## Lr

where r is the discount rate, expressed as a decimal value and t is the number of years into the future that the cash flow occurs.

Alternatively, and rather more easily, the present value of a cash flow may be calculated by multiplying the cash flow by the appropriate discount factor. A small table of discount factors is given in Table 3.3.

The NPV for a project is obtained by discounting each cash flow (both negative and positive) and summing the discounted values. It is normally assumed that any initial investment takes place immediately (indicated as year 0) and is not discounted. loiter cash flows are normally assumed to take place at the end of each year and are discounted by the appropriate amount.

Net present value (NPV) and internal rate of return (IRR) are collectively known as discounted cash flow (DCF) techniques.

Note that this example uses approximate figures - when you have finished reading this section you should be able to calculate the exact figures yourself.

Table 3.3 Table of NPV discount factors

Discount rate (ck)

Table 3.3 Table of NPV discount factors

Discount rate (ck)

 Year 5 6 S 10 12 15 1 0.9524 0.9434 0.9259 0.9091 0.8929 0.8696 More extensive or detailed 2 0.907 0.89 0.8573 0.8264 0.7972 0.7561 tables may be constructed 3 0.8638 0.8396 0.7938 0.7513 0.7118 0.6575 using the formula 4 0.8227 0.7921 0.7350 0.683 0.6355 0.5718 discount factor = —!— 5 0.7835 0.7473 0.6806 0.6209 0.5674 0.4972 (l+tf 6 0.7462 0.705 0.6302 0.5645 0.5066 0.4323 to* various values of r (the 7 0.7107 0.6651 0.5835 0.5132 0.4523 0.3759 discount rate) and 1 (the 8 0.6768 0.6274 0.5403 0.4665 0.4039 0.3269 number of years from now) 9 0.6446 0.5919 0.5002 0.4241 0.3606 0.2843 10 0.6139 0.5584 0.4632 0.3855 0.322 0.2472 15 0.481 0.4173 0.3152 0.2394 0.1827 0.1229 20 0.3769 0.3118 0.2145 0.1486 0.1037 0.0611 25 0.2953 0.233 0.1460 0.0923 0.0588 0.0304

Exercise 3.5 Assuming a lOtf discount rate, the NPV for project I (Table 3.2) would be calculated as in Table 3.4. The net present value for Project I. using a I0# discount rate is therefore £618. Using a 10** discount rate, calculate the net present values for projects 2, 3 and 4 and decide which, on the basis of this, is the most beneficial to pursue.

 Year Project / cash flow /or* Discounted cash flow {£) 0 -100.000 1.0000 -100.000 1 K).(KX) 0.9091 9.091 2 10,000 0.8264 8,264 3 10.000 0.7513 7.513 4 20.000 0.6830 13.660 5 100.000 0.6209 62.090 Net Profit: £50.000 NPV: £618

It is interesting to note that the net present values for projects I and 3 are significantly different - even though they both yield the same net profit and both have the same return on investment. The difference in NPV reflects the fact that, with project I. we must wait longer for the bulk of the income.

The main difficulty with NPV for deciding between projects is selecting an appropriate discount rate. Some organizations have a standard rate but, where this is not the case, then the discount rate should be chosen to reflect available interest rates (borrowing costs where the project must be funded from loans) plus some premium to reflect the fact that software projects are inherently more risky than lending money to a bank. The exact discount rate is normally less important than ensuring that the same discount rate is used for all projects being compared. However, it is important to check that the ranking of projects is not sensitive to small changes in the discount rate - have a look at the following exercise.

Calculate the net present value for each of the projects A, B and C shown in Exercise 3.6 Table 3.5 using each of the discount rates 8%. 10% and 12%.

For each of the discount rates, decide which is the best project. What can you conclude from these results?

Alternatively, the discount rate can be thought of as a target rate of return. If. for example, we set a target rate of return of 15% we would reject any project that did not display a positive net present value using a 15% discount rate. Any project that displayed a positive NPV would be considered for selection - perhaps by using an additional set of criteria w here candidate projects were competing for resources.

 Year Project A (£) Project B (£) Projected) 0 -8.000 - 8.000 - 10.000 1 4.000 1.000 2.000 2 4.0(X) 2.0ÎX) 2.(XX) 3 2,000 4.000 6.000 4 1.000 3.000 2.000 5 500 9.000 2.000 6 500 -6.000 2.000 Net Profit 4.000 5.000 6.000

Internal rate of return

One disadvantage of NPV as a measure of profitability is that, although it may be used to compare projects, it might not be directly comparable w ith earnings from other investments or the costs of borrowing capital. Such costs are usually quoted

The IRR may be estimated by plotting a series of guesses:

For a particular project, a discount rate of 8% gives a positive NPV of £7,898; a discount rate of 12% gives a negative NPV of -£5,829. The IRR is therefore somewhere between these two values. Plotting the two values on a chart and joining the points with a straight line suggests that the IRR is about 10.25%. The true IRR (calculated with a spreadsheet) is 10.167%.

as a percentage interest rate. The internal rate of return (IRR) attempts to provide a profitability measure as a percentage return that is directly comparable with interest rates. Thus, a project that showed an estimated IRR of 10% would be worthwhile if the capital could be borrowed for less than 10% or if the capital could not be invested elsewhere for a return greater than 10%.

The IRR is calculated as that percentage discount rate that would produce an NPV of zero. It is most easily calculated using a spreadsheet or other computer program that provides functions for calculating the IRR. Microsoft Excel and Lotus, for example, both provide IRR functions which, provided with an initial guess or seed value (which may be zero), will search for and return an IRR.

Manually, it must be calculated by trial-and-error or estimated using the technique illustrated in Figure 3.3. This technique consists of guessing two values

(one either side of the true value) and using the resulting NPVs (one of w hich must be positive and the other negative) to estimate the correct value. Note that this technique will provide only an approximate value but, in many cases that can be sufficient to dismiss a project that has a small IRR or indicate that it is worth making a more precise evaluation.

The internal rate of return is a convenient and useful measure of the value of a project in that it is a single percentage figure that may be directly compared with rates of return on other projects or interest rates quoted elsewhere.

Table 3.6 illustrates the way in which a project w ith an IRR of 10% may be directly compared with other interest rates. The cash flow for the project is shown in column (a). Columns (b) to (e) show that if we were to invest £ 100.000 now at an annual interest rate of 10% in, say, a bank, we could withdraw the same amounts as we would earn from the project at the end of each year, column (e), and we would be left with a net balance of zero at the end. In other words, investing in a project that has an IRR of 10% can produce exactly the same cash flow as lending the money to a bank at a 10% interest rate. We can therefore reason

Figure 3.3 Estimating the internal rate of return for project /.

Figure 3.3 Estimating the internal rate of return for project /.

11 12

Discount rate (%)

that a project with an IRR greater than current interest rates will provide a better rate of return than lending the investment to a hank. We can also say that it will be worth borrowing to finance the project if it has an IRR greater than the interest rate charged on the loan.

Table 3.6

A project cash flow treated as an investment at 10%

Year

Equivalent investment at l(fk

Project cash flow forecast (£) -KX).(XX) 10.000 10.000 10.000 20.000 99.000

Capital at Interest Capital at End of year start of year during year end of year withdrawal

100.000 100.000 100.000 100.000 90.000 0

10.000 10.000 10.000 10.000 9.000 0

110.000 110.000 110.000 110.000 99.000 0

### 10.000 10.000 10.000 20.000 99.000 0

£100.000 invested at 10% may be used to generate the cash flows shown. At the end of the 5-year period the capital and the interest payments will be entirely consumed leaving a net balance of zero.

One deficiency of the IRR is that it does not indicate the absolute size of the return. A project with an NPV of £100.000 and an IRR of 15% can be more attractive than one with an NPV of £10.000 and an IRR of 18% - the return on capital is lower but the net benefits greater.

An often quoted objection to the internal rate of return is that, under certain conditions, it is possible to find more than one rate that will produce a zero NPV. This is not a valid objection since, if there are multiple solutions, it is always appropriate to take the lowest value and ignore the others. Spreadsheets will normally always return the lowest value if provided with zero as a seed value.

NPV and IRR are not, however, a complete answer to economic project evaluation.

• A total evaluation must also take into account the problems of funding the cash flows - will we, for example, be able to repay the interest on any borrowed money and pay development staff salaries at the appropriate time?

• While a project's IRR might indicate a profitable project, future earnings from a project might be far less reliable than earnings from. say. investing with a bank. To take account of the risk inherent in investing in a project, we might require that a project earn a 'risk premium' (that is, it must earn. say. at least 15% more than current interest rates) or we might undertake a more detailed risk analysis as described in the following sections of this chapter.

• We must also consider any one project within the financial and economic framework of the organization as a whole - if we fund this one. will we also be able to fund other worthy projects?