Chapter10: Project Analysis
10-2 Sensitivity Analysis
Uncertainty means that more things can happen than will happen. Whenever you are confronted with a cash-flow forecast, you should try to discover what else can happen.
Put yourself in the well-heeled shoes of the treasurer of the Otobai Company in Osaka, Japan. You are considering the introduction of an electrically powered motor scooter for city use. Your staff members have prepared the cash-flow forecasts shown in Table 10.1. Since NPV is positive at the 10% opportunity cost of capital, it appears to be worth going ahead.
Before you decide, you want to delve into these forecasts and identify the key variables that determine whether the project succeeds or fails. It turns out that the marketing department has estimated revenue as follows:
The production department has estimated variable costs per unit as ¥300,000. Since projected volume is 100,000 scooters per year, total variable cost is ¥30 billion. Fixed costs are ¥3 billion per year. The initial investment can be depreciated on a straight-line basis over the 10-year period, and profits are taxed at a rate of 50%.
1. Investment is depreciated over 10 years straight-line.
2. Income is taxed at a rate of 50%.p. 244
These seem to be the important things you need to know, but look out for unidentified variables. Perhaps there are patent problems, or perhaps you will need to invest in service stations that will recharge the scooter batteries. The greatest dangers often lie in these unknown unknowns, or “unk-unks,” as scientists call them.
Having found no unk-unks (no doubt you will find them later), you conduct a sensitivity analysis Analysis of the effect on project profitability of possible changes in sales, costs, and so on. with respect to market size, market share, and so on. To do this, the marketing and production staffs are asked to give optimistic and pessimistic estimates for the underlying variables. These are set out in the left-hand columns of Table 10.2. The right-hand side shows what happens to the project's net present value if the variables are set one at a time to their optimistic and pessimistic values. Your project appears to be by no means a sure thing. The most dangerous variables are market share and unit variable cost. If market share is only .04 (and all other variables are as expected), then the project has an NPV of − ¥10.4 billion. If unit variable cost is ¥360,000 (and all other variables are as expected), then the project has an NPV of − ¥15 billion.
Value of Information
Now you can check whether you could resolve some of the uncertainty before your company parts with the ¥15 billion investment. Suppose that the pessimistic value for unit variable cost partly reflects the production department's worry that a particular machine will not work as designed and that the operation will have to be performed by other methods at an extra cost of ¥20,000 per unit. The chance that this will occur is only 1 in 10. But, if it does occur, the extra ¥20,000 unit cost will reduce after-tax cash flow by
It would reduce the NPV of your project by
putting the NPV of the scooter project underwater at +3.43 − 6.14 = − ¥2.71 billion. It is possible that a relatively small change in the scooter's design would remove the need for the new machine. Or perhaps a ¥10 million pretest of the machine will reveal whether it will work and allow you to clear up the problem. It clearly pays to invest ¥10 million to avoid a 10% probability of a ¥6.14 billion fall in NPV. You are ahead by − 10 + .10 × 6,140 = +¥604 million.
On the other hand, the value of additional information about market size is small. Because the project is acceptable even under pessimistic assumptions about market size, you are unlikely to be in trouble if you have misestimated that variable.
Limits to Sensitivity Analysis
Sensitivity analysis boils down to expressing cash flows in terms of key project variables and then calculating the consequences of misestimating the variables. It forces the manager to identify the underlying variables, indicates where additional information would be most useful, and helps to expose inappropriate forecasts.
One drawback to sensitivity analysis is that it always gives somewhat ambiguous results. For example, what exactly does optimistic or pessimistic mean? The marketing department may be interpreting the terms in a different way from the production department. Ten years from now, after hundreds of projects, hindsight may show that the marketing department's pessimistic limit was exceeded twice as often as the production department's; but what you may discover 10 years hence is no help now. Of course, you could specify that, when you use the terms “pessimistic” and “optimistic,” you mean that there is only a 10% chance that the actual value will prove to be worse than the pessimistic figure or better than the optimistic one. However, it is far from easy to extract a forecaster's notion of the true probabilities of possible outcomes.3
Another problem with sensitivity analysis is that the underlying variables are likely to be interrelated. What sense does it make to look at the effect in isolation of an increase in market size? If market size exceeds expectations, it is likely that demand will be stronger than you anticipated and unit prices will be higher. And why look in isolation at the effect of an increase in price? If inflation pushes prices to the upper end of your range, it is quite probable that costs will also be inflated.
Sometimes the analyst can get around these problems by defining underlying variables so that they are roughly independent. But you cannot push one-at-a-time sensitivity analysis too far. It is impossible to obtain expected, optimistic, and pessimistic values for total project cash flows from the information in Table 10.2.
If the variables are interrelated, it may help to consider some alternative plausible scenarios. For example, perhaps the company economist is worried about the possibility of another sharp rise in world oil prices. The direct effect of this would be to encourage the use of electrically powered transportation. The popularity of compact cars after the oil price increases in 2007 leads you to estimate that an immediate 20% rise in the price of oil would enable you to capture an extra 3% of the scooter market. On the other hand, the economist also believes that higher oil prices would prompt a world recession and at the same time stimulate inflation. In that case, market size might be in the region of .8 million scooters and both prices and cost might be 15% higher than your initial estimates. Table 10.3 shows that this scenario of higher oil prices and recession would on balance help your new venture. Its NPV would increase to ¥6.4 billion.
Managers often find scenario analysis Analysis of the profitability of a project under alternative economic scenarios. helpful. It allows them to look at different but consistent combinations of variables. Forecasters generally prefer to give an estimate of revenues or costs under a particular scenario than to give some absolute optimistic or pessimistic value.
When we undertake a sensitivity analysis of a project or when we look at alternative scenarios, we are asking how serious it would be if sales or costs turned out to be worse than we forecasted. Managers sometimes prefer to rephrase this question and ask how bad sales can get before the project begins to lose money. This exercise is known as break-even analysis Analysis of the level of sales at which a project would just break even..p. 246
In the left-hand portion of Table 10.4 we set out the revenues and costs of the electric scooter project under different assumptions about annual sales.4 In the right-hand portion of the table we discount these revenues and costs to give the present value of the inflows and the present value of the outflows. Net present value is of course the difference between these numbers.
You can see that NPV is strongly negative if the company does not produce a single scooter. It is just positive if (as expected) the company sells 100,000 scooters and is strongly positive if it sells 200,000. Clearly the zero-NPV point occurs at a little under 100,000 scooters.
In Figure 10.1 we have plotted the present value of the inflows and outflows under different assumptions about annual sales. The two lines cross when sales are 85,000 scooters. This is the point at which the project has zero NPV. As long as sales are greater than 85,000, the project has a positive NPV.5
Managers frequently calculate break-even points in terms of accounting profits rather than present values. Table 10.5 shows Otobai's after-tax profits at three levels of scooter sales. Figure 10.2 once again plots revenues and costs against sales. But the story this time is different. Figure 10.2, which is based on accounting profits, suggests a breakeven of 60,000 scooters. Figure 10.1, which is based on present values, shows a breakeven at 85,000 scooters. Why the difference?
When we work in terms of accounting profit, we deduct depreciation of ¥1.5 billion each year to cover the cost of the initial investment. If Otobai sells 60,000 scooters a year, revenues will be sufficient both to pay operating costs and to recover the initial outlay of ¥15 billion. But they will not be sufficient to repay the opportunity cost of capital on that ¥15 billion. A project that breaks even in accounting terms will surely have a negative NPV.
Operating Leverage and the Break-Even Point
A project's break-even point depends on the extent to which its costs vary with the level of sales. Suppose that electric scooters fall out of favor. The bad news is that Otobai's sales revenue is less than you'd hoped, but you have the consolation that the variable costs also decline. On the other hand, even if Otobai is unable to sell a single scooter, it must make the up-front investment of ¥15 billion and pay the fixed costs of ¥3 billion a year.
Suppose that Otobai's entire costs were fixed at ¥33 billion. Then it would need only a 3% shortfall in revenues (from ¥37.5 billion to ¥36.4 billion) to turn the project into a negative-NPV investment. Thus, when costs are largely fixed, a shortfall in sales has a greater impact on profitability and the break-even point is higher. Of course, a high proportion of fixed costs is not all bad. The firm whose costs are fixed fares poorly when demand is low, but makes a killing during a boom.
A business with high fixed costs is said to have high operating leverage Fixed operating costs, so called because they accentuate variations in profits (cf. financial leverage ).. Operating leverage is usually defined in terms of accounting profits rather than cash flows6 and is measured by the percentage change in profits for each 1% change in sales. Thus degree of operating leverage (DOL) The percentage change in profits for a 1% change in sales. is
Note: DOL is estimated as the median ratio of the change in profits to the change in sales for firms in Standard & Poor’s index, 1998–2008.
The following simple formula7 shows how DOL is related to the business's fixed costs (including depreciation) as a proportion of pretax profits:
In the case of Otobai's scooter project
A 1% shortfall in the scooter project's revenues would result in a 2.5% shortfall in profits.
Look now at Table 10.6, which shows how much the profits of some large U.S. companies have typically changed as a proportion of the change in sales. For example, notice that each 1% drop in sales has reduced steel company profits by 2.20%. This suggests that steel companies have an estimated operating leverage of 2.20. You would expect steel stocks therefore to have correspondingly high betas and this is indeed the case.
3If you doubt this, try some simple experiments. Ask the person who repairs your dishwasher to state a numerical probability that it will work for at least one more year. Or construct your own subjective probability distribution of the number of telephone calls you will receive next week. That ought to be easy. Try it.
4Notice that if the project makes a loss, this loss can be used to reduce the tax bill on the rest of the company's business. In this case the project produces a tax saving—the tax outflow is negative.
5We could also calculate break-even sales by plotting equivalent annual costs and revenues. Of course, the break-even point would be identical at 85,000 scooters.
6In Chapter 9 we developed a measure of operating leverage that was expressed in terms of cash flows and their present values. We used this measure to show how beta depends on operating leverage.
7This formula for DOL can be derived as follows. If sales increase by 1%, then variable costs will also increase by 1%, and profits will increase by .01 × (sales − variable costs) = .01 × (pretax profits + fixed costs). Now recall the definition of DOL: