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)	I1
		|___	____________________________________	___|
	(1)	1		1 + i	i
–	(2)	1 – d		1	d
=	(3)	d		i	i – d
	(4)	d		i	id
	(5)	v		1	1 – v
	(6)	iv		i	i(1 – v)

The first relation is found by comparing I₁ in lines (2) and (5):

d = 1 – vAmounts of interest discounted from 1 are equal

The second relation is found by comparing A(0) in lines (3) and (6):

d = ivAmount of discount equals discounted amount of interest

The third relation is found by comparing I₁ in lines (3) and (4):

i – d = idDifference in interest equals interest rate times difference in principal

Dividing the third relation by id yields:

1	-	1	= 1
d		i

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These relations are easily derived from the identity:

(1 + i)(1 – d) = 1

 

v = 1 – d

d = 1 – vThe first relation

 

(1 + i) – d(1 + i) = 1

d(1 + i) = i

d = ivThe second relation

 

d(1 + i) = i

i = d + id

i – d = idThe third relation

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