TECHNICAL | IDEC The gentle touch Bob Scarlett Sensitivity analysis is a useful decision-making tool that gives a feel for how a project’s results might be affected by changes to the values of critical variables ccording to the 2000 edition of CIMA Official Terminology, sensitivity analysis is a “modelling and risk assessment procedure in which changes are made to significant variables in order to determine the effect of these changes on the planned outcome”. Sensitivity analysis is a more general approach than its more specific, quantitative equivalents, and it can be used in many areas of business decision-making with varying levels of refinement. Consider the following simple example of a project proposal to make and sell units over three years: A Initial capital cost £4,000 Annual unit sales 100 Selling price per unit £60 Variable cost per unit £35 Fixed costs per year £900 You would normally evaluate this project using an appropriate discount rate – say, 6 per cent – which gives the following result: Year Cash flow Discount PV 0 – 4,000 1.000 – 4,000 1 1,600 0.943 1,509 2 1,600 0.890 1,424 3 1,600 0.840 1,343 Net present value (NPV): 277 1 Sensitivity to unit sales Project NPV (£) 1,200 1,000 800 400 200 0 -200 -400 -600 90 100 110 Annual unit sales 2 The impact of 2.5 per cent adverse variances in each project element Base case With 2.5% adverse variances Initial capital cost £4,100.00 Selling price per unit £58.50 Variable cost per unit £35.88 Fixed cost per year £922.50 Units sold per year 97.50 NPV (£) 277 177 – 124 43 217 110 % change — – 36.10 – 144.77 – 84.48 – 21.66 – 60.29 The positive NPV gives the appearance of a viable project, but in most practical situations there are uncertainties. You often find that the various elements aren’t sure figures. Instead, they represent the mean or most likely outcomes from a range of possibilities. For example, the figure of 100 unit sales per year is a forecast. Actual sales in any one year could be above or below that number. A sensitivity analysis would seek to give an impression of what the overall outcome of the project might be with a range of alternative annual unit sales results. For example, it might be judged that a worst-case scenario would be annual sales of 90 units and a bestcase scenario would be 110 units. The three alternative outcomes would therefore be: Case Unit sales NPV Worst 90 – 391 Forecast 100 277 Best 110 945 The sensitivity of the project to annual unit sales can be represented graphically (see figure 1, left). The graph provides the following insights: l Around 70 per cent of possible outcomes in the range of 90 to 110 annual unit sales give a positive NPV. l Annual sales of 96 units or greater are required to give a positive NPV. Presenting the sensitivity analysis in this way gives an impression of the dynamics of the situation, but it’s still an imperfect one. For one thing, it’s likely that results outside the range of 90 to 110 unit sales are possible. For another, it’s unlikely that annual unit sales are the only uncertain element. It is possible that all elements are uncertain. You might conduct a sensitivity analysis to get an idea of which of them gives rise to the greatest uncertainty in the overall outcome. For example, you could consider the impact on the project’s NPV of a 2.5 per cent adverse variance in each element in turn. So NPV is recalculated with an initial capital cost of £4,100 (ie, £4,000 x 1.025) with all other factors held constant and so on. The resultant NPVs can be seen in figure 2, left. This process gives the following insights: l A relatively small proportional change in any one of the elements produces a much larger change in the overall outcome. l The viability of the project is more vulnerable to some key variables than it is to others. The NPV of this project seems particularly sensitive to unit selling price, given that a 2.5 per cent adverse variance in this element causes a 144 per cent adverse variance in NPV. It may be possible to re-engineer a project in some way to alter its risk/return February 2003 CIMA Insider 17 TECHNICAL | IDEC profile. For example, customers might be prepared at the outset to contract for £59.50 as a fixed selling price, but guarantee to buy the units. In this scenario the expected project NPV would drop from £277 to £143. The expected return would fall, but a major source of uncertainty affecting its viability would be eliminated. You may also be faced with a choice between alternative methods of achieving given objectives. Where there are key variables, sensitivity analysis can help here as well. For example, say you need to provide a given standard of service for a five-year period and there are two ways of achieving this: l A high-capital approach involving the purchase of equipment that costs £14,000 and has a residual value of £1,400. This method uses 200 resource units annually. l A low-capital approach involving the purchase of equipment that costs £2,800 and has a disposal cost of £1,200. This uses 360 resource units. The cost of capital is 10 per cent. There is uncertainty surrounding the likely average cost of a resource unit over the term, but it will lie somewhere between £10 and £30. A sensitivity analysis will offer useful insights in this case. You can project the NPV of costs at alternative resource unit prices as follows: £ per unit High cap Low cap 10 20,712 17,192 15 24,503 24,015 20 28,294 30,839 25 32,085 37,662 30 35,875 44,486 The sensitivity of the project to resource costs can also be represented graphically (see figure 3, below). The insights revealed by this analysis include: l The high-capital approach offers a cheaper solution in about 70 per cent of the possible unit price outcomes, including the mean figure of £20 per unit. l The low-capital approach offers a cheaper solution only if the unit price is below £16. Even in the case of the lowest price (£10 per unit) the difference between the two approaches is only around £3,500 Purely on the basis of the information given, you would probably choose the highcapital approach, but the process of using sensitivity analysis may distort decisionmaking. When considering issues related to risk and uncertainty, decision-makers commonly make certain working assumptions. One of these is that, in conditions of uncertainty, the probability distribution of possible project outcomes is grouped symmetrically around a mean and most likely outcome. Indeed, many business projects are like that. The sensitivity analysis in this case uses a unit price of £20 as the median position, so you may be inclined to assume this is the expected result, and that outcomes close to this figure are more likely than ones remote from it. If you run projects often enough, you might expect the median to be the average outcome. Sensitivity analysis tends to focus on ranges of possible outcomes without considering the probabilities of different results within them. It may be that outcomes at one 3 Sensitivity to resource costs Cost NPV (£) 45,000 40,000 35,000 30,000 25,000 20,000 High cap 15,000 Low cap 10,000 5,000 10 15 20 25 Cost per resource unit (£) 18 CIMA Insider February 2003 30 extreme of the range are actually more likely than ones in the centre. In this case, the lowcapital approach may be preferable if there is a high probability that resource unit prices will be at the lower end of the range. Business decision-making is an art form. Sensitivity analysis is a general approach that can give decision-makers powerful insights into the problem they are confronting, but it is not a technique that can provide the solution to that problem. n Bob Scarlett is an accountant and consultant Sensitivity analysis exercise The following is an extract from a 1999 advertisement by a pensions company seeking investments in a “managed fund”: “Projected end values at alternative average annual yields for a £10,000 deposit placed with the fund for 18 years are as follows: 7 per cent (worst case), £33,799; 10 per cent (forecast), £55,599; 13 per cent (best case), £90,243. The 10 per cent forecast annual yield is based on results achieved by the fund over the past 25 years.” Set up a spreadsheet to calculate the end value of the investment, assuming an average annual 10 per cent yield in each of the following sensitivity cases: 1 Base case: 10% uniform yield each year. 2 Low variability: three-year cycles of 15 per cent (year one), 10 per cent (year two) and then 5 per cent (year three) – or 5 per cent, 10 per cent and then 15 per cent if you prefer. 3 Medium variability: three-year cycles of 20 per cent, 10 per cent and then 0 per cent. 4 High variability: three-year cycles of 25 per cent, 10 per cent and then – 5 per cent. Use the spreadsheet to calculate the expected end value of the investment, assuming a 10 per cent average annual return and equal probability for each of the four cases. Construct a diagram to illustrate the sensitivity of the end value to the degree of variability in the annual returns. Write a one-sentence comment on your findings. A model answer (in Excel) can be found in the Students section of the CIMA website (www.cimaglobal.com).