Periodic discount rate
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
Periodic yield
The periodic discount rate (d) is also related to the periodic yield (r), and each can be calculated from the other.
Conversion formulae (d to r and r to d):
r = d / (1 - d)
d = r / (1 + r)
Where:
r = periodic interest rate or yield
d = periodic discount rate
