DEFERRED TEMPORARY ANNUITIES.
137
By Davies’s formulae—
hi m-\-d+n *“
d: dt - ” k •
The present value of £l paid down, and of an annuity of ¿61 for
d years, subject to the existence of a life aged m, is (Art. 139)
N m _; — N OT+ct
d~
153. If the total number of payments be d, the first of which is paid
down, the present value will be
, N m —__ D m -f N m — N m+(i _i N m _!
+ D m " D m ~ D m
154. To find the annual premium necessary to secure an annuity for
ii years, to be entered upon at the expiration of d years, we must divide
the present value of the deferred annuity by unity added to the present
value of an annuity for d years, which gives
(Art. 139)
-h* m-\-d ~~~ hlm-f-rf-j-n * h;
D m N m _j—N m+ii N m _!—
155. When the total number of annual payments is d, w r e divide the
present value of the deferred annuity by unity added to the present
value of an annuity for d — 1 years, which gives
hIm-4-i? D,„ ,n-\-d A
D,„ Nm+rt-l N m ^_,
Example. Required the single premium to secure an annuity of ¿650
for 7 years, to be entered upon at the expiration of 9 years, subject to
the existence of a life now aged 40. (Carlisle 4 per cent.)
#(401 >—
19
'40
#49 =
7 + 9=16
4458x .702587x 13.15312
5075
— 8.11769
0 (40)
lia
/ r la
6 50« '
110
#56 —
4000 x.533908 X 10.96607
5075
— 4.61467
3.50302
50
£175 3 0 175.1510
Also,
N 40+9 —N 40+9+7 _N 49 -N 56 _8580.9492-4878.0207_3702.9285 ;
D 40 D 40 , “ 1057.0669 ~~ 1057.0669
3.50302
50
175.1510 =£175 3 0