The Internal Rate of Return (IRR)
There are two basic DCF models that compare the PV of future project cash inflows and outflows to an initial investment.Net present value (NPV) incorporates a discount rate (r) using a company's rate of return, or cost of capital, which reduces future net cash inflows (Ct) to a PV to determine whether it is greater or less than the initial investment (I0). Internal rate of return (IRR) solves for a rate, (r) which reduces future sums to a PV equal to an investment's cost (I0), such that NPV equals zero. Mathematically, given:
The IRR is a special case of NPV, namely a hypothetical return or maximum rate of interest required to finance a project if it is to break even. It is then compared by management to a predetermined cut-off rate. Individual projects are accepted if:
IRR > a target rate of return: IRR > the cost of capital or a rate of interest.
Collectively, projects that satisfy these criteria can also be ranked according to their IRR. So, if our objective is IRR maximisation and only one alternative can be chosen, then given:
Activity 6
A project costs £172,720 today with cash inflows of zero in Year 1, £150,000 in Year 2 and £64,900 in Year 3.Assuming an 8 per cent cut-off rate, is the project's IRR acceptable?
Using Equation (8) or DCF tables, the following figures confirm a break-even IRR of 10 per cent (NPV = 0). So, the project's return exceeds 8 per cent (i.e. NPV is positive at 8 per cent) more of which later.
Year |
Cash flows |
DCF Factor (10%) |
PV |
0 |
(172.72) |
1.0000 |
(172.72) |
1 |
- |
0.9091 |
- |
2 |
150.00 |
0.8264 |
123.96 |
3 |
64.90 |
0.7513 |
48.76 |
NPV |
Nil |
Unsure about IRR or NPV? Remember NPV is today's equivalent of the cash surplus at the end of a project's life. This surplus is the project's net terminal value (NTV). Thus, if project cash flows have been discounted at their IRR to produce a zero NPV, it follows that their NTV (cash surplus) built up from compound interest calculations will also be zero. Explained simply, you are indifferent to £10 today and £11 next year with a 10 per cent interest rate.
The Inadequacies of IRR and the Case for NPV
IRR is supported because return percentages are still universally favored performance metrics. Moreover, computational difficulties (uneven cash flows, the IRR is indeterminate, or not a real number) can now be resolved mathematically by commercial software. Unfortunately, these selling points overstate the case for IRR.
IRR (like ARR) is a percentage averaging technique that fails to discriminate between project cash flows of different timing and size, which may conflict with wealth maximisation in absolute cash terms. Unrealistically, the model also assumes that even if cash data is certain:
- All financing will be undertaken at a borrowing rate equal to the project's IRR.
- Intermediate net cash inflows will be reinvested at a rate of return equal to the IRR.
The implication is that capital cost and reinvestment rates equal the IRR, which remains constant over the project's life to produce a zero NPV. However, relax one or other assumption and IRR changes. So, why calculate a hypothetical IRR, which differs from real world cut-off rates that can be incorporated into the DCF model to determine whether a project's actual NPV or NTV is positive or negative?
The IRR is a "castle built on sand" without economic meaning unless we compare it to a company's desired rate of return or capital cost. Far better to discount project cash flows using one of these rates to establish a true economic surplus in absolute money terms as follows:
Individual projects are accepted if:
Collectively, projects that satisfy either criterion can also be ranked according to their NPV.
Of course, NPV, like IRR, still requires certain assumptions. Known investment costs, project lives, cash flows and whatever discount rate, must all be factored into the NPV model. But note this is more realistic. Capital cost and intermediate reinvestment rates now relate to prevailing returns, rather than IRR, so there are fewer margins for error. NPV is near the truth by representing the possible money surplus (NTV) you will eventually walk away with.
Review Activity
Using data from Activity 6 with its 8 per-cent cut-off rate and Equations 10-11, confirm that the project's NPV is £7,050 and acceptable to management because the life-time surplus equals an NTV of £8,881.
Summary and Conclusions
We can tabulate the objective functions and investment criteria of PB, ARR, IRR and NPV with respect to shareholder wealth maximisation as follows:
Model Payback |
Wealth Max. Rarely |
Objective Minimise Payback (Maximise liquidity) |
Investment Criteria Time |
ARR |
Rarely |
Maximise ARR |
Profitability percentage |
IRR |
Rarely |
Maximise IRR |
Profitability percentage |
NPV |
Likely |
Maximise NPV |
Absolute profits |
Capital Budgeting Models