Periodic discount rate: Difference between revisions
imported>Doug Williamson (Expand example) |
imported>Doug Williamson (Expand example) |
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Periodic discount rate is a cost of borrowing - or rate of return - expressed as: | |||
*The excess of the amount at the end over the amount at the start | *The excess of the amount at the end over the amount at the start | ||
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==Example 1== | ====Example 1==== | ||
GBP 1 million is borrowed. | GBP 1 million is borrowed. | ||
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d = (End amount - start amount) / End amount | d = (End amount - start amount) / End amount | ||
Which can also be expressed as: | |||
d = (End - Start) / End | d = (End - Start) / End | ||
''or'' | |||
= (1.03 - 1) | d = <math>\frac{(End - Start)}{End}</math> | ||
= <math>\frac{(1.03 - 1)}{1.03}</math> | |||
= 0.029126 | = 0.029126 | ||
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==Example 2== | ====Example 2==== | ||
GBP 0.97 million is borrowed or invested | GBP 0.97 million is borrowed or invested | ||
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The periodic discount rate (d) is: | The periodic discount rate (d) is: | ||
= (End - Start) / | = <math>\frac{(End - Start)}{End}</math> | ||
= (1.00 - 0.97) | = <math>\frac{(1.00 - 0.97)}{1.00}</math> | ||
= 0.030000 | = 0.030000 | ||
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==Example 3== | ====Example 3==== | ||
GBP 0.97 million is borrowed. | GBP 0.97 million is borrowed. | ||
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Calculate the amount repayable at the end of the period. | Calculate the amount repayable at the end of the period. | ||
'''''Solution''''' | |||
The periodic discount rate (d) is defined as: | The periodic discount rate (d) is defined as: | ||
d = (End - Start) / | d = <math>\frac{(End - Start)}{End}</math> | ||
d = | d = <math>\frac{End}{End}</math> - <math>\frac{Start}{End}</math> | ||
d = 1 - <math>\frac{Start}{End}</math> | |||
''Rearranging this relationship:'' | ''Rearranging this relationship:'' | ||
1 - d = <math>\frac{Start}{End}</math> | |||
Start | End = <math>\frac{Start}{(1-d)}</math> | ||
''Substituting the given information into this relationship:'' | |||
End = <math>\frac{0.97}{(1 - 0.030000}</math> | |||
= <math>\frac{0.97}{0.97}</math> | |||
= '''GBP 1.00m''' | = '''GBP 1.00m''' | ||
==Example 4== | ====Example 4==== | ||
An investment will pay out a single amount of GBP 1.00m at its final maturity after one period. | An investment will pay out a single amount of GBP 1.00m at its final maturity after one period. | ||
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Calculate the amount invested at the start of the period. | Calculate the amount invested at the start of the period. | ||
'''''Solution''''' | |||
As before, the periodic discount rate (d) is defined as: | As before, the periodic discount rate (d) is defined as: | ||
d = (End - Start) / | d = <math>\frac{(End - Start)}{End}</math> | ||
d = 1 - | d = 1 - <math>\frac{Start}{End}</math> | ||
''Rearranging this relationship:'' | ''Rearranging this relationship:'' | ||
<math>\frac{Start}{End}</math> = 1 - d | |||
Start = End x (1 - d) | Start = End x (1 - d) | ||
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''Substitute the given data into this relationship:'' | ''Substitute the given data into this relationship:'' | ||
Start = | Start = 1.00 x (1 - 0.030000) | ||
= '''GBP 0.97m''' | = '''GBP 0.97m''' | ||
Revision as of 14:13, 28 October 2015
Periodic discount rate is a cost of borrowing - or rate of return - expressed as:
- The excess of the amount at the end over the amount at the start
- Divided by the amount at the end
Example 1
GBP 1 million is borrowed.
GBP 1.03 million is repayable at the end of the period.
The periodic discount rate (d) is:
d = (End amount - start amount) / End amount
Which can also be expressed as:
d = (End - Start) / End
or
d = <math>\frac{(End - Start)}{End}</math>
= <math>\frac{(1.03 - 1)}{1.03}</math>
= 0.029126
= 2.9126%
Example 2
GBP 0.97 million is borrowed or invested
GBP 1.00 million is repayable at the end of the period.
The periodic discount rate (d) is:
= <math>\frac{(End - Start)}{End}</math>
= <math>\frac{(1.00 - 0.97)}{1.00}</math>
= 0.030000
= 3.0000%
Example 3
GBP 0.97 million is borrowed.
The periodic discount rate is 3.0000%.
Calculate the amount repayable at the end of the period.
Solution
The periodic discount rate (d) is defined as:
d = <math>\frac{(End - Start)}{End}</math>
d = <math>\frac{End}{End}</math> - <math>\frac{Start}{End}</math>
d = 1 - <math>\frac{Start}{End}</math>
Rearranging this relationship:
1 - d = <math>\frac{Start}{End}</math>
End = <math>\frac{Start}{(1-d)}</math>
Substituting the given information into this relationship:
End = <math>\frac{0.97}{(1 - 0.030000}</math>
= <math>\frac{0.97}{0.97}</math>
= GBP 1.00m
Example 4
An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.
The periodic discount rate is 3.0000%.
Calculate the amount invested at the start of the period.
Solution
As before, the periodic discount rate (d) is defined as:
d = <math>\frac{(End - Start)}{End}</math>
d = 1 - <math>\frac{Start}{End}</math>
Rearranging this relationship:
<math>\frac{Start}{End}</math> = 1 - d
Start = End x (1 - d)
Substitute the given data into this relationship:
Start = 1.00 x (1 - 0.030000)
= GBP 0.97m
