Last month, we showed how to calculate and use the internal rate of return (IRR) where interest is paid annually. This allows different investments with different associated cash flows to be compared. This month we examine IRR where regular cash flows arise either more or less frequently than once a year.

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

### Last month’s illustration – annual cash flows on an investment

Last month, we used the example of an investor who had the opportunity to purchase a bond for £250 now, which would give rise to cash flows of £10 at the end of year 1, £20 at the end of year 2, £30 at the end of year 3, £40 at the end of year 4 and £300 (£50 plus repayment of principal) at the end of year 5. We calculated the internal rate of return as 11.16%.

### Zero interest

This illustration and the HP12C calculator both assume that some interest is paid annually. If an IRR is being calculated using the HP12C, when no interest would be earned, the zero cash flow should be entered (and not ignored).

### Different interest periods

This method of calculation will work for any regular series of cash flows, whether more or less frequent. But, because the calculator assumes that all payments are annual payments, any result needs to be annualised.

### More frequent payments

For example, let’s assume that our investor has the opportunity to purchase the same bond for £250 now but that it will give rise to cash flows every six months. These cash flows rise each year as in the first example, but the payments are received every six months rather than annually.

In year 1, these two cash flows will be £5 at the end of six months and at the end of year 1. In year 2 the payments will be £10 every six months, in year 3 £15, in year 4 £20 and in year 5 the investor will receive a payment of £25 after six months and £275 at the end of the year (£25 plus repayment of principal).

We can calculate the internal rate of return using the same process as last month, which gives a result of 5.569%. Obviously this is wrong. The calculator has assumed that all the payments were made annually and so has underestimated the IRR. We need to adjust the result to reflect the fact that interest is paid more frequently, every six months

To do so, we need to perform the following calculation:

^{n}

So, for semi-annual payments:

The annual IRR = 1 + 0.05569^{2} – 1 = 0.1145 or 11.45%

### Less frequent payments

On the other hand, let us assume that the investor will receive no interim interest payments whilst holding the £250 bond, but will receive £400 on redemption at the end of year 5 (£150 interest plus repayment of principal).

We can still calculate the internal rate of return in the same way with the calculator but this gives a result of 60%. This is clearly an overestimate which has arisen because of the calculator’s assumption of annual payments. It has assumed the repayment was at the end of year 1 not year 5. Again we need to adjust the result to reflect this.

To do so, we need to perform the following calculation:

^{1 /n }

So, for payments every five years (in this case, no interim payments):

The annual IRR = 1 + 0.6^{1 /5}