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 - 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 I1 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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