IRR is rarely easy to compute and in exceptional cases is not a real number. But management often favor it because profitability is expressed in simple percentage terms. Moreover, when a project is considered in isolation, IRR produces the same acceptreject decision as an NPV using a firm's cost of capital or rate of return as a discount rate (r). To prove the point, let:
I0 = Investment; PVr and PVIRR = future cash flows discounted at r and IRR respectively.
By combining the NPV and IRR Equations from Chapter Two, projects are acceptable if they generate a lifetime cash surplus i.e. a positive net terminal value (NTV) since:
A project is unacceptable and in deficit if its IRR (breakeven point) is less than r, since:
But what if a choice must be made between alternative projects (because of capital rationing or mutual exclusivity). Does the use of IRR, rather than NPV, rank projects differently? And if so, which model should management adopt to maximise shareholder wealth?
We have already observed that the difference between IRR and NPV maximisation hinges on their respective assumptions concerning borrowing and reinvestment rates. Moreover, the former model only represents a relative wealth measure expressed as a percentage, whereas NPV maximizes absolute wealth in cash terms.
So, let us explore their theoretical implications for wealth maximisation and then focus upon the realworld application of DCF analyses that must also incorporate relevant cash flows, taxation and price level changes.
3.1. Ranking and Acceptance Under IRR and NPV
You will recall from Chapter One (Fisher's Theorem and Agency theory) that if a project's returns exceed those that shareholders can earn on comparable investments elsewhere, management should accept it. DCF analyses confirm this proposition.
If a project's IRR exceeds its opportunity cost of capital rate, or the project's cash flows discounted at this rate produce a positive NPV, shareholder wealth is maximised.
However, where capital is rationed, or projects are mutually exclusive and a choice must be made between alternatives, IRR may rank projects differently to NPV. Consider the following IRR and NPV £000 (£k) calculations where the capital cost (r) of both projects is 10 per cent
Project
Year 0
Year 1
Year 2
Year 3
Year 4
Year 5
IRR(%)
NPV
1
(135)
10
40
70
80
50
20%
45.4
2
(100)
40
40
50
40

25%
34.3
Consider also, the effects of other discount rates on the NPV for each project graphed below.
Figure 3.1: IRR and NPV Comparisons
The table reveals that in isolation both projects are acceptable using NPV and IRR criteria. However, if a choice must be made between the two, Project 1 maximizes NPV, whereas Project 2 maximizes IRR. Note also that IRR favors this smaller, shortlived project.
Activity 1
Figure 3:1 reveals that at one extreme (the vertical axis) each project NPV is maximised when r equals zero, since cash flows are not discounted. At the other (the horizontal axis) IRR is maximised where r solves for a zero breakeven NPV. Thereafter, both projects underrecover because NPV is negative. But why do their NPV curves intersect?
Between the two extremes, different discount rates determine the slope of each NPV curve according to the size and timing of project cash flows. At relatively low rates, such as 10 per cent, the later but larger cash flows of Project 1 are more valuable. Higher discount rates erode this advantage. Project 2 is less affected because although it delivers smaller returns, they are earlier. At 15 per cent, the relative merits of each project (size and time) compensate to deliver the same NPV. So, we are indifferent between the two. Beyond this point, Project 2 is preferred. Its shallower curve intersects the horizontal axis (zero NPV) at a higher IRR.
Projects with different cash patterns produce NPV curves with different slopes and indifference points (intersections). Thus, IRR and NPV maximisation rarely coincide when a choice is required. IRR is an average percentage break even condition that favors speedy returns. Unlike NPV, which maximises absolute wealth, IRR also fails to discriminate between projects of different size.
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