Performance

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

= (1.03 - 1) / 1.03

= 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:

(End amount - start amount) / End amount

= (1.00 - 0.97) / 1.00

= 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 = (End amount - start amount) / End amount

d = 1 - (Start amount / End amount)

Rearranging this relationship:

(Start amount / End amount) = 1 - d

Start amount = End amount x (1 - d)

Start amount / (1 - d) = End amount

End amount = Start amount / (1 - d)

Substituting the given information into this relationship:

End amount = GBP 0.97m / (1 - 0.030000)

= GBP 0.97m / 0.97

= 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 = (End amount - start amount) / End amount

d = 1 - (Start amount / End amount)

Rearranging this relationship:

(Start amount / End amount) = 1 - d

Start amount = End amount x (1 - d)

Substitute the given data into this relationship:

Start amount = GBP 1.00m x (1 - 0.030000)

= GBP 0.97m

See also

• Effective annual rate
• Discount rate
• Nominal annual rate
• Periodic yield
• Yield