So far the analysis has been in terms of two time periods; we now extend it to three. It will be assumed that there is a project capital cost in the current year (year 0) and a net benefit in each of years 1 and 2; a net benefit means a benefit less any project operating costs incurred in the year in question. As noted earlier, the assumption about the accrual date of benefits and costs is important.
Project benefits and costs generally accrue more or less continuously throughout the year. It would be possible to discount benefits and costs on a monthly, weekly or even daily basis using the appropriate rate of interest. However in practice such level of detail is unnec-essary. Instead it is generally assumed that all costs or benefits experienced during the year occur on some arbitrarily chosen date: it could be the first day of the year, the middle day or the last day. This assumption converts the more or less continuous stream of benefits and costs to a set of discrete observations at one-year intervals. The annual rate of interest is then used to discount this discrete stream back to a present value.
There is no general rule as to which day of the year should be chosen, or whether the calendar or financial year is the relevant period. However we suggest that costs or benefits should be attributed to the last day of the calendar year in which they occur. For example, fol-lowing this convention, the net cash flow of a project with a capital cost of $100 during the current calendar year, and net benefits of $60 in each of two subsequent calendar years would be treated as a cost of $100 on December 31 this year, and net benefits of $60 on December 31 in each of the two subsequent years. It should be noted that while this procedure tends to introduce a slight downward bias to a positive NPV calculation (assuming the firm has access to short-term capital markets), any other assumption about accrual dates also introduces bias.
In denoting the project years we generally call the current year “year 0” and subsequent 24 Benefit-Cost Analysis
years “year 1”, “year 2” etc. Since we do not discount benefits or costs occurring in the current year, we are effectively choosing December 31 of the current year as the referent date – “the present” in the net present value calculation. Benefits or costs occurring in year 1 (by assump-tion on December 31) are discounted back one period, those occurring in year 2 are discounted back two periods, and so on. Thus the number used to denote the year tells us how the appropriate discount factor is to be calculated. This discussion might seem a bit laboured and trivial but consistency in the choice of accrual date is an important practical issue.
Using our assumption about the accrual date, our illustrative project has a capital cost now (K at time zero), and a net benefit one year from now (at the end of the first year), and two years from now (at the end of the second year). The resulting sequence, –K, B1, B2,can be termed a net benefit stream as illustrated in Figure 2.4. To calculate the project NPV we need to bring B1and B2to present values, sum them and then subtract the project cost, K, which is already a present value (since it is incurred in year 0).
The net benefit B2can be brought to a present value by using the interest rate in year 2 to bring it to a discounted value at the end of year 1, and then using the interest rate in year 1 to bring that discounted value to a present value. B1can be discounted to a present value using the interest rate in year 1:
NPV = –K + B1/(1+r1) + B2/[(1+r1)(1+r2)]
In practice it is almost invariably assumed that the rate of interest is constant over time, so that r1=r2=r, in which case the NPV formula reduces to: NPV = –K + B1/(1+r) + B2/(1+r )2. To take a simple example, suppose the project costs $1.6 and yields net benefits of $10 after 1 and 2 years respectively, and that the rate of interest is 10%:
Investment Appraisal: Principles 25
+
–
year
0 1 2
Figure 2.4 Net Benefit Stream of a Two-period Investment Project
B1
B2
K
$
NPV = –1.6 + 10*0.909 + 10*0.826 = 15.75
The values of 1/1.1 and 1/(1.1)2are obtained from a Table of Discount Factors, or by using a pocket calculator. The benefit/cost ratio is given by 17.35/1.6 = 10.84 and the net benefit/cost ratio by 15.75/1.6 = 9.84. (The mechanics of deriving discount factors and using discount tables are discussed in more detail in Chapter 3.) It can be seen from the NPV formula that the NPV falls as the discount factors fall, and hence as the discount rate rises; this relationship is illustrated in Figure 2.5.
It was explained earlier that the internal rate of return is the discount rate which reduces the project’s NPV to zero, as denoted by IRR in Figure 2.5. It can be calculated by solving the following equation for rp:
–K + B1/(1+rp) + B2/(1+rp)2= 0.
Multiplying both sides of this equation by (1+rp)2gives an equation of the following form:
–Kx2+ B1x+ B2= 0,
where x = (1+rp). This is the familiar quadratic form, as illustrated in Figure 2.6, which can be solved for x and then for rp.
26 Benefit-Cost Analysis
Discount Rate (% p.a.) Net
Present Value
$
0
IRR
Figure 2.5 Net Present Value in Relation to the Discount Rate
More generally it can be noted that if we extended the analysis to include a benefit at time 3 we would need to solve a cubic, and if we further extended it to time 4 it would be a quartic equation, and so on; fortunately we will be solving only a couple of IRR problems by hand – the rest will be done by computer using a spreadsheet program.
To return to our example, we need to solve:
–1.6x2+ 10x + 10 = 0.
Recalling our high school algebra, this is a quadratic equation of the general form:
ax2+ bx + c = 0,
where a= –1.6, b=10, and c=10, and which has two solutions given by the formula:
x = [–b +/– SQRT(b2– 4ac)]/2a.
A bit of arithmetic gives us the solution values 7.13 and –0.88, which is where the function crosses the x-axis as illustrated in Figure 2.6 This means that rpis either 6.13 (613%) or –1.88 (–188%). The negative solution has no meaning in the context of project appraisal and is dis-carded. Hence the IRR is 613%, which is very healthy compared with an interest rate of 10%.
Let us now change the example by making B2= –10. For example, the project could be a mine which involves an initial establishment cost and a land rehabilitation cost at the end of the project’s life. When we calculate the IRR using the above method we find that there are
Investment Appraisal: Principles 27
X = (1 + rp) F(x)
0
7.13 –0.88
Figure 2.6 Calculating Internal Rates of Return – One Positive Value
two solutions as before, but that they are both positive: rp= 0.25 (25%) or 4 (400%). Which is the correct value for use in the project appraisal process? The answer is possibly neither. As discussed above, when we use the IRR in project appraisal we accept projects whose IRR exceeds the rate of interest. In the current example the IRR clearly exceeds an interest rate in the 0–24% range, and it is clearly less than an interest rate in excess of 400%. But what about interest rates in the 25–400% range? We cannot apply the rule in this case, and it is for this reason that it is not recommended that the IRR be used in project appraisal when there is more than one positive IRR.
What would have happened if we had asked the computer to calculate the IRR in this case? The computer’s approach is illustrated in Figure 2.7.
F(x) is the value of the quadratic function, and setting F(x)=0 gives the solution values – these occur where the function crosses the x-axis. In the previous example this occurred once in the negative range of x and once in the positive range, and the negative value was dis-carded. However in the present case there are two positive values of x for which F(x)=0, 1.25 and 5. The computer solves the quadratic equation by asking for an initial guess, say x = x1in Figure 2.7. It then calculates the first derivative of the function at that value of x (the equation of the slope of the function at that point) and works out the value of x, x2, that would set the first derivative equal to zero. It then uses x2as its guess and repeats the process until it converges on the solution value x = 5 (rp= 4) in Figure 2.7.
If you had given the computer xa,, for example, as the initial guess it would have con-verged on the solution value x = 1.25 (rp= 0.25). How would you know if there were two positive solutions? Some programs will report no solution or an error if there is more than one positive IRR. However a simple way of checking for the problem in advance is to count the 28 Benefit-Cost Analysis
X = (1 + rp) F(x)
0 1.25 5
Xa X2 X1
Figure 2.7 Calculating Internal Rates of Return – Two Positive Values
number of sign changes in the net benefit stream. In the initial example we had a well-behaved cash flow with signs “– + +”, i.e. one change of sign, hence one positive IRR. In the second example we had “– + –”, i.e. two changes of sign and two positive IRRs. This is a general rule which applies to higher order equations – the number of positive solutions can be the same as the number of sign changes2. If you are analyzing a proposed project which has a net benefit stream (including costs as negative net benefits) with more than one change of sign, it is better not to attempt to calculate the IRR. The NPV rule will always give the correct result irrespective of the sequence of signs.
As illustrated in Figure 2.5, the IRR is the discount rate which reduces the project NPV to zero. The case in which there are two positive IRRs is illustrated in Figure 2.8, where the project NPV is positive at first, and falling as the discount rate rises, and then, after reaching some negative minimum value, it starts to rise and eventually becomes positive again. There are two points at which NPV = 0 and these correspond to the two positive IRRs. The discount rate at which NPV is a minimum, 100% in the example, can be solved for from the NPV equation.
Finally we consider a case in which no IRR can be calculated. Suppose that in the original example we change the initial cost of $1.6 to a benefit of $1.6. The net benefit stream now consists of three positive values and there is no value of the discount rate which can reduce the NPV to zero and hence no IRR. This case is illustrated in Figure 2.9 where the NPV continually falls as the discount rate increases, but never actually becomes zero.
Investment Appraisal: Principles 29
Discount Rate (% p.a.)
$
0 Net Present Value
25
100
400
Figure 2.8 Net Present Value in Relation to the Discount Rate – the Two Positive Internal Rates of Return Case
2 In fact more than one change in sign is a necessary, but not a sufficient condition for the existence of more than one positive IRR. A sufficient condition is that the unrecovered investment balance becomes negative prior to the end of the project’s life. The reader is referred to advanced texts for a discussion of this concept.