## Bond pricing – The Internal Rate of Return

##### Published: Jul 2002

Last month, we showed how to translate a future value into a net present value (and vice versa), assuming a prevailing interest rate. This month we look at another way to calculate future value, the internal rate of return.

When contemplating an investment, we may be asked to pay a certain sum on the understanding that we will receive certain cash flows in the future. In other words, we know that we will have to pay a present value to receive certain future values. Before deciding whether to make the investment as described, we will want to compare it with alternatives.

One way to do that is to calculate the internal rate of return, the rate of return generated by the investment regardless of prevailing interest rates. This is the interest rate that translates the future cash flows into a net present value such that the sum of all the discounted future values is equal to the amount we propose to invest. We can then compare the internal rates of return of alternative possible investments, knowing that we are comparing ‘like with like’. This is particularly useful when investments yield different cash flows or the same or different cash flows at different intervals.

To perform this calculation accurately, we need the use of a calculator with a time value of money function. The alternative is to use iteration (trial and error) for the series of cash flows. This will be very time-consuming, particularly if they are irregular.

### Illustration

Assume that an investor can purchase a bond now for £250 which will 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 £50 plus repayment of principal (total cash flow of £300) at the end of year 5. What is the internal rate of return?

##### HP12C

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

• 250 CHS g CFo
• 10 g CFj
• 20 g CFj
• 30 g CFj
• 40 g CFj
• 300 g CFj
• f IRR
• = 11.16

This is the constant interest rate at which all five future cash flows are discounted to a present value equal to the initial £250 investment.

This can be checked using the following formula:

$$PV = {FV} {1 \: + \: i^n}\:\:$$ , where FV is the future cash flow, i is the constant interest rate and n is the number of years to the cash flow. In this case:

$$PV = {10}{1\: + \: 0.1116\:^1} \: + \: \frac{20} {1\: + \: 0.1116\:^2} \: + \: \frac{30} {1\: + \: 0.1116\:^3} \: + \: \frac{40} {1\: + \: 0.1116\:^4} \: + \: \frac{300} {1\: + \: 0.1116\:^5}= 250$$

This illustration has assumed the existence of annual cash flows. We will examine IRR where cash flows arise more or less frequently next month.

### What does this mean?

This means that an investment in the bond with the associated cash flows illustrated above is equivalent to any other investment with an internal rate of return of 11.16%. For example, it is equivalent to an investment where £250 is invested at a rate of 11.16% for five years with annual interest payments (which would result in a cash flow of £424.31 at the end of the five year period). It does assume that the four annual cash flows are reinvested at the same rate.

### The use of internal rate of return

Different investments (whether bonds or capital projects) with different associated cash flows can be compared using the internal rate of return. However it is important to remember that the calculation we have illustrated above assumes constant reinvestment rates. The timing and size of cash flows arising from investments should always be considered.

Unfortunately the Windows calculator is unable to make this calculation as it does not have this functionality but there is an IRR function in Excel and it is possible to perform these calculations using numerous resources available on the Internet. A search at any of the proprietary search sites will give you lots of options.