TEMPORARY ASSURANCES. 
165 
The number of annual payments will be t, consisting of an imme 
diate payment and of a temporary annuity for t— 1 years ; the single 
premium must therefore be divided by 
1 + « 
(nr, mi, r»2» &c.) 
= 1 —r*p 
+ a- 
~x\ 
(m, mi, ma, &c.), t (m, mi m2, &c.) 
n 
(m,mi.ma, &c.) . 
0* 
Art, 19*7. A L =r{ 1—r*p T -r4- 1—(1—r)a— 
Cm, mi, m2, &c.)^ 1 x (m, 
adding and subtracting 1 —r\'P (m t ’ we ^ ave 
1 — r*p AL — (1— r‘p- ~ ) + ?'( 1 —r l p — ) 
1 (m, mi, mg, &c.), I (m, m,, raj.&c.), t' ' 1 (m, Ac.), * y 
(1 ~ r ) a - t 
Cm, nip 77*2» 
1 — r' y? 
(m, nip m2» &c.)j £ 
« -(l-r)(l—r'y; 
(m, mp m2» 
(m, mj, mg, Ac.) 
1 r P(m,m l ,m 2 ,&c.),t (1 ?') { 1 r< P (m . mi> } * 
which divided by the quantity 
1 — r‘p 
•T n- 
1 - T~ LI .. — a 
(m, 77^2, &c.), t (m, mi, mg, &c.) ^ 
gives for the annual premium 
1-r'p 
(m, mq, mg, &c.), £ 
I — rp 
- O -r) 5 
(tn, nip mg, &c.), £ C m > m l> w 2» & c 0 . 
when there is only one life it becomes 
1 —r'.p^t 
1 + 
l—r*. 
(I-r) 
L 
I'm 
(1 -r). 
By Davies’s method- 
The divisor 1 + « (m) = 1 . (Art. 139); the formula for 
the annual premium is therefore (Art. 198) 
D„ 
—M 
m lvl m+i 
D m N m _i—N m+/ _i N m _i 
or, (Art. 199) 
r (N m _i—N,„ +< _i ) — (N m —N m+t ) D„ 
D.„ 
N m —N m+/ 
N,„_!—N, n+< _i N m _i N m+< _i 
201. Rule. To find the single premium. 
Midtiply the present value of £l, due at the end of the given period