Annual Premiums   57

Also   P(m) a(m) =P(m) ( 1

xnt' xn~   xn~

in

+a (m) 1 ) —A

x n x— m

(r)Px(m) a(m) =Ax(r)

(,),P(m) a(m) =A(r_> x~' xni   xtl

Px¢x=Ax and «Pxax=Ax

9. Again, the annual premium may be payable by instalments and the full number of instalments become payable for each year on which (x) enters. If such a premium is payable by m instalments it is written Pzml

so that Pzm).ail)=Px or

(n)Px[m) . a(m) = (n)P

— x

Pam] a(xm) + m — 1 PXm]Ax —Ax 2m

or

and (0D)PI'] • a(m) - - (O0)P

x   11   x

where a(!) may be obtained from the relation f(m)a i ~) =d.

11

When the annual premium is thus paid by instalments a deduction is made from the sum assured equivalent to the value of the unpaid instalments due in the year of death.

10. Since 1 = (Px+d)ax = (Px+1+d)ax+1

and   ax = ax -1= vpxax+1

1   1

we have Px+1=Px+(Px—vox) — =Px+(Px+1—z'4x)—

ax   ax

Px (1+i) =4x+px(Px+1—Px)ax+1•

Each of these has a meaning that can be expressed in words.

11. Consider two lives (x) and (y).

For an assurance payable on the first death, that is when the pair is broken, we have

Pxyax,, =Ax,,.