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.

It is often denoted by a lower case (r).

Example 1: Periodic yield (r) of 3%

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: Periodic yield of 3.09%

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: End amount from periodic yield

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: Start amount from periodic yield

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 (EAR)

The periodic yield (r) is related to the effective annual rate (EAR), and each can be calculated from the other.

Conversion formulae (r to EAR and EAR to r)

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

Periodic discount rate (d)

The periodic yield (r) is also related to the periodic discount rate (d), and each can be calculated from the other.

Conversion formulae (r to d and d to r)

d = r / (1 + r)

r = d / (1 - d)

Where:

d = periodic discount rate

r = periodic interest rate or yield

See also

• Effective annual rate
• Discount rate
• Nominal annual rate
• Nominal annual yield
• Periodic discount rate
• Yield
• Forward yield
• Zero coupon yield
• Par yield

Other resources

The Treasurer students, Simple solutions