Discounting the interest in money
| by Nigel Coulthurst
01 Dec 1998
The paper 3 Management Information syllabus includes an introduction to the use of discounting techniques (which are explored further in papers 8 and 14). Candidates for the paper 3 examination have requested an article in the Students’ Newsletter on the subject.
The paper 3 Teaching Guide specifies (on the subject of discounting) that candidates should be able to:
Discounting is often combined, in examination questions, with probability/expected value (another aspect of the paper 3 syllabus). This article seeks to cover the aspects of discounting specified in the Teaching Guide in this wider setting.
It is useful first to consider compounding, which is the opposite of discounting, because this is more likely to be familiar. Compounding is the addition of interest each period to a sum of money in order to determine its value some time in the future. It is a geometric progression with a ratio (1 + r) where r is the interest rate per period expressed as a decimal.
The future value (FV) of a single present sum (PV) after n periods can be expressed as:
FV = PV (1 + r)n
In contrast, discounting seeks to convert a future sum to a present value, which will depend upon the rate of interest per period and the period in the future when the sum is expected to arise. The present value (PV) of a single future sum (FV) received after n periods, at an interest rate of r per period can be expressed (rearranging the formula above) as:
Discounting, as with compounding, reflects the time value of money. £1 today is worth more than £1 at some future time because of its earning potential.
Discounting is applied, in business practice, to the evaluation of the worth of capital investment projects, typified by an up-front investment followed by a stream of earnings over several years of project life. The objective in using discounting, in capital investment project evaluation, is to relate the cash flows expected to arise from a project to a base year by means of discount factors. NB. It is cash flows, not accounting profits, in each period that are relevant to the evaluation.
It was stated above that the formula for converting a future sum to a present sum by means of discounting can be expressed as:
To illustrate, if for example £1,000 is expected to be received in one year’s time, at a prevailing annual interest rate of 10%, then the present value of that future sum of money can be calculated as:
If the £1,000 is, instead, expected to be received in two year’s time then its present value is:
In this context, ‘0.909’ and ‘0.826’ are referred to as discount factors and can be obtained from a mathematical table provided for candidates sitting the paper 3 examination. An extract from the discount table is shown in Table 1. The future period (row) and the period interest rate (column) combine to identify the appropriate discount factor
If it were required to calculate the total present value of £1,000 received each year for the next three years, at an interest rate of 10% per annum, this would be:
The formula for the present value of a sequence of cash flows becomes:
NB. It is normal, both in practice and in examinations, for cash flows and interest rates to be per annum (i.e., this is the interval between periods).
The interest rate
The interest rate to be used in the evaluation of capital investment projects will always be specified in paper 3 examination questions. Suffice to say that the interest rate represents the minimum acceptable return on investment that a business requires in order to satisfy the different providers of capital (i.e., shareholders and lenders). This is also termed the cost of capital. The calculation of this required interest rate is an element within the syllabus for paper 14.
Discounted cash flow (DCF) methods
There are two main discounting methods used for the evaluation of the worth of capital investment projects, collectively termed Discounted Cash Flow (DCF) methods. The two methods are Net Present Value (NPV) and Internal Rate of Return (IRR).
Net Present Value (NPV)
Using the NPV method, the interest rate which represents the minimum acceptable return on investment is used as the discount rate, in order to calculate the net present value of the cash inflows and outflows. The NPV method provides an absolute measure of the cash surplus or deficit in present value terms.
The test of the profitability of a project is the relationship between the total present value of the cash inflows and the present value of the cash invested in the project i.e. whether, the net present value is positive or negative. The decision rule is invest if NPV > 0 (i.e., positive).
Internal Rate of Return (IRR)
The IRR method of discounted cash flow involves finding the percentage rate of interest which, when used to discount the future cash flows, will produce a zero net present value for a capital investment project (i.e., where the total present value of the sequence of cash inflows is equal to the present value of the cash invested). The IRR thus provides a relative measure of project worth.
The interest rate is determined by trial and error to achieve a zero NPV. The higher the percentage used to discount (and thus the smaller the discount factors to apply to the future cash flow sums) the lower will be the present value (and vice versa). The test of the profitability of a project is the relationship between the internal rate of return percentage and the percentage that represents the minimum acceptable return for the business. The decision rule is invest if IRR % > minimum acceptable %.
The following example is adapted from a previous paper 3 examination question. “A company is considering the launch of a new product, for which an investment in equipment of £200,000 would be required. The project life would be limited to 5 years by the expected life cycle of the product. It is expected that the equipment would have no value at the end of five years.
Market research has indicated a 70% chance of demand for the new product being high and a 30% chance of demand being low. Cash flows are forecast as follows:
The minimum acceptable return required by the company is 10% per annum”.
Initially, in order to illustrate the two DCF methods, the evaluation will concentrate solely on the cash flows under conditions of high demand. The initial investment is usually identified with Year 0, the beginning of the project with a factor of 1.0 (i.e., not discounted), with the cash inflows from Year 1 onwards (i.e., on average a year later etc.)
Using the NPV method the evaluation of the cash flows, using the minimum acceptable return of 10% as the discount rate, would be as follows:
One would conclude that the project would be profitable if the high demand estimates were achieved.
Using the IRR method, it is obvious that the solution rate is above 10% because the project (at high sales demand) has a positive NPV when discounted at that rate. If a discount rate of 18% is chosen (see later for justification of this choice) the discounted cash sums are:
The positive value of £1,631 indicates a solution percentage rate in excess of 18%. At 19% the discounted cash sums are:
The IRR is, therefore, between 18 and 19%, but nearer 18% (smaller positive discounted sum at 18% than negative sum at 19%). To the nearest whole integer, the IRR is therefore 18%. In general, this is sufficiently precise given the fact that all cash flows are estimates. This confirms the conclusion reached earlier that, at high sales demand, the project would be profitable.
Shortcuts to the IRR
It was stated earlier that the calculation of the IRR is a trial and error process. This can be very time consuming in examinations (without access to computer software) unless careful thought is given to the process. A number of short-cuts are available, particularly the use of cumulative discount factors and interpolation/extrapolation, in order to minimise the iterative process.
Cumulative Discount Factors
The present value of the same cash flow sum received each period can be calculated by using a cumulative (multi-period) discount factor:
Like single period discount factors these are available in a table (but the table is not provided for paper 3 candidates in the examination). Instead, the cumulative factors can be calculated easily by the addition of single period factors. Thus, for example, the present value of 50,000 each year for five years, starting in a year’s time, at a discount rate of 10% per annum can be calculated as:
In such situations (i.e., where the same cash flow sum is received each period), the same principle can be applied to the calculation of the IRR. You may have noticed that the cash flows in the above illustration are those arising (in the project example used in this article) if there is low demand for the new product. The NPV is thus a negative £10,500 ((200,000) – 189,500).
The cumulative discount factor required for there to be a zero present value (i.e., at the IRR solution rate) can be calculated easily. The present value of the cash inflows needs to be £200,000 (i.e., equal to the investment) and therefore, the cumulative discount factor over 5 years, to multiply by the annual cash inflow of £50,000, is:
At 10%, the cumulative discount factor over 5 years is 3.790. As the percentage rate is reduced, the cumulative discount factor increases (i.e., the lower the rate the greater will be the present value of future cash flows), and vice versa. At 9%, the cumulative factor is 3.889 (add the single year factors for years 1 to 5 in the 9% column of Table 1), at 8% it is 3.993, and at 7% it is 4.100. Therefore, to the nearest whole integer, the IRR of the cash flows at low demand is 8%.
One would conclude that the project would not be profitable if the low demand estimates occurred.
It can be seen from the above that where cash flows are expected to be the same in each year of a project, calculation of the IRR (as well as the NPV at the minimum acceptable return on investment) is relatively straightforward.
Approximate Cumulative Discount Factors
In general, however, cash inflows in each year of a project are unlikely to be estimated to be the same. Nevertheless, an approximated cumulative discount factor, calculated from uneven cash flows, can indicate where the trial and error process should begin.
Returning to our example, and the estimated cash inflows for high demand, the total over 5 years is expected to be £327,000 (60,000 + 62,000 + 65,000 + 70,000 + 70,000), an average of £65,400 per annum. A cumulative discount factor of 3.058 can be calculated (200,000/65,400). Because the balance of cash flows is later in the project (i.e., the last two years > the first two years) the cumulative factor could be adjusted upwards to reflect a lower return (e.g., to 200,000 /65,000 = 3.08 ). At 18% the cumulative discount factor over 5 years is 3.127, and at 19% it is 3.057 (hence the use of these rates earlier in the calculation of the IRR).
Cash flows from a project may fluctuate significantly from year to year which makes the approximation more difficult. If the majority occur in the early years then the return will be higher (and the cumulative discount factor needs to be lower) than if the flows were evenly spread, and vice versa.
Interpolation and Extrapolation
It is useful to plot discounted values against discount rates on a graph, to illustrate the fact that there is not a linear relationship between the two. Using our earlier example, the discounted cash flows for high demand are plotted on Figure 1.
As the discount rate is increased, the present value falls. The NPV (i.e., the discounted net sums using the minimum acceptable rate of return as the discount rate) can be read from the graph. The IRR% is where the curve cuts the horizontal axis (i.e., zero present value).
The shape of the curve indicates that the present value reduces by an increasingly smaller absolute amount in response to successive changes of a certain amount in the discount rate i.e., as r (the discount rate) is successively increased, the discount factor (1/(1 + r)n) reduces by a smaller and smaller amount. This is due to the fact that although ‘1 + r’ is increasing by the same absolute amount, this is an increasingly smaller amount relative to the previous ‘1 + r’.
Nevertheless, a linear relationship can be assumed, between present value and discount rate, in order to calculate the approximate IRR%. This is called interpolation and extrapolation. Any two present values can be used (one of which could be the undiscounted present value (i.e., at 0%)). If the two points used lie either side ofthe IRR (i.e., one rate of discount above the IRR% with a negative present value, and one below with a positive present value), then it is referred to as ‘interpolation’ (the calculation of an intermediate term). ‘Extrapolation’ (the calculation of a term outside the known range) is equally valid, using either two positive present values or two negative ones.
The general formula for interpolation/extrapolation is: IRR = lower discount rate
For example, using the data obtained earlier regarding the cash flows at high demand discounted at 18% and 19%:
at 18% the present value is £1,631
Using instead the data at 0% (undiscounted) and 10% (discounted at the minimum acceptable rate of return):
at 0% the present value is £127,000
= 0+(10x 1.56)
The key question is — “when is it reasonable to approximate in this way?” As a general rule, any two values can be used as long as either the present value at each point is less than 20% of the value undiscounted, or the present value at one of the points is within 10% of the value undiscounted. It can be seen that the interpolation data used above easily satisfies this criteria, whereas the extrapolation data does not.
Let us assume now (again using the project cash flows under the high sales demand scenario) that the calculation of the IRR is required, but not the NPV. Let us also assume that 19% is our first estimate of the IRR (based on 200,000 – 65,400).
We thus have:
at 19% a present value of (£3,023).
By interpolation (reasonable because one of the values is in the required range):
In conclusion, it has been demonstrated that the calculation of a sufficiently accurate IRR% (both in practice and for examination purposes) can be achieved from only one discounted present value via the use of cumulative discount factors and interpolation/extrapolation.
Probability may be introduced, into examination questions on capital investment project evaluation, by the provision of a range of estimated values for different elements of a project, and with probabilities attached to each value.
There are two possible approaches to the incorporation of these probabilities into the evaluation. One approach is to calculate the expected value (average based on the probability distribution) of each year’s cash flow for each element, followed by the calculation of a single expected value for the NPV/IRR. The alternative, preferred but longer, approach is to calculate a range of NPVs/IRRs using the different possible outcomes and only then to combine them, using the probabilities, in the calculation of an expected value for the NPV/IRR.
Taking the first approach, and using the sales demand probabilities given in our example, the expected value for each year’s cash flow and the NPV/IRR evaluation is:
On this basis, with expected values of £28,943 for the NPV and over 15% for the IRR, the project is viable.
Using the second approach to the inclusion of probability data into project evaluation, the expected value for the NPV/IRR would only be calculated after separate evaluation of the high and low demand estimates. For example, the expected value for the NPV, using the values calculated previously under each separate scenario, becomes:
A more informed decision can be made, on the basis not only of the expected value but also of the range of estimates i.e., in this case the knowledge that, if the low demand estimates occurred, the return would be below the minimum acceptable.
It is important that ACCA students studying for paper 3 have a good grasp of discounting and its use in the calculation of the NPV and the IRR of an investment project.
This article has sought to cover those aspects of the topic that are set out in outline in the syllabus and teaching guide for the subject.
Unable to open a PDF document? To open a PDF you need Adobe Acrobat Reader, which can be downloaded for free from the Adobe website.