| Previous | An Introduction to the Theory of Life Contingencies (1931) | Next |
Annuities 37 Similarly for a deferred annuity-due,
I ax=vnnpsax+n=ax—axnl =ax —axn--T•
The following diagram which is drawn on the assumption that (x) will die after n years may be helpful:
|
|
|
|
|
|
|
|
|
|
year of death | |
|
|
| |||||||||
|
n years | ||||||||||
|
I |
I | |||||||||
|
I |
I I x+n |
|
I |
|
I | |||||
|
i x x+lx+2 | ||||||||||
|
ax = |
1 1 |
1 |
1 1 |
1 |
1 |
1 |
1 |
1 | ||
|
ax = 1 |
1 1 |
1 |
1 1 |
1 |
1 |
1 |
1 |
1 | ||
|
ax1 = |
1 1 |
1 |
1 1 |
|
|
|
|
| ||
|
ax~ = 1 |
1 1 |
1 |
1 |
|
|
|
|
| ||
|
nlax= |
|
|
|
1 |
1 |
1 |
1 |
1 | ||
|
ax= |
|
|
1 |
1 |
1 |
1 |
1 |
1 | ||
Note that ax —ax =1
axn~ —ax; =1—vnnpx nax —n ax=vnnpx•
If (x) has a whole life annuity of 1 per annum and if he should survive one year to age x+1, he will then have a property worth 1+ax+1, therefore ax=vpx(l+ax+1) =vpx ax F1•
This may be written ax+iax=px(1+ax+1)+gxXO
= (1+ax+1) —qx(l+ax+l), (
(7 1 \ ~F CI C E
or lxax(l+i) =1x+1(l+ax+1)+dxXO
=lx(1+ax+1) —dx(1+ax+1).
The student should note the meanings of these identities.
Suppose that the payments under a life annuity on (x) were to be, not 1 each year, but 1 at the end of the first year, 2 at the end of the second year, 3 at the end of the third year and so on, the total value would be
| Previous | An Introduction to the Theory of Life Contingencies (1931) | Next |