Let's assume that we are going to receive an amount x at a time t periods from now. t could be measured in any unit of time, and for some investment instruments days or half years is appropriate, but most commonly it is measured in years.

If we require a return of Y% then we can calculate what the value today is of receiving x at future time t. This is called the present value of x - the PV(x) - and can be thought of as today's fair price for buying or selling the right to receive x at time t.

The formula linking x and its present value is:

This is called the general discounting equation, because it discounts a future value to take into account the time value of money between now and the date of payment.

If we know the values of x, Y and t we can calculate PV(x). However, we can also use the equation in other ways.

If we know x, PV(x) and t, we can calculate Y. This is called the internal rate of return or IRR for short, because it is the interest rate implied by the price quoted today for a cashflow to be received at time t.

Of course bonds usually involve a series of promised cashflows set to arrive at known future dates. 

And to arrive at either the fair price for a bond, or to work out the rate of return it offers, the generalised DCF formula must be employed for this series of cashflows. 

All bond arithmetic is based on the generalised DCF formula for a series of cashflows. The formula is: