ACCA F9 Financial Management - 2010 - Study text - Emile Woolf Publishing
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Paper F9
Financial management
CHAPTER
10
DCF: risk and uncertainty
Contents
1 Riskanduncertaintyincapitalinvestment
appraisal
2Sensitivityanalysis
3 ExpectedvalueoftheNPV:assessmentofproject
risk
4 Othermethodsofriskanduncertaintyanalysis
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Risk and uncertainty in capital investment appraisal
The problem of risk and uncertainty
Methods of assessing risk and uncertainty
1 Risk and uncertainty in capital investment appraisal
1.1 The problem of risk and uncertainty
Investment projects are long-term projects, often with a time scale of many years. When
the cash flows for an investment project are estimated, the estimates might be incorrect.
Estimates of cash flows might be wrong for two main reasons:
risk in the investment, and
uncertainty about the future.
Risk
Risk exists when the actual outcome from a project could be any of several different
possibilities, and it is not possible in advance to predict which of the possible
outcomes will actually occur.
The simplest example of risk is rolling a dice. When a dice is rolled, the result will
be 1, 2, 3, 4, 5 or 6. These six possible outcomes are known in advance, but it is not
possible in advance to know which of these possibilities will be the actual outcome.
With risk assessment, it is often possible to estimate the probabilities of different
outcomes. For example, we can predict that the result of rolling a dice will be 1, 2, 3,
4, 5 or 6, each with a probability of 1/6.
Risk can often be measured and evaluated mathematically, using probability
estimates for each possible future outcome.
Uncertainty
Uncertainty exists when there is insufficient information to be sure about what will
happen, or what the probability of different possible outcomes might be. For
example, a business might predict that sales in three years’ time will be £500,000,
but this might be largely guesswork, and based on best-available assumptions about
sales demand and sales prices.
Uncertainty occurs due to a lack of sufficient information about what is likely to
happen.
It is possible to assess the uncertainty in a project, but with less mathematical
precision than for the assessment of risk.
Management should try to evaluate the risk and uncertainty, and take it into
account, when making their investment decisions. In other words, investment
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decisions should consider the risk and uncertainty in investment projects, as well as
the expected returns and NPV.
1.2 Methods of assessing risk and uncertainty
There are several methods of analysing and assessing risk and uncertainty. In
particular:
Sensitivity analysis can be used to assess a project when there is uncertainty
about future cash flows
Probability analysis can be used to assess projects in which there is risk.
Other methods of risk and uncertainty analysis include:
risk modelling and simulation
risk-adjusted discount rates
adjusted payback
discounted payback as one of the criteria for investing in capital projects.
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Sensitivity analysis
The purpose of sensitivity analysis: assessment of project uncertainty
Methods of sensitivity analysis
Estimating the sensitivity of a project to changes in the cost of capital
The usefulness of sensitivity analysis
2 Sensitivity analysis
2.1 The purpose of sensitivity analysis: assessment of project uncertainty
Sensitivity analysis is a useful but simple technique for assessing investment risk in
a capital expenditure project when there is uncertainty about the estimates of future
cash flows. It is recognised that estimates of cash flows could be inaccurate, or that
events might occur that will make the estimates wrong.
The purpose of sensitivity analysis is to assess how the NPV of the project might be
affected if cash flow estimates are worse than expected.
2.2 Methods of sensitivity analysis
There are two main methods of carrying out sensitivity analysis on a capital
expenditure project.
Sensitivity analysis can be used to calculate the effect on the NPV of a given
percentage reduction in benefits or a given percentage increase in costs. For
example, what would the NPV of the project be if sales volumes were 10% below
estimate, or if annual running costs were 5% higher than estimate? The
percentage variation in the expected cash flows should be an amount that might
reasonably occur, given the uncertainty in the cash flow estimates.
Alternatively, sensitivity analysis can be used to calculate the percentage
amount by which benefits must fall below estimate or costs rise above estimate
before the project NPV becomes negative. For example, by how much (in
percentage terms) would sales volumes need to fall below the expected volumes,
before the project NPV became negative? Or by how much (in percentage terms)
would running costs need to exceed the expected amount before the NPV
became negative?
There are also forms of ‘stress testing’, similar to sensitivity analysis, that might be
used to assess the investment risk. An assessment could be made to estimate the
effects of:
of an unexpected event occurring in the future that would make the cash flow
estimates for the project wrong, or
the effect of a delay so that the expected cash inflows from the project occur later
than planned.
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Example
A company is considering the following project.
Year
Equipment Income
Running
costs
Netcash
flow
DCFfactor
at10% PV
$ $ $ $ $
0 (55,000) 1.000 (55,000)
150,000 35,000 15,000 0.909 13,635
280,000 55,000 25,000 0.826 20,650
3100,000 70,000 30,000 0.751 22,530
440,000 30,000 10,000 0.683 6,830
NPV +8,645
Required
Estimate the sensitivity of the project to:
income being lower than estimate, and
running costs being higher than estimate.
Answer
We can calculate the PV of the equipment cost, the income and the running costs
separately, as follows.
Method 1
We can estimate the effect on the NPV of a percentage change for the worse in each
of the cash flow estimates. For example, suppose that running costs are 10% higher
than estimated? The NPV would be as follows.
Year
Equipment Income
Running
costs
Netcash
flow
DCFfactor
at10% PV
$ $ $ $ $
0 (55,000) 1.000 (55,000)
150,000 38,500 11,500 0.909 10,454
280,000 60,500 19,500 0.826 16,107
3100,000 77,000 23,000 0.751 17,273
440,000 33,000 7,000 0.683 4,781
NPV (6,385)
This analysis shows that if running costs are 10% higher than expected, the NPV
will be negative and the project would fail to provide a 10% return.
Method 2
An alternative method of sensitivity analysis is to calculate by how much cash flows
need to be worse that expected before the NPV becomes negative.
We can calculate the PV of the equipment cost, the income and the running costs
separately, as follows. (The original cash flows are discounted here.)
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Year Equipment Income Runningcosts
PV PV PV NPV
£ £ £ £
0 (55,000)
145,450 (31,815)
266,080 (45,430)
375,100 (52,570)
427,230 (20,490)
––––––––
–
–––––––
–
––––––– ––––––––
PV (55,000) 213,950 (150,305) +8,645
––––––––
–
–––––––
–
––––––– ––––––––
The sensitivity of the project to errors in the estimates can now be assessed as
follows.
The NPV will become negative if the PV of any cost is more than $8,645 above
estimate or if the PV of benefits is more than $8,645 below estimate.
Income. The project would cease to have a positive NPV if income is below the
estimate by more than (8,645/213,950) = 0.040 or 4.0%.
Running costs. The project would cease to have a positive NPV if running costs
are above the estimate by more than (8,645/150,305) = 0.058 or 5.8%.
Equipment cost. The project will cease to have a positive NPV if the cost of the
equipment in Year 0 exceeds the estimate by more than (8,645/55,000) = 0.157 or
15.7%.
An assessment of risk and uncertainty could then be made on the basis of the
likelihood that income will fall below estimate or running costs will increase above
the estimate by more than these amounts.
2.3 Estimating the sensitivity of a project to changes in the cost of capital
The sensitivity of the project to a change in the cost of capital can be found by
calculating the project IRR. This can be compared with the company’s cost of
capital.
Example
A company is considering the following project:
Year
Netcash
flow
DCFfactor
at15%
PV
DCFfactor
at20%
PV
$$$
0
(
55,000
)
1.000
(
55,000
)
1.000
(
55,000
)
1 15,000 0.870 13,050 0.833 12,495
2 25,000 0.756 18,900 0.694 17,350
3 30,000 0.658 19,740 0.579 17,370
4 10,000 0.572 5,720 0.482 4,820
–
––––––– ––––––––
NPV +2,410
(
2,965
)
–
––––––– ––––––––
IRR = 15% + [2,410 / (2,410 + 2,965) ] × (20 – 15)% = 17.2%
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The sensitivity of the project to changes in the cost of capital is quite small. The cost
of capital is 10% but the cost of capital would have to be over 17.2% before the NPV
became negative.
2.4 The usefulness of sensitivity analysis
Sensitivity analysis is useful because it directs management attention to the critical
variables in the project. These are the variables where a variation in the cash flows
by a fairly small amount – and certainly by an amount that might reasonably be
expected, given uncertainty about the cash flows – would make the NPV negative
and the project not financially viable.
If the project is undertaken, sensitive items of cash flow should be closely
monitored and action taken if they vary from plan.
If a project NPV is particularly sensitive to an item of cost or revenue,
management might decide to reject the project because of the investment risk
involved.
A major problem with sensitivity analysis is that only one variable is varied at a
time and it is assumed that all variables are independent of each other. In reality
variables may all vary to some extent and they may be interdependent. For this
reason simulation models may provide additional information when assessing risk.
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Expected value of the NPV: assessment of project risk
Definition of expected value
Calculating and using the EV of the NPV
Advantages and disadvantages of using expected values
3 Expected value of the NPV: assessment of project risk
3.1 Definition of expected value
An expected value is a weighted average value, calculated using probability
estimates of different possible outcomes. To calculate an expected value, the
probability of each possible outcome is estimated, and the mean (average) outcome
is calculated.
3.2 Calculating and using the EV of the NPV
Expected value (EV) =
∑ px
where:
p represents the probability of each outcome and
x represents the value of each particular outcome.
When expected values are used to assess the risk in capital investment appraisal, x
would be the NPV for each possible outcome, and the EV would be the expected
value of the project net present value (the EV of the NPV).
The basic decision rule is that an investment project should be undertaken if the
expected value of its NPV is positive.
However, a project with a positive EV of NPV might not be undertaken if the risk
involved seems too great in relation to the amount of the return expected.
Example
A company is considering an investment in a project. The project would be a five-
year project, and would cost $200,000. The actual returns from the investment are
subject to uncertainty, but the following estimates have been prepared for the
different possible outcomes:
Probability NPV
$
0.10 (8,000)
0.30 4,000
0.40 12,000
0.20 20,000
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The EV of the NPV is calculated as follows:
Probability NPV EV
p x Px
$ $
0.10 (8,000) (800)
0.30 4,000 1,200
0.40 12,000 4,800
0.20 20,000 4,000
–
–––––––
EVofNPV9,200
–
–––––––
The EV of the NPV is positive, +$9,200. The decision should therefore be to
undertake the investment, provided that the risk does not seem too great. In this
example there is a 10% probability that the NPV will be negative, –$8,000.
Management might therefore consider whether the investment is worth
undertaking, in view of this risk.
Exercise 1
A company is considering an investment project which would involve an
investment of $1,000,000. The expected returns will depend on economic conditions
over the next four years. The following estimates have been prepared. The cost of
capital is 10%.
Year
Strong
economy
Weak
economy
$ $
0 (1,000,000) (1,000,000)
1 400,000 100,000
2 600,000 300,000
3 400,000 200,000
4 300,000 50,000
Probability 0.75 0.25
Required
Calculate the EV of the NPV of the project, and recommend whether the project
should or should not be undertaken.
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3.3 Advantages and disadvantages of using expected values
The advantages of using expected values of the NPV are as follows:
It is a weighted average measure of all the possible outcomes. It is therefore,
arguably, a more appropriate measure of return than the most likely or most
probable EV of NPV.
It provides a single figure, not a range of different figures, for making an
investment decision.
The disadvantages of using expected values of the NPV are as follows:
The estimates of probabilities might be subjective, and based on judgement and
guesswork.
The EV of the NPV is not a value for any of the actual possible outcomes. In
other words, the EV itself will not happen. It is simply an average representing a
number of different possible outcomes.
An EV is much more reliable for estimating the average outcome from events
that will happen repeatedly, many times over. A weighted average is not nearly
as suitable for estimating the expected outcome for a once-only capital
expenditure project.
Most important of all, an EV does not provide any analysis of the project risk.
When capital investment projects are evaluated, and a decision is made whether
or not to undertake the investment, there should be a thorough analysis of the
risk as well as the expected returns.