56   Life Contingencies

Similarly Ax nl =mPx . ax ml and hence =,nPix . ax ml and hence and   Ax nl = mPx nl . ax ml and hence

dAx nl mPx n~ =   r

1—Ax~ dAx nl

mPx _

1—Ax'

dA mPx =   x

1—Axml

8. The premium may be payable half-yearly or quarterly, and the benefit may be payable at the end of the month or the quarter in which the life may fail, instead of at the end of the year of death.

P is always an annual premium payable in advance, but Pi') indicates that an mth of this premium is payable at the beginning of each mth part of a year, so long as (x) may live, and (n)Pxm) indicates further that the benefit will be paid at the end of that nth part of a year in which (x) may die.

P(m)a(m) =Pxlm) (-1 -Fazm)) =Ax,

m

Thus

= A

so that Px m)   x   Ax =

a(xm) ax —

m—1

Px   Px

1_m—1 1 1_    -1(Px+d) 2m ax

2m

Whence Pxm) [1— 2m1 (Px+d)] =Px or   Pxm)=Px+ 2m1Px(m)(Px-~d)

Thus PI') exceeds Px by the sum of the two terms [m2—m1 PI') 1Px and [m2-ml Pxm) id the meanings of which should be noted.