46   Life Contingencies but each becomes 1=Sax+Ax where,

(1+i) = (l+i(rn)m= C1 — {J (-))-'" = (1 —d)—1 =eb.

m   m   l

The corresponding statements in words are left as an exercise for the student.

8. Further, rewriting Ax =1—dax, we have ax =1 —Ax — 1   1 . Ax

—

d   d   d

This result for ax could have been deduced from first principles in a manner similar to that used to prove Ax =1—dax.

Expressing the relation in words, we may say that the value now of an annuity-due of 1 per annum on the life of (x) is the difference in value between two perpetuities-due each of 1 per annum. The first perpetuity-due will provide 1 at the beginning of each year starting at once, while the second perpetuity-due will withdraw 1 each year starting at the beginning of the year following the death of (x).

Or we may say, since 1—Ax =dax, that 1—Ax represents the value now of an annuity-due of d per annum so long as (x) may live, and in proportion 1—Ax will represent the value now of an

d

annuity-due of 1 per annum so long as (x) may live.

9. Similarly, we may write,

1—(1+i)Ax   1   1

ax =   

i   i   d

1— 1+I(m0)Ax(m)

also   ax(m) =    m   = 1 — 1 A (m)'

J(m)   .3(m)   f(m)

(m) — 1 —Ax m)   1   1   (m)

ax —   --•A

f(m)   f(m)   f(m)   x