
A and B respectively, aged m and and also during t years after the
death of B, provided A shall live so long.
The value of the annuity during the two joint lives is a,„, mi .
The remaining part consists of two portions, one during the next
t years, the value of which, by the last Article, is a m — a (m> _ ) , and
«1 _ ‘1
the other after t years; the value of any payment of which, as the ??th,
will be the present value of £l, due at the end of n years, multiplied by
the chance of A surviving that period, and of B having died within t
years of that time, viz.—
T • Pm,n (,Pm 1 ,n—t P m l i") — ? (-Pm,n • Vm y ,n—t P(m,m l ), n )
in which, p m ,n.p mi ,n-t
¿m+n
ilm+n Cnj+n—t \ Ci t —t P(m, wii—Q,n
Ci \ Ci C’-i—t / Pt
the expression therefore becomes r n (^ 0,w —P(m, m ^,n )>
\ Pwi—i, t /
the successive values of which, being found for every year after the ¿th,
will give for their sum
m, mx-0
n
Pmi—t,t
a (.m, mi) >
If
adding to which the value of the other portions, we obtain for the total
value required
5f m,mi't' a (m) a (m,mj) + ( ’ 1 () -
n n
Pm L —l, i
3y Davies’s Tables,— •
1 ^(m, rni-i).
^(m, mi) 1 — ®‘On)”j
n# in
N m+ ,
»(«) nV
Vhen m is greater than m l — t,
71 1 «1-0 I mi—t bijn+i, mi
]<
*mi—t, t D.
m, mi—t
Ci i f’m vni—i'
N
m+t,
N
m-H, w j