# 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).

## Calculating periodic yield from start and end cash

Given the cash amounts at the start and end of an investment or borrowing period, we can calculate the periodic yield.

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

= (1.03 / 1.00) - 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.

As before, the periodic yield (r) is:

r = (End / Start) - 1

= (1.00 / 0.97) - 1

= 0.030928

= 3.0928%

Check:

Amount at end = 0.97 x 1.030928 = GBP 1.00m, as expected.

## Calculating end cash from periodic yield

We can also work this relationship in the other direction.

Given the cash amount at the start of an investment or borrowing period, together with the periodic yield, we can calculate the end amount.

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

As before, the periodic yield (r) is:

r = (End / Start) - 1

Rearranging this relationship:

1 + r = (End / Start)

End = Start x (1 + r)

Substituting the given information into this relationship:

End = GBP 0.97m x (1 + 0.030928)

= GBP 1.00m

## Calculating start cash from periodic yield

We can also work the same relationship reversing the direction of time travel.

Given the cash amount at the end of an investment or borrowing period, again together with the periodic yield, we can calculate the start amount.

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:

r = (End / Start) - 1

Rearranging this relationship:

1 + r = (End / Start)

Start = End / (1 + r)

Substitute the given data into this relationship:

Start = End / (1 + 0.030928)

= GBP 0.97m

Check:

Amount at end = 0.97 x 1.030928 = GBP 1.00m, 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