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
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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).