Pike Robert, Neal Bill. Corporate finance and investment: decisions and strategy
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Chapter 8 Analysing investment risk 205
Table 8.5 UMK cost structure
Unit data £ £
Selling price 20
Less: Materials (6)
Labour (5)
Variable costs (1)
(12)
Contribution 8
Annual sales (units) 12,000
Total contribution 96,000
Less: Additional fixed costs (20,000)
Annual net cash flow 76,000
Present value (4 years at 10%)
240,920
Less: Capital outlay (200,000)
Net present value 40,920
Annual cash receipts
The break-even position is reached when annual cash receipts multiplied by the annu-
ity factor equal the investment outlay. The break-even cash flow is therefore the invest-
ment outlay divided by the annuity factor:
This is a percentage fall of
Annual fixed costs could increase by the same absolute amount of £12,909, or
Annual sales volume the break-even annual contribution is £63,091 £20,000 £83,091.
Sales volume required to break even is £83,091/£8 10,386, which is a percentage
decline of
Selling price can fall by:
a decline of
Variable costs per unit can rise by a similar amount:
Continued
£1.07
£12
100 8.9%
£1.07
£20
100 5.4%
£96,000 £83,091
12,000
£1.07 per unit
12,000 10,386
12,000
100 13.5%
£12,909
£20,000
100 64.5%
£76,000 £63,091
£76,000
17.0%
£200,000
3.17
£63,091
76,000 3.17
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206 Part III Investment risk and return
New risks put scenario planning in favour
FT
Who could have predicted the
horrific events of September 11,
2001? A 1999 US congressional
commission led by former sena-
tors Gary Hart and Warren
Rudman came close. It warned
that the US was ‘increasingly vul-
nerable to attack on our home-
land’ and that ‘rapid advances in
information and biotechnologies
will create new vulnerabilities’.
But perhaps more important
than the commission’s prophetic
messages was its approach.
Instead of forecasting a specific
future, it set out a collection of
possible attack scenarios. It then
evaluated national security by
analysing possible policies to pre-
pare for, or respond to them.
This approach – known as
scenario planning – has gained
renewed popularity among pub-
lic and private decision-makers.
In January this year, the New
England Journal of Medicine pub-
lished a scenario planning analysis
on whether US health workers or
the whole nation should be vacci-
nated against smallpox to counter
the threat of bio-terrorism. Presi-
dent George W. Bush decided to
inoculate 500,000 military person-
nel and 439,000 health workers.
Scenario planners face three
challenges. The first is construct-
ing meaningful scenarios. This
requires expert analysis of the
factors that affect the outcomes. A
second challenge is determining
the likelihoods of the scenarios.
Finally, planners must decide
on a good criterion for selecting
strategies. Most individuals and
institutions are risk-averse: they
value an uncertain reward at a
level significantly below the aver-
age level the reward in fact reach-
es. Strategies with higher average
pay-offs often entail greater risks.
Hence, scenario planning often
involves analysing the reward at
different levels of risk – much as
is done in financial planning.
What explains the recent inter-
est in scenario planning? For one
thing, we live in turbulent times.
Terrorism, political instability
and threats of war make scenar-
ios of extreme price fluctuations
in commodity and energy mar-
kets more likely. Severe acute res-
piratory syndrome, ‘mad cow
disease’ and foot-and-mouth dis-
ease have rekindled awareness of
the natural biological threats we
face. Accounting scandals force us
to second-guess what used to be
considered accurate information
about suppliers and customers. In
short, companies face far greater
risks than before. Indeed, when
Mattel used scenario planning to
formulate its 2002 strategy, it
considered scenarios with several
big customers (such as Kmart,
FAO and eToys) going bankrupt
and others (Wal-Mart) starting to
make their own toys.
Source: Awi Federgruen and Garrett Van Ryzin,
Financial Times, 19 August 2003, p. 11.
Sensitivity analysis, as applied in the above example, discloses that selling price and
variable costs are the two most critical variables in the investment decision. The deci-
sion-maker must then determine (subjectively or objectively) the probabilities of such
changes occurring, and whether he or she is prepared to accept the risks.
■ Scenario analysis
Sensitivity analysis considers the effects of changes in key variables only one at a time.
It does not ask the question: ‘How bad could the project look?’ Enthusiastic managers
can sometimes get carried away with the most likely outcomes and forget just what
might happen if critical assumptions – such as the state of the economy or competitors’
reactions – are unrealistic. Scenario analysis seeks to establish ‘worst’ and ‘best’ scenar-
ios, so that the whole range of possible outcomes can be considered. It encourages ‘con-
tingent thinking’, describing the future by a collection of possible eventualities.
Discount rate
The break-even annuity factor is £200,000/£76,000=2.63. Reference to the present value
annuity tables for four years shows that 2.63 corresponds to an IRR of 19 per cent. The error
in cost of capital calculation could be as much as nine percentage points before it affects the
decision advice.
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Chapter 8 Analysing investment risk 207
Market factors Investment factors Cost factors
Market size Investment outlay Variable costs
Market growth rate Project life Fixed costs
Selling price of product Residual value
Market share captured by the firm
Self-assessment activity 8.5
What do you understand by Monte Carlo simulation? When might it be useful in capital
budgeting?
(Answer in Appendix A at the back of the book)
■ Simulation analysis
An extension of scenario analysis is simulation analysis. Monte Carlo simulation is an
operations research technique with a variety of business applications. The computer
generates hundreds of possible combinations of variables according to a pre-specified
probability distribution. Each scenario gives rise to an NPV outcome which, along with
other NPVs, produces a probability distribution of outcomes.
One of the first writers to apply the simulation approach to risky investments was
Hertz (1964), who described the approach adopted by his consultancy firm in evaluat-
ing a major expansion of the processing plant of an industrial chemical producer. This
involved constructing a mathematical model that captured the essential characteristics
of the investment proposal throughout its life as it encountered random events.
A simulation model might consider the following variables, which are subject to
random variation.
Comparison is then possible between mutually exclusive projects whose NPV prob-
ability distributions have been calculated in this manner (Figure 8.6). It will be observed
that project A, with a higher expected NPV and lower risk, is preferable to project B.
Monte Carlo simulation
Method for calculating the
probability distribution of
possible outcomes
B
A
NPV
Frequency
0
Figure 8.6
Simulated probability
distributions
In practice, few companies use this risk analysis approach, for the following reasons:
1 The simple model described above assumes that the economic factors are unrelated.
Clearly, many of them (e.g. market share and selling price) are statistically interde-
pendent. To the extent that interdependency exists among variables, it must be speci-
fied. Such interrelationships are not always clear and are frequently complex to model.
2 Managers are required to specify probability distributions for the exogenous vari-
ables. Few managers are able or willing to accept the demands required by the sim-
ulation approach.
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208 Part III Investment risk and return
Two approaches are commonly used to incorporate risk within the NPV formula.
■ Certainty equivalent method
This conceptually appealing approach permits adjustment for risk by incorporating the
decision-maker’s risk attitude into the capital investment decision. The certainty equiv-
alent method adjusts the numerator in the net present value calculation by multiplying
the expected annual cash flows by a certainty equivalent coefficient. The revised for-
mula becomes:
Certainty equivalent method
where: is the expected net present value; is the certainty equivalent coefficient,
which reflect’s management’s risk attitude; is the expected cash flow in period t; i is
the riskless rate of interest; n is the project’s life; and is the initial cash outlay.
The numerator represents the figure that management would be willing to
receive as a certain sum each year in place of the uncertain annual cash flow offered
by the project. The greater is management’s aversion to risk, the nearer the certainty
equivalent coefficient is to zero. Where projects are of normal risk for the business, and
the cost of capital and risk-free rate of interest are known, it is possible to determine
the certainty equivalent coefficient.
Example
Calculate the certainty equivalent coefficient for a normal risk project with a one-year
life and an expected cash flow of £5,000 receivable at the end of the year. Shareholders
require a return of 12 per cent for projects of this degree of risk and the risk-free rate of
interest is 6 per cent.
The present value of the project, excluding the initial cost and using the 12 per cent
discount rate, is:
Using the present value and substituting the risk-free interest rate for the cost of
capital, we obtain the certainty equivalent coefficient:
The management is, therefore, indifferent as to whether it receives an uncertain
cash flow one year hence of £5,000 or a certain cash flow of £4,732 (i.e. £5,000 0.9464).
■ Risk-adjusted discount rate
Whereas the certainty equivalent approach adjusted the numerator in the NPV formula,
the risk-adjusted discount rate adjusts the denominator:
0.9464
a
1£4,4642
11.062
£5,000
a £5,000
1 0.06
£4,464
PV
£5,000
1 0.12
£4,464
1aX
t
2
I
o
X
t
aNPV
NPV
a
N
t 1
aX
t
11 i2
t
I
o
8.7 ADJUSTING THE NPV FORMULA FOR RISK
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Chapter 8 Analysing investment risk 209
Adjusting the discount rate: Chox-Box Ltd
Chox-Box Ltd is a manufacturer of confectionery currently appraising a proposal to launch
a new product that has had very little pre-launch testing. It is estimated that this proposal
will produce annual cash flows in the region of £100,000 for the next five years, after which
product profitability declines sharply. As the proposal is seen as a high-risk venture, a 12
per cent risk premium is incorporated in the discount rate. The risk-adjusted cash flow,
before discounting at the risk-free discount rate, is therefore £89,286 in Year 1 (£100,000/1.12),
falling to £56,742 in Year 5 (£100,000/1.12
5
).
To what extent does this method reflect the actual riskiness of the annual cash flows for
Years 1 and 5? Arguably, the greatest uncertainty surrounds the initial launch period. Once
the initial market penetration and subsequent repeat orders are known, the subsequent
sales are relatively easy to forecast. Thus, for Chox-Box, a single risk-adjusted discount rate
is a poor proxy for the impact of risk on value over the project’s life, because risk does not
increase exponentially with the passage of time, and, in some cases, actually declines over
time. The Eurotunnel project provides another illustration of this. By far the greatest risks
were in the initial tunnelling and development phases.
A deeper understanding of the relationship between the certainty equivalent and risk-
adjusted discount rate approaches may be gained by reading the appendix to this chapter.
246 810
Time (years)
Expected
cash flow
Cash-flow (£)
Figure 8.7
How risk is assumed to
increase over time
Risk-adjusted discount rate method
where k is the risk-adjusted rate based on the perceived degree of project risk.
The higher the perceived riskiness of a project, the greater the risk premium to be added to
the risk-free interest rate. This results in a higher discount rate and, hence, a lower net
present value.
Although this approach has a certain intuitive appeal, its relevance depends very
much on how risk is perceived to change over time. The risk-adjusted discount rate
involves the impact of the risk premium growing over time at an exponential rate, imply-
ing that the riskiness of the project’s cash flow also increases over time. Figure 8.7 demon-
strates this point. Although the expected cash flow from a project may be constant over
its ten-year life, the riskiness associated with the cash flows increases with time. However,
if risk did not increase with time, the risk-adjusted discount rate would be inappropriate.
NPV
a
N
t 1
X
t
11 k2
t
I
o
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210 Part III Investment risk and return
Table 8.6
Risk analysis in 100
large UK firms
1975 (%) 1980 (%) 1986 (%) 1992 (%) 1997 (%)
Sensitivity analysis 28 42 71 86 89
Best/worst case analysis n.a. n.a. 93 95 n.a.
Reduced payback period 25 30 61 59 11
Risk-adjusted rate 37 41 61 64 50
Probability analysis 9 10 40 47 42
Beta analysis – – 16 20 5
Source: Pike (1996), Arnold and Hatzopoulos (2000)
The big gamble: Airbus rolls out its new weapon in its battle with Boeing
The biggest bet placed by Europe’s aerospace industry was
officially launched in January 2005. The Airbus A380 – a
twin-decked behemoth with seats for 555 passengers – was
rolled out in Toulouse and gives the company a complete
range of models to challenge Boeing, ending the lucrative
monopoly Boeing has had in the very large aircraft market
for 35 years.
Noel Forgeard, Airbus chief executive, says the company
had made a ‘successful metamorphosis to a world leader’.
He claims the group is almost twice as profitable as
Boeing’s commercial airplanes division, helped by a ‘huge,
relentless effort to reduce unit cost and grow our produc-
tivity: it is the reason why we can gain market share and
grow profitably.’
Boeing has certainly looked increasingly vulnerable. The
group’s critics say its long years of success led to compla-
cency and it allowed the pace of product innovation to
slow as it prioritised short-term earnings over investment.
‘Boeing has struggled with the development work needed
to take the company into the 21st century,’ says Tim
Clark, president of Emirates, the Dubai-based airline that
is one of the world’s most important buyers of long-haul
aircraft and will be the biggest operator of the Airbus
A380.
Airbus’s A380 ‘will change the game for long-haul air-
lines and airports,’ says Chris Avery, aviation analyst at JP
Morgan. ‘With operating costs 15 per cent below the
B747-400, we believe A380 operations will have an advan-
tage on long-haul services in markets between Europe and
Asia, across the Pacific and across the Atlantic.’ Boeing,
however, thinks the A380 is a white elephant, designed for
a world that no longer needs aircraft of such great size.
Airbus has 149 orders so far, still short of the 250 that it
estimates are needed to give a profit. The $11 billion
investment of public and private money is a huge gamble.
Boeing and Airbus agree that air traffic over the next 20
years is expected to increase annually on average by about
5 per cent. But they differ greatly on how airlines will
accommodate that. Boeing’s vision is based on the ‘frag-
mentation’ of aviation markets, reflecting passengers’ pref-
erence for more point-to-point, non-stop services and more
frequent services instead of being routed to destinations via
connecting hubs. Airbus accepts that fragmentation, but it
also expects consolidation on the main trunk routes.
MINI CASE
8.8 RISK ANALYSIS IN PRACTICE
To what extent do companies employ the techniques discussed in this chapter? Table 8.6
shows changes over the past twenty-five years.
Sensitivity analysis and best/worst case (or scenario) analysis are conducted in
almost all larger UK companies. Approximately one half of larger firms adjust the dis-
count rate for risk. Less common, however, are techniques requiring managers to
assign probabilities to possible outcomes; managers prefer to assess the likelihood of
outcomes in a more subjective manner. Finally, there is little real evidence that Beta
analysis (based on the Capital Asset Pricing Model and discussed in Chapter 10) is
used extensively in industry.
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Chapter 8 Analysing investment risk 211
Airbus seems to be winning the argument. In recent
years, it has won over some operators that previously used
only Boeing aircraft. But Airbus still has plenty to prove. In
Japan, Boeing reigns supreme. There, the government and
the aerospace industry are backing Boeing’s 7E7.
Another challenge is the weakness of the dollar against
the euro. This has the potential to undermine its long-term
competitiveness. For most of 2005 and 2006 Airbus is pro-
tected, having hedged about $40 billion of revenues at
around It has also taken the precaution of pricing
most of its purchases in dollars – even in Europe –
thereby transferring exchange risk to suppliers. Gerald Blanc,
Airbus executive vice-president operations, warned, ‘This will
probably impair our ability to invest as much in research and
development as we have done so far.’
A third challenge is Airbus’s ability to show it will not be
thrown off course by the change of management at the
top. The tussle for supremacy in managing Airbus between
the parent company’s dominant French and German share-
holders means that it is currently without a chief executive
at a time when the A380 project is about to enter the cru-
cial phase of flight testing, certification and the build-up of
production before the first delivery in 2006.
Source: Based on Financial Times, 17 January 2005.
Required
Identify the strategic and financial risks in the Airbus
A380 project and suggest how they should be assessed
and managed.
;1>$1.
Risk is an important element in virtually all investment decisions. Because most peo-
ple in business are risk-averse, the identification, measurement and, where possible,
reduction of risk should be a central feature in the decision-making process. The evi-
dence suggests that firms are increasingly conducting risk analysis. This does not
mean that the risk dimension is totally ignored by other firms; rather, they choose to
handle project risk by less objective methods such as experience, feel or intuition.
We have defined what is meant by risk and examined a variety of ways of measur-
ing it. The probability distribution, giving the probability of occurrence of each possi-
ble outcome following an investment decision, is the concept underlying most of the
methods discussed. Measures of risk, such as the standard deviation, indicate the
extent to which actual outcomes are likely to vary from the expected value.
Key points
■ The expected NPV, although useful, does not show the whole picture. We need to
understand managers’ attitudes to risk and to estimate the degree of project risk.
■ Three types of risk are relevant in capital budgeting: project risk in isolation, the
project’s impact on corporate risk and its impact on market risk. The last two are
addressed more fully in the following two chapters.
■ The standard deviation, semi-variance and coefficient of variation each measure, in
slightly different ways, project risk.
■ Sensitivity analysis and scenario analysis are used to locate and assess the potential
impact of risk on project performance. Simulation is a more sophisticated approach,
which captures the essential characteristics of the investment that are subject to
uncertainty.
■ The NPV formula can be adjusted to consider risk. Adjustment of the cash flows
is achieved by the certainty equivalent method. The risk-adjusted discount rate
increases the risk premium for higher-risk projects.
SUMMARY
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212 Part III Investment risk and return
Further reading
A fuller treatment of risk is found in Levy and Sarnat (1994). Useful research studies on the use
of risk analysis are given in Pike (1988, 1996), Pike and Ho (1991), Mao and Helliwell (1969) and
Bierman and Hass (1973).
For simplicity, we have so far assumed single-period investments and conveniently
ignored the fact that investments are typically multi-period. As risk is to be specifical-
ly evaluated, cash flows should be discounted at the risk-free rate of interest, reflect-
ing only the time-value of money. To include a risk premium within the discount rate,
when risk is already considered separately, amounts to double-counting and typically
understates the true net present value. The expected NPV of an investment project is
found by summing the present values of the expected net cash flows and deducting
the initial investment outlay. Thus, for a two-year investment proposal:
where is the expected NPV, is the expected value of net cash flow in Year 1,
is the expected value of net cash flow in Year 2, is the cash investment outlay and
i is the risk-free rate of interest.
A major problem in calculating the standard deviation of a project’s NPVs is that
the cash flows in one period are typically dependent, to some degree, on the cash flows
of earlier periods. Assuming for the present that cash flows for our two-period project
are statistically independent, the total variance of the NPV is equal to the discounted
sum of the annual variances.
For example, the Bronson project, with a two-year life, has an initial cost of £500 and
the possible payoffs and probabilities outlined in Table 8.7. Applying the standard
deviation and expected value formulae already discussed, we obtain an expected NPV
of £268 and standard deviation of £206.
Assuming a risk-free discount rate of 10 per cent, the expected NPV is:
NPV
300
11.102
600
11.102
2
500 £268
I
o
X
2
X
1
NPV
NPV
X
1
1 i
X
2
11 i2
2
I
o
APPENDIX
MULTI-PERIOD CASH FLOWS AND RISK
Table 8.7
Bronson project pay-
offs with independent
cash flows
Probability Year 1 cash flow (£) Year 2 cash flow (£)
0.1 100 200
0.2 200 400
0.4 300 600
0.2 400 800
0.1 500 1,000
Expected value £300 £600
Standard deviation £109 £219
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Chapter 8 Analysing investment risk 213
The standard deviation of the entire proposal is found by discounting the annual
variances to their present values, applying the equation:
In our simple case, this is:
The project therefore offers an expected NPV of £268 and a standard deviation of
£206.
■ Perfectly correlated cash flows
At the other extreme from the independence assumption is the assumption that the
cash flows in one year are entirely dependent upon the cash flows achieved in previ-
ous periods. When this is the case, successive cash flows are said to be perfectly cor-
related. Any deviation in one year from forecast directly affects the accuracy of
subsequent forecasts. The effect is that, over time, the standard deviation of the prob-
ability distribution of net present values increases. The standard deviation of a stream
of cash flows perfectly correlated over time is:
Returning to the example in Table 8.7, but assuming perfect correlation of cash
flows over time, the standard deviation for the project is:
Thus the risk associated with this project is £280, assuming perfect correlation,
which is higher than that for independent cash flows. Obviously, this difference would
be considerably greater for longer-lived projects.
In reality, few projects are either independent or perfectly correlated over time. The
standard deviation lies somewhere between the two. It will be based on the formula
for the independence case, but with an additional term for the covariance between
annual cash flows.
■ Interpreting results
While decision-makers are interested to know the degree of risk associated with a
given project, their fundamental concern is whether the project will produce a positive
net present value. Risk analysis can go some way to answering this question. If a pro-
ject’s probability distribution of expected NPVs is approximately normal, we can esti-
mate the probability of failing to achieve at least zero NPV. In the previous example,
the expected NPV was £268. This is standardised by dividing it by the standard devi-
ation using the formula:
Z
X NPV
s
£280
s
£109
1.1
£219
11.12
2
s
a
N
t 1
s
t
11 i2
t
s
B
s
1
2
11 i2
2
s
2
2
11 i2
4
B
12,000
11.12
2
48,000
11.12
4
£206
s
B
a
N
t 1
s
t
2
11 i2
2
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214 Part III Investment risk and return
where X in this case is zero and Z is the number of standardised units. Thus, in the
case of the independent cash flow assumption, we have:
Reference to normal distribution tables reveals that there is a 0.0968 probability that
the NPV will be zero or less. Accordingly there must be a or 90.32 per cent
probability of the project producing an NPV in excess of zero.
It is probably unnecessary to attempt to measure the standard deviation for every
project. Even the larger European companies tend to use probability analysis sparingly
in capital project analysis. Unless cash flow forecasting is wildly optimistic, or the
future economic conditions underlying all investments are far worse than anticipated,
the bad news from one project should be compensated by good news from another
project.
Sometimes, however, a project is of such great importance that its failure could
threaten the very survival of the business. In such a case, management should be fully
aware of the scale of its exposure to loss and the probability of occurrence.
■ Probability of failure: Microloft Ltd
Microloft Ltd, a local family-controlled company specialising in attic conversions, is
currently considering investing in a major expansion giving wider geographical cov-
erage. The NPV from the project is expected to be £330,000 with a standard deviation
of £300,000. Should the project fail (perhaps because of the reaction by major competi-
tors), the company could afford to lose £210,000 before the bank manager ‘pulled the
plug’ and put in the receiver. What is the probability that this new project could put
Microloft out of business?
We need to find the value of Z where X is the worst NPV outcome that Microloft
could tolerate:
Assuming the outcomes are normally distributed, probability tables will show a 3.6
per cent chance of failure from accepting the project. A family-controlled business, like
Microloft, may decide that even this relatively small chance of sending the company
on to the rocks is more important than the attractive returns expected from the project.
£210 £330
£330
1.8
Z
X NPV
s
11 0.09682
1.30 standardised units
Z
0 £268
£206
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