Policy Values   73

8. Policy values can be written down for policies with premiums payable more often than once a year and/or policies where the sum assured is payable at the end of that nzth part of the year in which the death occurs.

Thus n VI') represents the reserve after n years on a whole life policy for 1 due at the end of the year of death where the premiums are payable m times a year, so that

n VP) = ) —   (m) ax+n

(m)

Ax+n — Px

=Ax+n —[Px + 4 2m1 Pzm) (px+d)] Lax+n 2m11 =Ax+n —Px•ax+n — n1 -1 pxm) (px+d)ax+n+P m) m -1

2m   2m

nT'x   2m1Pzm)•(1—nhx)+p1m)' 2m

Tx[1+ 2—1pxm)J

Again,   (n) v(m) =A (n) — (n)p(m) (m)

1   1

where AY+1X(1+i)2n =Ax+r=Ax+, (1 +i)2 approximately, az+i =ax+i — 2m1 approximately,

(I+ n — 1 i )A

(n)   x

and hence   (n)Pxm) = Ax   =    2n    approximately.

ax(m)      m—1

ax —

9. For a whole life policy with instalment premiums,

V x[m] = 1— m — P(m) A x+n —Pix   ~-,, m)• ax(m) approximately, 2m z

But   P[m] =Px+ nz -1 Pzm)• d approximately 2m

therefore nVxm]=nVx, which should be obvious if n is an integer.

2m