Periodic yield
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
