Pike Robert, Neal Bill. Corporate finance and investment: decisions and strategy

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Chapter 5 Investment appraisal methods 135

rationing may arise because of the following:

1 Management sets maximum limits on borrowing and is unable or unwilling to raise

additional equity capital in the short term. Investment is restricted to internally gen-

erated funds.

2 Management pursues a policy of stable growth rather than a fluctuating growth

pattern with its attendant problems.

3 Management imposes divisional ceilings by way of annual capital budgets.

4 Management is highly risk-averse and operates a rationing process to select only

highly profitable projects, hoping to reduce the number of project failures.

The capital budget forms an essential element of the company’s complex planning

and control process. It may sometimes be expedient for capital expenditure to be

restricted – in the short term – to permit the proper planning and control of the organ-

isation. Divisional investment ceilings also provide a simple, if somewhat crude,

method of dealing with biased cash flow forecasts. Where, for example, a division is

in the habit of creating numbers to justify the projects it wishes to implement, the insti-

tution of capital budget ceilings forces divisional management to set its own priorities

and to select those offering highest returns.

It is clear that capital rationing can be explained, in part, by imperfections in both

the capital and labour markets and agency costs arising from the separation of own-

ership from management. Of particular relevance are the problems of information

asymmetry and transaction costs.

Information asymmetry

Shareholders and other investors in a business do not possess all the information available

to management. Nor do they always have the necessary expertise to appreciate fully the

information they do receive. Capital rationing may arise because senior managers, con-

vinced that their set of investment proposals is wealth-creating, cannot convince a more

sceptical group of potential investors who have far less information on which to make an

assessment and who may be influenced by the company’s recent performance record.

Transaction costs

The issuing and other costs associated with raising long-term capital do not vary in

direct proportion to the amount raised. Corporate treasurers in large organisations will

not want to go to the capital market each year for relatively small sums of money if the

costs can be significantly reduced by raising much larger sums at less frequent intervals.

Capital rationing is therefore a distinct possibility in the intervening years, although this

usually means delaying the start date for investments rather than outright rejection.

■ One-period capital rationing in Mervtech plc

The simplest form of capital rationing arises when financial limits are imposed for a sin-

gle period. For that period of time, the amount of funds available becomes the limiting

factor. The manufacturing division of Mervtech plc has been set an upper limit on cap-

ital spending for the coming year of £20 million. It is not normal practice for the group

to set investment ceilings, and it is anticipated that the capital constraint will not extend

into future years. Assuming a cost of capital of 10 per cent, which of the investment

opportunities set out in Table 5.8 should divisional management select?

In the absence of any financial constraint, projects A–D, each with positive net pres-

ent values, would be selected. Once this information has been communicated to

investors, the total stock market value would, in theory at least, increase by £44 mil-

lion – the sum of their net present values.

However, a financial constraint may prevent the selection of all profitable projects.

If so, it becomes necessary to select the investment package that offers the highest net

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136 Part II Investment decisions and strategies

present value within the £20 million expenditure limit. A simple method of selecting

projects under these circumstances is the profitability index. Recall that this measure

is defined as:

Project selection is made on the basis of the highest ratio of present value to invest-

ment outlay. This method is valuable under conditions of capital rationing because it

focuses attention on the net present value of each project relative to the scarce resource

required to undertake it. Appraising projects according to the NPV per £1 of invest-

ment outlay can give different rankings from those obtained from application of the

NPV rule. For example, while in the absence of capital rationing, project A ranks high-

est (using the NPV rule), project B ranks highest when funds are limited, as shown in

Table 5.9. Assuming project independence and infinite divisibility, divisional manage-

ment will obtain the maximum net present value from its £20 million investment

expenditure permitted by accepting projects B and D in total and £7 million or 7/15 of

project A.

However, the profitability index rarely offers optimal solutions in practice. First,

few investment projects possess the attribute of divisibility. Where it is possible for

projects to be scaled down to meet expenditure limits, this is frequently at the expense

of profitability. Let us suppose that projects are not capable of division. How would

this affect the selection problem? The best combination of projects now becomes A and

B, giving a total net present value of £23 million. Project D, which ranked above A

using the profitability index, is now excluded. Even more fundamental than this, how-

ever, is the limitation that the profitability index is appropriate only when capital

rationing is restricted to a single period. This is not usually the case. Firms experienc-

ing either hard or soft capital rationing tend to experience it over a number of periods.

In summary, the profitability index provides a convenient method of selecting proj-

ects under conditions of capital rationing when investment projects are divisible and

independent, and when only one period is subject to a resource constraint. Where,

as is more commonly the case, these assumptions do not hold, investment selec-

tions should be made after examining the total net present values of all the feasible

Profitability index 

Present value

Investment outlay

Table 5.8

Investment opportuni-

ties for Mervtech plc

Cash flows (£m) Present NPV

Project Initial value at

cost (£m) Year 1 Year 2 at 10% (£m) 10% (£m)

A3015

B138

C219

D2012

E17

3101020

11128

121212

1055

171715

Table 5.9

NPV vs. PI for Mervtech

plc

Project Profitability index Outlay (£m) Outlay (£m) NPV (£m)

B 2.6 5 accept 5 8

D 2.5 8 accept 8 12

A 2.0 15 accept 7/15 7 7

C 1.7 12 reject – –

E 0.8 20 reject – –

60 38 27

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Chapter 5 Investment appraisal methods 137

alternative combinations of investment opportunities falling within the capital outlay

constraints.

■ Multi-period capital rationing

Many business problems have similar characteristics to those exhibited in the capital

rationing problem, namely:

1 Scarce resources have to be allocated between competing alternatives.

2 An overriding objective that the decision-maker is seeking to attain.

3 Constraints, in one form or another, imposed on the decision-maker.

As the number of alternatives and constraints increases, so the decision-making

process becomes more complex. In such cases, mathematical programming models are

particularly valuable in the evaluation of decision alternatives, for two reasons:

1 They provide descriptive representations of real problems using mathematical

equations. Because they capture the critical elements and relationships existing in

the real system, they provide insights about a problem without having to experi-

ment directly on the actual system.

2 They provide optimal solutions – that is, the best solution for a given problem

representation.

A mathematical programming approach to solving more complex capital rationing

problems is provided in Appendix II to this chapter.

Self-assessment activity 5.6

What do you understand by ‘soft’ and ‘hard’ forms of capital rationing? Give two

approaches available to resolve capital rationing problems.

(Answer in Appendix A at the back of the book)

We have examined a number of commonly employed investment appraisal techniques

and asked the question: to what extent do they assist managers in making wealth-

creating decisions? The primary methods advocated involve discounting the incre-

mental cash flows resulting from the investment decision, although non-discounting

techniques are useful secondary methods for evaluating capital projects.

Key points

■ The net present value (NPV) method discounts project cash flows at the firm’s

required return and then sums the cash flows. The decision rule is: accept all proj-

ects whose NPV is positive.

■ The internal rate of return (IRR) is that discount rate which, when applied to proj-

ect cash flows, produces a zero NPV. Projects with IRRs above the required return

are acceptable.

■ The profitability index (PI) is the ratio of the present value of project benefits to the

present value of investment costs. The decision rule is to accept projects with a PI

greater than 1.

Continued

SUMMARY

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APPENDIX I

MODIFIED IRR

138 Part II Investment decisions and strategies

Further reading

■ The NPV, IRR and PI methods give the same investment advice for independent

projects. But where projects are mutually exclusive, differences can arise in rankings.

■ The NPV approach is viewed as more sound than the IRR method because it

assumes reinvestment at the required return rather than the project’s IRR.

■ The modified IRR (MIRR) is that rate of return which, when the initial outlay is

compared with the terminal value of the project’s cash flows reinvested at the cost

of capital, gives an NPV of zero. This method provides a rate of return consistent

with the NPV approach. This is discussed in Appendix I.

■ Payback is a useful method, but ignores cash flows beyond the payback period.

Simple payback also ignores the time-value of money.

■ Accounting rate of return (ARR) compares the average profit of the project against

the book value of the asset acquired. Its main merit is that, as a measure of prof-

itability, it can be related to the accounts of the business. However, it takes no

account of the timing of cash flows or of the size and life of the investment.

■ Capital rationing, where it exists, tends to be of the ‘softer’ form where manage-

ment voluntarily imposes investment ceilings in the short term.

■ Single-period capital rationing is resolved by ranking projects according to their

profitability index. More complex multi-period capital rationing problems demand

a mathematical programming approach.

Most good finance texts cover the topic of investment appraisal well, including Brealey, Myers and

Allen (2005). These texts also address the capital rationing problem. More detailed treatment of

capital rationing is found in Pike (1983), Elton (1970), Lorie and Savage (1955) and Weingartner

(1977). For a fuller discussion on the modified IRR, see McDaniel et al. (1988).

Most managers prefer the IRR to the NPV method. The modified IRR seeks to adjust

the IRR so that it has the same reinvestment assumption as the NPV approach.

■ The modified internal rate of return (MIRR)

MIRR is that rate of return which, when the initial outlay is compared with the termi-

nal value of the project’s net cash flows reinvested at the cost of capital, gives an NPV

of zero.

This involves a two-stage process:

1 Calculate the terminal value of the project by compounding forward all interim

cash flows at the cost of capital to the end of the project.

2 Find the rate of interest that equates the terminal value with the initial cost.

We established earlier that the Lara proposal offered an NPV of £4,252 (Table 5.1)

and an IRR of approximately 19 per cent (Table 5.3). Table 5.10 shows that by com-

pounding the interim cash flows at 14 per cent to the end of Year 4, the project offers

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Chapter 5 Investment appraisal methods 139

Table 5.10

Modified IRR for Lara

Year Cash flow (£) Future value factor @ 14% Terminal value (£)

(a) Find the terminal value

1 16,000 23,704

2 16,000 20,794

3 16,000 1.14 18,240

4 12,000 1.00 12,000

74,738

(b) Find the rate of interest (denoted by x) which equates the terminal value with

initial cost:

Using tables (Appendix C) for four years we find that 17 per cent gives a PVIF of 0.534.

To check: 39,910

less initial investment (40,000)

NPV (90) i.e. close to zero

The modified IRR is approximately 17 per cent compared with the IRR of 19.3 per cent.

You might like to check this out using an Excel spreadsheet. In cells A1 to A5 type

in -40,000, 16,000, 16,000, 16,000, 12,000. In cell A6 click on the fx icon and select

Financial/MIRR. In the box enter for Values, A1:A5, and .14 for both Finance rate and

Reinvestment rate.

£74,738  0.534 

PVIF

1x%, 4yrs2

 £40,000>£74,738  0.535

11.142

a terminal value of £74,738. To find the Year 4 present value factor that comes clos-

est to equating the terminal value with the initial outlay, we divide the initial out-

lay by the terminal value and look up the interest rate that gives this factor in Year

4 (see Appendix B). For the Lara project, the MIRR is approximately 17 per cent, a

good 2 per cent below the IRR figure. For more profitable projects, the deviation

would be greater.

Self-assessment Activity 5.7

Describe how the modified IRR is calculated. What advantages does the MIRR have over

the IRR in assessing capital investment decisions?

(Answer in Appendix A at the back of the book)

APPENDIX II

MULTI-PERIOD CAPITAL RATIONING AND MATHEMATICAL

PROGRAMMING

Where an overriding financial objective exists (such as maximising shareholder

wealth) and financial constraints are expected to operate over a number of years, the

allocation of capital resources to investment projects is best solved by the mathemati-

cal programming approach.

Many programming techniques have been developed. We shall concentrate on

the most common technique: linear programming. The assumptions and limitations

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140 Part II Investment decisions and strategies

underlying the LP approach will be discussed in a subsequent section. Problem-

solving using the LP approach involves four basic steps:

1 Formulate the problem. This requires specification of the objective function, input

parameters, decision variables and all relevant constraints. Take a firm that pro-

duces two products, A and B, with contributions per unit of £5 and £10 respective-

ly. The firm wishes to determine the product mix that will maximise its total

contribution. The objective function may be expressed as follows:

maximise contribution: £5A + £10B

A and B are the decision variables representing the number of units of products A

and B that should be produced. The input values £5 and £10 specify the unit con-

tribution values for products A and B respectively. Constraint equations may also

be determined to describe any limitations on resources, whether imposed by man-

agerial policies or the external environment.

2 Solve the LP problem. Simple problems can be solved using either a graphical

approach or the simplex method. More complex problems require a computer-

based solution algorithm.

3 Interpret the optimal solution. Examine the effect on the total value of the objective

function if a binding constraint were marginally slackened or tightened.

4 Conduct sensitivity analysis. Assess, for each input parameter, the range of values for

which the optimal solution remains valid.

These four stages in the LP process are illustrated in the following example.

Example: multi-period capital rationing in Flintoff plc

Flintoff’s five-year planning exercise shows that the cost of its six major projects, forming

the basis of the firm’s investment programme, exceeds the planned finance available.

Flintoff is already highly geared and control is in the hands of a few shareholders who are

reluctant to introduce more equity funds. Accordingly, the main source of funds is

through cash generated from existing operations, estimated to be £300,000 p.a. over the

next five years. The six projects are independent and cannot be delayed or brought for-

ward. Each project has a similar risk complexion to that of the existing business. If neces-

sary, projects are capable of division but no more than one of each is required. The

planned investment schedule and associated cash flows are given in Table 5.11.

The six projects, if implemented, are forecast to produce a total NPV of £857,000.

However, the annual capital constraint of £300,000 means that for the next three years

the required investment expenditure exceeds available investment finance, i.e. there is

a capital rationing problem.

The solution sequence is as follows.

■ 1. Specify the problem

The objective function seeks to maximise the NPV from the given set of projects available.

Max NPV: 130A  184B  35C  42D  186E  280F

Table 5.11

Flintoff plc: planned

investment schedule

(£000)

Project outlays

Total Available

Year A B C D E F outlay capital

0 – – 300

1 – 300

2 300

3 300

NPV 130 184 35 42 186 280

Total NPV = £857,000

200200

6165005066

71848150100220220

55424110220200

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Chapter 5 Investment appraisal methods 141

However, given the capital expenditure constraints over the coming years, we must

express for each year the capital required for each project and the maximum capital

available each year (i.e. £ 300,000):

In addition to the capital constraints, we need to define the bounds for each vari-

able. As no more than one of each project is required and projects are divisible, we can

specify the bounds as:

This linear programming formulation tells us to find the mix of projects producing

the highest total net present value, given the constraint that only £300,000 can be spent

in any year and that not more than one of each project is permitted.

■ 2. Solve the problem

Using a linear program on the computer gives the solution in Table 5.12. Flintoff plc

should accept investment proposals B and E in full plus 14.5 per cent of project A and

46.8 per cent of project F. This will produce the highest possible total net present value

available, £520,000. This is significantly less than the £857,000 total NPV if no con-

straints are imposed.

■ 3. Interpret the optimal solution

Table 5.12 shows that only in Years 1 and 2 is the full £300,000 utilised. These years

then impose binding constraints – their existence limits the company’s freedom to pur-

sue its objective of NPV maximisation because it restricts the investment finance avail-

able to the firm in those years. Conversely, Years 0 and 3 are non-binding: they do not

constrain the firm in its efforts to achieve its objective. Hence while there is no addi-

tional opportunity cost (besides that already incorporated in the discount rate), for

non-binding periods there is an additional opportunity cost attached to the use of

investment finance in the two years where constraints are binding. These additional

opportunity costs are termed shadow prices (or dual values). Shadow prices show

how much the decision-maker would be willing to pay to acquire one additional unit

A, B, C, D, E, F  0  1

Year 0 200A  220C  110D  24E  300

Year 1 220A  220B  100C  150D  48E  300

Year 2 66B  50C  500F  300

Year 3 200E  300

Table 5.12

Projects accepted

based on LP solution

Capital outlay (£ 000)

Proportion NPV

Project accepted (£ 000) Year 0 Year 1 Year 2 Year 3

A 0.145 19 32 – –

B 1 184 220 66

C 0–

D 0–

E 1 186 –

F 0.468 131 – – –

520

20030030058

234

2004829

29

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142 Part II Investment decisions and strategies

of each constrained resource. In this case, computer analysis reveals that the shadow

prices are:

Year Shadow price (£) Constraint

0 0 non-binding

1 0.59 binding

2 0.56 binding

3 0 non-binding

A £1 increase (reduction) in capital spending in Year 1 would produce an increase

(reduction) in total NPV of £0.59. Similarly, for Year 2 a £1 change in investment

expenditure would result in a £0.56 change in total NPV. Because the capital con-

straints in Years 0 and 3 are non-binding, their shadow prices are zero and a marginal

change in capital spending in those years will have no impact on the NPV objective

function. Shadow prices, while of value in indicating the additional opportunity cost,

can be used only within a specific range. In addition, it is desirable to ascertain the

effect of changes in input parameters on the optimal solution. These issues require

some form of sensitivity analysis.

■ 4. Perform sensitivity analysis

The computer output provides two additional pieces of information. First, it tells the

decision-maker the maximum variation for each binding constraint. In our example,

the shadow price for the Year 1 constraint has a range of to In other words,

the shadow price of £0.59 would hold up to an increase in capital expenditure for that

year of £188,000, or a reduction of £36,000.

The program also indicates the margin of error permitted for input parameters

before the optimal solution differs. In our example, the actual NPV for the optimal

investment mix could fall as indicated below and still not change the optimal solution:

Maximum permitted

Project fall in NPV (£000)

This facility is particularly appropriate as a means of assessing the margin of error

permitted for risky projects under conditions of capital rationing.

LP assumptions

In order to assess the value of the basic linear programming approach, we must con-

sider the assumptions underlying its application. These are as follows:

1 All input parameters to the LP model are certain.

2 There is a single objective to be optimised.

3 The objective function and all constraint equations are linear.

4 Decision variables are continuous (i.e. divisible).

5 There is independence among decision variables and resources available.

Most, if not all, of these limiting assumptions can be relaxed by using more complex

mathematical programming. For example, uncertainty, multiple objectives and non-

linearity can be better addressed by other approaches such as stochastic LP, goal pro-

gramming and quadratic programming respectively.

For most businesses, the LP assumption that projects are divisible is unrealistic.

Even if a project could be operated on a reduced scale, it is unlikely that the NPV

280

158

17

68

188.36

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Chapter 5 Investment appraisal methods 143

would reduce pro rata because many of the fixed costs would remain while the bene-

fits of sale would be reduced. Integer programming is more appropriate when projects

are non-divisible. This is a special case of linear programming where variables can

take only the values 0 (reject the project) or 1 (accept it in toto).

Applying integer programming to Flintoff plc requires only one change in the prob-

lem specification. The bounds become:

The solution, provided by an integer programming computer application, shows

that only two projects should be accepted: projects B and E. These offer a combined

NPV of £370,000, which is the best available given the capital constraints. This is well

under half the £857,000 total NPV achievable in the absence of capital rationing (see

Table 5.11). Were the shareholders of Flintoff plc aware of these lost wealth-creating

opportunities, they might well be concerned and ask the chairman whether the capi-

tal constraints were really as fixed as they appeared to be!

A, B, C, D, E, F,  0 or 1

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144 Part II Investment decisions and strategies

Questions with a

coloured number have solutions in Appendix B on page 694.

1 Microtic Ltd, a manufacturer of watches, is considering the selection of one from two mutually exclusive

investment projects, each with an estimated five-year life. Project A costs £1,616,000 and is forecast to gener-

ate annual cash flows of £500,000. Its estimated residual value after five years is £301,000. Project B, costing

£556,000 and with a scrap value of £56,000, should generate annual cash flows of £200,000. The company oper-

ates a straight-line depreciation policy and discounts cash flows at 15 per cent p.a.

Microtic Ltd uses four investment appraisal techniques: payback period, net present value, internal rate of

return and accounting rate of return (i.e. average accounting profit to initial book value of investment).

Make the appropriate calculations and give reasons for your investment advice.

2 Mace Ltd is planning its capital budget for 19_7 and 19_8. The company’s directors have reduced their initial

list of projects to five, the expected cash flows of which are set out below:

QUESTIONS

Project 19_7 19_8 19_9 19_0 NPV

7,90040,00030,00010,00050,000

90055,00045,00080,000

8,30070,00060,00050,00040,000

1,30045,00025,00020,00030,000

1,60025,00025,00030,00060,000

None of the five projects can be delayed and all are divisible. Cash flows arise on the first day of the year.

The minimum return required by shareholders of Mace Ltd is 10 per cent p.a. Which projects should Mace Ltd

accept if the capital available for investment is limited to £100,000 on 1 January 19_7, but readily available at

10 per cent p.a. on 1 January 19_8 and subsequently?

3 The directors of Mylo Ltd are currently considering two mutually exclusive investment projects. Both projects

are concerned with the purchase of new plant. The following data are available for each project:

Project

1 (£) 2 (£)

Cost (immediate outlay) 100,000 60,000

Expected annual net profit (loss)

Year 1 29,000 18,000

2 (1,000) (2,000)

3 2,000 4,000

Estimated residual value 7,000 6,000

The company has an estimated cost of capital of 10 per cent and employs the straight-line method of depre-

ciation for all fixed assets when calculating net profit. Neither project would increase the working capital of

the company. The company has sufficient funds to meet all capital expenditure requirements.

Required

(a) Calculate for each project:

(i) the net present value

(ii) the approximate internal rate of return

(iii) the profitability index

(iv) the payback period

(b) State which, if any, of the two investment projects the directors of Mylo Ltd should accept, and why.

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