Derivative and Periodic yield: Difference between pages
imported>Doug Williamson (Remove surplus link.) |
imported>Doug Williamson (Expand to incorporate conversion formulae.) |
||
Line 1: | Line 1: | ||
__NOTOC__ | |||
Periodic yield is a rate of return - or cost of borrowing - expressed as the proportion by which the amount at the end of the period exceeds the amount at the start. | |||
====Example 1==== | |||
== See also == | GBP 1 million is borrowed or invested. | ||
* [[ | |||
* [[ | GBP 1.03 million is repayable at the end of the period. | ||
* [[ | |||
The periodic yield (r) is: | |||
r = (End amount / start amount) - 1 | |||
Which can also be expressed as: | |||
r = (End / Start) - 1 | |||
''or'' | |||
r = <math>\frac{End}{Start}</math> - 1 | |||
= <math>\frac{1.03}{1}</math> - 1 | |||
= 0.03 | |||
= '''3%''' | |||
====Example 2==== | |||
GBP 0.97 million is borrowed or invested. | |||
GBP 1.00 million is repayable at the end of the period. | |||
The periodic yield (r) is: | |||
r = <math>\frac{End}{Start}</math> - 1 | |||
= <math>\frac{1.00}{0.97}</math> - 1 | |||
= 0.030928 | |||
= '''3.0928%''' | |||
''Check:'' | |||
Amount at end = 0.97 x 1.030928 = 1.00, as expected. | |||
====Example 3==== | |||
GBP 0.97 million is invested. | |||
The periodic yield is 3.0928%. | |||
Calculate the amount repayable at the end of the period. | |||
'''''Solution''''' | |||
The periodic yield (r) is defined as: | |||
r = <math>\frac{End}{Start}</math> - 1 | |||
''Rearranging this relationship:'' | |||
1 + r = <math>\frac{End}{Start}</math> | |||
End = Start x (1 + r) | |||
''Substituting the given information into this relationship:'' | |||
End = GBP 0.97m x (1 + 0.030928) | |||
= '''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 yield is 3.0928%. | |||
Calculate the amount invested at the start of the period. | |||
'''''Solution''''' | |||
As before, the periodic yield (r) is defined as: | |||
r = <math>\frac{End}{Start}</math> - 1 | |||
''Rearranging this relationship:'' | |||
1 + r = <math>\frac{End}{Start}</math> | |||
Start = <math>\frac{End}{(1 + r)}</math> | |||
''Substitute the given data into this relationship:'' | |||
Start = <math>\frac{1.00}{(1 + 0.030928)}</math> | |||
= '''GBP 0.97m''' | |||
''Check:'' | |||
Amount at start = 0.97 x 1.030928 = 1.00, as expected. | |||
====Effective annual rate==== | |||
The periodic yield (r) is related to the [[effective annual rate]] (EAR), and each can be calculated from the other. | |||
'''Conversion formulae''' | |||
EAR = (1 + r)<sup>n</sup> - 1 | |||
r= (1 + EAR)<sup>(1/n)</sup> - 1 | |||
Where: | |||
EAR = effective annual rate or yield | |||
r = periodic interest rate or yield, as before | |||
n = number of times the period fits into a calendar year | |||
==See also== | |||
*[[Effective annual rate]] | |||
*[[Discount rate]] | |||
*[[Nominal annual rate]] | |||
*[[Periodic discount rate]] | |||
*[[Yield]] |
Revision as of 09:51, 1 November 2015
Periodic yield is a rate of return - or cost of borrowing - expressed as the proportion by which the amount at the end of the period exceeds the amount at the start.
Example 1
GBP 1 million is borrowed or invested.
GBP 1.03 million is repayable at the end of the period.
The periodic yield (r) is:
r = (End amount / start amount) - 1
Which can also be expressed as:
r = (End / Start) - 1
or
r = <math>\frac{End}{Start}</math> - 1
= <math>\frac{1.03}{1}</math> - 1
= 0.03
= 3%
Example 2
GBP 0.97 million is borrowed or invested.
GBP 1.00 million is repayable at the end of the period.
The periodic yield (r) is:
r = <math>\frac{End}{Start}</math> - 1
= <math>\frac{1.00}{0.97}</math> - 1
= 0.030928
= 3.0928%
Check:
Amount at end = 0.97 x 1.030928 = 1.00, as expected.
Example 3
GBP 0.97 million is invested.
The periodic yield is 3.0928%.
Calculate the amount repayable at the end of the period.
Solution
The periodic yield (r) is defined as:
r = <math>\frac{End}{Start}</math> - 1
Rearranging this relationship:
1 + r = <math>\frac{End}{Start}</math>
End = Start x (1 + r)
Substituting the given information into this relationship:
End = GBP 0.97m x (1 + 0.030928)
= 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 yield is 3.0928%.
Calculate the amount invested at the start of the period.
Solution
As before, the periodic yield (r) is defined as:
r = <math>\frac{End}{Start}</math> - 1
Rearranging this relationship:
1 + r = <math>\frac{End}{Start}</math>
Start = <math>\frac{End}{(1 + r)}</math>
Substitute the given data into this relationship:
Start = <math>\frac{1.00}{(1 + 0.030928)}</math>
= GBP 0.97m
Check:
Amount at start = 0.97 x 1.030928 = 1.00, as expected.
Effective annual rate
The periodic yield (r) is related to the effective annual rate (EAR), and each can be calculated from the other.
Conversion formulae
EAR = (1 + r)n - 1
r= (1 + EAR)(1/n) - 1
Where:
EAR = effective annual rate or yield
r = periodic interest rate or yield, as before
n = number of times the period fits into a calendar year