4 ON THE VALUE OF ANNUITIES. 
When money due at the expiration of a certain period is discharged 
hy the payment of an immediate sum, the party making it ought not 
to pay the whole sum, hut that portion of it only, which put out at 
interest, will amount at the expiration of the period to the sum due; 
for instance, £100 paid down when interest is 5 per cent, is equivalent 
to the payment of £ 105 at the expiration of a year. 
Finding the present value is therefore precisely the same case as 
that solved in Art. 6, and as p the present value in this case corresponds 
with p the principal in the former, s the sum due with s the amount, 
the notation for the time and rate being the same, we have by Art. 6, 
(Art. 4.) 
(Art. 7.) 
^ 1 + ia 
s = p (1 + ill) 
n — 
V 
ip 
The rules given in Articles 5, 6, 7, and 8, apply equally here, if we 
substitute the words present value, and sum due, for principal and 
amount. 
12. To find (d) the discount— 
This is found by taking the difference between the present value and 
the sum due. 
s 
cl = s — p ~ s — -—■—— • 
1 1 + in 
Example. What discount should be allowed for the present pay 
ment of a bill of £325, due at the end of 3 months, interest 5 per cent ? 
325 
n = vj = .25 
i ~ .05 
.05 
rol25 = 
1 + in 
_i 325 
1.0125)325 (320.988= 
... 30375 
1 + in 
2125 
20250 
4.012 = £4 0 3 
discount. 
1000 
911 
8 
The above is the true mode of finding the discount, but in the mer 
cantile world it is customary to take for the discount the interest of the 
sum for the time that elapses till it becomes due, by which mode more 
than the true discount is obtained. 
The formula for finding the interest by Art. 4, is ins, and therefore 
the discount received above the true discount is 
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