50   Life Contingencies

(m

2m 1) (1+i)Ax — mm2   1 6 Ax, the last term of which is

quite small.

Therefore, ax') —ax =m    -1 [iax+(1+i)Ax]   -1, approximately.

2m   2m

14. The annuity represented by a under which the last payment is made on the payment date preceding death is called a curtate annuity. If however a payment is to be made at death for the portion of the payment-period lived through before death, the annuity is called complete or apportionable and is written d. Thus ax =ax+IA =ax+2 (l +2i)A approximately

and dzm~ = axm) + 2m Ax = axm) +2m (1 +Zi)Ax approximately.

15. The connecting links for assurance and annuity symbols involving two lives would follow by the same methods as for one life.

Thus Axy = ly[v(lxy —lx+l : y+l) +v2(lx+l :y+l -1x+2 : y+2) +...1

1

[vi xy +v2lx+l : y+i + ... ] xy

1   7

—   [Vlx+l : y+l + V21x+2 : y+2 +   • }
txy

or   Axy =vaxy —axy. Also Azy =Ax+Ay—Axy

= (1 —dax) + (1 —day) — (1 —daxy) =1— d (ax +ay —axy) =1—daxy .