ACCA paper f9 financial management study materials F9FM session07 d08

SESSION 07 – PROJECT APPRAISAL UNDER RISK

OVERVIEW
Objective
To appraise investment projects where the outcome is not certain.

Definitions
Sources of risk

RISK AND
UNCERTAINTY

REDUCTION OF
RISK

SENSITIVITY
ANALYSIS

SIMULATION

STATISTICAL
MEASURES

Definition
Method
Advantages
Limitations

Use
Stages
Advantages

Limitations

Expected values
Standard deviation

0701


SESSION 07 – PROJECT APPRAISAL UNDER RISK

1

RISK AND UNCERTAINTY

1.1

Definition

1.1.1

Risk

A condition in which several possible outcomes exist, the probabilities of
which can be quantified from historical data.

1.1.2

Uncertainty

The inability to predict possible outcomes due to a lack of historical data

being available for quantification.

1.2

Sources of risk in projects

The major risks to the success of an investment project will be the variability of the future
cash flows. This could be the variability of income streams or the variability of cost cash
flows or a combination of both.

2

SENSITIVITY ANALYSIS

Definition
The analysis of changes made to significant variables in order to determine
their effect on a planned course of action.
The cash flows, probabilities, or cost of capital are varied until the decision changes, i.e. the
NPV becomes zero. This will show the sensitivity of the decision to changes in those
elements.
Therefore the estimation of IRR is an example if sensitivity analysis, in this case on the cost
of capital.
Sensitivity analysis can also be referred to as “what if?” analysis.

2.1

Method

Step 1

Calculate the NPV of the project on the basis of best estimates.

Step 2

For each element of the decision (cash flows, cost of capital)
calculate the change necessary for the NPV to fall to zero.

The sensitivity can be expressed as a % change.
For an individual cash flow in the computation

0702


SESSION 07 – PROJECT APPRAISAL UNDER RISK
Sensitivity =

NPV
× 100%
PV of flow considered

Commentary
For change in sales volume, the factor to consider is contribution. This may involve
combining a number of flows.

Example 1
Williams has just set up a company, JPR Manufacturing Ltd, and estimates its
cost of capital to be 15%. His 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 be used to produce 15,000 deluxe pairs of rugby boots per
annum generating a contribution of $2.75 per pair. He estimates that annual

fixed costs will be $15,000 per annum.

Required:
(a) Determine, on the basis of the above figures, whether the project is
worthwhile.
(b) Calculate what percentage changes in the factors would cause your
decision in (a) change.
(i)
(ii)
(iii)
(iv)
(v)

initial investment
volume
fixed costs
scrap value
cost of capital

Comment on your results.
Ignore tax.

0703


SESSION 07 – PROJECT APPRAISAL UNDER RISK

Solution
(a)

Time

0
1 − 15
1 − 15
15

Initial cost
Contribution
Fixed costs
Scrap value

Cash flow
$

DF @ 15%

PV
$

_______
_______

(b)

The sensitivity of the decision in (a) can be calculated by expressing the
NPV as a percentage of the various factors.
(i) Initial investment

Sensitivity =

(ii) Volume

The PV figure of contribution is directly proportional to volume.
Sensitivity =

(iii) Fixed costs

Sensitivity =

(iv) Scrap value

Sensitivity =
(v) Sensitivity to cost of capital

This can be found by calculating the project’s IRR:
Year

0
1-15
15
NPV

0704

Cash flow
$
(150,000)
26,250
15,000

factor

1

Present value
$

_______
_______


SESSION 07 – PROJECT APPRAISAL UNDER RISK
IRR

= r1 +

NPV1
(r2 − r1 )
NPV1 − NPV2

=

2.2

Advantages of sensitivity analysis
It gives an idea of how sensitive the project is to changes in any of the original estimates.
It directs management attention to checking the quality of data for the most sensitive
variables.
It identifies the Critical Success Factors for the project and directs project management.
It can be easily adapted for use in spreadsheet packages.

2.3

Limitations
Although it can be adapted to deal with multi-variable changes, sensitivity is normally
only used to examine what happens when one variable changes and others remain
constant.
Assumes data for all other variables is accurate.
Without a computer it can be time-consuming.
Probability of changes is not considered.

3

SIMULATION

3.1

Use of simulation

Simulation is a technique which allows more than one variable to change at the same time.
One example of simulation is the “Monte Carlo” method. Calculations will not be required
in the exam, an awareness of the stages is sufficient.

3.2

Stages in a Monte Carlo simulation

(1) Specify the major variables.
(2) Specify the relationship between the variables.
(3) Attach probability distributions to each variable and assign random numbers to reflect

the distribution.
(4) Simulate the environment by generating random numbers.

0705


SESSION 07 – PROJECT APPRAISAL UNDER RISK
(5) Record the outcome of each simulation.
(6) Repeat simulation many times to obtain a probability distribution of the likely outcomes.

3.3

Advantages
It gives more information about the possible outcomes and their relative probabilities.
It is useful for problems which cannot be solved analytically.

3.4

Limitations
It is not a technique for making a decision, only for obtaining more information about
the possible outcomes.
It can be very time-consuming without a computer.
It could prove expensive in designing and running the simulation, even on a computer.
Simulations are only as good as the probabilities, assumptions and estimates made.

4

STATISTICAL MEASURES

4.1

Expected values

The quantitative result of weighting uncertain events by the probability of their occurrence.

4.1.1

Calculation

Expected value

= weighted arithmetic mean of possible outcomes.
= x p(x)

∑

Where x = value of an outcome, p(x) = probability of that outcome , ∑ = sum

Example 2
State of market
Probability
Project 1
Project 2
Project 3

Diminishing
0.4
100
0
180

Static
0.3
200
500
190

Expanding
0.3
1,000
600
200

Figures represent the net present value of projects under each market state in
$m.

Required:
Determine which is the best project based on expected values.

0706


SESSION 07 – PROJECT APPRAISAL UNDER RISK

Solution
Project 1

Expected value =

Project 2

Expected value =

Project 3

Expected value =

The best project based on expected values is

4.1.2

Advantages

It reduces the information to one number for each choice.
The idea of an average is readily understood.

4.1.3

Limitations

The probabilities of the different possible outcomes may be difficult to estimate.
The average may not correspond to any of the possible outcomes.
Unless the same decision has to be made many times, the average will not be achieved;
it is therefore not a valid way of making a decision in “one-off” situations.
The average gives no indication of the spread of possible results, i.e. it ignores risk.

4.2

Standard deviation

A measure of variation of numerical values from a mean value.
A measure of spread i.e. it indicates the likely level of variation from an expected value.
Exam questions are more likely to provide standard deviation for interpretation, rather than
to require its calculation.

4.2.1

Calculation

σ = standard deviation

=

x

=

each observation

x

=

mean of observations

Prob (x)

=

probability of each observation

∑ (x − x )

2

prob ( x )

Note that variance = σ2

0707


SESSION 07 – PROJECT APPRAISAL UNDER RISK

Example 3
Using the information from Example 2, calculate the standard deviation for
each project.

Solution
Project 1
Project 2
Project 3

4.2.2

Advantages

It gives an idea of the spread of possible results around the average.
It can be used in further mathematical analysis, in particularly using Normal
Distributions.”

4.2.3

Limitations

The calculation of standard deviation can be difficult.
The exact meaning is not widely understood by non-financial managers.

5

REDUCTION OF RISK

Ways of reducing project risk:
Setting a maximum payback period.
Use of a higher discount rate − therefore reducing the influence of distant cash flows.
Select projects with a combination of low standard deviation and acceptable average
predicted outcome.
More effort directed to Critical Success Factors indicated by sensitivity analysis.

0708


SESSION 07 – PROJECT APPRAISAL UNDER RISK

Key points
Exam calculations on project risk are likely to focus on sensitivity analysis
i.e. finding the value of key variables at which NPV = 0.
Adjusting the discount rate to reflect a project’s risk is dealt with later in
the session on the Capital Asset Pricing Model (CAPM).

FOCUS
You should now be able to:
distinguish between risk and uncertainty;
evaluate the sensitivity of project NPV to changes in key variables;
explain the role of simulation in generating a probability distribution for the NPV of a
project;
apply the probability approach to calculating expected NPV of a project and the
associated standard deviation.

0709


SESSION 07 – PROJECT APPRAISAL UNDER RISK

EXAMPLE SOLUTIONS
Solution 1 — Sensitivity analysis
(a)

Time

0
1 − 15
1 − 15
15

Initial cost
Contribution
Fixed costs
Scrap value

Cash flow
$
(150,000)
41,250
(15,000)
15,000

DF @ 15%

1
5.847
5.847
0.123

PV
$
(150,000)
241,189
(87,705)
1,845
_______

5,329
_______
The project is worthwhile as NPV is positive
(b)

The sensitivity of the decision in (a) can be calculated by expressing
the NPV as a percentage of the various factors.
(i) Initial investment

If the initial investment rises by more than $5,329, the project would be
rejected.
Sensitivity =

5,329
× 100 = 3.6%
150 ,000

(ii) Volume

The PV figure of contribution $241,189 is directly proportional to
volume. If this PV is reduced by more than $5,329, the project would
be rejected.
Sensitivity =

5 ,329
× 100 = 2.2%
241,189

(iii) Fixed costs

Sensitivity =

0710

5,329
× 100 = 6.1%
87 ,705

SESSION 07 – PROJECT APPRAISAL UNDER RISK
(iv) Scrap value

Sensitivity =

5 ,329
× 100 = 289%
1,845

From the above calculations the decision to accept the project is
extremely sensitive to most of the figures given in the question. The
project will be rejected in the event of small rises in the initial
investment or fixed cost figures or falls in contribution or volume. It
could be seen, for instance, that the project just breaks even if fixed
costs become $15,000 × 1.06 = $15,900.
The scrap value is relatively irrelevant to the investment decision – we
would have to pay to have the plant taken away before the project
would be rejected.
(v) Sensitivity to cost of capital

This can be found by calculating the project’s IRR, which is probably
only marginally above 15%.
Year

Cash flow
$
(150,000)
26,250
15,000

0
1-15
15

16% factor

1
5.575
0.108

Present value
$
(150,000)
146,344
1,620
_______

NPV at 16%
IRR

= r1 +

(2,036)
_______
NPV1
(r − r )
NPV1 − NPV2 2 1

= 15% +

5,329
(16% − 15%)
5,329 + 2,036

= 15.7%
If the cost of capital rises from 15% to more than 15.7% the project
would be rejected.

Solution 2 — Expected values
Project 1

Expected value =

100 × 0.4 + 200 × 0.3 + 1,000 × 0.3

= 400

Project 2

Expected value =

0 × 0.4 + 500 × 0.3 + 600 × 0.3

= 330

Project 3

Expected value =

180 × 0.4 + 190 × 0.3 + 200 × 0.3

= 189

Therefore, based on expected values, Project 1 should be adopted.

0711


SESSION 07 – PROJECT APPRAISAL UNDER RISK

Solution 3 — Standard deviation
Project 1

( 100 − 400 ) 2 × 0.4 + (200 − 400) 2 × 0.3 + (1,000 − 400) 2 × 3

=
Project 2

( 0 − 330 )2 × 0.4 + (500 − 330)2 × 0.3 + (600 − 330) 2 × 0.2
=

Project 3

74 ,100 = 272
( 180 − 189 ) 2 × 0.4 + (190 − 189) 2 × 0.3 + (200 − 189)2 × 0.3

=

0712

156,000 = 395

69 = 8.3