EOY 1 EOY 2 EOY 3 This layout may also be written as:

Y 0 Y 1 Y 2 Y 3 . . .

Capital outlays which occur in any year after year 0 are subtracted from the relevant end- of-year cash flows. If a future capital outlay occurs at the immediate start of any year, it is usually included with the previous end-of-year cash flows to ensure it is correctly timed for discounting. In Project Alpha above, for example, we might have an upgrade outlay of $120 planned for the start of year 3. The $120 would be subtracted from the end-of-year flows for year 2. The cash flow table would be:

EOY 0 EOY 1 EOY 2 EOY 3

−900 300 280 600

The decision criterion in an NPV analysis is the change in a firm’s current wealth. Cash flowing into the firm increases wealth; cash flowing out of the firm decreases wealth.

Table 6.1. Delta Project: annual net cash flow

EOY 0 EOY 1 EOY 2 EOY 3 EOY 4 EOY 5 EOY 6 EOY 7 EOY 8

−1,002,000 195,510 200,087 −295,135 379,643 384,421 608,760 618,061 638,362

This rule holds for both capital and operational flows. In defining the cash flows, then, it is important to apply the correct sign, plus or minus, and to make sure the correct sign is employed in any computerized calculation. Some computer packages have implied cash flow signs within NPV formulae, but some do not.

The amount of an NPV represents the addition to a firm’s value. Rational management will accept all projects with a positive NPV and reject all those with a negative NPV. The assumption of rationality implies that projects with only $1 of positive NPV will be accepted along with projects of $1,000,000 of positive NPV. The assumption here is that capital for investment is freely available, and that projects do not have to be ranked in order of relative benefit to the firm. In conjunction with the assumption of perfect markets, this assumption will always be tenable. The ‘market’ will recognize the firm’s beneficial project and will provide the necessary capital to the firm.

Under the assumption of certainty, a project with an NPV of $1 is perfectly acceptable. Without certainty, this small NPV may not provide an adequate buffer against errors in forecasts.

The net present value model applied

In Chapter 2, the Delta Project example was introduced to illustrate the project cash flow analysis, with Table 2.2 showing the calculation of cash flows for the project. The essential final row of data from that table, the annual net cash flow, is reproduced here as Table 6.1.

The NPV of this project can be computed by evaluating the cash flows. They begin with the initial outlay of $1,002,000 at EOY 0 and continue until EOY 8, at which time the asset salvage value is included. The NPV computation is:

NPV= −1,002,000+195,510 (1.05)1 + 200,087 (1.05)2 + −295,135 (1.05)3 + 379,643 (1.05)4 + 384,421 (1.05)5 + 608,760 (1.05)6 + 618,061 (1.05)7 + 638,362 (1.05)8 =$1,049,852

With this positive NPV, the project is acceptable.

We have used the risk-free rate of 5% per annum as the required rate of return here, as the cash flows are assumed to be certain.

Workbook 6.1

This calculation is also shown in the Excel workbook 6.1, where the NPV function is applied.

Other project appraisal methods

While the NPV criterion is the most appropriate method in most cases, the other discounted cash flow technique, the internal rate of return (IRR), is also frequently used, sometimes as a supplementary measure to NPV. Non-discounted cash flow methods such as payback period (PP) and accounting rate of return (ARR) have a number of serious defects but are still being used in practice in some situations. Sometimes PP is used in conjunction with NPV, particularly in making risky investment decisions. It is useful to understand these methods and their drawbacks so that the most appropriate method can be used for investment evaluation.

Internal rate of return

The IRR is the rate of discount which returns an NPV of zero. The IRR can thus be defined as the highest rate at which the future cash flows can be discounted making the project’s NPV equal to zero. Since the IRR is a rate of return, the decision rule for project acceptance is: accept the project if its IRR is higher than the required rate of return.

Since the IRR cannot be expressed in terms of a solvable mathematical formula for projects, the economic life of which extends over a number of years, this rate is normally calculated by a trial-and-error process using a computer package. Using i to represent the IRR, the equation for the IRR calculation using the same cash flows as employed in the NPV calculation (Table 6.1) is:

$0 (as NPV )= −1,002,000+195,510 (1+i )1 + 200,087 (1+i )2 + −295,135 (1+i )3 + 370,643 (1+i )4 + 384,421 (1+i )5 + 608,760 (1+i )6 + 618,061 (1+i )7 + 638,362 (1+i )8 Workbook 6.1

The solution for i,calculated by the Excel IRR function, is 19.80%. Since this rate is above the required rate of 5%, the project is acceptable.

Accounting rate of return

The accounting rate of return uses accounting income data to calculate a ratio which is used as a decision variable. It is worth noting that, as discussed in Chapter 2, accounting income is different from cash flow. The ARR ratio is the annual average accounting income divided by an asset value. For example, an outlay of $1,000 may earn $200, $500 and $700

as accounting income over three years. The ARR is calculated as: ARR=[(200+500+700)/3]/1000

=46.66% Workbook

6.1

For the Delta Project example in Chapter 2, the accounting rate of return can be calcu- lated by dividing the average net income by the initial outlay of $1,002,000. The aver- age net income ($214,564) is obtained by summing the net income for the eight years (71,010+ · · · +399,162) and dividing the sum (1,716,509) by eight. The ARR for the Delta Project is approximately 21%. If this rate is higher than a pre-determined required rate, then the project is acceptable.

There is no standard way to calculate the ARR, and this makes the definition of the ratio ambiguous. This drawback of the ARR definition is discussed later in the chapter. Payback period

Payback period is the period of time over which the accumulated cash flows will equal the initial outlay. For example, an outlay of $1,200 may generate cash inflows of $820, $450 and $300 over three years. The total cash inflow to the end of year 2 is $1,270, so the payback period would be within two years. There is no objective time criterion associated with payback, but a period of two to three years would be generally acceptable.

Workbook 6.1

In the Delta Project example, the operating cash flows accumulate progressively to $864,526 at EOY 5 and $1,473,825 at EOY 6. As the initial outlay is $1,002,000, the payback will occur within year 6.

Suitability of different project evaluation techniques

The previous section illustrated four criteria for investment decision-making. This section briefly discusses their relative merits and demerits, and the discussion leads to the conclu- sion that NPV is the most suitable criterion; it can be applied in all cases and overcomes the problems of other criteria. This does not necessarily mean that the other criteria are completely irrelevant and useless. They may be used as supplementary measures to NPV to facilitate the relevant decision-making.

Net present value

The net present value model is the only decision technique which links the goal of the firm to the calculated output. The calculated NPV is the actual dollar amount by which the firm’s

current wealth will increase if the project is undertaken. Its calculation accounts for the time value of money at the required rate of return, and uses this as a data input, rather than as a decision output. The weaknesses and problems of the other three criteria, as discussed in the following three sections, demonstrate the superiority of the NPV criterion.

Internal rate of return

The IRR is the financial equivalent of an algebraic problem. The problem is: given a value for Y , what is the solution for x in the following equation?

Y = C (1+x)1 + C (1+x)2 + C (1+x)3 + · · ·

This geometric progression has the same structure as a set of discounted cash flows, where the numerator of the equation is the set of cash flows, and the x value is an interest rate. In algebra this equation has meaning. Unfortunately, when it is transferred to finance it is not economically relevant. In finance, the role of x cannot be clearly defined. In the NPV model, the NPV is clearly defined. In the IRR equation, however, it is difficult to define IRR in its own terms, because it effectively means something like: ‘the rate of return at which all funds, if borrowed at the IRR, could be repaid from the project, without the firm having to make any cash contribution’. The IRR criterion does not measure the project’s contribution to the firm’s value.

The IRR remains in use because decision-makers are used to dealing in ‘rates of return’ rather than the more esoteric NPV. The IRR measure is useful for easily comparing the rate of return from the project being considered with various alternative returns.

Workbook 6.2

There are a number of conceptual and computational problems with using IRR. For example, the IRR calculation implicitly assumes that cash earned can be reinvested at the calculated IRR. The NPV calculation employs this assumption too, but it is probably more tenable there as it is likely that investment opportunities will be available at the general required rate of return, more so than at the unique IRR. A ‘modified IRR’ (MIRR) has been developed to overcome this problem. The modified IRR computational measure allows for a ‘reinvestment rate’ to be input to the calculation to derive the MIRR. The use of Excel’s MIRR function is demonstrated in Workbook 6.2. With an assumed reinvestment rate of 8.25% per annum for the Delta Project example, the MIRR is 17.69%. This is, of course, less than the original IRR of 19.8%. The resulting MIRR figure may be defined as ‘the earning rate of the project if it is assumed that funds when received are reinvested at the forecast reinvestment rate’.

Workbook 6.3

There may be one or many solutions for the IRR. According to Descartes’ rule of sign, there can be as many positive solutions for the IRR as there are changes of sign in the cash flows. For example, a series of flows of−$190,+$455,+$270 can have at most one solution for the IRR, which is 189%. But, if the series is−$190,+$455,−$270, there can be up to two positive solutions for the IRR. There are, in fact, two solutions, and these are 8.49% and 31%. Only the first solution is given by the Excel calculation. The second one is read off from the created chart, shown in Workbook 6.3. The difficulty in using the IRR as a decision-making tool in this case is that both are correct calculations, and thus both are acceptable solutions. If the required rate of return, for example, is 14% per annum, then one IRR is below and the other is above the required rate; we can come to no sensible decision as to whether the project should be accepted or rejected.

Workbook 6.4

There may be no solution for the IRR. For example, given the set of cash flows,−$210,

+$455,−$270, there is no IRR solution, even though at an assumed required rate of 14% per annum, there is a valid NPV solution of −$18.63. As shown on the graph on Workbook 6.4, there is no IRR solution, as the NPV profile line does not intersect the x-axis and the NPV is always negative.

The IRR decision can also conflict with the NPV decision for certain projects. This conflict is especially important where only one from two or more mutually competing projects can be selected. For example, let us assume Project A has cash flows of−$2,000,

+$200,+$3,700 and Project B has cash flows of−$2,000,+$2,000 and+$1,480. The two projects differ only in the timing of the cash inflows; their initial outlays and overall lives are similar. Both have a required rate of return of 9% per annum. The outcomes from these two projects are:

A B

IRR 41.11% 49.50%

NPV at 9% per annum $1,297.70 $1,080.55

A decision conflict arises here. Using the IRR, Project B should be selected; using the NPV, Project A should be selected. In such cases the NPV rule should be used, as it alone measures the absolute contribution to the firm’s value made by the project. The IRR measures only the relative rates of return and not the projects’ contributions to the value of the firm. Additionally, the IRR must be compared to the required rate for a decision to be made. This implies that the required rate must be known, and if it is known, then the NPV can be calculated using that rate anyway. Under certainty, as is assumed here, the required rate is known, so the NPV is easily calculated.

The ranking conflict between the NPV and the IRR for competing projects can be high- lighted in an NPV profile chart (Figure 6.1). This chart shows the NPVs of both projects at various required rates of return (discount rates). The NPV schedules intersect at one point. This crossover point is known formally as the ‘Fisher Intersection’. In this particular case

−500 0 500 1,000 1,500 2,000 2,500 10 20 30 40 50 60

Required Rate of Return (%)

Net present value ($)

Project A Project B

Figure 6.1. Net present value profiles for projects A and B.

its value is given in Workbook 6.5 as approximately 23%. If the appropriate discount rate happens to be lower than this crossover rate, there is a conflict in project ranking between NPV and IRR. For example, a discount rate of 9% will produce conflicting rankings under the two criteria. The NPV criterion ranks A above B while the IRR criterion ranks B above A. If the appropriate discount rate happens to be higher than the crossover point, for example 35%, then both criteria produce the same ranking – B is preferred to A.

Workbook 6.5

Payback period

Payback period (PP) is a measure of the time taken to recoup the initial outlay. Sup- pose for example that Project C has the following yearly cash flows: −$280, +$120,

+$140,−$60, +$90. The progressive sum of the cash flows after the initial outlay is: $120, $260, $200, $290. The payback occurs in year 4. There are several problems with this measure:

r _{The cash flows are not discounted. As the time value of money is not taken into account,} the future cash flows cannot be related to the initial outlay.

r _{The data outcome ‘four years’ is not a decision variable. It does not relate to the firm’s} goal of wealth maximization.

r _{There is no objective measure of what constitutes an acceptable payback period.} Management may set an ad hoc target of say three years, but this value is not objectively related to the firm’s goal.

r _{Cash flows occurring after the payback period are ignored. In the case where large outflows} may occur on the termination of the project, such as the cost of rehabilitation of a mine site, a project may be erroneously accepted on the basis of a short payback term.

Payback is a very unsophisticated and misleading measure, and it is not recommended as a criterion for accepting or rejecting projects. It may be useful as a support measure to the NPV criterion, as an aid and comfort to some decision-makers when considering very risky projects. For example, suppose that a large natural resource project is to be established in a foreign country which is subject to unstable government and tribal fighting. In such a case a short payback period may be desirable to ensure that the capital expenditure is quickly recovered and repatriated so that if something goes wrong and the project has to be abandoned, at least the initial investment will have been recovered. In other words, PP may be an additional consideration for very risky foreign investments in politically and socially unstable countries. Additional risk factors in foreign investments are discussed in Chapter 16.

Accounting rate of return

The accounting rate of return (ARR) is the ratio of average accounting income to investment value. For example, suppose we have an initial outlay of $200, and subsequent annual accounting income figures of $80, $110, $70 and $120. The average annual accounting income would be (80+110+70+120)/4=$95, and the ARR would equal 95/200, or 47.5%.

Unfortunately, there are several variations on this simple measure. The divisor can take on several meanings and values. Examples of three of these are:

r _{Average of opening and closing book-values. With an opening book-value of $200, we} might assume a closing written down book-value of $40. The ‘average’ value thus com- mitted to the investment is (200+40)/2=$120. The ARR is thus 95/120, or 79.16%. r _{Average of net opening and closing book-values. Given the values of $200 and $40, the}

‘net’ average value is (200−40)/2=$80, and thus the ARR is $95/80, or 118.75%. r _{Average of progressive written down book-values. Written down book-values at the end}

of each year are: $160, $120, $80 and $40. The average is (160+120+80+40)/4= $100, and thus the ARR is 95/100, or 95%.

Each of the four calculated ARR values, 47.5%, 79.16%, 118.75% and 95% is ‘correct’. The ARR is obviously not a reliable measure.

It also suffers other conceptual drawbacks: it does not account for the time value of money; it uses accounting data which is not directly related to the wealth of the firm; and it has no objective decision criterion. The decision criterion usually employed is a comparison of the ARR with the required rate of return. As we have seen, the ARR is only an accounting ratio; it is not a time value of money measure. It should not be compared to the time value required rate of return.

ARR is a very unsophisticated, vague and misleading measure. ARR is not recom- mended as a capital budgeting decision-making criterion. The ARR may play a role as an aid and comfort measure for supporting a project which is acceptable under the NPV criterion.

Summary of the suitability of the four criteria

Both the payback period and the accounting rate of return ignore the time value of money. Neither of these criteria gives an acceptable approach to the investment decision problem. Both the NPV and the IRR acknowledge the time value of money and are worthy of consideration. Of these, the IRR technique suffers from both conceptual and computational drawbacks, and should not be the primary decision criterion. Only the NPV method relates the time value of money to the cash flows, and measures the project’s direct impact on the