Investment appraisal under uncertainty
1 Introduction
A major reservation of any investment appraisal decision is that the figures used in the calculations are only estimates and stand to be uncertain. Clearly if any of the cash flows used in the decision turn out to be different from what was estimated, the decision itself could be affected.
In this chapter we will look at four approaches that attempt to either reduce the problem or quantify the possible effect of the problem.
A requirement to discuss techniques for dealing with risk and uncertainty means these four. Net present value, internal rate of return and the accounting rate of return are appraisal methods and score nothing here, and nor does defining risk and uncertainty where the question has not asked for it. With two marks on offer for each technique, take each one to the project in front of you — which variable the net present value turns out to be most sensitive to, or what single volume the probabilities weight down to.
2 Sensitivity analysis
Sensitivity analysis analyses the effect of changes made to variables in the problem in order to determine their effect on the decision.
First we calculate the NPV of the project on the basis of the best estimates.
Then we calculate what % change (or sensitivity) in each of the variables would result in a NPV of zero (i.e. the breakeven position – any further change would change the decision).
By considering the sensitivity of each variable we can ascertain which variables are the most critical and therefore perhaps need more work confirming our estimates.
Daina has just set up a new company and estimates that the cost of capital is 15%.
Her first project involves investing in $150,000 of equipment with a life of 15 years and a final scrap value of $15,000.
The equipment will produce 15,000 units p.a. generating a contribution of $2.75 each. She estimates that additional fixed costs will be $15,000 p.a..
(a) Determine, on the basis of the above figures, whether the project is worthwhile
(b) Calculate the sensitivity to change of:
i. the initial investment
ii. the sales volume p.a.
iii. the contribution p.u.
iv. the fixed costs p.a.
v. the scrap value
vi. the cost of capital
(c) comment on the results
3 Simulation
Simulation is a technique which allows more than one variable to change at the same time.
You will not be required in the examination to actually perform a simulation, but you should be aware of the principle involved.
3.1 Essentially, the stages are as follows:
identify the major variables
specify the relationship between the variables
attach probability distributions to each variable and assign random numbers to reflect the distribution
simulate the environment by generating random numbers
record the outcome of each simulation
repeat the simulation many times to be able to obtain a probability distribution of the possible outcomes
4 Expected values
With this approach, we identify the various possible outcomes for each uncertain variable, together with the associated probability.
We then use for each uncertain variable the weighted average outcome (or expected outcome), and use these figures in our investment appraisal calculation.
Daiga plc is considering launching a new product.
This will require additional capital investment of $200,000.
The selling price of the product will be $10 p.u.. Daiga has ascertained that the probability of a demand of 50,000 units p.a. is 0.5, with a probability of 0.4 that it will be 20% higher, and a 0.1 probability that it will be 20% lower.
The company expects to earn a contribution of 50% and expects fixed overheads to increase by $140,000 per year.
The time horizon for appraisal is 4 years. The machine will be sold at the end of 4 years for $50,000.
The cost of capital is 20% p.a.
Calculate the expected NPV of the project
Assuming that the demand is certain at 50,000 units p.a. what is the NPV of the project if fixed overheads are uncertain as follows:
Fixed overheads | Probability |
|---|---|
100,000 | 0.20 |
140,000 | 0.35 |
180,000 | 0.25 |
220,000 | 0.20 |
5 Risk-adjusted discount rate
Although it is easy enough to identify that the estimated returns from a project are uncertain, it is not normally realistic in practice to identify the various possible outcomes and then attach probabilities to each of them.
The risk inherent in a project depends very much on the type of activity involved. For example, investing in a new project to sell solar-powered vehicles is perhaps more risky than a new project to sell accountancy services.
(Note that higher risk does not mean that the project is automatically worse – solar-powered vehicles might give a much higher return, but equally there is the possibility of them giving a much lower return).
Although all people have different attitudes to risk, it is generally the case that people will be prepared to accept projects with higher risk provided that the expected return is higher.
One approach to dealing with this is to discount higher risk projects using a higher rate of interest – effectively adding a premium to the interest rate for risk, or using a risk-adjusted discount rate.
We will discuss this approach in a later chapter - the idea forms the basis for a technique known as the capital asset pricing model.
Investment appraisal under uncertainty
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