REVERSIONS. 
141 
When т, — t is greater than m, 
Q'(rn l mi—О ^m+t, mi t ^m-H, Wi ^»ii—i 
П , 
J^m L —t,t l^m, m L —i im\ ^ ^ nl l 
•N”m+/, m L p 1 •■^m+t.mf 
/ / * г) 
L nn * c, m i • ' m, m\ 
162. To find the value of an annuity on a life A, aged in, after the 
failure of the joint existence of two other lives, P and Q, aged and 
m 2 years. 
The chance of receiving the annuity in the wth year is 
Pm, n ( 1 — — Pm, n P(m,m\, mg), n } 
the value of the reversion is therefore 
2 1 (.Pm,n P(m,mi,mi),n')— *hn — &m, т{,тц1 
and the annual premium = ^—j—• 
1 "P &т, m\, m2 
163. If A does not enter on possession until after the death of the 
survivor of P and Q, the chance of receiving the annuity in the nth 
year is 
Pm, n (1 "~Pm\, n )( 1 Pm-2, n)“ Pm, n~~P (in, nij), n p(m, mg), n "P P(m, mi,m 2 ),n) Ulld 
2 t (jPm, n P(m, m L ), n~P(m,m-z), n ~PP(m,m\, mg), n)—^m — O m> Ml ~ С1 т , m2 “b d m , mj) m2 
is the present value of the reversion. 
The annual premium, which is payable so long as A is in existence, 
with cither P or Q, is found by dividing the single premium by 
1 + 2 r n p m> „ (p mi, n “P Pmi, n P(.mi,m£), n) • 
1 *р 2 Г {P(m, mi), n~PP(m, mg),n P(m, mi, mg),n} 1 "P ' 
“P n m m ~ a„ 
164. The value of an annuity on the joint lives of A and B, aged m 
and m y , after the death of P, aged m 2 , is 
2?'"(1 p m , „ ) P( m , mi), „“2 1' Cp(m, mj.),n P(m, m\, mg), n ) — ^m, m\ ®m, mi, mg , 
the annual premium which is payable during the joint existence of A, 
B, and P, is found by dividing by 1 + a mi Bli >W-> . 
165. The present value of an annuity on the survivor of two lives, A 
and B, aged m and m l} after the death of P, aged m 2 , is 
2 7 ( 1 ”” Pmi, «) (pm, n’ [ Pmi. n ~~ P(m, mi), n) ' 
* {Pm,n “P Pmi,n ~ P(m,mi),n P(m,mg), « “ .P(mi,mg), n~P PCm.m,, mg),«}— 
”P mi ”” ¿hrti.mgH a.