Question 6.EX.4: Calculation of internal rates of return Clement plc is eva...

Project A
In the previous example we found that (all values in £000):

Where project cash inflows are identical, we can determine the cumulative present value factor for a period corresponding to the life of the project and a discount rate equal to the internal rate of return (r*). If we represent this by (CPVF_{r^*,7}), then from our above expression:

Rearranging:

From CPVF tables (see pages 482–3), looking along the row  corresponding to seven years, we find that the discount rate corresponding to this cumulative present value factor is approximately 12 per cent. Project A therefore has an internal rate of return of 12 per cent.
Project B
The cash flows of Project B are all different and so to find its IRR we must use linear interpolation. This technique relies on the fact that, if we know the location of any two points on a straight line, we can find any other point which also lies on that line. The procedure is to make an estimate (R_1) of the internal rate of return, giving a net present value of NPV_1. We then make a second estimate (R_2) of the internal rate of return: if NPV_1. was positive, R_2 should be higher than R1; if NPV_1. was negative, R_2 should be lower than R_1. We then calculate a second net present value, NPV_2, from R_2. The values of R1,R_2, NPV1 and NPV2 are then put into the following expression.

We calculated earlier that the NPV of Project B was £618,000 at a discount rate of 10 per cent. If we now increase the discount rate to 20 per cent, since 10 per cent was less than the internal rate or return, we can recalculate the NPV, as shown in Table 6.6. The earlier NPV calculation is included for comparison.
Interpolating, using the method discussed earlier:

So the internal rate of return of Project B is approximately 13.9 per cent.
We say ‘approximately’ since in using linear interpolation we have drawn a straight line between two points on a project NPV line that is in fact a curve. As shown in Figure 6.2, the straight line will not cut the x-axis at the same place as the project NPV curve, so the value we have obtained by interpolation is not the actual value of the IRR, but only an estimated value (and, for conventional projects, an overestimate). We would have obtained a different value if we had used a different  estimate for R_2; for example, if we had used R_1 = 10 per cent and R_2 = 15 per cent, we would have obtained a value for the IRR of 13.5 per cent. To determine the actual IRR the interpolation calculation must be repeated, feeding successive approximations back into the calculation until the value produced no longer changes significantly. A financial calculator or a computer spreadsheet can do this task easily.

Project C
The calculation of the NPV of Project C at Clement’s cost of capital of 10 per cent and a first estimate of the project IRR of 15 per cent is given in Table 6.7. Interpolating:

The internal rate of return of Project C is approximately 12.3 per cent.

The decision on project selection
We can now summarise our calculations on the three projects:

All three projects have an IRR greater than Clement’s cost of capital of 10 per cent, so all are acceptable if there is no restriction on capital. If the projects are mutually exclusive, however, it is not possible to choose the best project by using the internal rate of return method. Notice that, although the IRR of Project C is higher than that of Project A, its NPV is lower. This means that the projects are ranked differently using IRR than they are using NPV. The problem of mutually exclusive investment projects is discussed in Section 6.5.1.