Interest and Discount
The accumulation function for interest i paid at the end of the period is:
a(t) = (1 + i)^{t}
The accumulation function for discount d paid at the beginning of the period is:
a(t) = (1 - d)^{-t}
The important relations between i and d are shown in graphic form below:
A(0) | A(1) | I_{1} | ||
|___ | ____________________________________ | ___| | ||
(1) | 1 - d | 1 | ||
(2) | v | 1 | ||
1 | 1 + i | i | ||
1 - d | 1 | d | ||
+ | + | |||
(3) | d | i | i - d | |
(4) | iv | i | ||
(5) | d | i | id |
The first relation is found by comparing A(0) in lines (1) and (2):
v = 1 - d
The discounted values of 1 are equal
d = 1 - v
The second relation is found by comparing A(0) in lines (3) and (4):
d = iv
Discount-amount equals discounted interest-amount
The third relation is found by comparing I_{1} in lines (3) and (5):
i - d = id
Difference in interest equals interest-rate times difference in principal
Dividing the third relation by id yields:
1 | - | 1 | = 1 |
d | i |
These relations are easily derived from the equivalent discounted values of 1:
v = 1 - d
d = 1 - v
The first relation
d(1 + i) = i
d = iv
The second relation
d(1 + i) = i
i = d + id
i - d = id
The third relation
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