ACCA P4 Advanced Financial Management - 2010 - Study text

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Chapter 7: Other aspects of capital investment appraisal

Capital rationing

 The nature of capital rationing

 Single period capital rationing

 Multi-period capital rationing and the simplex method

 The simplex method and your examination

3 Capital rationing

3.1 The nature of capital rationing

Capital investment decisions might be affected by capital rationing. Capital

rationing occurs when there is not enough capital available to invest in every project

with a positive NPV that the company would like to undertake. Since capital is in

short supply, a decision has to be made about which projects to invest in with the

capital that is available.

A distinction is sometimes made between ‘hard’ and ‘soft’ capital rationing.

 Hard capital rationing occurs when there is a real shortage of capital for

investment. For example, a company might be unable to raise new capital in the

capital markets or borrow large amounts from a bank. Its capital for investment

might therefore be limited to the amount of capital it adds to the business each

year as cash flows from retained profits.

 Soft capital rationing occurs when there is sufficient capital to invest in every

project, but management has taken a policy decision that spending on capital

investment should be limited to a budgeted maximum amount. The policy

decision therefore sets a limit on the amount of capital available.

Several methods have been devised for indicating which projects should be selected

for investment when there is capital rationing. These methods are based on the

following assumptions:

 The choice of projects or investments should have the objective of maximising

total NPV.

 The projects that are available for investment are comparable in terms of risk.

The method used to identify which projects to select depends on how many years of

capital rationing there will be. Capital rationing might be for one year only (single

period capital rationing) or might be for several years, and the investment projects

will require some additional investment in each of the years when there will be

capital rationing (multi-period capital rationing).

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3.2 Single period capital rationing

When there is capital rationing in one year only (usually Year 0), the selection of

projects for investment is normally straightforward. The method used depends on

which of two assumptions should be used:

 Alternative assumption 1: The projects available for selection are indivisible. The

choice is between investing in 100% of a project or not investing in the project at

all.

 Alternative assumption 2: The projects available for investment are divisible,

although there is a maximum amount that can be invested in any project. This

second assumption might seem unusual, although it is likely to feature in an

examination question on capital rationing.

When alternative 1 is the assumption, the combination of projects should be

selected:

 for which there is sufficient capital and

 which will maximise the total NPV.

When alternative 2 is the assumption, the combination of projects should be

selected:

 which uses all the capital available for investment, and

 maximises the NPV per $1 invested in the year of capital rationing.

Example

A company has $20 million to invest. There are four projects available for

investment, all similar in terms of risk. The amount of investment required and the

NPV of each project are as follows:

Project Capital required

in Year 0

Capital required

in Year 1

NPV

$m $m $m

A 8 0 + 3.0

B 5 1 + 2.0

C 6 0 + 2.5

D 10 2 + 4.2

Required

(a) Which projects should be selected if they are not divisible?

(b)

Which projects should be selected if they are divisible?

Answer

(a) Projects are not divisible

If the projects are not divisible, the first step is to identify the combinations of

projects that may be selected with the capital available in Year 0. The total

Chapter 7: Other aspects of capital investment appraisal

NPV for each combination should be calculated and the combination with the

highest total NPV should be selected.

Combination

of projects

Total capital

required in Year

Total

NPV

$m $m

A + B + C 19 (3 + 2 + 2.5) + 7.5

A + D 18 (3 + 4.2) + 7.2

B + D 15 (2 + 4.2) + 6.2

C + D 16 (2.5 + 4.2) + 6.7

Here, the combinations are easy to identify, and total NPV will be maximised

by investing in projects A, B and C.

(b) Projects are divisible

If the projects are divisible and there is a maximum investment in each

project, the aim should be to maximise the NPV per $1 invested in Year 0 (the

year of capital rationing: Year 1 is ignored because there is no capital rationing

in Year 1).

Project NPV Capital

required in

Year 0

NPV per $1

invested in

Year 0

Priority for

investment

$m $m $m

A + B + C 3.0 8 0.375 4

A + D 2.0 5 0.400 3

B + D 2.5 6 0.417 2

C + D 4.2 10 0.420 1

The choice of investments should be as follows:

Project Priority Capital

required in

Year 0

NPV per $1

invested in

Year 0

Total

NPV

$m $m $m

D 1

10 0.420 4.2

C 2

6 0.417 2.5

B (balance) 3

4 0.400 1.6

8.3

Total NPV is maximised by sending the $20 million in Year 0 by investing in

100% of Projects D and C, and investing the remaining $4 million in 80% of

Project B. The total NPV will be (in $ million) 4.2 + 2.5 + (80% × 2) = $8.3

million.

The total NPV should always be higher when projects are divisible than when

they are not divisible.

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3.3 Multi-period capital rationing and the simplex method

When there is capital rationing in more than one year, and some or all of the projects

require additional finance in each year where there will be capital rationing, a

different method is needed to identify the combination of projects that will

maximise total NPV,

where the projects are divisible.

The mathematical technique used to identify the NPV-maximising combination of

projects is called

linear programming, and the technique of linear programming

you are most likely to encounter is the

simplex method.

For your examination, you are not required to solve a linear programming problem.

However, you should understand how to formulate a linear programming problem,

and how the simplex method works.

The method will be explained using a simple example. (The example is simple

because there is capital rationing in two years only, and there are only three

projects. The same technique can be applied to problems when there is capital

rationing in more than two years, and when there are many different divisible

projects available for selection.)

Example

A company has $9 million to invest in Year 0 and $6 million to invest in Year 1.

There are three projects available for investment, all similar in terms of risk. The

amount of investment required and the NPV of each project are as follows:

Project Capital required

in Year 0

Capital required

in Year 1

NPV

$m $m $m

X 6 3 + 4.0

Y 2 1 + 2.0

Z 5 2 + 2.0

Required

Which projects should be selected if they are all divisible?

Answer

There is capital rationing in both Year 0 and Year 1 because in each year the capital

available for investment is less than the total needed to invest in all three projects.

Formulating the linear programming problem

A linear programming problem is stated as an objective function that is subject to

certain constraints or limitations. The objective function is to maximise or minimise

something. With capital rationing, the objective function should be to maximise the

total NPV, and the objective function can be stated as follows:

Chapter 7: Other aspects of capital investment appraisal

Objective function:

Maximise 4x + 2y + 2z

where x, y and z are the proportion of Projects X, Y and Z respectively chosen for

investment.

There are five constraints on investment. These are the amounts of capital available

in Year 0 and Year 1, and the fact that it is impossible to invest in more than 100% of

a project.

The

five constraints can be stated as follows:

Year 0 capital 6x + 2y + 5z

≤ 10.2

Year 1 capital 3x + y + 2z

≤ 6.0

Maximum investment in X x

≤ 1.0

Maximum investment in Y y

≤ 1.0

Maximum investment in Z z

≤ 1.0

(Note: The symbol ≤ means ‘is less than or equal to’.)

There are also non-negativity constraints, which mean that the value of x, y and z

cannot be a negative amount.

If there is a more complex problem, with capital rationing in more years, there will

be a constraint for each year in which the capital rationing will occur.

Formulating and interpreting the initial simplex tableau

The simplex method can be used to find the optimal solution to the linear

programming problem. The method is to test feasible solutions to the linear

programming, one feasible solution at a time, until the optimal solution is found

that maximises (or minimises) the value of the objective function.

Each feasible solution is tested in a table or ‘tableau’. A tableau should have:

 one column for each variable in the problem, one column for each constraint,

and a total column

 one row for each variable in the solution and a row for the objective function.

There will always be exactly as many variables in the solution as there are

constraints in the linear programming problem. In our example, this is 5.

In addition, we need to introduce some new variables into the problem. There

should be one variable for each constraint.

 Let Y

= the number of unused capital in Year 0

 Let Y

= the number of unused capital in Year 1.

 Let U

= the proportion of Project X not invested in.

 Let U

= the proportion of Project X not invested in.

 Let U

= the proportion of Project X not invested in.

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The initial tableau for this problem is constructed as follows:

Variable in the

solution

Total

(million)

0 9

0 6

0 1

1 1

Objective function

- 4

- 2

0 0

Notice that the figures in each row of the x, y and z columns are taken directly from

the linear programme, and the objective function on the bottom row shows the NPV

of each project with a minus sign.

The initial tableau tests the feasible solution that there is 9 million of unused capital

in Year 0 and 6 million of unused capital in Year 1. The proportion of each project

not invested in is 1.0 (100%) Y

, Y

, U

, and

are therefore the five variables in

this feasible solution, and the values of x, y and z are 0. The total NPV (bottom row,

total column) is $0 million.

Testing other feasible solutions

This is clearly not the optimal solution, and the simplex method now produces

another feasible solution where the value of the objective function is higher. The

second tableau might be as follows:

Variable in the

solution

Total

(million)

- 6

0 3.0

- 3

0 3.0

Project X (x)

0 1.0

1 1.0

Objective function

- 2

4.0

0 4.0

This tableau has introduced x into the solution in place of U

In this solution, there will be an investment in 100% of project X (since x = 1.0 in the

tableau). This will leave unused capital of $3.0 million in Year 0 and $3.0 million in

Year 2. The total NPV will be $4 million.

This is not the optimal solution, because there are still some minus signs in the

objective function row, and for every unit of y or z that is introduced into the

solution, the total NPV can be increased by $2 million.

Chapter 7: Other aspects of capital investment appraisal

The final solution and its interpretation: dual prices or shadow prices

Variable in the

solution

Total

(million)

Project Z (z)

- 6

0 0.2

- 3

0 1.6

Project X (x)

0 1.0

Project Y (y)

0 1.0

1 0.8

Objective function

0.6

1.6

0.8

0 6.4

The final tableau is shown here, with the solution that maximises total NPV. The

problem in this example is fairly simple, so the solution is quite straightforward.

To maximise total NPV, the company should invest in 100% of Project X, 100% of

Project Y and 20% of Project Z (since x = 1.0, y = 1.0 and z = 0.2). This will leave

unused capital of $1.6 million in Year 1 (since Y

= 1.6). The proportion of Project Z

not invested in is 0.8.

Total NPV will be $6.4 million.

The solution also shows the dual prices or shadow prices of the variables that are

not in the solution. These are Y

, U

and U

 All the available capital in Year 0 is used up by the solution. The dual price

indicates that if $1 of extra capital could be made available in Year 0, the total

NPV could be increased by $0.4.

 The maximum investment is made in project X, so U

= 0 in the final solution.

The dual price for U

indicates that if the maximum investment in project X

could exceed 100%, the total NPV could be increased by $1.6 million for every

additional project X that is available.

 The maximum investment is made in project Y, so U

= 0 in the final solution.

The dual price for U

indicates that if the maximum investment in project Y

could exceed 100%, the total NPV could be increased by $0.8 million for every

additional project Y that is available.

Dual prices for projects are not particularly significant. However,

the dual price or

shadow price for capital is significant

. It shows by how much total NPV could be

increased if more capital could be made available in that year (given no change in

any other constraint in the problem).

For example, this solution indicates that if the available capital in Year 0 could be

increased by, say, $1 million, from $9 million to $10 million, there would be a

different optimal solution and the total NPV for this solution would be higher by

$400,000 ($1 million × 0.4).

Paper P4: Advanced Financial Management

3.4 The simplex method and your examination

The study guide for the examination syllabus indicates that you need to know about

‘multi-period capital rationing to include the formulation of programming methods

and the interpretation of their output’. This indicates that some knowledge is

required for the examination of linear programming and the use of the simplex

method for solving capital rationing problems.

You should therefore try to familiarise yourself with the nature of a simplex tableau,

and what the solution tells you (including the meaning of dual prices or shadow

prices).

Chapter 7: Other aspects of capital investment appraisal

Real options

 Definition of a real option

 Real options theory

 Real options: the theoretical framework

 Different types of real option

 The valuation of real options

4 Real options

4.1 Definition of a real option

Options are normally associated with financial market derivative instruments and

share options of companies. A different type of option is a real option, sometimes

called an ‘embedded real option’.

A real option is an alternative or choice that exists with an investment project or

investment opportunity. They are real choices that management will be able to

make at some stage during the life of the project. For example, if a company

undertakes a particular project, there might be a real option (a choice, and not an

obligation) to take one or more of the following courses of action at some stage

during the project’s life:

 Make a further incremental investment, and increase the total amount invested

in the project.

 Abandon the project early because it is losing money.

 Defer the investment in the project, instead of having to invest immediately.

For example, a mining company might be negotiating a ten-year lease on a mine,

and it might want to include a ‘break clause’ (an option) in the lease contract that

will enable it to cancel the lease after five years. The option to cancel the lease and

abandon the investment would be a real option. The company would have to

consider how much it would be prepared to pay for the break clause to be included

in the terms of the lease.

They are called ‘real’ options because they are generally associated with choices in

relation to ‘real’ tangible assets.

Real options provide flexibility by giving choice – the flexibility for example to

avoid losses and to take opportunities that unexpectedly emerge.

Paper P4: Advanced Financial Management

4.2 Real options theory

Real options theory can be used for investment appraisal, as an alternative to simple

net present value analysis.

 Like NPV analysis, investment appraisal with real options analysis calculates a

present value for a project by discounting future cash flows.

 Unlike NPV analysis, a real options approach also puts a financial value to the

real options that are embedded in the investment opportunity, and which have

some value. The value of the real options is taken into consideration in deciding

whether to invest, or which of two or more alternative projects to invest in.

There are two aspects to real options theory:

 providing a theoretical framework for the valuation of real options in investment

appraisal

 developing methods of measuring the value of real options for the purpose of

investment appraisal.

4.3 Real options: the theoretical framework

When a company decides to invest in a capital project, it faces the risk that actual

results will be different from what was expected when the investment decision was

made. The risk in a project comes from a combination of:

 the uncertainty about the future, and the different situations that might arise,

and

 the means that management have at their disposal to deal with any of these

situations if they arise.

There are many different sources of uncertainty in a project. They include the risks

from:

 changes in the condition of the overall economy

 technological change

 a change in regulations

 unforeseen actions by a competitor.

Any of these events or changes in situation could affect the value of the investment.

Management deals with uncertainty in two stages.

 Stage 1. At the project selection stage. When the investment decision is made,

management considers the different possible outcomes and assesses the risk in

the project. Different investment options are considered, and management

decides to invest in the projects that appear to offer the best prospects for a good

return.

 Stage 2. Project management. After the decision has been made to invest in a

project, the project is managed. Management takes action to reduce the risks and

uncertainty in the project, and seeks to enhance the positive effects of

unexpected events.

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

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