ACCA P4 Advanced Financial Management - 2010 - Study text

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Chapter 4: DCF: risk analysis

Fiscal risk varies between countries. Some countries have a reputation for fiscal

stability, so that any changes in tax are fairly insignificant. In other countries there is

a much higher risk of tax changes. Obviously, companies considering an investment

in a country, even their domestic country, should take fiscal risk into consideration

when deciding whether or not to invest in a project.

Example

The oil and natural gas industry is an example of an industry where fiscal risk has

had a significant impact on capital expenditure decisions.

Oil companies purchase concessions to explore for oil or gas, and extract any that

they find. If they fail to find any oil or gas, they will lose their investment. However,

they face a risk that if they are successful in finding and extracting oil, the

government will decide to tax their profits. One way of doing this is to charge a

windfall tax on the profits of firms in the industry that operate in the country.

In the past the UK government has charged a windfall tax on the profits of

companies in the North Sea oil business. The matter was debated in Parliament in

2002, where it was claimed that if the government raised the rate of tax on the

profits of offshore oil and gas extraction to 40% or 50%, further exploration by the

oil companies would cease to be viable.

Reducing fiscal risk

Fiscal risk is unavoidable for companies that pay tax on their profits. However, the

risk might sometimes be reduced for large companies. A large company can try to

negotiate with the government of a country before deciding whether to invest in the

country. They might argue that they will not invest unless the government gives an

assurance of fiscal stability.

The risk of an unexpected additional tax (windfall tax) cannot be removed simply

by a government’s promise, because promises might be broken. Another approach

that might be used by some companies is to offer to pay a royalty tax on their

output. For example, an oil company might undertake to pay a royalty on every

barrel of oil that it extracts under a concession agreement. If the company pays tax

in this way, so that the royalty payments increase with the volume of business, the

government might be less inclined to charge additional tax on what it sees as

excessive profits.

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Monte Carlo simulation

 The nature of Monte Carlo simulation

 Using computers for Monte Carlo simulation

 Monte Carlo simulation and capital investment appraisal

 The significance of the results from a Monte Carlo simulation

 Project volatility

2 Monte Carlo simulation

2.1 The nature of Monte Carlo simulation

Monte Carlo simulation is a method of measuring an expected outcome by means of

generating multiple trials or iterations, in order to determine the expected value of

the outcome and also to measure the variability or risk.

This method of determining the EV of an outcome gets its name from the large

number of random outcomes that are used to obtain the measurement, which can be

compared with the random outcomes that would be expected on the gambling

tables of a casino in Monte Carlo.

Monte Carlo simulation depends on the fact that by taking the outcomes of a large

number of individual random events (iterations), an accurate measurement of the

expected value and probability distribution of the outcome will be obtained.

A simple example might help to make this point more clear.

Example

Suppose that you roll two dice and calculate the total of the values that turn up on

each dice. Each dice has six sides numbered 1 to 6, so the total from rolling two dice

could be anything between 2 (1 +1) and 12 (6 + 6).

If you rolled the dice just once, the total could be anywhere in the range 2 to 12. If

you roll the dice again, the total might be different. If you roll the dice enough times,

you will get a large number of totals, all in the range 2 to 12. Some totals will appear

more frequently than others. For example, a combined total of 6 for the two dice

(which results from 1 + 5, 2 + 4 and 3 + 3) will occur more often than a total of 12

(where both dice must turn up as 6).

By rolling the two dice many times over, an average total value will be obtained. (It

should be 7, if the dice are rolled enough times. However there might be a slight

error, and the average obtained from your simulation might be slightly above or

below 7.0. The risk of error is reduced by rolling the dice more times – doing more

‘iterations’ in your simulation.)

Chapter 4: DCF: risk analysis

The simulation will also produce a probability distribution of total values. This

shows the proportion or number of times that each total value was obtained from

the two dice. The distribution will look something like this.

Number

percentage

of times

2 3 4 5 6 7 8 9 10 11 12

Total of the

two dice

The expected value and the probability distribution of the random outcome can be

measured. The usual method of calculating the variability in the value is to calculate

a standard deviation. In this example, we can calculate the expected value of the

total of the two dice (7.0) and we can measure the variability in the total above or

below 7.0 as a standard deviation of the expected value. (The standard deviation is

about 2.4, but the workings are not shown here.)

2.2 Using computers for Monte Carlo simulation

The example of the two dice is a very simple one, and in practice simulation models

are used for the analysis of more complex outcomes. The use of Monte Carlo

simulation modelling is made possible by computers, which can produce a large

number of iterations quickly, to produce a reliable expected value and probability

distribution of the outcome.

There are various software packages that might be used for Monte Carlo simulation.

Simulation models can be constructed, for example, in a spreadsheet.

Another simplified example might help to illustrate the construction of a simulation

model on spreadsheet software such as MS Excel.

Example

Suppose that you want to use a Monte Carlo simulation model to estimate the EV of

the total cost of a project, and the standard deviation of this expected value.

The project consists of six activities. You have estimated the minimum cost and the

maximum cost for each activity, as follows:

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Activity Minimum cost Maximum cost

$000

1 10

2 15

3 8

4 16

5 21

6 14

Total 84

124

The cost for each activity could be any value in the range between the minimum and

maximum cost, and there is an equal probability of each value for cost occurring.

The total cost of the project will therefore be a random variable with a value

between $84,000 and $104,000.

We can use simulation to calculate the EV of the total cost and a standard deviation

for this EV.

 The model should be used to generate random values for the cost of each

activity.

 The randomly selected values for the six activities should be added, to obtain a

total cost for the project.

 This process should be repeated through a large number of iterations.

 The EV of the total project cost is the average of the random total costs produced

by all these iterations.

 A standard deviation of the EV of total cost can also be calculated from the data.

A spreadsheet in Excel might be prepared as shown below. The example shows just

one iteration to produce a random total cost for the project.

Cell A

1 Activity 1

Total

2 Min cost ($000) 10

3 Max cost ($000) 20

124

4 Iterations

5 Iteration 1 12

102

The Excel spreadsheet package includes a random number selection function

=RAND( ). The formula in cell B5 (and B6, B7, B8 and so on for each successive

iteration) is =RAND ( )*(20000 – 10000) + 10000. This will produce a random number

anywhere in the range between 10,000 and 20,000.

The total column H is simply a total of the values in columns B to G for that row.

For example, the formula in cell H5 will be =SUM(B5:G5).

There should be a row in the spreadsheet for each iteration; therefore the number of

iterations required should be decided when the model is constructed. The model

can also be programmed to calculate an EV and standard deviation of the EV of the

total project cost.

Chapter 4: DCF: risk analysis

2.3 Monte Carlo simulation and capital investment appraisal

The previous example might help to suggest how Monte Carlo simulation might be

used in capital investment appraisal. There are many different cash flows in a

project whose value might be uncertain, but which can be estimated as a probability

distribution. A Monte Carlo simulation model can be created, and the model can be

used to calculate the NPV of the project for many iterations, in order to obtain an

expected value of the project NPV and a standard deviation of the expected NPV.

Example

Suppose that you want to use a Monte Carlo simulation model to estimate the NPV

of a proposed capital investment project, where there is some uncertainty about the

units of sale each year and the variable cost per unit sold.

Probability distributions might be prepared for each of these unknown variables,

and random numbers allocated to each possible value, as follows:

Units per

year

Probability

Random

numbers

Variable

cost

Probability

Random

numbers

5,000 0.15 00 – 14 $5.00 0.25 00 – 24

6,000 0.20 15

–

$5.50

0.60

–

7,000 0.30 35

–

$6.00

0.15

–

8,000 0.25 65

–

9,000 0.10 90

–

With a more complex model, a large number of different variables could be given

various possible values, with associated probabilities and random number

allocations.

The model is then used to produce a large number of possible outcomes and NPVs

for the project. For each possible outcome, random values are given to each of the

uncertain variables by generating random numbers and deciding the value in

accordance with the random number generated.

Alternatively, a model might be constructed based on the assumption that sales

units might be anywhere in the range 5,000 to 9,000 each year and the variable cost

per unit might be anywhere between $5.00 and $6.00. Random numbers can be

generated for both units of sale and variable cost per unit, in many iterations, using

the same method as described above for the project total costs.

For your examination, you are not required to construct a Monte Carlo simulation

model for capital investment appraisal and risk analysis, but you are expected to

know how a simulation model would be constructed, and to interpret the output

obtained from using the model.

2.4 The significance of the results from a Monte Carlo simulation

The output from a Monte Carlo simulation is the results from a large number of

iterations. These results are used to calculate an expected value (average value) for

the item that is being measured, and a standard deviation.

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The standard deviation provides a measure of risk. A large standard deviation

relative to the average or expected value indicates high risk. The actual result could

be significantly higher or lower than the expected value. A small standard deviation

relative to the size of the EV indicates lower risk and a greater probability that the

actual result will be close to the expected value.

Risk analysis using the standard deviation

A significant feature of Monte Carlo simulation is that the range of outputs from the

many iterations usually follow a normal distribution (provided that the number of

iterations is large enough). If it can be assumed that there is a normal distribution of

possible results around the expected value, normal distribution analysis can be used

to assess probabilities of what the actual outcome might be.

In the previous example of the project activity costs, if the outputs from the

iterations follow a normal distribution, the EV of total cost and the standard

deviation can be used to assess:

 the probability that the total project cost will not exceed a particular amount, say

$95,000

 what the maximum total cost of the project should be, at the 95% or 99% level of

probability.

Note

If you are not familiar with normal distribution and the analysis of risk with normal

distribution, you should read the final section of this chapter. This provides a

short

beginner’s guide to normal distribution analysis

How accurate is the estimate of the expected value?

It is also possible to estimate the accuracy of the estimate of the EV of a project NPV

using Monte Carlo simulation. The accuracy of the estimated EV can be measured

statistically as a total error, as follows:

Error = (3 ×

σ) / √ N

where

= the standard deviation of the expected value of the NPV, obtained from the

simulation model

N = the number of iterations used.

For example, if the EV of the NPV is + $500,000, the standard deviation of the EV of

the NPV is $10,000 and 400 iterations of the simulation model were made to

produce these results, the total error in the estimate of the EV of the NPV would be:

Error = (3 × 10,000) / √ 400 = $1,500.

For a larger value of N (the larger the number of iterations), the error is smaller. Not

surprisingly, this indicates that greater accuracy is obtained from simulation by

doing a larger number of iterations of the model.

Chapter 4: DCF: risk analysis

2.5 Project volatility

You might come across the term ‘project volatility’. Volatility in this context refers to

the possibility that a project’s NPV might be much higher or lower than the

expected value of the NPV. (In the financial markets, volatility refers to the extent to

which share prices, interest rates or exchange rates might fluctuate up and down.)

High volatility indicates much greater variability and so much greater risk.

In financial management, volatility is measured statistically, usually as a standard

deviation. The volatility of a capital investment project can be measured as either:

 the standard deviation of the expected NPV of the project, or

 the standard deviation of the PV of the annual net cash flows from the project.

If you are given the volatility of a project in an examination question, check

carefully to see whether volatility is defined. If it is not specifically defined in the

question, you should assume that it is the standard deviation of the expected NPV

of the project.

Example

Monte Carlo simulation has been used to calculate the expected value of the NPV of

a proposed capital investment project and the volatility of the project.

The EV of the NPV is + $106,000 and the project volatility, measured as the standard

deviation, is $80,000.

This indicates that the EV of the NPV is greater than $0 (‘break-even’) by a Z score

of:

106,000/80,000 = 1.325 standard deviations.

Normal distribution tables show that the probability of a value greater than 1.325

standard deviations below the mean of a normal distribution is 0.5000 – 0.4079 =

0.0921 – about 9.2%.

106,000

NPV

1.325 

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Normal distribution tables can also be used to calculate the following probabilities:

 At the 95% confidence level, the NPV will not be lower than 1.645 standard

deviations below the mean or EV.

N(0.95)

σ = 1.645 × $80,000 = $131,600.

At the 95% confidence level, the NPV will not be worse than $106,000 - $131,600

= - $25,600.

 At the 99% confidence level, the NPV will not be lower than 2.327 standard

deviations below the mean or EV.

N(0.99)

σ = 2.327 × $80,000 = $186,160.

At the 99% confidence level, the NPV will not be worse than $106,000 - $186,160

= - $80,160.

Chapter 4: DCF: risk analysis

Project value at risk

 Annual volatility of a project

 The meaning of value at risk and project value at risk

 Calculating project value at risk

3 Project value at risk

3.1 Annual volatility of a project

Volatility can be measured over differing periods of time. In the previous example

of project volatility, the volatility was measured over the entire life of the project.

Volatility can also be measured over shorter time periods, such as annual volatility.

In capital investment appraisal, annual volatility is the standard deviation of the PV

of annual cash flows from the project.

For any item with a variable value, for which volatility is measured, the statistical

volatility increases with the length of the time period over which it is measured. The

relationship is as follows:

= σ

× √T

where

is the volatility (standard deviation) over the longer time period

is the volatility (standard deviation) over the shorter time period

T is the number of short time periods that make up the long time period.

Example

Expected annual cash flows are $100,000 and the volatility of annual cash flows is

$15,000. Cash flows in any one year are independent of what the cash flows were in

the previous years.

Over a five year period, expected cash flows are $500,000 and the five-year volatility

of cash flows is:

$15,000 × √ 5 = $33,541.

3.2 The meaning of value at risk and project value at risk

The concept of Value at Risk (VaR) has been developed in the financial services

industry, mainly by banks. In banking, Value at Risk can be measured for items

such as the risk of losses from credit risk (bad debts) and the risk of losses from a

fall in the market value of investments.

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For a given period of time, such as one day, one week, one month, three months or

one year, and at a given confidence level, Value at Risk can be stated as a single

figure. For example, a bank might use its VaR computer model to calculate that on

any one day at the 95% confidence level, the Value at Risk is $10 million. This means

that at the 95% level of confidence, the maximum loss the bank will suffer on any

one day is $10 million.

The concept of VaR can be extended to capital investment appraisal, although at the

moment it is not widely used in practice.

A project value at risk is the maximum amount, at a given confidence level, by

which the actual NPV from a project will be worse than the expected value of the

NPV. It can therefore be used to assess the risk in a capital investment project,

which should help management to decide whether or not to invest in the project,

taking into consideration the risk as well as the expected return (expected NPV).

3.3 Calculating project value at risk

Project value at risk can be calculated in either of two ways:

 using the standard deviation of the project NPV: this was illustrated by the

example in the previous section of this chapter

 using the annual volatility of the PV of cash flows.

Example

A simulation model has been used to calculate the expected value of the NPV of a

project. This is $282,000. The project has an expected life of ten years, and the

volatility of the PV of the annual cash flows is $30,000.

Normal distribution tables can also be used to calculate the following probabilities:

 At the 95% confidence level, the project value at risk is:

N(0.95) 30,000 √10 = 1.645 × $94,868 = $156,058.

At the 95% confidence level, the NPV will not be worse than $282,000 – $156,058

= $125,942.

 At the 99% confidence level, the project value at risk:

N(0.99) 30,000

√10 = 2.327 × $94,868 = $220,758.

At the 99% confidence level, the NPV will not be worse than $282,000 - $220,758

= $61,242.

This suggests a low level of risk with this capital investment project, because the

NPV is expected to be positive even at the 99% confidence level for project value at

risk.

ACCA P4 Advanced Financial Management - 2010 - Study text - Emile Woolf Publishing

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