Description of the method
The internal rate of return (IRR) method, considered next, is largely analogous to the NPV method. Only two assumptions are modified - concerning the reinvestment of free cash flow surpluses and the balancing of capital tie-up and economic life differences. Also, a different target measure is considered: the internal rate of return.
Key Concept:
The internal rate of return is the rate that leads to a NPV of zero when applied as the uniform discount rate.
The internal rate of return represents the interest earned on the capital employed at specific points in time by the investment project under consideration. The following profitability criteria are applied in the IRR method, although it should be recognised at this point that the method is not applicable to all decision contexts:
Key Concept:
Absolute profitability is achieved if an investment project's internal rate of return is higher than the uniform discount rate.
Relative profitability: An investment project is preferred if it has a higher internal rate of return than the alternative investment project(s).
Accordingly, in assessing absolute profitability a comparison is made between the project's rate of interest and the cost of capital or the interest earned by an alternative financial investment, as represented by the uniform discount rate i. An investment project should be undertaken when its rate of interest is higher than the cost of capital and/or the interest that could be earned on alternative financial investments.
The meaningfulness of internal rates of return (and profitability assessments based on them) compared with that of the NPV or annuity methods depends on the cash flow profiles and, thus, on the type of investment. The following discussion concentrates on isolated investment projects. Such projects are characterised by the fact that the cash flow surpluses of the whole planning period only cover interest charges (at the internal rate of return) and repayment of the capital employed. That is, no reinvestment is made using the project's cash flow surpluses during the project's economic life. The investment is said to be 'isolated', and the internal rate of return is then independent of the interest rates that possible reinvestments might earn.
An isolated investment project has a project-specific cash flow balance that is negative for every period of its economic life if it is determined on the basis of the internal rate of return. This condition is fulfilled if:
• The sum of all net cash flows in the economic life is higher than or equal to zero:
And:
• For all periods t = 0,...,t*, with t* signifying the period in which the last cash outflow surplus appears, the cumulative net cash flows are smaller than or equal to zero:
These conditions are in any case fulfilled with a 'normal' investment, i.e. an investment with a cash flow profile, where, only once, does a change in algebraic sign take place (one or more initial investment outlays in the beginning followed by cash inflows only in the subsequent periods). Thus, the normal investment is a special case of an isolated investment.
Figure 3-3 depicts the NPV of two isolated investment projects A (IPa) and B (IPB) as a function of the uniform discount rate. With this type of investment (isolated investments), the results of the NPV and IRR methods are the same in regard to absolute profitability because the NPV is always positive when the IRR is greater than the uniform discount rate. The profitability comparison of mutually exclusive projects, however, can lead to contradictory results from the NPV and IRR methods, as also shown in Figure 3-3. In this case, the NPV method assesses project A as superior, since NPVa > NPVb (at the uniform discount rate i), whereas the IRR method favours project B, since rB > rA. The question of how to choose between mutually exclusive investments in such cases is taken up below.
From Figure 3-3 it can be deduced that, for an isolated investment project, only one positive internal rate of return exists, provided the sum of the cash inflows is greater than the sum of the outflows. For non-isolated (i.e. 'linked') investment projects, several internal rates of return may exist. The maximum number of these interest rates corresponds to the number of algebraic sign changes in accordance with Descartes' convention. It is also possible that no economically relevant internal rate of return exists.
Moreover, in the case of a linked investment project, a reinvestment assumption must be made. In contrast to the NPV model, the IRR method implicitly assumes that cash flows surpluses generated from a project (after covering interest charges and the repayment of the capital employed) can be used to make a financial investment that earns the internal rate of return. In general, this assumption is unrealistic and, thus, becomes a major disadvantage of the method. The application of the IRR method is therefore not meaningful for non-isolated investments - at least not without considering a specific supplementary investment or modifying the reinvestment assumption.
The internal rate of return (r) was defined above as the interest rate at which the NPV becomes zero. Therefore: T
NPV = X (CIFt - COFt) • (1 + r)"1 = 0 (3.16)
Only in special cases can the IRR be isolated from the above formula and determined accurately. This is possible with an acceptable effort when the time span encloses only one or two periods or, in the case of a longer time span, if only two cash flows take place or all the future cash flows are identical.
For projects spanning three or more periods, where none of the other special cases apply, approximation procedures should be applied to the determination of the IRR. Such approaches include the Newton's procedure and a method developed by Boulding (1936). Spreadsheet software usually also has programme functions that can be used to determine the IRR for multi-period projects.
In the following discussion an interpolation or extrapolation procedure is described for approximating the IRR of isolated investment projects.
For this procedure the net present value (NPV1) is determined for a discount rate (ij). If this is positive (negative), then a higher (lower) discount rate (i2) is selected and the net present value is re-calculated (NPV2). Then the two discount rates and their associated NPVs can be used to approximate the project's IRR. A linear interpolation (where a positive and a negative NPV have been determined) or an extrapolation (if both NPVs show the same algebraic sign) is executed. A formula supporting both the described interpolation and extrapolation can be derived with the help of the following figure.
Net present value A
npv1-npv2
npv1-npv2
12 Discount rate
Fig. 3-4: Interpolation to determine the internal rate of return
12 Discount rate
Fig. 3-4: Interpolation to determine the internal rate of return
In Figure 3-4 the IRR is represented by the point at which the NPV function and the abscissa intersect. This point can be approximated by drawing a straight line between the two determined NPV points (1 and 2 in Figure 3-4) and finding its intersection with the abscissa (r*). The interpolation formula is derived based on the theorem on intersecting lines.
NPV1 NPV1 - NPV2
NPV,
NPV1 - NPV2 NPV
NPV1 - NPV2
The exactness of the approximation depends on how big the differences are between the discount or interest rates i1 and i2, and between their associated net present values NPV1 and NPV2. This determines the deviation of the approximating straight line from the course of the true NPV function. For a 'good' approximation it might be necessary to use several discount rates and to calculate their associated NPVs in order to identify two discount rates suitable for the interpolation. Two discount rates are suitable when their difference is relatively small, and the range of their associated NPVs around zero is also small. It may be helpful to apply an r approximated IRR solution as a uniform discount rate and then use it for further interpolations. Thus, with an appropriate number of iterations, a reasonably accurate result can be reached for the IRR.
Example 3-3
For this example, the data from Example 3-1 is used again and the interpolation procedure is applied first to alternative A.
The NPV of project A with a uniform discount rate ii = 8% is already known (NPVai = €26,771.59). For a second chosen uniform discount rate = 18% the NPVA2 can be re-calculated as -€1,619.51.
Because the difference between i1 and i2 is relatively large and, thus, the exactness of the approximation would be poor, the NPV is re-calculated for another interest rate i1* = 17% to give an NPVa1* = €740.69. Now, the differences between the two discount rates and between the resulting NPVs are quite small.
Using the interest rates i1* (17% or 0.17) and i2 (18% or 0.18) together with their associated NPVs, the following interpolation result is obtained:
The IRR for project A is approximately 17.31%. Because this lies above the uniform discount rate of 8%, the project is profitable in absolute terms.
The fact that the IRR represents the interest on an investment project across different periods, based on the capital tie-up (remaining loan) in each period, can be illustrated by means of a finance and redemption plan. Such a plan is presented in the following table, which assumes the use of full debt financing at an interest rate of 17.31%.
Point in time t |
Net cash flow (€) (without initial investment outlay) NCFt |
It (= i • Vt-1) |
Change in capital tie-up (€) (redemption) AVt(= NCFt + It) |
Vt (= Vt-1 + A Vt) |
0 |
0 |
0.00 |
0.00 |
-100,000.00 |
1 |
28,000 |
-17,310.00 |
10,690.00 |
-89,310.00 |
2 |
30,000 |
-15,459.56 |
14,540.44 |
-74,769.56 |
3 |
35,000 |
-12,942.61 |
22,057.39 |
-52,712.17 |
4 |
32,000 |
-9,124.48 |
22,875.52 |
-29,836.65 |
5 |
35,000 |
-5,164.72 |
29,835.28 |
-1.37 |
Tab. 3-4: Financial budget and redemption plan (IRR method)
Tab. 3-4: Financial budget and redemption plan (IRR method)
At each period's end, the cash flow surpluses are used first to pay the interest on the capital tie-up at the beginning of the period. The remaining amount is available for the immediate redemption of capital. The net cash flows from the investment project enable the exact recovery of the capital and the payment of interest on capital employed at the internal rate of return (small rounding errors can be ignored). Correspondingly, the IRR equals the interests earned on the capital employed; this rate can also be interpreted as an upper limit for the cost of capital for the investment project. Since the amount of capital employed depends on the IRR, both must be determined simultaneously.
The results for project B are approximated in the same way. For discount rates of 25% and 26% the approximation of the internal rate of return is: 25.04%. Thus investment project B is also assessed as absolutely profitable.
However, in contrast to the NPV method, project B now appears to be relatively profitable owing to its higher IRR. This is because the IRR method uses the rate of interest earned on the capital that is tied-up as the target measure. Project B starts with a considerably lower initial investment outlay and, thus, its relative profitability increases. Absolutely profitable projects with lower capital tie-up often win in an IRR comparison of relative profitability, in contrast to the same comparison made using the NPV method.
Where competing projects differ in their capital tie-up or their economic life, the IRR method is of dubious suitability for assessing relative profitability. As with the NPV model, an implicit assumption is made in balancing such differences. It is assumed that a financial investment is made (or, alternatively, a corresponding financing alternative is realised), which yields an interest rate equal to the IRR of the investment project with the shorter economic life, or the smaller amount of capital tie-up. Among other things, this implies that financial investments yielding the IRR can be made without limit. This assumption is often unrealistic, because the IRR reflects the cash flows of the investment under consideration and not the opportunities in the capital market. Moreover, if in a comparison of two investment projects the capital tie-up is higher initially for one project and later, during the course of the economic life, for the other project, then it has to be assumed that the interest rate used to balance the capital tie-up changes with the IRRs. Finally, in any comparison of three or more different projects made by comparing two at a time, the inconsistency will be obvious as, in that situation, different rates will be used in the same decision process. In summary, the IRR method is not suitable for the assessment of (relative) profitability amongst two or more projects, since the assumption made for balancing capital tie-up and economic life differences at this very rate is unrealistic.
However, the IRR method can be used for the assessment of relative profitability when the IRRs are not taken from the investment projects themselves, but from differential investments (described in Section 3.2). An assumption in this case is that the differential investment is an isolated investment project.
The IRR of this differential investment corresponds to the discount rate that, used as a uniform discount rate, equalises the NPVs of the two projects under consideration (intersection IS of the NPV functions of projects A and B in Figure 3-4). This rate rD is always higher than the uniform discount rate (i) if the project with the higher initial investment outlay has the higher NPV (project A in Figure 3-4). Accordingly, the IRR leads to assessments of relative profitability identical to the NPV method if the following rule is applied to the comparison of two investment projects A and B with A having the higher initial investment outlay:
Key Concept:
An investment project A is relatively profitable compared to a project B if the IRR of the differential investment is higher than the uniform discount rate.
If the differential investment does not have a positive IRR, investment project B is relatively profitable for all positive uniform discount rates.
For the assessment of the IRR method the reader is referred to the corresponding statements about the NPV method. Both methods require the same data. The calculation of an approximated IRR is slightly more difficult than the NPV calculation but, as shown above, these approximations lead to satisfactory results.
The underlying assumptions are largely the same for both models. The only exceptions are the assumptions regarding the financial investments of free cash flow surpluses (i.e. the reinvestment assumption) and those pertaining to how capital tie-up and economic life differences are balanced when comparing projects. The IRR model assumes that free surplus cash flows yielded by an investment project can be reinvested to earn exactly the IRR rate, which is - in general - unlikely.
Therefore, the IRR method should not be used to assess the absolute profitability of linked (non-isolated) investment projects unless (i) explicit consideration is given to supplementary investments, or (ii) the reinvestment assumption is modified. As an additional problem, it is possible that several economically meaningful IRRs exist for non-isolated projects. This makes it difficult to interpret the results of the IRR analysis.
Similarly, assessments of relative profitability should not be made by comparing project IRRs, because of the unrealistic balancing assumption noted above. Instead, differential investments should be analysed. If they are isolated investment projects, their IRRs may be compared to the uniform discount rate to achieve a meaningful profitability comparison. If a differential investment is a linked (non-isolated) investment project, then using the IRR method does not make sense.
The applicability of the IRR method is not limited to the assessment of investment projects. The method is also suitable for calculating the effective interest rate of financial projects like loans. The IRR of the cash flow profile of a financial project indicates its effective rate of interest. In this case, the application of the IRR method for the comparison of projects is less problematic. The projects under consideration are typically similar in amount, in the structure of their interest and redemption payments, and in their general terms; balancing differences is seldom required. Thus, assessments of relative profitability will regularly not deviate from those of the net present value method.
As with the annuity method, the IRR method shows some advantages over the NPV method when it comes to interpreting the results. A project's IRR can be seen as representing the interest earned on the capital employed - an intuitive approach that makes the IRR method popular in company practice.
Additionally, another relationship between the NPV and IRR methods can be pointed out. The IRR is the uniform discount rate at which a project's NPV equals zero. If the NPV method is used and the 'certainty of data' assumption is discarded, the IRR can be interpreted as a critical rate of return which must not be surpassed by the real cost of capital if the investment project is to remain favourable.
Was this article helpful?
Do you have annuity you dont want? Discover When is it Time to Sell Your Annuity? What can I do? Where can I get the money I need? I have an annuity, but I dont know that I can sell it. Is there a good time to sell my annuity? I already have a home improvement loan, but it was used before the roof needed replacing.