Interest and Discount

When interest i is paid at the end of the period, the accumulation function is:

a(t) = (1 + i)t

When discount d is paid at the beginning of the period, the accumulation function is:

a(t) = (1 - d)-t

 

  Interest:   1 1 + i
    | ___________________ ___________________ |  
   
Discount:        1 - d 1  

The first important relation is found by accumulating 1 - d:

(1 - d)(1 + i) = 1

This is the important identity: a-1(t)a(t) = 1

v = 1 - d

The discount factors are equal

d = 1 - v

The second important relation is:

(1 - d)(1 + i) = 1 - d + i(1 - d) = 1 d = i(1 - d) d = iv

Discount-amount equals discounted interest-amount

The third important relation compares the events depicted in the above diagram:

d = i(1 - d) = i - id i - d = id

Difference in earnings equals interest-rate times difference in principal

As an alternative method, accumulate 1 in separate portions of 1 - d and d:

[(1 - d) + d](1 + i) = 1 + i 1 + d(1 + i) = 1 + i v + d = 1 d = 1 - v

The first relation

1 + d(1 + i) = 1 + i d(1 + i) = i

Accumulated discount-amount equals interest-amount

d = iv

The second relation

d(1 + i) = i

Accumulated discount-amount equals interest-amount

d + id = i i - d = id

The third relation

Dividing the third relation by id yields:

1  -  1  = 1
   
d i

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