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2 Example: Probability estimates of cash flows excluding discounting

# 2 Example: Probability estimates of cash flows excluding discounting

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Year 1 cash flow
\$'000
4,000
3,000
(1,500)

Probability

0.20
0.50
0.30

Year 2 cash flow
\$'000
2,500
1,000
(3,500)

Probability

0.10
0.65
0.25

Solution
Opening balance Year 1 cash flow
\$'000
\$'000
500
4,000
500
3,000
500
(1,500)

Closing balance
for year 1
\$'000
4,500
3,500
(1,000)

Probability

0.20
0.50
0.30

Expected value
\$'000
900
1,750
(300)
2,350

The expected value of the cash balance at the end of year 1 is \$2,350,000.
There is a 0.3 or 30% chance that the overdraft limit will be exceeded.
Year 1
closing
balance
\$'000
(a)
4,500
4,500
4,500
3,500
3,500
3,500
(1,000)
(1,000)
(1,000)

Year 2 cash
Probability
flow
Probability
\$'000
(b)
(c)
(d)
0.20
2,500
0.10
0.20
1,000
0.65
0.20
(3,500)
0.25
0.50
2,500
0.10
0.50
1,000
0.65
0.50
(3,500)
0.25
0.30
2,500
0.10
0.30
1,000
0.65
0.30
(3,500)
0.25

Year 2
closing
balance
\$'000
(a) + (c)
7,000
5,500
1,000
6,000
4,500
0
1,500
0
(4,500)

Joint
probability

(b)  (d)
0.020
0.130
0.050
0.050
0.325
0.125
0.030
0.195
0.075

Expected
value
\$'000

140.0
715.0
50.0
300.0
462.5
0.0
45.0
0.0
(337.5)
2,375.0

The expected value of the cash balance at the end of year 2 is \$2,375,000.
There is a 0.075 or 7.5% chance that the overdraft limit will be exceeded.

3.3 Problems with expected values
There are the following problems with using expected values in making investment decisions.

An investment may be one-off, and 'expected' NPV may never actually occur. For example, if there
is a 50% probability that the NPV will be + \$10,000 and a 50% probability that it will be \$(2,000),
the EV of the NPV is + \$4,000. On this basis the project will go ahead. However, an NPV of \$4,000
is not expected to happen. The NPV will be either plus \$10,000 or minus \$2,000.

Assigning probabilities to future events and outcomes is usually highly subjective.

Expected values do not evaluate the range of possible NPV outcomes.

4.1 Simulation
FAST FORWARD

Other risk adjustment techniques include the use of simulation models, discounted payback and riskadjusted discount rates.

Part D Investment appraisal  10: Project appraisal and risk

197

Simulation will overcome problems of having a very large number of possible outcomes, as well as the
correlation of cash flows (a project which is successful in its early years is more likely to be successful in
its later years).

4.2 Example: Simulation model
The following probability estimates have been prepared for a proposed project.
Year

Cost of equipment
Revenue each year

0
1-5

Running costs each year

1-5

Probability

1.00
0.15
0.40
0.30
0.15
0.10
0.25
0.35
0.30

\$
(40,000)
40,000
50,000
55,000
60,000
25,000
30,000
35,000
40,000

The cost of capital is 12%. Assess how a simulation model might be used to assess the project's NPV.

Solution
A simulation model could be constructed by assigning a range of random number digits to each possible
value for each of the uncertain variables. The random numbers must exactly match their respective
probabilities. This is achieved by working upwards cumulatively from the lowest to the highest cash flow
values and assigning numbers that will correspond to probability groupings, as follows.
Revenue

\$
40,000
50,000
55,000
60,000
*
**
***

Prob
0.15
0.40
0.30
0.15

Running costs
Random
numbers
00 – 14
15 – 54
55 – 84
85 – 99

*
**
***

\$
25,000
30,000
40,000
40,000

Prob
0.10
0.25
0.35
0.30

Random
numbers
00 – 09
10 – 34
35 – 69
70 – 99

Probability is 0.15 (15%). Random numbers are 15% of range 00 – 99.
Probability is 0.40 (40%). Random numbers are 40% of range 00 – 99 but starting at 15.
Probability is 0.30 (30%). Random numbers are 30% of range 00 – 99 but starting at 55.

For revenue, the selection of a random number in the range 00 and 14 has a probability of 0.15. This
probability represents revenue of \$40,000. Numbers have been assigned to cash flows so that when
numbers are selected at random, the cash flows have exactly the same probability of being selected as is
indicated in their respective probability distribution above.
Random numbers would be generated, for example by a computer program, and these would be used to
assign values to each of the uncertain variables.
For example, if random numbers 37, 84, 20, 01, 56 and 89 were generated, the values assigned to the
variables would be as follows.
Calculation

1
2
3

198

Revenue
Random number

37
20
56

10: Project appraisal and risk  Part D Investment appraisal

Value
\$
50,000
50,000
55,000

Costs
Random number

84
01
89

Value
\$
40,000
25,000
40,000

A computer would calculate the NPV many times over using the values established in this way with more
random numbers, and the results would be analysed to provide the following.
(a)
(b)

An expected NPV for the project
A statistical distribution pattern for the possible variation in the NPV above or below this average

The decision whether to go ahead with the project would then be made on the basis of expected return
and risk.

FAST FORWARD

The discounted payback period (DPP) is the time it will take before a project's cumulative NPV turns from
being negative to being positive.
The payback method of investment appraisal, discussed in Chapter 7, recognises uncertainty in
investment decisions by focusing on the near future. Short-term projects are preferred to long-term
projects and liquidity is emphasised.
The discounted payback period is the length of time before the cumulative PV of cash inflows from the
projects begins to exceed the initial outflow. It is similar to the payback method, but uses discounted cash
flows rather than non-discounted cash flows to measure the payback period.
Discounted payback uses discounted cash flows. This is also known as adjusted payback.

For example, if we have a cost of capital of 10% and a project with the cash flows shown below, we can
calculate a discounted payback period.
Year

0
1
2
3
4
5

Cash flow
\$

(100,000)
30,000
50,000
40,000
30,000
20,000

Discount
factor
10%
1.000
0.909
0.826
0.751
0.683
0.621

Present
value
\$

(100,000)
27,270
41,300
30,040
20,490
12,420
NPV = 31,520

Cumulative
NPV
\$

(100,000)
(72,730)
(31,430)
(1,390)
19,100
31,520

The DPP is early in year 4.
It may be approximated as 3 years + [1,390/(1,390 + 19,100)] × 12 months
= 3 years 0.8 months, say 3 years 1 month.
A company can set a target DPP, and choose not to undertake any projects with a DPP in excess of a
certain number of years, say, five years.

The approach has all the perceived advantages of the payback period method of investment appraisal: it
is easy to understand and calculate, and it provides a focus on liquidity where this is relevant. In addition,
however, it also takes into account the time value of money. It therefore bridges the gap between the
theoretically superior NPV method and the regular payback period method.
However, it does differ from NPV in that the discount rate used is the unadjusted cost of capital whereas
NPV often uses an adjusted rate to reflect project risk and uncertainty.
Because the DPP approach takes the time value of money into consideration, it produces a longer
payback period than the non-discounted payback approach, and takes into account more of the project's
cash flows.

Part D Investment appraisal  10: Project appraisal and risk

199

Another advantage it has over traditional payback is that it has a clear accept or reject criterion. Using
payback, acceptance of a project depends on an arbitrarily determined cut-off time. Using DPP, a project is
acceptable if it pays back within its lifetime.
DPP still shares one disadvantage with the payback period method: cash flows which occur after the
payback period are ignored (although, as the DPP is longer than the payback period, fewer of these are
ignored).
One way of dealing with risk is to shorten the payback period required. A maximum payback period can
be set to reflect the fact that risk increases the longer the time period under consideration. However, the
disadvantages of payback as an investment appraisal method (discussed in Section 3.2 of Chapter 7)
mean that discounted payback cannot be recommended as a method of adjusting for risk.

Investors want higher returns for higher risk investments. The greater the risk attached to future returns,
the greater the risk premium required. Investors also prefer cash now to later and require a higher return
for longer time periods.
In investment appraisal, a risk-adjusted discount rate can be used for particular types or risk classes of
investment projects to reflect their relative risks. For example, a high discount rate can be used so that a
cash flow which occurs quite some time in the future will have less effect on the decision. Alternatively,
with the launch of a new product, a higher initial risk premium may be used with a decrease in the
discount rate as the product becomes established.

200

10: Project appraisal and risk  Part D Investment appraisal

Chapter Roundup

Risk analysis can be applied to a proposed capital investment where there are several possible outcomes
and, on the basis of past relevant experience, probabilities can be assigned to the various outcomes and
estimated cash flows that could prevail.
Uncertainty analysis can be applied to a proposed capital investment where there are several possible
outcomes but there is little past relevant experience to enable the probability of the alternative outcomes to
be predicted.

There are a wide range of techniques for incorporating risk into project appraisal.

Sensitivity analysis assesses how responsive the project's NPV is to changes in the variables used to
calculate that NPV. Sensitivity analysis is one particular approach to uncertainty analysis. The certaintyequivalent approach is another; this involves the conversion of the expected cash flows of the project to
riskless equivalent amounts.

A probability analysis of expected cash flows can often be estimated and used both to calculate an
expected NPV and to measure risk.

Other risk adjustment techniques include the use of simulation models, discounted payback and riskadjusted discount rates.

The discounted payback period (DPP) is the time it will take before a project's cumulative NPV turns from
being negative to being positive.

Part D Investment appraisal  10: Project appraisal and risk

201

Quick Quiz
1

Give three examples of uncertain independent variables on which the NPV of a project may depend.

2

How are simulation models constructed?

3

Briefly list four ways in which managers can reduce risk.

4

Sensitivity analysis allows for uncertainty in project appraisal by assessing the probability of changes in
the decision variables.
True
False

5

Fill in the blanks.
The ........................................ is where expected cash flows are converted to riskless equivalent amounts.

6

Give two examples of ways that risk can be measured in probability analysis.

7

Expected values can help an accountant evaluate the range of possible net present value outcomes.
True
False

8

An investment project has the following discounted cash flows (\$’000).
Year

0
1
2
3
4

0%
(90)
30
30
30
30
30

Discount rate
10%
(90)
27.3
24.8
22.5
20.5
5.1

20%
(90)
25.0
29.8
17.4
14.5
(12.3)

The required rate of return on investment is 10% per annum.
What is the discounted payback period of the investment project?
A
B
C
D

202

Less than 3 years
3 years
Between 3 years and 4 years
More than 4 years

10: Project appraisal and risk  Part D Investment appraisal

1

Any three of:
(a)
Selling price
(b)
Sales volume
(c)
Cost of capital
(d)
Initial cost
(e)
Operating costs
(f)
Benefits

2

By assigning a range of random number digits to each possible value of each of the uncertain variables.

3

(a)
(b)
(c)
(d)

4

False. Sensitivity analysis does not assess probability.

5

Certainty-equivalent approach

6

Calculating the worst possible outcome and its probability
Calculating the probability that the project will fail to achieve a positive NPV

7

False

8

C

Set maximum payback period
Use high discounting rate
Use sensitivity analysis to determine the critical factors within the decision-making process
Use pessimistic estimates

At the end of year 3, \$74,600 has been 'paid back'. The remaining \$15,400 for payback will be

Now try the questions below from the Practice Question Bank

Number

Level

Marks

Time

Section A Q20

Examination

2

4 mins

Section A Q21

Examination

2

4 mins

Section C Q11

Introductory

N/A

29 mins

Part D Investment appraisal  10: Project appraisal and risk

203

204

10: Project appraisal and risk  Part D Investment appraisal

Specific investment
decisions

Topic list

Syllabus reference

D4 (a)

2 Asset replacement decisions

D4 (b)

3 Capital rationing

D4 (c)

Introduction
In this chapter, we consider specific investment decisions, such as whether to
lease or buy an asset, when to replace an asset and how to assess projects
when capital is a scarce resource.

205

Study guide
Intellectual level
D4

Specific investment decisions (lease or buy; asset replacement; capital
rationing)

(a)

Evaluate leasing and borrowing to buy using the before- and after-tax costs
of debt.

2

(b)

Evaluate asset replacement decisions using equivalent annual cost and
equivalent annual benefit.

2

(c)

Evaluate investment decisions under single period capital rationing,
including:

2

(i)

The calculation of profitability indexes for divisible investment projects

(ii)

The calculation of the NPV of combinations of non-divisible investment
projects

(iii)

A discussion of the reasons for capital rationing

Exam guide
You may be asked to calculate the results of different options and careful, methodical workings will be
essential. These calculations can be quite difficult and will need lots of practice.

FAST FORWARD

12/07, 12/09, 12/13

Leasing is a commonly used source of finance.
We distinguish three types of leasing:

Operating leases (lessor responsible for maintaining asset)
Finance leases (lessee responsible for maintenance)
Sale and leaseback arrangements

1.1 The nature of leasing
Rather than buying an asset outright, using either available cash resources or borrowed funds, a business
may lease an asset.

Key terms

Leasing is a contract between a lessor and a lessee for hire of a specific asset by the lessee from a
manufacturer or vendor of such assets.
The lessor has ownership of the asset and so provides the initial finance for the asset.
The lessee has possession and use of the asset on payment of specified rentals over a period.

1.1.1 Examples of lessors

Banks
Insurance companies

1.1.2 Types of asset leased

206

Office equipment
Computers
Cars
Commercial vehicles

11: Specific investment decisions  Part D Investment appraisal

Aircraft
Ships
Buildings

1.2 Operating leases
Key term

Operating lease is a lease where the lessor retains most of the risks and rewards of ownership.
Operating leases are rental agreements between a lessor and a lessee, for a relatively short period of
time. It is useful to think of operating leases as short-term rental agreements.
(a)

The lessor supplies the equipment to the lessee.

(b)

The lessor is responsible for servicing and maintaining the leased equipment.

(c)

The period of the lease is fairly short, less than the expected economic life of the asset. At the end
of one lease agreement, the lessor can either lease the same equipment to someone else and
obtain a good rent for it, or sell the equipment secondhand.

1.3 Finance leases
Key term

A finance lease is a lease that substantially transfers all the risks and rewards of ownership of an asset to
the lessee. It is an agreement between the lessee and the lessor for most or all of the asset's expected
useful life.
There are other important characteristics of a finance lease.
(a)

The lessee is responsible for the upkeep, servicing and maintenance of the asset.

(b)

The lease has a primary period covering all or most of the useful economic life of the asset. At the
end of this period, the lessor would not be able to lease the asset to someone else, because the
asset would be worn out or near the end of its useful life. The lessor must therefore ensure that the
lease payments during the primary period pay for the full cost of the asset as well as providing the
lessor with a suitable return on their investment.

(c)

At the end of the primary period the lessee can normally continue to lease the asset for an
indefinite secondary period, in return for a very low nominal rent, sometimes called a 'peppercorn
rent'. Alternatively, the lessee might be allowed to sell the asset on a lessor's behalf (since the
lessor is the owner) and perhaps to keep most of the sale proceeds.

Part D Investment appraisal  11: Specific investment decisions

207