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Interest and Discount
When interest i is paid at the end of the period, the accumulation
function is:
When discount d is paid at the beginning of the period, the accumulation
function is:
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Interest: |
1 |
1 + i |
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________________ |
________________ |
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| Discount: |
1 - d |
1 |
The first important relation is found by accumulating 1 - d:
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(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 |
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The second important relation is:
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(1 - d)(1 + i) = 1 - d + i(1 - d) = 1 |
| d = i(1 - d) |
| d = iv
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Discount-amount equals discounted interest-amount |
The third important relation compares the events depicted in the above diagram:
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d = i(1 - d) = i - id |
| i - d = id
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Difference in earnings equals interest-rate times difference in principal |
As an alternative method, accumulate 1 in separate portions of
1 - d and d:
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[(1 - d) + d](1 + i) = 1 + i |
| 1 + d(1 + i) = 1 + i |
| v + d = 1 |
| d = 1 - v |
The first relation |
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| 1 + d(1 + i) = 1 + i |
| d(1 + i) = i |
Accumulated discount-amount equals interest-amount |
| d = iv |
The second relation |
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| d(1 + i) = i |
Accumulated discount-amount equals interest-amount |
| d + id = i |
| i - d = id
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The third relation |
Dividing the third relation by id yields:
Copyright © 2004 The Stevens Computing Services Company, Inc. All rights reserved.
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