## Bond pricing: The Internal Rate of Return with Irregular Cash Flows

##### Published: Oct 2002

Over the last two months we have shown, firstly, how to calculate the internal rate of return (IRR) where interest is paid annually and, secondly, how to calculate IRR when it is paid regularly, but more or less frequently than once a year. This month we deal with irregular cash flows.

As with our previous illustrations, to perform this calculation accurately, we need the use of a calculator with a time value of money function.

### Irregular cash flows

This method of calculation is not really suitable for a bond that pays irregular cash flows. However, it is possible to ‘regularise’ a number of irregular cash flows by inventing a sufficient number of zero interest payments. This will have the effect of translating the irregular cash flows into a series of regular cash flows, albeit one where a few are zero returns. When this means that the interest periods are not annual the final result needs to be adjusted using one of the two formulae outlined last month.

### Illustration

Consider the following example. An investor can purchase a bond for £250 now, which will give rise to cash flows of £10 after six months, £20 after one year, £30 after two years and £340 (£90 plus repayment of the principal) after five years. How can we calculate the internal rate of return?

First of all, we have to regularise the cash flows. We do this by assuming that there are regular cash flows every six months, although many of them are zero. In other words, we input zero returns for the third and fifth to ninth periods in the calculation process. This is shown below:

This can then be entered into the calculator.

##### HP12C

Using an HP12C calculator (which has a time value of money function), we need to press the following keys:

This gives the result 5.535%. Having regularised the calculation by using six monthly payments, this result is not an annual return.

Therefore we have to adjust this result to an annual return in the same way as we did last month:

$$1 \: + \: result^n -1 = annual \: IRR, \: where \: n \:=\: number \: of \: regular \: interest \: periods\: per \: year.$$
$$So, \:the \: annual \: IRR = 1 + 0.05535\:^2 -1 = 0.1138\: or \:11.38\%$$

This shows that it is possible to regularise a set of irregular cash flows. However this can be difficult if the periods are very irregular as these will result in a large number of zero returns, which are difficult to manage.

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