SURVIVORSHIP ASSURANCES. 
173 
also 
p m , »—1 X Pmi, n ' 
‘ L +1 . 4n+l • Un v +2 o_ 
11 '' ll 
? - 3 + &c. 
MM • 
'm-1 
L ' 
L • 4 
I'm—l ^ffli ¿m—l+.t. 
I 
m—l, mj j 
if the present value of the insurance required be denoted by +„, jTOi , we 
have -4™,-»!,— -r^ r (1 + #m+l,*ni) + -7—X a„_ 1( „j') . 
(1) -s \ ‘»t I'm / 
214. This is the formula given by Mr. Baily in his treatise on Life 
Annuities : Mr. Milne’s formula is 
1 
2 
dm—1, m\ 
1, I 
dm, mi—I 
JPmi—l, I 
* 
which is more convenient than the other when we have tables showing 
the probability of a single life at every age living one year, and the 
reciprocal thereof, as in Table 5. 
215. As the failure of either of the lives will determine the event, the 
divisor for the annual premium must be l-f# m , m , . 
Example. What is the present value of an assurance of ¿£500 pay 
able on the death of a person aged 60, provided that event take place 
before the death of another aged 37 ? (Northampton, 3 per cent.) 
h A 
2 \ m > m i 
+-V( 1-HW, )4+ 
I'm 
X # m _ 
m—l, mi j — 
1 f m hi 1 + #61.37 
2 V 60 - 37 _ 4o 1.03 
+ "59 
#59.37 
¿60 
1 r A 
2 L eo ' 37 v 1 
( 1 +#61.37 ) 
1 - % #«, 
03 
1-.03X8.1539 
hÿ • #59.3 
.755383 
LI 
^G0 1 
1+Ï 
^61 ( 1 +#(61.37)) 
1.03 
ho • #(59.37) 
1.03 1.03 
1956x8.9619 
=.73338 
1. 03 
-2120x8.3407 
2038 
* This formula is of the same value as the other; for 
4nj+n-l 1m+n X Arn + n-1 . 1m\ — 1 P(m,m,—l),n 
Pm,n X pmi,n~l ^ X ^ 
-m- 4ii-l 
Pmi ~ I? 1 
2>” - n,B . ^ ; and in the same way it may be shown that 
V Pmi — l,i J 1 
When the assurance is for t years only, the 
G(m—1. wji n (m, mj—1) 
and 
l > n fm.n-l Xp mi ,— 1 
Pm-1, 1 
i»i) + 
»(m, mi-U| 
Pmi—Ul ’ 
expression wiil be 
Pm-1, i