Net Present Value Method
Description of the method
The net present value method focuses on selecting projects that maximise the 'net present value' (NPV) generated for the company:
Key Concept:
Net present value is the net monetary gain (or loss) from a project, computed by discounting all present and future cash inflows and outflows related to the project.
Using the NPV method, all future cash flows related to an investment project are discounted back to time 0 (i.e. t = 0), taken to represent the start of the investment project. The NPV represents a specific kind of PV. While it is possible to discount and compound all the cash flows to a later point in time, for example to the end of the investment project, it is more common to use t = 0 as this is the time at which the decision to invest (or not to invest) has to be made. It will be explained later that other methods relate cash flows to other points in time, e.g. to the end of the investment project.
The most rigid assumptions of the NPV method relate to the existence of a perfect (unrestricted) capital market. The single, or uniform, interest rate for this market represents the rate at which it is possible to borrow or invest without limits. Therefore this rate is used for discounting or compounding cash flows to any point in time.
Using the NPV method, the profitability of investment projects is assessed as follows:
Key Concept:
Absolute profitability is achieved if an investment project's NPV is greater than zero.
Relative profitability: An investment project is preferred if it has a higher NPV than the alternative investment project(s).
For the following examples it is assumed that the uniform discount rate (i) remains unchanged over the life of the investment. In that case, a project's NPV at t = 0 can be determined using the following equation: T
Where:
t = Time index
T = The last year when cash flows take place
Cash inflows in t Cash outflows in t
The difference between cash inflows and cash outflows (CIFt - COFt) is the net cash flow (NCFt). As shown in the following figure, all net cash flows after t = 0 are discounted back to this point in time.
Time
Period 1 Period 2
Period T
Fig. 3-2: Discounting net cash flows for the net present value method
The equation indicated above must be detailed if components of the net cash flows are to be considered in a differentiated way. A project's cash flows can be subdivided into: initial investment outlay, liquidation value(s), cash inflows (e.g. from sales) and cash outflows (e.g. from expenses). In addition, the following assumptions may be made when using the NPV method:
• Taxes and transfer payments can be ignored.
• Only one type of product is manufactured with the investment project.
• Production volume equals sales volume (i.e. no inventory produced).
• The cash flows are assigned to the following points in time:
- Initial investment outlay: beginning of the first period (t = 0).
- Cash inflows and outflows: end of each relevant period (t).
- Liquidation value: end of the project's economic life (t = T).
The following formula for the NPV then can be derived: T
NPV = -Io + X ((pt " cofvt) • xt - COFft) ■ q"1 + L • q"T
Where: t
T Io
= Time index
= The last year when cash flows take place (end of the project) = Initial investment outlay = Sales price in period t
cofvt = Production/sales dependent (variable) cash outflows per unit in period t xt = Production/sales volume in period t
COFft = Production/sales independent (fixed) cash outflows in period t q = (1 + i)
L = Liquidation value
In concluding this discussion, it should be noted that the calculation of the NPV can be simplified when cash flows remain constant throughout the economic life of a project. In this case, the NPV of the cash flows at t = 0 can be determined by multiplying the cash flow by the annuity factor (described at the beginning of this chapter). The NPV is then the sum of the net present values of initial investment outlay, net cash flows and liquidation value.
An interesting feature of the NPV method is the potential use of differential cash flow analysis. This approach helps to simplify comparisons between competing projects, since the identical aspects of the projects can be omitted and less project data is required. For example, if two competing project options are both expected to generate the same revenue streams (but with different costs), the revenue streams can be omitted from the analysis since they are the same for both projects and therefore, don't need to be compared, provided absolute profitabilities exist. Only the costs need to be compared, because they differ between the projects.
When comparing two alternative projects, the critical point is the differences between their cash flows - i.e. the differential cash flows, which can also be interpreted as the cash flows of a fictitious 'differential investment'. By calculating the NPV of these differential cash flows (i.e. NPVdiff), the difference between the NPVs of the two projects under consideration is determined as a measure of relative profitability is determined.
I.e. for two investment projects A and B it can be shown that:
NPVdiff = X((CIFtA -COFtA) -(CIFb -CO%)>q"1 t=0
= X (CIFtA - COFtA) • q-1 - X (CIFtB - CO%) ■ q t=0 t=0 = NPVa - NPVb (3.8)
If the net present value of NPVdiff is positive, then investment project A has a higher NPV than investment project B and is, therefore, relatively more profitable. This approach can be used to compare more than two competing projects, as long as care is taken to exclude only those cash flows from the analysis that are identical across all the project options. This can get complicated, so it is sometimes easier just to calculate the NPV for each project in its entirety and choose the project with the highest NPV. Note that absolute profitability cannot be assessed by calculating NPVdiff. To evaluate absolute profitability, the NPV calculation must include all the cash flows from a project. However, it may be useful to calculate NPVdiff to find the best project option, before calculating the NPV of this project (i.e. including all of its cash flows) to determine its absolute profitability.
A major assumption of the NPV method is that any net cash inflow that is not needed to cover a cash outflow will be reinvested at the relevant discount rate (i). This crucial assumption is further explained using an illustrative example.
Example 3-1
A company is considering an investment to expand their business. The necessary data for the two available investment options are listed in the following table:
|
Data |
Investment project A |
Investment project B |
|
Initial investment outlay (€) |
100,000 |
60,000 |
|
Economic life (years) |
5 |
4 |
|
Liquidation value (€) |
5,000 |
0 |
|
Net cash flows (€) | ||
|
t = 1 |
28,000 |
22,000 |
|
t = 2 |
30,000 |
26,000 |
|
t = 3 |
35,000 |
28,000 |
|
t = 4 |
32,000 |
28,000 |
|
t = 5 |
30,000 |
- |
|
Relevant discount rate (%) |
8 |
8 |
Tab. 3-1: Data for investment projects A and B (NPV method)
Tab. 3-1: Data for investment projects A and B (NPV method)
It is required to determine the absolute and relative profitability of the two projects by using the NPV method.
For investment project A the net present value (NPVa) can be calculated as follows:
NPVA = -€100,000 + €28,000 • 1.08-1 + €30,000 • 1.08-2 + €35,000 • 1.08-3 + €32,000 • 1.08-4 + €30,000 • 1.08-5 + €5,000 • 1.08-5
This result is interpreted as follows. Since NPVa is positive it can be deduced that investment project A is profitable in absolute terms. The NPV of €26,771.59 is the net monetary gain achieved if investment project A is undertaken. This assumes a discount rate of 8% for the interest (payments). This result would be achieved regardless of the source of finance: from internal funds, debt financing or any combination of these.
The investor could choose to use this monetary gain (NPVa) right now (t = 0) by taking out a loan of €26,771.59 for consumption purposes. Such a 'consumption loan', together with interest charges, could be paid back from the cash flow surpluses of the investment project. The project's cash flow surpluses would also cover the loan repayments (including interest) on the capital employed for the investment (€100,000 for project A).
This is illustrated by a financial budget and redemption plan that contains the cash flows and net monetary values associated with the above investment, assuming that:
• A 100% debt financing is used, i.e. €26,771.59 to be consumed at the beginning of the planning period plus €100,000 for the initial investment outlay.
• At the end of every period, interest is paid at the discount rate (8%) on the debt remaining.
• Net cash surpluses are used for immediate loan repayments.
|
Point in time |
Net project cash flow (€) (excluding initial investment outlay) |
Interest paid (used to repay the loan) |
Net monetary value (amount of loan outstanding) | |
|
t |
Nt |
It (= i • Vt-1) |
AVt (= Nt + It) |
Vt(= Vt-1 + AVt) |
|
0 |
0 |
0 |
0 |
-126,771.59 |
|
1 |
28,000 |
-10,141.73 |
17,858.27 |
-108,913.32 |
|
2 |
30,000 |
-8,713.07 |
21,286.93 |
-87,626.39 |
|
3 |
35,000 |
-7,010.11 |
27,989.89 |
-59,636.50 |
|
4 |
32,000 |
-4,770.92 |
27,229.08 |
-32,407.42 |
|
5 |
35,000 |
-2,592.59 |
32,407.41 |
-0.01 |
Tab. 3-2: Financial budget and redemption plan with 100% debt financing
Tab. 3-2: Financial budget and redemption plan with 100% debt financing
The financial budget and redemption plan shows that the initial loan of €126,771.59 is fully regained by the cash flow surpluses from investment project A.
The level of the NPV and, therefore, of the net monetary gain is the same regardless of the combination of internal funds and debt financing employed. This is a major outcome of the 'perfect capital market' assumption. To demonstrate this concept further, the financial budget plan is now given for a case where is financed by internal funds at 100%, in contrast to the 100% debt financing depicted above. In this case, the cash flow surpluses from the investment project are used for financial investments that - in accordance with the reinvestment assumption - are made in the capital market, and yield the relevant discount rate (8%).
|
(without initial investment outlay) |
Interest received (increase in internal funds) |
Net monetary value (total internal funds) | ||
|
t |
NCFt |
It(= i • Vt-1) |
AVt (= NCFt + It) |
V (= Vt.1 + AVt) |
|
0 |
0 |
0 |
0 |
0 |
|
1 |
28,000.00 |
0 |
28,000.00 |
28,000.00 |
|
2 |
30,000.00 |
2,240.00 |
32,240.00 |
60,240.00 |
|
3 |
35,000.00 |
4,819.20 |
39,819.20 |
100,059.20 |
|
4 |
32,000.00 |
8,004.74 |
40,004.74 |
140,063.94 |
|
5 |
35,000.00 |
11,205.12 |
46,205.12 |
186,269.06 |
Tab. 3-3: Financial budget plan with 100% financing by internal funds
Tab. 3-3: Financial budget plan with 100% financing by internal funds
As the financial budget plan shows, the investor has internal funds of €186,269.06 at the end of the investment project's life. This amount cannot be compared directly to the outlay of €100,000, however, since the amounts occur at different points in time. Yet it can be shown that the end value, discounted back to t = 0 at the interest rate of 8%, equals the sum of NPV and initial investment outlay:
The difference between this amount and the capital employed (€100,000) corresponds to NPVa (€26,771.59). Provided reinvestment at the discount rate is assumed, it can be seen that the internal funds-financed investment project yields the same net monetary gain of €26,771.59 as the debt-financed investment. This can also be shown by calculating the end value that an investor would hold if he or she invested capital of €100,000 over the economic life of the investment at an interest rate of 8%. The end value (EV) in this case is:
The difference between the end value of investment project A (€189,269.06) and the financial investment of €100,000 at the 8% discount rate (€146,932.81) is €39,336.25. If this amount is discounted back to t = 0 the difference becomes €26,771.59 which equals the project's NPV:
These examples illustrate the fact that an absolute profitability assessment made using the net present value model implies the operation of a uniform discount rate in a perfect capital market. This, is the alternative investment when an actual investment project is rejected.
Another interpretation of NPV is based on the possibility of investing in the capital market as an alternative to the investment project. The price of project A consists of the initial investment outlay. The alternative price is the financial investment amount required at the beginning of the planning period (t = 0) in order to generate the same expected cash flow surpluses (CIFt- COFt) in the future. This financial investment amount is the sum of separate investment amounts each arising from the cash flow surplus at time t discounted by the uniform discount factor:
Altogether the sum of all separate financial investments and with it the price of the future payment surpluses amounts to: T
Then the net present value represents the difference between this price and the initial investment outlay of the investment project under consideration: T
Or in the example:
Because the difference is positive, the investment project under consideration is superior to a financial investment in the capital market. Thus, it is absolutely profitable.
For the assessment of relative profitability the net present value of the alternative project B (NPVB) must also be calculated. It amounts to:
As NPVb is positive, investment project B is absolutely profitable. However, because NPVa (€26,771.59) is greater, investment project A appears to be relatively more profitable. Where both projects are mutually exclusive, investment project A will be preferred, provided net present value is the decision criterion.
As described above, the relative profitability of two investment projects can also be determined by calculating the NPV of the differential cash flows or a fictitious differential investment (NPVdff = NPVa - NPVb). This amounts in the example to:
NPVdiff = ^40,000 + €6,000 ■ 1.08-1 + €4,000 ■ 1.08-2 + €7,000 ■ 1.08-3 + €4,000 ■ 1.08-4 + €35,000 ■ 1.08-5
This implies that project A's net monetary gain exceeds the net monetary gain from investment project B by €1,302.27. Project A therefore is relatively profitable.
However, it should be noted that the capital tie-up of investment project B is considerably lower (initial outlay is €60,000 compared to €100,000 for project A). Moreover, the economic life of project B is shorter by one year and, therefore, subsequent investments may be realised at an earlier time.
This raises the important question: to what extent are the net cash flow profiles used to calculate investment projects' NPVs suitable for the assessment of relative profitability if differences exist in regard to:
• The capital tie-up at the beginning (i.e. different initial investment outlays)?
• The capital tie-up during the course of the investment (i.e. from different net cash flows over the life of the project)?
• The projects' economic lives?
Needless to say, if none of these differences exist the investment projects under consideration can be regarded as equivalent, since the analysis is limited to the comparison of cash flow profiles regardless of the nature of the actual project (e.g. a machine). If differences in capital-tie up and/or economic life are not explicitly included in the analysis, however, they need to be balanced by additional investments (or by financing alternatives).
Differences in projects' capital tie-up are easy to consider if perfect capital market conditions are assumed. This implies that the differences between projects' initial outlays can be balanced by assuming that a financial investment is made at the relevant uniform discount rate (or by the corresponding assumption that a financing alternative with this interest rate is achieved). The following example illustrates the use of such a financial investment (NPVF) (a fictitious investment project) to balance the initial capital tie-up difference between projects A and B for one period:
NPVf = -€40,000 + (€40,000 ■ 1.08) ■ 1.08-1 = €0
The NPV of this balancing financial investment is zero. Accordingly, the differences in capital tie-up do not need to be analysed explicitly for competing investments. They can be ignored if perfect capital market conditions are assumed. A similar argument can be applied to differences in economic life. Subsequent investments that may be undertaken at different points in time because of different economic lives, have no influence on the profitability of alternatives if it is assumed that all future investments yield the relevant uniform discount rate and therefore have an NPV of zero. Under the assumed perfect capital market conditions it is not necessary to consider supplementary investments explicitly. The NPVs of projects under comparison determined on the basis of the cash flow profiles are sufficient for the assessment of relative profitability. In spite of differences in capital tie-up and/or economic life, competing investments can be treated as comparable alternative projects.
But what if it is unrealistic to assume current and future (re-)investment at the uniform discount rate? A modified application of the NPV method can be used if specific information exists for other current and future investment projects that may be used to balance differences in capital-tie up and/or economic life, whose interest rates deviate from the expected uniform discount rate. This might be the case if, for example, subsequent projects are pursued at the end of the initial project's economic life. They constitute so-called chains of investments might consist of identical or non-identical projects, and refer to limited or unlimited time horizons. In such situations, the NPVs of the investment chains may be used to assess relative profitability. This is described in Chapter 5. It is important to note that the profitability of investment projects also depends on subsequent projects: in calculating an isolated net present value this fact is overlooked. Unless the investment project under consideration is the final project for the company - i.e. truly is an isolated project - the decision conditions assumed may be not realistic. Companies usually envisage an unlimited or, at least, a long term planning horizon, which limits the simplistic use of NPVs in investment decision-making.
Assessment of the method
The NPV method is one of the most widely known and used methods in both theory and company practice. To assess its usefulness (as for all the alternative methods) its computational ease, data collection requirements and, most important of all, model assumptions must be considered.
The computational effort is low, as simple arithmetical calculations are sufficient. The data collection, however, may cause problems because, as a rule, several forecasts are necessary. The NPV model requires forecasts of the initial investment outlay, all future cash flows, the project's economic life, the liquidation value at the end of the economic life and the relevant discount rate. Nevertheless, this challenge applies to all investment appraisal models.
The NPV model makes more realistic assumptions compared to the static models described in Chapter 2 because all years of the investment project's economic life are explicitly included. Thus, as the required computational effort is only slightly greater, the methods described in this chapter are more appropriate for real-world company practice. Yet the net present value model does make some assumptions, the potential effects of which should be evaluated against the real situation. The model's assumptions include the following:
a) A single target measure (the NPV) is considered adequate.
b) The economic life is pre-determined and appropriate.
c) Other associated decisions (such as financing and production decisions) are made before the investment decision in order to be able to forecast cash flows for separate investment projects.
d) The data is certain.
e) The cash flows can be allocated to specified time periods and points in time.
f) All current and future investments not explicitly considered (financial investments due to cash inflow surpluses and investments to balance up the different capital tie-up and/or economic life of investments) will yield at the relevant uniform discount rate.
g) A perfect capital market exists.
The profitability of investment projects often depends on several performance targets. In such cases a single target measure (assumption a) may be insufficient for decision-making and other methods should be used, e.g. multi-criteria methods (described in Chapter 6).
The economic life must be known before applying the NPV method (assumption b). In order to gain this information, models may be used to determine the optimum economic life as described in Section 5.3. These models may be based on the NPV model.
Decisions about other investment projects, as well as actions of other company divisions and/or functional areas (e.g. production, sales and financing), can influence the cash flows associated with the project under consideration and, therefore, the profitability of this project. In the model described above, it is assumed the NPV is calculated after these decisions are taken (assumption c). In reality, the two are interdependent since these other decisions rely on information about the investment project currently under consideration. Such interdependencies mean that investment decisions cannot normally be allocated unambiguously to a single investment project, as assumed here. Models for simultaneous decision-making that can lead to overall optimum solutions are described in Chapter 7.
It is highly unlikely that all necessary present and future data will be known with 100% certainty (assumption d). Therefore, an additional analysis of the effects of uncertainty on the forecasted data should be made simultaneously with the determination of NPVs - at least for more important investment projects. Methods and models that take account of uncertainty are described in Chapter 8.
The models described in this chapter assume that all cash flows can be allocated to a specified point in time - usually the beginning or end of a year (assumption e). In reality cash flows will occur more often and several will take place over a year. This may be accommodated in the NPV model easily by increasing the number of periods monitored and adjusting the interest rate from yearly to for example monthly or daily. This will improve the accuracy of forecasts but increases computational efforts. As an alternative, a continuous yield and constant cash flows can be assumed.
The implications of assumed yields at the relevant discount rate (assumption f) have been discussed for the above example. Generally speaking, this assumption is unrealistic. The significance of deviations from the assumed yield, and the availability of information required to include supplementary investment projects, should be discussed during the decision-making process. The assumption that future investments will yield interest at the relevant discount rate implies that more profitable subsequent investment projects will not be made (unless their consequences are already captured within the cash flow profile of the investment project under consideration, as discussed in Section 5.3).
Additionally, the impact that one current investment project may have on the feasibility and profitability of future investment projects can be disregarded provided assumption (f) holds true. In this case, the NPV of future investments is zero, and ignoring these investments has neither negative nor positive financial consequences. However, in reality, uncertainty exists about future investment possibilities. Among other things, technological advancements may result in future projects earning positive NPVs. Under such circumstances, the question arises as to whether a current investment project with a positive NPV should be undertaken now or renounced in favour of future investment projects. This can be answered by the explicit inclusion of future investment projects in the investment decision-making process as described in Section 5.3.
Also problematic is the assumption of a perfect capital market (assumption g) under which loans can be taken and financial investments made at any time and in any amount, all yielding the relevant uniform discount rate. Only in a perfect capital market can investment and finance decisions be made independently without endangering their optimality. Additionally, only in a perfect capital market can an investor at any point in time transfer funds generated by an investment project to different points in time using a financial investment, yielding the uniform discount rate, without affecting the original value of the investment project. This leads to the fact that the temporal distribution of income can be chosen at will and, thus, investment, withdrawal or consumption decisions are independent. Such a perfect capital market simply does not exist in reality. Among other things, interest rates differ between those receivable for financial investments and those payable for loans. Moreover, it is impossible to determine one uniform discount rate that simultaneously fulfils all these different purposes. This consideration becomes important where the level of the assumed uniform discount rate is a key determinant of the NPV. The identification of a suitable discount rate is discussed in a separate section (3.6) below. But first, other approaches to investment appraisal are discussed, starting with the annuity method.
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